How twisted and how weird, the way Rota – Rota’s writings, Rota’s ideas, even Rota’s recipe for pasta with tomatoes and garlic – and Rota’s Boston and Los Álamos – seems to be part of the background of my life ever since that first visit to Boston, back in 1993.
Around that time, we started dense, intense discussions with one of Rota’s MIT pupils – Mark E. – who was back then a student of Keisler. In 1993, we went with María Clara for the first time to Boston (I went to a logic meeting, she came with me and we both incredibly enjoyed that city, despite having had during the trip several glitches, including being almost thrown out of our first place because of a misunderstanding). Mark – who had been an MIT undergrad, opened the key to many Boston marvels to us. One of those marvels was the Rota way of looking at things – or at least, Mark’s version of it. Blended with incredibly passionate philosophical (anti-analytical, phenomenological, heideggerian) discussions (on the continuum, on art, on late Wittgenstein, on sufi and buddhist approaches to ego, or the lack of it, on Conway games, etc. etc.) was always a light (but deep) touch, a conciseness of expression, an essential playfulness. And an Italian obsession with things like cutting garlic the right way (crosswise, of course not the French way), sautéing the garlic at three different moments (so you get the flavor of roasted garlic plus the middle flavor plus freshly cut garlic at the same time).
Life changed for me after meeting Rota’s writings. Not just because of endless discussions with Mark – among the deepest and liveliest philosophy I have ever had the chance to glimpse – or the mixture with Rotesian pasta and wine (the fact that Mark was then so sensitive to philosophy and so utterly unsensitive to art was mind-boggling, yet triggered even better discussions – he actually forced us to read Heidegger for a tough argument we needed to pass through).
Rather, the fact that those writings have triggered so many things in myself and, I think, among some friends (certainly Alejo, Santiago and Fernando). Pondering phenomenological issues every single day (in and outside math), taking distance from things, objectivizing what should be subjective and learning to think in layers and Fundierung, as a kind of life process – never falling prey to reductionisms and always doing (without even noticing) eidetic variations on things that are presented to us in an opaque way, and are finally caught not by direct observation but by varying parameters, by putting them in different contexts.
I opened a book I had not looked at for years – Numbers, it’s called – by eight German authors including Ebbinghaus and Hermes – as I wanted to read some background for a second-semester class I am teaching on Numerical Systems, for math, statistics and physics majors. The book seems to contain deep insight on the way the natural numbers, the integers, the rationals, the reals, the complex numbers, quaternions, p-adics, Cayley numbers, surreals, infinitesimals arose [of course, I only get up to the complex numbers and some rings of polynomials in my class!].
In the book was a copy I made of Rota’s review of the book, in Advances. It contains true (explosive!) gems:
Perhaps the teaching of mathematics should be subjected to the strict and unbreachable subdivision of the ancient Greek mysteries. There is exoteric math, that we should and will teach to non-mathematicians, to their enjoyment and profit; and there is esoteric math, that should be reserved only for closed-door interaction among card-carrying mathematicians or would-be mathematicians.
In this otherwise laudable book, the two kinds of math are dangerously mixed and presented as if they were one and the same kind. This fatal mistake committed by the editors presents us with an opportunity to stress the expediency of books with strictly exoteric math and to avoid circulating among the wide public all esoteric parts of our subject.
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There is nothing more deadly for a non-professional mathematician than being taught a rigorous version of facts that he or she always held to be obvious.
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The two chapters on real and complex numbers are downright dangerous. Giving rigorous presentations of the obvious has always been, and always will be, the dark side of mathematics, the side that makes mathematics disliked and unwelcome among physicists and engineers, the side that from time to time threatens to do away altogether with the teaching of pure mathematics in schools and colleges.
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… speaking now of the exoteric side, the chapter on division algebras and topology tells every reader that mathematics has something new to contribute, something that the reader will find fascinating and did not even fathom.
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… the authors have missed the chance to introduce yet another generalization of the concept of number of which the public at large (even the mathematical public) is largely unaware, and that is the ring of ideles. This ring can be motivated starting with the p-adic numbers, which in turn can be motivated by the naive notion of “carrying to the right” … the fields obtained by carrying to the right are not homeomorphic as topological spaces for different bases p. The ring of ideles sets up a carrying algorithm (to the right) cleverly based on cyclic groups which does not play favorites among the p’s. Where else but in a book like this one would such a motivation for the ring of ideles be given? Certainly not in a book of algebra, where the author invariably does his or her best to conceal this fundamental motivation.
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A propos of Cantor, one does not understand why cardinal and ordinal numbers have been dropped in a book that (implicitly) pretends to treat “every” aspect of the concept of number; for example, a description of what is going on in the theory of large cardinals might have made fascinating reading.
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Who cares about the history of π?
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At a time when mathematics is suffering from a serious loss of status (and of faith), the circulation of this book among an already fed-up public is another time bomb to be defused, the sooner the better.
avn va - עולם 