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What is Logicism?

Philosopher's are sometimes more eager to refute what their adversaries write than to understand what they mean.

I sometimes test out my ideas using an internet newsgroup or mailing list. This is certainly a good way of discovering just how wide the difference can be between what you intend to say and what someone else takes you to mean.

In relation to logicism I have discovered by this means firstly that to make explicit distinction between logicism as a doctrine and logicism as a programme is desirable, since one is likely otherwise, while intending only to defend the view that mathematics is analytic, to be construed as defending the details of some unspecified historical logicist's failed attempt to demonstrate this claim.

Even when definitely discussing doctrine, beware of more or less similar accounts of the doctrine.

Statments by logicists about their beliefs: ! Statements about the meaning of logicism: *

  1. "the laws of arithmetic are analytic judgements" ([Frege80] p99) !
  2. mathematics is analytic
  3. the truths of mathematics are necessary truths
  4. the truths of mathematics are logically necessary
  5. "mathematics and logic are identical" ([Russell37] p v) !
  6. the truths of mathematics are reducible to logic
  7. the truths of mathematics are reducible to pure logic
  8. "the truths of mathematics are analytic and thus derivable from pure logic" ([Hatcher82] p73) *
  9. "all mathematics is derivable from universally valid logical principles" ([Hatcher82] p123) *
  10. the truths of mathematics are reducible to pure classical logic
  11. the truths of mathematics are reducible to truths derivable in a pure classical logic without use of an axiom of infinity
  12. the truths of mathematics are reducible to truths derivable in a pure classical logic in which no infinite collection can be shown to exist.

It may be that each of these statements means more or less the same as its predecessor, but they nevertheless proceed from a claim eminently worthy of defence to one which looks plainly false.


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