Two unrelated things I’ve enjoyed recently that can be shoehorned in here via their relation to language and/or communication:
1) Ildikó Enyedi has been one of my favorite directors ever since I saw her weird, brilliant My 20th Century (Az én XX. századom); the other day my wife and I made a rare excursion to an actual movie theater to see her new one, Silent Friend (Stiller Freund). It too is weird and brilliant, not to mention mind-bogglingly beautiful, and its main focus is on communication with plants in three periods, 2020, 1972, and 1908. You can read a synopsis here, and I’ll quote the Director’s Statement:
It is a film made by humans (by an amazingly open minded, dedicated team) for fellow humans — the spectators. We humbly acknowledge and embrace our specific perceptual limits. This film speaks, with the help of light and sound waves available for the human eyes and ear, of world perceptions outside these limits. We acknowledge that we are not the default — our is one of the many, equally valid worlds. What is it like to be a tree? We don’t know. So, we won’t show it. Instead, we show human curiosity, touchingly imperfect attempts of connecting, of acknowledging the “other” and accepting that for them we are the mysterious “other”. We show glimpses through more than 100 years of the botanical garden of a university. A place that was always (“universitas”) the hub of free and limitless human curiosity, of science. In times when it is so dangerously questioned and attacked, we would also like to draw attention not only to the importance, but also the beauty and the naive, daring force of scientific research. It can help us walk down from the frighteningly dizzy spot on top of the pyramid to a place better deserved and more homey — to be part of this world.
Also, it stars the great Tony Leung. Highly recommended.
2) Konstantin Kakaes writes for Quanta about Peter Scholze and Dustin Clausen, who “are taking the first step in a far bigger program to understand why numbers behave the way they do”; it’s one of the rare general-interest articles on math that a nonspecialist can understand, and it makes me nostalgic for the long-gone days when I myself wanted to be a mathematician (I was interested in number theory, topology, that kind of thing, none of your applied math). This is the bit that made me want to bring it here:
For two people who are reinventing a big chunk of 20th-century mathematics, Clausen and Scholze are unassuming. “To a large extent, what I am doing is rephrasing what others have done in my own words,” Scholze told the mathematician Maria Yakerson in a 2021 interview. “I’m not that much interested in theorems or proofs.” What he wanted to do, he said, was to come up with new definitions: “They must make it easy to state interesting theorems, and they must make it easy to prove them.” Scholze doesn’t see himself as creative. He is, he said, just “trying to give names to what is there.”
Clausen, for his part — as he told Yakerson in a separate interview around the same time — avoids publishing papers, because he believes that the scientific publishing industry is fundamentally flawed. He also largely avoids even informally writing up results, leaving that to collaborators. He just wants to focus on the math; like Scholze, he’s constantly looking for the right names, the right language. (At one point, in fact, he considered pursuing a career in literary translation.)
(Click through for links and more.)
The interesting piece about “condensed sets” in math will make many Americans of my age either smile or wince with its use of the phrase “Point of Know Return.” As far as the google books corpus knows (accounting for one hit w/ bad metadata), this four-word string had never been written down in English prior to the release (when I was in 7th grade) of https://en.wikipedia.org/wiki/Point_of_Know_Return
LOL.
There is no useful category theory in quantum field theory today—absolutely none. As a formal field theorist, I think category theory may well turn out to be useful in making rigorous sense of things* like path integrals and renormalization; however, no one has made any progress in that direction yet. One can try, of course, to describe quantum mechanical systems using category theory, but so far that has yielded only trivial translations into superficially different language. This may be fully rigorous in nonrelativistic or abstract quantum mechanics, but so far, it does nothing in quantum field theory to address the foundational questions that bedevil the field.
* Is a Feynman path integral a “thing.” Is a renormalization procedure? All threads are one.
As a physics undergrad, I took a brief summer course in math that touched on number theory and a few related topics. It extinguished for good any feeble interest I might have had in pure math, and reading the Quanta article brought back unpleasant memories of my inability to comprehend abstractions. I stared at the illustration of open sets for a little while and couldn’t make any sense of it at all.
I loved applied math, though. Integration by parts, Legendre polynomials, steepest descents, the whole nine yards. I mean, you can actually do stuff with it.
Point of Know Return
Oy, Kansas. I weep for your generation.
“Boomers of Jerusalem, weep not for me, but weep for yourselves, and for your children.” We’ve had almost a half-century to work through those traumas! We’re a resilient generation!
@dl
Just go to spaces with a metric. Open sets are sets that can be constructed by taking unions (or finite intersections) of the interiors of balls, in particular, a ball without the boundary points is a typical open set. From this you see that the interior of a cube, etc., is also an open set (just take the union of all the interiors of balls contained fully inside the cube, etc.).
Even in metric spaces, things can be tricky. In an infinitive-dimensional metric space (like a space of bounded, continuous functions), the closure of an open ball (all points less than d from a center point) is generally not the same as a closed ball (all points less or equal to than d from the center).
I recall learning the basics of point-set topology by reading the first half of what was then a current textbook (Dugundji). This was needed because I was taking a course in functional analysis, that, in turn, required some knowledge of measure theory, that, in turn, required knowing what an open set is.
I did manage to learn enough to get through, grateful for the whole sentence of motivation buried in the middle of chapter two…
@brett
https://math.stackexchange.com/questions/108010/when-is-the-closure-of-an-open-ball-equal-to-the-closed-ball
The takeaway: abjure evil function spaces that are not normed vector spaces. FEAR the seminorm, for it will lead you astray.
@PP, Brett: Thank you, I think, for your responses… Sad to say, my bafflement remains unabated.
Unabated Bafflement: Scourge of Our Times.
I wonder how much more I would get out of the condensed set explanations if I had stayed more in touch with people who found algebraic geometry inspiring. As it happens, when I did my doctorate I was surrounded by graph theorists, on of whom had several much repeated sayings, including “Maths is a language”. If you lean into that too much, you end up with people overstating the usefulness of simply translating something in category theory terms, as Brett says, but the idea has struck a chord with me, whether thinking about research or a lot of my work since, which seems to be as much about resolving competing interpretations of numbery things as anything else.
My bafflement is about the fundamental theorem of algebra. Is it really that difficult to prove? The proof that I know has a nice name the lady with the dog (right from Chekov, who was inspired by The lady of the camellias. Let’s just call it La traviata)
@D.O.: The difficulty with the fundamental theorem of algebra is that it cannot be proven solely using algebra.* Some analysis is absolutely necessary—in particular, the completeness of the real or complex field. From a purely algebraic standpoint, it is very difficult to distinguish R from Q; both are ordered fields. However, the fundamental theorem of algebra, that every polynomial has a root,** holds over C = R + iR but obviously not over Q + iQ.
It was not until the nineteenth century that analysis was developed with sufficient rigor for a proof to be possible. Moreover, that development was gradual enough that Gauss felt it was important to publish multiple proofs of the theorem, as the sophistication of the field increased. Almost everyone credits Gauss with finally proving “d’Alembert’s theorem,” but there is disagreement about which of Gauss’s proofs should count as the first complete one. The simple and picturesque “The Lady with the Dog” proof is one of the harder ones to make rigorous. The proof is based on the polar decomposition z = r exp(iθ) and the observation that when r is large, as θ ranges from 0 to 2π, the value of a polynomial of degree n in z will wrap n times around the origin, while for small r, it will not wrap around the origin at all. The completeness comes in with the claim that to interpolate between these two configurations, at some value of r, the orbit must actually pass through 0.*** The modern way to prove this is using basic algebraic topology, in the form of the fundamental group of the punctured plane.
* The other key theorem in algebra which is elementary to state but quite important and not trivial to prove is the Cayley-Hamilton theorem (that every matrix obeys its own characteristic equation). The Cayley-Hamilton theorem can be proven solely using algebraic methods. However, it is still much more natural to prove it using analysis (or, equivalently in this case, using topology): The theorem is trivial for diagonalizable matrices; the diagonalizable matrices are dense in the space of all matrices over C; and a continuous function that vanishes on a dense subset must vanish everywhere.
** That’s not really the entirety of the fundamental theorem, but it’s the part that is hard to prove. Once you can find a single root, it follows fairly straightforwardly, by synthetic division, that you can factor the polynomial as a product of linear terms.
*** Note that the difficulty here is essentially the same as the one that makes Euclid’s very first proof in The Elements incorrect. With the axioms Euclid uses, there is no justification for concluding that two curves that cross must actually have a point of intersection.
What does “cross” mean, then?
Cross isn’t the best word, but I couldn’t think of a better one. (Intersect, for example, is definitely a worse choice.) The standard vocabulary basically assumes the same thing that Euclid did—that two planar curves cannot pass, from one being on the left and one on the right to vice versa, without having an (intersection) point in common. Have a look at book I, proposition I in The Elements. (Heath’s edition with commentary is quite good.) There is an implicit assumption that the two circles drawn during the construction will meet at at least one point. Even long before Proclus (who gave the most complete and important commentary on The Elements in antiquity, including filling in many missing steps that Euclid had elided from his proofs) it was recognized that this was an additional assumption not derivable from Euclid’s postulates and common notions.
Heath’s edition with commentary is quite good.
Euclid proper starts on p. 153 (pdf 166); the proof in question (On a given finite straight line to construct an equilateral triangle) is on 241 (254). On the following page we have:
I like this, from the Preface (p. vi [13]):
He forgot to add: “And every year, every such book would be factitiously ‘updated’ so that students would be forced to buy an expensive new edition.”
Brett, well yes, the completeness of R (and of C) is rather the point. My general attitude is that real numbers are not mandated by Almighty, but people found something to be named “number” where the intermediate value theorem always works. The interesting point then is that not only the roots of the polynomials (algebraic numbers) have to be included, but a good deal more on top. In other words, axioms are the result of figuring out what gives interesting (and useful) theorems, not the starting point, as in textbooks.
Compare Imre Lakatos, Proofs and Refutations, p. 5:
He forgot to add: “And every year, every such book would be factitiously ‘updated’ so that students would be forced to buy an expensive new edition.”
“And eventually, a way would be found to deprive students of access to their books after a year or so.”
It was some time during my undergraduate education that I came to understand that we find the right definitions in order to be able to prove things from them. I probably hadn’t quite grasped that fully when I took 18.901 (Introduction to Topology) as a sophomore, but I think a lot of the best examples come from topology. Tychonoff gave the first definition of the product topology in the 1930 paper that proved the first version of his eponymous theorem—that the product of compact spaces (in the original paper, closed intervals specifically, although he asserted the result was actually more general) is compact. It was evidently the right way to define a topology on an infinite Cartesian product (as opposed to the more obvious-looking box product topology or the uniform metric topology), because the theorem was true with that definition. This also served to show that the modern definition of compactness was the right generalization from metric spaces (where there are a whole bunch of equivalent definitions of compactness) to arbitrary topological spaces; previously, the most commonly used definition was that a space was compact if every infinite set had a limit point.
Subsequently, we have learned more about both definitions and what makes them natural. For example, Gödel showed that the modern (Alexandrov-Urysohn) topological definition of compactness was also the correct one in logic. The product topology on a function space was recognized to be the very natural topology of convergence on points. Category theory subsequently showed that it was possible to unify discussions of product topology with other natural topologies as initial objects. The product topology, subspace topology, and topology of convergence of continuous functions on compact sets can all be described categorically, as the coarsest topologies that make some natural functions continuous. The categorical treatment actually makes the proof of Tychonoff’s theorem rather trivial, but at the cost of giving up essentially all information about specific compact spaces like the closed interval.
I’m too tired to follow any of this, but…
That has happened, hasn’t it?
Doubtless because Euclid’s proofs and arrangement are no longer required from candidates at examinations. QED!
@Brett: Is that “lady with the dog” proof equivalent to the proof visualizable with a picture of a “dog on a leash” at https://en.wikipedia.org/wiki/Rouch%C3%A9%27s_theorem ?
@David Marjanović
Reddit discussion in which your question is not answered:
https://www.reddit.com/r/math/comments/1njem0t/what_the_hell_is_geometry/
@Jerry Friedman: It’s a similar type of argument, but the geometric proof of Rouche’s theorem is perhaps more complicated, since it uses the Jordan curve theorem and Cauchy’s argument principle—meaning it requires more development of complex analysis to make use of the geometric argument.
Note that the Jordan curve theorem is another theorem about which there has been considerable debate regarding who gave the first rigorous proofs of which parts of the full theorem. You don’t need the full strength of the theorem to make the argument in Euclid’s first proposition complete. The full theorem states that any simple closed curve (in other words, any topological circle) divides a plane into two regions—the interior, which is homeomorphic to an open disc, and the exterior, which is homeomorphic to a punctured plane. Proving the homeomorphisms is quite a bit trickier than just demonstrating that the curve separates the plane into an interior and an exterior. Moreover, the result is specific to the plane. In Euclidean space with three or more dimensions, the analogous result does not hold. A topological sphere separates R³ into an interior and exterior, and the interior is homeomorphic to an open ball; however, the exterior can have a much more complicated topology than R³ minus a point. The topological spheres that have nontrivial exterior topologies are called “horned spheres”; the best known one was discovered by Alexander.
Reminds me of the
https://en.wikipedia.org/wiki/Jordan_curve_theorem
which is intuitively obvious, but the proof of which is, erm, not straightforward.
Heh.
well, where would you hear about that other than the hattery?
for what it’s worth I’ve read (un?)reasonable amounts of the standard literature of algebraic geometry (Hartshorne/Vakil/EGA/SGA), but what Scholze and company are doing is not at all transparent from that point of view to me. my impression is rather that some solid knowledge in p-adic analysis and the more advanced realms of homotopy theory could maybe provide a more concrete motivational background, but it’s just a guess
by and large yes, what people used to call “synthetic geometry” went more or less extinct somewhere towards the end of the 19th century at the latest. Felix Klein in his famous Erlanger Programm put it less bluntly (“Den Unterschied zwischen neuerer Synthese und neuerer analytischer Geometrie hat man zur Zeit nicht mehr als einen wesentlichen zu betrachten, da der gedankliche Inhalt sowohl als die Schlussweise sich auf beiden Seiten allmählich ganz ähnlich gestaltet haben.”), but the upshot is more or less that since all other areas of mathematics by that time were also being put on axiomatic ground there was not much motivation to introduce new and separate axioms for geometry when you could just build it on top of what you already had (not that out of this would have come a unified notion of what geometry is, mind you, algebraic and differential geometers are usually interested in quite different things)
also, it’s hard to stay interested in only straight lines and circles once you’ve read Gauß and Riemann… and besides, why would Euclid have more of a claim to be doing Geometry than the poor old Landvermesser?
Ha, I didn’t even notice!
To boldly go …
I noticed my typo when I came back to this thread. Frankly, I’m surprised that my fingers defaulted to “infinitive” instead of “infinite.” Maybe it means I spend too much time reading and commenting here, as opposed to doing physics and math.
Don’t call it a typo, call it a harbinger of an entirely new mathematico-linguistic form of analysis. “There are three types of triangles: perfective, ergative, and aoristic…”
I have a vague memory of a discussion we once had of a Russian crank* who solved linguistics by topology, but my google-fu is too weak to rediscover it. I do remember that I attempted to read a paper by the gentleman in question. I think pole dancing may have been involved in some way, but there may be alternative explanations for that belief in terms of the Psychology of the Individual.
* The best kind of crank.
I saw a copy of the German DVD yesterday, and it used the English title.
do not start putting ideas into people’s minds… “Among other results, C established the existence of a functorial equivalence UG between triangulated categories…”
a fair description of what LLMs are doing (to my understanding anyway)
shots fired at drasvi?!
Not drasvi.
Pole-dancing is what to of my female freinds once wanted to study. … P.S. I seriously considered joining them:)
Ah! Shaumyan. Thanks.
And pole-dancing was involved …
Wikipedia: “Sebastian Konstantinovich Shaumyan (Armenian: Սեբաստյան Շահումյան; February 27, 1916 – January 21, 2007) was a Georgian-born Armenian-American theoretical linguist and an outspoken adherent of structuralist analysis.[1]”
Born in Tbilisi. Clearly the equal-and-opposite crank to Marr.
…and note the հ; that’s a h, traditionally dropped in Russian transcription (as for the historical linguist known to the West as Djahukian).
So what’s the etymology of Shahumyan?
Only Xerîb knows.
https://en.wikipedia.org/wiki/Shahumyan_Province
A relation got a province named after him.
Shahumyan?
I should think this is formed in usual way that Armenian surnames are formed, in this case from the noun շահում šahum ‘winning, win; prize, winnings; gain, profit’, which is also used as a masculine personal name, Šahum. The noun շահում šahum is built with the suffix -ում -um that forms action nouns (‘X-ing’ ‘X-ation’ ‘X-ment’ to a verb ‘X’) from verbal stems. In this case, the base would be շահել šahel, šah- ‘to gain, win, earn, make a profit’, ultimately derived from Old Armenian շահ šah ‘gain, profit, benefit’, as in Philippians 1:21:
Old Armenian շահ is generally said to be of Iranian origin and specifically compared to Avestan xšaθra- ‘power, command, rule, lordship, dominion’, belonging to the group of Avestan xšā- ‘be able; rule, be lord of’, Parthian šh- ‘to be able’, Persian شایستن šāyistan ‘be suitable, appropriate’, Bactrian þιι- ‘can, be able’, etc., and note especially further Vedic kṣáyati ‘possess, be master of, rule’, kṣatrá- ‘dominion, power, government’. For remarks on the possible further Proto-Indo-European root etymology of the Indo-Iranian forms, see the Lexikon der indogermanischen Verben under h₃ekʷ- ‘ins Auge fassen, erblicken’, p. 297, note 3 here, and *tek- ‘die Hand ausstrecken, empfangen, erlangen’, p. 619, note 1 here (on the old proposal to connect the Indo-Iranian forms with the group of Greek κτῶμαι ‘get, obtain, acquire’, κτήματα ‘possessions, property, wealth, heirlooms’, etc.).
For the final -h of շահ šah from original Iranian *-θr-, note for example պահ pah ‘watch, guard; hour, moment’ beside Parthian pāhr ‘guard’ (< Old Iranian *pāθra- or the like); the Armenian male name Վահագն Vahagn beside the Persian name بهرام Bahrām and Avestan Vərəθraγna-, the spirit/divinity of victory (lit., ‘resistance-smiter’); and զոհ zoh ‘sacrificial victim, offering, sacrifice’ beside Middle Persian zōhr ‘libation, offering’, Avestan zaoθra- (cf. Vedic hotrá- ‘offering, oblation, sacrificing’), etc., etc. On that note—sacrifices—I’ll stop here for lack of time because it is almost Eid and I have to get on the road.
Thanks, our confidence in you was not misplaced!