We had a sweeping discussion touching upon many issues in the philosophy of mathematics, including the nature of mathematical truth, mathematical abstraction, the nature of mathematical existence, the meaning and role of proof in mathematics, the completeness theorem, the incompleteness phenomenon, infinity, and a discussion about the motivations that one might have for studying mathematics.
This will be an online Zoom talk for the Boston Computaton Club, a graduate seminar in computer science at Northeastern University, 16 June 12pm EST (note change in date/time). Contact the organizers for the Zoom link.
Abstract: Many familiar finite games admit natural infinitary analogues, which may captivate and challenge us with sublime complexity. Shall we have a game of infinite chess? Or how about infinite draughts, infinite Hex, infinite Wordle, or infinite Sudoku? In the Chocolatier’s game, the Chocolatier serves up an infinite stream of delicious morsels, while the Glutton aims to eat every one. These games and others illustrate the often subtle strategic aspects of infinite games, and sometimes their downright logical peculiarity. Does every infinite game admit of a winning strategy? Must optimal play be in principle computable? Let us discover the fascinating nature of infinitary strategic thinking.
This will be a talk for the CUNY Set Theory Seminar on May 5, 2023 10am. Contact the organizers for the Zoom link.
Abstract. The standard set-theoretic distinction between sets and classes instantiates in important respects the Fregean distinction between objects and concepts, for in set theory we commonly take the universe of sets as a realm of objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is commonly held that in a very general manner, there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework, it turns out that we can provide a completely deflationary account of this and other Fregean abstraction principles. Namely, there is a mapping of classes to objects, definable in set theory in senses I shall explain (hence deflationary), associating every first-order parametrically definable class $F$ with a set object $\varepsilon F$, in such a way that Basic Law V is fulfilled:
$$\varepsilon F =\varepsilon G\iff\forall x\ (Fx\leftrightarrow Gx).$$
Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument. A central goal of the project is to highlight the issue of definability and deflationism for the extension assignment problem at the core of Fregean abstraction.
This is a talk for the MidWest Computability Seminar conference held May 2, 2023 at the University of Chicago. The talk will be available via Zoom at https://notredame.zoom.us/j/99754332165?pwd=RytjK1RFZU5KWnZxZ3VFK0g4YTMyQT09.
Abstract: I shall explore several senses in which set-theoretic forcing can be seen as a computational process on the models of set theory. Given an oracle for the atomic or elementary diagram of a model (M,∈M) of set theory, for example, there are senses in which one may compute M-generic filters G⊂ℙ∈M over that model and compute the diagrams of the corresponding forcing extensions M[G]. Meanwhile, no such computational process is functorial, for there must always be isomorphic alternative presentations of the same model of set theory that lead by the computational process to non-isomorphic forcing extensions. Indeed, there is no Borel function providing generic filters that is functorial in this sense. This is joint work with myself, Russell Miller and Kameryn Williams.
The paper is available on the arxiv at https://arxiv.org/abs/2007.00418.
The talk took place in “The Barn” in the upper space between the Reyerson Laboratory and Eckhart Hall, where the University of Chicago Department of Mathematics is located:
This was an interview with Robinson Erhardt on Robinson’s Podcast, part of his series of interviews with various philosophers, including many philosophers of mathematics and more.
We had a wonderfully wide-ranging discussion about the philosophy of mathematics, the philosophy of set theory, pluralism, and many other topics. The main focus was the topic of infinity, following selections from my new book, The Book of Infinity, currently being serialized on my substack, joeldavidhamkins.substack.com, with discussion of Zeno’s paradox, the Chocolatier’s Game, Hilbert’s Grand Hotel and more.
Robinson compiled the following outline with links to special parts of the interview:
- 00:00 Introduction
- 2:52 Is Joel a Mathematician or a Philosopher?
- 6:13 The Philosophical Influence of Hugh Woodin
- 10:29 The Intersection of Set Theory and Philosophy of Math
- 16:29 Serializing the Book of the Infinite
- 20:05 Zeno of Elea, Continuity, and Geometric Series
- 39:39 Infinite Games and the Chocolatier
- 53:35 Hilbert’s Hotel
- 1:10:26 Cantor’s Theorem
- 1:31:37 The Continuum Hypothesis
- 1:43:02 The Set-Theoretic Multiverse
- 2:00:25 Berry’s Paradox and Large Numbers
- 2:16:15 Skolem’s Paradox and Indescribable Numbers
- 2:28:41 Pascal’s Wager and Reasoning Around Remote Events
- 2:49:35 MathOverflow
- 3:04:40 Joel’s Impeccable Fashion Sense
Read the book here: joeldavidhamkins.substack.com.
I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.
The interview concludes with a public comment and question & answer session.
This will be a talk for the Oxford Seminar in the Philosophy of Mathematics, 1 November, 4:30-6:30 GMT. The talk will be held on Zoom (contact the seminar organizers for the Zoom link).
Abstract. The standard treatment of sets and classes in Zermelo-Fraenkel set theory instantiates in many respects the Fregean foundational distinction between objects and concepts, for in set theory we commonly take the sets as objects to be considered under the guise of diverse concepts, the definable classes, each serving as a predicate on that domain of individuals. Although it is often asserted that there can be no association of classes with objects in a way that fulfills Frege’s Basic Law V, nevertheless, in the ZF framework I have described, it turns out that Basic Law V does hold, and provably so, along with other various Fregean abstraction principles. These principles are consequences of Zermelo-Fraenkel ZF set theory in the context of all its definable classes. Namely, there is an injective mapping from classes to objects, definable in senses I shall explain, associating every first-order parametrically definable class $F$ with a set object $\varepsilon F$, in such a way that Basic Law V is fulfilled: $$\varepsilon F =\varepsilon G\iff\forall x\ (Fx\leftrightarrow Gx).$$ Russell’s elementary refutation of the general comprehension axiom, therefore, is improperly described as a refutation of Basic Law V itself, but rather refutes Basic Law V only when augmented with powerful class comprehension principles going strictly beyond ZF. The main result leads also to a proof of Tarski’s theorem on the nondefinability of truth as a corollary to Russell’s argument.
Anyone can learn to count in the ordinals, even a child, and so let us learn how to count to $\omega^2$, the first compound limit ordinal.
The large-format poster is available:
Some close-up views:
I would like to thank the many people who had made helpful suggestions concerning my poster, including Andrej Bauer and especially Saul Schleimer, who offered many detailed suggestions.
I was interviewed 26 August 2021 by mathematician Daniel Rubin on his show, and we had a lively, wideranging discussion spanning mathematics, infinity, and the philosophy of mathematics. Please enjoy!
2:11 Joel’s background. Interaction between math and philosophy
9:04 Joel’s work; infinite chess.
14:45 Infinite ordinals
22:27 The Cantor-Bendixson process
29:41 Uncountable ordinals
32:10 First order vs. second order theories
41:16 Non-standard analysis
46:57 The ZFC axioms and well-ordering of the reals
58:11 Showing independence of statements. Models and forcing.
1:04:38 Sets, classes, and categories
1:19:22 Is there one true set theory? Are projective sets Lebesgue measurable?
1:30:20 What does set theory look like if certain axioms are rejected?
1:36:06 How to judge philosophical positions about math
1:42:01 Concrete math where set theory becomes relevant. Tarski-Seidenberg on positive polynomials.
1:48:48 Goodstein sequences and the use of infinite ordinals
1:58:43 The state of set theory today
2:01:41 Joel’s recent books
Go check out the other episodes on Daniel’s channel!
This will be a talk for the conference Fudan Model Theory and Philosophy of Mathematics, held at Fudan University in Shanghai and online, 21-24 August 2021. My talk will take place on Zoom on 23 August 20:00 Beijing time (1pm BST).
Abstract. Tennenbaum famously proved that there is no computable presentation of a nonstandard model of arithmetic or indeed of any model of set theory. In this talk, I shall discuss the Tennenbaum phenomenon as it arises for computable quotient presentations of models. Quotient presentations offer a philosophically attractive treatment of identity, a realm in which questions of identity are not necessarily computable. Objects in the presentation serve in effect as names for objects in the final quotient structure, names that may represent the same or different items in that structure, but one cannot necessarily tell which. Bakhadyr Khoussainov outlined a sweeping vision for quotient presentations in computable model theory and made several conjectures concerning the Tennenbaum phenomenon. In this talk, I shall discuss joint work with Michał Godziszewski that settles and addresses several of these conjectures.
A proof of the infinitude of primes, in meter.
Let’s prove the primes’ infinitude
they do exist in multitude
of quite astounding magnitude
we count the primes in plenitude!
‘Twas proved by Euclid way back when
in Elements he argued then
He proved it with exactitude
in his Book IX he did conclude
for any given finite list
there will be primes the list has missed
to prove this gem let number N
be when you multiply them then
multiply the list for fun,
multiply, and then add one
If into N we shall divide
the listed primes seen in that guide
remainder one each leaves behind
so none as factors could we find
but every number has some prime factor
with our N take such an actor
since it can’t be on the list
we thus deduce more primes exist
No finite list can be complete
and so the claim is in retreat
By iterating this construction
primes do flow in vast deduction
as many as we’d like to see
so infinite the primes must be
Thus proved the primes’ infinitude
in quite enormous multitude
of arb’trarly great magnitude
we count the primes in plenitude!
‘Twas known by Euclid way back then
he proved it in the Elements when
in modern times we know some more
math’s never done, there’s new folklore:
For Bertrand, Chebyshev had said
about the puzzle in his head
and Erdős said it once again:
there’s always a prime between n and 2n.
We proved the primes’ infinitude
they do exist in multitude
of inconceiv’ble magnitude
we count the primes in plentitude!
The primes are plenitudinous
they’re truly multitudinous!
Musical video production of Plenitudinous Primes by the supremely talented Hannah Hoffman!
I was so glad to be involved with this project of Hannah Hoffman. She had inquired on Twitter whether mathematicians could provide a proof of the irrationality of root two that rhymes. I set off immediately, of course, to answer the challenge. My wife Barbara Gail Montero and our daughter Hypatia and I spent a day thinking, writing, revising, rewriting, rethinking, rewriting, and eventually we had a set lyrics providing the proof, in rhyme and meter. We had wanted specifically to highlight not only the logic of the proof, but also to tell the fateful story of Hippasus, credited with the discovery.
Hannah proceeded to create the amazing musical version:
The diagonal of a square is incommensurable with its side
an astounding fact the Pythagoreans did hide
but Hippasus rebelled and spoke the truth
making his point with irrefutable proof
it’s absurd to suppose that the root of two
is rational, namely, p over q
square both sides and you will see
that twice q squared is the square of p
since p squared is even, then p is as well
now, if p as 2k you alternately spell
2q squared will to 4k squared equate
revealing, when halved, q’s even fate
thus, root two as fraction, p over q
must have numerator and denomerator with factors of two
to lowest terms, therefore, it can’t be reduced
root two is irrational, Hippasus deduced
as retribution for revealing this irrationality
Hippasus, it is said, was drowned in the sea
but his proof live on for the whole world to admire
a truth of elegance that will ever inspire.