# [FOM] Formalization Thesis: A second attempt

rgheck rgheck at brown.edu
Thu May 27 15:14:09 EDT 2010

```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

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