Abstracts of Talks
John Playfair, the Scottish Enlightenment, and “Progress” in the History and Philosophy of Mathematics
Amy Ackerberg-Hastings
University of Maryland University College
aackerbe@verizon.net
The thinkers of the Scottish Enlightenment were fascinated by the concept of progress, both in human history and in the development of the natural world. Commentators, especially popular writers, have noted that the Scots’ optimism about progress established a foundation for 21st-century assumptions about the nature of this concept. Toward the end of the Scottish Enlightenment, John Playfair (1748-1819), a younger contemporary of the principal Edinburgh figures of the movement, joined the conversation. In particular, in 1816 he penned "Dissertation Second: On the Progress of Mathematical and Physical Science Since the Revival of Letters in Europe" for Encyclopaedia Britannica. This talk provides some historical background for the session by describing how this University of Edinburgh professor of mathematics and then of natural philosophy understood the meaning and role of progress in mathematics and its history. The talk will also suggest how paying attention to the complexities of this concept can benefit today's teachers.
Analysis and Synthesis in Geometry Textbooks: Who Cares?
Amy Ackerberg-Hastings
NMAH/UMUC
aackerbe@verizon.net
Thirteen years ago, I completed a history of technology and science degree by writing a dissertation on how early 19th-century college teaching in the United States was shaped in part by two ubiquitous terms, analysis and synthesis, and three distinct but interrelated definitions for the terms: as mathematical styles, as directions of proof, and as educational approaches. To the best of my knowledge, however, the hardy few who read the dissertation were more interested in my biographies of Jeremiah Day, John Farrar, and Charles Davies than in the claims I made about the interactions between mathematics, philosophy, and pedagogy in these men's cultural context.
Now, I am rewriting the dissertation, rearticulating these intellectual connections, and, ultimately, reaffirming their historical significance. This talk will report on this process of rethinking in order to highlight the importance of philosophy in intellectual and cultural approaches to history. I will also discuss how an awareness of this interplay between philosophy and history can positively influence how we present mathematics to students.
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How My Philosophy of Teaching Affects My Communication of Mathematics
Amy Ackerberg-Hastings
MAA Convergence
aackerbe@verizon.net
I have been a believer in articulating one’s philosophy of one’s academic discipline since my graduate studies in the history of technology and science. To this day, undergraduate history majors at University of Maryland Global Campus prepare their philosophies of history because I introduced the assignment into the curriculum in 2006. Although I do not currently hold a teaching position and I spend most of my time around mathematicians, I still find myself drawing upon my philosophy of history as a researcher and editor. Even more so, I utilize my philosophy of teaching as I polish my own writing and coach mathematicians in communicating their mathematical and pedagogical ideas. In this talk, I’ll share some observations from my experiences with MAA Convergence, CSHPM’s Bulletin, and CSHPM Notes, a regular column in Notes of the CMS. I hope to invite cross-disciplinary conversations about the intersections between our philosophies of mathematics, of education, and of history.
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From Intuition to Esoterica
Deborah C. Arangno
University of Colorado-Denver
deborah.arangno@ucdenver.edu
Wisdom is not mere
knowledge nor the ability to acquire and synthesize a body of apparently useful
facts. Since antiquity wisdom has been
valued as an insight into truth; which itself transcends wisdom. When we study mathematics we begin to
understand the intrinsic relationship between these two hierarchal realms, and
the revelations that can be gleaned from them. I will argue that the methods and information
discovered from the process of Science is ultimately approximative and protean.
On the other hand, the transcendent
arena – which is the domain of mathematical principles – enjoys a kind of
perdurition through time. Therefore the
very methods and devices of science alone are inadequate to the task of examining
it. However there should never be any
disparity between the facts, gleaned by science, and the insights, revealed by
mathematics, which in turn transcend mere knowledge. Indeed, Mathematics has always given us insight
into the reality of things – even those which elude us empirically – from
imaginary numbers to black holes, so that even when we lack the faculty to
observe things we can know their existence simply because they ought
to exist, Mathematically.
Mathematical Thinking - From Cacophony to Consensus
Sean F Argyle
Kent State University
sargyle1@kent.edu
What does it mean to do mathematics? What counts as mathematics? Who decides? These sorts of fundamental questions about the nature of the discipline have not yet been answered such that there is general agreement on the matter. Without these answers, how can we trust in our derivations and proofs? More importantly, how can we train the next generation of mathematicians if we can’t even agree what it means to be a mathematician? What little research on the subject exists is disjointed and dissenting, leading some researchers to lament the possibility of ever coming to an agreement on how to define “mathematical thinking” as a viable construct. Rather than add one more voice into the cacophony of competing definitions, this presentation seeks to discuss the results of a meta-analysis of the term’s use in an appropriately titled journal Mathematical Thinking and Learning. This synthesis of more than a decade of research provides cognitive model of the internal process of doing mathematics utilizing a post-epistemological stance that relies on a compromise between the Platonist and Formalist extremes. Only when researchers and philosophers can agree on a vocabulary can we begin to “stand on the shoulders of giants.”
When is a Proof a Proof?
Joseph Auslander
University of Maryland (Emeritus)
jna@math.umd.edu
Why does "the mathematician in the street" believe a proof is correct? I note three reasons: certification, explanation, and exploration. I, not a number theorist, accept Wiles' proof as correct mainly because number theorists I respect have "certified" it. Explanation means we understand why a result is correct; here we look at the proof in detail. Related is exploration; writing out a proof may lead to new insights and results, as brilliantly developed in Bressoud's book "Proofs and Confirmations", on the alternating sign matrix conjecture.
Another topic is changing standards of proof, e.g., "Poincare's last theorem". For years, it was believed that Birkhoff's proof was incorrect but when Brown and Neumann looked at it carefully, they found that it was essentially correct. Also, a fake one line "proof" of the ergodic theorem appears in Halmos' book, where he asks "Can any of this nonsense be made meaningful?" Some thirty years later, a correct proof was given along these lines, probably the best proof of the ergodic theorem.
I also touch on computer proofs, e.g., the four color theorem and Hales' proof of the Kepler conjecture. Computer proofs are here to stay, but there are problems with them.
I draw on work of Rota, Hersh, Kitcher, and Thurston, among others.
The Design of Mathematical Language
Jeremy David Avigad
Carnegie Mellon University
avigad@cmu.edu
Formal languages provide informative descriptions of informal mathematical language, but there is a sense in which they specify too much, and there is a sense in which they specify too little. The fact that an ordinary mathematical text can be equally represented in any of a number of foundational languages suggests that these languages should be viewed as alternative implementations of mathematical vernacular. It is therefore reasonable to look for descriptions of mathematical language and patterns of inference that clarify the specifications that the implementations are designed to meet.
Formal language specify too little in the sense that features of ordinary mathematical language and inference that are essential to its function are not addressed by a formal specification. Instantiating the formal foundation is only the first step to implementing a mathematical proof assistant, and the bulk of the work goes into supporting the interactions with users that makes them usable in practice. The design of a proof assistant requires making countless engineering decisions that bear on the system’s usability, and we might optimistically seek a broad view of mathematical language that can help us evaluate the choices and assess their merits.
In this talk, I will make a start on developing a design specification for mathematical language that clarifies the challenges that proof assistants have to meet, and I will describe some of the mechanisms that contemporary proof assistants use to meet those challenges.
The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof
Jody Azzouni
Tufts University
Jody.Azzouni@tufts.edu
The relationship between formal derivations, which occur in artificial languages and mathematical proof, which occurs in natural languages is explored. The suggestion that ordinary mathematical proofs are abbreviations or sketches of formal derivations is rejected. The alternative suggestion that the existence of appropriate derivations in formal logical languages is a norm for ordinary rigorous mathematical proof is explored and qualified.
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Speaking mathematics in natural and formal languages
Jody Azzouni
Tufts University
Jody.Azzouni@tufts.edu
Although natural language, supplemented with technical, even formal, language tools is the language of professional mathematics--of "informal rigorous mathematics" as it’s often called--the working model of the language(s) of mathematics among philosophers is the formal language, either some version of first-order logic or some version of higher-order logic, depending on the philosopher in question. Accompanying the emergence of formal languages, among the pioneer mathematicians/philosophers, Frege, Hilbert, Skolem, Zermelo, and others, was also the emergence of a new model for language, one which sharply separates syntax and meaning (semantics). This model of meaningful language fully came into its own only with the postwar assimilation of the work of Alfred Tarski. This presentation discusses how hard that sharp division is to make—psychologically speaking--because it is so alien to how we understand and experience the languages we speak; keeping the distinction clear bedeviled not only the pioneer mathematicians/philosophers who were grappling with the impact of formal methods on mathematics but still bedevils even contemporary philosophers. This presentation illustrates the difficulties with ordinary (mathematical) talk of the parallel postulate and informal discussion of the historic discovery that there are geometries in (and of) which it is false.
A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics
Mark Balaguer
Cal State LA
In this lecture, I will attempt to clarify some things about the methodology of the philosophy of mathematics – about the kinds of theories that philosophers of mathematics can be seen as primarily putting forward and about the kinds of arguments that they use to justify these theories. I will also try to clarify the relationship between mathematics and the philosophy of mathematics.
Back to JMM 2013 Guest Lecture.
Philosophical implications of the paradigm shift in model theory
John BaldwinUniversity of Illinois at Chicago
Traditionally, logic was thought of as ‘principles of right reason’. Twentieth century philosophy of mathematics focused on the problem of a general foundation for all mathematics. In contrast, the last 70 years have seen model theory develop as the study and comparison of formal theories for studying specific areas of mathematics. For example, in a rough sense, algebraic geometry is the study of the first order definable subsets of the complex numbers. Moreover, syntactical information about the theories for different areas can uncover common strains. Thus, Abraham Robinson found a common framework for the Artin-Schreier theory of ordered fields, Hilbert’s nullstellensatz and differentially closed fields. Shelah's stability theory leads to a classification of such theories that makes more precise the idea of a ‘tame structure’. Thus, logic (specifically model theory) becomes a tool for organizing and doing mathematics with consequences for combinatorics, diophantine geometry, differential equations and other fields.
Back to Mathfest 2017 Guest Lecture.
Category theory and Model Theory: Symbiotic Scaffolds
John BaldwinUniversity of Illinois at Chicago (Emeritus)
baldwinj@sbcglobal.net
A scaffold for mathematics includes both local foundations for various areas of mathematics and productive guidance in how to unify them. In a scaffold the unification does not take place by a common axiomatic basis but consists of a systematic ways of connecting results and proofs in various areas of mathematics. Two scaffolds, model theory and category theory, provide local foundations for many areas of mathematic including two flavors (material and structural) of set theory and different approaches to unification. We will discuss salient features of the two scaffolds including their contrasting but bi-interpretable set theories. We focus on the contrasting treatments of ‘size’ in each scaffold and the advantages/disadvantages of each for different problems.
Why should we care about "foundations"?
Timothy Bays
University of Notre Dame, Department of Philosophy
tbays@nd.edu
Philosophers typically pay a lot of attention to the so-called "foundations" of mathematics. Working mathematicians…not so much. This talk will lay out the reasons some of us on the philosophical side of things care so much about foundations.
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Unfair Gambles in Probability
John Beam
University of Wisconsin Oshkosh
beam@uwosh.edu
In adopting the axioms from one mathematical discipline for another, one runs the risk of generating misleading results. The interplay between measure theory and probability provides a nice illustration of this. In the 1930’s, Kolmogorov borrowed the axiomatic system of the Lebesgue measure as a foundation for what is now the standard theory of probability. De Finetti argued that many of the modern analytic developments are devoid of meaning in the context of probability. In particular, he believed the assumption of countable additivity to be unjustified. He proposed a broader alternative theory of probability, consistent with Kolmogorov’s, but requiring neither countable additivity of the measure nor any sort of structure on its domain. A probability can thereby be interpreted as an assignment of fair odds for a bet. I shall demonstrate that if one attempts to use an analogous notion to include the axiom of countable additivity, grossly unfair bets may result.
long version:[Note: The general theme of my talk is that the axioms from one theoretical area of mathematics could lead to meaningless results, not only in the "real world," but even in other theoretical areas. Probability provides an interesting backdrop, because this subject is regarded by most mathematicians as a (semi-applied) sub-discipline of measure theory. In fact, probability was not generally held to be "legitimate" mathematics until Kolmogorov placed it on the foundations of measure theory. De Finetti believed these to be the wrong foundations. He provided many illustrations of this; I will describe some of these and conclude with my example of an unfair bet. I will not assume the audience to have any formal knowledge of measure theory or probability.]
In the 1930's, Kolmogorov borrowed the axiomatic system of the Lebesgue measure as a foundation for what is now the standard theory of probability. De Finetti argued that many of the modern analytic developments are devoid of meaning in the context of probability. In particular, he believed the assumption of countable additivity to be unjustified. He proposed a broader alternative notion of a "coherent" probability, consistent with the Lebesgue theory, but requiring neither countable additivity of the measure nor any sort of structure on its domain. A coherent probability can be interpreted as an assignment of fair odds for a bet. I shall demonstrate that if one attempts to use an analogous notion to include the axiom of countable additivity, grossly unfair bets may result.
To illustrate, I will consider the Lebesgue measure P on the unit interval. A "payoff function for a bet" is a function, the sum of a countable number of terms of the form a[I(A) P(A)], where a is "a" real number, "A" is a Borel set and "I" is the indicator function. "P(A)" is the cost of a $1-payoff bet on A. I will let c > 0 be given, and by exploiting the conditional nature of the convergence of the alternating harmonic series, I will construct a payoff function which is everywhere greater than c. The clever gambler can be assured of winning an arbitrary amount of money. This example does not rely on typical contrivances; for instance, only a finite amount of money is required of either the gambler or the house.
Remark: A number of notable mathematicians (including Lester Dubins and Bill Sudderth) subscribe to de Finetti's interpretation of a probability. Recently, the theory has been expanded to include a theory of integration and laws of large numbers.
Payoff: Here's an example of a simple bet: A gambler pays the house P({Heads}) dollars (in our example, 50 cents) for the promise of a one-dollar payoff if the outcome of the toss is Heads (and 0 payoff if the outcome is Tails). The payoff is described mathematically by
This is a real-valued function defined on the sample space. For instance, applied to the outcome Heads, the function takes the value 0.5 (the gambler netted 50 cents, because he got his dollar payoff and paid 50 cents for it). Applied to the outcome Tails, the function takes the value -0.5 (the gambler got no payoff but paid 50 cents for the opportunity).
An example of a more complex bet: A gambler pays the house 3 P({Heads}) dollars for 3 one-dollar-payoff bets on {Heads}, and SELLS to the house a one-dollar-payoff bet on {Tails} for the price of P({Tails}). The payoff function is
3[I({Heads}) - P({Heads})] + (-1)[I({Tails}) - P({Tails})].
Here, if the outcome is Heads, the payoff is 3[1 - 0.5] + (-1)[0 - 0.5],
or 2 dollars. (The gambler nets 2 dollars.)
If the outcome is Tails, the payoff is 3[0 - 0.5] + (-1)[1 - 0.5],
or -2 dollars. (The gambler loses 2 dollars.)
This payoff function is a linear combination involving two simple bets. De Finetti demonstrated that a finitely-additive probability measure is equivalent to an assignment P of odds for which it is NOT possible for a gambler to arrange any finite number of simple bets such that the resulting payoff function would be everywhere-positive over the sample space. (In other words, a clever gambler could not guarantee himself a win.)
A natural conjecture would be: A countably-additive probability measure is equivalent to an assignment of odds for which it is not possible for a gambler to arrange any COUNTABLE number of simple bets such that the resulting payoff function would be everywhere-positive over the sample space. This conjecture turns out to be false, and that is what my example will illustrate.
Not many probabilists, even, are aware of these sorts of examples. Much of modern probability theory is dependent on the axiom of countable additivity. So many standard results of the theory are meaningless, at least in the sense that probability should be something more than a branch of measure theory.
[To clarify some of the technical details. A "sample space" in probability is the set of possible outcomes of some experiment. It could be any type of set -- for instance, to model a single coin toss, the sample space could be {Heads,Tails}.
I: The indicator function "I" is the function which is 1 on outcomes in the set, 0 on other outcomes -- in the coin-tossing example, "I({Heads})" would take the value 1 if the outcome of the toss was Heads, and 0 if the outcome was Tails.
P: The probability "P" is a real-valued function defined on some collection of subsets of the sample space -- for example, a fair coin would be modeled by P({Heads})=1/2, P({Tails})=1/2, P({Heads,Tails})=1.]
The meaning of existence in mathematics
Michael Beeson
San Jose State University
Does the number two exist in the same way that electrons exist, or in a different way? What do we mean when we say, “There exists a number having such-and-such properties”? The talk will examine these questions in the light of twentieth-century science: Are we in a better position to answer these questions now than our predecessors were in 1907?
Back to Mathfest 2007 Guest Lecture.
What Are Mathematical Objects? An Empiricist Hypothesis
Carl E. Behrens
5107 Cedar Rd., Alexandria, VA 22309
cbehrens@crs.loc.gov
Many current philosophical problems may be simplified by approaching mathematics, and other mental activity, as purely physical phenomena that occur in the brains of human beings. The purpose of the presentation is not to determine whether the hypothesis is or can be true, but to explore the consequences for the philosophy of mathematics if it were true. Questions to be examined include: What are numbers and other mathematical objects? What is the relationship between mathematical laws and physical phenomena? What is the nature of mathematical knowledge? This topic was recently the subject of an extended discussion on the POMSIGMAA ListServe.
Why do we all get the same answers? Kitcher’s anti-apriorism and the problems of social constructivism
Carl E. Behrens
5107 Cedar Rd., Alexandria, VA 22309
cbehrens@crs.loc.gov
Philip Kitcher’s 1983 study, The Nature of Mathematical Knowledge, challenged the widely held principle that mathematical laws and methods are true a priori. Instead, he argued, they are developed in evolutionary fashion by mathematicians building on the work of previous generations. But if mathematics is constructed by human beings, why do they all agree on the results? Physical constants, such as gravity or the charge on the electron, are determined by observing the behavior of the external physical world, but mathematics is primarily, or completely, the product of the human mind. If mathematical laws and methods are not true a priori, why do all human minds produce the same answers? An empiricist response to this question will be discussed.
Are Euclid's Postulates Really Essences?
Carl E. Behrens
5107 Cedar Rd., Alexandria, VA 22309
cbehrens@crs.loc.gov
The Greek theory of Essences, say Lakoff and Nunez, holds that every thing is a kind of thing; that kinds, or categories, exist in the world; that everything has essences that make it the kind of thing it is, and that these essences are causal. They also argue that Euclid's postulates are the essence of plane geometry, and further, that all mathematical subjects, by which a few axioms lead to all other truths, are example of the theory of essences. The idea that categories have an existence of their own has persisted in many forms. Hersh, for example, identifies what he calls “social objects” in this way. Sonatas, the Supreme Court, and numbers, are examples of such objects, which he says have causal roles in society. Empiricists, on the other hand, reject the theory. J. S. Mill wrote: “A class, a universal, is not an entity per se, but neither more nor less than the individual substances which are placed in the class. There is nothing real in the matter except those objects, a common name given them, and common attributes indicated by the name.” Such generalizations exist as concepts in human minds, but their causality is only that of the individual objects aggregated. This talk will explore the influence on mathematical philosophy of the theory of essences.
John Stuart Mill's "Pebble Arithmetic" and the Nature of Mathematical Objects
Carl E. Behrens
5107 Cedar Rd., Alexandria, VA
22309
cbehrens@crs.loc.gov
The empiricist claim that all human knowledge rests on observation of physical events has always stumbled over phenomenon of abstract thought. David Hume tried to avoid the problem by defining two types of knowledge, which he called "matters of fact" and "relations of ideas," which latter he accepted as true in themselves. John Stuart Mill, however, insisted that even statements of abstract thought, including mathematical laws, were assumed to be true in general because they were observed to be true in single instances. To make this claim plausible Mill declared that "all numbers are numbers of something." This "pebble arithmetic," as his critics termed it, led to the disparagement of empiricism in the 20th Century, but it is no longer necessary to tie abstract mathematical objects to the external world. Whatever else they are, mathematical thoughts, along with all other thoughts, may be viewed as physical states of the brains of human beings, and thus as physical objects that may be observed as sources of empirical knowledge.
Mathematics Is a Science in its Own Right
Carl E. Behrens
5107 Cedar Rd., Alexandria, VA 22309
behrenscarl@yahoo.com
Wigner, like most physicists, viewed mathematics as a tool: as a means of exploring the physical world, of “discovering the laws of inanimate nature.” But mathematicians since the middle of the 19th Century have made it clear that theirs is a discipline that is more than a tool, a language for decoding the laws of the inanimate universe. It is a science aimed at discovering the laws that govern the part of the physical universe that is comprised of the human mind. This talk will explore the characteristics of the science of mathematics, viewed from this mission.
How does the mind construct/discover mathematical propositions?
Carl E. Behrens
5107 Cedar Rd., Alexandria, VA 22309
behrenscarl@yahoo.com
Recent discoveries in cognitive science probe deeply into the mental processes of mathematicians as they practice their art. George Lakoff and Rafael Nunez have focused most extensively on the roots of mathematical subjects, proposing that much advanced mathematics derives from schemas and conceptual metaphors used and developed for more common purposes. But other cognitive scientists, among them Antonio Damasio, Stanley Greenspan, and Stuart Shanker have directed their attention to the role of emotions in the practice of rational thought. Greenspan and Shanker argue that the ability to create symbols and to reason is not hard-wired in the human brain, but is actually learned through emotional signaling beginning in the first year of life. This presentation will attempt to tie together these various threads from cognitive science into a view of how mathematics develops and is practiced.
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A Conjecture about... Feminist Mathematics?
Sarah-Marie
Belcastro
Xavier University
smbelcas@cs.xu.edu
While there's a fairly well-developed literature on "feminist science," most of the literature focuses on biological and/or social science; there has been very little work done on the physical sciences in this regard. So, what might "feminist mathematics" mean? For me, a feminist science must revise the content or methodology of a science. I think it is plain to mathematicians that feminism cannot contradict the present content of physical sciences or mathematics. Thus, the only effect a feminist physical science could have on the content of a science is to influence in which directions that content might develop. A feminist physical science, if it exists, would have a constitutive rather than contextual influence (see Longino (1990, 1994) for definitions) on the development of content in the physical science.
I plan to argue that because inclusivity and non-hierarchicalness can be considered feminist values, improving the accessibility of mathematics is a feminist aim. Further, making mathematics more accessible could change the relative concentrations of people in mathematical subfields. That becomes a constitutive change in mathematics.
My conjecture is that writing proofs clearly and understandably could be a constitutive influence of feminism on mathematics. (My purpose in giving this talk is to open this conjecture to scrutiny and discussion.) Because known content in mathematics is defined by that which is communally understood, the language used in communicating content affects what is understood and how it is understood. Thurston points out that there are often many mathematically equivalent ways of framing, defining, and explaining a mathematical concept, and that ?one person?s clear mental image is another person?s intimidation? (Thurston, 1994). This phenomenon is well-known in the pedagogical sense, and carries through to the research sense as well?after all, the purpose of publishing research is to disseminate it, and to be truly disseminated some communication of results must occur. In fact, a proof is not verifiable if mathematicians as a whole cannot understand it (Tymoczko, 1979, 58-59; Tymoczko, 1986, 267).
We write proofs as we understand them, and as we wish others to view the material, rather than in such a way that others will understand them. Often, we mathematicians find it unrewardingly difficult to explain our new work in a way meaningful to many others. Clarity in proofwriting is an excellent example of a constitutive value which is also contextual: by virtue of being feminist, the value is contextual, though because of its influence on content and how it is understood, it is also constitutive.
Most mathematicians will agree that clearer proofs are better. Clearer proofs are more convincing (Resnik, 1992, 324) and appear to be simpler than obfuscating proofs; simplicity is prized by mathematicians for many reasons (De Millo/Lipman/ Perlis, 1979, 274). This leads to the question of whether my suggestion is one of feminist mathematics, or merely of good mathematics. Generally, feminist values are among those that mathematicians would say are part of ideal mathematics. But in reality, mathematicians do not practice ideal mathematics. So, the contribution feminism makes to mathematics is to remind it that feminist ideals are among the true ideals of mathematics. (A. Flaxman, August 2001) This resonates with Longino's charge to consider what a feminist viewpoint can bring to scientific (in this case mathematical) practice (Longino, 1987 and 1990).
How the way we 'see' mathematics changes mathematics
Sarah-Marie Belcastro
Xavier
University
smbelcas@cs.xu.edu
In the philosophy of science, there are theories which mediate between social constructivism and realism. I will adapt an aspect of one such theory, agential realism, to mathematics; my primary metaphor will be that of windows in a room as a limiting factor on our visualizations of mathematical ideas. In this same vein of adaptation, I will compare some aspects of taxonomic systems in science with classification systems in mathematics. Finally, I will draw these two seemingly unrelated threads together in order to describe how our internal conceptualizations function together with our choices of mathematical priorities to influence which mathematics is known and which remains unknown.
Epistemological Culture and Mathematics
Sarah-Marie Belcastro
Hadley, MA
smbelcas@toroidalsnark.net
After Evelyn Fox Keller, we define epistemological culture to mean the standards used by members of an academic discipline to achieve explanatory satisfaction. As mathematicians, we have a distinct epistemological culture (consider the use of the word “proof” in mathematics vs. its usage in other contexts).
We will argue that the epistemic privilege generally accorded to mathematics is inextricably linked with mathematical practice, and that both mathematical practice and epistemic privilege are intertwined with and inform mathematical epistemological culture. That is, mathematics is viewed as having a more powerful claim to truth than many other fields; our practice as mathematicians contributes substance to this view; and our standards for deciding validity are deeply related to our methods of producing/disseminating knowledge.
The epistemological culture of mathematics differs, in sometimes surprising ways, from the epistemological cultures of laboratory and social sciences. We posit that these differences partially explain vexing phenomena such as the inappropriate usage of mathematics in social science or cultural theory research, and the overgeneralization of feminist critiques of biological and social sciences to the physical sciences and mathematics.
Does Inclusivity Matter in Mathematical Practice?
sarah-marie belcastro
smbelcas@toroidalsnark.net
Many in the mathematical community believe that it is important to welcome participation from people with a variety of backgrounds and in particular from members of underrepresented groups. From a philosophical point of view, inclusivity is a broader concept than welcoming a diversity of human experience; for example, it includes welcoming a variety of mathematical perspectives (epistemic diversity).
Are there implications of encouraging inclusivity for the production of mathematical knowledge, and if so, what are they? And are they positive or negative? Conversely, are there implications of our mathematical practices, in terms of producing knowledge (theorems, proofs, etc.), on inclusivity?
We will carefully describe inclusivity as framed in the literature on scientific values, and restrict our discussion to epistemic values, and then to mathematical epistemic values. We will then examine what impacts the epistemic value(s) of inclusivity may have on mathematical practices, and what impacts current mathematical practices may have on inclusivity, and include specific examples. Finally, we will pose changes/actions that individuals or the community might make/take, in accordance with common mathematical values, and evaluate their impact relative to inclusivity.
If you're hoping for discovery, put away the handouts!
Steven R Benson
Lesley University
sbenson@lesley.edu
I have observed that students seem to take fundamentally different philosophical approaches to what we might consider identical problem “set-ups”, depending on whether or not the problem is given from the text/handout. When given written problems – no matter how open-ended – students tend to treat them as exercises, whereas problems that appear to be “spur of the moment” or that emerge from a classroom discussion are treated in a more exploratory way. I will present specific instances where deep mathematical insights have occurred in a variety of course levels.
The Pedagogical Challenges of One to One Correspondence
Satish C. Bhatnagar
University of Nevada-Las Vegas
bhatnaga@unlv.edu
If there is one concept that is a linchpin of entire engine of mathematics then it is the concept of one to one correspondence. The concept of limit in mathematics is the profoundest in the history of human thought. But it impacts only the analytic half of mathematics. Discrete mathematics is not affected by it.
While teaching students who are not math majors but have applied interest in math a few visual paradoxes are seen. We as math instructors establish one to one correspondence between points on two different line segments, or on two circles of different radii. What clearly registers in the minds of students is that the ‘numbers’ of points on two segments are the same. A student then naturally wonders as to what make the two segments of different lengths? In other words, what is a length, and what does it measure, or its contents? I have no satisfying answer. Obviously, the paper has deep philosophical overtones in it.
Sheaves of Probability
Owen Biesel
Southern Connecticut State University
owenbiesel@gmail.com
What does it mean for multiple agents' credence functions to be consistent with each other, if the agents have distinct but overlapping sets of evidence? Michael Titelbaum’s rule, called Generalized Conditionalization (GC), sensibly requires each pair of agents to acquire identical credences if they updated on each other's evidence. However, GC allows for paradoxical arrangements of agent credences that we would not like to call consistent. We interpret GC as a gluing condition in the context of sheaf theory, and show that a slightly stronger version of GC eliminates these paradoxes and gives a satisfying equivalent interpretation of consistency.
The unexpected usefulness of epistemological skepticism
Katalin Bimbó
University of Alberta
bimbo@ualberta.ca
David Hilbert believed that mathematical problems have definite answers. Some philosophers of mathematics concentrate on metaphysical questions such as "Do numbers (or sets, triangles, etc.) exist?" However, epistemological problems are probably more important for mathematical practice than taking a stance in an ontological debate. I will illustrate that moderate skepticism can help us to produce a definite answer to a precisely formulated mathematical problem. The example comes from theoretical computer science, which I take here to be a (relatively) new branch of mathematics. Objects in theoretical computer science are often more structured and complicated than an equilateral triangle, but at the same time, they are more abstract than an app or an OS. Occasionally, our intuitions come up short in reasoning about these kinds of objects. I will conclude that a certain skepticism together with insistence on more formal definitions and proofs can be fruitful.
Mathematics is a Meme(plex)
Andrew G. Borden
Converse, TX
aborden|@wireweb.net
A meme is a cultural pattern of activities or beliefs which is replicable and which can be propagated among contemporaries and from one generation to another. It is sometimes volatile in the early generations of propagation. Religious beliefs and practices are examples of memes. Memes occur and survive because they satisfy certain human needs. Memeplexes are clusters of related memes. We have done a simulation of memeplex robustness and survivability. Pure mathematics receives a high score from our model. It is clearly a robust memeplex and exists independent of meaning or truth. Among the different possible philosophical characterizations of mathematics, we consider it to be a social construct. We use a category theoretical argument to explain the relationship between pure and applied mathematics and to attempt to explain the “Unreasonable Usefulness of Mathematics in the Natural Sciences”.
Philosophical Implications of Experimental Mathematics
Jonathan Borwein
Dalhousie University
Professor Borwein is co-author of two recent books on experimental mathematics. Professor Borwein will discuss the philosophical implications of his work in experimental mathematics. Philosophers have frequently distinguished mathematics from the physical sciences. While the sciences were constrained to fit themselves via experimentation to the ‘real’ world, mathematicians were allowed more or less free reign within the abstract world of the mind. This picture has served mathematicians well for the past few millennia but the computer has begun to change this. The computer has given us the ability to look at new and unimaginably vast worlds. It has created mathematical worlds that would have remained inaccessible to the unaided human mind, but this access has come at a price. Many of these worlds, at present, can only be known experimentally. Work in experimental mathematics challenges the standard view of mathematics as a subject in which proof is the sole pathway to knowledge.
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Defining Mathematical Esthetics within the NCTM Standards
Michael J. Bossé
Indiana University of Pennsylvania
mbosse@iup.edu
The history of mathematics education within the United States from the New Math Movement (1950s-1970s) through the NCTM Standards (1989-2002) has been punctuated by distinct esthetic philosophic positions, While few would deny that the New Math Movement recognized the beauty of mathematics as an axiomatic system, many would have some difficulty defining NCTM’s esthetic position. However, NCTM’s esthetic position is defined within their publications and can be clearly recognized through philosophic analysis. This paper analyzes NCTM’s reform publications and reports the esthetic philosophic position found therein.
The NonEuclidean Revolution Makes Relativism Available to the Rest of the World
Michael J. Bossé
Morgan State University
mbosse@moac.morgan.edu
The accepted nature of truth has undergone significant change since before the NonEuclidean Revolution. Worldwide, absolutes have been replaced by relativism.The role of the NonEuclidean Revolution in this process cannot be underestimated. This paper discusses how mathematics opened the door to relativism to many other fields of science, sociology, and personal beliefs.
Object and Attribute: the case of Curves and Equations
Robert E. Bradley
Adelphi University
bradley@adelphi.edu
Does a curve have an associated equation, or does an equation have an associated curve (its graph)? Engaging this question can shed light on the nature of mathematical objects and the evolution of mathematical practice. There was a time in the history of mathematics when the answer would not have been subject to debate. In the mid 17th century, the curve was the object and its equation was the attribute. We will argue, however, that by the late 18th century the point of view had been reversed. In fact, the paradigm shift seems to have taken place in the years between the publication of L'Hôpital's Analyse des Infiniments Petits and Euler's Introductio in Analysin Infinitorum, as is indicated by the treatment of singular points of curves. This change in point of view concerning mathematical objects is a reflection of the success of differential calculus, which in this period amounted to a collection of algorithms operating on algebraic expressions.
Searle’s Metaphysics of Computation and Alternative Logics: A Surprising Connection
Jeff Buechner
Rutgers University
buechner@rci.rutgers.edu
There is a surprising connection between John Searle’s views on the metaphysics of computation and the view that logic is true by convention and that choice of a logic is choice of a convention. I’ll develop this connection in some detail, and then show how Quine’s argument (in his well-known essay “Truth by Convention”) against the view that logic is true by convention and Kripke’s (unpublished) arguments against the view that there are alternatives to classical logic can be used to undermine Searle’s views. Since Searle’s views on the metaphysics of computation underlie triviality arguments – the claim that any object can compute any function – which are devastating to the computational view of the mind, the interest here is in showing that work in the philosophy of mathematics can be usefully employed in the philosophy of mind.
Ignoring the Obvious in Philosophical Applications of the Gödel Incompleteness theorems
Jeff Buechner
Rutgers University
buechner@rci.rutgers.edu
The Gödel incompleteness theorems have been famously recruited in the philosophy of mind in arguments that claim human minds have no wholly computational description. What those applications – and other kinds of applications as well – ignore is a fundamental feature of the incompleteness theorems: the epistemic modality of the proof relation in a system of formal logic. I will describe some surprising consequences for such applications when proper attention is paid to the epistemic modality of the proof relation.
Using the Philosophy of Intuitionistic Mathematics to Strengthen Proof Skills
Jeff Buechner
Rutgers University
buechner@rci.rutgers.edu
There are several issues within the philosophy of intuitionistic mathematics that are useful for developing proof skills in undergraduate mathematics majors. This talk will examine the role of classical and intuitionistic logic in constructing proofs, the intuitionistic proscription of proof by contradiction, and the nature of constructive existence proofs and how attention to these issues can foster proof skills.
Mathematical practice and the philosophy of mathematics
Jeff Buechner
Rutgers University
buechner@rci.rutgers.edu
If the philosophy of mathematics had never existed, would contemporary mathematical practice be different from what it now is? I'll argue that it would be quite different in several respects, some of which are hardly controversial, having to do with (i) the developments in set theory that were a reaction to the discovery of the set-theoretic paradoxes and (ii) with the intuitionistic critique of classical mathematics. There are also respects in which it would not be different, and these respects are important, since they underscore a point that philosophers of mathematics need to explain: there are properties, structures and objects in mathematics that are immune to philosophical questioning of the foundations of mathematics. The question is why this is so. I'll attempt an explanation that develops an analogy between natural kind terms in the empirical sciences and mathematical inscriptions, although the analogy breaks down at a certain point, which (I claim) characterizes the difference between the empirical sciences and mathematics.
Mathematical Understanding and Philosophies of Mathematics
Jeff Buechner
Rutgers University
buechner@rci.rutgers.edu
I will argue that there are theorems in mathematics whose understanding (both in a psychological and a philosophical sense) depends upon holding a certain philosophy of mathematics. Are there any theorems common to all philosophies of mathematics which can be understood within any mathematical philosophy? Yes: there are theorems of elementary number theory that we understand only when we have a de re attitude toward natural numbers, regardless of which mathematical philosophy one holds. However, if we have only a de dicto attitude toward the natural numbers, we might not understand those theorems. This suggests a pedagogical strategy for both teaching and learning mathematics and also creates a philosophical problem: how can we explain those areas of mathematical practice on which all mathematical philosophies agree and then show how in extensions of that practice different mathematical philosophies differ as to the content of the set of theorems of those extensions. Finally, are there any theorems common to all philosophies of mathematics which can only be understood within a particular mathematical philosophy? I provide an example of one theorem, which draws on the work of Harvey Friedman's program of Boolean Relation Theory.
Formal Mathematical Proof and Mathematical Practice: a New Skeptical Problem
Jeff Buechner
Rutgers University
buechner@rci.rutgers.edu
There are several problems in the philosophy of mathematics concerning the notion of mathematical proof, at least one of which serves as the primary motivation for experimental mathematics. But there is a new problem which appears to have no easy fix; moreover, it is a skeptical problem. The problem is that one can construct a proof (in some cases by an algorithm) which conforms to the definition of a formal mathematical proof, which no mathematician would regard as a legitimate mathematical proof. Indeed, there are some constructions that even a layman with no knowledge of mathematics would regard as an illegitimate mathematical proof. Appeal to the informal notion of proof used by mathematicians is circular: to justify the formal notion, one needs to appeal to the informal notion, which, in turn, is justified in terms of the formal notion. The skeptical problem is: which proofs are genuine and provide mathematical knowledge, and which do not? It is worthless to appeal to the notion of a formal mathematical proof to resolve the skeptical issue.
Understanding the Interplay between the History and Philosophy of Mathematics in Proof Mining
Jeff Buechner
Rutgers University and the Saul Kripke Center, CUNY Graduate Center
buechner@rci.rutgers.edu
What is the nature of the relationship between the history of mathematics and the philosophy of mathematics? We conjecture one particular aspect of this relationship (which we take to be a necessary condition) contextualized to the field of proof mining: understanding issues in the philosophy of mathematics is needed to properly understand episodes and developments in the history of mathematics, and episodes and developments in the history of mathematics are needed to properly understand issues in the philosophy of mathematics. Hilbert’s program, which is a precursor of proof mining, cannot be properly understood without understanding the philosophical problem of theoretical terms, their explanatory role in mathematics, their role in questions of mathematical realism, the crisis in the foundations of mathematics, the change from classical to modern mathematics, and the nature of mathematical understanding. Some philosophers misunderstand Hilbert’s epistemology because they neglect the history of mathematics and some historians misunderstand Hilbert’s program because they neglect the philosophy of mathematics. We illustrate the symmetrical relation between the philosophy of mathematics and the history of mathematics in Hilbert’s original formulation of his program, how Gödel’s second incompleteness theorem eliminated certain aspects of Hilbert’s program and motivated the revision of other aspects, Kreisel’s re-interpretation of the program in terms of proof transformations needed to extract information from proofs such as effective bounds and algorithms for computing witnesses to ineffectively specified existential formulas, Kreisel’s no-counterexample interpretation, Kriesel’s notion of unwinding proofs, Gödel’s Dialectica (functional) interpretation, and some of Kohlenbach’s recent work in proof mining.
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A New Look at Wigner's ‘The Unreasonable Effectiveness of Mathematics in the Natural Sciences’
Jeff Buechner
Rutgers University and the Saul Kripke Center, CUNY Graduate Center
buechner@rci.rutgers.edu
There are several problems in the philosophy of mathematics that are intertwined in Wigner's elucidation of the unreasonable effectiveness of mathematics in the natural sciences. One problem is that of irrelevant inferences in mathematical proofs – that is, the question of when a proof of a mathematical theorem is genuine. Another problem is Kripke's skeptical problem for functionalist accounts of the mind, which gains traction from the way in which abstract objects are imperfectly realized in the real world. A third problem is that of the underdetermination of theory by data. That is, there are infinitely many incompatible functions each of which will (i) provide the same finite set of successful predictions and (ii) accord with the finite set of data points. This provides a reason for why the accuracy of a mathematical theory of the real world cannot be taken as a criterion of its truth – of reality and shows how the Kripke skeptical problem for functionalism is also a problem about the nature of physical reality.
What is an Adequate Epistemology for Mathematics?
Jeff Buechner
Rutgers University and the Saul Kripke Center, CUNY Graduate Center
buechner@rci.rutgers.edu
If we accept a mathematical epistemology in which we can know mathematical propositions with less than mathematical certainty, new possibilities become available for what counts as mathematical knowledge. For instance, if there are formal systems susceptible to the Gödel incompleteness theorems in which the consistency of Peano arithmetic is proved with less than mathematical certainty and the epistemic modality in which it is proved satisfies a reasonable notion of justification, then the limitations of the Gödel theorems will have been dramatically circumvented. In a 1972 paper, Georg Kreisel parenthetically remarks on the cogency of such an epistemology, but without developing it, while subsequent literature simply ignores it.
A stumbling block for a mathematical epistemology that licenses knowing mathematical propositions with less than mathematical certainty is the necessity of mathematical propositions. But work by Saul Kripke in his epochal Naming and Necessity severed the connection between the metaphysical notion of necessity and the epistemic notion of certainty, which opened the possibility of knowing a mathematical proposition in a different epistemic modality than mathematical certainty.
In my talk I will examine various conceptions of mathematical proof that answer to different views of what is an adequate epistemology for mathematics, as well as different mathematical epistemologies. I’ll argue that the resulting framework allows one to provide different characterizations (each relative to a different mathematical epistemology) of the difference between informal and formal mathematical proofs, and the difference between informal and formal rigor.
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Computers, mathematical proof, and the nature of the human mind: a surprising connection
Jeff Buechner
Rutgers University and the Saul Kripke Center, CUNY Graduate Center
buechner@rci.rutgers.edu
Although the use of computers in mathematical proofs antedates the Haken-Appel proof of the four-color theorem in 1976, it was Haken and Appel's proof that created a stir among mathematicians, philosophers, and computer scientists. Was their proof of the four-color theorem a genuine mathematical proof? At that time, Thomas Tymoczko established a conceptual framework for thinking about this issue, and subsequent discussion employed his framework, although some argued that it was deficient. I will argue that a line of thought mentioned (but not developed) by some commentators is necessary for understanding the use of computers in mathematical proofs. In particular, the consensus view of how computers work (accepted by computer scientists, mathematicians, and philosophers) makes it impossible to understand how computers function in mathematical proofs. I will show why this is so by connecting the consensus view of how computers work with a consensus philosophical view about the nature of the human mind. I will close with a speculation about how we might make progress in understanding how computers work, mathematical proof, and the nature of the human mind.
What makes a notation for the natural numbers a good notation?
Jeff Buechner
Rutgers University and the Saul Kripke Center, CUNY Graduate Center
buechner@rci.rutgers.edu
Decimal notation is just one among many distinct notations for the natural numbers. Binary and stroke notation are well-known alternatives. Those who use decimal notation experience a feeling that there is no additional computation to make when, say, they add the positive integers 14 and 17, and obtain the result 31. But that result in binary would require, for someone not versed in binary, an additional computation--into decimal--in order to see that the result is correct. One view is that this experience is relative to one's culture. In a culture in which binary notation is used, the experience would be that no additional calculation is required when the result in binary is obtained. In the early 1990s, in his Princeton seminar, Saul Kripke argued that the cultural relativism view cannot be wholly correct. He conjectured that decimal notation mirrors the "logical structure" of the natural numbers--as presented in the analyses of Russell and Frege--better than other notations. I will discuss Kripke's conjecture, and some problems that it raises.
Are mathematical explanations interest-relative?
Jeff Buechner
Rutgers University and the Saul Kripke Center, CUNY Graduate
Center
buechner@newark.rutgers.edu
Hilary Putnam introduced a wrinkle in the philosophical literature on explanation when he argued that explanations are interest-relative. What counts as an explanation for one set of interests might not count as an explanation for another set of interests. Suppose that some mathematical proofs do provide an explanation of what is proved. Are such explanations interest-relative, or are mathematical explanations via proofs immune to the interest-relativity of explanations? Certainly there can be different explanations of the same theorem--because there are different mathematical proofs of that theorem. For example, the interests of a topologist are satisfied by a topological proof of theorem A, while the interests of a number-theorist are satisfied by a number-theoretic-proof of theorem A. Can there be a topological proof of theorem A which explains A for, say, one topologist but not for another topologist (where both topologists are equally competent)?
What are mathematical objects, and who cares?
John Burgess
Princeton University
The questions philosophers have been asking about mathematics since the end of the period of foundational controversies in the first half of the last century - questions about the nature of mathematical objects and whether there even are such things - are ones that may seem strange to the working mathematician. I will try to explain why these questions have rightly or wrongly seemed natural to philosophers, and indicate something about the directions in which answers are now being sought.
Strict Finite Foundations of Mathematics
John R BurkeRhode Island College
jburke@ric.edu
Strict Finitism is a minority view in the philosophy of mathematics. In this talk, we will give a historical overview of the development of the strict finite philosophical stance. We will review some proposals for strict finite arithmetic and strict finite logic and discuss its viability as a foundation for mathematics. We will then turn our attention to finitism in geometry and announce new research proposing a strict finite foundation for geometric constructions which applies classical logic. Topics which will also be touched on are fictionalism, operationalism, and automated theorem provers.
Mathematical Explanation as an Aesthetic.
Jeremy Case
Taylor University
jrcase@taylor.edu
A traditional view of knowledge is that knowledge is justified true belief. Assuming a mathematical result is true, a person may not believe, or feel satisfied, with a valid argument. Beginning mathematicians often want examples to be convinced. Seasoned mathematicians may need explanations since most have seen counterexamples of seemingly solid proofs. Furthermore, a skeptic could challenge every deduction or claim. When does a proof become accepted? We propose that Mathematics at its core is a creative act, and every creative act has at its core an aesthetic. Mathematical aesthetics provides a necessary guide of mathematical knowledge or at least the acceptance of mathematical knowledge.
The Trapezoid Debate as a Vehicle for Discussing Definitions in Mathematics
John Chase
john.chase@mcpsmd.net
Montgomery County Public Schools (MD)
In 2018, John Chase and Will Rose staged a debate at the national NCTM conference that sparked a math war. The topic? The definition of trapezoids. Many mathematicians staunchly defend the more traditional "exclusive" definition – a quadrilateral with *exactly* one pair of parallel sides. But in recent decades, there has been a groundswell of support for the "inclusive" definition – a quadrilateral with *at least* one pair of parallel sides. After exhaustively summarizing the debate, we will look at how this debate can open larger conversations about the nature of definition in mathematics and weigh the costs and benefits of the "openness" of definitions. By bringing students into this debate, they have a chance to see that mathematics is not finished but is a conversation that welcomes all to participate.
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What is Philosophy of Mathematics? A Case Study of Fictionalism.
Charles Chihara
Philosophy Department
University of California at Berkeley
charles1@socrates.berkeley.edu
This talk explains what philosophy of mathematics is in terms of its goals. The talk then provides an overall assessment of a particular account of mathematics called “Fictionalism” from the perspective of the general account of philosophy of mathematics provided earlier in the talk.
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Fictionalism and Mathematical Practice
Matthew Clemens
Keene State College
mclemens@keene.edu
In a prominent critique of mathematical fictionalism, John Burgess has argued that there is no version of the view that can preserve the desideratum that a philosophy of mathematics be philosophically modest, i.e., non-revisionary with respect to mathematical practice. Several advocates of mathematical fictionalism have recently offered defenses of their views against this critique from Burgess. In this paper, I consider a number of such defenses of fictionalism, and argue that none are compelling solutions for the philosopher of mathematics who aims to respect mathematical practice. By contrast, I suggest that given a significant broadening of the definition of mathematical fictionalism, a fictionalist view might be articulated which is genuinely non-revisionary with respect to mathematical practice. Such a view retains the fictionalist analogy between the mathematical and the fictional, but maintains that the entities of such realms exist as abstract artifacts; call this artifactual fictionalism. As this new view departs radically from traditional fictionalism, I offer some remarks relating artifactual fictionalism to traditional versions of mathematical fictionalism.
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Sally Cockburn
Hamilton College
scockbur@hamilton.edu
A course in naive and axiomatic set theory provides a natural springboard for introducing students to many questions in mathematical philosophy: What is the ontological status of numbers, and does it depend on whether the numbers are finite or transfinite? What criteria should be used to determine the validity of a new mathematical concept, truth or expediency? How do humans, with fallible brains, have access to infallible mathematical truth? Is there any semantic content to mathematics, or it is purely syntax? Does mathematics reside inside human heads, or does it have some sort of external existence? At Hamilton College, I offer a senior seminar in which students spend the first two months learning the technical aspects of set theory using a Moore method approach, and the last month reading papers that address the issues and questions this material inspires. This has proved particularly successful as a “capstone experience” for the concentration.
An unorthodox Philosophy of Mathematics
Alejandro Javier CuneoUniversity of Cordoba, Argentina
alejandro.cuneo@unc.edu.ar
Corresponding: Ruggero Ferro
University of Verona, Italy
ruggero.ferro@univr.it
The classroom experience in teaching mathematics, with its primitive notions, tells us something about how we know mathematics. The development of student’s knowledge puzzles us about the idea of mathematics either as a set of inborn notions or of notions taken from an unreal world existing somewhere. The empirical attitude is closer to the basic way of knowing, but its traditional presentation has difficulties in justifying non-concrete notions. These notions are part of an evolving model of reality created by humans, clashing with a static, preexistent, not experienceable world. Such models of reality become more complex to face the complication of reality, yielding to the impossibility of grasping at once all its features, features that will have to be discovered by future scholars.
We are proposing to upgrade the traditional empiricism by considering not only the usual five senses but also a few well determined internal senses. These, in addition to memory and to specific mental operations producing further perceptions, are adequate to construct mathematical notions.
This position was presented in the chapter "From the Classroom: Towards A New Philosophy of Mathematics" that we contributed to the book MAA Notes 86 "Using the Philosophy of Mathematics in Teaching Mathematics".
"You cannot solder an Abyss with Air" - the Role of Metaphor in Mathematics
Lawrence D'Antonio
Ramapo College
ldant@ramapo.edu
Mathematical discourse is usually seen as being fundamentally different from literary discourse. Both types of discourse must, of necessity, be expressed in terms of a language, but the language of the mathematician seems to have little in common with that of the poet. This paper critically examines that received view by considering examples of figurative language in both mathematics and poetry. To bridge the gap between the familiar and the unfamiliar, the tangible and the intangible, both the poet and the mathematician resorts to a condensed form of speech in which metaphor plays a crucial role.
Molyneux's Problem
Lawrence D'Antonio
Ramapo College
ldant@ramapo.edu
On July 7, 1688, the Irish natural philosopher William Molyneux wrote a letter to John Locke posing the following question. Suppose a person, being blind from birth, having learned to distinguish between a sphere and a cube of equal size by touch, where to suddenly acquire sight; would that person then be able to distinguish the sphere and cube by sight alone? This problem, having philosophical, psychological and mathematical aspects, has been a source of interest and dispute up to the present day. Besides Locke, thinkers such as Berkeley, Leibniz, Voltaire, Diderot, and Helmholtz have discussed the problem (with no consensus as to what the correct answer should be). This talk will discuss the history of this problem and address the issue of the conceptual basis of our perceptions of geometric form.
Euler and the Enlightenment
Lawrence D'Antonio
Ramapo College
ldant@ramapo.edu
The Swiss mathematician and scientist Leonhard Euler is also a key figure in the philosophical discourse of the Enlightenment. In this talk we will take a detailed look at Euler’s contributions to the metaphysics of his era. For example, the theory of causality found itself under attack from the skepticism of Hume and also from philosophers who tried to reconcile Newtonian physics with role of God in the universe. The primary theories of causality in the early 18th century were that of pre-established harmony as put forth by Leibniz and Wolff and the theory of occasionalism as supported by the Cartesians. Against these theories, Euler in his Letters to a German Princess, argued for the interaction of substances known as the theory of physical influx. Euler’s theories of causality, the nature of forces, the divisibility of space, and the general nature of space and time, are important influences on the work of Immanuel Kant.
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The role of the untrue in mathematics
Chandler Davis
University of Toronto
davis@math.utoronto.ca
We reason with untrue statements in the same way as with true ones, else we would have to do without proofs by contradiction. This is our habit since antiquity. Nevertheless the issue assumed different forms in the twentieth century, and I will look into some of them: the recognition of alternative notions of truth and falsity, and the acknowledgement that a statement may sometimes have the same import if untrue as if true.
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The Decline, Fall, and Current Resurgence of Visual Geometry
Philip Davis
Brown University
Consideration of the changing attitudes towards the visual in mathematics raises questions as to the historical reasons for these ups and downs as well as philosophical questions such as what is mathematics, what is proof, what is the relation between mathematics and common sense.
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Some ways of reasoning productive for the logic of mathematical reasoning
| Paul Christian Dawkins, | Kyeong Hah Roh, | Derek Eckman, |
| Texas State University | Arizona State University | Arizona State University |
| pcd27@txstate.edu | khroh@asu.edu | dceckman@asu.edu |
| Steven Ruiz, | Anthony Tucci |
| Arizona State University | Texas State University |
| slruiz3@asu.edu | aat80@txstate.edu |
Our research team has been exploring the relationship between mathematical thinking and logic by exploring whether and how undergraduate students can reinvent key logical concepts by reflecting on mathematical statements and proofs. Observing that modern formalized logic developed quite late historically (in comparison to much of the rest of the undergraduate curriculum) and that logic primarily arose among mathematicians, we were inspired by the hypothesis that logic is formulated on the foundation of mathematical reasoning. By observing novices trying to abstract logic in our experiments, we have come to recognize some important ways of reasoning that we think constitute preconditions for abstracting logic for proof-based mathematics. We will briefly summarize some insights we have gained from eight years of such experiments. While these important ways of reasoning may not be surprising (e.g., associating to any property the whole set of objects that have the property), we think they are insightful for two reasons. Pedagogically, they help us understand what students need to develop to learn mathematical practice. Psychologically, they help identify some of the essential practices of mathematics that may be so familiar we take them for granted. (supported by NSF DUE#1954768 & 1954613)
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Will the real philosophy of mathematics please stand up?
Keith Devlin
Center for the Study of Language and Information
at Stanford University
devlin@stanford.edu
The Philosophy of Mathematics is a fascinating and well-established area of scholastic activity with a long history. Yet is has virtually nothing to say about mathematics as practiced by the majority of individuals who do mathematics on a regular basis, nor of the layperson’s understanding of what mathematics is. There’s nothing wrong in that, but it does mean the discipline’s name does not accurately reflect the meanings of its constituent words, putting it in the same category as “World Series” or “Miss Universe.” What would a “philosophy of mathematics” look like in order to fully justify that title?
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A Unifying Principle Describing How Mathematical Knowledge Unfolds
M. Anne Dow
Maharishi University of Management
mdow@mum.edu
At Maharishi University of Management, we seek fundamental principles unifying various branches of mathematics in order to help students appreciate how the topics they are studying relate to the whole discipline, to themselves, and to knowledge in general. One of the principles we have explored involves a universal pattern or dynamics, by which each theory of mathematics arises from a profound understanding of a particular fundamental concept. Examples are the development of the theory of analysis during the 19th century based on an understanding of the limit process, or the development of the theory of the continuum based on an understanding of the quantification of the continuum by the real numbers. This principle is articulated in a key verse of the Vedic literature [Rig-Veda I.164.39], which, according to the founder of Maharishi University of Management, Maharishi Mahesh Yogi, describes the fundamental dynamics giving rise to and governing the entire universe, and which should be expressed in the fundamental theories of every discipline. We have located these dynamics in several of the major branches of mathematics. In this talk, I will describe Maharishi's interpretation of this key verse and relate it to the theory of the continuum.
Long version:At Maharishi University of Management, we seek fundamental principles unifying various branches of mathematics in order to help students appreciate how the topics they are studying relate to the whole discipline, to themselves, and to knowledge in general. This means that we are concerned with general principles governing the way in which mathematical knowledge unfolds.
The founder of Maharishi University of Management, Maharishi Mahesh Yogi, has stated that the fundamental dynamics of consciousness governing the entire universe, expressed in a key verse of the Vedic literature, are also necessarily expressed in the fundamental theories of every discipline.
In the mathematics department at Maharishi University of Management, we have located these dynamics in the major branches of mathematics: set theory, logic, the theory of the continuum, algebra, analysis, topology, category theory, and many others.
The key verse comes from the Rig-Veda:
"The verses of the Veda exist in the collapse of fullness in the transcendental
field, in which reside all the laws of nature responsible for the whole
manifest universe. He whose awareness is not open to this field, what can
the verses accomplish for him? Those who know this level of reality are
established in evenness, wholeness of life." [Rig-Veda I.164.39]
What does this mean? The idea of consciousness is fundamental to Maharishi's interpretation of this verse. Up to now the term "consciousness" has been excluded from scientific discussion largely because its meaning has been too vague and indefinite. Psychology has mainly dealt with isolated aspects of conscious experience. Maharishi, in his Vedic Science, has provided a highly coherent theoretical account of what consciousness is and a reliable, systematic method by which it can be isolated and directly experienced in its most fundamental state. In this account, consciousness is primary, not an emergent property of matter that comes into existence through the functioning of the human nervous system. It is a vast, unbounded, eternal, unified field, which gives rise to and pervades all manifest phenomena including the human mind, nervous system, and behavior. The method of experiencing it is the practice of Transcendental Meditation, which allows the mind to be drawn beyond being conscious of perceptions, thoughts, feelings, and even individual identity, to identify itself with this field, a state in which consciousness alone is.
According to Maharishi, "fullness" in this verse refers to the unbounded, all-pervading nature of the unified field of consciousness. By virtue of being conscious, this field eternally experiences itself as knower (of itself), as the process of knowing, and as known (the object of its knowing). "Collapse" refers to the flow of attention (knowing) from itself as "fullness" to itself as a single object of knowing. The "transcendental field" in which this collapse takes place is therefore the field of consciousness itself. Further interactions of the three: knower, knowing, and known, each of which is none other than consciousness itself, are reverberations within consciousness. These reverberations were experienced by the ancient seers of the Vedic tradition as a sequence of sounds within the deep silence of their minds, which were recorded as the "verses of the Veda". These reverberations of consciousness constitute the deepest aspect of the innumerable laws of nature, those which govern the eternal, unchanging, unified field of consciousness, and those which science uncovers in the phenomenal world. The final two sentences of the verse tell us that knowledge and direct experience of the unified field of consciousness are necessary for success and fulfillment in life.
Where is this pattern found in mathematics? Let's take, for example, the following view of the theory of the continuum. The theory of the continuum unfolds as a sequence of definitions, theorems, and proofs. These may be perceived as analogous to the "verses of the Veda". Veda means pure knowledge, pure in the sense of consciousness knowing only itself. And these definitions, theorems, and proofs constitute knowledge of a slightly less abstract kind. The theory is based on ("exists in") the quantification of the continuum by the real numbers. This quantification takes place by means of nested infinite sequences of intervals, whose intersections are single points. Each of the sequences of nested intervals gives rise to an infinite decimal expansion that represents the point. Thus we identify the continuum itself with "fullness" and its quantification into points with "collapse". The phrase "in the transcendental field" may be taken to correspond to the fact that this quantification is properly described and supported by the abstract field of set theory. Set theory permits one to define and manipulate the infinite sequences of intervals, and to collect the resulting infinite decimals together into an uncountable set with an algebraic structure and a natural ordering. Then "in which reside all the laws of nature" can be taken to mean, in this context, that set theory enables us to deal with the rules ("natural laws") governing the diverse structures of the continuum: its algebraic structure of addition and multiplication and its geometric structure based on the natural ordering and the topological completeness of the real numbers. If one then thinks of applications of the theory of the continuum, the phrase "responsible for the whole manifest universe" corresponds to the fact that the integration of algebraic operations and geometric continuity in the continuum of numbers makes it possible to represent and completely quantify any continuous process using transformations (functions) of the continuum within itself.
The remaining two sentences of the verse are clear if one considers the history of calculus in the 19th century. At the beginning of the 19th century the foundations of calculus were very shaky. For example, Fourier's seminal paper on heat propagation was rejected because of the limited understanding of the concept of a function and because of basic unanswered questions about convergence of series of functions. By classifying and formalizing the concept of the infinite, set theory provided a foundation for the theory of the continuum and hence for modern analysis.
There are many other examples of this principle to be found in the theories of mathematics. Students express appreciation for the unity of mathematical thought when they see this same fundamental principle operating in the different areas of mathematics they study.
Why Plato was not a Platonist
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Platonism is one of the terms most widely used in discussion of the philosophy of mathematics. It might be assumed that this approach is based on insights to be found in the works of Plato. A quick check of a recent volume devoted to Platonism in the philosophy of mathematics of about 200 pages locates one reference to Plato, in a footnote. If this is the case, there is room for the suspicion that Plato's own views of mathematics have been lost in the course of the philosophical programme known as Platonism.
Plato's works span many years, and their dialogue form can make it difficult to determine which views were his and which only stalking-horses. Certainly the discussion of abstract objects and their centrality in Plato's view of human knowledge are elements that Platonism has not abandoned. On the other hand, Plato is reluctant to give the title of 'knowledge' to much that passes under that name in ordinary usage. If Platonism seeks to understand how so much mathematical knowledge is possible, Plato himself was perhaps more concerned with its fallibility.
One can argue that Platonism involves more than a tincture of Aristotle in addition to the Platonic elements. Aristotle introduced 'formal logic' to the scholarly community, even if logic in some form had scarcely needed to be invented. The basic assumption of formal logic (the notion of logical validity and arguments being true by virtue of their form) were not part of the Platonic arsenal. With the tools of formal logic, Platonism has gone well beyond what Plato would have recognized.
If there is one school of mathematical philosophy of the twentieth century that Plato might have recognized, it was the intuitionism of L.E.J. Brouwer. Brouwer crucially felt that mathematics preceded logic, and with that Plato would have felt at home. Brouwer claimed that language did not adequately capture mathematics, another claim that Plato could have endorsed. Brouwer found the essence of mathematics in the mind of the mathematician, and with that Plato would have quarreled. However, the similarity between Brouwer's notions and Plato's views of mathematics suggests that Plato's legacy may be more alive in philosophical perspectives not bearing his name.
Fictionalism and the interpretation of mathematical discourse
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
One of the popular ways to provide an understanding for mathematical discourse has been via fi ctionalism, the notion that mathematical objects have the same kind of existence that characters do in fiction. This approach suffers from a number of problems in detail, but there is a fundamental issue about the way in which mathematics is carried on that differs from the kind of narratives with which it is compared. Story-telling, if successful, generates a suspension of disbelief. Mathematics needs to achieve a higher level of both involvement and assent. This paper tries to distinguish the standards required in mathematics, and draws on some Platonic distinctions between different sorts of craft.
Dummett Down: Intuitionism and Mathematical Existence
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Michael Dummett's views on global anti-realism were shaped by his technical work on intuitionism. In particular, his criteria for existence are based on an intuitionistic view of truth. From this has sprung a whole array of anti-realisms that are discipline-specific. Whether that anti-realism fits the issue of the existence of mathematical objects particularly well is not resolved by this account of its origins. There was, after all, intuitionism before the formalization created by Heyting and pursued by many others. Here the history of intuitionism will be used to separate the Dummettian programme in general from the contribution intuitionism can make to understanding statements about mathematical objects.
Putting Content into a Fictionalist Account of Mathematics for Non-Mathematicians
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Non-mathematicians will often take a course in mathematics and literature with a much greater degree of comfort with the literary side than with the mathematical side. This comes partly from their sense of mathematics as a collection of rules handed down to them in classrooms of years past. One way to try to bridge the gap is not just to look at the mathematical aspects of literary structure and the representation of mathematical ideas in literature. Instead, one can explain the notion of fictionalism as a positive characterization of mathematical objects. Old-style fictionalism took mathematics as simply a tissue of useful lies. A more constructive fictionalism takes seriously the resemblance to fiction, especially for those who have put some time into trying to understand statements in fiction and their truth values. The repudiation of literalism on both sides of the divide (mathematical and literary) leads to a rapprochement of understanding the statements in mathematics, literature, and perhaps other disciplines as well.
Thought in Mathematical Practice
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Palle Yourgrau has recently argued that mathematics as currently practiced is a domain from which thought is absent. His claim is that philosophers who have tried to carry mathematical techniques over into metaphysics have fallen short because the questions that arise in philosophical discussions require thought and not just the application of technique. He points to a thread of criticism of mathematics that goes back to Plato. In this paper an attempt will be made to characterize stages in the doing of mathematics that require thought on the part of those performing them. While there are aspects of mathematical practice that are formulaic enough to appear not to require thinking, it is throwing babies out with bathwater to abandon what mathematics has to offer to the practice of metaphysics.
Mathematical Progress via Philosophy
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Mathematicians complain about the extent to which questions in the philosophy of their subject remain unaltered after thousands of years, while the discipline of mathematics itself seems to make indubitable progress. This talk looks at some of the issues in the philosophy of mathematics, from Aristotle to the twentieth century, that have led to advances within mathematics itself. The philosophical questions do not have to be resolved in order for work on them to contribute to mathematical advancement. While there may be no general agreement among the mathematical community about answers to certain philosophical questions involving the foundations of mathematics, there is no doubt that reflecting on foundations has led to interesting and important mathematics.
Zeno Will Rise Again
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
The adage that history is written by the victors has been as true in mathematics as elsewhere. When one looks at texts in the history of mathematics, there is more attention paid to the developments of the past that can be construed as leading to what mathematicians do today than to avenues that have proved to be dead ends. It is not surprising that mathematicians are interested in the roots of what they do, and the Whig interpretation of history cuts across many disciplines. Texts in the philosophy of mathematics are more catholic in their accounts of the past. This may be the result of the sense that no philosophical position, however unfashionable, is incapable of resuscitation by later hands and arguments.
Mathematicians are willing to relegate pieces of the past to a footnote, while philosophers do not readily inter those pieces. When one looks at the history of the philosophy of mathematics, it looks more like a spiral than a chronicle of progress. This talk will look at particular examples of the revival of philosophical positions and the difference in attitude toward the past between historians and philosophers.
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Explanatory and Justificatory Proofs.
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Michael Dummett has pointed to the difference between explanatory and justificatory proofs. It is also a distinction familiar to those who have to explain to a class that mathematical induction does not give the user a way to discover what is to be proved, but only to justify a particular result. As students proceed in their studies of mathematics, proofs that may originally have seemed purely justificatory take on an explanatory structure. This talk will look at Dummett’s distinction to see if it is more than a reflection of the level of mathematical experience of the prover.
Role of Real Numbers in an Introduction to Analysis
Thomas Drucker
University
of Wisconsin-Whitewater
druckert@uww.edu
Most of the courses a student will have taken up to an introduction to analysis will not address in any depth the question of what sort of objects the numbers are which appear in calculations. By the time students have finished an introduction to analysis, one would like them to be mildly familiar with what numbers are. Of course, that can be accomplished by presenting them with an axiomatization of, say, a real closed field. It makes more sense to look at what kinds of properties one needs in order to be able to prove familiar results. By this stage in a student’s career, there should be no danger of the student’s believing that axioms were handed down from a mathematical Mount Sinai. Instead, it is both more appropriate and exciting for the student to see how much has to be built into an axiom system in order for a user to be able to prove what is needed.
Why Can't Those With Conflicting Views on the Foundations of Mathematics Just Get Along?
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
There has been ongoing strife over the issue of whether set theory or category theory is the appropriate foundation for mathematics. Claims have been made as to the relative merits of one or the other with regard to certain branches of mathematics. For many mathematicians the issue of foundations is irrelevant, but that has not stopped the arguments. Can mathematicians do some, most, or even all mathematics without worrying about the choice of foundation? One can do arithmetic in different bases, but somehow different logics affect the content of mathematics rather more profoundly. The content of this talk will continue earlier investigations along the lines of Henle's 'The Happy Formalist'.
Why is There a Question About Why There is Philosophy of Mathematics At All?
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Ian Hacking has observed that philosophizing about mathematics is haunted by Plato’s ghost. Reuben Hersh was part of a movement to lay Plato’s ghost in understanding how mathematics works. His approach was based on observing mathematical practice and eliminating some of the traditional problems raised in a philosophical setting. This talk suggests that, while Hersh’s approach has some distinctive features, it’s not as though it suffices to lay Platonic issues entirely at rest.
Mathematics, Bivalence, and Alternative Logics
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Alternative logics have been studied as ways of understanding different branches of mathematics. This has led to the notion of competition between the different logics to earn the title of the 'correct' foundation. This talk will suggest that there is more cooperation than competition between the alternatives.
Three Hundred and Sixty-Four, Of Course: Foundations from Philosophy to Program
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
In 'Through the Looking-Glass and What Alice Found There' Lewis Carroll introduced the character of Humpty-Dumpty as an example of a philosopher without any technical understanding of mathematics. By the early twentieth century philosophical programs like intuitionism were emerging out of technical considerations. In his contribution to the Oxford Handbook of Philosophy of Mathematics and Logic, the late David McCarty makes the notorious comment, "...contemporary intuitionism is not a philosophy of mathematics." This talk makes the claim that it is possible to be both a philosophical position and a mathematical program.
Lewis Carroll and Mr. B*rtr*nd R*ss*ll: P.E.B. Jourdain on the State of Mathematical Philosophy in 1918
Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu
Philip E.B. Jourdain's 'The Philosophy of Mr. B*rtr*nd R*ss*ll' (1918) has generally been taken as a humorous assault on the writings of philosophers having to do with mathematics. He is working in the tradition of Lewis Carroll with entertaining results. There is also, however, a serious thread running through the book, as illustrated by Jourdain’s ongoing correspondence with Bertrand Russell. This talk looks at the philosophical points made by Dodgson, even in his humorous writings, and Jourdain’s demonstration that some of those points had not been satisfactorily addressed in the half-century since Dodgson wrote.
A Priori Concepts in Euclidean Proof
Peter Epstein
Brandeis University
pepstein@brandeis.edu
For over two millennia, Euclid's Elements was taken to be a paradigm of a priori reasoning. With the discovery that there are consistent non-Euclidean geometries, the status of our geometrical knowledge was radically undermined. In the wake of this upheaval, philosophers proposed two revisionary interpretations of Euclidean proof. Some, following Hilbert, suggested that we understand Euclidean proof as a purely formal system of deductive logic -- one not concerned with specifically geometrical content at all. Others suggested that Euclidean proof is in fact empirical, employing concepts derived, in some fashion, from sensory experience. In recent decades, this second interpretation has been endorsed by a wave of philosophers who, eschewing the traditional analysis of mathematics by way of "rational reconstructions"--i.e., attempts, like Hilbert's, to understand what rigorously logical underpinnings a mathematical system might have, without regard for whether the mathematicians who developed the system were actually driven by such considerations—have sought instead to achieve a deeper understanding of mathematical practice. In assessing how Euclid’s proof system actually operates, they suggest, we must attend to the crucial role of Euclid’s diagrams: according to this second interpretation of Euclidean proof, the diagrams, and our visual or imaginative mental representations of them, fill in the logical gaps in Euclid’s system. In this talk, I will argue that both interpretations fail to capture the true nature of our geometrical reasoning. Euclidean proof is not, as Hilbert would have it, a purely formal system of deductive logic; rather, it is one in which our grasp of the content of geometrical concepts plays a central role. These geometrical concepts, however, are not, as the philosophers of mathematical practice suggest, ones we derive from diagrams, visual experience, or imagination. They are, instead, genuinely a priori.
What's So Special about Deductive Proof?
Don Fallis
Northeastern University
d.fallis@northeastern.edu
Mathematics is famously a deductive science. That is, in order to establish that a mathematical claim is true, mathematicians prove it. And this is not just a matter of stylistic preference. We tend to think that deductive proof is better at securing mathematical knowledge than inductive evidence. The traditional view is that, although mathematicians sometimes make mistakes even when they restrict themselves to deductive proof, deductive proof is more reliable than inductive evidence.
However, in the last fifty years or so, mathematicians have developed many Monte Carlo methods for checking mathematical claims (see Rajeev and Raghavan 1996). For example, there are "probabilistic proofs" that can establish with arbitrarily high probability that a number is prime (e.g., Rabin 1980). As a result of these developments, some mathematicians (e.g., Zeilberger 1993, Borwein 2008) claim that mathematicians should use probabilistic proofs to establish that mathematical claims are true when such extremely reliable inductive evidence is available. And a few philosophers (e.g., Fallis 1997, Paseau 2015) have wondered whether there really is any epistemic (i.e., knowledge-related) value of deductive proof that inductive evidence lacks.
In response, several philosophers (e.g., Easwaran 2009, Smith 2016, 48-50, Berry 2019, Hamami 2022, Hamami forthcoming) have subsequently tried to identify the distinctive epistemic value of deductive proof. These proposals can definitely help us to understand why deductive proof is as reliable as it is. But as I will argue, since the identified properties of deductive proof are only valuable as a means to reliability, these proposals are ultimately unsatisfying. They do not provide us with any reason not to use inductive evidence when it is just as reliable as deductive proof in establishing that a mathematical claim is true. In other words, we have yet to identify the distinctive epistemic value of deductive proof.
From an Analysis of Definitions to a View of Mathematics
Ruggero Ferro
Univ Degli Studi di Verona
ruggero.ferro@univr.it
It is impossible to give a meaning to all words through explicit definitions. Indeed one would be bound either to vicious circles or to infinite descents. Hence mathematics assume certain words as primitive, i.e. words the meaning of which is assumed to be known even without definitions. An attempt to specify the meaning of a primitive word using the language could consist in describing the properties, the behavior and the characteristics of the meaning of that word (these descriptions may use the word the meaning of which is being looked for). This attempt would succeed if a rich enough description can be obtained such that it is satisfied only by that meaning. In mathematical logic, it is shown that, no matter how rich a language could be, even if the description consists of all the sentences in the rich language that are true of a certain notion, there are non isomorphic notions that satisfy the same description. Thus even the axiomatic approach cannot specify the meaning of a primitive word. Hence the language is not adequate to identify the primitive notions of mathematics. But, is there a meaning of the primitive words? If so, how to specify it, and how to communicate it?
Remarks about the notion of EXISTENCE in mathematics
Ruggero FerroUniv Degli Studi di Verona
ruggero.ferro@univr.it
There are different situations in which we use the word “exist” and the meaning meant in each one may not be exactly the same.
Here is a list of sentences in which the word “exist” occurs. I exist. The pain that I feel exists. This pen exists. Kangaroos exist. Different objects exist. The reality exists. The parenthood relation among humans exists. The order relation among natural numbers exists. A certain relation exists. An event exist. A procedure exists. A project exists. A model exists. A need exists. An opportunity exists.
What is the relationship between “it exists” and “there is?” Examples of uses of there is: there is in my fantasy; there is in my hopes; there is among my projects; there is in my dreams. Examples of existence in mathematics: existence of a specific number system; existence of a single number; existence of a solution (and the solution may be either a way of behaving or an object with adequate properties).
The presence of all these different situations requires a closer attention to what is meant by it exists. In my presentation, I plan to make same remarks and comments on the above hinted difficulties.
An analysis of the notion of natural number
Ruggero Ferro
Univ Degli Studi di Verona
ruggero.ferro@univr.it
I would like to address the theme of this meeting selecting the notion of natural number. I will try to point out the human problems and needs that motivate the elaboration of the notion of natural numbers, and to illustrate the steps and the choices made to arrive to a solution of the problems. The main problem is to compare quantities of elements. A procedure that could solve the problem in some difficult cases is that of counting. By counting we associate to each finite collection an ordered collection of iterations of the mental acts of considering a further element. These ordered collections could be viewed as the natural numbers. At this point we have two possible line of development. One, we can examine the structure of the collection of the entities that were introduced and the problem of infinity that it is raised. Two, one can consider the steps taken along the way of constructing the proposed notion of natural number, and analyze what it is needed to perform them. Most of the steps require introspections. This is due to the fact that we have to use internal perceptions. This notion of natural numbers somehow answers the question about their nature; and their existence is similar to the existence of plans, projects, organization, and mental activity.
Abstraction and objectivity in mathematics
Ruggero Ferro
Univ Degli Studi di Verona
ruggero.ferro@univr.it
I would like to read the theme of this conference the other way around: which problems in the philosophy of mathematics are raised from the teaching/learning perspective? For example. Why can we learn and understand mathematics? How do we learn mathematics? We cannot appeal to general philosophical principles and derive answers from them, because we would fall into a vicious circle: how do we know that a proposed philosophy is correct and can justify the deriving theory of knowledge? To avoid this, one has to investigate the ways of knowing and learning mathematics without any reference to a preconstrued theory. But this process is internal to the human being. From outside we can only observe consequences and results of having acquired a notion. Even a description of what it is being done, is just a description in a language and should be interpreted. Being impossible to analyze the process from outside, why not trying to look at it from inside through introspection? The conclusions would be subjective! Why so? We are just talking about learning and understanding mathematics. I would like to show that, along this way, something could be said, for instance about abstraction, and the conclusions should be considered objective, according to a reasonable notion of objectivity.
How Do I (We) Know Mathematics
Ruggero Ferro
Univ Degli Studi di Verona
ruggero.ferro@univr.it
I view the philosophies of knowledge divided into three broad groups. Some of them deduce their position about the process of knowing from general ideas about the nature of humankind, with the difficulty of justifying how do they know the correctness of their views. Some others want to be experimental, observing what other people do during the process of knowing, forgetting that they have to interpret and guess what in happening inside them, since the language is not as transparent as it is often assumed to be. Noting the difficulties faced by the other positions, a third group reverts to a mysterious unborn human capability to know. Knowledge is a personal endeavor: not only with respect to the acquired knowledge, but also the process of coming to know is very personal. Thus a fourth position can be imagined according to which a central role is played by introspection, i.e. I have an idea of what it is to know by analyzing within myself the way I come to know. To support this position one should make explicit what is seen by analyzing the process of knowing within oneself, how it relates to other people knowledge, and one should show how we can reach our actual knowledge (of mathematics in particular) through the detected process. My exposition will develop these points.
Mathematics vs Philosophy
Ruggero Ferro
Univ Degli Studi di Verona
ruggero.ferro@univr.it
I claim that the mismatch between the progress in mathematics and in philosophy is not surprising.
- Philosophy’s desire to answer the most fundamental questions of humankind is perhaps too ambitious.
- OK Scire per causas. But how to detect the causes of the situation that we experiment?
- Philosophy touches upon very sensitive topics such as personal beliefs, morality. Here the arguments to reach an agreement are not only deductive.
- Epistemological views are introduced within a theoretical system, and not beforehand to justify it.
Can a philosophy accept that we cannot justify everything, due to the human limitations?
On the other hand mathematics is more humble, if not coward.
- No one claims to know exactly the meaning of the axioms.
- Various principles are used, but don’t ask why they should be accepted.
- Proofs should be easily checked, but no one cares how they were devised.
- Mathematics is a good organization of multiplicity: by dropping information, a situation becomes manageable.
- “What is mathematics?” is a question dismissed as non-mathematical.
The role of language is central to many of these points.
To face some previous point, the internal non-physical experience is needed.
No surprise for the effectiveness of mathematics in the natural sciences
Ruggero FerroUniv Degli Studi di Verona
ruggero.ferro@univr.it
There are views of mathematics for which it is obvious that pure and abstract mathematics has to be efficient in application. I claim that mathematics is a human attempt to tame the complication of multiplicity. Complication is the main limit to understanding. Thus we abstract, from the available data, those that we deem relevant. We also idealize (introducing aspects not present in the data) and generalize. These three mental operations lead us to build, on experienced data, a sufficiently manageable model of the situation (reality) differing from the situation analyzed, but approximating it well enough, even though introducing complexity. This is true not only of mathematics, but also of physics and of each of the other natural sciences: they develop theories describing models. Since models may become very complex, ingenuity is needed to understand them, making models object of scrutiny, comparisons and evaluations. It should be no surprise that advanced mathematical results are useful, because, since the beginning, they were meant to tame the complication of multiplicity, possibly even the kind of multiplicity present in a specific application. The presentation will try to justify the claims proposed and to answer more directly to the theme of this meeting.
An analogy to help understanding Discovery, Insight and Invention in Mathematics
Ruggero Ferro
Universita' di Verona
ruggero.ferro@univr.it
An analogy with the discovery of how life would be evolving in a town to where one is moving in may help us to understand what could be meant by discovery, insight and inventing in mathematics. The key common features of these two environments that I will try to point out range from 1) the realization that anything observed is contingent; to 2) the very reasonable hypothesis that anything that was build responded to some need, requirement, convenience or development; 3) what was previously constructed has some influence and bearing on what is done afterwards; 4) an understanding of the motivation of what was done and of the manner in which it was realized are needed to continue the construction; 5) the needs and requirements are continuously evolving and newly invented artifacts or improvements should be added to face them; to 6) not every invented addition meets the situation and the requirements with the same short range and long range convenience, thus a preventive evaluation is convenient according to criteria to be established. I will also try to underline the difference between the attitude proposed and the one claiming that in mathematics everything ought to be so, it can't be but so, due to an a-priori mental evidence, since this is the truth.
Title: Trust but verify: What can we know about the reliability of a computer-generated result?
Nicolas Fillion
Simon Fraser University
Department of Philosophy
nfillion@sfu.ca
Abstract: Since the Second World War, science has become increasingly reliant on the use of computers to perform mathematical work. Today, computers have justifiably become a trusted ally of scientists and mathematicians. At the same time, there is a panoply of cases in which computers generate demonstrably incorrect results; and there is currently no reason to expect that this situation will change. This prompts the careful user to verify computer-generated results, but it is clear that we are often not in a position to review the work of computers as we would traditionally review a putative derivation or calculation. In this sense, computational processes are epistemically opaque.
Since Humphreys introduced the phrase `epistemic opacity' in the philosophical literature in 2004, the concept of opacity has been developed along different lines; furthermore, many incompatible claims have been advanced---be they about what opacity is or about whether we should worry about it---leaving this field of the philosophy of computing in a state of confusion. In this paper, we propose a framework that disentangles three core questions (1. What kinds of epistemic opacity are there in scientific computing? 2. Should we worry about epistemic opacity? 3. Should we seek greater transparency whenever possible?) and systematically survey how their answers inter-relate.
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The Square Root of 2, Pi, and the King of France: Ontological and Epistemological Issues Encountered (and Ignored) in Introductory Mathematics Courses
Martin E. Flashman
Humboldt State University and Occidental College
flashman@axe.humboldt.edu
Students in many beginning college level courses are presented with proofs that the square root of 2 is irrational along with statements about the irrationality and transcendence of pi. In Bertrand Russell’s 1905 landmark article ”On Denoting” one of the central examples was the statement, ”The present King of France is bald.” In this presentation the author will discuss both the ontological and epistemological connections between these examples in trying to find a sensible and convincing explanation for the difficulties that are usually ignored in introductory presentations; namely, what is it that makes the square root of 2 and pi numbers and how do we know anything about them?
If time permits the author will also discuss the possible value in raising these issues at the level of introductory college mathematics.
Dedicated to the memory of Jean van Heijenoort.
What Place Does Philosophy Have in Teaching Mathematics?
Martin E Flashman
Humboldt State University
flashman@humboldt.edu
In recent years discussion of the history of mathematics has grown in its treatment in mathematics courses from precalculus through advanced courses such as number theory, algebra, geometry and analysis. The speaker will address the question of what role the philosophy of mathematics might take in these and similar undergraduate courses.
Which Came First? The Philosophy, the History, or the Mathematics?
Martin E Flashman
Humboldt State University
flashman@humboldt.edu
The author will give examples from instruction where mathematics interacts with its history and philosophy in the context of a content based course illustrating how this interaction can enhance learning.
The Articulation of Mathematics - A Pragmatic/Constructive Approach to The Philosophy of Mathematics
Martin E Flashman
Humboldt State University
flashman@humboldt.edu
The philosophy of mathematics has often taken mathematics as a realm of discourse that is fixed. The investigation of this realm is what working mathematicians take as their task. This work leads to results and reports on what they have ascertained. Accompanying communications allow others to achieve comparable experiences of understanding or to accept the results for further investigations. The author will discuss an alternative "constructive" view: The mathematical realm is dynamic and changing while the work of mathematicians involves the articulation of this realm as a pragmatic work in progress.
Square Roots: Adding Philosophical Contexts and Issues to Enhance Understanding
Martin E Flashman
Humboldt State University
flashman@humboldt.edu
The nature of numbers can be confusing to students in a variety of learning contexts. One frequently encountered area of confusion surrounds numbers described as square roots, such as the square root of 2 and the square root of -1. The author will examine how illuminating some philosophical approaches to the nature of numbers (ontology) and knowledge about numbers and their properties (epistomology) can help students avoid some possible confusion. Time permitting the author may suggest possible empirical studies for (college level) students to provide evidence for the utility of introducing more philosophical approaches to pedagogy.
Logic is Not Epistemology: Should Philosophy Play a Larger Role in Learning about Proofs?
Martin E Flashman
Humboldt State University
flashman@humboldt.edu
Many courses designed to provide a transition from lower division computational courses to upper division proof and theory courses start with a review or introduction to what is often described as "logic". The author suggests that many students would be better served with an alternative approach that connects notions of proof with philosophical discussions related to ontology and epistemology. Some examples will be offered to illustrate possible changes in focus based on the author's experiences teaching such courses over the past 25 years.
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Is Philosophy of Mathematics Important for Teachers?
Martin Flashman
Humboldt State University
flashman@humboldt.edu
There has been much interest in recent years on what mathematical preparation is important for future teachers at all levels. Recommendations from the MAA CUPM on Undergraduate Curriculum and the Common Core in Mathematics are silent on the issue of what role the philosophy of mathematics can play. The author will suggest examples where a discussion of some issues from the philosophy of mathematics in courses taken by future teachers can enrich their backgrounds and training.
Do We Need a Separate Philosophy of Geometry?
Martin Flashman
Humboldt State University
and University of Arizona
flashman@humboldt.edu
The philosophy of mathematics has long been focused primarily on topics such as the ontology of numbers and sets and the epistemology of results in the theory of numbers (arithmetic) and sets through issues of axioms and proofs for these theories. Though much of mathematics today seems to involve geometry in one form or another, the philosophical issues of geometry seem to receive little attention, treated as subservient to the general philosophy of mathematics or considered a part of the philosophy of physics. The author will consider why the issues of geometry could use a distinct discussion of its philosophical issues for both intrinsic and pedagogical reasons.
Turing and Wittgenstein
Juliet Floyd
Boston University
jfloyd@bu.edu
On 30 July 1947 Wittgenstein penned a series of remarks that have become well-known to those interested in his writings on mathematics. It begins with the remark “Turings ‘machines’: these machines are humans who calculate. And one might express what he says also in the form of games.” Though most of the extant literature interprets the remark as a criticism of Turing's philosophy of mind (that is, a criticism of forms of computationalist or functionalist behaviorism, reductionism and/or mechanism often associated with Turing), its content applies directly to the foundations of mathematics. For immediately after mentioning Turing, Wittgenstein frames what he calls a "variant" of Cantor's diagonal proof. We present and assess Wittgenstein's variant, contending that it forms a distinctive form of proof, and an elaboration rather than a rejection of Turing or Cantor.
Back to 2012 Invited Paper schedule
Wittgenstein, Turing and 'Surveyability'
Juliet Floyd
Boston University
jfloyd@bu.edu
Recent debates about the philosophical status of formalization and mechanization of proof may be illuminated by considering the mutual impact Wittgenstein and Turing had on one another around issues concerning the evolution of notations in symbolic logic. When Wittgenstein remarked in 1937 that 'a proof must be surveyable' he was reworking ideas of Frege, Hilbert and Turing. "Surveyability" for Wittgenstein was neither a verificationist requirement nor a refutation of the claim that all proofs must have corresponding formal proofs, much less a refutation of logicism. Instead, it placed front and center what mathematicians do, i.e., it explores what logicism comes to in an everyday sense. The idea, consonant with certain trends in so-called "philosophy of mathematical practice", is not to provide or ask for a "foundation" for mathematics in any ordinary sense, but rather to take a pragmatic approach to the very idea of "foundations". For Wittgenstein, as for Frege and Hilbert, communicability is a central norm of proof, and the idea of "surveyability" furthers it. Turing’s everyday machine model of computation in his 1936 paper explicitly incorporates this ideal. Wittgenstein's path toward his greatest work, Philosophical Investigations, shifted in response to Turing's work on the Entscheidungsproblem: he concertedly developed the notions of mathematical "technique" and "form of life" beginning in 1937, going beyond the notions of "practice" and "language-game". In 1939 he and Turing discussed these ideas in his Cambridge lectures on the foundations of mathematics, sparking some of Turing’s subsequent work on types. The relevant ideas here draw out new ways of looking at Turing's 1936 paper, as well as his more speculative writings in the late 1940s about "intelligent machinery" and his 1950 "Turing Test".
Is the Proof in the Picture? Seeing, Believing, and Proving
Janet Folina
Macalester College
folina@macalester.edu
What is the role of visual information in mathematics? Can diagrams justify inferences in traditional, verbal proofs? Can pictures, or diagrams, be proofs on their own? There is much disagreement on these issues among both mathematicians and philosophers; part of the reason for the disagreement is confusion. The aim of this talk is to clarify some of the philosophical issues underlying disputes over the role of visual information in proofs. Diagrams can be highly convincing, useful for explaining, they can efficiently depict mathematical information. But that does not mean they are proofs. This talk will appeal to some general considerations in epistemology to explain the view that pictures fall short of being genuine mathematical proofs. But proofs are just one tool in the mathematician's toolbox; we will also aim to clarify why pictures can be so useful, convincing, and even justifying!
Back to Mathfest 2012 Guest Lecture.
The Rigour of Proof
Michele Friend
George Washington University
Philosophy Department
What is a rigorous proof? When is a proof sufficiently rigorous? What is the importance of rigour in a mathematical proof?
To answer the first question, we begin with a comparison between a formal proof and a rigorous proof. A rigorous proof need not be formal, but it needs to be possible, in principle, to make it formal. We might even have an informal proof to this effect, or suspect we could give one if called upon to do so.
To answer the second, we start with the very obvious looking distinction between sufficiently rigorous for acceptance by other mathematicians, sufficiently rigorous to establish a result and sufficiently rigorous to elicit further questions. Of course the latter does not come only from the rigour of the proof, but also its originality and importance. Nevertheless, rigour does play a role because it ensures a degree of transparency.
The importance of rigour in a proof has several answers. A realist about the ontology of mathematics might well accept a non-rigorous proof since it establishes a truth guaranteed by the ontology of mathematics, in this case rigour is of psychological or epistemological importance at best. It can be used to assuage doubt, or it can help a mathematician to know the truth of the conclusion by another means than by simply intuitively grasping the truth or it can help with understanding why the conclusion is true.
Some constructivist philosophers would assert that the term `rigorous proof' is redundant, since for them, a proof lacking in rigour is not a proof, it is at best a purported proof.
There is a less categorical stance than either of the above that we can take. We take a more nuanced view when we consider mathematical practice, purpose, meaning and theoretical context.
Slides from the talk are here. For more details, see her book, Pluralism in Mathematics; A New Position in Philosophy of Mathematics.
Back to JMM 2019 guest lecture.
Related Rates and Right Triangles: Developing Intuition in a Calculus Course
Benjamin Gaines
Iona College
gaines.benjamin@gmail.com
Often one area where first year Calculus students run into significant difficulties is in setting up word problems. For instance, there are many types of related rates questions involving right triangles that are approached in different ways. These problems provide an interesting interplay between the firm logical rules students are used to working with to that point, and intuition in choosing how to approach a problem that they have not seen before. In this talk we discuss our observations of how the surface similarities of these problems can confuse students, and methods we've used to help students develop an intuition for deciding which method will be appropriate.
Back to MathFest 2022 schedule
Hermann Minkowski - Einstein’s Metaphysician
Steven Gimbel
Gettysburg College
Hermann Minkowski was Albert Einstein’s professor in college, but the real lesson he taught Einstein came years later when he framed the geometric interpretation the special theory of relativity and claimed that its truly revolutionary result was not its non-Newtonian mechanics, but what it said about the unified nature of space and time. Minkowski was a mathematician whose use of late 19th century geometric tools to endow physical theory with an ontological significance was a radical departure from the formalist picture of mathematics set out by Minkowski’s life-long best friend David Hilbert, but anticipated views connecting mathematics, science, and philosophy by half a century.
Back to JMM 2014 guest lecture.
What is Mathematics I: The Question
Bonnie Gold
Monmouth University
bgold@monmouth.edu
The question, “What is mathematics?” can have many meanings. It can mean, “What are the subjects which are called mathematics?” In some sense, it was this question which Courrant and Robbins’ book, “What is Mathematics?” was answering. It can mean, “What is the nature of the objects of mathematics?” This, primarily, was the topic of Reuben Hersh’s “What is Mathematics, Really?” It can mean, “What is special about how we reason in mathematics, or about how we do mathematics?” There is yet another interpretation of this question, however, which this talk will begin to discuss: “What is the common nature of those subjects which are called mathematics which causes us to lump them together under this common name?”
I plan in this talk to examine some answers which have been given in the past to the question, “What is mathematics?” and why I believe they are not adequate. I shall then discuss some criteria which a good answer to the question should have, and why these are important criteria.
More Details Back to 2003 schedule
What Is Mathematics II: A Possible Answer
Bonnie Gold
Monmouth University
bgold@monmouth.edu
At my last talk at a POMSIGMAA contributed paper session two years ago, I tried to define a version of the question, "What is Mathematics?" The version I would like to answer is, "What is common to all those subjects we classify as mathematics, and not common to most things we don't classify as mathematics, by virtue of which we classify those subjects as mathematics." I discussed various answers which have been given, and suggested why none of these is an adequate answer. I also indicated various criteria a good answer should have. In this talk, I will propose one (or possibly two) answers to the question, and discuss the extent to which they meet my criteria.
Mathematical objects may be abstract, but they're NOT acausal
Bonnie Gold
Monmouth University
bgold@monmouth.edu
Although many mathematicians are closet Platonists, they are hesitant to embrace platonism openly because of the challenges philosophers have issued to the view. The problem is, if mathematical objects are outside of spacetime and have no causal interactions with people, how can people gain mathematical knowledge. In my talk I will challenge the view that mathematical objects are acausal, even though I agree that they are abstract. I accept that we cannot act in a causal way on mathematical objects - that is, I can't make four be prime or the Klein-four group be cyclic. But mathematical objects DO have causal-type effects on the world, of a variety of types. Some involve their effects on human thinking, but others involve physical objects. This talk is a very preliminary version of an article I hope eventually to publish, and I will be very interested in audience response.
Philosophical Questions You DO Take a Stand on When You Teach First-year Mathematics Courses
Bonnie Gold
Monmouth University
bgold@monmouth.edu
Most mathematicians have no interest in the philosophy of mathematics, and, when asked about their philosophical views, reply that they leave that to philosophers. However, in fact, in the process of teaching undergraduate courses, we DO take a stand on a range of philosophical questions, in most cases unconsciously. I've become aware of more and more of these as I've gotten involved in the philosophy of mathematics. They range from the well-known – the Intermediate Value Theorem is not a theorem from an intuitionistic perspective – to the more subtle. Some of them are closely related to errors students persistently make or misunderstandings they have. I will discuss several that we all take stands on when we teach first-year mathematics courses such as calculus and introduction to proof, and how I have begun to alert students to the subtleties involved.
Philosophy (But
Not Philosophers) of Mathematics Does Influence Mathematical
Practice
Bonnie Gold
Monmouth University
bgold@monmouth.edu
Since most philosophers of mathematics tend to ignore current mathematical practice outside of foundations, it is not surprising that mathematicians tend to ignore current philosophy of mathematics. I will argue, however, that in a deeper sense a mathematician's philosophy of mathematics, even if not coherently articulated, does affect his/her mathematical activities: the types of questions considered, whether (s)he focuses more in individual problems or on mathematical structures, the general direction of mathematical work over a time interval. Further, investigating, rather than suppressing, these underlying motivations can lead to interesting philosophical questions.
George Polya on methods of discovery in mathematics
Bonnie Gold
Monmouth University
bgold@monmouth.edu
George Polya, in his book How to Solve It and more so in his later two-volume Mathematics and Plausible Reasoning discussed methods of discovery in mathematics in considerable detail. This talk will examine both the methods he explicitly discussed, as well as some that are, I believe, implicit in his writing.
Is school mathematics ‘real’ mathematics?
Bonnie Gold
Monmouth University
bgold@monmouth.edu
One objection mathematicians often have to the work of philosophers of mathematics is that they don't ever discuss “real” mathematics. Many philosophers don't know mathematics beyond the school level (arithmetic and geometry); or if they know any advanced mathematics, it's often logic or set theory, which one can argue are somewhat anomalous. I've been interested for many years in trying to answer “what is mathematics?” by looking at topics that are on the boundary -- some would say they're mathematics, others would say they're not really mathematics. So I'd like to consider one such topic: is school mathematics “real mathematics” (as in David Corfield's Towards a Philosophy of Real Mathematics, or as what each of us means by the term)?
Back to JMM 2016 guest lecture
Melding realism and social constructivism
Bonnie Gold
Monmouth University
bgold@monmouth.edu
My own philosophical viewpoint has always been something of a blend of realism (platonism) and social constructivism: realism about mathematical objects, and social constructivism (Reuben Hersh's version, not Paul Ernest's) about our knowledge of those objects. More recently, while reading José Ferreirós's Mathematical Knowledge and the Interplay of Practices, I have been working on how to integrate his approach, which seems to me a more sophisticated version of social constructivism, with my viewpoint. I will discuss this version of pluralism, and briefly comment on the main topic, whether mathematicians need philosophy.
What makes proofs explanatory? Let's look at some examples.
Bonnie Gold
Monmouth University (Emerita)
bgold@monmouth.edu
Both in mathematics and in philosophy, a few good examples can help clarify a concept. So, to try to make some progress understanding the idea of explanatory proofs, I will look at several proofs that the sum of the first n positive integers is n(n+1)/2, and examine to what extent, and why, some of them seem more explanatory than others.
Henri Poincaré: Mathematician, Physicist,
Philosopher
In conjunction with CSHPM, the Kenneth O. May Lecture
Jeremy Gray
University of Warwick and the Open University
Henri Poincaré held strong views about human knowledge that animated his work in both mathematics and physics. He held views on the possibly non-Euclidean nature of space, on the foundations of mathematics, on the fundamental 'laws' of physics, on why the basic equations of mathematical physics are linear, on space and time, and on theory change in science. These views, and the debates they generated, give a rich insight into the frontiers of research a century ago.
Back to Mathfest 2013 guest lecture.
Teaching a Choose Your Own Adventure Gened Math Course
Jessie Hamm
hammj@winthrop.edu
Winthrop University
Despite what most of my students say, I believe we're all "math people". I believe that math is an art and it can be appreciated by everyone if seen in the right light. Because of these views, I have started using a new approach to teaching my general education Joy of Mathematics course--a "Choose Your Own Adventure" approach. Students are allowed to choose the topics they want to learn from a variety of given options and the course operates in a flipped fashion. In this talk I will discuss my motivations for this approach, the logistics of how the course runs, and some preliminary results on the outcomes.
Back to Mathfest 2024 schedule
Intuition: A History
Kira Hylton Hamman
Penn State University
kira@psu.edu
What is intuition, and what is its role in mathematics? I don't know, and neither do you, but many a distinguished scholar has speculated on these questions. We trace the trajectory of our understanding of mathematical intuition from the Greeks through the Enlightenment and into the present day. And while you probably will not emerge from this talk with answers to these compelling questions, you will at least be prepared to approach them with an understanding of where we have been.
Philosophy of Mathematics: What, Who, Where, How and Why
Charles R. Hampton
The College of Wooster
Hampton@wooster.edu
For more than fifty years mathematicians have largely abandoned the Philosophy of mathematics while a renaissance in this area has occurred among philosophers. With little notice by those in our profession, the past seven years have seen the publication of more than two dozen books devoted to philosophy of mathematics. In the context of a brief sketch of what the issues are in the ongoing discussion among philosophers, I will explore the reasons for mathematicians' abandonment of the field in the middle of the 20th century and discuss why mathematicians find it so difficult to get back into the discussion. I will also propose a route by which interested mathematicians might proceed in order to become part of the discussion and where they can find entrée into the current literature.
Applied Mathematics---A Philosophical Problem
Charles R. Hampton
The College of Wooster
Hampton@wooster.edu
Eugene Wigner was not the first mathematician or philosopher to ask himself why mathematics works so well in applied areas. But the title of Wigner's 1960 essay, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", captures the philosophical issue. The applicability of mathematics is a philosophical problem whether one is a Platonist or a formalist. It should not be ignored either by the realist or the advocate of fictionalism. This talk will survey what contemporary philosophers of mathematics are saying on this subject.
Strands in the history of geometry and how they affect our views as to what geometry is
David W. Henderson, Daina Taimina
Cornell University
dwh2@cornell.edu
We argue that the main aspects of geometry emerged from four strands of human activity, which seemed to have occurred in most cultures: art/patterns, navigation/stargazing, motion/machines, and building/structures. These strands developed more or less independently into varying studies and practices that from the 18th and 19th century on were woven into what we call geometry. Axiomatic mathematics developed (through Euclid) within the Building/Structures Stand and this strand has been emphasized (sometimes to the complete exclusion of the other three) in most discussions of the history and meaning of mathematics. This has distorted our understandings of mathematics and placed obstacles in paths of people trying to understand what mathematics is. This has led to confusing and ofttimes-incorrect statements in many expository descriptions and textbooks of geometry. This is true even in works written by well-known research mathematicians. These include answers to questions such as: What is geometry? What was the first non-Euclidean geometry? Jow and why was it investigated? Can you construct the trisector of any angle? What does "straight" mean in geometry? Why was spherical geometry in curricula and then mostly disappeared 100 years ago? What is the shape of our physical universe?
Propensities and the Two Varieties of Occult Qualities
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
In 1982, Keith Hutchison laid the foundations for the historical treatment of occult qualities, those that cannot be directly observed. They are insensible, as opposed to manifest. The case will be made that occult qualities naturally break themselves into two varieties, occult qualities of the first and second kinds. Occult qualities of the first kind may be analyzed in terms of other, more basic entities while occult qualities of the second kind are not amenable to such analysis. They are, in a sense, atomic in character. It will further be argued that propensities, whether long-term or short-term, as a basis for probability are occult qualities of the second kind.
Catching the Tortoise: A Case Study in the Rules of Mathematical Engagement
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
Many responses to Zeno’s paradoxes rely heavily on Cantor’s work on infinity and the work of Weierstrass and Dedekind on limits; this is certainly the case with Bertrand Russell’s resolution of these puzzles. It is interesting to note that Russell believed there is no reason to accept the idea that the spacetime structure of the universe is continuous rather than discrete since if the universe is not continuous, arguments of this sort are irrelevant. As he points out in another context, simply postulating continuity has all the advantages of theft over honest toil. Russell could not have missed the fact that his argument had a hole in it if read physically. Evidence suggests he took Zeno’s paradoxes as purely mathematical in nature, but the historical context of Zeno’s writings make this conclusion questionable at best. The current project proposes a model for motion in a discrete spacetime to complement Russell’s response to Zeno, but it also addresses another question: How do we properly use mathematics in debates in which the issues may be read as purely mathematical or as pertaining to the physical world?
What Does It Mean for One Problem to Reduce to Another?
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
In “Argumentations and Logic,” John Corcoran says that one of the jobs of deduction is to reduce new, unsolved problems to old, solved ones, but what does it mean for one problem to reduce to another? This can happen in a number of ways. First, it might be that one problem immediately reduces to another. For example, “No square number is twice another square number” (A) straightaway becomes a demonstration of the irrationality of the square root of two (B). Here B clearly implies A. In a more complicated second case, several lemmas may need to be demonstrated before the soughtafter theorem may be deduced. For instance, assume to show result C, preliminary results 1, 2, and 3 are needed. The relationship between C and 1, 2, and 3 may be thought of in two ways: either “1, 2, and 3 imply C” (just a more detailed version of the simple case) or, for instance, “Given 1 and 2, 3 implies C.” Both cases will be examined. Another complication is that a result may be demonstrated non-trivially in more than one way. Further, the meaning of ‘implies’ must be made clear. Obvious candidates include material implication, logical implication, and formal implication. Each of these, and others, will be considered.
What Is the Character of Mathematical Law?
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
The proposition that mathematics may be treated as just another empirical science has its origin in the writings of John Stuart Mill and is still defended by some philosophers of mathematics to this day: Lakatos argues that, like those of the physical sciences, mathematical investigations are quasi-empirical in nature; Maddy has said that sometimes axiom adoption in set theory "has more in common with the natural scientist's hypothesis formation than the caricature of the mathematician writing down a few obvious truths; " Goodman has gone as far as to say that "mathematics is no more different from physics than physics is from biology." If one assumes Mill's position, which I will call "naturalized mathematics," it seems not unreasonable to use the extant literature on laws of nature as a starting point for an investigation into the nature of the laws of mathematics, though, of course, this is not the originally intended application. Versions of laws of the physical sciences include the regularity, necessitarian, universals, systems, anti-realist, and anti-reductionist accounts. This presentation will assume the naturalized-mathematical position and consider which account best fits the laws of mathematics.
Causation and Explanation in Mathematics
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
One of the trickier issues in teaching a statistics course is making clear the causation/correlation distinction. Consider: (1) There is 100% correlation between mammals and animals having three bones in the middle ear. This all-and-only parallel seems to be a completely accidental evolutionary happenstance. (2) There is a very strong positive correlation between the shoe size and reading-comprehension scores of children. Here shoe size and reading scores are the twin effects of the common cause of aging. (3) Any free hydrogen atom is capable of bonding with any free fluoride atom. This stems directly from the atomic structures of hydrogen and fluoride. The subject of causality may seem far removed from the objects of mathematics, but Bernard Bolzano formulated a theory of cause and effect for (among other things) mathematical propositions in his 1837 Theory of Science. I will consider what Bolzano has to say about causality in mathematics and see what implications there are for the related subject of mathematical explanations.
Progress in Mathematics: The Importance of Not Assuming Too Much
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
John Stuart Mill took mathematics to be just another natural science. Exploiting this point of view, one may give a Mill-style analysis of the progress in mathematics in light of the literature written on the progress of other natural sciences. There is no more influential work on the progress of science than Thomas Kuhn’s The Structure of Scientific Revolutions. One of the planks in Kuhn’s platform is that after a scientific revolution the new paradigm is incommensurable with the old one. In part this means the new theory is not simply a generalization of the old theory. Kuhn claims that this is due to the fact that the terms of the old theory are grandfathered into the new one, but some of them are used in different ways. I argue that at least some post-revolutionary mathematical theories are incommensurable with pre-revolutionary theories, but for a different reason – because important operating assumptions of the old paradigm are dropped. Mill would not have been surprised when physicist David Bohm observed that dropping assumptions was the key to scientific advancement, providing another parallel between mathematics and (other) natural sciences.
The Mathematics of Quantum Mechanics: Making the Math Fit the Philosophy
James R Henderson
University of Pittsburgh-Titusville
henderso@pitt.edu
Nowhere in the history of science is it clearer than in the case of the development of a mathematical formalism for quantum mechanics that mathematics is the language of the scientist, if not science. In the mid-1920s, Schrödinger and Heisenberg had different visions of quantum mechanical systems and chose different mathematical tools to describe them. As far as making predictions are concerned, the two formulations are of course equivalent, but it is interesting that each man adopted a mathematical model that matched his own vision of microscopic systems. Schrödinger believed his continuous, deterministic, time-dependent wave function gave a realistic picture of the evolution of quantum mechanical systems (his view would change considerably over time). Heisenberg had adopted what would in the 1950s come to be called the Copenhagen interpretation and denied systems evolved between measurements (indeed, to say even that much may be a category error); his matrix mechanics makes for a tight fit for this view. Though the stories of Schrödinger's evolving viewpoint and Heisenberg's defining the dominant interpretation are interesting in their own right, I will discuss how mathematics and philosophy developed organically in the exciting period at the outset of the quantum revolution.
Kepler's Mysterium Cosmographicum
James R Henderson
Penn State University
jrh66@psu.edu
In 1596, Johannes Kepler completed his Mysterium Cosmographicum (MC), a bizarre text that “explains” the relative distances from the sun to the six then-known planets in terms of the five Platonic solids (astronomy was, in Kepler's day, largely a mathematical enterprise). It is remarkable that Kepler's utterly misguided model should have produced results as accurate as they were. I will argue that Kepler's reasoning springs from three propositions: (1) Kepler, deeply religious, believed god designed the universe with a necessary, specific, understandable plan; (2) Kepler believed that Copernicus was right about heliocentricity; (3) Kepler believed that mathematics can give rise to knowledge of the physical world. I will discuss these propositions in more depth, trace Kepler's motivation in writing MC, investigate the inspiration of the central idea of the book (the first spark was a single picture), and talk about how the propositions described above shaped MC. Kepler’s writing, in which he lays out his thought processes, the false leads he followed, and his missteps along with his successes, makes for fascinating (and lengthy) reading.
Strange Bedfellows: Thomae’s Game Formalism and Developmental Algebra
James R Henderson
Penn State University
jrh66@psu.edu
In a developmental math class, learning about manipulating mathematical entities can sometimes grind to a halt when questions about the entities themselves arise. This usually doesn’t happen with, say, whole numbers because students can understand them in terms of a simplistic Platonism. Trying to bring these students around to a different way of thinking may be a case of fixing something that isn’t broken. But consider, as a single example, when imaginary numbers are introduced. What is a beginner to make of a number that is neither positive, nor negative, nor zero, and when squared produces a negative? Since, to the uninitiated, imaginary numbers are mysterious in a way that whole numbers are not, I ask my students to adopt a formalist approach like that of Johannes Thomae in which math is purely a game with specific rules of play and the background assumption that no mathematical symbol has any meaning outside the game. In particular, i has no meaning, so the job is not to understand it. Rather, the job is to eliminate higher powers of i and square roots of negatives, and it can all be done with techniques familiar to the students. In this way, the puzzling nature of imaginaries never comes into play and new problems are reduced to old ones.
Otavio Bueno's Mathematical Fictionalism
James R Henderson
Penn State University
jrh66@psu.edu
Mathematical Platonists claim that mathematical objects actually exist (though not spatio-temporally) and that, therefore, there is but one mathematical Truth corresponding to those objects and the relations that hold between them. Nominalists, on the other hand, claim mathematical objects do not exist and, therefore, almost all of mathematics is false. (This includes any proposition involving an existence claim; universal propositions are taken to be vacuously true.) Not surprisingly, both Platonism and Nominalism have their own particular strengths and weaknesses. The question for the practicing mathematician or philosopher of mathematics is this: Does one have to "pick a side"? Fictionalism, at least as advocated by Otavio Bueno, takes its lead from Bas van Fraassen's Constructive Empiricism, a version of scientific anti-realism. Bueno's Fictionalism is a middle position where the sticky problem of the existence of mathematical entities is simply not addressed (or not completely, at any rate), and it gives a non-standard version of mathematical truth. Whether this brand of Fictionalism can provide a sound basis for mathematics as practiced by professional mathematicians will be discussed.
When Physicists Teach Mathematics
James Henderson
Penn State
jrh66@psu.edu
The teaching of mathematics is one of the field's most straightforward applications. Because the math lessons 10-year-olds absorb are of a very different sort than those mathematicians-in-training endure while studying functions of a complex variable in graduate school, it is not surprising that the material varies quite a bit in many settings in which math is taught, but the presentation of the material does, too, depending on the professor, the audience, and the purpose of the class. Further, it's not just mathematicians who teach math. Sometimes, for instance, physicists teach math to physics majors explicitly for use in physics courses. This was most famously done in 1961 and 1962 by Richard Feynman at Caltech. (Math was not all he taught, but algebra did warrant a chapter in his celebrated The Feynman Lectures on Physics [1963].) How does the presentation differ when a mathematician teaches math to math students and a physicist teaches math to physics students? What are the primitive terms and rules of inference in each case? How do these differences define the process of teaching? To answer these questions, I will rely on Feynman's lecture on algebra, input from a physicist friends, and my own experiences as a graduate student in a mathematics program.
Multiplicity of Logical Symbols: Why Is That a Thing?
James Henderson
Penn State
jrh66@psu.edu
It is not surprising that, over the years, different authors have suggested different symbols to stand in for different logical connectives and quantifiers. After all, different languages have various sounds for the same concept or object, and, perhaps more closely related, mathematicians sometimes have different ways of symbolizing the same operation or transformation. (Easy example: Newton and Leibniz famously used different notations for the derivative.) It is one thing, however, for two mathematicians to independently develop a rather sophisticated idea and use dissimilar notation, and quite another to have multiple symbols for very simple connectives like negation or disjunction, even after the development of a decades-old literature concerning truth tables. It seems a bit excessive. Why hasn't there developed, for instance, one canonical way to join two propositions into a conjunction rather than the use of the ampersand, the dot, the inverted wedge, or simply placing characters adjacent to each other? It is as if authors of elementary arithmetic books had half a dozen ways to express the notion of "two plus four." This talk concerns the motivations of authors to affect the way logical concepts are envisioned by readers, which may lie at the root of this phenomenon.
Explanatory proofs
James Henderson
Penn State
jrh66@psu.edu
When investigating the character of mathematical explanation, it is not unreasonable to begin with the literature concerning scientific explanation. If this approach is taken, the discussion must begin with Hempel and Oppenheim's 1948 paper, "Studies in the Logic of Explanation." Here a template for explanation is laid out, and a body of writing, some critical and some supportive, develops. The pump thus primed, new formulations of "scientific explanation" were produced in the ensuing years. Not surprisingly, some explanatory schemes designed for a scientific setting are a better fit for a mathematical context than others. I will stick to definitions of 'proposition', 'argument', and 'proof' used in this literature to assure a sound analysis. Study of equivalent theorems (for instance, the Mean Value Theorem and Rolle's Theorem) will offer interesting insights into this undertaking. It may also be useful to try to address the question of whether it is coherent to describe an explanation as successful and, if so, whether the success of an explanation is subject to degree.
Realism and Undetermination in Mathematics and the Physical Sciences
James Henderson
Penn State
jrh66@psu.edu
It is well known that, as a group, physicists who count themselves as scientific realists are completely untroubled by the existence and practice of different (and mutually incompatible) scientific theories. For example, general relativity and quantum mechanics are irreconcilable, yet brawls rarely break out between physicists specializing in the study of the very large and very small at APS conferences. This devil-may-care approach to the diversity of theoretical bases for physics doesn’t exist because physicists are wholly incurious about things that are of great interest to philosophers (though there are those who aren’t in the least interested in these matters); instead, there are solid reasons for this forbearance ranging from the practical to the theoretical. I will concentrate on the notion of underdetermination (strictly, the underdetermination of theories by experimental facts) and argue that, at least for some readings of 'realism', mathematicians of a realist orientation would be justified in adopting the same "Don’t worry, be happy" attitude with respect to different (and incompatible) mathematical foundations for analogous reasons.
Hempel's Ravens and the Goldbach Conjecture
James Henderson
University of Pittsburgh at Bradford
jrh258@pitt.edu
Hempel's Raven Paradox involves the principle of induction, intuition, simple logic, the nature of evidence, and a host of other issues. First published in 1965, it has given rise to much lively debate and no shortage of proposed resolutions. The Goldbach Conjecture, originally posed in a letter from Goldbach to Leonard Euler in 1742, is a proposition concerning number theory that has likewise engendered much discussion (and a lot of number crunching!), but at this writing a definitive proof still eludes the mathematical community. In 2013, Goldbach’s conjecture, that every even integer greater than two may be expressed as the sum of two (not necessarily distinct) primes, was verified for all even integers up to 4×10^{18}. That’s still a bit less than one one- hundred-thousandth of Avagadro’s number, but in everyday terms, it’s a lot. A whole lot. Even though we ordinarily restrict ourselves to deductive reasoning when proving mathematical propositions, perhaps 4×10^{18} is big enough to make us forgive someone who begins thinking in terms of induction and muses, "Given that no exceptions have been found in the first two- quintillion-minus-one attempts, I'd be surprised if Goldbach turned out to be wrong." I will apply the Raven Paradox to the Goldbach Conjecture (in particular the inductive approach to it just sketched) and consider how proposed resolutions to the paradox stand up in this context.
Subversive essays on the nature of mathematics
Reuben Hersh
University of New Mexico
rhersh@gmail.com
Quite a few people--mathematicians, philosophers, historians, cognitive scientists, sociologists and others--have recently written interesting, provocative things about the nature of mathematics. I have collected some of them into a new anthology.
Back to Mathfest 2006 Guest Lecture.
Mathematicians’ proof: “The kingdom of math is within you”
Reuben Hersh
University
of New Mexico, emeritus
rhersh@gmail.com
A mathematician’s informal proof works by enabling others to perceive internally what he/she is trying to show them. I give a simple example, that Sp(n), the sum of the pth powers of the first n integers, is a polynomial in S1(n), if p is an odd number. (Experiencing mathematics, starting on page 89.)
English philosopher Brendan Larvor asks, “What qualifies mathematicians’ informal proofs as proofs?” A mathematician seeking a proof is working with internal mental models of mathematical entities (numbers, spaces, algebraic structures and so on). You have direct access to your own internal mental models. You observe some properties of theirs, you manipulate them, you relate them to each other and to other mathematical entities. Your separate individual internal mental models match mine well enough that we communicate about them successfully. In mental struggle with your internal mental models, you notice something interesting. Then you want me to “see” what you “see”. You hunt for a sequence of steps which will lead me to share your insight. That sequence of steps, which enables me to “see” what you “see”, is what mathematicians call “a proof”.
Generalised likelihoods, ideals and infinitesimal chances - how to approach the "zero-fit problem"
Frederik S. Herzberg
University of Oxford
herzberg@maths.ox.ac.uk
The "zero-fit problem" is crucial for any investigation into the epistemological limitations of statistics, for it asks which methods there are to compare two atomless probability spaces (the canonical situation for infinite state spaces) - both of which have to assign probability ("fit") zero to the actual observation. Adam Elga claims to be able to reject David Lewis' suggestion of considering nonstandard probability measures - which can also attain infinitesimal values - as one way of tackling the zero-fit problem. We will indicate two major flaws in his general argument and apply our critique to the "toy problem" he employs to illustrate it. We will construct a nonstandard probability measure that solves the zero-fit problem in this particular case and give hints how to proceed in more general situations. In an appendix to his said paper, Elga also argues that a generalisation of the maximum likelihood technique (most common in theoretical statistics) is not suitable to solve the zero-fit problem. However, the mathematical reason behind any possible failure of that approach (the Radon-Nikodym Theorem) actually provides another interesting possibility to attack the zero-fit problem: the comparison of the ideals of null sets associated with the probability measures in question.
Mathematics: the divine madness
Morris W. Hirsch
University of Wisconsin
mwhirsch@chorus.net
All human activities are informed by myths, which may be true, false, or meaningless. We will discuss some of the ancient and modern Myths of Mathematics, including:
- The Myth of Goodness: Mathematics is a Good Thing.
- The Myth of Measurability: The world can be described and explained in mathematical terms.
- The Myth of Certainty: Mathematical knowledge is the most certain form of knowledge.
- The Myth of Existence: Mathematical objects exist independently of minds, time, space, energy, and physical reality.
- The Myth of Truth: Every mathematical statement is either true or false.
- The Myth of Computers: Computer computations are reliable.
- The Myth of Proof: There is a clear concept of proof.
Back to Mathfest 2008 Guest Lecture.
Possible and Impossible Infinities
Michael Huemer
University of Colorado--Boulder
owl232@earthlink.net
The infinite gives rise to many paradoxes. Some are aptly resolved by declaring certain infinite quantities impossible. But which infinities are possible, and which are impossible? On an Aristotelian view, there can be no "actual infinities", only "potential infinities". This view is wrong; there are many obvious examples of actual infinities. I draw three distinctions: between cardinal numbers and magnitudes, between intensive and extensive magnitudes, and between natural and artificial magnitudes. I then propose a new theory of the impossible infinite: there are infinite cardinal numbers, extensive magnitudes, and artificial magnitudes, but there can be no infinite natural, intensive magnitudes. This view rules out most of the scenarios appearing in the paradoxes of the infinite.
Back to JMM 2020 Guest Lecture.
Pure and Applied Computational Mathematics: Some Philosophical Morals
Paul Humphreys
University of Virginia
paul.w.humphreys@virginia.edu
Much of the attention given to computer assisted mathematics (CAM) has focused on proofs as a route to establishing mathematical truth. I shall emphasize instead the effects that CAM has on our understanding of results and the consequences of these effects for the practice of mathematics as an essentially human activity. Limitations arising from the finitistic nature of CAM and the epistemic opacity of its processes will be discussed. Its effects on applied mathematics will also be explored.
Back to JMM 2006 Guest Lecture.
Metaontology in Light of the Frege-Hilbert Controversy
Jared M. IflandUniversity of California, Davis
jifland@ucdavis.edu
Insights from the philosophy of mathematics have recently been brought to bear upon metaphilosophical disputes pertaining to a priori knowledge, especially by both Justin Clarke-Doane and Jared Warren. My aim is to demonstrate how the Frege-Hilbert controversy over the role of axioms in mathematical theories sheds light upon contemporary debates in metaontology that involve the philosophy of mathematics to any significant degree.
Clarke-Doane argues for a pluralistic construal of mathematical realism to guarantee that we could not have easily had false mathematical beliefs. Purportedly, this vindicates a metaphysically realist construal of Rudolf Carnap-inspired philosophical pragmatism. Contrary to Clarke-Doane, Warren argues that mathematical objects are merely byproducts of our use of mathematical language and nothing more. Consequently, Warren’s account bears more affinity to Carnap’s own pragmatism.
Like Frege and Hilbert, Clarke-Doane and Warren adopt different stances toward the relationship between method and metaphysics. For Frege and Clarke-Doane, the reliability of method must be certified by a complementary metaphysical account, whereas for Hilbert and Warren, no appeal to more fundamental principles is needed to justify method. I argue in favor of the stance adopted by Hilbert and Warren, and contra Clarke-Doane, submit that Carnapian pragmatism is in no need of a realist construal and that the more traditional construal suggested by Warren’s approach stands on its own.
What does mathematical terminology say about linguistic determinism?
Kevin Iga
Pepperdine University
kiga@pepperdine.edu
The question of whether our notation determines our understanding of mathematical concepts is part of a more general controversy in the study of linguistics: that of linguistic determinism, sometimes called the Sapir-Whorf hypothesis, which posits that features of language can have an effect on what kinds of thoughts are possible, or which are more easily accepted or understood. I will survey the history of this hypothesis and summarize a range of current views. Then I will look at a few cases in the history of development of mathematical notation, and explore how these examples can contribute to our understanding of the Sapir-Whorf hypothesis.
The Unreasonable Effectiveness of Mathematics: Dissolving Wigner's Applicability Problem
Arezoo Islami
San Francisco State University
In a 1980 Monthly article, "The Unreasonable Effectiveness of Mathematics", Richard Hamming discussed what he took to be Wigner's problem (from 1960) of "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" and offered some partial explanations. My goal is to show that Hamming's reading misses Wigner's highly original formulation of the problem. Rereading Wigner's work more closely, we are led in new directions in addressing and solving the applicability problem.
Back to JMM 2024 guest lecture
Wittgenstein and Social Constructivism
Ilhan M. Izmirli
George Mason University
iizmirl2@gmu.edu
In this paper our main objective is to interpret the major concepts in Wittgenstein's philosophy of mathematics, in particular, language games and forms of life, from a social constructivist point of view in an attempt to show that this philosophy is still very relevant in the way mathematics is being taught and practiced today. In the first section we briefly introduce the social constructivist epistemology of mathematics--a perspective that reinstates mathematics, and rightfully so, as "...a branch of knowledge which is indissolubly connected with other knowledge, through the web of language" (Ernest 1999), and portrays mathematical knowledge as a process that should be considered in conjunction with its historical origins and within a social context. In section two, we give a telegraphic overview of the main points expounded in Wittgenstein's two books, Tractatus Logico-Philosophicus and Philosophical Investigations, as well as in his "middle period" that is characterized by such works as Philosophical Remarks, Philosophical Grammar, and Remarks on the Foundations of Mathematics. In the third and last section, we highlight the connections between social constructivism and Wittgenstein's philosophy of mathematics.
Is Mathematics the Language of Physics?
Arthur M Jaffe
Harvard University
Jaffe@math.harvard.edu
We explore whether modern mathematics is an adequate tool to describe the natural science of our world. In other words, to what extent is "mathematics is the language of physics."
Back to 2012 Invited Paper Session
Mathematics as a human creation
Jeff Johannes
SUNY Geneseo
johannes@geneseo.edu
I deeply believe that mathematics is the most universally agreed upon human creation. It is both our agency and our unanimity that give mathematics its power. As a consequence meaning is central to all that I do, and classes emphasize constructivism over rule adherence. Furthermore, I have a deep respect for the power and role of gestures and physical expression of mathematics, often as an inspiration toward verbal communication. I will explore several examples of the influence of human-centered mathematics across the undergraduate curriculum.
Back to Mathfest 2024 schedule
Precalculus from an Ontological Perspective
| Whitney Johnson | Bill Rosenthal |
| University of Maryland, College Park | LaGuardia Community College, CUNY |
| wjohnso7@umd.edu |
Universities are struggling with the large number of students who place into classes below calculus and perform poorly. We posit that one factor underlying this problem is inattention to ontological issues - in particular, the very existence of the mathematical objects under study. A tacit assumption in most mathematics textbooks is that a clear definition of an object is ontologically sufficient for conceptualizing and operating with that object. We wish to disinter and examine this assumption. It is reasonable to conjecture that, to someone who questions the existence of an object, a definition of utter clarity may not suffice. Definitions address the question, "What is it?" - presupposing that, indeed, it is. Definitions do not address the primal question, "Is it?" Professional mathematicians, who hold the power to create objects of study by sheer will, need not bother with the latter query. Students fresh out of high school are unlikely to be so philosophically fortunate.
Each of us teaches precalculus, one at an urban community college, the other at a research university. As a case in point, we consider the ontological content of precalculus as inferred from textbooks, cross-referencing the findings with insights drawn from studies of our students' work.
Mathematical Authority and Inquiry-Based Learning
Matt Jones
California State University Dominguez Hill
This talk will give describe Inquiry-Based Learning (IBL) and its interaction with notions of mathematical authority. The talk will begin with a brief history of IBL, from Moore to current practice, and include a working definition of IBL. The talk will also explore the interplay between IBL and notions of mathematical authority, centering on the questions of, When is a proof considered valid? and, Who can validate a proof? There will also be a discussion of the research on students’ views of mathematical authority and the impact of these on student achievement.
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Gian-Carlo Rota and the Phenomenology of Mathematics
Chandra Kethi-Reddy
chan.dra@knights.ucf.edu
The celebrated combinatorialist Gian-Carlo Rota arguably produced a philosophy of mathematics more faithful to the actual practice of mathematics than any other American mathematician or philosopher of his time. While among his peers at MIT or the Los Alamos Research Laboratory, even while he was Vice President of the AMS, Rota had to fight the hegemonic and exclusionary institution of analytic philosophy in order to justify the intelligibility and practicality of his unique phenomenology of mathematics. In this presentation, I will take the audience through Gian-Carlo Rota's “Phenomenology of Mathematical Proof” in order to demonstrate the place of phenomenology in any rigorous philosophy of mathematics. I will also go through a selection of his advice from “Ten Lessons I Wish I Had Been Taught” and “Ten Lessons for the Survival of a Mathematics Department” to show how his philosophy can be lively, humorous, and close to life. I hope that this presentation will reinvigorate research in the phenomenology of mathematics and interest in the life and work of this titan.
Definitions in Their Developmental Stages: What should we call them?
| Firooz Khosraviyani | Terutake Abe | Juan J Arellano |
| Texas A&M International University | South Texas College | Texas A&M International University |
| FiroozKh@TAMIU.edu | tabe@southtexascollege.edu | juan.arellano@tamiu.edu |
In an axiomatic system, such as in much of modern Mathematics, terminologies such as definitions, theorems, lemmas, corollaries, propositions, and conjectures have specific uses to more effectively communicate their purpose. They fall into two categories depending on the role they play: organizational and developmental (related to the process of creating). The organizational terms further fall into two subcategories: definitional (definitions) and consequential (theorems, etc.). Correspondingly, the developmental terms should also fall into these two subcategories. But, though the existing mathematical literature has made an extensive use of the theorems in their developmental stages (conjectures), it has not done the same with definitions in their developmental stages. In these notes, we discuss ways to address this observable deficiency in the set of standard terminologies in Mathematics.
The Poetic View of Mathematics
Jerry P. King
Lehigh University
jpk2@lehigh.edu
A philosophical framework for mathematics is described. The framework shows mathematicians as mirror images of poets and provides a model for mathematics analogous to the manner in which mathematics itself models reality. Moreover, the poetic view gives a set of consistent answers to the classical questions: What is the nature of mathematical objects? How is mathematics related to the real world? What is the role of aesthetics in the creation of mathematics? In the classroom the poetic framework gives applications-weary students a new look at what seems to them a tired, old subject.
Mathematical Intuition and the Secret of Platonism
Sergiy Koshkin
University of Houston Downtown
koshkins@uhd.edu
We will look into the interplay between modern perceptions of mathematical objects, and of the roles of intuition and proof in mathematics, and the structural and notational changes in algebra and formal logic in 19th century. The received view, with its emphasis on formal proof, is arguably in tension with the traditional mathematical platonism, which perceives intuition as a primary means of interacting with mathematical objects. We will then discuss an alternative, diagrammatic, view of a mathematician's work developed by C.S Peirce, and supported by his diagrammatic notation for predicate calculus, and argue that it gives a plausible resolution to this tension. Namely, a theory of intuition that explains both the indispensability of formal proofs, and the secret of the continued appeal of platonism to mathematicians.
Mathematical Structuralism and Mathematical Applicability
Elaine Landry
University of California, Davis
emlandry@ucdavis.edu
I argue that taking mathematical axioms as Hilbertian is not only better for our account of mathematical structuralism, but it yields a better account of mathematical applicability. Building on Reck’s [2003] account of Dedekind, I show the sense in which, as mathematical structuralists, we ought to dispense with metaphysical/semantic demands. Moreover, I argue that it is these problematic demands that underlie both the Frege/Hilbert debate and the current debates about category-theoretic structuralism. At the heart of both debates is the metaphysical/semantic presumption that structures must be constituted from/refer to some primary system of elements, either sets or collections, platonic places or nominalist concreta, so axioms, as truth about such systems, must be prior to the notion of structure. But what we ought have learned from Dedekind [1888] and Hilbert [1899], respectively, is that we are to “entirely neglect the special character of the elements”, and so axioms are but implicit definitions, and, consequently “every theory is only a scaffolding or schema of concepts together with their necessary relations ... and the basic elements can be thought of in any way one likes ... [A]ny theory can always be applied to infinitely many systems.” The first thing to note is that no primitive system is necessary, the second is that any system, be it mathematical or physical, can be said to have a structure. Thus, applicability is just the claim that a physical system has a mathematical structure, i.e., that it satisfies the axioms, in certain respects and degrees for certain physical purposes.
Back to MathFest 2015 schedule
As-if Mathematics were True
Elaine Landry
University of California, Davis
emlandry@ucdavis.edu
When we shift our focus from solving philosophical problems to solving mathematical ones, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of both the practice and the applicability of mathematics whilst avoiding the conflation of mathematical and metaphysical considerations. Time for discussion with the audience will be included. This talk should be accessible to mathematicians at all levels with some interest in the philosophy of mathematics. The session is sponsored by POMSIGMAA, the special interest group of the MAA for the philosophy of mathematics.
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Why Is Proof the Only Way to Acquire Mathematical Knowledge?
Marc Lange
University of North Carolina at Chapel Hill
mlange@email.unc.edu
It has become increasingly common for philosophers and mathematicians to argue for the non- standard view that a mathematical proposition can (at least in principle) be known without anyone’s proving it – namely, by various sorts of non-deductive arguments from mathematical evidence. These arguments include not only enumerative induction (as when Goldbach’s conjecture has been verified for all even numbers below 4 × 10^{18}), but also “probabilistic proofs” (such as Rabin’s primality test). This paper defends the standard view by proposing an explanation of why proof is the only way to acquire knowledge of some mathematical proposition’s truth.
Admittedly, non-deductive arguments can justly raise our confidence in a given mathematical proposition to arbitrary heights. Why, then, can we not attain mathematical knowledge by non- deductive means? Here (I argue) we should take a lesson from the "lottery paradox", which demonstrates that knowledge requires more than justified, high confidence in a given proposition (such as that my lottery ticket will lose). In the lottery case, if my ticket ends up winning, despite my strong evidence (that the lottery is fair, there were n tickets,...) that it will lose, then this would just be by luck; there would be no reason why it won. This is why we cannot know in advance that my ticket will lose (even though we can have justified, high confidence that it will lose).
For the same reason, I argue, we cannot know a mathematical proposition without a proof of it. To know that p, we must justly expect that if p is actually false, despite the evidence for its truth, then this fact has an explanation. This necessary condition for knowledge can be satisfied by putative proofs (and proof sketches) in mathematics, as well as by non-deductive arguments in science, but not by non-deductive arguments for mathematical propositions from mathematical evidence.
Category Theory as an Explanatory Foundation
Chanwoo Lee
Dept. of Philosophy
University of California, Davis
chanwoo.lee.phil@gmail.com
Can category theory be a foundation of mathematics? I argue that category theory can serve as a foundation of mathematics in an explanatory sense. The explanatory sense of foundation is both historically situated and philosophically motivated. Based on the recent scholarship on mathematical explanation, I explain how CT can be explanatory and how it ties in with the more traditional debates on category theory’s foundational status.
How applied mathematics became pure
Penelope Maddy
Department of Logic and Philosophy of Science
University of California at Irvine
This talk traces the evolution of thinking on how mathematics relates to the world -- from the ancients, though the beginnings of mathematized science in Galileo and Newton, to the rise of pure mathematics in during the nineteenth century. The goal is to better understand the role of mathematics in contemporary science.
Back to JMM 2008 Guest Lecture.
Avoiding the Traps of Induction
Tom Mahoney
Emporia State University
tmahoney@emporia.edu
Mathematical Induction is a powerful and invaluable tool. But students often struggle to master it because many templates and examples used to teach induction lead to several errors, notably the "induction trap." One such error leads to a "proof" of the Four Color Theorem. We will look at a different approach to teaching induction that avoids this error. The associated "map" for the induction step provides a clearer, deeper understanding for how to apply induction beyond the familiar examples.
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Does the Indispensability Argument Leave Open the Question of the Causal Nature of the Mathematical Entities?
Alex Manafu
University of Paris-1 Pantheon-Sorbonne
Colyvan has claimed that the indispensability argument leaves open the question of the causal nature of mathematical entities (2001, p. 143). He defended this position by arguing that not all explanations are causal, and that some mathematical entities may play important explanatory roles even though they are causally idle in the ontology (in the sense that they do not interact with the particulars posited by that ontology).
I argue that Colyvan cannot maintain such an open attitude. I formulate an argument which shows that even if one grants the existence of mathematical entities which are explanatorily indispensable but causally idle in the ontology, Colyvan’s conclusion still doesn’t follow. If sound, the argument I offer shows that the question of whether the indispensability argument delivers causally active entities becomes settled. This result rehabilitates an argument offered previously by Cheyne and Pigden (1996).
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Structuralist Mathematics and Mathematical Understanding
Kenneth Manders
University of Pittsburgh
mandersk@pitt.edu
An increasingly wide range of mathematics advances strikingly by “structuralist” approaches: studying structured totalities of objects, often axiomatically characterised, and their mappings and constructions. It is felt that this gives the most fruitful understanding of mathematical issues that can be so treated. We will try to isolate and develop some strands of this claim.
Representations in Knot Classification
Kenneth Manders
University of Pittsburgh
mandersk@pitt.edu
We sketch roles and characteristics of basic representations in the knot classification program. Based on this, we argue (i) that the challenge is as much that of finding suitably behaved representations as that of proving theorems, and (ii) that the intellectual unity of the project resides not in the nature of the objects of study but in the intellectual motivating theme (knottedness types).
Both of these go against standard presumptions in the philosophy of mathematics.
A Philosophical Account of Mathematics that Won't Make You Hate Philosophers
Russell Marcus,
Hamilton College
rmarcus1@hamilton.edu
It is commonly believed that our best ways of knowing about anything require sense experience. Mathematics, as our most secure and conclusive knowledge, seems to eschew sense experience. Thus, we should either revise our views about how we know or revise our confidence in mathematics. Revising our views about how we know (our epistemology) in ways that accommodate a natural objectivity in mathematics, especially by appealing to intuition, seems problematic. I show how to invoke intuition in mathematical epistemology unproblematically, and thus that we need not impugn our confidence in mathematics. I start by describing objectivity about mathematics as a constraint on a satisfying epistemology. Then, I characterize and defend a version of intuition that I call thin. Lastly, I present an epistemology for mathematics, called autonomy platonism, that should be satisfying for both mathematicians and philosophers.
Canonical Maps: Where Do They Come From and Why Do They Matter?
Jean-Pierre Marquis,
Université de Montréal
jean-pierre.marquis@umontreal.ca
The term “canonical" is now common in mathematics and the term “canonical map" finds its way many mathematical contexts. However, there is no definition of what a canonical map is in general. In this talk, I want to sketch some of the roots of the terminology and explore why canonical maps are important mathematically and philosophically. I will focus on its progression in the literature and how this progression is intimately linked to the growth of category theory.
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Designing Mathematics: The Role of Axioms
Jean-Pierre Marquis
Université de Montréal
jean-pierre.marquis@umontreal.ca
The use of axioms in mathematics was more or less reintroduced in the 19th century and became a central tool at the end of that century and at the beginning of the 20th century. Already during this period, axioms had different functions. For Hilbert, it is first a tool for conceptual clarification and then, a more general tool for conceptual analysis. The American postulationists used axioms as logical knives and cutters. Noether and others introduced the axiomatic method and a way of abstracting, unifying and simplifying large portions of mathematics. My claim in this talk is that some mathematicians started using the axiomatic method not only in a new context, namely the context of categories, but that they also put the axiomatic method to a new usage. I will concentrate on Grothendieck's introduction of a host of types of categories, e.g. abelian categories, derived categories, triangulated categories, pretoposes, toposes, etc., in his quest to prove Weil's conjectures. In Grothendieck's head, the abstract character of the concepts involved is taken for granted and the purpose of the axiomatic method is primarily to construct the proper context for some tools, namely cohomological theories, to be used properly. Although Grothendieck's work marks a radical shift in mathematical style and some might even want to talk about a paradigm shift, he was soon followed by others who showed how this could be done for other problems. I will argue that this usage of the axiomatic method must be seen as an instance of conceptual design. The latter expression underlines the artifactual dimension of these parts of mathematics, as emphasized by Grothendieck himself, and allows us to contrast mathematical knowledge from scientific knowledge.
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In Praise of Cranks: Are You Thinking What I’m Thinking?
Andy D. Martin
Kentucky State University
andrew.martin@kysu.edu
Underwood Dudley’s books on Mathematical Cranks illustrate a chief feature of cranks: their refusal to accept(or, often, to even consider) the truth of statements whose proofs they do not understand. But why should they? Despite numerous cases of erroneous published proofs, mathematicians and mathematics teachers generally accept as established (for eternity!) theorems in fields outside of their own, established by proofs beyond their knowledge and, perhaps, beyond their ability to understand. Should they? Can my understanding of even the mere FACT of Fermat’s Last Theorem be the same as Andrew Wiles’s? Judith Grabiner famously asked, ”Is Mathematical Truth Time Dependent?” We would ask, ”Is Mathematical Truth personal?”
Claims Become Theorems, but Who Decides?
Andy D. Martin
Kentucky State University
andrew.martin@kysu.edu
How do claims of proofs become theorems and join the patchwork tapestry of mathematics we admire and try to describe to our students? Who actually decides for us when a very difficult proof is correct? Surely something like a vote occurs, with acknowledged experts the electors. Is my telling my students the Poincare Conjecture is settled more like a newscast or the Emperor's New Clothes (where I stand at the roadside as the naked ruler passes and assert that I, too, see his beautiful garments)? How can I KNOW that it is true? Reuben Hersh's view of mathematics as a socio-cultural construct certainly seems to describe this aspect of the mathematical world well. What then is the wisest attitude to have when hearing unverifiable claims? Surely not naive acceptance. What attitude should we foster in our students?
Why is it plausible?
Barry Mazur
Harvard University
mazur@math.harvard.edu
We have handy ways of discovering what stands a chance of being true. Any such way that is systematic, and that has been successful so far, goes under the catch-phrase heuristic method. They abound, these methods - explicitly formulated, or not. They lead us, perhaps, to a mere hint of a possibility that a mathematical statement might be plausible. At that point we might go about garnering other shades of plausibility arguments (as Polya wrote inspiringly about) and evidence of different colors, such as: analogies with things that are indeed true, computations, special case justifications, etc. Perhaps our thinking will reach the stage of some title of commitment such as conjecture. The end-game here is proof, of course. I want to focus on the beginning game, though, and spend the hour thinking of the nature of our current heuristic methods, and their fine structure.
Back to JMM 2012 Guest Lecture.
Measuring Imprecisely Defined Quantities: Degrees of Truth and Classical Finitary and Infinitary Logic
Vann McGee
MIT
vmcgee@mit.edu
A problem in the metatheory of applied mathematics: The concepts employed in pure mathematics aren't vague – there are no borderline cases of "prime number" – but we apply the methods of pure mathematics to solve problems in science and engineering and commerce, where the concepts employed are almost never precise. We can understand how the methods of pure mathematics are usable, say within sociology, by supposing that there are degrees of truth, and the degrees form a Boolean algebra. The conclusion of a good argument has to be at least as true as the premises. If we want to take account of infinitary rules of proof, like Shoesmith and Smiley's "cut for sets," the Boolean algebra will need to be complete and atomic, a requirement that makes van Fraassen's method of supervaluations nearly inevitable.
Reality never has just one correct foundation
Colin McLartyCase Western Reserve University
colin.mclarty@case.edu
No matter what foundation for mathematics you like – or even if you like none of them – mathematical reality today includes huge numbers of categories and functors, and huge numbers of variant models of Zermelo Fraenkel set theory. Algebraists, topologists, number theorists, and others routinely use categories and functors ranging from the intuitive (say, the category of Abelian groups) to the technical (say, the category of unfoldings of some singularity). Set theorists from Gödel through the latest work on determinacy, large cardinals, and inner models, work with many different models of Zermelo Fraenkel (say, models where the Continuum Hypothesis is true and others where it is false). All these things are mathematical realities. And because of they are realities, every leading candidate foundation already interprets all of them. These different foundations do not describe different mathematical worlds. They offer different basic views of the mathematical world.
On the Nature of Mathematical Thought and Inquiry: A Prelusive Suggestion
Padraig M. McLoughlin
Morehouse College
pmclough@morehouse.edu
The author of this paper submits mathematicians must be active learners. We must commit to conjecture and prove or disprove said conjecture. Ergo, the purpose of the paper is to submit the thesis that learning requires doing and that the nature of mathematical thought is one that is centred on 'positive scepticism.' 'Positive scepticism' is meant to mean demanding objectivity; viewing a topic with a healthy dose of doubt; remaining open to being wrong; and, not arguing from an a priori perception position. Hence, the nature of the process of the inquiry that justification must be supplied, analysed, and critiqued is the essence of the nature of mathematical enterprise: knowledge and inquiry are inseparable and as such must be actively pursued, refined, and engaged. The major philosophical influences of the thesis are Idealism, Realism, and Pragmatism. The author rejects an idea of mathematics rooted in a disconnected incidental schema where no deductive conclusion exists or can be gleaned and entrenched in a constricted schema, which is stagnant, simple, and compleat. The nature of mathematical thought is essentially the process of deriving an argument, supplying a refutation, constructing an adequate model of some occurrence, or providing connection between concepts.
Mathematics as an Emergent Feature of the Physical Universe
Ronald E. Mickens
Clark Atlanta University
rmickens@cau.edu
The elementary aspects of what came to be called “mathematics” were created to aid in the analysis, understanding, and prediction of those features of the physical universe of particular importance for human survival. Thus, mathematics had its genesis as a “help-aid” in exploration of human understanding and control over processes and events in the physical universe. We extend this argument to show that mathematics is not unreasonable effective as applied to the physical sciences; it is doing what it was constructed to do, i.e., function as a language, useful to the formation, analysis, and generalization of physical theories. The validity of this view does not preclude mathematics evolving (at a later time) into a separate discipline. A collection of essays on this subject is R.E. Mickens, (editor), Mathematics and Science (World Scientific, London, 1990)
On Godel's Proof and the Relation Between Mathematics and the Physical World
G. Arthur Mihram* and Danielle Mihram
Princeton, NJ
dmihram@usc.edu
Mathematician Devlin reviewed [SCIENCE 298: 1899, 2003] Godel's Proof: an effort to justify the naming of Godel as one of 20th Century's foremost thinkers. Yet, Godel's Proof is still 'read': there are conclusions in Science that could not be "proven"; and, there are questions about the physical/natural world which Science could not hope (within its role of providing the very explanation for---i.e., for the truth about---any particular naturally occurring phenomenon) to be able to answer. Earlier literature in mathematics had established that Godel's Proof could never support either conclusion: the proof deals with mathematical statements which could be proved (true) as a result of being based on arithmetic, mistakenly presumed to be (because of arithmetic's elementary nature) at the foundation of Science. Yet, infants and young children develop, before any awareness of arithmetic, logical constructs (e.g., combinations of if/then connexions), more at the root of knowledge than is arithmetic. Furthermore, since mathematics is neither necessary nor sufficient for Science (to wit: Darwin and K. Lorenz), Godel's Proof should never be construed so as to place any constraint on the certainties to be established via the Scientific Method [AN EPISTLE TO DR. BENJAMIN FRANKLIN, 1975(1974)].
Mathematical Platonism and the Five Components Model of Leadership
Mark Miller
millerr@marietta.edu
Marietta College
This talk examines the relationship between mathematics and leadership, from both epistemological and pedagogical perspectives, drawing from McManus & Perruci's Five Components Model of Leadership. Specifically, the talk will explore the ways that mathematical philosophy and leadership theory can inform the way we engage our students with an eye toward ethics, belonging, and justice – particularly in the context of first year mathematics courses with diverse student populations.
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The Philosophical Status of Diagrams in Euclidean Geometry
Nathaniel G. Miller
University of Northern Colorado
nat@alumni.princeton.edu
This talk will discuss a forthcoming book, Euclid and His Twentieth Century Rivals: Diagrams in the Logic of Euclidean Geometry, which considers the philosophical status of diagrams in Euclidean geometry. Euclid's Elements was viewed for most of its existence as being the gold standard of careful reasoning and mathematical rigor, but by the end of the nineteenth century, developments in mathematical logic had led many people to view Euclid's proofs as being inherently informal, in large part because of their use of diagrams. The work discussed in this talk seeks to show that the use of diagrams in Euclidean geometry can, in fact, be made as rigorous as other modes of proof, and that formalizing the use of diagrams in this way can shed a lot of light on the history and practice of Euclidean geometry.
CDEG: Computerized Diagrammatic Euclidean Geometry
Nathaniel G. Miller
University of Northern Colorado
nathaniel.miller@unco.edu
The use of diagrams in Euclidean geometry is an area in which most informal mathematical practice does not align well with most formal logical and philosophical accounts of geometry. Most people giving informal geometric proofs rely on diagrams as part of their proofs; this tradition, in fact, goes back to Euclid. However, most formal accounts of geometry developed over the last 150 years do not rely on diagrams, and it is often claimed that diagrams have no proper place in rigorous mathematical proofs. CDEG is a free computer proof system for manipulating and giving proofs with diagrams in Euclidean geometry that seeks to bridge that gap. It is based on a rigorously defined syntax and semantics of Euclidean diagrams. This talk will include a demonstration of CDEG, and a discussion of some of the mathematical, philosophical, and educational implications of such a diagrammatic computer proof system for Euclidean geometry.
On the Value of Doubt and Discomfort
Sheila K. Miller
United States Military Academy, West Point
sheila.miller@colorado.edu
To present mathematics as completely devoid of any of its relevant philosophical issues is to detach it from one of its principal sources of power to captivate and persuade. Deep learning is uncomfortable; when something causes us to question a belief we hold about the universe, our minds struggle and shift to find resolution. Many students come to college convinced that mathematics is a static subject safe from doubt and uncertainty. One gateway to an improved understanding of the field of mathematics is the discovery that there is a difference between truth and provability. These notions are definable in any course (with varying degrees of rigor), and the reality that there are limits to what can be known mathematically can be shocking, unsettling, and compelling. This talk will address how and why I discuss Gödel's Incompleteness Theorems and Cantor's Diagonalization argument in every course I teach.
Measurement and Truth in Set Theory
Sheila Miller Edwards
University of New Mexico, Taos
sheila.miller@colorado.edu
The work of set theorists is sometimes divided into two categories: studying the universe of sets and studying models of set theory. We present a perspective of truth in set theory that might allow these two categories to be reconciled into one. Joint work with Shoshana Friedman.
Rational Discovery of the Natural World: An Algebraic and Geometric Answer to Steiner
Robert H C Moir
Western University
rmoir2@uwo.ca
Steiner (1998) argues that the mathematical methods used to discover successful quantum theories are anthropocentric because they are “Pythagorean”, i.e., rely essentially on structural analogies, or "formalist", i.e., rely entirely on syntactic analogies, and thus are inconsistent with naturalism. His argument, however, ignores the empirical content encoded in the algebraic form and geometric interpretation of physical theories. By arguing that quantum phenomena are forms of behaviour, not things, I argue that developing a theory capable of describing them requires an interpretive framework broad enough to include geometric structures capable of representing the forms, which set theory provides, and strategies of algebraic manipulation that can locate the required structures. The methods that Steiner finds so suspect or mysterious are entirely reasonable given two facts:
- discovering new theories requires algebraic and structural variation of old theories in order to access new forms of behaviour; and
- recovering the (algebraic and geometric) form of the prior theories is necessary to retain their empirical support.
Accordingly, I argue that the methods used to discover quantum theory are both rational and consistent with naturalism.
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Some proofs and discoveries from Euler and Heaviside
Tom Morley
Georgia Tech
morley@math.gatech.edu
We give several examples of proofs that would not be considered proofs by contemporary Mathematicians, of correct theorems and calculations of Euler and Heaviside, including Euler’s remarkable approach to zeta of even integers, and Heaviside’s solution of the age of the earth partial differential equation. Although both of these examples can be “rigorized” by modern techniques, that is not the point. We pose more questions than answers.
Back to 2015 scheduleFeynman's Funny Pictures
Thomas D Morley
Georgia Tech
calcprof@gmail.com
"They were funny-looking pictures. And I did think consciously: Wouldn't it be funny if this turns out to be useful and the Physical Review would be all full of those funny looking pictures. It would be very amusing."--Richard Feynman
In this talk we cover briefly several ways of looking at one of the most important new notations of the twentieth century--the Feynman diagram. Feynman Diagrams were originally invented to keep track of terms in a perturbation series (Wick's expansion of a Dyson series) of the scattering matrix of a physical process with interaction(s). Yet they seem to say much more.
"The Feynman graphs and rules of calculation summarize quantum field theory in a form in close contact with the experimental numbers one wants to understand."--Bjorken, J. D.; Drell, S. D. (1965). "Relativistic Quantum Fields". New York: McGraw-Hill
We briefly look at the original use of Feynman diagrams, what they say seem to say mathematically and physically, and talk about some examples of how the notation has influenced physics and mathematics.
How my philosophy of mathematics and Asian-American upbringing Influenced my teaching and content of a "Women's Studies and Global Perspectives Course"
Junalyn Navarra-Madsen
Texas Women's University
jnavarramadsen@twu.edu
This paper focuses on the decade-long experience of the author as a mathematician teaching a specially created course at Texas Woman’s University (TWU), the largest university in the US focused on women and is also classified as a Hispanic Serving Institution (HSI). The course is Women and Minorities in STEM. This course fulfills both the baccalaureate degree requirements for graduation (i) "women’s studies (3 credit hours)" and (ii) "global perspectives (3 credit hours)" at TWU. The author will describe how her philosophy of mathematics and her Asian American background have shaped the evolution of how this course has been taught for the last few years.
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Philosophy of Mathematics in the 21st Century: Why does it need the Sciences of the Mind
Rafael Núñez
UCSD
rnunez@ucsd.edu
Mathematics is about abstract concepts, precise idealizations, relations, calculations, and notations, all of which are made possible by the amazing (albeit limited) workings of the human mind and the biological apparatus that supports it. Over the past 50 years the scientific study of mental phenomena has made enormous progress in understanding their psychological, linguistic, and neurological underpinnings. Traditional approaches in Philosophy of Mathematics such as Platonism, Formalism, Logicism, and Intuitionism - developed many decades, if not centuries prior to these developments - could not benefit from these findings. I argue that today, in the 21st century, philosophical investigation - e.g. What is mathematics? What is it for? How does it work? - should be informed by, and be compatible with findings in the sciences of the mind. I'll illustrate my arguments with research addressing issues in hyperset theory, infinitesimal calculus, and mathematical induction.
Re-Imagining Theorem-and-Proof in a Guided-Inquiry Geometry Course for Future K-8 Teachers
Chris Oehrlein
Oklahoma City Community College
cdoehrlein@gmail.com
What does it mean for elementary and middle school students to “prove” something about shapes and measurements? What does “geometric proof” mean for those students’ teachers? What role can dynamic software or apps play in developing future K-8 teachers’ concepts of evidence and proof related to geometric formulas, facts (theorems), and constructions? Reflections written by and discussions among these future teachers in a geometry course reveal what and how they comprehend pattern and evidence.
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Boundary Conditions: Numeric Representation and the Boundary of Pure and Applied Mathematics
Donald G. Palmer
Ossining, NY
dgpalmer@nul1.com
Being so basic to mathematics, numeric representational systems touch many areas of the discipline. Numeric systems could be said to lie on the boundary of pure and applied mathematics, providing the means to apply mathematics to the real world and to define how we manipulate numbers. The numeric system we use today is the decimal numeric system, which is a key system underlying current science and technology. This system is over one thousand years old, predating much of our science and technology. It can be argued that this mathematical tool, missing for the ancients, prevented them from attaining technology close to ours. In an analogous manner, discovering an expanded, more powerful numeric system would affect both sides of the pure/applied boundary. Being so basic to measurements, expanding the power of our numeric system could expand what we are able to measure, and hence provide a quantum leap in many areas of science. Being so basic to the concept of number, the expansion could provide new areas for theoretic mathematical investigation, potentially expanding what we think of as a number. This talk will consider characteristics of numeric representation and the potential for expanding our current systems, considering directions to work on and where this might lead.
Bridging the Boundary of Applied and Pure Mathematics: A Philosophical Argument for Expanding Mathematics and Scientific Disciplines via a New Numeric System
Donald G. Palmer
Ossining, NY
dgpalmer@nul1.com
This presentation is about new directions of study and discovery in both Mathematics and Science. Over the last several centuries, science has discovered objects in the world along a continuum of scale. In one direction, we have found planets and stars, galaxies and galaxy clusters. In the other direction we have found cells and proteins, atoms and neutrinos. In order to locate and model this world, we use the 3 traditional directions of length, width and height. However inherent in all our measurements is the scale of the objects we are measuring – a continuum we do not directly see with our eyes. A key reason we do not include this direction as part of our scientific models is that we do not have the appropriate mathematical tools to take measurements along this continuum. New mathematical tools may require a numeric representational system with more power than our traditional decimal or positional based numerals. Such a more powerful system may provide a single value for complex numbers able to measure across scale and adds to its structure the reversing operations of integration and differentiation. The ability to calculate integration and differentiation results would be a powerful mechanism for applied and for pure mathematics. The author presents some opening remarks on what is anticipated to be a much larger discussion, looking at a model of reality where objects at all levels of scale can be located and considers directions for generating more powerful mathematical tools than we have today.
Fictionalism, Constructive Empiricism, and the Semantics of Mathematical Language
Jae Yong John ParkNew York, NY
parkjohn0109@gmail.com
In Field's fictionalism, good mathematical theories do not need to be true, but rather must be consistent and conservative. Likewise, Van Fraassen views science to be nothing more than a study of obtaining truths about the observable phenomena of the world, so good scientific theories need not be true, merely empirically adequate. The takeaway from this comparison is that the concepts of acceptance and empirical adequacy of constructive empiricism can be used to better understand fictionalism. At first glance, constructive empiricism does not seem to help the case of fictionalism because the constructive empiricism is based on the possibility of empirical verification, which is also the basis of indispensability argument. This paper argues, however, that the empirical adequacy of scientific theories is comparable to the conservative nature of mathematical theories. By understanding that a scientific theory need not be entirely true, the falsity of mathematical statements in the fictionalist view becomes more graspable.
Why the Universe MUST be Complicated
G. Edgar Parker,* James S. Sochacki, David C. Carothers
James Madison University
parkerge@jmu.edu
Models in physics for the interaction of forces routinely consist of coupled systems of differential equations, and the Newtonian paradigm, based upon the interaction of forces, yields locally analytic solutions. From a philosophical perspective, if differential equations are to be used for such mathematical models, this coupling appears to be essential to capturing the effect of forces acting on each other. Granted that a mathematical model for physical interaction must take this coupled form, we argue that the functions that solve the system, to be analytic, must exhibit severe pathology. A heuristic argument is offered that indicates the plausibility that functions that model such physical interactions cannot be even C1. Basic ideas driving the pertinent mathematics that supports the arguments will be presented and sources for the mathematics referenced.
Structuralism and its Discontents
Charles Parsons
Harvard University
parsons2@fas.harvard.edu
By "structuralism" I mean primarily the structuralist view of mathematical objects, different versions of which have been developed and defended by several philosophers, although the underlying ideas come from much older views at least implicit in the writings of Dedekind, Hilbert, Bernays, and probably others. My own version tries to stay closer to the usual language of mathematics than some others, so that although the basic mathematical objects are "only structurally determined," no new ontology is needed to develop this idea. (For details see Mathematical Thought and its Objects (Cambridge 2008), chapters. 2-4.)
Structuralist views have been subjected to various objections. To the extent that time permits, I will try to canvass some of them and suggest replies.
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Does a proof exist if nobody has read it?
Klaus Peters
AK Peters LTD
In a practical variation of Bishop Berkeley's thesis "Esse est percipi" I will argue that mathematical results are only real when published. After some philosophical and anecdotal exploration of the statement I will argue from the point of view of a mathematician turned publisher to convince the audience of the necessity to accept this special case of Berkeley's philosophy even if one disagrees philosophically.
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A group theory perspective of mathematical constructs in physics
Horia I. Petrache
IUPUI
hpetrach@iupui.edu
In physics, mathematical constructs such as Fourier transforms and complex numbers are regarded as useful tools: they are used because they work as needed to model physical systems and their behavior. But are these tools unique or even necessary? Can we do physics without Fourier transforms, or without trigonometric functions? To answer this odd question, one would need to try to reconstruct physics without these mathematical ingredients, a very impractical task to say the least. One could also reason that if these mathematical ingredients were not necessary, physics would have likely eliminated them already! It is suggested here that a more unified "rediscovery" of mathematical constructs can be useful to address the question of their uniqueness and necessity in physics. An example is provided based on an investigation of the differential operator within group theory at elementary level. The framework of group theory is appropriate at this point in time because physics theories fundamentally are group theories. By doing this, we do not discover new mathematical constructs or new properties. Rather, the purpose of this exercise is to see how a number of mathematical constructs appear as consequences of fundamental physics principles.
Removing bias: the case of the Dirac equation
Horia I Petrache
Department of Physics, IUPUI
hpetrach@iupui.edu
I will argue that inherent human bias is often in the way of discovery. However such bias becomes obvious only in retrospect, after discovery is made. The Dirac equation for electrons and positrons is one such example of the interplay between mathematical insight and discovery. By attempting to reconcile Schrodinger equation with spacetime invariance, Dirac has used the insight that the four dimensions in spacetime needed to be put on equal footing. Although this requirement was obvious, the mathematical approach was not: it required relinquishing the natural bias that coefficients appearing in the equation must be simple commuting numbers. Once this bias was removed, Dirac equation led to new discoveries involving spinors and bispinors as the appropriate mathematical construction for fermions. It also predicted the existence of positrons, the antiparticle of electrons, as the "extra" solutions of the equation. I will discuss briefly the Dirac equation and how it further led to the use of covariant derivative in the standard model of interactions.
Philosophy Etched in Stone: The Geometry of Jerusalem's 'Absalom Pillar'
Roger Auguste Petry
Luther College at the University of Regina
Built in the first century C.E., the “Absalom Pillar” is an impressive 20 metre monument in Jerusalem's Kidron Valley noted for its unusual archaeological and geometric features. Over many years scholars have debated the meaning and function of the pillar, especially what portions serve as a sepulchral monument and what (if any) as a tomb. This paper makes use of a practical philosophical approach employed mathematically to identify external geometric features of the pillar and from these features derive principles that seem to inform its construction. In doing so, the paper draws upon (and constrains itself) to geometric knowledge available to builders in the first century C.E. A complex geometry seems to underlie the monument's construction with seeming allusions to Archimedes' works "Measurement of a Circle" and "On the Sphere and the Cylinder". Possible philosophical interpretations of these geometric findings are also explored through the writings of the Jewish philosopher, Philo of Alexandria (20 B.C.E. - 50 C.E.). The Pillar's geometry is shown to be readily intelligible through Philo's symbolic interpretations of mathematics including numeric symbolism he draws from Hebrew Scriptures. The paper concludes that the upper portion of the Pillar is likely a tomb marker and the lower portion a tomb on the basis of a possible geometric allusion to Archimedes' famous tomb marker in Syracuse.
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Defining Abstraction
| Rahmat Rashid | Mark Anderson |
| Rollins College | Rollins College |
| rrashid@rollins.edu | manderson@rollins.edu |
While there is a long philosophical tradition of examining abstract objects, along with their ontological status, definitions, and epistemic standing, the discussion takes on a uniquely important role in mathematics. Mathematicians might care to define abstraction for a better view of the connections between sub-fields and for precision of language. We provide an account of the process of abstraction, both on the micro-level of a student learning a new, abstract mathematical concept, and on the macro-level of the mathematical community spanning decades. This description gives us a better understanding of the importance of intuition in the creation of abstract mathematical objects, and the need for logic in their reification. It also allows us to compare the abstract concepts and structures that are formed, in order to delineate levels of abstraction.
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Transcendence and Emptiness
Agustín Rayo
MIT
arayo@mit.edu
I argue that the notion of a logical truth can be naturally extended to the notion of a transcendent truth. (Roughly, a sentence is transcendentally true if its truth at a world can be established by one’s metatheory without relying on information about that world.) Whether or not the transcendental truths go beyond the logical truths depends on subtle questions concerning the relationship between our language and the world it represents. I develop a picture on which arithmetical truths count as transcendentally true and use it to defuse a stubborn problem in the philosophy of mathematics.
Some Problems and Solutions in Contemporary Philosophy of Mathematics
Michael Resnik
University of North Carolina, Chapel Hill
resnik@email.unc.edu
This talk will begin by surveying some of the major problems and positions in contemporary philosophy of mathematics. This will provide the background for sketching his own approach to these problems—mathematical structuralism—and some of the important objections to his view.
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Developing mathematical intuition with a History of Math course
Raul Rojas-Gonzalez
University of Nevada Reno
rojas@inf.fu-berlin.de
We have been offering a course on History of Mathematics at UNR. The course focuses on the development of mathematical concepts and techniques through different epochs. Many of the methods discussed in the course were conceived by mathematicians working before rigorous proofs were found in later centuries. In the talk, I will review some of the beautiful intuitive proofs developed by Archimedes, Arab mathematicians, and even Isaac Newton. I will argue that the course has shown to be a very useful tool for building mathematical intuition in math majors and provide them with an overview of the fields of mathematics and their interrelationships. Some of the intuitive proofs are also useful for teaching a diverse variety of subjects.
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How logic is presented may obscure or enlighten.
Susan Ruff
Massachusetts Institute of Technology
ruff@mit.edu
Whether a proof enables readers to recognize why a statement is true may depend on both the proof's logic and how that logic is presented. This talk focuses on the latter. If a proof's logic is sufficiently amenable, a somewhat obvious strategy for making the proof explanatory is to add text that explicitly draws readers' attention to why the statement is true or to salient aspects of the proof. This strategy may increase the length of the proof or of the surrounding exposition. A less obvious strategy is to use Known->New structure to craft the proof so it flows well, thus revealing the flow of the underlying logic while keeping the proof concise. These two strategies are complementary. I will explain Known->New structure and provide examples to illustrate how combining the two strategies can help readers to follow a proof and, assuming the underlying logic is sufficiently amenable, to recognize why a statement is true.
The Tension and the Balance Between Mathematical Concepts and Student Constructions of It
Debasree Raychaudhuri
California State University at Los Angeles
draycha@calstatela.edu
The ability to abstract is imperative to learning and doing meaningful mathematics. Yet, in the preliminary stages of concept acquisition, learners of advanced mathematics are found to reduce abstraction in levels more than one, in their attempts to grasp the complex mathematical concept. Evidently, there is a tension between the way mathematics is and the level the learner is trying to reduce it to --to facilitate his or her accommodation of the concept. It is argued that the assimilation (or the balance) cannot be complete without attaining success in the last level, namely, situation of the new concept in learner's existing cognitive structure. In this presentation we will offer new data to validate this conjecture and show that it holds for mathematical concepts of any level, elementary or advanced.
A Trivialist Account of Mathematics
Agustin Rayo
Massachusetts Institute of Technology
agustin.rayo.fierro@gmail.com
I sketch an account of mathematics according to which the truths of
mathematics are not unlike the truths of logic. I argue that just like
nothing is required of the world to satisfy the demands of a truth of
logic, nothing is required of the world to satisfy the demands of a
truth of pure mathematics.
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Linguistic Relativity in Applied Mathematics
Troy D. Riggs
Union University
triggs@buster.uu.edu
Some "laws" that appear in applications of mathematics say far more about our goals, our mathematical tools and our conventions in applying those tools than they say about the nature of the universe itself. Benford's Law, the Buckingham Pi Theorem and Heisenberg's Uncertainty Principle are discussed.
The Interpretation of Probability Is Perhaps an Ill-Posed Question
Paolo Rocchi
IBM Research and Development, Italy
paolorocchi@it.ibm.com
After the Laplace definition, eminent authors put forward different approaches to the probability. The axiomatic theory, the frequency interpretation and the subjective understanding of the probability, the logical methods emerged in succession. Each view has strong and weak points, and none has been definitively accepted. The foundations of the probability calculus still remains as one of the most outstanding and unresolved mathematical question. What is worse, each school works on its own and mathematicians find hard to cooperate. Even if each stance has interesting sides, even if the subjective and frequentist interpretations appear very reasonable, theories appear irreconcilable and fire vehement debates.
Why these controversial reactions about the probability? In my opinion, we cannot directly address the probability; instead we have to scrutinize which assumptions influence the researches in the field, notably we bring the probability argument to attention in advance of the probability itself [1].
References
[1] P. Rocchi ---The Structural Theory of
Probability --- Kluwer/Plenum, N.Y. (2003).
Algebraists' Metaphors for Sameness: Philosophies, Variety, and Commonality
Rachel Rupnow and Eric Johnson
Northern Illinois University
rrupnow@niu.edu and ejohnson13@niu.edu
In order to understand how mathematicians relate mathematical concepts to sameness, surveys were sent to algebraists throughout the United States. The relevant portion of the data set for this talk focuses on open-ended responses to three questions on the meaning of sameness in math, the meaning of sameness in abstract algebra, and similarities or differences between sameness in algebra and other branches of math. Responses from 197 participants were analyzed using concep- tual metaphors (Lakoff, G., & Nunez, R. (1997). The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics. In Lyn D. English (Ed.), Mathematical reasoning: Analogies, metaphors, and images (pp. 21-89). Mahwah, NJ: Erlbaum.). Conceptual metaphors connect a target domain, in this case sameness, to a source domain that provides another way of reasoning (e.g., Sameness is a concept shown by structure-preservation.). Algebraists’ metaphors grouped into clusters including philosophical stances toward sameness, discipline-based instantia- tions of sameness, and informal language for sameness. Implications for discussing sameness-based conceptual connections across courses are discussed in light of the connections made by some of the mathematicians.
Mathematical Rigor in the Classroom
Laura Mann Schueller
Whitman College
schuellm@whitman.edu
We have become accustomed to hearing students describe their primary dislike of mathematics as the necessity to “always get the right answer.” It is not surprising, then, that our current educational climate has “redefined” the mathematics curriculum to include a wide array of activities that, while being reminiscent of traditional mathematics, no longer require exact answers or the level of rigor traditionally required in mathematics. While the trend of “softer, gentler math” may have inspired some students to study more math, this trend is not without cost.
In this talk, guided by actual and modified examples from existing curriculum, I will discuss the pros and cons of re-introducing rigor into the elementary, secondary, and post-secondary curricula.
The Eroding Foundation of Mathematics
David M. Shane
Methodist University
dshane@student.methodist.edu
The very foundation of the physical sciences is mathematics, which is arguably the most fascinating conversation in the philosophical arena. Paradoxically, the majority of mathematicians in modernity decline this engagement, and instead, set their cross hairs primarily on physical science and technological applications; which is to say that mathematicians are neglecting to develop new theoretical frameworks for their trade. This tendency may be directly observed within even the purest of mathematical branches, number theory, where the tip of the spear points to every direction, yet seldom a philosophical one. I contest that mathematicians need to acknowledge this shortcoming and reunite themselves with the philosophical network. While we may not agree completely with the assertions made by Pythagoras, Fermat, Russell, or Gödel, one cannot deny that their mathematical practices are interwoven in the fabric of their philosophical tapestry. In short, mathematics needs to spawn its own philosophers in order to facilitate the growth of new branches: the community needs avant-garde theories which are “for us, by us” and that pioneer into the wilderness of abstract thought.
Potential infinity: a modal account
Stewart Shapiro
The Ohio State University
shapiro.4@osu.edu
Beginning with Aristotle, almost every major philosopher and mathematician before the nineteenth century rejected the notion of the actual infinite. They all argued that the only sensible notion is that of potential infinity. The list includes some of the greatest mathematical minds ever. Due to Georg Cantor's influence, the situation is almost the opposite nowadays (with some intuitionists as notable exceptions). The received view is that the notion of a merely potential infinity is dubious: it can only be understood if there is an actual infinity that underlies it.
After a sketch of some of the history, our aim to analyze the notion of potential infinity, in modal terms, and to assess its scientific merits. This leads to a number of more specific questions. Perhaps the most pressing of these is whether the conception of potential infinity can be explicated in a way that is both interesting and substantially different from the now-dominant conception of actual infinity. One might suspect that, when metaphors and loose talk give way to precise definitions, the apparent differences will evaporate.
As we will explain, however, a number of differences still remain. Some of the most interesting and surprising differences concern consequences that one's conception of infinity has for higher-order logic. Another important question concerns the relation between potential infinity and mathematical intuitionism. We show that potential infinity is not inextricably tied to intuitionistic logic. There are interesting explications of potential infinity that underwrite classical logic, while still differing in important ways from actual infinity. However, we also find that on some more stringent explications, potential infinity does indeed lead to intuitionistic logic. This is joint work with Øystein Linnebo.
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Complementary foundations for mathematics: when do we choose?
Michael A ShulmanUniversity of San Diego
shulman@sandiego.edu
If one foundation for mathematics were clearly the most true or useful, there would be no debate. Instead, each foundation is well-suited for some purposes and ill-suited for others, as when we compare quantum mechanics and general relativity, or C++ and Python. Thus the question is not how to choose a foundation once and for all, but when to choose one foundation and when to choose another; and we can only answer this with examples. For instance, ZF set theory is good at studying well-foundedness; categorical set theory is good at relating different mathematical universes; constructive mathematics is good at continuity and computability; and homotopy type theory is good at invariant higher structures.
Structural Proof Theory: Uncovering capacities of the mathematical mind
Wilfried Sieg
Carnegie Mellon University
What is it that shapes arguments into mathematical proofs that are intelligible to us, and how is it that we can find proofs efficiently? – These are the informal questions I intend to address. Two aspects play a significant role: on the one hand, the abstract ways of the axiomatic method in modern mathematics and, on the other hand, the concrete ways of proof construction suggested by modern proof theory.
The subtle interaction between understanding and reasoning, i.e., between introducing concepts and proving theorems, is crucial and suggests principles for structuring proofs conceptually. These partly historical and partly theoretical investigations are complemented by experimentation with a strategically guided proof search algorithm.
It is Hilbert’s work that weaves these strands into a fascinating intellectual fabric and connects, in novel and surprising ways, classical themes with deep contemporary problems. These problems reach from proof theory through computer science to cognitive science and back. – At the very end, applications of these considerations to mathematics education are suggested.
Back to Mathfest 2010 Guest Lecture.
Methodological Frames: Mathematical structuralism and proof theory
Wilfried Sieg
Carnegie Mellon University
ws15@andrew.cmu.edu
The juxtaposition of mathematical structuralism (as practiced by Dedekind in the 1870s and 1880s) and proof theory (as articulated by Hilbert in the 1920s and Bernays in the 1930s) indicates a programmatic goal. I want to turn our attention from choosing between different "foundations" to focusing sharply on two central tasks in the philosophy of mathematics; namely, to understand the role of abstract structures in mathematical practice and the function of accessibility notions in methodological frames. Proof theory is playing a mediating role in such investigations.
Are Mathematical Objects Inside or Outside a Human Mind?
Roger Simons
Rhode Island College
rsimons@ric.edu
An important aspect of the nature of mathematics is the question of what a mathematical object is. A key component of this question is whether mathematical objects are thoughts inside people's minds or are entities external to human beings. Some arguments will be given for the internal case. But the position taken here is that mathematical objects are external to human minds. This helps account for the usefulness of mathematics in physics, engineering, and other applied fields. External objects also allow for the possibility of other species discovering and using mathematics. Denying such a possibility is an anthropocentric position analogous to assuming that our Earth is the center of the universe. Mathematical objects are certain patterns, relationships, classifications, and organizing schemes which may be perceivable in the real world. Mathematicians conceptualize these patterns to explore their properties and develop other concepts in terms of the more basic ones. But the concepts inside a mathematician's mind are analogous to integers represented inside a computer. In both cases, the internal representation is for the purpose of calculations or deductions about the external objects.
Peirce, Zeno, Achilles, and the Tortoise
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
In most of his writings, C. S. Peirce writes disparagingly of those
who fall for the arguments of Zeno. For example, in a note entitled
“Achilles and the Tortoise” [The New Elements of
Mathematics, ed. Carolyn Eisele, Mouton Publishers: The Hague, 1976,
pp. 118-120], Peirce writes:
If he [Zeno] really conducted his attack on motion so feebly as he is represented to have done, he is to be forgiven. But that the world should continue to this day to admire this wretched little catch, which does not even turn upon any particularity of continuity, but is only a faint rudimentary likeness to an argument directed against an endless series, is less pardonable.
Yet, as the quote suggests, Peirce recognized that what we read
about Zeno in Aristotle and Simplicius may not be an accurate account
of Zeno's intentions. In the same note, in the guise of a dream, he
tells us Zeno once
expounded to me those four arguments; he showed me what they really had been, and why just four were needed. Very, very different from the stuff which figures for them in Simplicius. He recognized now that they were wrong, though not shallowly wrong; and he was not a little proud of having rejected the testimony of sense in his loyalty to reason.
Although he does not reveal what he learned from Zeno, Peirce does
suggest the following alternative version of the paradox of Achilles
and the Tortoise:
Suppose that Achilles and the Tortoise ran a race; and suppose the tortoise was allowed one stadium of start, and crawled just one stadium per hour. Suppose that he and the hero were mathematical points moving along a straight line. Suppose that the son of Peleus, making fun of the affair, had determined to regulate his speed by his distance from the tortoise, moving always faster than that self-contained Eleatic by a number of stadia per hour equal to the cube root of the square of the distance between them in stadia.
In other words, suppose the Tortoise starts with an initial lead of 1 unit and let u and v be the positions of Achilles and the Tortoise, respectively, at any time t. Let x = u – v and suppose Achilles paces himself so that
(1)Peirce sees two possible outcomes, namely, either Achilles overtakes the Tortoise and wins the race or Achilles catches the Tortoise but never passes him, and asks if the arguments leading to these conclusions contain any fallacies.
His question is motivated by the mathematical nature of the problem: equation (1) is a standard example of a differential equation which does not have a unique solution in any region containing x = 0. Explicitly, with the initial condition x(0) = -1, the function
satisfies (1) for any x between 3 and infinity, inclusive. But the point Peirce wishes to make is philosophic, not mathematical; indeed, he asks not “what the answer to the mathematical problem is,” but rather where the fallacy lies in arguing to either of the proposed solutions. As such, he reminds us that finding a mathematical solution does not necessarily solve the underlying logical problem. Whereas the infinite series solution of the standard version of the Achilles paradox does provide a mathematical explanation of why there is no contradiction between our reason and our senses, in this case we are still left with a problem: does Achilles pass the Tortoise or not?
Realism and Mathematics: Peirce and Infinitesimals
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
The 19th century philosopher and mathematician C. S. Peirce well understood the importance of the work of Cauchy, Weierstrass, and others in creating a foundation for analysis in a logically sound understanding of limits. Nevertheless, he found what he called the doctrine of limits unsatisfactory because he saw it as a nominalistic solution to the problem. Peirce felt that, in the light of the work of Cantor on the infinitely large, one could develop a consistent theory of the continuum using infinitesimals. Moreover, he thought such a theory necessary to an understanding time and consciousness. In this talk, I will discuss how Peirce's commitment to scholastic realism and his own pragmaticism led him to the position of accepting infinitesimals as an essential reality of the continuum.
The De Continuo of Thomas Bradwardine
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
This talk will briefly explore Thomas Bradwardine’s view of the composition of continua. Bradwardine was a 14th century philosopher, logician, mathematician, theologian, and, shortly before his death, the Archbishop of Canterbury. Chaucer ranks him with Augustine and Boethius for his most famous work, De Causa Dei, a treatise on free will. C. S. Peirce states that Bradwardine “anticipated and outstripped our most modern mathematico-logicians, and gave the true analysis of continuity.” Bradwardine’s De Continuo is a careful analysis of the nature of geometric, physical, and temporal continua in 24 definitions, 10 suppositions, and 151 conclusions. Many of the conclusions investigate the logical difficulties faced by those of his contemporaries who held that continua could be built up from either a finite or an infinite number of indivisibles. Indeed, in his 151st conclusion, Bradwardine concludes not only that a line is not a mere aggregation of points, a surface a collection of lines, or a solid a union of surfaces, but that points, lines, and surfaces do not exist.
Should My Philosophy of Mathematics Influence My Teaching of Mathematics?
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
My short answer to the title question is, of course, "Yes, no, and it depends." There are times when the answer should be "yes," such as when one is considering how much time to devote to the Bayesian approach, or that of Neyman and Pearson, or Fisher, when teaching a course in statistics. At other times, it seems to me the answer is clearly "no." For example, I would argue that if you do not, on philosophical grounds, accept arguments from contradiction or the law of the excluded middle, you nevertheless would be doing your students in an analysis course a serious disservice not to introduce them to results depending on these principles. Mathematics without the laws of noncontradiction and the excluded middle is very different from mathematics with them. Yet in other cases the choices may not be so radical. For example, the approaches of nonstandard analysis and standard analysis lead to the same results by different routes. Would students be harmed if you chose the route you found most philosophically attractive?
Being a Realist Without Being a Platonist
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
We consider something to be real if its properties do not depend on what any collection of people might think of it. Many mathematicians tend to think this way about mathematical objects: as G. H. Hardy puts it, "317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is so, because mathematical reality is built that way." To explain their realism, Hardy and others often rely on the language of Platonism. As Alain Connes says, the object of mathematics is "not material, and it is located in neither space nor time," but nevertheless "has an existence that is every bit as solid as external reality, and mathematicians bump up against it in somewhat the same way as one bumps into a material object in external reality." Yet Platonism brings with it significant philosophical baggage, to the point that the world of ideas becomes in some way more real than the world of individuals, and knowledge involves peeking into a world of which we have no physical contact. But to be a realist does not necessitate being a Platonist. We can hold both "dogs" and "dog" to be real without a dog becoming a mere shadow. This talk will draw upon the thought of C. S. Peirce to formulate how one may be a realist without being a Platonist.
The Consequences of Drawing Necessary Conclusions
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
Benjamin Peirce defined mathematics to be “the science which draws necessary conclusions.” His son, Charles Peirce, pointed out a significant consequence of this definition: mathematics, out of all the sciences, relies upon no other science. A mathematician seeks out the consequences of given hypothetical relationships. In doing so, he need not concern himself with either the nature of the objects involved, or how it is that we come to know them.
In particular, mathematics is independent of philosophy. Yet this does not lessen the importance of the work of the philosopher of mathematics: an account of the nature of mathematical knowledge is of fundamental importance to our understanding of the nature of human knowledge as a whole. As G. H. Hardy pointed out, anyone "who could give a convincing account of mathematical reality would have solved many of the most difficult problems of metaphysics." In attempting to find this account, philosophers need to pay close attention to exactly what it is that mathematicians do. Although the philosophy of mathematics need not have any influence on mathematical practice, it is a matter of vanity for mathematicians to think that the philosophy of mathematics is worthwhile only if it were to have some such influence.
Philosophical and mathematical considerations of continua
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
What is a continuum? How is one composed?
Are these mathematical or philosophical questions? Over the years, mathematicians have conceived of continua in various ways. For the most part, modern mathematics considers a linear continuum to be anything homeomorphic to the real line (the real numbers endowed with a certain topology). Is this progress, or just consensus around one of many possible conventions?
Philosophical considerations of the nature of continua go back to at least Zeno. Over the last 2500 or so years, philosophers have given careful thought to the consequences of differing hypotheses concerning the makeup of continua without ever reaching anything close to a consensus. Is this lack of progress?
This talk will provide a brief historical overview of how philosophers and mathematicians have thought about continua and then address the question of whether or not philosophers have anything to contribute to how mathematicians conceive of them. In particular, we will look at some criticisms which C. S. Peirce directed at the identification of linear continua with the real numbers.
Insights Gained and Lost
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
Insights and discoveries in mathematics are seldom superseded or replaced in the course of further development. Our understanding of a certain mathematical concept or theory may increase with time, and may even undergo significant reformulation, yet the objects and relations remain, in most cases, unchanged. In contrast, the objects to which theories in the natural sciences refer have changed significantly over time. Even more, the discovery of a new object in modern physics is now a statement of statistics, a reference to a set of observations with a very small p-value. As G. H. Hardy observed, the difference appears to be that “the mathematician is in much more direct contact with reality.” This talk will consider the implications of this difference between mathematics and the natural sciences, and then consider one significant exception: how early insights on the nature of a linear continuum, from Aristotle to Bradwardine, have given way to the modern view of the real line, and what may have been lost in the process.
Making Philosophical Choices in Statistics
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
Most of us tend to believe we are agnostic as to our philosophical convictions when we are in the classroom. For much of what we teach, there is some truth to this belief: although choices have been made, they are so far in the background that we tend not to think much about them. However, the story is not as simple when we teach statistics. There we are confronted with at least three competing philosophical approaches from which to choose: the frequentist realist view of R. A. Fisher, the frequentist behaviorist perspective of Jerzy Neyman and Egon Pearson, or the subjective view of a Bayes/Laplace development. No philosophy of statistics has a claim to be the standard approach; indeed, some textbooks will present all three of these. Moreover, unlike, for example, an analysis course where the choice between a standard and a nonstandard development influences only the presentation, the philosophical choices we make in statistics influence our conclusions as well. In this talk, I will discuss these three schools of thought, with particular emphasis on the differences between the two frequentist approaches.
Hardy, Bishop, and Making Hay. Preliminary report.
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
I find many parts of constructive mathematics appealing. For example, I find Kronecker's dictum “[d]ie ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk,” an attractive starting point. Philosophically, I am intrigued by Errett Bishop's addition that “mathematics belongs to man” and “[i]f God has mathematics of his own that needs to be done, let him do it himself.” Yet, when confronted with, say, the version of the Hahn-Banach theorem in Bishop's Constructive Analysis, I find myself turning back to the cleaner statement in classical (“God's”?) mathematics. And I recall G. H. Hardy's remark that he was willing to accept the Axiom of Choice, in part, because to deny it “seems to make hay of a lot of the most interesting mathematics.” In this talk, I will discuss the context of Hardy's comment and how the creation of interesting and aesthetically pleasing mathematics influences one's acceptance of axioms, and hence one's philosophy of mathematics.
What is a measure?
Daniel C. Sloughter
Furman University
dan.sloughter@furman.edu
Most mathematicians first see measures as functions defined on sets: given a sigma algebra Σ of subsets of R, a measure is a function μ : Σ → R satisfying certain conditions. On the other hand, a functional analyst may find it more useful to think of a measure as a linear functional on the space Cc(R), the set of all continuous functions on R with compact support. The notation changes accordingly: ∫ f dμ becomes <μ f>. With the change of notation comes a change in view: for example, the latter motivates generalizations to linear functionals on related function spaces, such as distributions, or "generalized functions", as linear functionals on Cc∞ (R). What, then, is a measure? Poincaré wrote that "mathématique est l'art de donner le même nom à des choses différentes." But the other side of this: sometimes it's advantageous to give a different name to one thing. Or is there only one thing? How would we know?
Logic, Intuition, and Infinity
Rick Sommer
rickcorysommer@gmail.com
Stanford University
Mathematics survived, even flourished, through the late 19th Century while for the most part the mathematical community rejected the concept of an actual infinity. Then following the inspirational work of Cantor, the existence of infinite sets as completed totalities has become accepted to the point mathematicians now view properties of infinite sets as part of their mathematical intuition. But how can this be justified when we have no direct experience with infinite sets? Can we explain reasoning about infinite sets in a finitary way (in the spirit of Hilbert’s program)? Results in proof theory, briefly described in this talk, provide a finitary, or even finite, interpretation of the infinitary mathematics of number theory and analysis. We explain that our mathematical domains can be interpreted as finite approximations, that capture the content and meaning of their infinitary counterparts, allowing us to formally justify use of infinite sets in mathematics without having to change our use of “infinity” in the language of mathematics. This prompts the question of how can logical reasoning about infinite sets be understood? What is the relationship between our intuitions of infinite sets and the results of our formal theory of infinite sets? In this talk we explore these questions and offer an outline on approaching this problem through the apparatus of proof theory.
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Mathematics: An Aesthetic Endeavor
Sam Stueckle
Trevecca Nazarene University
sstueckle@trevecca.edu
In this talk I will briefly outline several views of aesthetic theory, including imitations/representation, form, expression, intrinsic beauty, and objective vs. subjective criteria. I will then examine mathematics as an aesthetic endeavor from these perspectives.
Mathematics as Representational Art
Sam Stueckle
Trevecca Nazarene University
sstueckle@trevecca.edu
There are several models of aesthetic value in the philosophy of aesthetics, including imitation/representation, formalism, and expressionism. In this talk I intend to examine the ways in which mathematics can be seen as having a representational aesthetic. Many forms of representation in aesthetics, from those that are very realist, where the aesthetic value is in how accurately the aesthetic object represents the real world, to the more general forms, where the aesthetic value is in how well the aesthetic object represents some abstract world, can be applied to mathematics. In Works and Worlds of Art, Nicholas Wolterstorff emphasizes the fundamental role of representation in art. He argues that although representation is not essential, it is both pervasive and fundamental in art. Also, representation is not merely about symbols and their relationship to entities that they symbolize; rather, it fundamentally involves the human activity of “world projection.” From this viewpoint mathematics is art at its best, from how well an applied mathematics model fits the real world to how well a mathematical theory represents an underlying mathematical structure.
Kalmár’s Argument Against the Plausibility of Church’s Thesis
Mate Szabo
Carnegie Mellon University
mszabo@andrew.cmu.edu
In his famous paper, “An Unsolvable Problem of Elementary Number Theory,” Alonzo Church (1936) identified the intuitive notion of effective calculability with the mathematically precise notion of recursiveness. This proposal, known as Church’s Thesis, has been widely accepted. Only a few papers have been written against it. One of these is László Kalmár’s “An Argument Against the Plausibility of Church’s Thesis” from 1959, which claims that there may be effectively calculable functions which are not recursive. The aim of this paper is to present Kalmár’s argument in detail, and to give an insight into Kalmár’s general views on the foundations of mathematics. In order to do this, first I will survey Kalmár’s papers on the philosophy of mathematics, “The Development of Mathematical Rigor from Intuition to Axiomatic Method” (1942) and “Foundations of Mathematics – Whither Now?” (1967). Then I will present his argument against Church’s Thesis in detail. After that, I will attempt to make his argument more appealing drawing on the core views he expresses in his other papers on the philosophy of mathematics.
The Roots Of Kalmár's Empiricism
Mate Szabo
Carnegie Mellon University
mszabo@andrew.cmu.edu
According to Kalmár, mathematics always stems from empirical facts and its justification is, at least in part, an empirical question. The idea that mathematics has empirical origins appears already in his first philosophical paper, The Development of Mathematical Rigor from Intuition to Axiomatic Method from 1942. By that time Kalmár's view was influenced by Sándor Karácsony, a Hungarian linguist and educationist. Karácsony had his own version of a picture theory of language. In his view people represent everything by "inner pictures" and communication works in the following way: the aim of the speaker is to describe their "inner pictures" for the listener in a way that the listener can access the same "inner picture." In Karácsony's view, these "inner pictures" always stem from experience. For Kalmár, these "inner pictures," originated in our experiences, are indispensable for mathematics. We use the pictures to "read off" the properties of mathematical concepts, not only on an intuitive level but even on the most abstract, axiomatic level. In my talk I will to explain Kalmár's view in detail, touching upon Karácsony's inuence.
Natural logicism for mathematics
Neil Tennant
Department of Philosophy of The Ohio State University
tennant.9@osu.edu
Logicism was the view that the truths of number theory and analysis were ‘logical truths in disguise’. Logicism had fallen out of favor by the 1930s. Mathematical foundationalists turned to first-order set theory, which had no pretensions of being 'just logic'. Set theory employed powerful existence assumptions that enabled it to serve as a unifying theory for all branches of mathematics.
Natural deduction was invented by Gerhard Gentzen in the mid-1930s. It is a system of logical proof based on so-called introduction and elimination rules of inference for the usual logical expressions---connectives, quantifiers and the identity predicate.
A mathematical foundation of set theory, developed strictly formally in the system of first-order natural deduction, has various drawbacks. Mathematical statements have to be laboriously transcribed into set-theoretical notation. And significant portions of one's formalization of any ordinary mathematical proof end up having more to do with unpacking arbitrarily chosen set-theoretic definitions than with pursuing the mathematical 'line of thought' within the mathematical proof itself.
The contemporary natural logicist is concerned to re-visit the issue of logicism, equipped now with more advanced methods of natural deduction. The aim is to regiment mathematical reasoning as one finds it, in the native terms of the various branches of mathematics. New rules are given, for introducing and eliminating important mathematical concepts and constructions. Natural logicism calls, in effect, for a theory of natural deduction for logico-mathematical reasoning. Formalizations of mathematical proofs should capture their endogenous ‘lines of thought’, and not be cluttered with the extra moves occasioned by an extraneous foundation. Natural logicism is a research program calling for a clearer understanding of both the conceptual constructs and the abstract structures of mathematics.
Back to Mathfest 2011 Guest Lecture.
Some remarks on collective sense-making
Theodore V Theodosopoulos
The Nueva School, San Mateo, CA
ttheodosopoulos@nuevaschool.org
What is sense-making? How do we recognize it? How might it become collective? We propose a view of sense as a story, populated by characters and actions, arising from consistently imputable motives. Analogies play a catalytic role in what feels like a cascade of simplifications. What are obstacles to this sense-making cascade and when are we willing to accept something manifestly true as reasonless? When do we entertain the possibility that a claim is genuinely neither true nor false, at least not in a forced way, where it feels like we have the freedom to choose? We explore the discontinuous nature of "aha" moments and the sense of agency this engenders. We embrace the irreducibly dialectic nature of sense-making and demonstrate how collaboration facilitates keeping ambiguity from collapsing. It feels like delaying as much as possible the answer to the question "what is being said," while simultaneously saying as much as possible. The resulting contingency across the facets of the emerging story encodes a great variety of logics, which render the whole incommensurate with the parts.
Assimilation in Mathematics and Beyond
Robert S D Thomas
University of Manitoba
thomas@cc.umanitoba.ca
“Assimilation” is my term for the operation of assigning something to a class, whether others would do so or not, and for the formation of classes in that way. This is an ordinary-language phenomenon; one sees a chipmunk and recognizes it as a chipmunk. One has available one’s personal class of chipmunks based on acquaintance with past chipmunks and what one knows of mammalian species or just pictures. This operation has an interesting relation to mathematics. Poincaré goes so far as to say “Mathematics is the art of giving the same name to different things.” It has been done successfully, and it has failed. It is avoided, and it can be done well (formation and representation of equivalence classes). But there is not even a standard term for it. It is the method of my essay, “Extreme Science: Mathematics as the Science of Relations as such” in the Gold/Simons MAA anthology, where I assimilate mathematics to the sciences. In the paper, I discuss assimilation in a historical way.
Back to MathFest 2013 schedule
Persecution of Nikolai Luzin
Maryam Vulis
NCC and York College CUNY
maryam@vulis.net
This presentation will discuss the life and work of the Russian mathematician Nikolai Luzin, who was denounced by the Soviet Government over his adverse views on the philosophy of mathematics. Luzin was involved in the early 20 century crisis of philosophical foundations of mathematics. He built on L. E. J. Brouwer’s intuitionist work. In particular, their rejection of the Law of Excluded Middle was condemned as contrary to Marxist dogma that every problem is solvable. Luzin was accused of following the traditions of the Tsar Mathematical School which among other transgressions promoted religion. Many important details of Luzins case came to light only recently. Even his famous students, Kolmogorov, Aleksandrov, and Pontryagin joined the vicious campaign, however despite the danger he faced, Luzin never renounced his position.
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On the hierarchy of natural theories
James WalshCornell University
jameswalsh@cornell.edu
Gödel famously showed that there is no universal axiomatic theory for the development of arithmetic. Instead, we are left with a vast array of ever stronger competing axiomatic theories. Perhaps the main tool for comparing these axiomatic theories is by measuring their relative consistency strength. It is a well-known empirical phenomenon that it is always possible to compare natural axiomatic theories by relative consistency strength, even though axiomatic theories are not generally comparable according to consistency strength. Why are the natural axiomatic theories linearly ordered by consistency strength? Without a precise mathematical definition of naturalness, it is unclear how to study this question mathematically. I will discuss some strategies for addressing this problem that have been developed recently.
Logic in the Integers
Yale Weiss
co-Director of the Saul Kripke Center
City University of New York, The Graduate Center
Since at least 1679, logicians have been interested in arithmetical interpretations of formal systems of logic, that is, in ways of interpreting given logics in natural arithmetical structures (or, identifying logics over such given structures). Thus, for example, Leibniz developed interpretations of the syllogistic in the divisibility lattice (N,|) and, more recently, logicians have investigated the tense logic of (Z,<). In this talk, I will survey some of these results, both historical and contemporary, with a special focus on (N,|) and non-classical logics exactly characterizable therein. Emphasis will be given both to philosophically suggestive features of arithmetical structures and to how certain metalogical results logicians have been independently interested in can be given elegant new proofs by exploiting elementary properties of the numbers.
Philosophy of Mathematics in Classical India: an Overview
Homer S. White
Georgetown College, KY
Homer_White@tiger.georgetowncollege.edu
We will offer an overview of the little that is currently known about the philosophical reflection on mathematics in classical India, comparing typical Indian views on the nature and purpose of mathematics with those that have tended to prevail in the West. We will raise, and to some extent suggest answers to, a variety of questions: were Indian views of mathematics overridingly empiricist? Why does mathematics appear to have played no role in classical Indian philosophy, even in technical disciplines such as logic and ontology? Were rigorous proofs important to Indian mathematicians? What were the criteria for an acceptable proof?
Beyond Practicality: George Berkeley and the Need for Philosophical Integration in Mathematics
Joshua B. Wilkerson
Texas A&M University
jbwilkerson@tamu.edu
“When am I ever going to use this?” As a math teacher, this is the number one question that I hear from students. It is also a wrong question; it isn’t the question the student truly intended to ask. The question they are really asking is “Why should I value this?” and they expect a response in terms of how math will solve their problems. But should we study math only because it is useful? Or should we study math because it is true?
It is my contention that valuing mathematical inquiry as a pursuit of truth is a better mindset in which to approach the practice of mathematics, rather than exalting practicality. This paper will demonstrate one unexpected reason to support such a philosophical view: it actually leads to more practical applications of mathematical endeavors than would otherwise be discovered.
Support for this theory may be found in the life of George Berkeley. This paper will examine the historic mathematical implications of Berkeley's philosophical convictions: the refinement of real analysis and the development of nonstandard analysis. Berkeley not only answers the question of why we need philosophical integration in mathematics, but also how we approach such integration. This paper will close by examining the latter.
Gardens of Infinity: Cantor meets the real deep Web
Luke Wolcott
Lawrence University
luke.wolcott@lawrence.edu
The real deep Web – curated, visceral, profound – is an antidote to oversaturated webpages of words and mindless viral videos. The content complements logical arguments with stories and meaningful prompts to contemplate. The format moves away from walls of text towards high-concept design that encourages deep thought.
The Gardens of Infinity project is a collaboration between a mathematician, an interaction designer and a programmer. We present five provocative statements from Cantor’s set theory (for example, of course, ||Z|| < ||R||), and the translation between rigorous mathematics and metaphor is carefully articulated. Each statement branches down four paths: the user can read a rigorous proof of the statement, a shorter more accessible summary argument of the statement, the story of the people and events surrounding the statement, or a philosophical discussion of what it might mean. These last sections – sometimes presenting conventional philosophical interpretations, sometimes unapologetically metaphorical – are in a sense the real meat of the project, leading the user to contemplate infinity in new ways. My talk will explain and demo this web project, which may or may not be up at gardensofinfinity.com by the time of the conference.
Explanation and Existence
Stephen Yablo
Massachusetts Institute of Technology
stephen.yablo@gmail.com
Platonists hold that mathematical objects "really exist." Nominalists deny this. The standard argument for platonism, which emphasizes the indispensability of mathematics to physical science, has fallen on hard times lately. Why should calculus have to be true, to help with the representation of facts about the motion of bodies? Platonists have responded that math also plays an *explanatory* role - e.g. honeycomb has a hexagonal structure because that is the most efficient way to divide a surface into regions of equal area. Two questions, then. Can physical outcomes occur for mathematical reasons? If so, how does this bear on debates about the existence of mathematical objects?
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Statistics as a Liberal Art
Brian R Zaharatos
University of Colorado
brian.zaharatos@colorado.edu
Statistics is often classified as a branch of mathematics or as one of the “mathematical sciences.” For example, the Department of Statistics at the Florida State University claims that “Statistics is the mathematical science involved in the application of quantitative principles to the collection, analysis, and presentation of numerical data.” (italics added) Such classifications give the impression that statistics is essentially about numerical manipulation, calculation, and procedure. But at the same time, such classifications conceal a number of important philosophical issues in statistical theory and practice. In this paper, I argue that (1) a number of philosophical issues arise in statistical theory and practice; (2) in part because of these philosophical issues, statistics is better classified as a branch of philosophy, and thus, a liberal art; and (3) classifying statistics as a liberal art would be beneficial for attracting students that are otherwise not initially attracted to the mathematical sciences.
Math-Speak: Syntax, Semantics, and Pragmatics
Paul Zorn
Saint Olaf College
zorn@stolaf.edu
Mathematics is famously difficult, especially for students first seriously encountering theory and proofs. The problem is not just that “math is hard,” but that the special language of mathematics is especially hard.
This is not surprising: communicating technical ideas and fine distinctions naturally requires extra linguistic effort. This difficulty stems, I'll argue, only partly from the genuinely complicated syntax and semantics of mathematical language. It arises also from linguistic “pragmatics”: what's “heard” depends not only on what's said but also, crucially, on what “hearers” bring to the “conversation”. I'll illustrate with examples connecting the pragmatics and the syntactical and semantic issues, and, perhaps, suggest some possible strategies.
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Proofs that explain, proofs that don't, and proofs of the obvious.
Paul Zorn
Saint Olaf College
zorn@stolaf.edu
Proof and explanation are fundamentally different mathematical and psychological activities. For example, careful mathematicians (e.g., journal referees) will mainly agree with each other that a proposed proof is or isn't valid, but they may disagree strongly on its explanatory value. Proofs are rule-bound and formal; explanations may appeal mainly to psychology, or even to aesthetics.
Yet proof and explanation are closely tied in practice, especially in pedagogy. Standard proofs mathematics majors encounter may explain a lot, a little, or almost nothing. And, surprisingly, proofs of ostensibly obvious facts can illuminate unexpected and deeper properties of ostensibly familiar objects. I'll illustrate with simple examples, mainly from set theory.
Watching our language: syntax, semantics, and pragmatics in teaching
Paul Zorn
zorn@stolaf.edu
Saint Olaf College (retired)
A broad philosophy of mathematics entails many hard questions: Is mathematics constructed or discovered? Formal or experimental? Really "true" or only agreed-upon? Related, and perhaps easier, are "linguistic" questions: How are mathematical claims constructed syntactically? How effectively do syntactically correct claims actually convey meaning? What do human readers and hearers, especially students, make of mathematical claims, many of which are grammatically and logically complicated? How can we teachers help students develop ease, precision, and (eventually) deep understanding of mathematical claims? How can mathematical texts help students navigate these rough waters?
I will argue--with concrete examples--that mathematical syntax, semantics, and pragmatics are all difficult, but that concrete examples, chosen with due attention to language, have a special role in motivating and helping students engage the abstract ideas and theories of higher mathematics.
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