When Hilbert devised his philosophy, it looked like there was a serious possibility that Brouwer's intuitionism might win. Hilbert's philosophy can be read as an attempt to justify classical mathematics to the doubters, especially, the intuitionists.

He said, "No one shall throw us out of the paradise Cantor has created for us."

In order to convince doubters, Hilbert had to argue from common ground, that is, from things on which everyone agreed. His chosen common ground is finitist mathematics. After all, on a finite domain, classical logic applies.

Hilbert admits that the trend in the foundations of mathematics has been toward eliminating the infinite. It is only because Weierstrass eliminated the infinite from analysis, that analysis became rigorous.

However, while Weierstrass eliminated the potential infinite from mathematics, he brought the actual infinite in the back door. Hilbert therefore needs to show how we can have the benefits of Cantor's actually infinite collections without postulating anything more than finite collections.

Some sentences that make finitistic sense do not have finitistic negations: 'There is an even number that is not the sum of two primes' makes finitistic sense. Its negation does not.

When we prove a statement that apparently makes no finitistic sense, we often do it by proving a stronger, finitist statement:

There are infinitely many primes.

For any %$n$% there is a prime between %$n%$% and %$n!+1$%, inclusive. I'm using "any" instead of "every" in a special way.

Thus, sometimes we can find a finitist replacement for the negation of a finitist statement. But not always. That is a lot like, Hilbert thinks, parallel lines and solutions to equations. To add ideal negations, we need an axiomatized notion of proof. We then need to check that adding in negations is cool. It will be cool if the resulting system is consistent since then any finitary statement that can be proved will already have a finitary proof.

Hilbert already thought that consistency is all that is required within mathematics to justify talking about existence, and so this coheres nicely.

Hilbert's program would justify classical mathematics to doubters in some sense, either that one could pretend to employ classical mathematics in obtaining results or that classical mathematics is shown on a finitist basis to be correct. It isn't clear which version Hilbert prefers.

Intuitionists did not accept Hilbert's program, but it failed for a more central reason: it is a consequence of Gödel's incompleteness theorem that no consistent sufficiently complicated axiom system can prove its own consistency. Since Hilbert's ideal systems included finitistic mathematics, finitistic mathematics cannot prove them consistent.

The failure of Hilbert's program was an important nail in the coffin of foundationalism.

Why does Hilbert think that finitistic mathematics is a necessary basis for the ideal elements? "the infinite is nowhere to be found in reality, no matter what experiences, observations, and knowledge are appealed to" 191.

-- ShaughanLavine - 11 Sep 2008