UNORTHODOXIES: Most mathematicians have had the following experience
and those whose activities are somewhat more public have had it often:
an unsolicited letter arrives from an unknown individual and contained
in the letter is a piece of mathematics of a very sensational nature.
The writer claims that he has solved one of the great unsolved
mathematical problems or that he has refuted or that he has refuted
one of the standard mathematical assertions. In times gone by. circle
squaring was a favourite activity: in fact, this activity is so old
that Aristophanes parodies the circle squarers of the world. In more
recent times, proofs of Fermat's "Last Theorem" have been very
popular. The writer of such a letter is usually an amateur, with very
little training in mathematics. Very often he has a poor understanding
of the nature of the problem he is dealing with, and an imperfect
notion of just what a mathematical proof is and how it operates. The
writer is usually male, frequently a retired person with leisure to
pursue on his mathematics, often he has achieved considerable
professional status in the larger community and he exhibits his status
symbols within the mathematical work itself.
Very often the correspondent not only 'succeeds' in solving one of the
great mathematical unsolvables, but has also found a way to construct
an antigravity shield, to interpret the mysteries of the Great Pyramid
and of Stonehenge, and is well on his way to producing the
Philosopher's Stone. This is no exaggeration.
If the recipient of such a letter answers it, he will generally find
himself entangled with a person with whom he cannot communicate
scientifically and who exhibits many symptoms of paranoia. One gets to
recognize such correspondents on sight, and to leave their letters
unanswered, thus unfortunately increasing the paranoia.
I have on my desk as I write a paper of just this sort, which was
passed on to me by the editor of one of the leading mathematical
journals in the United States. For self-protection I shall change the
personal details, retaining the flavor as best I can. The paper is
nicely and expensively printed on glossy stock and comes from the
Philippines. It is written in Spanish and purports to be a
demonstration of Fermat's Last Theorem. There is a photograph of the
author, a fine-looking gentleman in his eighties, who had been a
general in the Philippine army. Along with the mathematics, there is a
lengthy autobiography of the author. It would appear that the author's
ancestors were French aristocrats, that after the French Revolution
the cadet branch was sent to the East, whence the family made its way
to the Philippines, etc. There are also included in this paper on
Fermat's Last Theorem, nice engravings of the last three reigning
Louis of France and a long plea for the restoration of the Bourbon
dynasty. After page one, the mathematics rapidly wanders into
incomprehensibility. I spent ten minutes with this paper; your average
editor would spend less. Why? The Fermat "Last Theorem" is at the time
of this writing a great unsolved problem. Perhaps the man from the
Philippines has solved it. Why did I not examine his work carefully?
There are many types of anomalous or idiosyncratic writing in
mathematics. How does the community strain out what it wants? How does
anyone recognise brilliance, genius, crankiness, madness? Anyone can
make an honest error. Shortly after World War II, Professor Hans
Rademacher of the University of Pennsylvania, one of the leading
number theoreticians of the world, thought he had proved the famous
Riemann Hypothesis. The media got wind of this news and an account was
published in Time magazine. It is not often that a mathematical
discovery makes the popular press. But shortly thereafter, an error
was found in Rademacher's work. The problem is still open as these
words are being written.
This is an example of incorrect mathematics produced within the bounds
of mathematical orthodoxy - and detected there as well. This happens
to the best of us every day of the week. When an error is pointed out,
one recognises it as an error and acknowledges it. This type of
situation is dealt with routinely.
At the opposite pole, there is the type whose psychopathology has just
been described above. This type of writing is usually dismissed at
sight. The probability that it contains something of interest is
extremely small and it is a risk that the mathematical community is
willing to take. But it is not always easy to draw the line between
the crank and the genius.
Davis and Hersh, The Mathematical Experience (1981)
The authors go on to talk about Ramanujan and Hardy, Grassman and
Wronski, but clearly believe that these examples are exceptions. I
mention this article because in recent weeks and months we appear to
have had a number of Philippine generals wander by. I believe that
they fit the profile described quite well.
Apart from this, I recommend the book as a thoughtful approach to the
background of mathematics, including more on the unorthodoxies above
and a description of the "ideal mathematician". The link below is to
the UK Amazon decription of the book. Unfortunately it only seems to
be available in hardcover for £60 (over $100) but I am sure libraries
and second-hand shops will be able to help you.
http://www.amazon.co.uk/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.co.uk%2FMathematical-Experience-Philip-J-Davis%2Fdp%2F0817637397%2Fsr%3D8-1%2Fqid%3D1161455819%3Fie%3DUTF8%26s%3Dbooks&tag=thesmirks-21&linkCode=ur2&camp=1634&creative=6738
Ian
--
Ian W Halliday, BA Hons, SA Fin, ATMG, CL
+44 772 546 2965 (GMT+1)
https://www.linkedin.com/in/ianwhalliday
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