Meanwhile, if you're an energetic youngster with some compelling vision of how certain areas or problems should workan areaof mathematics, it may
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Over the years, I've had many different thoughts about this perspective. For me personally, it was truly decisive, in that I hadn't been very interested in number theory until I realized, almost with a shock, that the study of solutions to equations had been 'reduced' to the study of maps between spaces of a quite rigid sort. In recent years, I think I've also reconciled myself with the more classical view, whereby numbers are some kinds of algebraic gadjetsgadgets. That is, thinking about matters purely algebraically does seem to provide certain flexible modes of thought that can be obscured by the insistence on geometry. I've also discovered that there is indeed a good deal of variation in how compelling the inner picture of a fiber bundle can be, even among seasoned experts in arithmetic geometry. Nevertheless, it's clear that the geometric approach is important, and informs a good deal of important mathematics. For example, there is an elementary but important key step in Faltings' proof of the Mordell conjecture referred to as the 'Kodaira-Parshin trick,' whereby you (essentially) get a compact curve $X$ of genus at least two to parametrize a smooth family of curves
Now, I've mentioned already that this is far from a personal image of a mathematical object. But it still seems to be a good example of a very basic picture that you refrain from putting into words most of the time. If it really had been only a personal vision, it may even have been all but maddening, the schism between the clarity of the mental image and what you're able to say about it. Note that the process of putting the whole thing into words in a convincing manner in fact took thousands of pages of foundational work.
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Professor Thurston: To be honest, I'm not sure about the significance of competing mental imagesin this context. If I may, I would like to suggest another possibility. It isn't too well thought out, butI don't believe it to be entirely random either.
Many people from outside the areaseem to have difficulty understanding the picture I mentioned becausethey are intuitively suspicious of its usefulness. Consider a simplerpicture of the real algebraic curve that comes up when one studies cubic equations like$$E: y^2=x^3-2.$$There, people are easily convinced that geometry is helpful,especially when I draw the tangent line at the point $P=(3,5)$to produce another rational point. What is the key difference from the other pictureof an arithmetic surface and sections? My feeling is it has mainly to do with the suggestion thatthe point itself has a complicated geometry encapsulated by the arrow$$P:Spec(\Bbb{Z})\rightarrow E.$$That is, spaces like$Spec(\Bbb{Q})$ and $Spec(\Bbb{Z})$ are problematic and, after all, are quite radical.
In $Spec(\Bbb{Q})$, one encounters the absurdity that the space $Spec(\Bbb{Q})$ itself is just a point. So one has to go intothe whole issue that the point is equipped with a ring of functions,which happens to be $\Bbb{Q}$, and so on. At this point, people's eyes frequently glaze over, but not, Ithink, because this concept is too difficult or because it competes with some other view.Rather, the typical mathematician will be unable to see the point oflooking at these commonplace things in this way. The temptation arises to resort to persuasion by authority then (suchand such great theorem uses this language and viewpoint, etc.), but it's obviously betterif the audience can really appreciate the ideas through some first-hand experience, evenof a simple sort. I do have an array of examples that might help in this regard,provided someone is kind enough to be still interested. But how helpful they really are, I'm quite unsure.
At the University of Arizona, we once had a study seminar on random matrices and number theory, to whichI was called upon to contribute a brief summary of the analogous theory over finite fields.Unfortunately, this does involves some mention of sheaves, arithmetic fundamental groups, and some other strange things. Afterwards,my colleague Hermann Flaschka, an excellent mathematician with whom I felt I could speak easily about almost anything, commented thathe couldn't tell if the whole language just consisted of word associations or if some actualgeometry was going on. Now, I'm sure this was due in part to my poor powersof exposition. But further conversation gave me the strong impression that the questionthat really went through his mind was: 'How could it possibly be useful to think aboutthese objects in this way?'
To restate my point, I think a good deal of conceptual inhibition comes froma kind of intuitive utilitarian concern. Matters are further complicated by the important fact that this kind of conceptual conservatism is perfectly sensible much of the time.
By the way, my choice of example was actually motivated by the fact that it is quite likely to be difficult for people outside of arithmetic geometry, including many readers of this forum. This gives it a different flavor from the situations where we all understand each other more or less well, and focus therefore on pedagogical issues referring to classroom practice.
