# Philosophy of Mathematics

## Contributed Paper Sessions

## AMS-MAA joint meetings and MAA MathFests

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
Back to main POMSIGMAA
homepage

## San Diego, CA, January 12, 2018

### Philosophy of Mathematics as Actually Practiced

The philosophy of mathematics has often failed to account
for actual mathematical practice, concentrating only on the finished
product, theorems and proofs, and even then, not proofs as
mathematicians give them, but the formal proofs by which they could be
replaced. In the last quarter of the 20th century, many philosophers
of mathematics began to be interested in considering mathematics as it
is actually developed, leading to the formation, in 2009, of the
Association for the Philosophy of Mathematical Practice. This kind of
approach requires that the philosopher of mathematics have a good
understanding of mathematics, how it develops, and how it is taught
and learned. It therefore requires a significant interaction with the
mathematical community. This session invites contributions that
discuss philosophical issues involved with mathematics as it is
actually practiced. Papers that bring out issues that have not yet
been discussed by philosophers but that involve philosophical issues
with current mathematical practice are especially welcome. Other
topics in the philosophy of mathematics will be considered as time
allows.

8:00 a.m. 1135-A5-144
David M. Shane, "The Eroding
Foundation of Mathematics"

8:30 a.m. 1135-A5-362
James Henderson, "When Physicists Teach Mathematics"

9:00 a.m. 1135-A5-609
Daniel C. Sloughter, "Hardy, Bishop, and Making Hay"

9:30 a.m. 1135-A5-301
Jae Yong John Park, "Fictionalism, Constructive Empiricism, and the Semantics of Mathematical Language"

10:00 a.m. 1135-A5-814
Chandra Kethi-Reddy, "Gian-Carlo Rota and the Phenomenology of Mathematics"

10:30 a.m. 1135-A5-1883
sarah-marie belcastro, "Does Inclusivity Matter in Mathematical Practice?"

## 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?"

Back to main POMSIGMAA homepage## 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?"

Back to main POMSIGMAA homepage

##
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!”

Back to main POMSIGMAA homepage

## 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'"

Back to main POMSIGMAA homepage

## 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?"

Back to main POMSIGMAA homepage

## 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”

Back to main
POMSIGMAA homepage

## 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

Back to main POMSIGMAA homepage

## 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

Back to main POMSIGMAA homepage

## 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

Back to main POMSIGMAA homepage

## 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

###
Does Inclusivity Matter in
Mathematical Practice?

sarah-marie belcastro

smbelcas@toroidalsnark.net

Many in the mathematical community believe that it is important to
welcome participation from people with a variety of backgrounds and in
particular from members of underrepresented groups. From a
philosophical point of view, inclusivity is a broader concept than
welcoming a diversity of human experience; for example, it includes
welcoming a variety of mathematical perspectives (epistemic
diversity).
Are there implications of encouraging inclusivity for the production
of mathematical knowledge, and if so, what are they? And are
they positive or negative? Conversely, are there implications
of our mathematical practices, in terms of producing knowledge
(theorems, proofs, etc.), on inclusivity?
We will carefully describe inclusivity as framed in the
literature on scientific values, and restrict our discussion to
epistemic values, and then to mathematical epistemic values. We will
then examine what impacts the epistemic value(s) of inclusivity may
have on mathematical practices, and what impacts current mathematical
practices may have on inclusivity, and include specific
examples. Finally, we will pose changes/actions that individuals or
the community might make/take, in accordance with common mathematical
values, and evaluate their impact relative to inclusivity.

Back to 2018 schedule

If you’re hoping for discovery, put away the
handouts!

Steven R Benson

Lesley University

sbenson@lesley.edu

I have observed that
students seem to take fundamentally different philosophical approaches to what
we might consider identical problem “set-ups”, depending on whether or not the
problem is given from the text/handout. When given written problems – no matter
how open-ended – students tend to treat
them as exercises, whereas problems that appear to be “spur of the moment” or
that emerge from a classroom discussion are treated in a more exploratory way.
I will present specific instances where deep mathematical insights have
occurred in a variety of course levels.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

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

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.itThere
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?

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

###
When Physicists Teach Mathematics

James Henderson

Penn State

jrh66@psu.edu

The teaching of mathematics is one of the field's most straightforward
applications. Because the math lessons 10-year-olds absorb are of a
very different sort than those mathematicians-in-training endure while
studying functions of a complex variable in graduate school, it is not
surprising that the material varies quite a bit in many settings in
which math is taught, but the presentation of the material does, too,
depending on the professor, the audience, and the purpose of the
class. Further, it's not just mathematicians who teach
math. Sometimes, for instance, physicists teach math to physics majors
explicitly for use in physics courses. This was most famously done in
1961 and 1962 by Richard Feynman at Caltech. (Math was not all he
taught, but algebra did warrant a chapter in his celebrated The
Feynman Lectures on Physics [1963].) How does the presentation differ
when a mathematician teaches math to math students and a physicist
teaches math to physics students? What are the primitive terms and
rules of inference in each case? How do these differences define the
process of teaching? To answer these questions, I will rely on
Feynman's lecture on algebra, input from a physicist friends, and my
own experiences as a graduate student in a mathematics
program.

Back to 2018 schedule

### Mathematicians’ proof:
“The kingdom of math is within you”

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 *S*_{p}(*n*), the sum of the *p*th
powers of the first *n* integers, is a
polynomial in *S*_{1}(*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

###
Gian-Carlo Rota and the
Phenomenology of Mathematics

Chandra
Kethi-Reddy

chan.dra@knights.ucf.edu

The celebrated combinatorialist Gian-Carlo Rota arguably produced a
philosophy of mathematics more faithful to the actual practice of
mathematics than any other American mathematician or philosopher of his time. While among his peers at MIT or the Los Alamos Research Laboratory, even while he was Vice President of the AMS, Rota had to fight
the hegemonic and exclusionary institution of analytic philosophy in
order to justify the intelligibility and practicality of his unique
phenomenology of mathematics. In this presentation, I will take the
audience through Gian-Carlo Rota's “Phenomenology of Mathematical
Proof” in order to demonstrate the place of phenomenology in any
rigorous philosophy of mathematics. I will also go through a selection
of his advice from “Ten Lessons I Wish I Had Been Taught” and “Ten
Lessons for the Survival of a Mathematics Department” to show how his
philosophy can be lively, humorous, and close to life. I hope that
this presentation will reinvigorate research in the phenomenology of
mathematics and interest in the life and work of this titan.

Back to 2018 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

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?

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

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
###
Fictionalism, Constructive Empiricism, and the Semantics of Mathematical Language

Jae Yong John Park

New York, NY

parkjohn0109@gmail.com

In Field's fictionalism, good mathematical theories do not need to be
true, but rather must be consistent and conservative. Likewise, Van
Fraassen views science to be nothing more than a study of obtaining
truths about the observable phenomena of the world, so good scientific
theories need not be true, merely empirically adequate. The takeaway
from this comparison is that the concepts of acceptance and empirical
adequacy of constructive empiricism can be used to better understand
fictionalism. At first glance, constructive empiricism does not seem
to help the case of fictionalism because the constructive empiricism
is based on the possibility of empirical verification, which is also
the basis of indispensability argument. This paper argues, however,
that the empirical adequacy of scientific theories is comparable to
the conservative nature of mathematical theories. By understanding
that a scientific theory need not be entirely true, the falsity of
mathematical statements in the fictionalist view becomes more
graspable.
Back to 2018 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 PetryLuther 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

###
The Eroding Foundation of Mathematics

David M. Shane

Methodist University

dshane@student.methodist.edu

The very foundation of the physical sciences is mathematics, which is
arguably the most fascinating conversation in the philosophical
arena. Paradoxically, the majority of mathematicians in modernity
decline this engagement, and instead, set their cross hairs primarily
on physical science and technological applications; which is to say
that mathematicians are neglecting to develop new theoretical frameworks for their trade. This tendency may be directly observed within even the purest of mathematical branches, number theory, where the tip of the spear points to every direction, yet seldom a philosophical one. I contest that mathematicians need to acknowledge this shortcoming and reunite themselves with the philosophical network. While we may not agree completely with the assertions made by Pythagoras, Fermat, Russell, or
Gödel, one cannot deny that their mathematical practices are interwoven in the fabric of their philosophical tapestry. In short, mathematics needs to spawn its own philosophers in order to facilitate the growth of new branches: the community needs avant-garde theories which are “for us, by us” and that pioneer into the wilderness of abstract thought.

Back to 2018 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* = *u* – *v* and suppose
Achilles paces himself so that

(1)
Peirce sees two possible outcomes, namely, either Achilles
overtakes the Tortoise and wins the race or Achilles catches the
Tortoise but never passes him, and asks if the arguments leading
to these conclusions contain any fallacies.

His question is motivated by the mathematical nature of the
problem: equation (1) is a standard example of a differential
equation which does not have a unique solution in any region
containing *x *= 0. Explicitly, with the initial
condition *x*(0) = -1, the function

satisfies (1) for any* x* between 3 and infinity,
inclusive. But the point Peirce wishes to make is philosophic,
not mathematical; indeed, he asks not “what the answer to the
mathematical problem is,” but rather where the fallacy lies in
arguing to either of the proposed solutions. As such, he reminds
us that finding a mathematical solution does not necessarily
solve the underlying logical problem. Whereas the infinite
series solution of the standard version of the Achilles paradox
does provide a mathematical explanation of why there is no
contradiction between our reason and our senses, in this case we
are still left with a problem: does Achilles pass the Tortoise
or not?

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

###
Hardy, Bishop, and Making Hay. Preliminary report.

Daniel C. Sloughter

Furman University

dan.sloughter@furman.edu

I find many parts of constructive mathematics appealing. For example,
I find Kronecker's dictum “[d]ie ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk,” an attractive starting point. Philosophically, I am intrigued by Errett Bishop's addition that “mathematics belongs to man” and “[i]f God has mathematics of his own that needs to be done, let him do it himself.” Yet, when confronted with, say, the version of the Hahn-Banach theorem in Bishop's
Constructive Analysis, I find myself turning back to the cleaner
statement in classical (“God's”?) mathematics. And I
recall G. H. Hardy's remark that he was willing to accept the Axiom of
Choice, in part, because to deny it “seems to make hay of a lot
of the most interesting mathematics.” In this talk, I will discuss the context of Hardy's comment and how the creation of interesting and aesthetically pleasing mathematics influences one's acceptance of axioms, and hence one's philosophy of mathematics.

Back to 2018 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

“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.Back to MathFest 2013 schedule

### 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?

Back to 2005 schedule

### 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.
Back
to 2012 schedule

### 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

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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