[FOM] mathematics as formal

[Vaughan Pratt]

catarina dutilh wrote:
> I am not sure whether I am comfortable with the idea of 'the nature
> of mathematics': even if there is such a thing, it is so elusive (and
> perhaps subjective) that it is nearly impossible to talk about it.

To paraphrase William Goldman ("The Princess Bride"), get used to 
discomfort.

Should not "the nature of mathematics" be one of FOM's main questions? 
We can't answer P =? NP, but that shouldn't stop us from trying.

What are the best answers to date to the question "what is mathematics?" 
Do these answers need improvement?
If so, in what respects?
And what are the current obstacles to making such improvements?

 > Mathematical practice, now as well as in the past,
> displays enough heterogeneity so as not to allow for global
> qualifications such as the one implied in my original phrasing of the
> distinction.

An upper bound on the dimension of that heterogeneity might serve to 
contain it.  Is it more then say 3?  For example there may be only those 
who publish without regard for formality, those who feel more strongly 
the need for formality, and computers, whose modes of thought are quite 
unlike ours.

> Going back to the distinction proposed by Vaughan Pratt: the fact
> that there is such a distinction between mathematical practice and
> foundational work does not (should not!) mean that either of them is
> more important vis-a-vis the other.

I fully agree.  Importance is a function of the weighting function, 
which we can easily adjust to produce any desired ranking.

 > In fact, the
> tradition of foundations of mathematics has become a branch of
> mathematical practice in itself -- which was in some sense what
> Hilbert had intended all along.

To the proprietors of a birds' nest, the trunk of the tree may well 
appear as just a branch of the branch supporting their nest.  Consumers 
of mathematics such as physicists and engineers may be reluctant to 
regard foundations as a branch of mathematics since they would not know 
what to do with it.  Not all physicists: the mathematical physicist John 
Baez greatly enjoys foundations, though in his case it is a good 
question as to how his loyalties are divided between physics and 
foundations -- I'd say somewhere around 20/80.

Vaughan