[FOM] Reply to Insall on "100-year thesis"
joeshipman@aol.com
joeshipman at aol.com
Mon Aug 20 01:35:10 EDT 2007
Insall:
****
In the year 2100, will proofs of the four-color theorem, the
classification of finite simple groups, and Fermat's Last Theorem be
accessible to undergraduates, by your thesis?
****
Shipman (excerpted):
****
Yes, unless there is another counterexample (Dirichlet's theorem is the
only unarguable counterexample so far, though I am currently
researching this to see how much of it can be made accessible).
The proof of the 4-color theorem is already accessible to
undergraduates now in the same sense it is accessible to any other
mathematician ...
Wiles's proof is being rapidly absorbed into an increasingly rich
theory, I fully expect it to be accessible to undergraduates by 2100.
Finite group theory is a good example of a field of mathematics with
lots of long messy unilluminating proofs, and I have no idea HOW the
proofs will eventually be simplified, but I discussed this with an
expert on the subject, John Conway, the other day, and he said he feels
sure that much much better proofs exist and that we have just not,
collectively, been smart enough to find them so far....
****
Insall (excerpted):
****
I apparently misinterpreted your 'thesis' as a claim of (almost)
universal simplicity of what mathematicians can do, and of an elevation
in conceptual ability of up and coming undergraduates. Perhaps by
'anything the best human mathematicians can do will be doable by
undergraduates in 100 years',
you really meant
'anything the best human mathematicians HAVE DONE UP TO NOW will be
doable by undergraduates in 100 years''
Sorry, but I take a 'thesis' to be like the ones attributed to Turing
and Church, especially when they refer to what CAN be done. That is,
'CAN BE DONE by the best mathematicians' would, I say, parallel 'CAN BE
COMPUTED by a machine'. In the theses of Turing and Church, 'CAN BE...'
is interpreted, as far as I can tell, as indicating an upper bound on
what 'EVER WILL BE...' done by a (Turing) machine.
******************************
Matt, I think you're still missing my point.
I am observing that almost anything mathematicians prove, historically,
gets "boiled down" and simplified and embedded in general theories in
such a way that it becomes much easier to learn over time. I'm not
claiming that today's undergraduates are smarter than the
undergraduates of previous centuries, but they have the benefit of
learning mathematics in a way that has been polished and improved so
that they can "reach" particular results much quicker than when the
original discoverers of those results hacked their way out to what was
then the mathematical "frontier".
This observation, framed as a "100-year rule", is sufficiently
well-supported, even though there may be a couple of exceptions, that
it says something general about the structure of (humanly accessible)
mathematics, and I feel justified in proposing it as a "thesis",
meaning a proposal I will defend but which I encourage attempts to
refute.
This has nothing to do with PA-theorems that have only extremely long
proofs which human mathematicians would probably never reach anyway (if
they reach them aided by computers, as in the case of the 4-color
theorem, I am only proposing that the part that humans understand will
be boiled down to undergraduate level within 100 years, NOT that the
part that only computers can do will be eliminated!).
If future mathematics changes in such a way that my thesis is
frequently wrong, that is not a good thing, it means that mathematics
will become more esoteric, fragmented, and difficult to achieve new
results in, because it will take longer for researchers to "reach the
frontier" than it historically has, and fewer will be willing to
attempt it. But my impression is that it does not take much longer to
reach the frontier than it ever has; graduate students are generally
there by their 4th year, and good ones get there sooner, and this seems
to have been true for a long time.
Of course, the total amount of mathematics is increasing without limit,
while there must be a limit to how much the more elementary parts of
the subject can be compressed, so one would expect SOME increase in the
"time to reach the frontier" over time, which is only partially reduced
by increasing specialization -- but I don't see that happening any time
soon.
-- JS
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