On proofs and more

October 17, 2013

This is a transcript of an exchange on Twitter on what mathematicians and others expect from proofs. (A previous exchange on a different topic is here. Twitter produces surprisingly nice results sometimes. What follows is a bit meandering, but interesting points are made.)

It began at 7:56 am – 27 Jun 13, with the twitter account of Republic of Mathematics (a website started by Gary Davis) quoting from Bill Thurston‘s great essay On proof and progress in mathematics. The quoted sentence was a short excerpt from the following:

The question is not even “How do mathematicians make progress in mathematics?”
Rather, as a more explicit (and leading) form of the question, I prefer “How do mathematicians advance human understanding of mathematics?”
This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.

To this, the account of The True Beauty of Math replied with “[Which is] why computer proofs [are of] little use.” Republic of Mathematics objected to this position, and quoted from an article by Sara Billey titled Computer Proofs. What is the value of computer assisted proofs?. The quoted sentence is an excerpt from:

Some mathematicians have tried to protect their egos by proposing that human proofs are superior to computer assisted proofs. They claim that we don’t learn as much from computer assisted proofs as we do from a human proof. They claim computer proofs can be difficult to verify. They claim computer proofs are less elegant. I find these complaints to be naive.

This was retweeted by Guy Longsworth, which is how I saw it.

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On CH (After Hamkins)

April 2, 2012

(Wherein I quote from Twitter.)

This is my excuse to put this page to use. It all started, more or less, around here:

Moreno, Javier (bluelephant). “Es muy bueno este artículo de Hamkins sobre algo así como sociología de la teoría de conjuntos: http://arxiv.org/abs/1203.4026” 2 April 2012, 3:39 p.m. Tweet.

Villaveces, Andrés (gavbenyos). “@bluelephant lo siento bastante crudo – me parece que Joel simplemente pone en versión impresa cierto consenso, pero falta argumentar” 2 April 2012, 4:32 p.m. Tweet.

(Loosely translated: “This is a very good article by Hamkins on something akin to sociology of set theory” “I find it weak — I think that Joel is simply putting in writing a known consensus, but needs to argue for it more”) Long story short, I feel Andrés is right, but I thought I should elaborate my view, at least somewhat. Originally I considered writing a blog entry, but it quickly became apparent it would grow longer than I can afford time-wise. So, I used twitter instead.

(But there is a serious caveat, namely, it seems that the paper is intended for a general mathematical and philosophical audience, so the omission of technical issues is most likely intentional. Javier even remarked as much.)

What follows is the series of Tweets I posted. It is not a transcription; to ease reading, I have added a couple of links, reformatted the posts rather than continuing with the MLA suggested approach, very lightly edited the most obvious typos, and added a couple of phrases where I felt more clarity was needed. [Edit, April 22: I also removed a line on MM^{++} vs Woodin’s (*), as the proof of the underlying claim has been withdrawn.]

I started at 10:44 p.m., with a warning: “(Technical pseudo-philosophical thoughts for a few posts.)”

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