Re: Sketches and Platonic Ideas

["F. William Lawvere" <wlawvere@hotmail.com>]

Certainly I did not mean to suggest that either John or Andree were
supporting platonism as a philosophy of mathematics. In fact I had
momentarily even forgotten that John had used the term. In my 1972
Perugia Notes I had made an attempt to characterize the relation between
these sorts of mathematical considerations and philosophy by saying that
while platonism is wrong on the relation between Thinking and Being,
something analogous is correct WITHIN the realm of Thinking. The relevant
dialectic there is between abstract general and concrete
general.
Not concrete particular ("concrete" here does not mean
"real").There is another crucial dialectic making particulars
(neither abstract nor concrete) give rise to an abstract
general; since experiments do not mechanically give rise to theory, it is
harder to give a purely mathematical outline of how that dialectic
works, though it certainly does work. A mathematical model of it can be
based on the hypothesis that a given set of particulars is somehow itself
a category (or graph), i.e., that the appropriate ways of comparing the
particulars are given but that their essence is not. Then their
"natural structure" (analogous to cohomology operations) is an
abstract general and the corresponding concrete general receives a
Fourier-Gelfand-Dirac functor from the original particulars. That
functor is usually not full because the real particulars are infinitely
deep and the natural structure is computed with respect to some
limited doctrine; the doctrine can be varied, or "screwed up or down" as
James Clerk Maxwell put it, in order to see various
phenomena.

From: baez@math.ucr.edu 
>To: categories@mta.ca (categories) 
>Subject: categories: Sketches and Platonic Ideas 
>Date: Mon, 3 Dec 2001 19:42:40 -0800 (PST) 
>
>Toby Bartels writes: 
>
>> There could be multiple ideas that generate the same sketch; 
>> how do we decide which is the correct idea among equivalent ones? 
>> OTOH, if we take equivalence classes of ideas, then we're taking sketches. 

...

>> who has the right idea? 
>
> I'm confused: in my understanding, a sketch basically amounts to 

...

>By the way, in response to Lawvere's comments: 
>
> My use of the term "Platonic idea of X" for the free 

...

>versus concrete particulars. 

>Best, 
>John Baez