[FOM] How much of math is logic?
joeshipman@aol.com
joeshipman at aol.com
Mon Feb 26 14:31:38 EST 2007
Responding to Raatikainen:
>> ... but there are alternative
>> formulations of number theory which are built up from pure logic,
going
>> back to Russell. It is possible to define a mapping from sentences
of
>> arithmetic to statements of logic, in such a way that the axioms and
>> theorems of Peano Arithmetic map to logical validities.
>
>It is possible to effectively map (almost) any r.e. set to any other
r.e.
>set, but that does not make an arbitrary r.e. set of formulas a set of
>logically valid sentences. (I am simplifying a bit, but nothing hinges
on that)
This is a silly objection, the point is that the mapping **can be
seen** to connect arithmetical statements to logical ones in a
truth-preserving way. What do you think Russell thought he was doing?
>> However, this route does not seem to transcend Peano Arithmetic; as
I
>> understand it, arithmetical statements whose proof (in ZF) requires
the
>> use of the axiom of Infinity will not be reachable in the versions
of
>> this setup that correspond in some way to "first order logic" ...
>There are two different senses of "an axiom of infinity".
>...
>The axioms of ZFC without this
>axiom already make the domain infinite, but it is this
>axiom which gives ZFC its extreme power. It is much stronger
assumption
>than an axiom of infinity in the first sense.
>Now I don't think that simple type theory and such can provide us much
>mathematics without the addition of an axiom of infinity in the first
sense.
>And the claim that such an assumption is logically true is quite
controversial.
This is exactly what I was trying to say -- I was careful to
distinguish the ZFC axiom of Infinity as something that was NOT
"logical", in order to argue that those parts of math which do not need
it CAN be thought of as simply logical.
>Almost all axioms (well, not extensionality) of ZFC are set existence
>axioms, and even a few of them constitute together an axiom of
infinity in
>the first sense. Hence, they are not logically true.
Here we disagree; I take it as a logical truth that "something exists",
namely (if nothing else) concepts, and in particular that concepts have
certain basic properties (Pairing, Union, etc.). (By the way that does
NOT mean I identify all concepts with sets, only that I claim some
concepts are sets.) Thus, "the empty set" is a valid concept, and so
"exists". If you refuse to grant logic any ontology whatsoever, then OF
COURSE logicisim is false, but only in a trivial way unworthy of
further discussion.
>> My third point, that there does not seem to be any interesting open
>> question outside of set theory which is not equivalent to the
validity
>> of a sentence of second-order logic with standard semantics,
supports
>> the slogan "almost all math is just (second-order) logic".
>Well, this takes us back to the old dispute on whether second-order
logic
>really is logic. I think not, but I would not like to open that worm
can again.
Why not? In my opinion, many people who say that second-order logic is
not logic are prejudiced because they find the view that statements
like CH do not have an absolute truth value congenial; that view allows
one to maintain the Tennantian anti-realist position that there do not
exist truths which are in principle unknowable.
To be fair, Tennant himself does not share this prejudice; his attitude
towards CH is that it is no different from other statements independent
of earlier mathematical systems, which mathematical investigation and
reflection eventually led to acceptance or rejection of. Discussing
this near the end of chapter 6 of "The Taming of the True", he says "We
have no way of foreclosing on future extensions of our collective
intellectual insight."
-- JS
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