Hilbert

Logicism Review

The logicist program is to reduce math to logic. This will, for example, settle debates about what math can and cannot do by getting agreement on obvious foundations.
Frege's system required Axiom V to get arithmetic, but Axiom V allows a version of Russell's paradox.
This is the first of two knock-out blows to the dream of both logicism, and to Hilbert's dream (see below); the second is Godel's Incompleteness Theorems.

Hilbert background

David Hilbert was born January 23, 1862 in Konigsberg, in what was then Prussia. Died 1943.
Took Ph.D. at Konigsberg in 1884.
Taught at University of Gottingen.
Gave monumental address to Internation Mathematical Congress in 1900, laying out 23 problems for the new century.
Did important work in mathematics and mathematical physics.
Founded formalist school of the philosophy of mathematics.

Formalism

Mathematics consists of two things: a core of logical propositions which are to be universally admitted, and the formal manipulation of symbols (which can be but which need not be interpreted as referring to something).
When we reason about something like infinite sets, according to the formalist, what we are doing is (or at least in part is) manipulating symbols (this is like saying much of math is at its foundations just syntax; or maybe like saying that when in doubt, we can remain neutral about whether the formal operations represent some kinds of mathematical objects).
In at least one extreme interpretation of formalism, the manipulated symbols are real, physical objects (like scratches on a board). It looks like many formalists don't take it this far.
To practice formalism, we just attempt to define as clearly as possible the syntax of what we are doing, and the syntactic operations that we are undertaking.
Formalism is a living program, but it has been very controversial. Objections have included:

Incredulity that math and related kinds of reasoning can be, when controversial, about nothing or about formal manipulations.
Claim that we secretly rely on mathematic intuitions in organizing the real-world symbols that we manipulate in doing math.
Claim that we still assume certain non-formal principles (such as mathematical induction -- see Poincare's criticism).
Recognition that physical symbols must be of a kind for this to work (this is only a criticism of the extreme form of formalism, and as noted above there may not be any of these extreme formalists).

The Problems

Hilbert was invited to address the Mathematical Congress in 1900
Seeking a monumental theme to set the tone for a new century, Hilbert described 23 problems.
One problem was to develop formal systems which allowed us to check any proof for correctness.
Another problem was to prove that mathematics was consistent.
These goals became the new standards for a rigorous exploration of the abilities (and ultimately the limits) of reason.
Note: a system that allowed us to check any proof, and which was consistent, would allow us to mechanically prove anything. Can you see why this is?

Intuitionism

Founded by Luitzen Brouwer (1881-1966), intuitionism asserts that mathematics does not require any foundations or justification, but is given in a special indubitable intuition.
Those elements of mathematics that are not given in this intuition are to be rejected. For this reason, intuitionists reject any non-constructive proofs (that is, they require that one actually show that some sentence is true in order to conclude that it is true).
This commitment to constructivism requires rejection of the principle of the excluded middle for any consideration of infinite sets (e.g., for any consideration about numbers).
Intuitionist mathematics has been very fruitful, but in general the denial of the principle of the excluded middle is seen as too costly.

Next: Brouwer, Russell, Godel.