[FOM] Corfield : Simpson :: front end : back end ?
Timothy Y. Chow
tchow at alum.mit.edu
Mon Oct 6 09:50:02 EDT 2003
Bill Taylor's example of complex analysis is one of a host of "unreasonably
effective" principles at the "front end" (fumbling and groping for a result)
rather than the "back end" (formally writing it up) of mathematics. Quantum
field theory and string theory have led to mirror symmetry, a new proof of
the Atiyah-Singer index theorem, and a whole host of other mathematical
bonanzas. Why should such nonrigorous methodology lead to such a banquet
of rigorously verifiable results? Treating primes as randomly distributed
numbers with a certain density leads to many accurate predictions in number
theory. Why does this work? The list goes on and on. Why do holomorphic
functions behave a lot more like polynomials than like C-infinity real
functions (cf. Serre's GAGA theorem)? Why is it that the simple advice
"When in doubt, differentiate" works so well?
It is not only nonrigorous heuristics that are unreasonably effective. The
concept of "finding the right level of generality" pervades mathematics.
Adeles and ideles were initially regarded by some as an unnecessary and
pretentious abstraction---why not stick to ideal theory? But they've long
since won the battle for acceptance. Ditto for schemes versus varieties.
Slightly more controversial are matroids---do they strike a resonant
frequency or are they a superfluous generalization of graphs and vector
spaces? More generality isn't always better; group theory is far more
interesting than semigroup theory.
The mental image that mathematicians have at the front end is much more
like a web than a tree. Vertical connections (a la the hierarchy of
concepts) are often less important than horizontal connections between
seemingly unrelated fields, e.g., Stanley finds a connection between toric
varieties and polytopes, and sparks fly. Unreasonably effective
principles play a key role in driving the discovery of these connections.
In case the philosophical interest, or "general intellectual interest,"
of the front end is not clear, let me suggest that the fact that the web
and the tree coexist peacefully may provide a model for finding a successful
synthesis of foundationalism and coherentism (using these terms the way
philosophers typically use them) and give insight into the structure of
human knowledge in general. Mathematical knowledge has the advantage of
being highly precise and is thus suitable for an initial case study.
The coexistence of the tree and the web might also provide some room for
dialogue between list 1 and list 2. List-2 folks emphasize the importance
of the web. If you show them a foundational treatment of their subject,
they will judge it based on how well they think it illuminates the web
structure. In other words, what interests them is not so much what atoms
you choose for building the universe, but whether you successfully
reconstruct the major concepts and crowning achievements. In that sense
the high-level phenomena are more "basic" than the low-level constructs
that they're built from.
I end with a disclaimer that I also haven't read Corfield's book and
hence it's likely that I am merely adding to the cacophony of uninformed
loudmouths. For this I apologize.
Tim
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