# C[omp]ute

Welcome to my blog, which was once a mailing list of the same name and is still generated by mail. Please reply via the "comment" links.

Always interested in offers/projects/new ideas. Eclectic experience in fields like: numerical computing; Python web; Java enterprise; functional languages; GPGPU; SQL databases; etc. Based in Santiago, Chile; telecommute worldwide. CV; email.

© 2006-2013 Andrew Cooke (site) / post authors (content).

## A Century of Controversy Over the Foundations of Mathematics

From: Manuel Johannes Simoni <csae4529@...>

Date: Mon, 15 Dec 2003 21:16:46 +0100

Historical Introduction --- A Century of Controversy Over the Foundations of
Mathematics

G J Chaitin

Link: http://www.cs.auckland.ac.nz/CDMTCS/chaitin/cmu.html

G.J. Chaitin's 2 March 2000 Carnegie Mellon University School of Computer
Science Distinguished Lecture. The speaker was introduced by Manuel Blum. The
lecture was videotaped; this is an edited transcript which appeared on pp. 12-
21 of a special issue of Complexity magazine on Limits in Mathematics and
Physics'' (Vol. 5, No. 5, May/June 2000).

...

Now I don't think that Hilbert really wanted us to formalize all of
mathematics. He didn't say that we should all work in an artificial language
and have formal proofs. Formal proofs tend to be very long and inhuman and
hard to read. I think Hilbert's goal was philosophical. If you believe that
mathematics gives absolute truth, then it seems to me that Hilbert has got to
be right, that there ought to have been a way to formalize once and for all
all of mathematics. That's sort of what mathematical logic was trying to do,
that's sort of what the axiomatic method was trying to do, the idea of
breaking proofs into smaller and smaller steps. And Leibniz thought about
this, and Boole thought about this, and Frege and Peano and Russell and
Whitehead thought about this. It's the idea of making very clear how
mathematics operates step by step. So that doesn't sound bad. Unfortunately it
crashes at this point!

So everyone is in a terrible state of shock at this point. You read essays by
Hermann Weyl or John von Neumann saying things like this: I became a
mathematician because this was my religion, I believed in absolute truth, here
was beauty, the real world was awful, but I took refuge in number theory. And
all of a sudden Gödel comes and ruins everything, and I want to kill myself!

So this was pretty awful. However, this

This stmt is unprovable!''

is a very strange looking statement. And there are ways of rationalizing,
human beings are good at that, you don't want to face unpleasant reality. And
this unpleasant reality is very easy to shrug off: you just say, well, who
cares! The statements I work with normally in mathematics, they're not
statements of this kind. This is nonsense! If you do this kind of stupidity,
obviously you're going to get into trouble.
But that's rationalizing too far. Because in fact Gödel made this

This stmt is unprovable!''

into a statement in elementary number theory. In its original form, sure, it's
nonsense, who ever heard of a statement in mathematics that says it's
unprovable? But in fact Gödel made this into a numerical statement in
elementary number theory, in arithmetic. It was a large statement, but in some
clever way, involving Gödel numbering of all arithmetic statements using prime
numbers, he was writing it so that it looked like a statement in real
mathematics. But it really indirectly was referring to itself and saying that
it's unprovable.

So that's why there's a problem. But people didn't really know what to make of
this. So I would put surprising'' here, surprising, a terrible shock!

...

Summary of Gödel's Theorem:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/eesti.html#23

G J Chaitin's homepage:
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/index.html

Manuel