[FOM] How much of math is logic?
Timothy Y. Chow
tchow at alum.mit.edu
Wed Feb 28 10:13:44 EST 2007
JS: This is a silly objection, the point is that the mapping **can be
JS: seen** to connect arithmetical statements to logical ones in a
JS: truth-preserving way. What do you think Russell thought he was doing?
TC: Why is it silly? It seems to me that the burden is on you to
TC: elucidate what sort of things "can be seen" in this context.
TC: Otherwise one can "sneak in" arbitrarily strong non-logical
TC: assumptions (that nevertheless "can be seen" to be true in some
TC: sense) and claim that all kinds of non-logical assertions "can be
TC: seen" to be "equivalent" to logical ones.
JS: I would elucidate it, except that it has already been done by Russell
JS: and others.
Well, I haven't studied the Principia itself, so maybe you, or someone
else on this list, can educate me, but my impression was that to get the
"interesting" arithmetical truths, they had to introduce things like the
axiom of reducibility, which you say is "non-logical." It would be
helpful if someone could summarize the relevant proof-theoretic facts.
Also, maybe somebody could clarify whether Russell thought he was carrying
out Frege's logicist program as Frege originally envisioned it (reducing
mathematics strictly to pure logic) or whether he was carrying out a
slightly modified program in which some non-logical assumptions were
permitted.
In any case, I still don't believe that you can dismiss Raatikainen's
objection so easily. Your assertion seems to be:
If S is a set of arithmetical statements, and L is a set of logical
statements, and f is a mapping from S to L that "can be seen" to be
truth-preserving, then S has, from the point of view of the logicist
program, been "reduced to pure logic."
Asked to elucidate what "can be seen" means, your response seems to be:
Whatever "can be seen" might mean in general, there exists a mapping
from theorems of PA to logical statements that "can be seen" to
preserve truth. But replace PA with ZFC and no known mapping "can
be seen" to have the analogous property.
If I've accurately captured your claims, then surely, when I put it this
way, it is not at all silly to press you to be more precise about what
exactly can and can't "be seen"?
Tim
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