Re: Sketches and Platonic Ideas

Re: Sketches and Platonic Ideas

[F. William Lawvere]

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 

Re: Sketches and Platonic Ideas

[Michael Barr]

There are a number of definitions of sketch around, some of which require
it to be a category with finite products.  In one of Ehresmann's (and
Bastiani's, I believe) there is mentioned the possibility of its being
what they called a quasicategory (or some such substructure term) in which
composition is a partly defined multi-ary operation (in other words, fgh
could be defined without fg or gh being defined).  Charles and I realized
that this was equivalent to what we called a graph with diagrams, which
seemed a more useable notion.  So what we called a sketch was a graph with
diagrams as well as certain cones and cocones that were singled out to be
taken to limits and colimits, resp.  Peter Johnstone criticized us for
doing the equivalent of replacing groups by generators and relations,
which is correct, but it was a conscious decision and there were reasons
for it.  I had never heard the term "idea" in this connection or we might
have used it.  But anyway, "sketch" is used in different ways and I guess
Charles and I contributed to this, but didn't create it. 

On Mon, 3 Dec 2001 baez@math.ucr.edu wrote:

> 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.
> > For example, one could define the idea of multiplication in a monoid
> > as a binary operation and a nullary operation
> > or alternatively as an operation on finite tuples.
> > The former is more common, but I prefer the latter;
> > who has the right idea?
> 
> I'm confused: in my understanding, a sketch basically amounts to
> a way of giving generators and relations for a category with products, 
> Different sketches give the same category with products, not vice versa.  
> Your example gives two sketches, but one category with products.  In
> this sense, a sketch is more like an "idea" than you seem to be giving
> it credit for.
> 
> By the way, in response to Lawvere's comments:
> 
> My use of the term "Platonic idea of X" for the free 
> category/category with products/monoidal category/2-category/whatever 
> on an X was not meant as an endorsement of "Platonism" in the philosophy
> of mathematics - especially since "Platonism" means many things to
> many people.  It was also not meant to suggest that Plato had this idea.
> It was basically meant to get people thinking about abstract generals
> versus concrete particulars.
> 
> Best,
> John Baez
> 
> 
> 
> 
> 
>