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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

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