[FOM] Math vs. Logic

[Vaughan Pratt]

The Edwards letter Dana posted prompts me to ask, do the foundations of 
produced mathematics differ in any essential way from those of its 
consumed counterpart?

The produced-consumed distinction I have in mind corresponds to the 
hard-easy distinction in roughly the same sense that coal mining as 
production is hard while coal burning as consumption is easy.  Pure 
mathematicians, and some applied mathematicians, bend over backwards to 
solve impossibly hard problems. (The Davis-Hersh article mentioned in 
passing by Edwards seems to be describing that subset of pure 
mathematicians that are creatively expanding the domain of 
barely-solvable problems by inventing new concepts that may or may not 
find actual applications some day.  They're the ones likely to have the 
hardest time motivating their work at a cocktail party or a PR 
interview, as D&H point out, perhaps with tongue in cheek.  But I would 
say they are valuable nonetheless, on average.)

My general impression is that mathematics is conducted along a spectrum 
from logic to algebra.  Mathematics becomes easy when certain functions 
have been identified, permitting dropping into the vernacular of 
algebra.  When the going gets tough, the tough have to turn to first 
order logic, which at heart is a way of naming functions by their 
properties.  When you say "for all x there exists y such that P(x,y)," y 
is your name for some functional dependency on x having P as its 
input-output relation, whose properties are as yet insufficiently well 
understood as to warrant giving that dependency a name.  Once all 
relevant dependencies have been named and adequately understood, one can 
shift from first order logic to algebra.

Physicists don't in general produce mathematics, they consume it. 
Physics tends to be packaged algebraically: this equals that using 
various functions.  Rarely will you find "for every x there exists a y" 
because in physics most such y's have already been given names. 
Physicists don't waste time with unskolemized mathematics.  (I can 
relate to Skolem because he was my great-grand-advisor.)

So where does category theory fit in that spectrum?  I would say that CT 
emphasizes functions over values, without however excluding the latter. 
  A morphism of the form f: X --> Y for arbitrary X and Y is a 
functional dependency on elements of X taking values in Y.

In the special case where X exhibits signs of rigidity, e.g. by having a 
narrowly prescribed set of endomorphisms (exactly one when X is the 
final object 1 of a topos, making f an atom), or the ring R as its set 
of endomorphisms (in the setting of abelian categories, with scalar 
multiplication realized as composition with those endomorphisms), such 
morphisms are what a first order logician would think of as elements of Y.

X need not be 1 itself, if it were 1+1 then one would view the elements 
of Y as pairs, but 1+1 would still exhibit signs of rigidity by having 
only four endomorphisms.

What CT has that first order logic lacks is a Duality Principle. 
Descartes' vision of a universe founded on dualism proved hard to 
swallow, leading to a monistic view admitting only body (Hume) or mind 
(Berkeley).  Hume maintained we are the sum of our parts, Berkeley that 
we are the product of our properties.

Whereas algebra is very much aligned with Hume, both first order logic 
and category theory strike a happy medium, albeit by quite different means.

* FOL quantifies over the same material entities as algebra, but unlike 
algebra admits the explicit notion of property, which is to be composed 
with entities to yield truth values chosen from a set of cardinality at 
least two.  "Explicit" in my mind is synonymous in this context with 
"kludge."

* CT exploits its intrinsic Duality Principle to do for the codomain Z 
in h: Y --> Z what element-hood of f: X --> Y did for X.  Whereas 
rigidity of X makes X a generic element and f: X --> Y an element of Y, 
rigidity of Z makes Z a generic property and h: Y --> Z that property 
enumerated over all elements of Y, what we customarily think of as a 
state vector.  The composite hf: X --> Z is ambiguously a generic 
element of the generic property Z and a generic property of the generic 
element X.  I find the evident duality here less of a kludge than first 
order logic's separate treatment of elements and properties, along with 
its unwarranted commitment to two truth values, or even the notion of 
truth itself!  Nature (or our understanding thereof) may make Hom(X,Z) 
small, but does not limit it to two morphisms.

In a topos, the setting for logic adopted by category theorists, X = 1 
(so values are atoms) while Z = Omega (the subobject classifier or 
generic property, whose atoms are truth values).  The topos Set of sets 
has two truth values forming the prototypical Boolean algebra, while the 
topos Grph of graphs has three truth values forming the smallest Heyting 
algebra not a Boolean algebra.  These constitute the truth values for 
the internal logic of respectively sets and graphs.

What does physics use?

Up to and including the first quarter of the 20th century, physicists 
would seem to have based their mathematics more on algebra than first 
order logic.

But then came quantum mechanics with its bras and kets and their 
complex-valued inner product.  This is neither first order logic nor 
algebra, given the weird contravariance of the bras.  And how should a 
complex number be understood as a truth value?

CT lets us make sense of this unprecedented mixed variance by thinking 
of X and Z as the ring (in fact field) of complex numbers, kets as 
elements (morphisms from X), bras as states (morphisms to Z), and their 
inner product as their composite yielding a complex number as an element 
of Z (or ambiguously a state of X).  Quantum mechanics makes truth a 
complex notion, as you will no doubt have already noticed, along with 
Richard Feynman who warned against believing that you understand quantum 
mechanics.

Disclaimer: this is not a widely held view of the relationships between 
applied mathematics, pure mathematics, physics, philosophy, foundations 
of mathematics, set theory, algebra, and category theory.  (The talk in 
this vein that I gave at the 2011 International Conference on Category 
Theory in Vancouver convinced me that Michael Barr was the only category 
theorist at the conference thinking along even vaguely similar lines!) 
It is just my idiosyncratic way of looking at all these things when 
taken together in order to view them as a coherent whole, without which 
I would assuredly go nuts.  You may go nuts yourself thinking this way 
if you try to reconcile the world with classical first order logic. 
Zap, that does not compute. ;)

Vaughan Pratt