*Indiscrete Thoughts*

That is a splendid 1996 book on mathematics and mathematical researchers, by Gian-Carlo Rota.  I found philosophical, mathematical, and also managerial insights on most of the pages.  It is playful and yet earnestly serious at the same time.  Here is one bit:

He [Alonzo Church] looked like a cross between a panda and a large owl.  He spoke in complete paragraphs which seemed to have been read out of a book, evenly and slowly enunciated, as by a talking machine.  When interrupted, he would pause for an uncomfortably long period to recover the thread of the argument.  He never made casual remarks: they did not belong in the baggage of formal logic.  For example, he would not say “It is raining.”  Such a statement, taken in isolation, makes no sense.  (Whatever it is actually raining or not does not matter; what matters is consistence.)  He would say instead: “I must postpone my departure for Nassau Street, inasmuch as it is raining, an act which I can verify by looking out the window.”

It is full of the sociology of everyday life, in mathematical communities that is, for instance:

How do mathematicians get to know each other?  Professional psychologists do not seem to have studied this question; I will try out an amateur theory.  When two mathematicians meet and feel out each other’s knowledge of mathematics, what they are really doing is finding out what each other’s bottom line is.  It might be interesting to give a precise definition of a bottom line; in the absence of a definition, we will give some typical examples.

…I will shamelessly tell you what my bottom line is.  It is placing balls into boxes, or as Florence Nightingale David put it with exquisite tact in her book Combinatorial Chance, it is the theory of distribution and occupancy.

The author fears the influence of philosophy on mathematics, which led to this paragraph:

Philosophical arguments are emotion-laden to a greater degree than mathematical arguments and written in a style more reminiscent of a shameful admission than of a dispassionate description.  Behind every question of philosophy there lurks a gnarl of unacknowledged emotional cravings which act as a powerful motivation for conclusions in which reason plays at best a supporting role.  To bring such hidden emotional cravings out into the open, as philosophers have felt it their duty to do, is to ask for trouble.  Philosophical disclosures are frequently met with the anger that we reserve for the betrayal of our family secrets.

Definitely recommended, the book also has some of the best and most concrete discussions of Husserl’s philosophy I have seen, along with a meta-account of such, and also there is a discussion of the exoteric and esoteric readings of cosmology and black holes and indeed mathematics too.  Here is further information on Gian-Carlo Rota the author.

For the pointer to the book I thank Patrick Collison.


'The author fears the influence of philosophy on mathematics'

Starting with Pythagoras, undoubtedly. Then there was that Descartes guy - wouldn't want him wandering over any Cartesian landscape, messing up the purity of essence of the X and Y axis. And the less said about Pascal, the better - mechanical calculators are clearly the bane of pure mathematicians.

My thoughts exactly. Apparently only "continental" philososphers are allowed wherever it is that he teaches/preaches.

Pascal was, with Descartes, the greatest 17-th century geometer in the world.
He was the first mathematician ever to prove a highly non-trivial theorem in projective geometry, namely he gave a necessary and sufficient condition for six points in the projective plane to lie on a conic ("Pascal's mystic hexagram"). It was only 150 years later that Poncelet found new results in projective geometry, while being a war prisoner in Russia, courtesy of Napoleon. Pascal also invented probability theory, a whole new branch of mathematics: not bad for a "mechanical calculator who is the bane of pure mathematicians".

this is deep stuff and my guess is there will not be many comments. Perhaps someone will wonder how incredibly fill in the blank someone has to be to think they can put a paragraph in a book they write which "ranks" "philosophical arguments" against "mathematical arguments." What kind of comment is that? Probably not (i.e. probably no one will wonder) , though, because there is no real partisan gain to asking such questions, and a lack of perceived partisan gain usually totally depresses the number of comments around here, no fault of Professor Cowen's or Professor Tabarrok's, o course. Professor Rota pops up every once in a while in various contexts - he was incontestably bright and quotable, and I have heard about his approach to money-making analyses and similar issues off and on for the last couple of decades. But even an ur-Rota like Hume knew no more about how real humans think about how other humans think about real questions than poor short Moe understood about what Larry and Curly were going to do next and why, so there's that. Smart guy, though (not Moe, Rota - Moe was just a clown - hilarious balls into boxes or not). Still, a little too much of a calculus groupie, if you know what I mean (Rota, not Moe) - and if you don't, well that is all right too. Nothing wrong with calculus but seriously how much of one's lifetime efforts should go into very very small incremental improvements in calculus, of all things? Even if they are slightly better than vanishingly incremental? The question answers itself. (Or the question was never competently asked - maybe I thought until 10 minutes ago that Gian Carlo Rota was a composer of motion picture scores. Or a puppeteer. Nobody knows anything much, except when they do, and we almost never do, as I have been told again and again).

If I had posted that on Slate Star Codex or the Shtetl website I would expect an avalanche of two or three prissy negative comments. Not here, though. I like this place.

Your timing is off - the rhythm of these posts is likely product managed so that they have sufficient time to diffuse across the various informational channels to reach a broader audience. Wait until say 9am EST, and not 6 hours earlier.

Though you never know what attracts quantifiable attention and comments, of course. Which is either a philosophical or a mathematical observation, depending on one's mood affiliation or lack of emotional loading.

I'm not sure what part of your comment would provoke negative reactions. Honestly, it's hard to find anything to react to.

FG: Thanks for reading my comment, even if only once, and for your interesting rhetorical deployment of uninterested superiority! I was not expecting that! Mathoverflowesque, however, and not quite the Slate Star Codex and Shtetl flavor of comfortably intellectual blasé disinterest I was looking for. I would not post anything remotely like that on 'Mathoverflow', of course. Anyway, my guess is there would be no reactions at those sites (Codex and Shtetl) to one comment among thousands that had similar allusions to Moe Howard, Giancarlo Menottii, Nino Rota, puppeteers, or even Gian-Carlo Rota. I think it was the negative Hume allusion that would get the avalanche of one or two prissy "corrections" regarding the true value of (the mostly fraudulent, bless his heart) Hume. I could be wrong.

Great excerpts!

I am almost certain that I read, in a biography of Nikola Tesla several years ago, that he insisted on cutting all his meals into small cubes before eating them.

I can't find a corroborating source, though :-(

Thanks, I will have to read this.

Next door in the "what's it like to be a physicist" department, I'd like to recommend to our host (and others) Graham Farmelo's biography of Dirac, 2009:


As a mathematician I want to rebut several of those statements.

1) When mathematicians meet they behave just like everyone else...only with more socially awkward people involved. Like everyone else they ask each other what they do. This just tends to be a more common topic because it is often the only thing of common interest, it the least threatening topic for those who are socially isolated (it is something they know they are objectively good at), and a good number of mathematicians seem to simply have less interest in normal gossip/chatty topics. While most mathematicians are normal, well adjusted people who interact the same way everyone else does a small fraction of the population can give certain topics greater frequency. I have no idea what this determining the bottom line buisness is.

2) Mathematicians do philosophy all the time anyway. The just do it in an amateur and often highly misguided way.

3) I'm sure that is an exaggeration about Church. Certainly about the reason why he did it. I've heard enough stories about him to know he wasn't enough of an idiot to deny the existence of the possibility of context playing a role in sentence meaning or to think communication required forcing verificationism into every sentence. The stories I've heard suggested he certainly had some quirks so maybe he talked like that but not for the reasons suggested.

"2) Mathematicians do philosophy all the time anyway. The just do it in an amateur and often highly misguided way."

As an undergrad, I took several math / logic courses from Alonzo Church when he was teaching at UCLA. By far the smartest guy I've ever met; a giant who inspired awe in everyone. His short-term memory was sharply degraded circa 1980, yet it didn't seem to impair his intellectual capacity in the slightest. From close observation, I'd say his consciousness cache was limited to about a 30 second interval. In a sense, he was a living Turing Machine at this point in his life. He coped with this limitation by writing out on the blackboard numbered topic sentences from a pad of paper he held in his hand. After writing out such a sentence, he'd turn to the students and lecture for awhile, then turn back to the board to remind himself which topic he'd just covered and write out the next sentence from his pad in his neat cursive script. This worked well, with one memorable exception. In one class (I think it was scheduled for 3-6 pm, once per week) he just kept on lecturing. When he finally turned back around to the blackboard he was surprised by the wall clock, which read 8:30. He apologized profusely but needn't have; no student had left because his insights were so highly prized.

Another anecdote: a young rising-star philosopher made a seminar presentation in a large lecture hall at UCLA. At one point the presenter launched into a criticism of Church's approach to logic, darting nervous glances to the back of the hall, where Church was seated not far from me. The critique seemed strong. Looking over, I saw Church with his head tilted to one side, slack jawed, fiddling with hearing aids in his lap. A wave of pity came over me. Here was the old lion of logic dithering helplessly, apparently unable to hear or understand the attack being leveled against him. But in class the very next day, Church began in his characteristically humble way: "I hope you won't mind if we depart from the slated lecture today. Some of you may have attended So-and-So's seminar yesterday. Although I'm not qualified to discuss his ideas, not having understood them, there are a few observations I'd like to make." There followed the most devastating dismantling of a seemingly cogent set of ideas that I've ever witnessed, delivered with quiet humility, even puzzlement, in neatly numbered topic sentences.

Oh, come on. It's been long enough that you should be able to tell us who the "rising star" was without embarrassing them.

"Mobius draws attention to the rarity of a simultaneous propensity for medicine and mathematics, which exactly fits in with our own investigations. While, on the other hand the double disposition for mathematics and philosophy is quite common."

Ernst Kretschmer, Physique and Character, 1922

I had the privilege of taking two courses with Prof. Rota in college -- combinatorics and phenomenology. The latter philosophy course was so oversubscribed that he scheduled it for 7-10 PM on Friday to discourage students (this did not work!).

He was a delightful, puckish man who knew how to keep a room of students completely spellbound even while handling often very dry material; I wish that the median lecturer were even half as skilled.

His approach to the probability, and to how to understand probability and determinism together, remains with me even two decades later.

"His insight about the low productivity growth in services also helped explain why overall growth in an economy increasingly dominated by services can stagnate." That's from the NYT obituary for Professor Baumol. https://www.nytimes.com/2017/05/10/business/economy/william-baumol-dead-economist-coined-cost-disease.html? Are mathematicians subject to the cost disease? Are economists? Technology is applied to machinery to make it produce more efficiently and, hence at less costs. The machinery doesn't complain that it may be compensated less. When technology is applied to services to make it produce more efficiently and, hence, at less costs, somebody's gonna complain. Do advancements in mathematics make it produce more efficiently and, hence, at less costs? Or is it the opposite, each advancement giving rise to ever more advancements, with each advancement conferring economic benefits on the one responsible for the advancement. I work in an industry, legal services, that is the apex of inefficiency. Why is it inefficient? Is it because being a lawyer is a pretty good gig, and one doesn't wish to be so efficient that there's not enough work to go around. The same could be said for a college professor. Or a banker. I began my career as a lawyer before word processing equipment, documents typed (and retyped) on typewriters. When word processing equipment (does anyone remember mag card machines) came along, many if not most of the older lawyers resisted the advancement, believing that it promoted mistakes and inefficiency: if mistakes were easily corrected, mistakes would happen more often. Today, desk top publishing allows lawyers to create, improve, modify, and, yes, correct documents quickly and whenever they wish. Are lawyers more efficient as a result? If they are, why are there so many more lawyers charging ever higher rates for their "work"? It doesn't take a mathematician to explain the paradox. Baumol already did. Is America, with an economy increasingly dominated by services, doomed?

For maths I very much enjoyed this.


Alonzo Church sounds like he'd be good on Conversations with Tyler. Why hasn't he been on yet?

maybe because he died 22 years ago?

He can still take the Implicit Association Test though.

Well, you'd sold a copy until I saw $69.99 on Kindle! (Marked down from $89.99.) Ouch! But the hardcover is just $59.40. (First time I've seen that reversal.)

This looks also like a splendid book for a revised edition at a more accessible price—the sort of thing one could possibly imagine the New York Review of Books press doing, for example.

+1! Had the same reaction when I tried to find a copy of "Audubon: A Vision".

I knew Alonzo Church and my late father knew him far better than I did. He was most taciturn, speaking rarely but always very carefully, as has been stated, I believe. Another detail about him was that his eyesight was quite poor, and he would sometimes walk around with his eyes closed as looking at the world distracted him from the more important matter of thinking. This was how he came to meet his wife. One time he was crossing a street with his eyes closed and was hit by a car and ended up in the hospital. She was a nurse there, and it was said that she continued to take care of him after that, at least as long as she was alive, which did not last as long as he lived. When they were together he spoke especially infrequently.

On the matter of philosophy and math, well this partly depends on what one means by philosophy. Most people consider mathematical logic to be a branch of philosophy, or the area where the two most frequently overlap, especially with regard to foundational issues where one gets into the differences between different schools of math that accept or reject different fundamental axioms. There is a certain parallel with economics in that there is an orthodoxy, the math that is based on the ZFC (Zermelo-Frenkel-Axiom of Choice) set of axioms, which make it easier to prove many nice theorems that most mathematicians accept than many of the constructivist and other alternatives, even though many of the mathematicians who use ZFC admit that they are not all that confident in the actual truth, such as mathematical truth is (one of those philosophical issues), of some of those axioms. The parallel with econ is obvious in that standard general equilibrium theory is mighty handy for a lot of economists, but formally relies on axioms that few of us think are really true.

Where I have more sympathy with Rota is when he inveighs against woolier sorts of philosophizing that gets all worked up about such things as the "true meaning of infinity," and indeed there is also the unresolvable matter of what is true mathematical reality and whether it is a Platonic ideal, and so on and so forth.

Another mathematician who had bad eyesight and walked around with his eyes closed was Norbert Weiner. I have it from someone who saw it that one time he walked into a classroom whose door was open. He had been walking down the hall feeling the wall, and when he got to the open door he entered the room. The professor who was lecturing there immediately stopped and remained silent while Weiner went all the way around the classroom feeling the wall (including, I gather, the blackboard), and then out the door and on down the hall before he resumed lecturing.

Wiener, not Weiner (which means cry-baby).

Right. Thanks. Another story about his eyesight is that one time his family moved to a new house, and, being absent-minded as well as poor eyesight, he took bus to the neighborhood and encountered a little girl whom he asked "Do you know where the Wieners live?" to which she supposedly replied, "I'll take you home, Daddy." That is not from a primary source, and I suspect it is apocryphal, as they say, or at least exaggerated.

I've actually read some of the Rota book, I remember he recommended "The Mathematical Experience" by Davis and he also suggested Introduction to Phenomenology by Sokolowski.

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