Too technical for Nature. Just about right for People.
"General-audience journals" like Nature and Science are in fact journals for biology and neighboring fields. There's a big disconnect between the level of biology and chemistry literacy they assume and the level of mathematics or physics literacy they assume.
I think just in general, you can get away with knowing nothing of physics that was developed in my lifetime and be considered more or less current. On biology, you have to keep up with new discoveries, like genetics and the clitoris.
General audience? I have a subscription to Nature atm and it's unintelligible. I read about 3 pages of it. I didn't buy this substription, my dad did. Last year it was National Geogrpahic which got read much more than fucking Nature. I'm slightly drunk so didn't read it properly but too fucking mathsy? no such thing.
Damn, I was trying to correct typos.
Surely if you don't know about the Higgs Boson you are a social outcast.
One upvote for slightly drunk asilon.
Worky christmas lunch. Trying to sober up so I can do some shopping (only free time before the weekend) and then go out for dinner tongiht.
3.1 Twas ever thus. My father studied maths and physics to the age of 18 in 1930 and Relativity was never even mentioned.
Mrs y had hers today too, at 12:30. Still no sign; not worried yet.
It's just more evidence, along with literally any math Wikipedia page, that math is uniquely incapable (compared to other sciences) of being described in words roughly comprehensible to a general reader. It just is. A few paragraphs in, they start with non-defined terms that are totally non-intuitive and make no sense, making the obit incomprehensible.
Math terms are hard to understand, but I think I got the main point. He's definitely dead, right?
Biology papers have more numbers in them than maths papers.
Medicine papers have lots of numbers in them and would have even more if they weren't total Nazis about the number of tables allowed and whether something that runs for six pages is really a single table.
13: Sure, but "32 teeth" is a pretty accessible concept.
But numbers are well-defined terms and basically intuitive in a way that a 'communicative ring' isn't.
15: I only have 28 and thanks for rubbing it in.
[H]e came into his own when he took up algebraic geometry. This is the field where one studies the locus of solutions of sets of polynomial equations by combining the algebraic properties of the rings of polynomials with the geometric properties of this locus, known as a variety.
I don't think the problem is that the authors overestimate the mathematical literacy of Nature's readership, I think the problem is that the authors significantly overestimate the clarity of their own writing. They think they have written an article that should be accessible to a non-mathematical audience, but they failed in this effort. The article doesn't need to be "dumbed-down" further in the sense of losing any of its substantive mathematical content (of which there is very little), but it needs to be dumbed down significantly in terms of more carefully explaining (in a way that is intuitive!) mathematical terms that have no meaning to a general audience.
Oh, I see that I took too long to compose comment 18. Oh well. It was an original thought when I started typing it.
I think the part quoted in 18 isn't going to over the heads of Nature's readers. Or not by much. The next sentence is where I went into the weeds.
Grothendieck is a particularly tough person to write about for a general audience. Given the likely word limit I think they did quite well. I would have tried to avoid the words "commutative ring" and used "number system" instead. I also would have given an example of a variety. But the editor's comments are ridiculous. Complex numbers are something scientists should know, and I think the editors shouldn't have asked them to write this if they weren't willing to let them mention complex numbers.
Agree with 21. The article is terribly unclear, but not for the reasons the editors seem to be complaining about.
I don't actually understand what complex numbers are used for in a practical sense, except that if you don't get an even number of is, you almost certainly fucked up somewhere.
Is there something going on where mathematicians are peculiarly bad at picking names for highly technical concepts? Words like 'variety' and 'group' are awful when you're looking at a layperson-level explanation of anything, because they're the sort of word that it's hard to rule out as having come up in their ordinary English sense. (Not that that's the whole problem, but it contributes to the feeling I have of "I'm not going to even bother trying to puzzle through that" I get when I'm reading descriptions of serious math.)
The article is terribly unclear, but not for the reasons the editors seem to be complaining about.
Yes, this was part of my point. I think "our readers are too stupid for you" is being used merely as a more polite way to reject the article than "you have done a bad job of writing in a way that would be comprehensible for a non-mathematical audience."
I think in an obituary some of the opaque summary of other contributions at the end is fair. Readers don't need to know what Gal(Qbar/Q) is, they just need to know that he did other cool stuff.
Thinking about it more, if I were going to write this I might try to talk less about what schemes are and instead aim for giving a rough understanding of the Weil conjectures. But that's super hard too...
I think the part quoted in 18 isn't going to over the heads of Nature's readers. Or not by much. The next sentence is where I went into the weeds.
Huh, that's weird--I didn't have any problem with the next sentence.
27: I don't understand what is different about doing using arbitrary fields or what makes a field arbitrary.
And I'm pretty sure what I don't understand has something to do with what LB was talking about in 25.
18: Tim only respects your intellectual property when you're paying him for it.
It's true that they didn't do a particularly good job, but if they had a tight word limit, it would be hard to elaborate in the space allotted.
Part of the problem here is that they have to ask people who are famous. Among famous people they asked two people who are very good mathematical writers (Tate wrote probably my favorite math textbook) and would be great choices for a broad mathematical audience. But if you want something at a Slate level the ask Jor/dan Ellen/berg and be done with it.
if they had a tight word limit, it would be hard to elaborate in the space allotted
This is true, and I hadn't been considering this point.
I suspect having a firm understanding of what a "field" is would be some kind of a key divide between normal people and math people. I think maybe somebody mentioned in it Calc, but it wasn't on the test.
24: In other subject areas it's sometimes possible to make a decent guess at what a specialized term means from the context of it's use. That's very rarely the case with math beyond a pretty basic level. If you haven't seen the definition of a term you're never going to guess the meaning based on the rest of the sentence it's in.
That's a big part of what makes explaining math so difficult. There's a big ladder of definitions and if you try to skip rungs things become confusing in a hurry.
Field: You can add, subtract, multiply, and divide. I claim you are familiar with at least one example.
11, 24. Contemporary mathematicians see generality, abstraction, and logical elegance as universal virtues. This basically aesthetic judgement is not necessary for thinking about large portions of math. Those who like this style basically do not want to communicate with the outside world.
People who write about math but do not think this way include Ivars Peterson, Terry Tao, John Baez. Hardy actually writes pretty clearly also, so does Ulam.
Some mathematicians (the snobs) hate examples in finished work, others include them.
37: Grothendieck was the snob's snob, though. Famously someone said to him "Let's take a specific prime as an example." Forced to come up with a specific prime number, he said "27".
36: If there's not more to it than that, mathematicians are assholes.
"Arbitrary" means "without restriction." Or more specifically "without the usual restriction" or "without the restriction mentioned above." In this case if you work over the real numbers or complex numbers than solutions to polynomials are *geometric* gadgets like surfaces or higher dimensional analogues of surfaces. But if you work with numbers 0,1 with 1+1=0 (characteristic 2) there's no "geometry" intuitively but Grothendieck says there's a generalized version of geometry anyway!
40: They're assholes.
I have the story wrong. It's 57, which is slightly less embarrassing.
They should call more things Beta, like normal people.
Would you prefer a multi-syllabic made-up Latin word? Having short words for very common notions like "group" is pretty nice since you say it a lot. I think it was originally "a group of substitutions". It's also just harder to turn names into nouns than adjectives. So "Euclidean Space" is fine but you can't just call it "Euclid."
Would you prefer a multi-syllabic made-up Latin word?
Yes. So much harder to get confused and so much easier to look up.
Group or field isn't hard to look up, just google together with the word math.
That brings up Wolfram Math pages. I suppose those are more reliable than Wolfram Alpha pages.
Also wikipedia, right?
Yes. I mentioned Wolfram out of sheer churlishness.
Yeah, it's just impossible to write anything about mathematics that non-mathematicians will understand. I think the obituary is fine, in that it looks like an obituary and is shaped like an obituary. Mathematicians will already know everything in it, and non-mathematicians will learn nothing from it, but it's fine, if we think of the obituary as a form of performative speech where the content is secondary.
46: Yes, because it's visible as a technical term. A made-up word can be short (quark?), or you can do long Latin words and suffer through the extra couple of syllables, but common English words being used in a sense that's really unrelated to their common English sense are stumbling blocks to people who haven't internalized them all. (Like, I know group and field are technical terms, but variety is new to me, and I initially read it as English rather than math.)
(Obviously, people who are actually working in math aren't troubled by this, and it's not going to change, but it is a slight added layer of difficulty.)
and is shaped like an obituary
An obituary is an n-dimensional shape defined by an arbitrary array and the set of people in slightly dated suits who aren't lawyers.
I think just in general, you can get away with knowing nothing of physics that was developed in my lifetime and be considered more or less current. On biology, you have to keep up with new discoveries, like genetics and the clitoris.
And with math you can stick with what was known in the early 18th century, except statistics.
There's a big ladder of definitions and if you try to skip rungs things become confusing in a hurry.
This seems to me to be the biggest problem; obviously it's also true in other sciences but you can often fudge the definitions in a way to make things more comprehensible.
But, also, math has a uniquely absent bottom line. It's all just interesting conclusions of formal logic on top of other interesting conclusions of formal logic, so there's ultimately no there there to explain. By contrast, for example, I'm vaguely aware that physicists think that the universe is composed in some deep way of vibrating strings in multiple dimensions above the fourth dimension. Now, I have no idea at all why one should think that (other than reading some Discover-magazine level book), or what that really means, or even if it's quite right, and it's obviously totally counterintuitive, but at some level there's a concrete bottom line that can be explained in terms other than the formal logic that got you to the bottom line. For math it seems like this isn't true.
This is another episode of deep thoughts on the philosophy of science, by Tim "Ripper" Owens.
I'm somewhere in-between the math knowledge of some of you.
The problem with giving examples is that non mathematicians seem to expect something concrete, with all sort of irrelevant details. A vector in an Euclidean space and all those functions that can be formed from linear combinations of sines and cosines share a common structure. Does that help clarify the notion of a vector space?
Sometimes a concrete example can be plenty abstract. For me, permutation groups and the set of invertible matrices (of a specified size) are canonical examples of non-Abelian groups. I'm never sure how much this clarifies.
The obit is atrocious -- the paragraph that starts "to illustrate how revolutionary this was" is especially terrible, since I understand most of the words in it but have zero idea how they connect to schemes and how revolutionary they are.
I think the right approach would be to focus on discussing some results that can be discussed in plain English, and then hinting at the mathematical machinery necessary to get there, and what some especially creative and fruitful ideas are. In other words, provide some motivation for studying someone's work. This obit reads more like someone going over some Chapter 1 concepts really slooowwlly as if that's going to help a non-expert make any sense of it.
Hold on: mathematicians use "an" in front of Euclidean? No wonder I can't understand this stuff.
Yeah, it's just impossible to write anything about mathematics that non-mathematicians will understand.
Furthermore, the thing that made Grothendieck obituary-worthy is that he did things that were at the cutting edge of mathematics. Mumford and Tate can hardly be faulted for failing to explain these complex concepts at the layman-level within the space of a few paragraphs—if it were possible to do that, someone would have done it before, and then everybody would already understand these concepts. So you're faced with a choice: either describe Grothendieck's accomplishments in terms that will be hard to comprehend for people not already equipped to understand them, or simply say, "He did some really amazing shit with numbers."
This seems like a better attempt at explaining Grothendieck for the layperson. Maybe someone more mathy than I can offer an opinion.
With that last name, if they said nothing but "he did some really amazing shit with numbers," I'd be convinced. There's no way a guy named Grothendieck didn't do something impressive in some field that there's no way I could actually read it.
Assuming he wasn't a serial killer.
, because it's visible as a technical term. A made-up word can be short (quark?), or you can do long Latin words and suffer through the extra couple of syllables, but common English words being used in a sense that's really unrelated to their common English sense are stumbling blocks to people who haven't internalized them all.
Illustrating this point is the way the authors use the word "field" in its ordinary English sense right up against discussion of a topic in mathematics in which "field" is a central technical concept. Too bad they didn't work in some sentences like "he developed a new scheme for dealing with this problem" and "people had suggested a variety of different ways of handling this analysis."
Also, Snark and I just vigorously disagreed about the merits of the explanation of infinitesimals in paragraph 5. I find it dreadful. No! Don't suddenly start bringing in imaginary imperfect measuring instruments and "there's no harm if we set it to zero"! I remember that we're talking about math, and this way of talking about things really muddies the water for me.
62: From my layperson perspective, yeah, that's a hell of a lot better. Of course, it's a lot longer than the Mumford and Tate piece, but even so it gives me a much better sense of why Grothendieck's work was revolutionary.
Landsburg appears to be vastly better at explaining higher math than he is at discussing social science.
The fourth paragraph is awkwardly written, but the fifth paragraph was about where I started to think the editor was right to reject the obituary (probably not for the reasons they gave the writers, but I think 25 is probably right about that). I could probably drag my way through up till that point if I really wanted to put a lot of effort into figuring out what all those words meant, though I'm not sure because I'm not going to do it. But when you start out by saying "to illustrate how revolutionary this was" the example really, really should be something that people who don't already know what this was should have some grasp on. The people with enough math experience to see why that was revolutionary (let alone what it means in the first place) already damn well know why it was revolutionary. The point of an example, for non-expert readers at least, is to make things less technical, not more. Also it probably shouldn't be followed up by "In effect...", because that would actually be the thing the example was supposed to do. And it can't really do that well once the complicated example is sitting right there above it.
That the next paragraph starts out "going further into abstraction" really just adds insult to injury at that point.
it's just impossible to write anything about mathematics that non-mathematicians will understand
Strongly disagree with this--it's not impossible, it's just hard. There are a good generally-accessible books that explain complex mathematical concepts.
62: yes, much better. Although, as 31 noted, maybe over the word limit for Nature.
to focus on discussing some results that can be discussed in plain English
The problem is that with Grothendieck there are literally no such results. With Tao or Jones or Wiles or Perelman you'd have a chance, but you're wildly underestimating how incredibly hard this is to do with Grothendieck.
69: Could you give an example? I'm pretty sure whatever you're talking about is just way way way way less complex than anything Grothendieck did.
I'm under no illusions that I even vaguely understand Grothendieck's work from reading the piece linked in 62, but I think it does convey a sense of "this is why the people who do understand are so impressed".
I was thinking about why math uses common words for things instead of long latin/greek words as in other scientific fields. I think a lot of it is just that math has relatively few "fundamental" nouns. If you take a word like mitochondria, there are probably thousands of equally fundamental notions in biology. But if you take the word "group" I could probably list off all the similarly fundamental nouns. I think that may be part of why there's no need to come up with weird names and so it wasn't done.
(Set, group, ring, field, vector space, algebra, module, representation, manifold, topological space, function, category, functor, variety, scheme, stack, operator, measure, model, bundle, sheaf. Probably another half-a-dozen.)
At any rate I see what you're saying about it being nice to have a "this is a technical word" indicator, and I think it's an interesting question to sort out why that didn't happen historically.
There's a secondary problem, which is the abuse of a few common adjectives like normal, regular, simple, etc. Everyone acknowledges that's bad.
The problem is that with Grothendieck there are literally no such results
62 uses the Weil conjectures as an example of something derivable from Grothendieck's work. Seems relatively accessible to me.
I think the link in 62 is good, but the description of the Weil conjectures is oversimplified to the point of being totally incomprehensible. For Nature it'd be nice if in addition to vague analogies about nuts (which are great and should have been there) at least a tiny bit of actual mathematics.
I mean, the Weil conjectures are what I'd aim for if I were writing this (see 26), but if you think 62 told you what the Weil conjectures were I have a bridge I'd like to sell you.
Can't we legitimately prefer the comforting illusion that we might have an idea of what the Weil conjectures are about?
In other words, tell me more about the bridge.
Can one explain schemes to biologists
Jeez, Mumford. Let's start with proper punctuation?
Okay, fuck it, I'm going to explain what a scheme is, and what the Weil conjectures are about. One reason why mathematicians are terrible at explaining things for popular audiences is that simplifying involves a certain amount of bullshitting over technical points. Since thinking about technical points in great detail is the day job of mathematicians, they find it hard to stop.
Since Descartes, a major topic of mathematics research is understanding the solutions to polynomials equations. Descartes observed that while finding solutions is a matter of algebra, that when you view all of the solutions together, you enter the realm of geometry. For example, the set of solutions to X2 + Y2 = 1 is a circle.
The set of solutions to one or more polynomial equations is called a variety, and the study of such things is called algebraic geometry.
What's wrong with calling it the "set of solutions to some polynomial equations"? Aside from the extra syllables.
If it's an obit for the layperson, they should probably just use a different measuring stick: "Grothendieck was so good at math, he could do long division with 10 digit numbers without using a scratch pad."
This also helps to set up a hierarchy, as maybe Perelman can only do 9 digit numbers, or he could do 10 digits, but only using his fingers.
||
If the university wants to schedule exams in 3-hour blocks, you'd think they could at least make sure there's a somewhat comfortable chair at the front for the proctor to sit in. It's only one hour in and my back is killing me.
|>
The proctor supposed to pace menacingly down the aisles.
84: Um, it's a truth universally acknowledged that a warehouse worker in possession of good sense must be in want of a foreman/enforcer. What's the problem?
Sounds like the guy had an interesting 1969:
In 1969, for reasons not entirely clear to anyone, he left the IHES where he had done all this work and plunged into an ecological/political campaign that he called Survivre. With a breathtakingly naive spririt (that had served him well doing math) he believed he could start a movement that would change the world. But when he saw this was not succeeding, he returned to math, teaching at the University of Montpellier.
83. Fine, I understand that, but you haven't said what a scheme is. Also "variety" appears redundant; "set of solutions" is clear, simple and unambiguous. Why complicate your jargon?
""Grothendieck was so good at math, he could knock Chuck Norris unconscious using only the first 10 digits of pi."
"Grothendieck was so good at math, there are some numbers you can't use any more, because he used them up."
"Grothendieck was so good at math, he could divide by zero."
Originally, algebraic geometry involved solutions in real or complex numbers. (Usually the complex numbers, because that turns out to be much easier, since you can freely take square roots, etc., without having to worry about signs.) But the only things you need for the definitions to work is that you can add, subtract, and multiply. (A set where you can add, subtract, and multiply is called a ring, to punish Moby for his sins.) There are lots of rings.
So Grothendieck set out to generalize algebraic geometry to arbitrary rings. His generalization of a variety to this setting is called a scheme. Interestingly, if you start with a variety (over the complex numbers), there's a standard way to associate a ring with it, and in that case Grothendieck's construction doesn't give you anything new. It's for the other kinds of rings that you get something new. So there's a partial dictionary between varieties and rings, and schemes are missing entries in the dictionary.
Terminology is hard. I (or rather, two collaborators and I) introduced a useful technical term of the form "common English adjective + already existing technical term" to refer to something very specific, and now people have started using it all the time for things that are not instances of the technical thing we had in mind but share some vague properties associated with the common English adjective. It's getting confusing. I still think we made about the best choice we could have, though.
98 to, uh, lots of stuff halfway upthread since I haven't read the rest yet.
From the point of view of a non-math audience, is there any benefit to explaining the Weil conjectures in much greater detail than Landsburg's "The Weil conjectures make some very precise quantitative claims about how the number of (approximate) solutions grows as you become increasingly liberal"?
I mean, the best you can do in such a piece is create an illusion of understanding and provide inspiration to do some more reading.
You could give a very technical description; or you could take the technical description and "translate" the technical terms into more familiar language, which will generally add zero comprehension but make the description longer. Or you could take a few steps back and explain the view from 10,000 feet away.
Walt, you are a marvel of clarity. Please keep going!
|| So, our stand your ground shooter convicted. |>
Another example of a ring is the integers -- you can add, subtract, and multiply integers. Here the idea of schemes captures a weird idea that goes back to the nineteenth century. The scheme for the integers consists of one point for each prime number. So you can picture the integers as points on a straight line at 2, 3, 5, 7, ... and nowhere else. (Physicists would put an extra point at 9, and Grothendieck himself would put an extra point at 57.) So schemes are naturally related to number theory, and in fact have helped proved theorems in number theory such as Fermat's Last Theorem.
Allow me to acquaint you with something we like to call "the real world", Mr. Armchair Economist.
As I think was mentioned recently in the archives, this is a guy who had a real hard time figuring out why people walk up stairs instead of standing there like they do on escalators. (Though in fairness after a week or so he managed to work out a theory.)
Walt's doing a great job so far. I'm willing to forgive only talking about affine schemes (though it's a bit awkward since the Weil conjectures aren't about affine schemes, but there's probably a way to fudge it). That's the first spot where I would have tried to add a tiny bit more technicality.
Finite fields are coming next... There will be clocks.
At any rate, by talking about Spec Z instead of Spec C[x]/x^2 Walt has made a better choice than the article. Though he's going to end up with something at least twice too long for Nature (once you add back in the biographical stuff).
"There are only two hard things in Computer Science: cache invalidation and naming things."
98. Maybe a dark commenter that nobody perceives directly could explain how this kind of thing spreads.
On to the Weil conjectures. Think of clockwork arithmetic. You can add, subtract, and multiply hours or minutes on a clock face. In each case, you do the arithmetic with ordinary numbers, and then you throw away multiples of 12 (for hours), or 60 (for minutes). This operation of throwing away multiplies is called the "modulo" operator. So 7 times 2 modulo 12 is 2.
There are a couple of other instances of the modulo operator that you've probably used without knowing about it. Taking the last digit in a number is the same as that number modulo 10. So 1234 modulo 10 is 4. Adding up the digits of a number is the same as modulo 9. If you ever learned the trick to check if a number is a multiple of 3 by adding up the digits and checking that, you are actually working modulo 9.
(The mod 9 thing is a nice touch which I wouldn't have thought of.)
The court room erupted in applause when the verdict was read. People have been taking up collections, and cooking food for the kid's parents, here now for 3 weeks already.
The fact that when you add the digits repeatedly you get the number mod 9 might be the first fact about math that I ever figured out for myself instead of reading in a book. I was inordinately pleased with myself.
Sounds like the guy had an interesting 1969
In 1969, for reasons not entirely clear to anyone, he left the IHES where he had done all this work and plunged into an ecological/political campaign that he called Survivre.
"Grothendieck was so good at math, one time he was at a party and an amateur magician asked him to think of a number between 1 and 100, and so Grothendieck closes his eyes for a second, then smiles and nods at the magician, and the magician's head explodes! Bits of brain and skull fragments are everywhere! And all the party guests are screaming and climbing over each other trying to get out of the room, but Grothendieck is just standing there calmly, smiling and nodding."
"That was New Year's Eve, 1968."
Numbers modulo N give you another ring -- you can add, subtract, or multiply modulo N, and that gives you another number modulo N.
What's nice about numbers modulo N is that there are finitely many of them. They're also useful in number theory. Let's say that you want to know there are solutions to some polynomial equation over the integers -- say X3 + Y3 = Z3. One easy check is see if there are any solutions modulo N. If there aren't, then there aren't any solutions at all. So an interesting question for number theory is how many solutions are there modulo N?
Andre Weil (who's sister was Simone Weil) conjectured a kind of formula for the number of solutions modulo N. He did so via a far-fetched analogy with topology.
Take a disk (a filled-in circle), and consider a continuous map of the disk to itself. One example of a continuous map is a rotation, where you spin the disk around its middle. The point you spin it around is a fixed point -- it doesn't move. You can prove (and it's a difficult theorem) that every continuous map has to have at least one fixed point. There is a more general formula, called the Leftschetz fixed point formula, that allows you to count the number of fixed points in general (for shapes more complicated that disks).
113: That's the basis for two puzzles I wrote: Math Reunion and Father of the Brides. The first one is simpler, but I'm rather proud of the latter.
We had a puzzle thread a while back where the solution relied on modular arithmetic. (Something to do with prisoners who were not allowed to communicate with each other?)
For the integers modulo N, you can add, subtract, and multiply, but you can't always divide, and you can't always do things like take square roots. (Here, x is the square root of y modulo N if x*x is y modulo N. So 3 is the square root of 2, modulo 7. Pretty weird, huh?)
The division problem is easily fixed -- just make N be a prime. The root problem is harder to solve, since some numbers don't have square roots, cube roots, etc. even ifN is a prime. The solution is to add "imaginary numbers" modulo N, the same way that we add i, the square root of -1 to get the complex numbers. The complex numbers have an operation defined on them, called conjugation, that sends i to -i. There's a similar operation modulo N, called the Frobenius automorphism.
Weil said that we pretend that working modulo N was a kind of space, then we could apply the Lefshetz fixed point theorem, and count the number of solutions. This is a completely far-fetched anology, because there's no geometry here.
That's where schemes come in. Schemes supply the missing geometry. Grothendieck showed how to generalize the topological techniques to this setting so that a version of the Lefschetz fixed point theorem could be proven to settle the Weil conjectures. The proof is absurdly hard and abstract, but it is related to a relatively concrete question. (Unfortunately, the formula the conjectures give you is it itself a bit hard to use, so I don't know any easy explanation of what it means, but I think it does have some real-world applications in coding theory and cryptography.)
Okay, I don't know anything from math, but I have to wonder if Ursula Le Guin knew of this fellow (not totally impossible given her academic background and general social contacts). The Wikipedia article about him (which makes his politics sound much more comprehensible than the obituary does) reminds me strongly of Shevek in The Dispossessed.
105: I figured you would object. I think this is the kind of thing you have to bullshit about when you're explaining to a non-mathematical audience. People just want the general flavor.
I'm not sure the discussion of the Frobenius automorphism adds much and I'd cut it. Otherwise great job! Though I do think it's really really hard to cut that down a whole lot further to the length Nature wanted.
Oh wait, sorry, you need it later on of course. I got confused and thought you were trying to use it to explain algebraic closure and that didn't seem necessary to me.
I was improving the time during a faculty meeting trying to come up with an explanation of "variety" and "scheme" that wasn't, unlike the obit's, totally useless to non-mathematicians. Now that I come back to the thread, though, I see that Walt is on fire, and I will just bask in the light.
I thought I had written an exam that would take about an hour and a half, maybe two for the slow students. Two hours and forty minutes in, only one student has turned it in. Eek.
My first semester of grad school I decided to sit in on an algebraic geometry class. The whole first semester more or less was spent defining "scheme" and giving a few totally trivial examples and one incomprehensible one. That was my last attempt at attending math classes.
130: You should have left out the question about the Frobenius automorphism.
General rule the lecturer I worked for used: Students take 6X as long to work an exam as the prof or TAs. So, if you write it, then work through it writing out every answer completely like your students have to, it should take you 15 min or so.
Optimism: Grading the blank parts will be fast and easy!
This really is fantastic. Walt should right a math popularizing book. Working title: "The Flavor of Bullshit".
Thanks everybody! It was my pleasure.
Moby, for you there will be a test. Everyone else gets an A.
Bravo, Walt! That was super. (It was surprising to me but shouldn't have been that the Lefschetz fixed-point theorem is a hard result, given how central some other fpts are to undergraduate-level mathematics.)
"Grothendieck was so good at math that his Erdös number was -1"
This has the advantages of being
Some may think these advantages are actually disadvantages.
137: I suggest "Grothendieck was so good at math, Erdos check his Grothendieck number" as a modest improvement.
Loads of non-mathematicians know what an Erdös number is!
Yeah, I read a whole book about Erdős! Didn't learn anything about math, but it was great.
I hate to say it but Nature is right (albeit perhaps not for the articulated reasons). The obit is written like a math paper - start of strong - general, accessible, etc. and then run directly into jargon. It goes off the rails here:
"the locus of solutions of sets of polynomial equations by combining the algebraic properties of the rings of polynomials with the geometric properties of this locus, known as a variety."
Jibber jabber to most nature readers despite their training in mathematics. "Locus" is too technical - how about collection? Rings? Seriously - why should biologists, chemists, etc. have come across the definition of a ring unless they are in specific sub-disciplines. The label of "variety" is useless in the context of what they write about - in fact it never appears again.
It gets worse from there.
Now, I have great respect for Mumford an Tate but the writing here completely misses the mark. How can they not see this?
Is this another member of Judas Priest?
Cryptic ned gets extra points for doing the correct Hungarian accented-umlaut thing.
I only read the first 60 comments or so, but I like to think I'm still a math person. Grothendieck is weird. Very weird. The obit was poorly written and feels stilted even when talking about his very interesting non-mathematical life. I understand they might not have a lot a space, but they should have tried harder. On the other hand, Nature's response about complex numbers is ridiculous.
LB, as usual, really hits the nail on the head in 24 and such. It is sometimes very difficult to tell which words are jargon, and even then the jargon isn't consistent. I was taught that a ring always has a multiplicative identity ("a 1"), but that is not always the case. Thus, every book talking about rings has to define them and you need to familiarize yourself with the local dialect. For another example, I have a vague sense that the kernel of an algebraic object is something that's kind of, but not, quite, like nothing (i.e. in some sense it maps to nothing), but it's defined slightly differently for each sort of interesting algebraic object and I've since mixed them all up.
The absolute worst case I've experienced was in a logic class I was grading, where (forgive my metasyntactic variables) a foo of bar was defined as something and lower on the same page the foo of bar was defined as a particularly important foo of bar.
In my work as a programmer mostly using object-oriented languages, there's an alternative scheme for making jargon: simple English words with jargony meanings strung together. A FooFactory is a thing that makes Foos. a DelegatingBar is a Bar that takes another Bar and depends upon that Bar's behavior for some or all of the DelegatingBar's behavior. Thus a DelegatingFooFactory is a thing that makes Foos by asking another Foo factory to do it, possibly with some modifications in the process. So, you get the simpleish English words at the root, but a clear jargon out of it by being agglutinative.
Sometimes these worlds collide; a friend pointed me to the code for an audio library that had interfaces with names like AbelianGroup. I suspect they were over abstracting.
I only read the first 60 comments or so, but I like to think I'm still a math person.
You really should read the rest of the thread, especially Walt's comments starting with 83.
Thanks. Did so. Very good read. "Schemes are varieties over [arbitrary] rings" isn't something that I got from the obit.
Grothendieck was so good at math, his Erdos number is a differentiable fiber bundle over the Erdos numbers.
The division problem is easily fixed -- just make N be a prime.
I know this makes me such a Nature reader, but I don't get how 3/2 mod N is an integer mod N, even if N is prime. I also don't get how the square root of 2 is an integer mod N after you get finished waving i at it.
Grothendieck was so good at math, his obituary led to discussions pf the comprehensibility of mathematical language.
Grothendieck was so good at math, his obituary led to discussions pf the comprehensibility of mathematical language.
Grothendieck was so good at math, his obituary led to discussions pf the comprehensibility of mathematical language.
JP was so good at bureaucracy, he posted his comments in triplicate.
I don't get how 3/2 mod N is an integer mod N
Wouldn't 3/2 just be the integer that when multiplied by two yields three? So if N is five, 3/2 is four.
Geez, LB, they won't even leave the poor / symbol alone.
(Quick and mildly mathy; apologies for being a real Nature obituary reject) 150: We want to show that if N is prime, for every x in {1, N-1}, there's a y such that x*y = 1 mod p. (That is, y is the multiplicative inverse of x.) If so, then we can treat y as the reciprocal of a (mod p). We do this by showing that for non-zero x, for each z, w in {0,N-1} x*z != x*w mod p. If those N values are all unique, one of them must be equal 1.
So suppose that that doesn't hold, that x*z = x*w mod p. Then:
x*z-x*w = 0 mod p
x*(z-w) = 0 mod p
Thus, the left side is a multiple of p. p is prime (important!) so either x or z-w must be a multiple of p (as p has no other factors). By assumption x is in {1, N-1} mod p so it isn't a multiple of p, so then z-w must be. But then z = w mod p, which is what we wanted to show--multiplying the modular residues of p by x doesn't reduce the number of things you have. And thus every value in {0,N-1} can be obtained by multiplying x by something, including 1, and so we've found our multiplicative inverse.
As a counterexample for why N must be prime, suppose instead N=6. The multiples of 2 mod 6 are:
2*0 = 0 mod 6
2*1 = 2 mod 6
2*2 = 4 mod 6
2*3 = 0 mod 6
2*4 = 2 mod 6
2*5 = 4 mod 6
So 2 doesn't have a multiplicative inverse. Note that the multiples of 2 are all even (err, obviously, I suppose) which can lead us to fun things like quotient groups and whatnot.
Amusement for the innumerate: Andre & Simone Weil as kids, in the Petrement bio of Simone:
She had great admiration and a warm feeling of friendship for her brother. She considered it an honor to follow him in all his boyish pranks. One day they both went around, holding each other by the hand, and knocked at the doors of some of the neighboring villas. To those who came to the door they said, "We're dying of hunger; our parents are letting us die of hunger." "Poor children!" the people exclaimed and immediately gave them cake and candy. They came back home stuffed and very satisfied. When their parents found out what they had done, they were overwhelmed with shame and indignation. They tried to keep the children busy and amused, for they were absolutely unbearable when they were bored. (11)
It was probably at this time that André decided not to wear knee-socks. He wanted to become tougher. Simone imitated him. In the street people would stare at the two children with astonishment. Mme. Weil, terribly embarrassed by this, tried to put knee-socks on Simone at least. But the children had learned how to cry at will . . . Then they invented the trick of making their teeth chatter while they were on the streetcar, saying over and over, "I'm cold, I'm cold. Why don't our parents want to buy us knee-socks?" The other passengers would stare at Mme. Weil with fury in their eyes. (15)
Couldn't they have just written:
"Grothendieck has died. You wouldn't understand."
and now people have started using it all the time for things that are not instances of the technical thing we had in mind but share some vague properties associated with the common English adjective.
Quantum.
162 would be a truly great obituary.
Grothendieck is dead. Your ignorance drove him into hiding for 20 years, and ultimately killed him.
Following up on 159 with a specific example:
If you work in mod 5, you have the following situation:
Multiples of 1: 1, 2, 3, 4
Multiples of 2: 2, 4, 1, 3 (corresponding to 2 4 6 8)
Multiples of 3: 3, 1, 4, 2 (corresponding to 3 6 9 12)
Multiples of 4: 4, 3, 2, 1 (corresponding to 4 8 12 16)
Every row contains each of 1, 2, 3, 4 exactly once. That means every number has a multiple equal to 1 and thus we can always find the "reciprocal" of a number. In particular 1*1=1, 2*3=1, 3*2=1, and 4*4=1.
The key is that for any prime modulus, the multiples will show the pattern above (every number appearing once). If that wasn't the case, some number would appear at least twice in a row and the difference between those multiples would be 0. That would imply that 0 is a multiple as well (because the difference of two multiples is a multiple). But that's not possible when you are working with a prime: two numbers multiply out to 0 only if at least one of them is 0 (this is not true with non primes, see 160). So that proves that each number appears once in each row, which means division is well defined.
He developed my favorite scheme, you probably haven't heard of it.
Since Nature wouldn't publish their Grothendieck obituary, maybe they should send it to Field and Scheme.
Actually, this reddit math thread I linked right after he died has some decent explanations.
Sometimes these worlds collide; a friend pointed me to the code for an audio library that had interfaces with names like AbelianGroup. I suspect they were over abstracting.
Was it be Ed Kmett?
In a funny moment of "Why common English words are bad choices for technical terms", I came into the office this morning to find a letter in a very sensitive case on my chair (actually, one of a group of precisely similar letters we've gotten in similar cases, that are sensitive as a group). The prior letters were addressed to me, and we've been discussing an approach for responding to them. This one was addressed to a line attorney, and she left it on my chair with a note saying that I should be aware of the letter, and "I did answer."
And I was horrified. You answered the letter? What on earth did you say? You didn't talk to anyone about it? This is really important -- how on earth did you think it was okay to just answer?
I stormed into her office, and started biting her head off. And then it became apparent that she had done nothing wrong, in fact had done nothing at all, and her note meant to convey that the letter was addressed to her because she had been responsible for the answer in the case back when it was filed.
Bit of an adrenaline rush there.
In a funny moment of "Why common English words are bad choices for technical terms", I came into the office this morning and discovered my coworkers loudly talking about "energy".
I have forgotten all the details of that argument, but still maintain vociferously that I was right and not even a little confused.
174: In a funny moment of "Why common English words are bad choices for technical terms", I came into the office this morning to find a letter in a very sensitive case on my chair ...
I take it that you do not mean that the letter was not all caps (i.e., not all upper case, which would not have been very sensitive at all)
177: It was in a very sensitive case and when she touched it, it exploded.
It was in a very sensitive case and when she touched it, it exploded.
So that's why they call it the "ablative."
Grothendieck Grottlesex would be an excellent name for a something or other. Mervyn Peake character? British actor playing Sherlock Holmes on TV?
The math portions of this thread sound like this to me: "Math starts out simple, but mathematicians are always adding stuff to make it more complicated. Then they need someone to come along and explain how all the complicated stuff is actually simple. The deceased was one of those." Is that about right?
Grothendieck generally made things more complicated, rather than simpler. The literal definition of variety is basically "solutions to some polynomial equations". The definition of scheme is so complicated that essear had an entire semester-long class on just describing the definition. A large portion of Grothendieck's output was in the form of thousand-page manuscripts.
Grothendieck's talent is that he could identify the exact right technical definition that would allow you to prove a result in much greater generality than anyone would have imagined possible. There is a certain geometric picture that explains why some functions have fixed points. The geometric picture is pretty simple. Grothendieck figured out that almost all of that geometric picture is totally irrelevant, and that certain complicated abstract facts were all you need.
Walt, you obviously understand algebra better than I have been exposed to. I once tried to explain to one of my coworkers why a Rubik's cube was a problem in algebra, that is, group theory. Your explanation seemed to me to always have letters standing for numbers of some sort or another. Is there a sensible application where they do not?
Thanks.
182: Letters can stand for all kinds of mathematical objects, not just numbers. The Wikipedia article on the Rubik's Cube group starts off referencing the letters G, F, B, U, D, L, R, and E (for the group, the six basic moves, and the empty move), and none of them is standing for a number in the ordinary sense of that phrase. But it is still perfectly good algebra and group theory.
Thanks. I was unclear.
I wondered if varieties and schemes had a sensible example outside of numbers, in some sense.
182: Good question. While schemes can be defined over rings that are not numbers, they all vaguely look like functions on numbers. (You can add, subtract, and multiply functions in the obvious way, i.e. f + g is the function given by f(x) + g(x).)
There's the example from the obituary: you allow an extra "infinitely small number" dx such that dx times dx is zero.
I stand in the perfect intersection of two sets here -- Nature reader, and bewildered schmuck. But you get a higher class of bewilderment on unfogged.
an extra "infinitely small number" dx such that dx times dx is zero
And then you can make those numbers anticommute and we can talk about fermionic dimensions and supersymmetry.
Now I'm wrestling with the mathematics of how to assign vastly different learning outcomes a compressed range of grades between B- and A.
You should do like the financial rating companies. I think they have seven or at grades between those.
Am I allowed to skip directly from B- to C and bypass C+? Because I feel like two of these people really deserve to get a C-. I assume I'd be fired immediately if I gave out a D or an F.
Or is C and F only for a major and C is a D for a nonmajor?
an extra "infinitely small number" dx such that dx times dx is zero
Thinking about this gives me vertigo.
Or is C and F only for a major and C is a D for a nonmajor?
C, E, and G are for major, and C, E flat, and G are for minor.
I was thinking that maybe to translate a grade at my institution to what they really should mean you pretend ours is a minor scale and pick the relative major, but that would map our C to an E flat instead of an F.
Also it would mean I'd have to give the really good students an F-sharp, right?
I still have the vertigo. At the edge of the bottomless pit there's a decimal point, and a 1 has fallen in, and as it plummets eternally down towards nothingness it leaves an infinitely long trail of zeroes in its wake. At some point in its descent it will reach the infinitely small number dx, but when?
"Zero" seems like a really big number right now. Impossibly big.
I had a similar feeling with calculus when all the little rectangles measuring the area below the curve were suddenly supposed to get so small that the little corners of the curve above the lines disappeared. I decided to just take Newton's word for it.
Speaking of which, can anyone recommend a good teach-yourself-calculus book? Say, aimed at someone who was very good at math in high school but then hit a brick wall when they encountered calculus in college, but years later feels like having another go at the beast?
202: Check if Grothendieck wrote anything on the subject. I hear he's pretty accessible.
Really, if you can't follow Philosophiæ Naturalis Principia Mathematica, you'll never understand it properly.
202: MIT's freshman calc course online? It's got the lectures and the problem sets, tells you what textbook to buy, and so on.
Khan Academy is pretty good. I liked it for a while but I also needed basic trigonometry and algebra before calculus and it was just taking too long so I got bored and lazy and moved on to another leisure time activity, meaning I'll probably never learn calculus. Also when I did my initial diagnostic test with them I think they thought I was at about a 6th grade level. But the lectures are clear and the problems are fun and you can do it all on your computer.
The MIT course looks good, but given my low level of commitment to the project I think I'll probably give Khan Academy a try first.
so I got bored and lazy and moved on to another leisure time activity, meaning I'll probably never learn calculus
This is the probable outcome for me as well.
202: Calculus Made Easy by Silvanus P. Thompson. I am totally serious, but I also haven't re-read it since I learned calculus from it.
Hey, Silvanus P. Thompson taught me calculus too! For me it was an introduction not only to calc but to a certain kind of donnish humor.
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.
Some calculus-tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics--and they are mostly clever fools--seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.
Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.
Oh, italics don't persist across paragraphs. That's Silvanus calling himself a stupid fellow, not me, though it would be no harm if it were otherwise.
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Help with fonts.
Can anyone tell me the name of this font?
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Which font? The font of the site? The font of the name on the first set of cards? On the second set? . . . ?
I believe (with moderate confidence) the cards linked in 212 are printed in Calibri.
(On preview: I'm answering about the font on the cards labeled "Modern Business Card 024".)
215 was what I was asking about. The Modern Business Card 024.
I'm also interested in the fonts in this card.
216: That appears to be something in the ITC Blair family.
