[FOM] How much of math is logic?

[joeshipman@aol.com]

Chow:
>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, you have correctly captured my claims; the reason I was not more 
precise was that I did not want to rehash technical details that people 
on this list could be presumed to be familiar with -- I am rather 
surprised at the number of replies that seem to feel Frege, Russell,  
etc. did not actually accomplish anything (I am not claiming they fully 
reduced mathematics to logic, just that a fragment equivalent to PA was 
so reduced without needing the more powerful axioms of less clear 
"logical" status that they needed for the rest of mathematics.)

Raatikainen quibbles about how "logical" assumptions such as EmptySet 
and Extensionality are; but if you are going to be able to talk about 
classes nontrivially, then, as Pollard points out, you get the empty 
set for free. As for extensionality, it is more in the nature of a 
definition than an axiom, and is clearly dispensable (you end up 
proving theorems that say certain sets have the same members rather 
than that they are equal, but there is no loss of expressive or 
deductive power).

Raatikainen points out that with EmptySet, Extensionality, and the 
capacity to adjoin an element to a set, one can already interpret 
Robinson's Q; but in doing this he makes my point for me. I have been 
saying all along that these and similar operations are so fundamental 
and independent of "subject matter" that they deserve to be called 
"logical", even if they require extending the predicate calculus 
slightly. Any reasonable way of formalizing "concepts" or "classes" 
will be at least this strong. As a result, for any theorem provable in 
Robinson's Q, there will be a corresponding statement of pure logic 
whose validity is equivalent. This is the sense in which Robinson's Q 
can be reduced to logic.

(Lindauer gets off the bus at this point, he is not interested in 
describing anything going beyond the pure FOL predicate calculus as 
"logical").

Going from Q to PA here is nontrivial; I believe Friedman has a similar 
interpretability result for PA but I don't recall its exact form. The 
fact that PA has full induction is not a serious obstacle, since for 
any PA-theorem we will be able to eliminate the specific uses of 
induction and construct an equivalent logical statement about inductive 
sets (rather than "all sets") which is proved using only EmptySet, 
Extensionality, and other principles sufficiently fundamental to count 
as "logical" (binary union and pairing are enough).

Raatikainen objects to my explication of logical validity as "true in 
all models" -- I did not mean this in the precise sense he interprets 
it, but I won't try to repair that "definition" here since we are 
really arguing about what should count as "logical". If we carry out my 
program by augmenting the predicate calculus with either a membership 
relation or a formalization of classes or of concepts, then we have to 
define logical validity in the metalanguage anyway, but in a 
straightforward fashion.

The serious objections to set theory only kick in when we posit 
actually infinite sets; but just having an empty set or being able to 
adjoin an element to a set or take the union of two sets or a pair of 
sets does not involve any conceptual vagueness or incoherence, and if 
the concept of "set membership" is still problematic then we can redo 
the work in terms of classes with some very weak second-order axioms. I 
repeat, I am not interested in this type of low-level objection. What I 
really want to know is, if we are willing to accept "finite 
mathematics" as determined by pure logic, does the existence of an 
infinite set then give us the rest of (ZFC-)mathematics.

The technical issue here is, that although the axioms of ZFC except 
AxInf are TRUE in the hereditarily finite sets, we are not getting 
"finite mathematics" by starting with ZFC/AxInf. We are starting from 
much weaker principles which are sufficiently fundamental that they 
deserve to be called "logical". Maybe PowerSet or Replacement are too 
"settish" to be acceptable as "logical" in the way that Pairing is. In 
that case, "(ZFC-)mathematics is logic plus the Axiom of Infinity" is 
NOT necessarily true, and I would like to hear arguments against it. I 
don't really want to quibble about the status of finite mathematics.

-- JS


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