Re: pragmatic foundation

[Vaughan Pratt <pratt@cs.stanford.edu>]

Colin McLarty wrote:
> [responding to Manin thoughts] I myself am also confident that people will calm down and notice that
> axiomatic categorical foundations such as ETCS and CCAF work perfectly
> well, in formal terms, and relate much more directly to practice than
> any earlier foundations.

Thanks, Colin.  There I was nicely calmed down and then you got me all
worked up again.  :)

I prefer the Euclidean plane over sets as a suitable starting point for
understanding mathematics.  What advantage is there to making geometry
rest on set theory as opposed to vice versa?

What is wrong with starting from a geodesic space as a place where it is
always determined, given two points, what is the next one, subject to
some simple equational principles?  This is a common basis for the
second postulate of Book I of Euclid's *Elements*, Newton's first law
of motion, Einstein's theory of general relativity that a falling body
is merely following a geodesic in a space curved by a nearby mass, and
the notion of Hamiltonian flow of a vector field for an energy function
defined on the cotangent space of a manifold as an expression of the
principle of least action.

In this framework a *set* is simply a geodesic space where the next
point after x and y is x.  (So if I ask what is the next element in the
sequence 3,4,... the answer is 3, not 5.)

More on this at http://boole.stanford.edu/pub/consgeom.pdf .  A geodesic
space or geode, aka kei, is related to a quandle (see
http://en.wikipedia.org/wiki/Quandle ), the difference being that for
abelian groups, quandles are merely sets whereas flat geodes (those
satisfying Euclid's 5th postulate) form a symmetric monoidal closed
category fully and reflectively extending Set (properly of course).
Moreover its subdirect irreducibles are those of Ab except for those of
even order as per the last slide.  Quandles are for knot theory, not
geometry.

The difference between sets and geodesic spaces in foundations is like
the difference between scales and Fur Elise for piano students.  Both
are good ways to get started but the second is more interesting.
(Apologies again to Eduardo for my impenetrable writing, in this case I
can only counsel patience since these ideas seem to come with a certain
viscosity that inhibits any royal road of the kind Eduardo would like.)

Best,
Vaughan


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