In 1920, Hilbert proposed a mathematical research program that would show that all of mathematics follows from a finite set of axioms, and that the resulting axiomatic system would be provably consistent.

Of Hilbert's program Huber-Dyson writes:

*Wanted was a proof that the system using, what Hilbert called "ideal elements" — reasoning about infinite sets the way we are used to reason about finite ones — was not going to lead beyond the realm of what is justified by finitary reasoning, in other words, that the make believe world of predicate logic was a mere conservative extension of finitism. Alas, he wishfully thought that all he needed was a proof of simple consistency as defined above. That is why the demand for a finitist proof of the consistency of any of the current formal systems was his major concern.*

Of course, Gödel showed that Hilbert's program was not feasible:

*...Gödel showed how to construct, in any formal system that encompasses a minimal fragment of elementary arithmetic, a sentence, which is true if and only if it cannot be proved, and therefore is true but not provable, unless the system is unsound.*

For me, the most interesting part of the paper is something called mathematical intuitionism.

*In a nutshell: intuitionist reasoning is a refinement of classical thinking. Identifying notnotA with A only when there is good reason to do so is much closer to everyday reasoning than the practice of the classical law of double negation. With a bit of care most of the popular but devious proofs of positive claims by reductio ad absurdum can be replaced by direct arguments...*

claim (A or B) only if you have a method for deciding which one of them is the case; if you want to insist on the existence of an object with a certain property, be sure you have a method for producing such an object when called for.

claim (A or B) only if you have a method for deciding which one of them is the case; if you want to insist on the existence of an object with a certain property, be sure you have a method for producing such an object when called for.

Wikipedia says this about intuitionism:

*For example, to say A or B, to an intuitionist, is to claim that either A or B can be proved. In particular, the law of excluded middle, A or not A, is disallowed since one cannot assume that it is always possible to either prove the statement A or its negation.*

I like the sound of this idea (though I will have to do a lot more reading on this subject before I accept it). You see, I read this as a mathematical analogue of logical positivism. If there is no mechanism by which proposition A can be verified, we are not justified in assigning it a truth value at all.

## 1 comment:

A good place to start is Michael Dummett's

Elements of Intuitionism.Post a Comment