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December 23, 2012

This is what is wrong with contemporary mathematics

Erik Poupaert:

The 1925 publication of Principia Mathematica caused something of a stir in academic circles back then. By firmly vesting precise mathematical notation and insisting on the precise use of language, Betrand Russell and Alfred Whitehead convinced approximately everybody in science and mathematics that it was time to stamp out ambiguity in scientific claims. Expressions and symbols like this became commonplace: φx ≡x ψx .⊃. (x): ƒ(φẑ) ≡ ƒ(ψẑ).

The year 1925 precedes a lot of change that has hit the world since then. The most important change is the advent and widespread use of computing devices, that is, computers.

One problem that we have run into is that a mathematical problem phrased in the Russell-Whitehead notation is not particularly well executable on a computing device. Besides making a particular, precise claim, the Russell-Whitehead notation does not allow you to verify it on a computer. That characteristic of the Russell-Whitehead approach is not particularly efficient. I will show in this blog post why it is even dangerous.

The Curry-Howard correspondence, on its side, claims that every mathematical "proof" is a program and the other way around. Therefore, instead of using a non-executable notation for "proof", it should always be possible to use an executable one.

Posted by Jim Zellmer at December 23, 2012 2:01 AM
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I have not found any information about Erik Poupaert's qualifications, and if he is qualified to give his opinions about proofs, Goedel's Incompleteness theorem, Church's Lambda calculus, etc.

I'm certainly not and before accepting Poupaert's opinions, it would be wise to consult with a math expert in this area.

However, from the materials I have read by bonafide experts giving summary opinions on this subject in lay terms, Poupaert's opinions are significantly off the mark.

For example, he claims the successor function is wrong, but we haven't found the bug yet. Though I've never read a mathematician's view on the successor function, I would guess that the successor function has no such problem because the mathematics of integers with single operation plus is consistent and complete.

Euclid's geometry is consistent and complete, as I believe is the propositional calculus. So, there are some formal axiomatic systems for which Goedel's theorem are not applicable.

Poupaert's says "In fact, Gödel's 1931 Second Incompleteness Theorem insists that it is strictly forbidden to ever claim the truth of any scientific or mathematical statement:"

As I understand it in lay terms, Goedel's theorem talks about the characteristics of certain formal axiomatic mathematical systems. Poupaert's attack on the validity of science justified using Goedel's theorem is false. Science is not an formal axiomatic system, therefore Goedel's theorem is irrelevant to the discussion of science.

Posted by: Larry Winkler at December 24, 2012 1:19 AM

On second thought, I think Poipaert's comments should give us some needed positive reinforcement that the US educational system is not so bad compared to, say, the Belgian.

I did find a reference to this blogger's education. Magna Cum Laude MS in computer science and engineering. So educated on paper and yet so wrong.

Posted by: Larry winkler at December 24, 2012 2:00 AM
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