[FOM] Axioms of reducibility and infinity
mlink at math.bu.edu
mlink at math.bu.edu
Sun Aug 7 15:59:41 EDT 2011
I´m working on Tractatus; Wittgenstein rejects
the axiom of reducibility (see
<tel:%286.1232-6.1233>6<tel:%286.1232-6.1233>.1232-6.1233),
the axiom of infinity (5.535) and even the even
the set theory (6.031). First, I ´d like to know
more about those axioms. Second, I´d like to know
why/how does Wittgenstein reject all of them?
Francisco, I am not skilled with these topics, but let me try to
start things off. From what I've heard (from Burton Dreben,
Warren Goldfarb, and Juliet Floyd), Wittgenstein's presentation
of the axioms of reducibility and infinity in the Tractatus were
part of his larger philosophical project rejecting the logicism
of Russell and Frege. The logicism of Frege and Russell was the
unfulfilled hope that all of mathematics, indeed, all of rational
discourse, could be reduced to pure logic.
What Wittgenstein says at 6.031 is that set theory (the "theory
of classes") is "superfluous in mathematics." This has been
interpreted in a number of different ways, but at the minimum it
might simply mean that there is not only one approach to the
foundations of mathematics. Russell himself had tried to avoid
appealling to classes as unwelcome entities. Frege wanted to
base arithmetic and mathematical induction on concepts of logic.
But the ramified theory of types, which Russell developed in
order to avoid a paradox (viz. the class of all classes that are
not members of themselves, which results from Frege's Basic Laws
of Arithmetic), entailed in the Principia Mathematica the appeal
to the axiom of reducibility. This axiom reduces or flattens the
typed hierarchy all to one level. Ramsey in 1925 showed that
once the axiom of reducibility is called on, extensionality is
fully in play, making Russell's intensional proclivities
extraneous. Ramsey in making this argument appealed to
formulations he saw in the Tractatus.
The axiom of infinity was another piece of the puzzle that
Russell had hoped to avoid but could not find a way around.
Whitehead and he used it in the Principia to develop the real
numbers, for example. Wittgenstein seems to have had a different
understanding of logic than did Russell and Frege. Perhaps he
did not see logic as representing (4.0312) a distinct or separate
realm which could explain mathematics and rationality. Instead,
it might be that in the Tractatus Wittgenstein understood logic
as the form common to any thoughtful presentation. But that is a
simile.
You might check Wikipedia or the Stanford Encyclopedia of
Philosophy for more substantive information. Also, I can provide
you with references to some of the main primary and secondary
sources, if you contact me directly. Finally, there may be a
number of mistakes above, but fortunately there are real experts
on these matters who will be forthcoming with corrections where
required.
--ml
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