[FOM] Math vs. Logic
pratt at cs.stanford.edu
Sun Mar 4 02:19:16 EST 2012
The Edwards letter Dana posted prompts me to ask, do the foundations of
produced mathematics differ in any essential way from those of its
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
* 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
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
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. ;)
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