our attention to the beauty of the theo-
ries posited to date, drawing a parallel
to Maxwell’s equations from electro-
magnetism. Those equations have an
underlying physical underpinning, but
they also can be derived exclusively
from mathematical first principles. De-
spite the complexity of the concepts
they describe, they are, from the per-
spective of calculus, surprisingly simple
and straightforward. At the time of the
formulation of the equations, the un-
derlying instinct that simplicity plus
beauty coexists with truth motivated
and encouraged Maxwell and his peers.
Frenkel sees a parallel perspective with
the Langlands Program and projects that
it is headed to pay dirt.
I would have gone in a different direc-
tion. These discoveries and their residue
in pure mathematics (complex numbers,
sets, groups, symmetry, dimensions,
loops) are sufficiently exciting on their
own to justify a book. But they also are
avatars of truth.
In his monthly column in Natural His-
tory, the late paleontologist Stephen Jay
Gould frequently mentioned the concept
of a spandrel in evolving animal spe-
cies. The original meaning of the word
comes from the (often colossal) pieces
of architecture that held up a medieval
cathedral in the preliminary phases of
its construction but were superfluous
in the final product. Our appendix is an
evolutionary spandrel—necessary for an
ancestral species but unnecessary resi-
due for us.
The spandrel of the mathematical
voyage of discovery that Frenkel de-
scribes is fundamental pieces of math
itself. Godspeed to pioneers like Frenkel,
but not just because the final result will
be beautiful. Their leftovers are what the
rest of us use every day.
PAUL CONLIN is a health actuary at
Aetna in Hartford, Conn., and works at
home in Lake Zurich, Ill.
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