[FOM] Formalization Thesis: A second attempt

[rgheck]

On 05/26/2010 03:05 AM, Vaughan Pratt wrote:
>
> On 5/25/2010 8:57 AM, Paul Budnik wrote:
>    
>> I see mathematics as formalizing what conclusions (theorems) are logically determined by assumptions (axioms).
>>      
> Very interesting.  I see logic as formalizing the conclusions of
> mathematics.  I wonder how we arrived at such different viewpoints.
>
> Actually I'm not sure anyone I know personally sees mathematics as
> formalizing the conclusions of logic, but those that do would surely
> have to be logicians rather than mathematicians.  Those mathematicians
> that care at all about axioms usually work backwards from what they just
> proved to see what axioms they had been assuming.  "Reverse mathematics"
> is a misnomer: mathematics is reverse logic and logic is reverse
> mathematics.
>
>    
I sympathize with the viewpoint expressed here, in many ways, but I 
think the perspective on logic is insufficiently general, as well as 
historically suspect. Logic, the discipline, has always been understood 
as (in some sense) concerned to characterize valid inference (where the 
term "inference" might mean a sequence of sentences or propositions or a 
certain sort of mental act, and the question which it ought to be is 
important). Since Aristotle, mathematics has been understood as a 
critical test case, since, in some sense, mathematical reasoning seems 
ideally to involve only inferences that are supposed to be deductively 
(not inductively, not abductively) valid---though of course some have 
disagreed, over the centuries. So, in that sense, the ideal of 
formalization---of being able in some sense to evaluate mathematical 
reasoning in terms of the characterization of valid inference delivered 
by logic---has been in place since Aristotle, too, though of course this 
could not actually be done, except in a very limited way, until the late 
1800s, for well known reasons.

On the other hand, mathematics is equally a source of (putatively) valid 
argument, and the standards mathematicians use in evaluating actual 
reasoning form part of the data for logic. An account of valid inference 
that declares invalid a great deal of mathematical reasoning that is 
accepted as valid has a whole lot of explaining to do, though of course 
the conflict doesn't by itself show the account to be false, since logic 
is not descriptive but prescriptive. Still, the model John Rawls 
suggested for the theory of justice serves one well here (and, I 
believe, may actually have had its roots here): we want to reach a 
"reflective equilibrium" between logical practice and logical theory, 
and neither is immune from the effects of the other.

I am aware, of course, that "logic" here largely means "mathematical 
logic", but it is, in my view, a serious mistake to loose mathematical 
logic from its historical roots.

Richard Heck