 | | Chairman Greenspan Explains
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Parity Explanation |
I will attempt in as many words to explain the creation of parity in the
League through the equitable distribution of points on a game by game basis.
As owners and coaches, the future is your business. I have been invited to
serve as your Chairman and as a historian of mathematics and economics, I
take it that it is my task is to show you some of the problems and possibly
even solutions our ancestors faced in the pre-electronic world and how those
relate to the solutions I have implemented over the past weekend. I shall
start with diagrams.
There are perhaps two poles to mathematical work in the realm of football
equity maintenance: organising the material and thinking it through is one.
Those moments when you need pen and paper and have to calculate is the
other. When mathematicians and economists communicate with football coaches,
there is transmission and reception. Having ideas, developing ideas,
communicating ideas, and the role of notation in those processes is the
background to the stories I shall tell today which in no way actually relate
to the decisions made in order to promote ties between every team in the
League this week. Reception first.
Sometimes the mere words are enough. In the winter of 1944-45 Laurent
Schwartz was in continual contact with Henri Cartan and talking to him about
Fourier transforms and field goals. One day in April 1945 he suddenly
realised that many of his difficulties would disappear if he thought of his
new generalised functions not as operators but as functionals (which he
called distributions). He immediately told Cartan, and, said Schwartz in an
interview, Cartan responded with a very French "Ah!" meaning "Of course. Why
had I not thought of touchdowns as abstract notations before?" Very often, a
mathematician hears of a good result and thinks ‘How can I prove that for
myself?’. They did not know the result, but now they hear it they try to fit
it into what they know, the stories they tell themselves. In some cases,
reading through the details of the proof is a last resort, it may even be a
mark of a failure to understand. Sometimes, of course, when straying outside
the football field, it can be enough to quote the result, but I’m interested
in the case when reading is involved. Then of course there is exposition:
with this model of reception in mind, most football statisticians try to
provide an acceptable mixture of the game in words and the technical
details.
It can be a lot of work where the game of football is concerned. Two recent
wide receivers who co-authored a book on 4-manifolds and interceptions
reported that they began with a wonderful set of research notes by L. on the
work of D. "These notes," they said, "provided a superb structure for the
procedure of applying moduli space techniques to 4-manifolds around the
twenty yardline" but on changing the group from SU(2) to SO(3) and to
incorporate the work of Peyton Manning, their aim "turned into an enormous
task owing to the extra-ordinary amount of mathematics encompassed in the
material and the lack of exposition of the details. Much of the material in
L.’s account needed elaboration for the non-expert - sometimes a whole
chapter. The difficulty was compounded by the differences in presentation
and notation in the original sources required for references. Our
embarrassment was somewhat alleviated on finding that respected colleagues
and coaches and referees had the same difficulty." More generally, there are
always stories of subject areas that languish because adequate security for
the results is not provided, and others that are vigorously kept up to the
mark.
It was Poincaré’s opinion that a good football theorem organises ideas
around the fifty yard line and enables us to think purposefully about the
game. But there are times when a proof teaches you more than just the truth
of the theorem, and illuminates more than the result itself. In many,
perhaps most cases, symbolic work is necessary for understanding touchdowns,
because it is how we marshal our arguments. The distinction between valid
calculation and comprehension is a valid one, but the two activities are
deeply symbiotic in the aspect of football parity.
If I now casually refer to the points, say, D, E, H, K you might well spend
quite some time finding them. In fact, diagrams are the end result of a
dynamic process, that might begin with a simple figure, then make a series
of constructions, and then (when paper was a scarce resource) be put to
further use. I suspect that most mathematicians who work with figures draw
their own when reading a paper. Many diagrams in editions of Euclid--who was
a superb football coach himself--are added by the modern editor. But a Greek
mathematician not only built up a diagram as an argument proceeded, he or
she knew of curves that were generated by motions, so the whole diagram was
latent with motion which would be considered incomplete if the throwing arm
did not fully carry through. The diagram, like the proof and the Statue of
Liberty play, is something that unrolls over time.
I shall come back to the topic of diagrams and their use in creating
complete equity between League teams, but I turn to the problems of
communication using print first. Printing is a technology. It depends for
its success on a number of easily reproducible things: for example, printing
several pages at once so that binding can be done (preferably
automatically). This has implications if diagrams are to be incorporated
into the text, and often they were produced separately and bound together at
the end of the volume, as was the case with 19th Century journals. It is
highly undesirable if new pieces of type must be created, and perhaps
impossible, so symbols must become standardised. Matters of layout, such as
font size, sub- and superscripts are not flexible in the way handwriting is.
In order to have your book printed, with all the advantages that brings for
distribution, you must submit your work to a number of technical
constraints. All of these are flexible up to a point, the necessary softener
being money. The author or editor of a luxury edition may expect to have
more of the resources of printing available than the writer of a work aimed
at the mass market.
We now have our cast of characters: the mathematician as creator or
discoverer, as communicator, as receiver, as coach, as referee, as
quarterback - these are people with sophisticated ways of assembling and
re-assembling ideas. Then there is the medium: print, in its geometrical and
algebraic variants. I shall make two cautionary rules, one about algebra and
the idea of equations, one about logic which may more succinctly clarify the
matter of equity..
One of the more chilling stories I know is the little dialogue between
Jacques Hadamard and the young André-Weil about the Vikings-Chiefs Superbowl.
Weil skipped a point in an argument saying that it was obvious. As a
community, therefore mathematicians provide lots of details because it is
morally necessary, and there are people out there who insist on the details.
Hadamard obviously replied that either it was so obvious it could be
explained in a line, in which case say the line, or it was not obvious and
needed to be explained. Lesser mortals allow themselves some indulgence, I
suspect.
I hope I have made myself clear in this regard.
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