Philosophy of Mathematics

Contributed Paper Sessions

AMS-MAA joint meetings and MAA MathFests

January 2017, Atlanta, GA
January 2016, Seattle, WA
August 2015, Washington, DC
January 2015, San Antonio TX

January 2014, Baltimore, MD

August 2013, Hartford, CT

January 2013, San Diego CA

January 2012, Boston MA
January 2011, New Orleans LA
January 2010, San Francisco CA
August 2009, Portland OR
January 2008, San Diego CA

January 2007, New Orleans LA
January 2006, San Antonio TX
January 2005, Atlanta GA
January 2004, Phoenix AZ
January 2003, Baltimore MD

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Atlanta GA, January 7, 2017

Do Mathematicians Really Need Philosophy?

Nobel physicist Steven Weinberg famously declared that philosophers were useful to him only to defend him from other philosophers. Weinberg was complaining mostly about logical positivists, who don't seem to deal with mathematics much. But the philosophy of mathematics is a battleground for a number of warring schools, most prominently Platonists and constructivists. Does a practicing mathematician have to choose which school to join? Philosophical questions have been shown to have a huge positive effect in the teaching of mathematics, but need they come up during the development of a new branch of mathematics? Philosophy of mathematics has recently seen a movement toward "pluralism": let's accept everyone's philosophies, no matter how contradictory they may be. Is that a useful trend? Is it just a futile attempt to sweep a problem under the rug? This session, sponsored by POMSIGMAA, will give a forum for views from all sides of the issue, whether from the perspective of doing mathematical research, teaching mathematics, or more general philosophical fruitfulness. Other topics in the philosophy of mathematics will be considered as time allows.

1:00 p.m.    1125-C5-248      James R. Henderson, "Otavio Bueno's Mathematical Fictionalism"

1:30 p.m.    1125-C5-2521      Thomas Drucker, "Why Can't Those With Conflicting Views on the Foundations of Mathematics Just Get Along?"

2:00 p.m.    1125-C5-1670      Katalin Bimbó, "The unexpected usefulness of epistemological skepticism"

2:30 p.m.    1125-C5-507      Bonnie Gold, "Melding realism and social constructivism"

Seattle WA, January 7, 2016

Using Philosophy to Teach Mathematics

Courses in the philosophy of mathematics are rare, but philosophical questions frequently arise in the regular curriculum, often presenting difficulties to teachers who haven’t prepared to respond to them. In recent years a growing number of teachers of mathematics are discovering that addressing philosophical issues deliberately in their courses not only eases the strain but also enhances students’ ability to grasp difficult mathematical concepts. The upcoming MAA Notes volume, Using the Philosophy of Mathematics in Teaching Collegiate Mathematics, illustrates the ways a wide variety of teachers have found to introduce philosophical questions as an exciting part of presenting standard mathematical material. This session invites teachers at all levels to discuss ways they have found to include philosophy in the mathematics classroom. Papers on other topics in the philosophy of mathematics will be considered as time permits.

8:00 a.m.    1116-T5-103      Daniel C. Sloughter, "Making Philosophical Choices in Statistics"

8:30 a.m.    1116-T5-608      Sally Cockburn, "Senior Seminar in Set Theory as a Springboard for Mathematical Philosophy"

9:00 a.m.    1116-T5-2355    Thomas Drucker, "Role of Real Numbers in an Introduction to Analysis"

9:30 a.m.    1116-T5-2376    Brian R Zaharatos, "Statistics as a Liberal Art"

10:00 a.m.  1116-T5-177      James R Henderson, "Strange Bedfellows: Thomae's Game Formalism and Developmental Algebra"

10:30 a.m.  1116-T5-2300    Luke Wolcott, "Gardens of Infinity: Cantor meets the real deep Web"

11:00 a.m.  1116-T5-2556    Martin Flashman, "Is Philosophy of Mathematics Important for Teachers?"

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Washington DC August 6, 2015

MathFest Centennial joint with Canadian Society for the History and Philosophy of Mathematics

Special Session on Philosophy of Mathematics

2:30 p.m. Elaine Landry, "Mathematical Structuralism and Mathematical Applicability"

3:00 p.m. Jean-Pierre Marquis, "Designing Mathematics: The Role of Axioms"

3:30 p.m. Alex Manafu, "Does the Indispensability Argument Leave Open the Question of the Causal Nature of the Mathematical Entities?"

4:00 p.m. Carl Behrens, "How Does the Mind Construct/Discover Mathematical Propositions?"

4:30 p.m. Jeff Buechner, "What is an Adequate Epistemology for Mathematics?"

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 San Antonio TX January 13, 2015

Discovery and Insight in Mathematics

One new development in the philosophy of mathematics that mathematicians should welcome is an interest in the philosophy of mathematics as actually practiced by mathematicians.  This session invites talks addressing philosophical issues concerning two related topics: how mathematics is discovered, and the role of insight in mathematical understanding and discovery.  Epistemology studies how we come to know things.  A distinction has been made between methods of discovery and methods of justification:  that is, the way one discovers a mathematical truth – a conjecture, for example – may be quite different from how it is later justified (by a proof).  What are the methods and grounds for such discoveries?  What is the role insight plays in these discoveries?  How do interconnections between mathematical concepts or subjects lead to discoveries?  Talks addressing any of these issues within the philosophy of mathematics are appropriate for this session.  Papers on other topics in the philosophy of mathematics will be considered as time permits.

Organizers: Dan Sloughter and Bonnie Gold


Session 1

8:00 a.m. 1106-C5-583 Daniel C Sloughter, “Insights Gained and Lost”

8:30 a.m. 1106-C5-443 James R Henderson, “Kepler's Mysterium Cosmographicum”
9:00 a.m. 1106-C5-2535 Horia I Petrache, “Removing bias: the case of the Dirac equation”

9:30 a.m. 1106-C5-1961 Ruggero Ferro, “An analogy to help understanding Discovery, Insight and Invention in Mathematics”

10:00 a.m. 1106-C5-1803 Carl E. Behrens, “How does the mind construct/discover mathematical propositions?”


Session 2
1:00p.m. 106-C5-1594 Reuben Hersh, “Mathematicians’ proof: ‘The kingdom of math is within you’”
1:30 p.m. 1106-C5-1714 Thomas Drucker, “Explanatory and Justificatory Proofs”

2:00 p.m. 1106-C5-249 Bonnie Gold, “George Polya on methods of discovery in mathematics”

2:30 p.m. 1106-C5-2391 Tom Morley, “Some proofs and discoveries from Euler and Heaviside”

3:00 p.m. 1106-C5-2763 Kira Hylton Hamman, “Intuition: A History”

3:30 p.m. 1106-C5-838 Steven R Benson, “If you’re hoping for discovery, put away the handouts!”

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Baltimore MD  January 16, 2014

Is Mathematics the Language of Science?

In 1960 physicist Eugene Wigner published an article entitled: “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” in which he raised the question of the relationship between mathematics and the empirical sciences. Discussions of Wigner’s article often reflect the assumption that mathematics has relevance only as a means of exploring the physical world: as Wigner puts it, “in discovering the laws of inanimate nature.” Many mathematicians would find this an unacceptable restriction on the definition of their pursuits and activities. This session will explore the extent to which Wigner’s approach defines the role of mathematics, and entertain alternative or additional functions and purposes.

Organizers: Carl Behrens, Tom Drucker, Dan Sloughter

1:00 p.m. 1096-G1-1195  Carl E. Behrens, "Mathematics Is a Science in its Own Right"
1:30 pm: 1096-G1-149   James R Henderson, "The Mathematics of Quantum Mechanics: Making the Math Fit the Philosophy"
2:00 pm: 
1096-G1-2359 Horia I. Petrache, "A group theory perspective of mathematical constructs in physics"
2:30 pm: 1096-G1-842   Mate Szabo, "The Roots Of Kalmar's Empiricism"
3:00 pm: 1096-G1-2034 
Ruggero Ferro, "No Surprise for the Effectiveness of Mathematics in the Natural Sciences"
3:30 pm: 
1096-G1-743   Ronald E. Mickens, "Mathematics as an Emergent Feature of the Physical Universe"
3:55 pm: 1096-G1-1414  Jeff Buechner, "A New Look at Wigner's `The Unreasonable Effectiveness of Mathematics in the Natural Sciences'"
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Hartford CT August 1 - 3, 2013

There will be two types of sessions in Hartford: a session running throughout the meeting, of talks in either the history or philosophy of mathematics, and a session of talks on the interactions between history and philosophy of mathematics.  Of the former, only the specifically philosophical talks during the daily sessions are listed:

Friday, August 2, 9 a.m. Matthew Clemens, "Fictionalism and Mathematical Practice," Keene State College

10 a.m. Robert H C Moir "Rational Discovery of the Natural World: An Algebraic and Geometric Answer to Steiner"

11 a.m. Jean-Pierre Marquis "Canonical Maps: Where Do They Come From and Why Do They Matter?"

3 p.m. Martin E Flashman "Logic is Not Epistemology: Should Philosophy Play a Larger Role in Learning about Proofs?"

Interactions Between History and Philosophy of Mathematics

This session is geared specifically to interactions between the history and philosophy of mathematics. Talks will be expected either to approach specifically how each discipline informs the other in particular or general contexts, or to discuss issues and episodes that have implications for both the philosophy and the history of mathematics.

Organizers:  Thomas Drucker, University of Wisconsin-Whitewater and Glen Van Brummelen, Quest University

Saturday, August 3
10:30 a.m.
Thomas Drucker, Zeno Will Rise Again
11:00 a.m.
Amy Ackerberg-Hastings, Analysis and Synthesis in Geometry Textbooks: Who Cares?
2:30 p.m.
Robert S D Thomas, Assimilation in Mathematics and Beyond
3:00 p.m.
Lawrence D'Antonio, Euler and the Enlightenment
3:30 p.m.
Maryam Vulis, Persecution of Nikolai Luzin
4:00 p.m.
Roger Auguste Petry, "Philosophy Etched in Stone: The Geometry of Jerusalem's 'Absalom Pillar'"
4:30 p.m.
Jeff Buechner, Understanding the Interplay between the History and Philosophy of Mathematics in Proof Mining

San Diego CA January 9 - 12, 2013

Contributed Paper Session Friday, January 11

Philosophy, Mathematics and Progress.  Description:  Mathematics as a discipline seems to make progress over time, while philosophy is often taken to task for not having made such progress over the millennia.  When philosophy comes to tackle issues related to mathematics, one natural topic is how mathematics succeeds in making progress while philosophy does not.  One question to be addressed in this session is whether philosophy can help to explain the apparent progress displayed by mathematics.  Another is whether the mismatch in progress between the disciplines is more apparent than real.  As currents of mathematical change gather speed, perhaps a philosophical perspective is needed to make sure that current practitioners do not lose their footing.  Papers addressing issues of progress in mathematics and philosophical ways of understanding that progress will help to continue conversations between mathematicians and philosophers.

Organizers:  
Thomas Drucker, University of Wisconsin-Whitewater and Daniel Sloughter, Furman University

1:00 p.m. 1086-L5-459  Deborah C. Arangno, “From Intuition to Esoterica
1:30 p.m. 1086-L5-71 Sean F. Argyle, “Mathematical Thinking: From Cacophony to Consensus

2:00 p.m. 1086-L5-618 Thomas Drucker, “Mathematical Progress via Philosophy

2:30 p.m. 1086-L5-1597 Daniel Sloughter, “Philosophical and Mathematical Considerations of Continua

3:00 p.m. 1086-L5-45 Amy Ackerberg-Hastings, “John Playfair, the Scottish Enlightenment, and ‘Progress’ in the History and Philosophy of Mathematics

3:30 p.m. 1086-L5-365 Ruggero Ferro, “Mathematics versus Philosophy

4:00 p.m. 1086-L5-209 James R. Henderson, “Progress in Mathematics: The Importance of Not Assuming Too Much

4:30 p.m. 1086-L5-874 Mate Szabo, “Kalmar’s Argument Against the Plausibility of Church’s Thesis

Boston MA January 4-7, 2012

Invited Paper Session Wednesday, January 4

Organizers:  Thomas Drucker, University of Wisconsin-Whitewater, Bonnie Gold, Monmouth University, and Daniel Sloughter, Furman University

2:15 p.m. 1077-AJ-71, Arthur M Jaffe, "Is Mathematics the Language of Physics?"
3:00 p.m.
1077-AJ-1755, Charles Parsons, "Structuralism and its Discontents"
3:45 p.m. 1077-AJ-59, Stephen Yablo, "Explanation and Existence"
4:30 p.m. 1077-AJ-83, Agustin Rayo, "A Trivialist Account of Mathematics"
5:15 p.m. 1077-AJ-1994, Jody Azzouni, "The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof"
6:00 p.m. 1077-AJ-1045, Juliet Floyd, "Turing and Wittgenstein"

Contributed Paper Session Friday, January 6

Philosophy of Mathematics and Mathematical Practice.  Description: Philosophers search for insights into the most general epistemological and ontological questions: How do we know, and what is it that we know? Since mathematical knowledge is a significant piece of what we know, an explanation of the nature of mathematics plays an important role in philosophy.  To this end, a philosopher of mathematics must pay careful attention to mathematical practice, what it is that mathematicians claim to know and how they claim to know it.  A philosopher's explanation of mathematics cannot be a local explanation: it must fit within the larger picture of knowledge as a whole. A mathematician may have an account of mathematics which suffices for her work, but unless this account fits coherently into a larger epistemological and ontological picture, it will not suffice as a philosophy of mathematics.  This session will address questions concerning the relationship between the philosophy and the practice of mathematics. We encourage papers to address questions such as: Should the philosophy of mathematics influence, or be influenced by, the practice of mathematics? Is it necessary for the philosophy of mathematics to influence the practice of mathematics for it to be relevant to mathematicians?

Organizers:  Daniel Sloughter, Furman University and Bonnie Gold, Monmouth University

1:00 p.m. 1077-K1-1196 Daniel C. Sloughter, "The Consequences of Drawing Necessary Conclusions"

1:30 p.m. 1077-K1-383 Thomas Drucker, "Thought in Mathematical Practice"

2:00 p.m. 1077-K1-85 Joshua B. Wilkerson, "Beyond Practicality: George Berkeley and the Need for Philosophical Integration in Mathematics"

2:30 p.m. 1077-K1-2002 Bonnie Gold, "Philosophy (But Not Philosophers) of Mathematics Does Influence Mathematical Practice"

3:00 p.m. 1077-K1-1287 Sarah-Marie Belcastro, "Epistemological Culture and Mathematics"

3:30 p.m. 1077-K1-904 Ruggero Ferro, "How Do I (We) Know Mathematics"

4:00 p.m. 1077-K1-800 Jeff Buechner, "Formal mathematical proof and mathematical practice: a new skeptical problem"

4:30 p.m. 1077-K1-1744 Nathaniel G. Miller, "CDEG:  Computerized Diagrammatic Euclidean Geometry"

New Orleans LA January 8, 2011, 1 - 5:25 p.m.

The topic of this contributed paper session is "Philosophy of Mathematics in Teaching and Learning." Description:  Mathematicians tend not to think about philosophical issues while teaching. Yet we are making ontological and epistemological commitments in much of what we do in the classroom. Every time we use a proof by induction or contradiction, discuss the existence or non-existence of a mathematical object, or refer to the discovery or creation of some piece of mathematics, we are endorsing some philosophical view of our subject.

The focus of this session is on the recognition and use of the philosophy of mathematics in the teaching and learning of mathematics. Can we understand mathematics without a philosophical context? Papers are encouraged to address questions such as: What philosophical issues (such as the nature of mathematical objects, the method of mathematical proof, and the nature of mathematical knowledge) should be treated in a mathematics course? How? In which course(s)? In what ways does the consideration of philosophical issues enhance a mathematics, or mathematics related, course? What does a learner gain by contact with issues from the philosophy of mathematics?  Other papers of a philosophical nature will be considered for inclusion as time permits.

Organizers: Dan Sloughter, Furman University, and Martin E. Flashman, Humboldt State University

1:00 p.m.  1067-T1-1395, Martin E. Flashman, "Square Roots: Adding Philosophical Contexts and Issues to Enhance Understanding"
1:30 p.m.  1067-T1-2224, Whitney Johnson and Bill Rosenthal, "Precalculus from an Ontological Perspective"
2:00 p.m.  1067-T1-1766, Thomas Drucker, "Putting Content into a Fictionalist Account of Mathematics for Non-Mathematicians"
2:30 p.m.  1067-T1-2223, Sheila K. Miller, "On the Value of Doubt and Discomfort"
3:00 p.m.  1067-T1-712, Jeff Buechner, "Mathematical Understanding and Philosophies of Mathematics"
3:30 p.m. 1067-T1-1527, Ruggero Ferro, "Abstraction and objectivity in mathematics"
4:00 p.m.  1067-T1-159, James R. Henderson, "Causation and Explanation in Mathematics"
4:30 p.m.  1067-T1-2327, Andy D. Martin, "Claims Become Theorems, but Who Decides?"
5:00 p.m.  1067-T1-2300, Firooz Khosraviyani, Terutake Abe and Juan J Arellano, "Definitions in Their Developmental Stages: What should we call them?"
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San Francisco CA January 15, 2010, 1 - 5 p.m.

Schedule

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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San Diego CA January 7, 2008, 8:30 - 11:55 a.m.

Schedule

Organizers:
Kevin Iga, Pepperdine University, kiga@pepperdine.edu
Bonnie Gold, Monmouth University

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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New Orleans, LA Saturday, January 6, 2007, 8:00 a.m. - noon, 1:00 - 4:00 p.m.

Schedule


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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San Antonio TX Thursday, January 12, 2006, 8:00-10:55 A.M.

Schedule


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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Atlanta GA January 7, 2005, 1:00-4:40 P.M.

Schedule

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

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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Phoenix AZ January Friday January 9, 2004, 2:00-6:30 P.M.

Schedule


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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Baltimore MD January 17, 2003, 1 - 6 p.m.

Schedule

All talks are 15 minutes, followed by 10 minutes of discussion.  Links are to abstracts, below.  From the abstracts, as they become available, there may be links to fuller versions of the talk.

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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Abstracts of the 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.

Back to 2013 schedule

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.
Back to MathFest 2013 schedule

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.

Back to 2013 schedule

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.”

Back to 2013 schedule

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.

Back to 2004 schedule
 

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.

Back to 2012 schedule

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.

More details          Back to 2003 schedule
 

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.

Back to 2006 schedule
 

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.

Back to 2007 schedule
 

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.

Back to 2008 schedule
 

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.

Back to 2010 schedule

 

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. 

Back to 2014 schedule

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.

Back to MathFest 2015 schedule

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 ).

Back to 2003 schedule
 

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.

Back to 2006 schedule
 

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.

Back to 2012 schedule

If youre 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.

Back to 2015 schedule

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.

Back to 2003 schedule
 

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.

Back to 2017 schedule

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”.

Back to 2008 schedule
 

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.

Back to 2003 schedule
 

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.

Back to 2004 schedule
 

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.

Back to 2005 schedule
 

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.

Back to 2007 schedule
 

 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.

Back to 2008 schedule
 

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.

Back to 2009 schedule

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.

Back to 2010 schedule

 

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.

Back to 2011 schedule

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.

Back to 2012 schedule


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.
Back to MathFest 2013 schedule

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.
Back to 2014 schedule

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.


Back to MathFest 2015 schedule

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.


Back to JMM 2017 schedule

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.

Back to MathFest 2013 schedule

Senior Seminar in Set Theory as a Springboard for Mathematical Philosophy

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.

Back to 2016 schedule

"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.

Back to 2004 schedule
 

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.

Back to 2010 schedule

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.

Back to MathFest 2013 schedule

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.
More details          Back to 2003 schedule
 

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.

Back to 2003 schedule
 

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.

Back to 2005 schedule
 

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.

Back to 2010 schedule

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.

Back to 2011 schedule

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.

Back to 2012 schedule

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.

Back to 2013 schedule

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.

Back to MathFest 2013 schedule

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.

Back to 2015 schedule

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.
Back to 2016 schedule

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'.
Back to 2017 schedule

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?

Back to 2007 schedule
 

Remarks about the notion of EXISTENCE in mathematics

Ruggero Ferro
Univ 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.

Back to 2008 schedule
 

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.

Back to 2010 schedule

 

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.

Back to 2011 schedule

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.

Back to 2012 schedule

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.

1) Philosophy’s desire to answer the most fundamental questions of humankind is perhaps too ambitious.

2) OK Scire per causas. But how to detect the causes of the situation that we experiment?

3) Philosophy touches upon very sensitive topics such as personal beliefs, morality. Here the arguments to reach an agreement are not only deductive.

4) 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.

a) No one claims to know exactly the meaning of the axioms.

b) Various principles are used, but don’t ask why they should be accepted.

c) Proofs should be easily checked, but no one cares how they were devised.

d) Mathematics is a good organization of multiplicity: by dropping information, a situation becomes manageable.

e) “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.

Back to 2013 schedule

No surprise for the effectiveness of mathematics in the natural sciences

Ruggero Ferro
Univ 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.

Back to 2014 schedule


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.

Back to 2015 schedule

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.

Back to 2006 schedule
 

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.

Back to 2007 schedule
 

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.

Back to 2009 schedule

 

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.

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.

Back to 2011 schedule

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.

Back to MathFest 2013 schedule

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.
Back to 2016 schedule

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 schedule

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.

Back to 2005 schedule
 

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.

Back to 2006 schedule
 

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.

Back to 2009 schedule

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.

Back to 2012 schedule

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.
Back to 2015 schedule

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.
Back to 2017 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.
Back to 2015 schedule

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.

Back to 2004 schedule
 

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.

Back to 2005 schedule
 

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?

Back to 2005 schedule
 

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.

Back to 2006 schedule
 

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?

Back to 2007 schedule
 

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.

Back to 2008 schedule
 

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.

Back to 2010 schedule

 

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.

Back to 2011 schedule

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.

Back to 2013 schedule

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.

Back to 2014 schedule

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.

Back to 2015 schedule

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.

Back to 2016 schedule

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.

Back to 2017 schedule

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”. 

Back to 2015 schedule


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.

Back to 2005 schedule
 

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 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.

Back to 2011 schedule
 

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.

Back to 2011 schedule

 

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.

Back to 2004 schedule
 

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

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).

Back to MathFest 2015 schedule

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.

Back to 2003 schedule
 

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.

Back to 2007 schedule
 

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.

Back to MathFest 2013 schedule

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.

Back to MathFest 2015 schedule

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?”

Back to 2007 schedule
 

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?

Back to 2011 schedule

 

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.

Back to 2004 schedule
 

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) 

Back to 2014 schedule

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)].

Back to 2004 schedule
 

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.

Back to 2007 schedule
 

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.

Back to 2012 schedule

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.

Back to 2011 schedule

 

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:

1. discovering new theories requires algebraic and structural variation of old theories in order to access new forms of behaviour; and

2. 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.

Back to MathFest 2013 schedule

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 schedule

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.

Back to 2007 schedule
 

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 premits, I will try to canvass some of them and suggest replies.

Back to 2012 schedule

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.

Back to 2014 schedule

 

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.

Back to 2015 schedule

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.

Back to MathFest 2013 schedule

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.

Back to 2004 schedule
 

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.

Back to 2012 schedule

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.

Back to 2005 schedule
 

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).

Back to 2004 schedule
 

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.

Back to 2008 schedule
 

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.

Back to 2003 schedule
 

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 = uv  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?

Back to 2003 schedule
 

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.

Back to 2005 schedule
 

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.

Back to 2008 schedule
 

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?

Back to 2009 schedule

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.

Back to 2010 schedule

 

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.

Back to 2012 schedule

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.

Back to 2013 schedule

 

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.

Back to 2015 schedule


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.

Back to 2016 schedule


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.

Back to 2006 schedule
 

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.

Back to 2007 schedule
 

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.

Back to 2013 schedule

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.
Back to 2014 schedule

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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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?

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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.

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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.

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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.

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