A
philosophical theory of numbers
. Consisting of the
thesis that the propositions of
mathematics are
reducible to, or are no more informative than, what is found in the axioms and principles of
logic. You can get an idea of what motivates logicism if you think through this (an example reminiscent of, though perhaps not strictly representative of,
Gottlob Frege's pioneering thinking):
To what do we refer when we use the symbol '2'? Perhaps we refer to a set; most obviously (i.e., after some thought), this would be the set of all sets with two members.
Now set theory is the modern boundary between mathematics and logic: it suggests, for the logicist, the unity of the two areas. For consider the statement,
'2 plus 2 equals 4.'
Let
2 and
4 be sets, and the statement be rephrased as
'Any set that is a member of 2, in disjoint union with another such set, forms a set equinumerous with any set that is a member of 4.'
The long-winded reconstruction can be said to use just
logical axioms, concepts of sets and
quantification to achieve a logical
formalization of the first statement.
(Just as long as set theory can properly be considered part of logic.)
These ideas of were an important part of Bertrand Russell's essentially positivist philosophy in Principia Mathematica, an attempt at the definitive formalization of mathematics -- the placing of mathematics on a sound footing, as it were. He believed that
- The theorems of mathematics constitute a proper subset of those of logic.
- The vocabulary of mathematics is constructible, by mere sequences of definitions, out of logical vocabulary.
As is frequently noted in discussion, Russell's own paradox helped to defeat the bold aims of the Principia. The complexity introduced by the need to cope with these paradoxes of set theory forced increasingly ad hoc-seeming modifications. Furthermore, Gödel's Theorem, a 'death knell' for the axiomatic method, denied any prospect of finding logical proofs (within a single system) of all mathematical truths. Because there is no satisfactory proof of arithmetic's consistency, there is no meeting the intuitive demand for the logicist to reduce mathematics to something 'less complex' than itself -- something consistent, axiomatized and complete.
However, it may be salutary to consider how things look if the problems revealed by the Principia in set theory (not just the mere presence of paradoxes but the forced addition of questionable axioms such as AC and the axiom of infinity) are set aside. That is, what if logicism's motivation was correct? Could it be that mathematics is just logic?
It seems to me that even without the paradoxes inherited from set theory, logicism says crazy things about what mathematics is. Accepting its argument -- mathematical truths are analytic, because mathematical concepts are logical concepts -- just doesn't seem valid or sound. There is a flaw (a flaw perhaps exposed in the writings of Wittgenstein, and Quine) in the positivist, which is to say Russellian, denial of the existence of the synthetic a priori. The meanings of mathematical terms don't just reduce to their formal role or to some particular embedding into logic; they have substantive meaning in their own right.