[FOM] How much of math is logic?

[joeshipman@aol.com]

-----Original Message-----
From: dennis.hamilton at acm.org

  I'm sorry, I'm still baffled.

I understand that the absence of finite models for PA is a metatheorem, 
if
you will.  Of course it is.  So is that a purely mathematical 
metatheorem,
or is it indeed "logical?"

I just don't see granting of existence to a finite model, which is 
certainly
external to what PA provides, as somehow more logical than that.  That
observation is hardly "in the system" either, it seems to me.
****

The original question, "How much of math is logic", is intended to 
investigate which mathematical statements are so fundamental and 
independent of "subject matter" that they can properly be regarded as 
"logical".

It is an elementary confusion to think that, because statements in the 
language of arithmetic or set theory involve operation and relation 
symbols (+, *, membership), they cannot be "logical", or that finite 
models are relevant here. The point is that, in certain natural systems 
(simple extensions of the standard predicate calculus, or theories of 
"classes" or "concepts" that involve weak forms of comprehension or 
basic operations like union and pairing), some "ordinary mathematics" 
can be INTERPRETED (in a manner that, metatheoretically, is transparent 
and semantically straightforward).

The import of such an interpretation is that there can be no question 
of the validity of the mathematical result, because it does not require 
any controversial ontology or assumptions about a mathematical "subject 
matter". I claim this is the most satisfactory "foundation" possible 
for a mathematical result.

This is not a "precise" project because we are not defining "logical". 
However, reverse-mathematical investigations reveal equivalences (over 
weak theories) between mathematical results and stronger theories. A 
general theme is that logical power is added via "set existence 
axioms".

Certain set existence axioms are so weak (for example, the existence of 
the empty set, or the existence of a set adjoining a given element to a 
given set) that they can be considered "laws of thought", as long as 
"set" is given a plausible philosophical interpretation (many such 
interpretations, corresponding to words such as "class" or "concept" or 
"element of Von Neumann's hierarchical universe V", are adequate here 
since they only need to satisfy very weak axioms). All that is 
necessary is that SOME mental scheme is admitted to satisfy those 
axioms, and we have a metatheoretical consistency proof and 
interpretation of the corresponding mathematics.

I would argue that some additional set existence axioms, such as binary 
union, pairing, and binary intersection, are sufficiently tame that  
there is a coherent conceptual scheme obeying those axioms, which can 
be meaningfully presented as a "logic".

I also propose, more controversially, that this can be done for a 
system strong enough to interpret the theory of hereditarily finite 
sets, which is equivalent in a strong sense to Peano arithmetic (where 
"interpret" refers to a process that is transparent and semantically 
straightforward).

I next ask, if sufficiently many principles are accepted as "logical" 
that we can interpet PA (and thereby provide a "logicist" foundation 
for it), how much additional mathematics is a consequence of an axiom 
of infinity (in the "strong sense")? If your "logical" principles are 
the axioms of ZFC/AxInf (which are all true in the HF sets), then 
adding AxInf gets you all of ZFC, but this may be going too far for 
some (for example, those who are not comfortable with a general 
conceptual scheme in which the Powerset or Replacement Axioms hold). On 
the other hand, it certainly gets you far enough to obtain Con(PA) or 
the Paris-Harrington theorem.

In some formulations of "2nd order logic", the existence of an infinite 
set follows as a logical truth. How acceptable can such formulations be 
made?

The systems I have been discussing so far can reasonably be supposed to 
have a complete deductive calculus (meaning that semantic entailment is 
no stronger than deduction), but certain strong versions of 2nd order 
logic can be formulated in which very difficult mathematical questions 
(such as any sentence of arithmetic, or the Continuum Hypothesis) are 
equivalent to the validity of purely logical statements. If you are 
willing to accept such a strong logic, there is no hope of DECIDING 
mathematics by reducing it to logic, but philosophical questions about 
the MEANING of mathematics are answered in a definitive way. My final 
question is, in this case, how much of mathematics is firmly grounded 
as meaningful?

-- Joseph Shipman

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