This is the material which-to use a metaphor from computer science-is in the high speed memory or storage cells. What is done, created, practiced, at any given moment of time can be viewed in two distinct ways: as part of the larger cultural and intellectual consciousness and milieu, frozen in time, or as part of a changing flow of consciousness.
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What was in Archimedes' head was different from what was in Newton's head and this, in turn, differed from what was in Gauss's head. It is not just a matter of "more," that Gauss knew more mathematics than Newton who, in turn, knew more than Archimedes. It is also a matter of "different. Newton knew the statement as a deduction and as application, but he might also have pondered the question of whether the statement is so true, so bound up with what is right in the universe, that God Almighty could not set it aside.
Gauss knew that the statement was sometimes valid and sometimes invalid depending on how one started the game of deduction, and he worried about what other strange contradictions to Euclid could be derived on a similar basis. Take a more elementary example.
Counting and arithmetic can be and have been done in a variety of ways: by stones, by abacuses, by counting beads, by finger reckoning, with pencil and paper, with mechanical adding machines, with hand-held digital computers. Each of these modes leads one to a slightly different perception of, and a different relationship to, the integers.
If there is an outcry today against children doing their sums by computer, the criers are correct in asserting that things won't be the same as they were when one struggled with pencil and paper arithmetic and all its nasty carryings and borrowings. They are wrong in thinking that pencil and paper arithmetic is ideal, and that what replaces it is not viable. To understand the mathematics of an earlier period requires that we penetrate the contemporary individual and collective consciousness. This is a particularly difficult task because the formal and informal mathematical writings that come down to us do not describe the network of consciousness in any detail.
It is unlikely that the meaning of mathematics could be reconstructed on the basis of the printed record alone. The sketches that follow are intended to give some insight into the inner feelings that can lie behind mathematical engagement.
Rather, we mean to describe the most mathematician-like mathematician, as one might describe the ideal thoroughbred greyhound, or the ideal thirteenth-century monk. We will try to construct an impossibly pure specimen, in order to exhibit the paradoxical and problematical aspects of the mathematician's role. In particular, we want to display clearly the discrepancy between the actual work and activity of the mathematician and his own perception of his work and activity.
The ideal mathematician's work is intelligible only to a small group of specialists, numbering a few dozen or at most a few hundred. This group has existed only for a few decades, and there is every possibility that it may become extinct in another few decades.
However, the mathematician regards his work as part of the very structure of the world, containing truths which are valid forever, from the beginning of time, even in the most remote corner of the universe. He rests his faith on rigorous proof; he believes that the difference between a correct proof and an incorrect one is an unmistakable and decisive difference. He can think of no condemnation more damning than to say of a student, "He doesn't even know what a proof is.
In his own work, the line between complete and incomplete proof is always somewhat fuzzy, and often controversial. To talk about the ideal mathematician at all, we must have a name for his "field," his subject. Let's call it, for instance, "non-Riemannian hypersquares. He studies objects whose existence is unsuspected by all except a handful of his fellows. Indeed, if one who is not an initiate asks him what he studies, he is incapable of showing or telling what it is.
It is necessary to go through an arduous apprenticeship of several years to understand the theory to which he is devoted.
Only then would one's mind be prepared to receive his explanation of what he is studying. Short of that, one could be given a "definition," which would be so recondite as to defeat all attempts at comprehension. The objects which our mathematician studies were unknown before the twentieth century; most likely, they were unknown even thirty years ago. Today they are the chief interest in life for a few dozen at most, a few hundred of his comrades. He and his comrades do not doubt, however, that non-Riemannian hypersquares have a real existence as definite and objective as that of the Rock of Gibraltar or Halley's comet.
In fact, the proof of the existence of nonRiemannian hypersquares is one of their main achievements, whereas the existence of the Rock of Gibraltar is very probable, but not rigorously proved. It has never occurred to him to question what the word "exist" means here. One could try to discover its meaning by watching him at work and observing what the word "exist" signifies operationally.
In any case, for him the non-Riemannian hypersquare exists, and he pursues it with passionate devotion. He spends all his days in contemplating it. His life is successful to the extent that he can discover new facts about it. He finds it difficult to establish meaningful conversation with that large portion of humanity that has never heard of a non-Riemannian hypersquare.
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This creates grave difficulties for him; there are two colleagues in his department who know something about non-Riemannian hypersquares, but ope of them is on sabbatical, and the other is much more interested in non-Eulerian semirings. He goes to conferences, and on summer visits to colleagues, to meet 39 Varieties of Mathematical Experience people who talk his language, who can appreciate his work and whose recognition, approval, and admiration are the only meaningful rewards he can ever hope for.
At the conferences, the principal topic is usually "the decision problem" or perhaps "the construction problem" or "the classification problem" for non-Riemannian hypersquares. This problem was first stated by Professor Nameless, the founder of the theory of non-Riemannian hypersquares.
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It is important because Professor Nameless stated it and gave a partial solution which, unfortunately, no one but Professor Nameless was ever able to understand. Since Professor Nameless' day, all the best non-Riemannian hypersquarers have worked on the problem, obtaining many partial results. Thus the problem has acquired great prestige. Our hero often dreams he has solved it. He has twice convinced himself during waking hours that he had solved it but, both times, a gap in his reasoning was discovered by other non-Riemannian devotees, and the problem remains open.
In the meantime, he continues to discover new and interesting facts about the non-Riemannian hypersquares. To his fellow experts, he communicates these results in a casual shorthand. There he piles up formalism on top of formalism. Three pages of definitions are followed by seven lemmas and, finally, a theorem whose hypotheses take half a page to state, while its proof reduces essentially to "Apply Lemmas to definitions A-H.
It gives the impression that, from the stated definitions, the desired results follow infallibly by a purely mechanical procedure. In fact, no computing machine has ever been built that could accept his definitions as inputs. To read his proofs, one must be privy to a whole subcul- 40 The Ideal Mathematician ture of motivations, standard arguments and examples, habits of thought and agreed-upon modes of reasoning.
The intended readers all twelve of them can decode the formal presentation, detect the new idea hidden in lemma 4, ignore the routine and uninteresting calculations of lemmas 1, 2, 3, 5, 6, 7, and see what the author is doing and why he does it. But for the noninitiate, this is a cipher that will never yield its secret. If heaven forbid the fraternity of non-Riemannian hypersquarers should ever die out, our hero's writings would become less translatable than those of the Maya.
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The difficulties of communication emerged vividly when the ideal mathematician received a visit from a public information officer of the University. I appreciate your taking time to talk to me. Mathematics was always my worst subject. That's O. You've got your job to do. I was given the assignment of writing a press release about the renewal of your grant. The usual thing would be a one-sentence item, "Professor X received a grant of Y dollars to continue his research on the decision problem for non-Riemannian hypersquares. First of all, what is a hypersquare? I hate to say this, but the truth is, if I told you what it is, you would think I was trying to put you down and make you feel stupid.
The definition is really somewhat technical, and it just wouldn't mean anything at all to most people. Would it be something engineers or physicists would know about?