- 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.
- 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.
- 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
- 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
- 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
- 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
- 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).
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?
- Hilbert was invited to address the Mathematical Congress
- 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.
- 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
- 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
Next: Brouwer, Russell, Godel.