FOM: I thought this exchange would interest this list.

[Vaughan R. Pratt]

>Gardner review of Hersh, forwarded by Martin Davis:
|      Hersh grants that there may be aliens on other planets who do 
| mathematics, but their math could be entirely different from ours. The 
| "universality" of mathematics is a "myth." "If little green critters 
| from Quasar X9 showed us their textbooks," Hersh thinks it doubtful that 
| those books would contain the theorem that a circle's area is pi times 
| the square of its radius. Mathematicians from Sirius might have no 
| concept of infinity because this concept is entirely inside our skulls. 
| It is as absurd, Hersh writes, to talk of extraterrestrial mathematics 
| as it is to talk about extraterrestrial art or literature. 
|     With few exceptions, mathematicians find these remarks incredible. 

A pity Martin Gardner is not on the FOM mailing list, where he could
read such postings as

>I have never really understood
>the way category theorists seem to view the rest of mathematics, not
>to mention the rest of human knowledge.  I know that some topos
>theorists such as Peter Freyd have sometimes asserted that category
>theory or topos theory is or can be turned into a good foundational
>theory for all of mathematics or at least a good portion of it.  I
>have never understood how this can be.  It seems wildly exaggerated.
>Could somebody please try to explain it clearly?  (I.e., no
>mumbo-jumbo.)

and learn that little green critters, with textbooks full of mysterious
definitions and theorems purporting to be the true view of mathematics,
are already among us and even holding conferences in nice places like
Montreal and Vancouver where they engage in extraterrestrial
mathematics.

As a grue critter, so to speak, I find the FOM view of mathematics just
as extraterrestrial.  A lot of you seem to believe that everything is a
set, and claim not to understand the point of view of those who deny
this.  As an interesting corollary, size (the only property a set can
have up to isomorphism) looms large among what you regard as the most
important questions of foundations.

The corresponding point of view in physics would be to say that
everything is a gas, and that liquids and solids are simply gases with
metaphysical structure that can be understood in terms of laws about
certain gases.

Clearly there is a perspective from which one *can* say that everything
massive is the sum of its quarks, leptons, and gluons, and that light
consists entirely of photons.  But physicists collectively no longer
view this as *the* right primitive basis for the reduction of physics.
Everything, whether its rest mass is zero or nonzero, has a dual nature
that allows it to be equally well understood as consisting of particles
or waves, and optimally as some blend thereof.

The corresponding point of view in mathematics *should* be that
everything can be understood equally well as consisting of sets or
Boolean algebras, and optimally as a blend thereof.  In reducing
mathematics to sets you commit the same error a physicist would make in
reducing physics to particles.  Interestingly, both errors can be
documented by essentially the same underlying machinery.

MG is a wonderful expositor of elementary mathematics.  But he is not a
mathematician, complex numbers are just a tad beyond what he is willing
to expound on, and his pronouncements on both the directions of
mathematics today and its philosophers are without basis.  Even on
earth, let alone Quasar X9, mathematicians disagree wildly among
themselves as to "was sind und was sollen die Zahlen".

Mathematics is no different than physics in this regard.  Were a 19th
century physicist to try to make sense of today's QM texts, he would
think he had been transplanted to Quasar X9.  Today's mathematicians
are at just as much risk of finding a brave new world of mathematics if
they time travel a century or two forward.  Relatively stable concepts
with a long half-life like pi and e will remain, but there will be many
new, simple, and widely understood structures that are as unknown to
today mathematicians as matrices (not to be confused with determinants)
were to 19th century mathematicians, and that will have displaced many
by-then-forgotten concepts that we teach today.

Mathematics is like art and fashion, it is defined by its producers and
consumers and it has its trends and staples.  Everything changes in
mathematics, some faster than others.

Vaughan