William Reinhardt Memorial Lecture

The William Reinhardt Memorial Lecture in the Philosophy of Mathematics was founded to commemorate the life of William Reinhardt, Professor of Mathematics at the University of Colorado from 1967 until his death in 1998. He did important work in set theory, logic, and the foundations of mathematics. A central focus of his work was the search for new axioms of mathematics. The study of new axioms in mathematics requires the ability to stand back from mathematical practice and ask questions about the general principles that guide and justify it -- questions that engage philosophical as well as mathematical issues. For this reason, Professor Reinhardt was considered a philosopher as well as a mathematician. He made profound contributions to the philosophy and foundations of mathematics.

The Reinhardt Lecture, which is co-sponsored by the Reinhardt Fund and the Department of Philosophy, brings a leading contemporary philosopher of mathematics to Boulder to give a talk on some topic in set theory, logic, or the foundations of mathematics.

2025 Reinhardt Lecture

Joel David Hamkins, University of Notre Dame
"How we might have taken the Continuum Hypothesis as a fundamental axiom, necessary for mathematicsâ€
March 11, 2025, 1:45 PM, KOBL 342
Ìý
Abstract: I shall describe a simple historical thought experiment showing how our attitude toward the continuum hypothesis might easily have been very different than it is. If our mathematical history had been just a little different, I claim, if certain mathematical discoveries had been made in a slightly different order, then we would naturally have come to view the continuum hypothesis as a fundamental axiom of set theory, necessary for mathematics, indispensable even for the core ideas of calculus.
Ìý

Past Reinhardt Lectures

Elaine Landry,ÌýUniversity of California, Davis
"As If Mathematics Was Trueâ€
April 26, 2024, 3:15 PM, ECON 117
Ìý
Abstract: An ongoing, and seemingly unending, philosophical debate is the realism versus anti-realism debate. On the one side are realists, claiming that objects exist in some realm that is independent of us and that it is in virtue of reference to these objects that our statements about them are true. Those making such assertions about the metaphysical realm we call metaphysical realists or platonists. Those making such assertions about the physical realm we call physical or scientific realists. On the other side are anti-realists, claiming that such realist positions are untenable and so talk of such objects should be interpreted ideally, nominalistically, formally, or fictionally.
Ìý
In this talk, I argue for a version of mathematical realism that cuts a midpoint between these two philosophical poles. I first show that Plato himself keeps a clear distinction between mathematical and metaphysical realism and the knife he uses to slice the difference is method. The philosopher’s dialectical method requires that we tether the truth of hypotheses to the existence of metaphysical objects. The mathematician’s hypothetical method, by contrast, takes hypotheses as if they were true first principles and so no metaphysical account of their truth is needed. Thus, we come to Plato’s methodological as-if realism: in mathematics, we treat our hypotheses as if they were true first principles, and, consequently, our objects as if they existed, and we do this with the purpose of solving mathematical, metamathematical and physical problems.
Ìý
I turn next to develop my own methodological as-if structuralism by comparing structuralist views to if-thenist views; I show that while these latter push us back to the same realist versus anti-realist debates, my structural as-ifist approach yet survives. Taking the road suggested by Plato, I argue that some of our methodological commitments to taking our axioms as if they were true first principles, will be made in light of mathematical practice (with the goal of solving mathematical problems); some will be made in light of mathematical applicability (with the goal of solving physical problems); and some will be made in light of logical/philosophical considerations (with the goal of solving meta-mathematical problems). Yet, none of these commitments will be made with the goal of solving metaphysical problems.
Ìý
Finally, I use my as-ifist account to break Benacerraf’s [1965] dilemma. Recall Benacerraf’s dilemma: we either must choose a shared (Tarskian) semantics between mathematics and ordinary discourse (and adopt a realist view of mathematical objects and thereby forego a reasonable epistemology) or we must choose a reasonable epistemology (and give up mathematical realism and thereby forego a shared semantics). I will argue that we do have a Tarskian shared semantics, but for ordinary discourse, truth is taken as a consequence of existence, whereas for mathematics, existence is taken as a consequence of truth. Thus, one can have a shared semantics without adopting mathematical realism.

Steve Awodey, Carnegie Mellon University
"Intensionality, Invariance, and Univalence"
September 13, 2019, 3:15 PM, HUMN 150

Abstract: What does a mathematical proposition mean? Under one standard account, all true mathematical statements mean the same thing, namely True. A more meaningful account is provided by the Propositions-As-Types conception of type theory, according to which the meaning of a proposition is its collection of proofs. The new system of Homotopy Type Theory provides a further refinement: The meaning of a proposition is the homotopy type of its proofs. A homotopy type may be seen as an infinite-dimensional structure, consisting of objects, isomorphisms, isomorphisms of isomorphisms, etc. Such structures represent systems of objects together with all of their higher symmetries. The language of Martin-Löf type theory is an invariant of all such higher symmetries, a fact which is enshrined in the celebrated Principle of Univalence.

John Bell, University of Western Ontario
"Infinitesimals and the Labyrinth of the Continuum"
October 5, 2018, 3:15 PM, HLMS 199

Professor James Robert Brown, University of Toronto
“Pure and Applied: Mathematics and Ethicsâ€
October 6 2017, 3:15-5:00pm

Abstract: Empiricism has trouble with both math and ethics. ÌýThis talk will be about different conceptions of pure and applied math and what it means for doing epistemology. ÌýAnalogies with ethics will be developed. ÌýThe continuum hypothesis will be used as an example. ÌýThe technicalities will be minimal, so the talk should be accessible to all.Ìý

Ìý

W. D. Hart, University of Illinois at Chicago
'Orayen’s Paradox’
Friday, December 6, 2013 3:15-5pm, HUMN 150

Abstract: A central application of sets is the standard theory of truth, Tarsi’s. But that view of truth fits set theory only awkwardly; that awkwardness is Orayen's Paradox.

W.D. Hart (A.B. scl, Harvard College 1964; PhD Harvard University 1969) is Professor Emeritus of Philosophy at the University of Illinois at Chicago, where he was chair of the philosophy department from 1994 until 2006 and from which he retired in June 2011. He previously taught at the University of Michigan (1969-74), University College London (1974-91), and the University of New Mexico (1992-93). His primary interests are logic,
philosophy of mathematics, metaphysics, and epistemology. His book The Engines of the Soul (Cambridge 1988, 2009 pbk) is an argument for dualism as a solution to the mind-body problem. The Evolution of Logic (Cambridge 2010) is a critical history of the relations between logic and philosophy over the last 130 years, and it reflects the core of his teaching over his career. Readings in the Philosophy of Mathematics (Oxford 1996), which he edited with an introduction, is a successor to the old Hintikka volume (the two have no overlap, the newer volume being a collection of philosophy papers, not mathematics).

, University of Miami
"What Does a Mathematical Proof Really Prove?"
Friday, May 3, 2013
3:00 PM, President's Room, Old Main Heritage Center

, University of Cambridge
"Does Mathematics Need Replacement (and Is It Even True)?"
Friday, March 16, 2008
3:15 PM, Eaton Humanities 150

, UCLA
"Is Set Theory about Sets?" ()
Friday, March 12, 2004
6:00 PM, Humanities 250

, University of Chicago
"Frege's Three Methodological Principles" ()
Monday, March 17, 2003
7:00 PM, UMC 289

, MIT
"Private Meanings, Shared Truths" ()
Monday, September 11, 2001
6:30 PM, British Studies Room, Norlin LIbrary