Wednesday, October 8, 2014

Jewish Mathematics

The other day, following my introductory post on Arnold's proof of the fifth degree, one V. I. Kennedy posted a link to this remarkable lecture by a prof at the University of Toronto:

http://drorbn.net/dbnvp/AKT-140314.php

I didn't get a chance to look at the link until yesterday, because I was away. I wish Mr. Kennedy had given a little more information when he posted, because it turns out I was in Toronto  at the time and if I had known, I definitely would have called on the professor, one Dror Bar-Natan. I think we would have had an interesting conversation.

What makes me think that Dror and I would have anything in common? Well, for one thing both of us had the same reaction to this video by Boaz Katz: namely, both of us, as soon as we saw the video, had to drop whatever else we were doing and tell the world about this fantastic "new" proof of fifth degree ("new" in the sense that it was just in the 1960's that Arnold came up with it.)

But that's not all. When Dror tells his class why he felt driven to talk about this proof, he explained it in almost the same words I used twenty years ago when I was the math guy on community access TV here in Winnipeg: how, when he was a student, he took a whole course in abstract algebra, sitting through a labyrinth of theorems, lemmas, and corroloraies, until one day near the end of the course the professor announced "...and therefore the fifth degree is unsolvable"; and how it had always bothered him that for all that effort, he still never really understood why. If you watched Dror's video, you might check out this old Math with Marty episode where I say the same thing.

The other funny thing about Dror's lecture is he goes off on a little tangent about how you can understand that the square root of two is irrational. It's funny that the "alternate" proof he shows is almost the same as the one I did on this Math with Marty episode, except Dror refines it down to the bare bones while I work it through using numerical examples. Dror also has a hilarious punch line where he shows a "simple" one-line proof of the irrationality of any higher-order nth root of two: the joke is that his proof relies on the fact that a rational nth root of two would be contradicted by Fermat's Last Theorem, whose "proof" of course takes about nine hundred pages.

But beyond all that there's one more thing that you just can't avoid noticing through all this: it's the Jewish presence. All of us here in this discussion are Jews. Dror is a Jew. Boaz Katz is a Jew. I'm a Jew. Even V.I. Arnold is a Jew. Yes, Balarka Sen is a thirteen-year-old kid from West Bengal, but he's the exception that proves the rule. What's going on here?

Yes, we know the Jews are smart. But there's something more going on. We're brought up to believe we're smarter than everyone else, and maybe we are just a little, but it turns out there's a lot of smart goyim out there too. And by the way, if we believe that we're so smart, we also have to be prepared to accept that we have other distinguishing characteristics, which aren't always of the positive persuasion, so to speak. I wrote about this a couple of years ago in this blogpost, "Jewish Lightning", which asks the question: do Jews burn down their stores to collect the insurance money? But I digress.

Quite apart from the hypothetical question of how smart the Jews are, I think the present discussion illustrates another reason why Jews are high achievers in math and physics. I think we have a different aesthetic sense of math than your average white person. Dror is clearly excited about Arnold's proof. Boaz Katz is excited about it. If you look up physics lectures by people like Feynaman or Walter Lewin of MIT, they are clearly excited by their topics.

But it's more than just that. What we (the participants in the present discussion) all have in common is that we are excited by the beauty and elegance of an explanation which allows us to understand these things in human terms. I'm not saying that your average white person isn't also capable of experiencing the same emotional response to a math proof. It just seems to me that they aren't driven that way to the same extent. It's like you can teach a dog to walk on its hind legs, but it isn't exactly natural.

I've met enough regular white people who are "smarter" than me in my travels as a university student to be fully aware of my own limitations and the potential abilities of others. But when I compare the very smart people who I've known to the Jews like Boaz and Dror (also smarter than me), not to mention Feynmann and Lewin, I think I can pinpoint a qualitiative distinction. Most of the really smart white people I've met in math and physics seem to have an ability, incomprehensible to me, to be able to read a mathematical proof the way you or I read a newspaper article. I don't begin to know how that is possible. I can only "read" a proof line by line if I already figured out in my head what it means and where it's going.

I'm not saying some Jews don't have that purely analytical ability as well. Obviously Feynmann did. I'm just saying that I'm able to function at a pretty high level in math and physics working on a purely intuitive level. The conclusion I'm drawing is that the over-representation of the Jews at the very highest levels isn't necessarily because we're smarter than regular white people, but because we produce individuals who combine a high analytical ability with an intuitive approach driven by out unusual aesthetic sense: the same aesthetic sense that produces a Barbara Streisand or a Leonard Cohen.

In the old country, we even had a word for that quality: we called it the pintele Yid, the Jewish spark or the "point of light". I don't think we even knew exactly what we meant by the expression, but we knew there was something there that needed its own word.

EDIT: I sent Dror an email to tell him about the discussion we've been having here, and he wrote back the next day to tell me he'd read my blogpost, but he had a small correction to make: namely, he says he's not Jewish. I guess it's possible, but I still think he might be just a little bit Jewish. In fact, I think he looks quite a lot like the guy in The Princess Bride. What do you think?

POST-SCRIPT: I've been going over the video proofs and I think both Boaz and Dror have left out something important. I still think Arnold's proof is valid, but I don't think either of these guys have presented it correctly. I'm going to email them and ask them to respond to my criticism and we'll see what they say.

Here is my problem: I've watched both their videos again, and if I follow their logic exactly, it seems they've both proven that you can't solve the basic quintic equation:

x^5 - 2 = 0

Look at the roots. There are five of them. You can easily generate closed-loop excursions of the coefficients of the above equations in the complex plane which have the effect of reshuffling the five roots in any arbitrary order. We then look at arbitrary combinations of commutators of these permutations, and we find that it is possible to construct infinitely high towers of commutators that fail to return the roots to their original order. But we can also prove that if the roots are expressible as complicated expressions of nested radicals ("solvable in radicals"), that any sufficiently high tower of commutators must return the roots to their original order. This is a contradiction, and therefore the equation is not solvable in radicals.

Am I wrong, or do Boaz and Dror both fail to explain this case in their videos? Yes, it's a mistake that can be corrected, but I don't think it's a trivial mistake. Because to fix this mistake I think you have to invoke some Galois theory, and it seems that avoiding Galois theory was one of the big "selling points" of Arnold's method in the first place.

Someone tell me I've got this wrong...

Sunday, September 21, 2014

V. I. Arnold's Topological Proof

I just came across the strangest thing. This Israeli guy put up a youtube video about how you can prove the fifth-degree equation is unsolvable, based on the ideas of a Russian mathematician named V. I. Arnold. I haven't worked it all out yet, but it starts off with the strangest idea. You take an equation, and then plot out the roots in the complex plane. Then you also plot out the coefficients  in the complex plane. Why the complex plane, you ask? The cooefficients are all integers. Why not just plot them on the number line? Answer: because we're about to mess with them.

This is the funny thing. Take the simplest possible equation, like:

x^2 - 2 = 0.

The coefficients are 1 and 2, and the roots are +/- sqrt(2). Here is what we are going to do. We are going to take the "2" (the constant coefficient) from the equation and move it slowly along a circle about the origin. And we're going to observe what happens to the solutions of the equation as it changes.


Try it yourself! You'll see that when you complete the circle, so that the equation returns to its original form, that the roots of the equation also go back to their original values. But they are reversed! You have to go around the circle twice to put the roots back where they started. It's the strangest thing.

In the video, the Israeli guy claims (without really explaining it all that clearly) that in general, you can construct loops in the map of the coefficient such that by dragging the coefficients around those loops, you can arbitrarily force every possible permutation of the roots. I've shown you what happens when you drag the "2" about a loop in the complex plane - in the map of the roots, the two square roots of two switch places. Arnold's idea is that in general, you can force every permutation of the roots by dragging the coefficients around the complex plane.

I haven't yet figured out why this must be so (EDIT: Okay, I've thought about it and it's true: Boaz explains it around 4 minutes into his video), but if  since it's true then it has consequences. The idea is that if there is a solution to the fifth degree, it has to be written in terms of the coefficients arranged somehow within a complicated nested system of radicals...but for any such representation, there are restrictions as to where the roots can go when you mess with the coefficients.

How does this help us? Well, for one thing, in the example I've just shown, it proves that the solutions of x^2 - 2 cannot be rational. (EDIT: No, that's not quite right: it only shows that you need to write them with a formula that includes a square root sign). Why? Because the loop we constructed flips them around. But it's not so hard to see that if the roots are given by rational expressions, then any loop of the coefficients in the complex plane has to bring each of those rational expressions (for the roots) right back to where it started. So rational expressions don't flip around with each other, the way we did with the square roots of two.

Anyhow, that's the basis of Arnold's proof, which I'm not able to go much farther into right now. But it's something to think about.

If you're a follower of my blog, you know I've written a lot about the fifth degree equation. I think I explain it pretty well here at Why You Can't Solve The Qunitic. But I've never seen anything like Arnold's method before.,

Thursday, September 4, 2014

How Light and Sound are Different

The other day I started doing the Doppler shift as a relativity problem. It starts off looking a lot like the Doppler shift for ordinary sound in air. I had Alice sending a series of sound/light pulses to a train travelling at 4/5 the speed of sound/light; and I had Bob on the back of the train with a mirror reflecting the sound/light back to Alice. It's not too hard to calculate that if Alices pulses are 1 microsecond apart, then when they come back to Alice they are 9 microseconds apart, for a Doppler shift of 900%. You can see it easily from the picture below. The calculation is correct for light, and it's correct for sound:

What gets funny if we asked how much Doppler shift Bob sees. It's not hard to do the calucation for sound in air. We just drop vertical lines from Bob's pulse detection points down to the time axis. It's not hard to see that Bob detects the pulses 5 microseconds apart, for a Doppler shift of 500%:


Here's the thing: Bob is sending out pulses 5 microseconds apart, and Alice is measuring them 9 microseconds apart. So Alice's Doppler Shift is 180%. That's not the same as Bob's. So by analyzing who has a greater doppler shift, they can figure out that Alice is stationary and Bob must be moving.

And that's not how relativity works. In relativity we're not allowed to distinguish the stationary from the moving observer, so both Alice and Bob have to measure the same doppler shift. It's almost impossible to see how they can do that...unless we realize that time is moving slower for Bob than for Alice.

The total doppler for the reflected pulses is 900%. And the only way to make it the same for both observers is for them both to see 300%. Bob sees the pulses 3 microseconds apart, and Alice sees them 9 microseconds apart.

We did the caluclation for the special case of the train going at 4/5 the speed of light. But if we let the speed vary from 0 to 100%, and trace the contour defined by the equal-interval ticks, we will find that they are the hyperbolas defined by the equation x^2 - t^2 = constant:



Tuesday, September 2, 2014

The Doppler Shift in Relativity

It's been quite a while since I've posted any new physics, but I had a visit on the weekend from an old physics buddy from univeristy days. Richard is teaching at Waterloo these days, and relativity is his thing. I've mentioned once or twice that it's not really my territory, but last year I thought I did a pretty nice series on how to do those first-year compound velocity problems with pictures. I didn't assume any formulas except for the fact that x^2 - t^2 is invariant (the relativistic Law of Pythagoras.) You can see what I wrote here. Near the bottom of the page you can see how hyperbolas on the x-t diagram are the same "distance" from the origin. So someone moving along the orange line counts of his seconds "one, two, three..." accorcing to when he intersects those hyperbolas:


Well, on the weekend Richard and I were talking physics, and he told me about a problem he gave his students to calculate the doppler shift in relativity. He wanted them to do it without using the formula, just deriving it from basic principles. Naturally I wanted to try it for myself - and without assuming anything about x^2-t^2. I think I got it right, and as a consequence, basically derived the fact that x^2-t^2 is invariant. Here is what I did.

Now, what you normally do is have an observer on the ground (Alice) and an observer in the train (Bob). Alice stands on the tracks after the train has passed and broadcasts a radio signal to Bob, say 900 kHz. Bob measures the frequency and finds it is slower than what Alice sent out. This is normal: it's just the same way sound works. If Alice blows a whistle at 900 hz, Bob hears it at a lower pitch. If Bob blows a whistle, Alice hears the pitch change from high to low as the train passes by her. That's how sound works, and that's how light works. Either way, the picture looks like this:

The orange line is the moving train (which I'm going to take as going 4/5 the speed of sound (or light) in this picture, and the blue lines represent the "pulses" passing between the two observers. You can think of them as pulses or you can think of them as wavefronts, where I've drawn the waves in purple. I've done something a little bit odd: instead of Alice blowing one whistle and Bob blowing another one, I have Bob holding up a mirror so that Alices waves get reflected right back to her. The sound is reflected back at a lower pitch, and so is the light.

It's not hard to do the calculation for Alice. It's not so hard to see (from the basic geometry) that if she blows a whistle at 900 Hz, the echo that relfects back reaches her ear at 100 Hz. And it's the same for light waves: if she radios Bob at 900 kHz,  it comes back at 100 kHz.


(It's actually a little more natural to work with periods instead of frequency. If alice sends out pulses (of sound or light) one microsecond apart, they come back to her nine microseconds apart.)

But where it gets interesting is when we consider what Bob hears. For Alice, it's all the same if she's sending out light pulses or sound pulses. But for Bob it makes a difference, and there are big imlications that flow from that difference. That's what we'll talk about when I return.

Wednesday, July 16, 2014

Sex and Yiddish



This is one of my favorite Jewish Post articles ever.

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Last year I mentioned the legend whereby the Eskimos supposedly have dozens of words for “snow”. Whether or not this is strictly true, it illustrates the tendency of a language to develop in areas which are important to the culture of its speakers. In this regard, one has to wonder if any language has a more fully developed (or faster growing) vocabulary for matters sexual than our own English language. 

One of the newest entries to the extended sexual lexicon is “twerking”, a dance move which rocketed to prominence with the wrecking-ball video of Miley Cyrus. I’m not sure if the word is totally brand-new, but the dance move surely goes back a good few years. Myself, I give Christina Aguilera credit for permanently imprinting the “twerk” in my consciousness with her 2002  video “Dirty”. If you’re a man, you know what I’m talking about.

Of course there are any number of innovative words used to describe women as sexual objects, from the derogatory skanks and cougars to the self-consciously neutral posslq, but I think my favorite has to be the milf. I’ll never forget in Season 3 of “The Apprentice” when Donald Trump told TanaGoertz, a popular 40-ish contestant from the midwest, that people were saying she was a milf. “Do you know what a milf is?” he asked her. “Yes”, she answered, “it’s a mother I’d like to fool around with.” Yes Tana, you certainly were, with your cornfed Iowa wholesomeness. But I digress. 

If sexuality is front and center in our North American culture, then what more can we expect from the Yiddish language than the paucity of expressions for such things? I’m not even totally sure how I would say “girlfriend” in Yiddish…there is khaverte, which in North America would surely be understood as girlfriend by analogy with the English usage, but I don’t think that was the connotation in the old country. (It should be admitted that Yiddish is not alone in having difficulty here…even in English, it is not so easy to distinguish the case of a simple female friend, never mind the awkwardness of an unmarried elderly gentleman having to introduce his female companion as a “girlfriend”.)

At the other extreme of the relationship spectrum, we have the prostitute. Yiddish eschews the German hure in favor of either the Hebrew zoyne or the Slavic kurve (Polish kurwa). I don’t have a very good feel for the distinction in nuance between these, but I think it would have been consistent with the natural ironic bent of the language to apply the Hebrew term to the Gentile prostitute and vice versa.
And finally, in between the girlfriend and the prostitute, we have the mistress. In Yiddish she is the kokhanka, borrowed from the Polish. Once again, it is somewhat beyond my expertise to determine if the meanings correspond with exactitude. Even in English, we have to ask…just what is a mistress? A married man who keeps an apartment for a lover on the side surely has a mistress. But what if he just sneaks around with her on a regular basis? Is she his mistress or just his girlfriend? I’m not sure. I’m not sure if a single man in North America can be said to have a mistress (even whether or not he pays her) but I think in the old country he could have had a kokhanka….probably because there’s nothing illicit nowadays in sleeping with your unmarried girlfriend, as there would have been in der alter heim.  

In Mein Zikhroynos, the memoir of Yekhezkel Kotik, the author remembers from his childhood (around 1860) the jealousy of the poor Orthodox priest in his village, comparing his lot with that of the local Catholic priest, whose lifestyle was lavishly supported by the wealthy Polish squires. Hear what the Orthodox galakh though of his counterpart’s four beautiful sisters, who lived together with him in luxury:

“Nur der ârimer Russischer galakh, welcher flegt platzen far kinah (envy) vun dem reichen luxus-leben vun dem Kathòlischen galakh, hât var seine pauerim, die poretzische leib-knecht (the squire’s serfs) geschwòren, as die schöene Fräuleins seinen gâr nischt seine schwester, séi seinen ihm wild-fremde (total strangers), kokhankes seinen séi ihm, nur asõ wie a Kathõlischer galakh tor doch kein weib nischt hâben, hât er araus-gelâsen a shem (let it be known) as séi seinen schwester. Men mus moydah sein (one must admit) as der ârimer Pravoslavner galakh hât gehat recht: séi seinen wirklich geween kokhankes, un nischt seine schwester.”

Yes, it was a very different world. No one was twerking on MTV, and a woman was (sadly) old at forty, not milfy in the slightest. And yet some things were the same…

Thursday, July 10, 2014

John Q. Public



A few weeks ago I wrote about the word öffentlechkeit, which I claimed was a mistranslation of the American concept of “the public”. Yes, as an adjective öffentlech means “open” in the sense of “public”. And –keit changes an adjective to a noun. But surely the noun which results from adding –keit to öffentlech should have more to do with the attribute of something being in the public domain than a literal translation of the phrase “the public” in the sense of the man in the street.

I took up this discussion with a German Language forum on the internet. True, the nuance in Yiddish will sometimes be different from the equivalent in German, but it’s a good place to start. Except I got shot down in flames. It turns out that in German, according to several very reliable correspondents, die Öffentlichkeit is exactly “the public” in the same sense we  use it in English. To be sure, there is are secondary meaning which carry the connotation of  either “the public discourse” or the forum where that discourse takes place, but in common usage it is simply “the public”.

I still think that regardless of the facts on the ground, I should have won that argument. The English “public” comes from the Latin publicus, which was literally the public. The concept of the public as a body of citizenry distinct from the rulership or the slave class was an novel idea of the Roman system, so it is only fitting that the word has come down to us in that sense today. Its use as an adjective is clearly derivative from the noun…something is “public knowledge” because it is open to the public. Whereas in German you start with the adjective/adverb “openly”, tack on an ending, and it becomes…the public? It doesn’t make sense.

Oddly enough, in German (and Yiddish) das publikum is a perfectly good word, but it’s used to describe the audience, as in a theater performance or a lecture. (I don’t think it would be used for a sporting event.) In that sense it’s very similar to Yiddish der oylam, except that the Yiddish word also serves for the congregation in a synagogue, and I don’t think das Publikum would be used for churchgoers.

We still have one more noteworthy expression for “the public” in Yiddish: die gass, literally “the street” from the German Gasse, small lane. Oddly enough the German’s have Strassen und Gassen, big and small streets, but in Yiddish we have only gassen: our highway is die chaussée. No, that’s not evidence that Yiddish had its roots in Old France…like cauchemar (nightmare), trottoir (sidewalk), and even crêpelach (!) these French words came to us via upper-class Russian society. Not to be outdone by the Germans, mind you, we can still rhyme “streets and roads”; but instead of Strassen und Gassen, we have weggen un steggen.

What is interesting about die gass is that there was really no such thing as “the public” in the old country. There was literally no occasion when you would speak of “the public” as comprising both Jews and Gentiles; you could talk about die Yiddische Gass or die Goysiche Gass, but never just die Gass. There was no Ivan Q. Public in Minsk or Vilna; there were Jews and Christians, Catholics and pravoslavne (Russian Orthodox), serfs and landholders, Poles and Ukrainians…but no coherent “public”. So the Western European concept of die Öffentlichkeit would have been pretty much a non-starter.

Monday, June 30, 2014

The Odessa File

From my Jewish Post series:

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As I get older, I find I have less patience for reading fiction. I don’t feel entirely comfortable knowing that my emotions are being manipulated with made-up stuff. Oh, I still enjoy a good read, if perhaps not quite as much any more. But I haven’t forgotten the feeling of heart-pounding, cold-sweating excitement I used to get from a good thriller/mystery. The Odessa File by John Forsythe was one such book. Maybe you saw the movie starring Jon Voight as a young German journalist named Peter Miller who makes it his business to track down a notorious S.S. commandant named Eduard Roschmann (a true historical figure by the way), also known as the Butcher of Riga. 

I especially remember the thrilling confrontation, near the very end of the book, where Miller confronts Roschmann with the photograph of a man he had murdered. That man was….Miller’s father!…a Wermacht officer whom, in the last days of the war, Roschmann had shot in the back in order commandeer  a ship to facilitate his own escape the advancing Russian army. Of course, everyone had assumed that Miller was an idealist who was motivated by outrage at the Holocaust. “So…it wasn’t the Jews after all?” asks an astonished Roschmann? “Oh, I felt sorry for the Jews all right,” answers Miller. “But not that sorry.”

To be perfectly honest, I also remember feeling a distinct tinge of resentment at Miller…that he placed a higher priority on avenging the personal murder of his father over the much more enormous crime of what Roschmann had done to the Jews. Oh, it was a brilliant dramatic twist, no doubt…but did Forsythe have to give the Jewish tragedy a back seat to Miller’s personal family tragedy? It bothered me more than a little.

But that isn’t why I bring up the book today. No, I wanted to tell my our readers that I’ve started buying free-run eggs. Not all the time, but fairly often. Not when the regular eggs go on sale of course…I can’t pass up the chance to stock up at $2.00 a dozen…but when they’re not on sale, if it only costs 50 cents more to buy free run, I’m down with that. The fact is, I feel sorry for the poor little chickens, cooped up in those tiny cages, forced to sit in one place their whole life. Not that sorry, but sorry enough to cough up an extra 50 cents now and then. (And for some reason, every time I do, I think of Jon Voigt.)

Which brings me to the real point of today’s article: cruelty to animals. To what extent are we, as human beings, entitled to bring suffering to the animals we eat, simply for the sake of a slightly cheaper cut of meat? Don’t we have a duty to pay just a little more at the meat counter if it means that an innocent animal might have enjoyed a better quality of life?

This is not a joke. You should watch some of the documentary footage available of the appalling suffering in large-scale commercial pig barns. It is especially heart-wrenching when you see how playful and spirited pigs can be when allowed their freedom. Yes, in the end they are all going to be slaughtered for meat…but aren’t they entitled to a decent life up to that point?

I know what your thinking…what does this have to do with the Jews? Well, I’m wondering what would happen if we could find 100 orthodox rabbis to come forward and say that for the sake of tzaar bale-khayim, that they were going to sit down to a Sabbath meal of free-run pork.

What in the world would be the point of that, I hear you ask? I’m thinking it would send a pretty powerful message to the rest of the world. After all, we are famous for regarding khazer-fleisch as an abomination. Wouldn’t it say something to the world if all those rabbis were willing to break one of our most sacred commandments in order to encourage more humane treatment of animals?
For two thousand years our people have distinguished themselves by the extraordinary lenghts to which we have gone to follow God’s laws to the letter, even in the face of the greatest difficulties. Many of our religious leaders have preached that if only by reaching perfection in our obedience to those commandments can we hasten or bring about the coming of the Messiah. Now, I can’t pretend that I am personally a believer, but what if…what if  the real reason we were given those laws was to test us, not to see how obedient we could be, but to see if we had the courage and the clarity of vision to understand when we ought to break those rules instead of following them?

It’s something to think about.