Somewhat belatedly, here’s the expository paper I wrote for the ICM Proceedings: a short overview of my work with Malabika Pramanik, Vincent Chan and Kyle Hambrook on harmonic analytic estimates for singular measures supported on fractal sets.
The connection between Fourier-analytic properties of measures and geometric characteristics of their supports has long been a major theme in Euclidean harmonic analysis. This includes classic estimates on singular and oscillatory integrals associated with surface measures on manifolds, with ranges of exponents depending on geometric issues such as dimension, smoothness and curvature.
In the last few years, much of my research has focused on developing a similar theory for fractal measures supported on sets of possibly non-integer dimension. This includes the case of ambient dimension 1, where there are no non-trivial lower-dimensional submanifolds but many interesting fractal sets. The common thread running through this work is that, from the point of view of harmonic analysis, “randomness” for fractals is often a useful analogue of curvature for manifolds. Thus, “random” fractals (constructed through partially randomized procedures) tend to behave like curved manifolds such as spheres, whereas fractals exhibiting arithmetic structure (for instance, the middle-thirds Cantor set) behave like flat surfaces. There is a clear connection, at least on the level of ideas if not specific results, to additive combinatorics, where various notions of “randomness” and “arithmetic structure” in sets of integers play a key role.
The paper discusses three specific questions that I have worked on: restriction estimates, differentiation estimates, and Szemeredi-type results. I’ve also mentioned some open problems. At this point, I feel like we’re only started to scratch the surface here; there is much more left to do, for example optimizing the exponents in some of the estimates I’ve mentioned and, perhaps more importantly, figuring out what properties of fractal measures determine such exponents.
If you haven’t yet read this classic essay by Linda Nochlin on the question of why there have been no great women artists, I recommend it very highly. The essay is from 1971, but Nochlin’s points remain very much relevant to today’s arguments about why there have been so few great women philosophers, or mathematicians, or whatever.
Nochlin starts out by questioning the common notion of a “great artist” as a singularity that exists independently of society and history. The truth is, it takes at least a village. Great artists are enabled by the society they live in, draw on its artistic traditions, engage in a dialogue with other practitioners. Indeed, if artistic greatness depended only on innate talent, it would be very difficult to explain what Nochlin calls “conditions generally productive of great art,” such as must have existed, for instance, in the 15th century Florence and Rome, or in France in the second half of the 19th century. (We’ll note here that much of the same can be said of mathematics.)
The society also establishes standards for what qualifies as “great art,” and what does not. In the pre-impressionist Europe, historical painting– understood broadly so as to include biblical scenes, Greek and Roman mythology– was considered the highest and most prestigious form of art. Landscapes, still-lifes, portraits, and other suchlike were deemed less worthy. To wit:
Until the 20th century, Mona Lisa was one among many and not the “most famous painting” in the world as it is termed today. Among works in the Louvre, in 1852 its market value was 90,000 francs compared to works by Raphael valued at up to 600,000 francs.
“Great art,” going back to ancient Greece and Rome and then again starting with Renaissance, more often than not depicted naked and partially naked human bodies. Think Michelangelo, Raphael, Titian, Botticelli, Rubens. Even when the figures are clothed, the paintings still display a thorough knowledge of human anatomy. Such knowledge was usually gained through extensive study of the nude model, a practice that continues to be a mainstay of art programs. And yet, as Nochlin explains in detail, nude models (both male and female) were forbidden to women painters before the end of the 19th century. That right there explains completely why there has been no female Michelangelo or Raphael.
Nochlin cites many other ways in which the society refused to enable women artists: the apprenticeship system, access to academic educational institutions such as the Ecole des Beaux-Arts, opportunities to establish suitable relationships with art patrons, and more.
But the part I want to highlight here is the prevailing attitude to “the lady’s accomplishment”:
Every now and then, I’m instructed to have more faith in the progressive tendencies of humanity. Racism and sexism, I’m told, are relics of the past, and especially so in science and tech. Progressive, open minded people are against discrimination. Scientists and tech geeks are open minded almost by definition, therefore progressive, therefore against racism and sexism, which therefore are no longer a problem. I should just look around and see how many Chinese and Indian immigrants work in tech and science; clearly, this means that the field is not racist. And if there aren’t so many women around, that’s obviously because they’re not interested – or, as the progressive feminist Steven Pinker explains, maybe it’s just the innate differences. Progress! We’re all in this together! Let’s forget our differences, join our hands and work together for a better future.
Except I’ve seen it before: the unquestioning conflation of all possible good causes, the expectation that good intentions alone guarantee progress and enlightenment. Take, for example, the Western New Left in the 1950s and 60s. Leszek Kolakowski in “My correct views on everything: A rejoinder to Edward Thompson’s ‘Open letter'” from 1974:
[S]ocialist writing [...] amounts to saying that the world should be good, and not bad, and I am entirely on your side on this issue. I share without restrictions your (and Marx’s, and Shakespeare’s, and many others’) analysis to the effect that it is very deplorable that people’s minds are occupied with the endless pursuit of money, that needs have a magic power of infinite growth, and that the profit motive, instead of use-value, is ruling production. Your superiority consists in that you know exactly how to get rid of all this and I do not.
Eastern Europe, meanwhile, was being subjected to a practical implementation of that wondrous dream of progress and unity. If the Party represents everything that’s good and progressive, why would anyone ever want to oppose it? Who would need other political parties? Why, indeed, should any organizations be independent of the Party and government – surely, progressive and well-meaning people would want to associate themselves with the historic forces of good? Who could possibly be against world peace? The logical consequences are obvious. Kolakowski again:
… we got rid of this fraudulent bourgeois device of the division of powers and we achieved the socialist dream of unity, which means that the same apparatus has all legislative, executive and judicial power in addition to its power of controlling all means of production; the same people make law, interpret it and enforce it; king, Parliament, army chief, judge, prosecutor, policeman and (new socialist invention) owner of all national wealth and the only employer at one and the same desk-what better social unity can you imagine?
Clearly, if some Eastern Europeans were unhappy with this arrangement, this was just because they didn’t understand. They had “false consciousness,” in the language of the Marxist-Leninist theory, and therefore needed to be told what was really good for them, and beaten into submission if necessary. Not to say that the system was perfect, of course:
… you, not unlike most of both orthodox and critical communists, believe that everything is all right in the Communist system as long as the leaders of the party are not murdered. This is, in fact, the standard way of how communists become “critical”: when they realize that the new alternative socialist logic does not spare the communists themselves and in particular party leaders. Did you notice that the only victims Khrushchev mentioned by name in his speech of 1956 (whose importance I am far from underestimating) were the Stalinists pur sang like himself, most of them (like Postychev) hangmen of merit with uncountable crimes committed before they became victims themselves? Did you notice, in memoirs or critical analyses written by many ex-communists (I will not quote names, excuse me) that their horror only suddenly emerged when they saw communists being slaughtered?
I’m thinking of the progressive folks who believe that the main focus, always and in all matters, should be on them. I’m thinking of the men testifying on feminist blogs that they, too, have to prove their merit all the time, and there was even this committee meeting three years ago when someone interrupted them twice. I’m thinking of the journalism genre that “treats race as an intellectual exercise – a low-stakes cocktail party argument between white liberals and white conservatives over their respective racial innocence.” I’m also thinking of how the same people, when asked to stop and listen for a moment, respond with “I’m listening” followed by yet another barrage of words on how their arguments are really superior. Which of course they must be, seeing how eagerly they are accepted and applauded by the important target audience of other like-minded progressives. If someone like me continues to dissent, that’s my false consciousness speaking. Or maybe I’m simply too emotional and pessimistic. I should just continue to do my good work and we’ll all benefit, as I’ve been told many times. Kolakowski again:
… the spontaneous and almost universal mistrust people from Eastern Europe nourish towards the Western New Left. By a strange coincidence the majority of these ungrateful people, once they come to or settle in Western Europe or in the US, pass for reactionaries. These narrow empiricists and egoists extrapolate a poor few decades of their petty personal experience (logically inadmissible, as you rightly notice) and find in it pretexts to cast doubts on the radiant socialist future elaborated on the best Marxist-Leninist grounds by ideologists of the New Left for the Western countries.
I lived through two phases in Paris. In 1950, I was an attaché of the Polish embassy and attended parties with Paul Éluard and Pablo Neruda. The following year, after breaking with the Polish Communist regime, I came to live there as a refugee. At that time, French intellectuals were completely in love with Communism and Stalin. Anyone who was dissatisfied and who came from the East like myself was considered a madman or an agent of America. The French felt that their so-called ideés générales were valid for the whole planet—beautiful ideas, but hardly realistic. At that time the political climate of Europe was dismal; millions of people were in gulags; their suffering contaminated the aura, the air of Europe. I knew what was going on. The West had to wait for Solzhenitsyn to write The Gulag Archipelago to learn about it.
I’ve never been a fan of sloppy comparisons to communism, and I want to be very specific here. My beef is with those who say, “I’m a nice, progressive person, therefore I can’t be doing anything wrong and your complaints are not valid.” It’s with those who believe in the theory and refuse to see the evidence. It’s with those progressives who feel that every “good” cause, by virtue of their self-identification with it, is about them; and that their opinions trump everyone else’s experience because they, sensitive and enlightened as they are, would obviously notice any signs of injustice or discrimination; and that, when such experience is presented to them, the proper answer is to point to the bright future that is sure to descend on us soon like a state of grace.
Progressive minds such as those like to be unburdened by history and evidence. For all the talk of “inevitable historical forces” in Marxist theory, socialist writing rarely respects history as history. Instead, it invokes history as the future, the promise and the fairytale. “Historical determinism” was then, as “progress” is now, the magic wand that would forge a perfect world out of the fairy dust of good intentions. Forgive me if I’m a little bit skeptical.
This happy and cheerful period on my blog would not be complete without some mention of Anton Makarenko’s “Pedagogical poem,” aka “The Road to Life.” (Full text available here.) Makarenko, in case you don’t know, was one of the founders of Soviet pedagogy, best known for his work at the Gorky colony and the Dzerzhynsky colony, and for the book in question.
The Soviet government was chaotic and disorganized through much of the 1920s, with multiple factions and doctrines competing for dominance. On a practical level, the Bolsheviks had little if any experience with actual governance and running of the state institutions, and whatever blueprints they might have thought they had rarely survived confrontation with reality, so they made it up as they went along. Makarenko was one of such improvisers, controversial at first for his methods. Evidently, not everyone – even among Bolsheviks – shared his ideas, especially as they pertained to child labour and military-style organization of educational institutions. Stalin’s rule put an end to those voices in the 1930s. Makarenko was vindicated in a Resolution of the Central Committee of the Party on “pedological distortions” in 1936, became respected and imitated, and was awarded the Order of the Red Banner of Labour in 1939. (Warning: that link is to a Soviet-era hagiography piece in PDF.)
In 1920, the Bolshevik authorities tasked Makarenko, a 32 year old teacher at the time, with establishing a new colony for juvenile delinquents in the countryside near Kharkiv, Ukraine. (This eventually became known as the Gorky colony.) Initially, the staff consisted of a manager and two more teachers; the buildings were in a state of disrepair, the furniture, equipment and almost everything else of value having been stolen. After two months or so of preparations, the colony welcomed its first six charges, who immediately set about ignoring their supposed superiors, undermining their authority, and demanding food and service while refusing to do any work. One was arrested for robbery and murder soon after his arrival. The colony continued in that manner for a few months, until Makarenko finally found a way into their hearts, which he describes thusly.
And then, one day, the storm broke. I suddenly lost my footing on the tight rope of pedagogical practice. One wintry morning I asked Zadorov to chop some wood for the kitchen stove, receiving the usual cheerfully insolent reply: “Do it thyself! God knows there are plenty of you here!”
It was the first time any of the boys addressed me with the familiar ‘thou.” Desperate with rage and indignation, driven to utter exasperation by the experiences of the previous months, I raised my hand and dealt Zadorov a blow full in the face. I hit him so hard that he lost his balance end fell against the stove. Again I struck him, seizing him by the collar and actually lifting him off his feet. And then I struck him the third time.
I saw to my astonishment that he was simply aghast. Pale as death, he kept putting on and taking off his cap with trembling hands. Perhaps I would have gone on hitting him, if he had not begun to whimper out: “Forgive me, Anton Semyonovich!”
And then… they happily lived ever after. When Makarenko ordered the boys to work that day, they complied:
A few weeks ago, I finally got around to reading “Between the Lord and the Priest”, a book-length conversation between Adam Michnik, Jozef Tischner and Jacek Zakowski. I came for the historical content, but stayed in part for certain disputes in Catholic theology in Poland in the 1960s and 70s. It’s not my usual cup of tea; Tischner himself acknowledges that all this was of very limited interest to the general public while trailing well behind contemporary Western European philosophy. It nonetheless describes beautifully some of the disagreements I’ve had with my fellow mathematicians with regard to life in general and social issues in particular.
The framework for the dispute is provided by the long-standing dichotomy between St. Augustine and St. Thomas Aquinas. The way Tischner explains it, Thomism posits eternal, unchangeable truths that must be accepted as dogma and followed in life. It prescribes synthesis, universality, vast generalizations, logical chains of cause and effect all the way back to deity. Augustine, on the other hand, is less sure of himself. Even if the truth, somewhere out there, might be eternal and unchanging, our understanding of it is grounded in history, tradition and experience, and in the end that understanding is all we can ever access. In practice, this is a more bottom-up approach to religion, starting with personal, individual existential questions and then seeking guidance in the Scriptures and the church’s intellectual tradition.
Now, here is where things get interesting. Tischner goes on to say that Thomism, in its methodology and spirit, is actually quite similar to Marxism. Marxists, too, had their axioms of class struggle and dialectical materialism. They presumed to shape human consciousness through class awareness, much as Thomists presumed to shape it through philosophical and religious dogma, with little regard to individual experience and understanding.
That was why Michnik, an atheist and a leftist at odds with communism, tuned into Tischner’s polemics with Thomists. Thomism, like Marxism, represented codified, linear thinking where “one thing always follows from another, and everything is perfectly arranged and therefore very simple.” Tischner found that he could not talk like that to his parishioners – people who’d fought in the war, lived through the horrors of Nazi occupation, made choices that most of us wouldn’t want to think about. Their experience defied the scheme. Michnik, then in his twenties and already a veteran of protests and prisons, trying to graduate from university before his next arrest, had no love for simple explanations of everything, either. He’d rejected Marxism already; he would go on to consider religion, but not if it offered no escape from the same kind of closed-minded thinking, not if it were perfectly arranged with one thing always following from another.
At times, Marxist-Leninist philosophy was almost comical in its straightforwardness. Michnik cites Lenin’s theory of cognitive reflection, asserting that
(1) a world exists “independent” of and “external” to consciousness, and (2)
knowledge consists of approximately faithful “reflections” of that world in consciousness.
The second part of that, understood literally as Lenin indeed intended, is, on a very basic level, at odds with science, and I could say much more along these lines just based on my experience with photography. What’s less funny is the underlying Thomist assumption that there can only be one intellectually correct interpretation and only one right set of conclusions, namely those espoused by the bearer of the dogma, and that any departure from that must be a result of either misinformation or bad faith. When communists censored dissenting opinions, part of it was a genuine conviction that such opinions were obviously nonsensical and therefore there was no reason to disseminate them. When they lost the 1946 referendum in Poland, they blamed it on “confused thinking” and “complete ignorance” among the population. In a similar vein, but centuries earlier, the Catholic Inquisition might first try intellectual arguments, but if the accused were not persuaded, that constituted proof that they were possessed by the devil, because how else could they not agree? Thomists responding to Tischner informed him on a regular basis that he did not really know St. Thomas, because had he known him, he’d love him.
I started drawing my own analogies long before the point where Tischner actually uses the word “mathematical.” Like a good Augustinian, I’ll start with specifics. A couple of weeks ago, in a comment section far away, a mathematician proposed to “solve racism” by generalizing it (to something he never quite defined) so that racism itself would follow easily as a special case. In a different comment section last year, several mathematicians insisted on a purely mechanistic solution to sexism in mathematics. They accepted it as a self-evident axiom that mathematicians were progressive and well-intentioned people who would automatically eliminate sexism from their ranks if it only were pointed out to them. One might of course wonder why it hasn’t worked yet; but one would then be doubting an axiom, an act that’s not only morally reprehensible but, worse, logically inexplicable.
I’m thinking of mathematicians who’ve argued with my blog posts by taking shots at some sentence pulled out of context, the way they might point to an incorrect formula in a math paper. I’m thinking of one person who started a discussion with me, then allowed reluctantly in response to my arguments that he might not be able to change my mind after all, because convincing people is hard in general. Apparently, the possibility of me convincing him had never been on the table. I’m thinking of those who expected I’d stop believing in that gender bias thing if they only could explain it all to me, almost like religious evangelists. Sorry, no. I’ve heard your arguments many times already. I disagree with them, not because I don’t understand them well enough, but because I do. They don’t address my experience, and they never will if you keep starting from your own axioms instead.
It became clear to me over the last couple of years that I’m not, and probably never have been, part of a “mathematical community” of any kind. Sure, some of my best friends are mathematicians. I do my expected share of “service to the community.” But after hours, I’d rather kick off my shoes with people who at least share my logic. I’d rather discuss experiences, not axioms. I’d rather debate someone who’s actually listening to me, not just building his own castles of abstraction.
It’s been claimed by some of those in question that mathematics itself supports Thomist thinking. (As in, “I’m a mathematician and therefore this is how I approach this problem.”) I’m not so clear on that. To some extent, sure: we’re all trained in binary logic and deductive reasoning, as we should be. But in my own research practice, I often work in the Augustinian direction, starting with specific examples and then working towards something more general. Freeman Dyson’s “Birds and Frogs” article comes to mind, except that I’ve met froggy types like me who are incredibly dogmatic on social issues, and birds who are not.
If I were a Thomist, I’d try for a diagnosis, conclusion, and a list of recommendations for my peers. Maybe the problem is when mathematicians Act Like Mathematicians, showing off their smarts where wisdom is called for. Maybe, too, it’s unexamined purchase into the letter of the deductive philosophy of Russell and Bourbaki, without stopping to consider the actual practice of mathematical research; but that argument becomes circular right there. So, instead, I’d rather just leave you with something to think about, and excuse myself from Math Overflow once again.