Post

More from @JDHamkins

Joel David Hamkins

@JDHamkins

Mar 14

On the cultural differences between math and philosophy in regard to Q & A after a seminar or colloquium talk. A thread. Please feel free to comment. 🧵

Philosophy
Philosophers typically have extended Q & A after a talk, often as long or even longer than the talk itself. A standard format would have a one hour talk or 50 minutes, perhaps a short break of 5 minutes, and then another hour or 50 minutes of questions and discussion. Another more compressed standard format calls for a 20 minute talk and 40 minutes of Q & A. In truth, there is never enough time—at almost every talk, more people want to ask questions or contribute to the Q & A discussion than is possible, and it is common for the moderator to announce that time is up, alas, even though many people still remain on the list to ask questions. At conferences, many talks are arranged into small sessions with a main speaker and then several commentators who give talks criticizing the main talk, and then rebuttals and then Q & A with that panel but also eventually audience participation.

Mathematics
At a math talk, in contrast, usually the talk itself is about an hour or 50 minutes, after which there will be just a short amount of time for questions, sometimes just 5 or 10 minutes, or at most 15 minutes, which in truth is entirely sufficient. For usually there are just a few questions, perhaps one or two, and quickly addressed, and this situation is not at all embarrassing in the way it would be in philosophy. Sometimes there are no questions, which is only slightly embarrassing or even not a problem at all, although often the chair will feel called upon to ask to nominal question of some kind to save face (and many session chairs plan for this contingency during the talk). I have never seen the extended Q & A format at a math talk, and I have never seen the talk/commentary/rebuttal format at a math conference.

Read 7 tweets

Joel David Hamkins

@JDHamkins

May 11, 2024

Take my Philosophy of Mathematics final exam!
Post your answers in response to each question.

What is logicism? Which philosophers are associated with this philosophical project? What were the central ideas and major issues or developments arising in connection with this program? Was the program ultimately realized successfully?

Explain Dedekind’s categoricity argument for the arithmetic of natural numbers and explain how it relates to the philosophy of structuralism.

Read 18 tweets

Joel David Hamkins

@JDHamkins

Nov 29, 2023

An imaginary mathematical history.🧵
I should like to sketch an imaginary mathematical history, an alternative history by which the continuum hypothesis (CH) might have come naturally to be seen as a core axiom of set theory and one furthermore necessary for ordinary mathematics.

Let us imagine that in the early days of calculus, the theory was founded upon infinitesimals, but in a clear manner that identified the fundamental principles relating the real field ℝ to the hyperreals ℝ*, identifying the transfer principle and saturation.

And let us further imagine that in the heady developments at the end of the 19th and into the early 20th centuries providing the categorical characterizations of ℕ and ℝ, that a Zermelo-like figure provided also a categorical characterization of the hyperreal field ℝ*.

Read 10 tweets

Joel David Hamkins

@JDHamkins

Aug 19, 2023

Three variations on Tarski.
Let me present three variations of Tarski's theorem on the non-definablity of truth. The first is proved in Gödelian style using Gödelian methods; the second is Russellian; and the last is purely Cantorian. But all express the nondefinability of truth.

First, the sentential variation, in Gödelian style. Truth is not definable, in the sense that there is no arithmetically expressible formula T(x) such that T('σ')⟷σ for every arithmetic sentence σ.

The proof amounts to the reasoning of the Liar paradox, since the fixed-point σ would assert its own falsity, if T did indeed have the stated truth-predicate property. Because it appeals to the fixed-point lemma and the coding of syntax, this proof is Gödelian.

Read 8 tweets

Joel David Hamkins

@JDHamkins

Aug 16, 2023

The countable random graph.
I realized something today about the countable random graph. Namely, although this graph is highly homogeneous and has numerous automorphisms, it is nevertheless definably rigid, meaning that it has no definable automorphisms. Let me explain.

The countable random graph arises from a countable set of vertices by flipping independent fair coins to determine whether there is an edge between any two nodes.

Almost surely such a process fulfills what is called the finite pattern property: for any two finite disjoint sets of vertices A,B, there is a node x adjacent to every node of A and to no node of B.

Read 11 tweets

Joel David Hamkins

@JDHamkins

Aug 1, 2023

Consider several structures on the natural numbers:
• Successor ⟨ℕ,0,S⟩
• Order ⟨ℕ,0,<⟩
• Addition ⟨ℕ,0,+⟩
What I claim is that each structure is definable in the next, but not conversely.

The positive claim is easy, for we can easily define the successor function from the order—the successor Sx is simply the least element that is larger than x. And similarly, we can define the order from +, since
x < y if and only if ∃z (z≠0 and x+z = y).

The converse nondefinability claims are more subtle.
To see that we cannot define < from successor, consider a nonstandard elementary extension
⟨ℕ,0,S⟩ ≺ ⟨ℕ*,0,S*⟩, which must consist of an ℕ-chain growing out of 0 together with a number of dissconnected ℤ-chains.

Read 16 tweets

Share this page!

Enter URL or ID to Unroll

Joel David Hamkins

Try unrolling a thread yourself!

More from @JDHamkins

Joel David Hamkins

Joel David Hamkins

Joel David Hamkins

Joel David Hamkins

Joel David Hamkins

Joel David Hamkins

Did Thread Reader help you today?

Don't want to be a Premium member but still want to support us?

Send Email!