[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 

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 

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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