## 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.

## 502 – Formal systems

August 28, 2009

1. Formal systems

Before introducing first order logic, I want to present a “toy example” of the issues we will face, and the results we are after.

In a formal system we present a set of rules about manipulation of finite strings of symbols, and attempt to study which strings can be obtained through these manipulations.

Informally, this corresponds to the syntactic part of logic, and the beginning of proof theory.

I will follow an example from Richard Kaye’s book The Mathematics of Logic.