[FOM] Brouwer on mathematical operations
Vaughan Pratt
pratt at cs.stanford.edu
Sun Dec 28 00:33:17 EST 2008
The largely speculative nature of our understanding of the mathematical
activity of the mind notwithstanding, I for one find Brouwer's position
eminently reasonable. Whatever clues the relayed thought processes of
others give us, and whatever our conscious insights into our own
thinking, we have no way of judging the full depth of those mental
activities ultimately responsible for problem solving. But the odds
that deep down the solving process mimics the eventual solution, as some
automated proof discovery procedures seem to picture the process, seem
pretty small to me.
Chess masters sometimes review their just-won game orally, commenting on
why they made this move and that. But one is left with the feeling that
absorbing these reasons cannot possibly make one a better chess player,
because the master has not revealed the real rationales behind those
moves, not so much because of any proprietary interest in them as for
lack of any personal insight into them. The real reasons seem likely to
be buried far beyond conscious access.
Likewise even if the successful mathematician can explain his or her
methods in terms of the written proof, one is left with a similar
feeling: maybe he's just making up a rationale as he reads back his
written proof. (This was my reaction to Polya's "How to Solve It" when
I read it in 1962 - I was unable to see how it could help a good problem
solver become a better one. The techniques seemed far too shallow, much
as we might view a resolution theorem prover's techniques shallow today.)
Do we even know anything at all about mathematical thought processes?
One thing one can observe is that there seems to be a spectrum
(dichotomy?) of mathematicians. At one end are the symbolic thinkers,
who might solve problems by matching formulas to patterns and reducing
them via rules. At the other are the visual thinkers who seem able to
simply "see what's going on." Certainly in algebra we are taught the
former mode of thought, and even visual thinkers may start out in life
solving quadratic equations via the square root of the discriminant
before realizing that (for them anyway) it helps to picture a parabola
floating around somewhere near the X-axis, with one plus the sign
(-1,0,1) of the discriminant giving the number of points of intersection
therewith.
If genuine subconscious mathematical activity really can be divided
along symbolic and visual lines, that right there constitutes an insight
into mathematical thought, namely that not all mathematicians depend on
the same intuitions. To try to collect the mathematical activities of
the mind under a single umbrella of intuitionism (if that was Brouwer's
original hope) would seem to ignore that dichotomy. At the very least
one should ask which kind of mathematical thought various flavors of
intuitionism claim to cater for. And do all visual thinkers work from a
similar stock of images?
If we assume with Brouwer that the "mathematical activity of the mind"
is a significantly different activity from written mathematics, the only
point left to argue about is which of the two is more deserving of the
epithet "essence of mathematics." To me the written solution is
essential in its own way; mathematics would never have arrived at its
present state had it evolved as something each of us has to work out in
private without both the obligation and the ability to share our
findings. The essence of mathematics as a discipline is embodied in the
documentation of its successes as much as in the discovery process.
Interestingly the following paper seems more directly informed by Polya
than Brouwer, yet indirectly more by Brouwer through its preference for
analysis over Polya's wider range of interests. (Polya was a noted
skeptic of Brouwer, and famously accepted a bet by Brouwer's defender
Weyl, that in 20 years Polya and a majority of representative
mathematicians would admit that the propositions "every bounded nonempty
sets of reals has a precise least upper bound," and "every infinite set
of numbers has a denumerable subset," contain vague concepts, but that a
natural interpretation makes them both false.
Wasburn-Moses, J. M. , 2004-10-21 "Solving and Solution: The Interplay
of Multiple Discourses of Mathematics" Paper presented at the annual
meeting of the North American Chapter of the International Group for the
Psychology of Mathematics Education, Delta Chelsea Hotel, Toronto,
Ontario, Canada Online <.PDF>. 2008-10-10 from
http://www.allacademic.com/meta/p117689_index.html
While the paper doesn't dive very far below the surface of mathematical
thought, I liked the title's distinction between "solving" and
"solution," one that is likely to remain wide open for fruitful
investigation for many more decades.
Vaughan Pratt
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