[FOM] How much of math is logic?

[joeshipman@aol.com]

>>
>> 1) A theorem of Peano Arithmetic which is not equivalent to a 
logical validity
>>
>The obvious answer would be, say: ~(Ex)(0 = Sx). That's not logically
>valid and is therefore not equivalent to any logical validity, given
>what is usually meant by "logically valid". Or (to channel for George
>Boolos): the conjunction of ~(Ex)(0 = Sx) and (x)(y)(Sx = Sy --> x=y)
>has no finite models and so is not logically valid on any understanding
>of "logically valid" presently available to us.
>> 2) A theorem of ZF without the axiom of Infinity which is not 
equivalent to a
>logical validity
>>
>And of course similar answers can be given here.
>
>So, obviously, you mean something else by "logically valid", but then

What I mean is this: if you add the standard symbols for arithmetic 
(0,1,S,+,*, etc.) to the language, of course the axioms of Peano 
Arithmetic will not be logically valid; 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. (Russell and 
Whitehead famously took hundreds of pages to get to 1+1=2, but much 
more streamlined versions are possible.)

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" (I am not 
considering Russell's Axiom of Reducibility as a "logical" axiom here, 
and I don't know how much proof-theoretic power it adds to Russell's 
system -- I just want to avoid it so that there is no question about 
the status of the axioms used as "logical".)

I would like someone to either confirm or correct my understanding 
here.(By the way, I use "logical truth" and "logical validity" 
interchangeably to mean sentences true in all models; in the case of 
first-order logic there is an enumeration of such sentences, but I'm 
not restricting my attention to first-order logic.)

My second point is that apart from the Axiom of Infinity, which is an 
existence postulate that is necessary because pure first-order-logic 
does not entail such a rich ontology, it can be argued that the axioms 
of ZF (or of an equivalent system like VNBG) appear to deserve the 
status "logical" rather than merely "mathematical". This supports the 
slogan "mathematics is just (first-order) logic plus the axiom of 
infinity".

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".

There is a little tension between these two slogans, because of an 
equivocation -- the "is" in the first slogan is deductive, and assumes 
(ignoring some technicalities about AC) that mathematics consists of 
finding consequences of ZF, while the "is" in the second slogan is 
semantic, and is concerned with what is true rather than what is 
provable.

So if you want to treat mathematics formally, as a deductive calculus, 
I can say that except for the Axiom of Infinity, math is just logic; 
and if you want to treat mathematics semantically, I can say that 
almost all math is just logic.

I further say, as my fourth point, that our inability to decide all 
arithmetical truths or CH simply means we don't have a complete 
understanding of logical validity, and I'm not troubled by this because 
I see no reason why we should expect logical validities to be 
enumerable -- it does NOT mean that second-order logic is not really 
"logic".


> Boolos, who was as
>interested in the status of second-order logic as anyone, more or less
>abandoned the claim that second-order logic was "really logic" late in
>his life, not because he became convinced of the contrary view---though
>the essentially G"odelian worry that second-order validity encodes too
>much mathematics bothered him a lot---but rather because he ceased, he
>said, to understand the notion of "logic" at issue. Is it supposed to 
be
>epistemological? metaphysical? or what? I don't know that anyone really
>has a good view about this.

Encoding "too much mathematics" seems a silly worry -- if it was a 
valuable advance for Russell to show that logic could encode 
arithmetic, why isn't further progress in this direction also valuable?

I'm willing to entertain a variety of notions of "logic", but what they 
have in common is that there is no need for a "subject matter" -- logic 
is purely conceptual, and logical truth applies to every domain we can 
talk about consistently.

-- JS
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