Philosophy of Mathematics

The topic of this Contributed Paper Session is "Philosophy of Mathematics for Working Mathematicians."  Description:  Philosophers have a wide range of views on the nature and existence of mathematical objects. How is it that mathematics continues to flourish, year after year, when philosophical questions about the fundamental nature of mathematical objects remain controversial and unsettled? This session invites papers that address, and clarify the relevance of, this issue, and propose views of mathematical objects that are consistent with mathematical practice.

Organizers:  Carl Behrens, Alexandria, Virginia, and Bonnie Gold, Monmouth University

1:00  1056-M5-259 James R. Henderson, “What Is the Character of Mathematical Law?”

1:30  1056-M5-596 Carl E. Behrens, “John Stuart Mill's "Pebble Arithmetic" and the Nature of Mathematical Objects”

2:00  1056-M5-1635 Thomas Drucker, “Dummett Down: Intuitionism and Mathematical Existence”

2:30  1056-M5-1770 Martin Flashman, “The Articulation of Mathematics-A Pragmatic/Constructive Approach to The Philosophy of Mathematics”

3:00  1056-M5-445 Lawrence A. D’Antonio, “Molyneux's Problem”

3:30  1056-M5-1015 Jeff Buechner, “Mathematical practice and the philosophy of mathematics”

4:00 1056-M5-444 Daniel C. Sloughter, “Being a Realist Without Being a Platonist”

4:30  1056-M5-1918 Ruggero Ferro, “An analysis of the notion of natural number”

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Portland OR August 7, 2009, 1 - 2:15 p.m.

Schedule

Note:  this session, "The History of Mathematics and its Philosophy, and Their Uses in the Classroom" was joint with HOMSIGMAA.  Most of the talks were history of mathematics; the talks listed below involved either only philosophy of mathematics, or both history and philosophy.

Organizers:  Janet Beery, University of Redlands; Bonnie Gold, Monmouth University; Amy Shell-Gellasch, Pacific Lutheran University; Charlotte Simmons, University of Central Oklahoma

1:00 – 1:15  "Which Came First? The Philosophy, the History, or the Mathematics?" Martin E Flashman, Humboldt State University

 1:20 – 1:35 "Should My Philosophy of Mathematics Influence My Teaching of Mathematics?" Daniel Sloughter, Furman University

 1:40 – 1:55 "Philosophical Questions You DO Take a Stand on When You Teach First-year Mathematics Courses" Bonnie Gold, Monmouth University

  2:00 – 2:15 "Using the Philosophy of Intuitionistic Mathematics to Strengthen Proof Skills" Jeff Buechner, Rutgers University

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8:30 - 8:55 a.m.   Mathematical Rigor in the Classroom, 1035-Q1-1936
Laura Mann Schueller, Whitman College

9:00 - 9:25 a.m.  Mathematics is a Meme(plex), 1035-Q1-25
Andrew G. Borden, Converse, TX

9:30 - 9:55 a.m.  Are Euclid’s Postulates Really Essences? 1035-Q1-1360
Carl E. Behrens, Alexandria, VA

10:00 - 10:25 a.m.  The De Continuo of Thomas Bradwardine, 1035-Q1-181
Daniel C. Sloughter, Furman University

10:30 – 10:55 a.m.   Ignoring the Obvious in Philosophical Applications of the Gödel Incompleteness theorems,  1035-Q1-1461
Jeff Buechner, Rutgers University

11:00 – 11:25 a.m.  What Does It Mean for One Problem to Reduce to Another?  1035-Q1-94
James R Henderson, University of Pittsburgh-Titusville

11:30 – 11:55 a.m.  Remarks about the notion of EXISTENCE in mathematics,  1035-Q1-1722
Ruggero Ferro, Univ Degli Studi di Verona

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Organizers:
Bonnie Gold, Monmouth University, bgold@monmouth.edu
Charles Hampton, The College of Wooster

8:00 – 8:35:  What Place Does Philosophy Have in Teaching Mathematics?  1023-N1-1867
Martin Flashman, Humboldt State University

8:40 – 9:15:   Mathematics as Representational Art, 1023-N1-1392
S. Stueckle, Trevecca Nazarene University

9:20 – 9:55:  From an analysis of definitions to a view of mathematics, 1023-N1-637
Ruggero Ferro, University of Verona

10:00 – 10:35:  Searle’s Metaphysics of Computation and Alternative Logics: A Surprising Connection,  1023-N1-973
Jeff Buechner, Rutgers University

10:40 – 11:15: Why do we all get the same answers?  Kitcher’s anti-apriorism and the problems of social constructivism,  1023-N1-882
Carl E. Behrens, Alexandria VA

11:20 – 11:55:  In Praise of Cranks: Are You Thinking What I’m Thinking?  1023-N1-292
Andy D. Martin, University of Kentucky

1:00 – 1:35:  Why the Universe MUST be Complicated,  1023-N1-243
G. Edgar Parker,* James S. Sochacki, David C. Carothers, James Madison University

1:40 – 2:15:  Catching the Tortoise: A Case Study in the Rules of Mathematical Engagement,  1023-N1-133
James Henderson, University of Pittsburgh-Titusville

2:20 – 2:55:  The Philosophical Status of Diagrams in Euclidean Geometry,   1023-N1-459
Nathaniel Miller, University of Northern Colorado

3:00 – 3:55:  Representations in Knot Classification  1023-N1-1387
Kenneth Manders, University of Pittsburgh

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Organizers:
Roger A. Simons, Rhode Island College, rsimons@ric.edu
Satish C. Bhatnagar, University of Nevada

8:00 - 8:25    What Are Mathematical Objects? An Empiricist Hypothesis, 1014-A1-1158
Carl E. Behrens, Alexandria, VA

8:30 -8:55    Mathematical objects may be abstract, but they're NOT acausal, 1014-A1-276
Bonnie Gold, Monmouth University

9:00 - 9:25    How the way we `see' mathematics changes mathematics, 1014-A1-1352
Sarah-Marie Belcastro, Xavier University

9:30 - 9:55    The Square Root of 2, Pi, and the King of France: Ontological and Epistemological Issues Encountered (and Ignored) in Introductory Mathematics Courses, 1014-A1-1010
Martin E. Flashman, Humboldt State University and Occidental College

10:00 -10:25    Mathematics: An Aesthetic Endeavor, 1014-A1-626
Sam Stueckle, Trevecca Nazarene University

10:30 - 10:55    Propensities and the Two Varieties of Occult Qualities, 1014-A1-500
James R Henderson, University of Pittsburgh-Titusville

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1:00-1:20    What Is Mathematics II:  A Possible Answer, 1003-P1-488
 Bonnie Gold, Monmouth University

1:25-1:45    Strands in the history of geometry and how they affect our views as to what geometry is, 1003-P1-652
 David W. Henderson, Daina Taimina, Cornell University

1:50-2:10    Object and Attribute: the case of Curves and Equations, 1003-P1-830
 Robert E. Bradley, Adelphi University

2:15-2:35    Philosophy of Mathematics in Classical India: an Overview, 1003-P1-522
 Homer S. White, Georgetown College, KY

2:40-3:00    Realism and Mathematics: Peirce and Infinitesimals, 1003-P1-655
 Daniel C. Sloughter, Furman University

3:05-3:25    Fictionalism and the interpretation of mathematical discourse, 1003-P1-405
 Thomas Drucker, University of Wisconsin-Whitewater

3:30-3:50    Linguistic Relativity in Applied Mathematics, 1003-P1-409
 Troy D. Riggs, Union University

3:55-4:15    Applied Mathematics---A Philosophical Problem, 1003-P1-708
 Charles R. Hampton, The College of Wooster

4:20-4:40    Generalised likelihoods, ideals and infinitesimal chances - how to approach the "zero-fit problem," 1003-P1-597
 Frederik S. Herzberg, University of Oxford

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2:00-2:20    Philosophy of Mathematics: What, Who, Where, How and Why, 993-U1-528
 Charles R. Hampton, The College of Wooster

2:25-2:45    On the Nature of Mathematical Thought and Inquiry: A Prelusive Suggestion, 993-U1-1331
 Padraig M. McLoughlin, Morehouse College

2:50-3:10    The Interpretation of Probability Is Perhaps an Ill-Posed Question, 993-U1-185
 Paolo  Rocchi, IBM Research and Development, Italy

3:10-3:35      RECEPTION

3:40-4:00    When is a Proof a Proof?, 993-U1-1266
 Joseph Auslander, University of Maryland (Emeritus)

4:05-4:25    The Poetic View of Mathematics, 993-U1-1080
 Jerry P. King, Lehigh University

4:30-4:50    "You cannot solder an Abyss with Air" - the Role of Metaphor in Mathematics, 993-U1-376
 Lawrence D'Antonio, Ramapo College  (Note: this talk wasn't presented due to speaker's illness)

4:50-5:15      RECEPTION

5:20-5:40    The NonEuclidean Revolution Makes Relativism Available to the Rest of the World, 993-U1-294
 Michael J. Bossé, Morgan State University

5:45-6:05    The Tension and the Balance Between Mathematical Concepts and Student Constructions of It, 993-U1-1564
 Debasree Raychaudhuri, California State University at Los Angeles

6:10-6:30    On Godel's Proof and the Relation Between Mathematics and the Physical World, 993-U1-408
G. Arthur Mihram* and Danielle Mihram

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1:00 p.m.  Why Plato was not a Platonist, 983-S1-49
Thomas Drucker, University of Wisconsin-Whitewater

1:30 p.m.  Peirce, Zeno, Achilles, and the Tortoise, 983-S1-351
 Daniel C Sloughter, Furman University

2:00 p.m.  Structuralist Mathematics and MathematicalUnderstanding, 983-S1-544
 Kenneth Manders*, University of Pittsburgh

2:30 p.m.   Are Mathematical Objects Inside or Outside a HumanMind?983-S1-546
 Roger A. Simons*, Rhode Island College

3:00 p.m. What is Mathematics I: The Question, 983-S1-341
 Bonnie Gold*, Monmouth University

3:30 p.m.   A Conjecture about... Feminist Mathematics?  983-S1-429
 Sarah-marie Belcastro*, Xavier University

4:00 p.m.   Defining Mathematical Esthetics within the NCTMStandards, 983-S1-137
 Michael J. Bossé*, Indiana University of Pennsylvania

4:30 p.m.   Unfair Gambles in Probability (Preliminary Report), 983-S1-482
 John E Beam*, University of Wisconsin-Oshkosh

5:00 p.m. The Pedagogical Challenges of One to One Correspondence, 983-S1-551
 Satish C Bhatnagar*, University of Nevada-Las Vegas

5:30 p.m.   A Unifying Principle Describing How MathematicalKnowledge Unfolds, 983-S1-545
 M Anne Dow*, Maharishi University of Management

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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.

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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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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.

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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.

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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.

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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.

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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

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Carl E. Behrens
5107 Cedar Rd., Alexandria, VA 22309

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. 

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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 ).

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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.

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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.

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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.

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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.

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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”.

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        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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.
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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.

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

Abstract: 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.

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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)?

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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.

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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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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.

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Molyneux's Problem

 Lawrence D'Antonio
Ramapo College
ldant@ramapo.edu

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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.
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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.

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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.

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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.

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Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu

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Thomas Drucker
University of Wisconsin-Whitewater
druckert@uww.edu

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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?

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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.

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An analysis of the notion of natural number

Ruggero Ferro
Univ Degli Studi di Verona
ruggero.ferro@univr.it

Back to 2010 schedule

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.

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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.

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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.

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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.

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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.

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The Articulation of Mathematics - A Pragmatic/Constructive Approach to The Philosophy of Mathematics

Martin E Flashman
Humboldt State University
flashman@humboldt.edu

Back to 2010 schedule

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.

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Turing and Wittgenstein

Juliet Floyd
Boston University
jfloyd@bu.edu