Recent events

Joint Math Meetings 2020, Denver, CO, January 15-18, 2020

Guest Lecture
Possible and Impossible Infinities
Michael Huemer, University of Colorado--Boulder
Friday, January 17, 2020 at 6:15 pm
after reception at 5:30 pm and business meeting at 6:00 pm
Location: Room 504

Abstract:

The infinite gives rise to many paradoxes. Some are aptly resolved by declaring certain infinite quantities impossible. But which infinities are possible, and which are impossible? On an Aristotelian view, there can be no "actual infinities", only "potential infinities". This view is wrong; there are many obvious examples of actual infinities. I draw three distinctions: between cardinal numbers and magnitudes, between intensive and extensive magnitudes, and between natural and artificial magnitudes. I then propose a new theory of the impossible infinite: there are infinite cardinal numbers, extensive magnitudes, and artificial magnitudes, but there can be no infinite natural, intensive magnitudes. This view rules out most of the scenarios appearing in the paradoxes of the infinite.

Slides from the talk. For more details, see his book, Approaching Infinity.

Contributed Paper Session
Role of Explanation in Mathematical Proofs
Date: Saturday, January 18, 2020, 8 am to 11:55 am
Location: Room 502
Organizers: Jeff Buechner, Kevin Iga, Sally Cockburn

Description:

Mathematical proofs are a form of argument. We can say of arguments in general--and mathematical proofs specifically--that, when sound, they show us that the claim made is true. But often some arguments--and this includes some mathematical proofs--do more. They also explain to us why it is true. Proposed talks might discuss (but are not limited to) the following topics. What is it to explain why a mathematical theorem is true? Which mathematical proofs explain why the theorem proved is true? Some doubt that proofs by mathematical induction can. Is mathematical explanation different from, say, scientific explanation (which does not involve the use of mathematics)? Is it different from historical explanation? Or is there a unified sense of 'explanation' which is common to its use in all subjects, including mathematics? Are there instances of mathematical theorems which have multiple proofs, some of which are elegant and simple, but not explanatory, while the others are neither elegant nor simple, but explanatory? If so, does being explanatory count as a good reason to prefer one kind of proof over the other? If a mathematician finds a shorter proof of some theorem, will the shorter proof be more explanatory than the longer proof? All paper proposals which discuss the role of explanation in mathematical proofs will be considered.

8:00 a.m.1154-O1-487James Henderson, "Explanatory Proofs"
8:35 a.m.1154-O1-2227Jeremy Case, "Mathematical Explanation as an Aesthetic"
9:10 a.m.1154-O1-2672Paul Zorn, "Proofs that explain, proofs that don't, and proofs of the obvious"
9:45 a.m.1154-O1-716Bonnie Gold, "What makes proofs explanatory? Let's look at some examples"
10:20 a.m.1154-O1-2716Susan Ruff, "How logic is presented may obscure or enlighten"
11:00 a.m.1154-O1-1421Jeffrey Buechner, "Are mathematical explanations interest-relative?"

Past POMSIGMAA activities

POMSIGMAA sponsored our first ever invited paper session at the 2012 AMS/MAA Joint Meeting in Boston.  The speakers were Arthur Jaffe at Harvard University on "Is Mathematics the Language of Physics" and Charles Parsons at Harvard University on "Structuralism and its Discontents", Jody Azzouni at Tufts University on "The Relationship of Derivations in Artificial Languages to Ordinary Rigorous Mathematical Proof", Juliet Floyd at Boston University on "Turing and Wittgenstein", Agustin Rayo at MIT on "A Trivialist Account of Mathematics" and Stephen Yablo at MIT on "Explanation and Existence."

POMSIGMAA has sponsored contributed paper sessions at

the 2003 AMS/MAA Joint Meeting in Baltimore,

the 2004 AMS/MAA Joint Meeting in Phoenix,

the 2005 AMS/MAA Joint Meeting in Atlanta,

the 2006 AMS/MAA Joint Meeting in San Antonio,

the 2007 AMS/MAA Joint Meeting in New Orleans,

the 2008 AMS/MAA Joint Meeting in San Diego,

MathFest 2009 in Portland, OR, jointly with HOMSIGMAA, "The History of Mathematics and its Philosophy, and Their Uses in the Classroom",

the 2010 AMS/MAA Joint Meeting in San Francisco, "Philosophy of Mathematics for Working Mathematicians,"

the 2011 AMS/MAA Joint Meeting in New Orleans, "Philosophy of Mathematics in Teaching and Learning,"

the 2012 AMS/MAA Joint Meeting in Boston, "Philosophy of Mathematics and Mathematical Practice."

the 2013 AMS/MAA Joint Meeting in San Diego, "Philosophy, Mathematics, and Progress."

the 2013 MathFest, in conjunction with CSHPM and HOMSIGMAA, daily sessions on history of mathematics and philosophy of mathematics, and one session on the Interactions Between the History and Philosophy of Mathematics.  

the 2014 AMS/MAA Joint Meeting in Baltimore, "Is mathematics the Language of Science?"

the 2015 AMS/MAA Joint Meeting in San Antonio, "Discovery and Insight in Mathematics."

the 2016 AMS/MAA Joint Meeting in Seattle, "Using Philosophy to Teach Mathematics."

the 2017 AMS/MAA Joint Meeting in Atlanta, "Do Mathematicians Really Need Philosophy?"

the 2018 AMS/MAA Joint Meeting in San Diego, "Philosophy of Mathematics as Actually Practiced"

the 2019 AMS/MAA Joint Meeting in Baltimore, "Philosophy of Mathematics: Do choices of mathematical notation (and similar choices) affect the development of mathematical concepts?"

the 2020 AMS/MAA Joint Meeting in Denver, "Role of Explanation in Mathematical Proofs"

Full list of all these sessions with their abstracts

At 2020 AMS/MAA Joint Meetings in Denver CO, January 16 - 19, Michael Huemer, University of Colorado--Boulder gave the POMSIGMAA Invited Address, "Possible and Impossible Infinities" Abstract . Slides. For more details, see his book, Approaching Infinity.

At 2019 AMS/MAA Joint Meetings in Baltimore, MD, January 16 - 19, Professor Michele Friend, of George Washington University gave the POMSIGMAA Invited Address, "The Rigour of Proof" Abstract . Slides. For more details, see her book, Pluralism in Mathematics; A New Position in Philosophy of Mathematics.

At 2018 AMS/MAA Joint Meetings in San Diego, CA, January 10 - 13, Professor Rafael Núñez, of UCSD gave the POMSIGMAA Invited Address, "Philosophy of Mathematics in the 21st Century: Why does it need the Sciences of the Mind?" Abstract

At MathFest 2017 in Chicago, IL, July 26 - 29, John Baldwin, Emeritus Professor at the University of Illinois at Chicago, gave the POMSIGMAA Invited Address, "Philosophical implications of the paradigm shift in model theory." Slides: pdf

At 2017 AMS/MAA Joint Meetings in Atlanta, GA, January 4 - 7, Jeff Buechner, of the philosophy department at Rutgers University Newark and the Saul Kripke Center, CUNY Graduate Center, gave the POMSIGMAA Invited Address, "Computers, mathematical proof, and the nature of the human mind: a surprising connection." Slides available in long or short version: long(pdf) short(pdf)

At MathFest 2016 in Columbus, OH, August 3 - 6, Stewart Shapiro, of Ohio State University, gave the POMSIGMAA Invited Address, "Potential infinity: a modal account." Full text (pdf)

At the 2016 AMS/MAA Joint Meetings in Seattle, WA, January 6 - 9, Bonnie Gold, of Monmouth University, gave the POMSIGMAA Invited Address, "Is School Mathematics 'Real' Mathematics?" Full text (pdf)

At MathFest 2015 in Washington, DC, August 5 - 8, John Burgess, of Princeton University, gave the POMSIGMAA Invited Address, "What are mathematical objects, and who cares?"

At the 2015 AMS/MAA Joint Meetings in San Antonio, TX, January 10 - 13, Matt Jones, of California State University Dominguez Hills, gave the POMSIGMAA Invited Address, "Mathematical Authority and Inquiry-Based Learning."

At MathFest 2014 in Portland, OR, August 6 - 9, Paul Zorn of St. Olaf College gave the POMSIGMAA Invited Address, “Math-Speak: Syntax, Semantics, and Pragmatics.”

At the 2014 AMS/MAA Joint Meetings in Baltimore, MD, January 15 - 18, Steve Gimbel of Gettysburg College gave the POMSIGMAA Invited Address, "Hermann Minkowski  The Quiet Genius."

At MathFest 2013 in Hartford, CT July 31 - August 3, the MAA met jointly with the Canadian Society for the History and Philosphy of Mathematics.  POMSIGMAA cosponsored contributed paper sessions running throughout the meeting, as well as a plenary talk (the Kenneth O. May Lecture) on Saturday, August 3 by Jeremy Gray, of the University of Warwick and the Open University, on "Henri Poincaré: Mathematician, Physicist, Philosopher."

At the 2013 AMS/MAA Joint Meetings in San Diego, CA, January 9 - 12, Mark Balaguer of California State University, Los Angeles, gave the POMSIGMAA invited address “A Guide for the Perplexed: What Mathematicians Need to Know to Understand Philosophers of Mathematics.”  Full text (pdf)   Handout (pdf)

At MathFest 2012 in Madison, WI August 2 - 4, Janet Follina, of Macalester College, gave the POMSIGMAA invited address, "Is the Proof in the Picture? Seeing, Believing, and Provings."

At the 2012 AMS/MAA Joint Meeting in Boston, MA, January 4 - 7, Barry Mazur of Harvard University gave the POMSIGMAA invited address, "Why is it plausible?"  Full text (pdf)   Slides (pdf)

At MathFest 2011 in Lexington, KY, August 4 - 6, Neil Tennant, of the Department of Philosophy of The Ohio State University gave the POMSIGMAA invited address, "Natural Logicism for Mathematics.Full text (pdf)

At the 2011 AMS/MAA Joint Meeting in New Orleans, LA, January 6 - 9, Keith Devlin of the Center for the Study of Language and Information at Stanford University gave the invited address, "Will the real philosophy of mathematics please stand up?"

At MathFest 2010 in Pittsburgh, PA, August 5 - 7, Wilfried Sieg, of Carnegie Mellon University, gave the POMSIGMAA invited address, "Structural Proof Theory: Uncovering capacities of the mathematical mind."  Full text (pdf) PowerPoint slides

At the 2010 AMS/MAA Joint Meeting in San Francisco, CA, January 13 - 16, Charles Chihara (Philosophy Department, University of California at Berkeley) gave the invited POMSIGMAA address, "What is Philosophy of Mathematics? A Case Study of Fictionalism."

At MathFest, August 6-8, 2009 in Portland, OR, POMSIGMAA held a Reception and Moderated Discussion led by Martin Flashman, Humboldt State University, Past-Chair POM SIGMAA, on "The Role of The Philosophy of Mathematics in Teaching and Learning."

At the 2009 AMS/MAA Joint Meeting in Washington, DC, Chandler Davis (University of Toronto) gave an invited address, jointly sponsored by HOMSIGMAA (the SIGMAA for the History of Mathematics), "The Role of the Untrue in Mathematics".  It will be published in the Mathematical Intelligencer; a preliminary version is available here as a Word document, but it is copyrighted by the Mathematical Intelligencer, all rights reserved.  At that meeting, there was also a panel jointly with HOMSIGMAA (the History of Mathematics SIGMAA), "The Intersection of the History and Philosophy of Mathematics" with panelists Thomas Drucker (University of Wisconsin-Whitewater, current POMSIGMAA treasurer), Kenneth Manders (University of Pittsburgh) and Daniel Sloughter (Furman University).

At MathFest 2008, in Madison, WI, Morris W. Hirsch (University of Wisconsin) gave an invited POMSIGMAA address, "Mathematics:  the divine madness."

At the 2008 AMS/MAA Joint Meeting in San Diego, Penelope Maddy, of the Department of Logic and Philosophy of Science of the University of California at Irvine, gave an invited address at the POMSIGMAA session, "How applied mathematics became pure."  Charter changes were approved at the business part of this session.  Our 2007-2009 Chairperson, Martin Flashman also gave a minicourse, "Teaching and the Philosophy of Mahtematics," at these joint meetings.

At Mathfest 2007, in San Jose, Michael Beeson, of San Jose State University, gave an invited address at the POMSIGMAA session, "The meaning of existence in mathematics."

At the 2007 AMS/MAA Joint Meeting in New Orleans,  Klaus Peters, of AK Peters LTD, gave an invited address at the POMSIGMAA session, "Does a proof exist if nobody has read it?"

At Mathfest 2006, in Knoxville, TN, Michael Resnik, of the University of North Carolina, Chapel Hill, gave an invited address at the POMSIGMAA session, "Some Problems and Solutions in Contemporary Philosophy of Mathematics."

At the 2006 AMS/MAA Joint Meeting in San Antonio, Paul Humphreys, of the University of Virginia, gave an invited address on "Pure and Applied Computational Mathematics: Some Philosophical Morals".

At MathFest 2005 in Albuquerque, NM, Reuben Hersh gave an invited address, "Subversive essays on the nature of mathematics".

At the 2005 AMS/MAA Joint Meeting in Atlanta, Jonathan Borwein, of Dalhousie University, gave an invited address, "Philosophical Implications of Experimental Mathematics"

At MathFest 2004 in Providence, RI, Philip Davis, of Brown University, gave an invited address sponsored by POM SIGMAA.  Professor Davis was introduced by Roger Simons (Chair-elect) and his talk was entitled "The Decline, Fall, and Current Resurgence of Visual Geometry." It was followed by a brief POM SIGMAA business meeting led by POM SIGMAA Chairperson, Bonnie Gold.

At the MathFest 2003 in Boulder, CO  there was an 80 minutes open discussion involving approximately 25 people, on the topic, "What is Mathematics?"  Bonnie Gold moderated this discussion and it included representatives of most of the modern points of view:  social constructivism, formalism, versions of platonism.  The discussion ebbed and flowed, sometimes being fairly carefully focused, sometimes wandering a bit off the topic.
 

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