Examples First highlights

Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.

Rereading a bit, that blog post comment section is probably the original source of my minor obsession with the role-of-intuition-in-maths literature.

(This isn’t actually a particularly major real-life obsession, just a secondary one that I like writing about/have worked out how to write about/have bored everyone I actually know with so I need to go talk about it somewhere else where people can opt out easily)

The only thing I can’t find in there is the Vladimir Arnold rant, which I must have picked up somewhere else. And actually Rota doesn’t make an appearance either. But there’s a load of the standard lore, the usual quotes from Thurston and Poincaré (it’s always the same quotes because this subfield is tiny). 

I’d kind of forgotten that, because I think of it more as my starting point in figuring out how to learn differential geometry. Following the references in the comments taught me more than any maths course I took in undergrad. 

Anyway, some highlights:

  • “I have a colleague (in CS, not math) who reads papers as follows: First he skims the paper by skipping all English and reading only formulas, then he reads the introduction, and then he reads again forcing himself to read some of the English too.”
  • that story about Grothendieck thinking that 57 was prime
  • “One day I realized it was all a lot clearer if I specialized the arguments. As a simple example, a theorem about differentiable real-valued functions on an interval might reduce to the case of the behavior, at 0, of a differentiable function f satisfying f(0) = 0 and f'(0) = 0. Cosmetic assumptions like these simplify the difference quotient and make the key issues clearer (to a novice anyway). The “general case” of such a theorem is often the result of composing the specific proof with an affine transformation. The symbols implementing this transformation play no essential role in the argument.”
  • Fields Medallist admits they never really understood what all that Sylow subgroup stuff was on about
  • ”My undergraduate days left me afraid of many subjects: complex analysis, measure theory, most of algebra and almost all geometry, for example”
  • link to John Baez on normal subgroups. I gave up with trying to understand group theory when I didn’t understand what the definition of a normal subgroup was on about. Unfortunately this is like week 3 of an intro to group theory course, so that was kind of it for me and abstract algebra. I clicked on the link but never read it properly and still don’t really get what a normal subgroup is. Even so, after reading this I felt better for realising it wasn’t some completely obvious thing I ought to ‘just get’ and didn’t.
  • “Dualization is a rather simple idea but I think it is perhaps one of, if not the, most powerful tools in mathematics, especially in the modern era. There is, I’m sure, a good story about why. Perhaps someone can explain or tell me where to find an explanation?”
  • someone asks how to start learning differential geometry and a student of Chern turns up to answer
  • ‘I like to call differential geometry “nonlinear linear algebra”.’
  • a long involved interesting argument about whether you should identify the tangent and cotangent space when you can to save bothering to keep track of a distinction you don’t need right now, or whether you’ll confuse yourself more in the long run
  • anecdote about helping a six-year-old who “could do 3+2 with no problem whatsoever. In fact, she had no trouble with addition. She just couldn’t get her head around all these wretched apples, cakes, monkeys etc that were being used to “explain” the concept of addition to her.”
  • “I was talking to two students about conjugation and talked about how gfg^{-1} is the function that takes g(x) to g(y) if f takes x to y. I then asked them to come up with a function from the reals to the reals that takes x^3 to (x+1)^3 for every x. After a while, one of them had the idea of taking the cube root, adding one, and cubing. But it was clear that he did that by forgetting all about my discussion of conjugation and just looking at the example. Only afterwards, when I pointed it out, did he realize that he had just done a conjugation.”
  • “Kazhdan’s advice to my friend: You should know everything in this book but don’t read it.”

deep paths of mathematics

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That education thing made me remember I wrote something about my maths teacher a few months ago as setup for a complex point that I never wrote down and have now completely forgotten. I’m probably not going to remember so let’s post it as is:

So I’m digging through some differential equation stuff trying to fill a few gaps in my knowledge, mostly arsing around ‘doing some prerequisites’ for quantum field theory instead of just jumping in. This time it’s the Fredholm Alternative. It looks like one of those bits of arcane mathematical methods textbook lore with the silly names, like the bilinear concomitant or Rayleigh’s quotient or ‘the method of undetermined coefficients’, which I always thought was just called ‘guessing’. This looks pretty useful and general though, looking at some inner product to see whether boundary value problems have one solution or no solutions or infinitely many solutions.

Actually it looks a bit like… oh, yeah, look, there’s even a matrix version. In fact,…

I’m in a classroom with the other Further Maths nerds. It doesn’t fit on the timetable so we’re stuck in there again after school, eating vending machine sweets and solving systems of simultaneous equations using Gaussian elimination. As always, our teacher has gone beyond the rote work of the syllabus and is making sure we understand what’s going on geometrically, using three planes as an example. We’re systematically working through the possibilities: all three planes parallel, two parallel and one crosses them (‘British Rail logo’), all three intersect along a single line, all the rest. We can end up with no solutions, or a unique solution at a single point, or a whole line or plane of solutions.

Then someone looks out the window. It’s someone’s brother in Year 8, mucking about on the flat roof across the playground. He thinks everyone’s gone home.

Our teacher opens the window, still caught up in his system of equations. “GET OFF THAT PLANE!!! NOW! YOU’RE IN DETENTION TOMORROW!”

And afterwards, as the kid complies: “…did I just say ‘plane’ instead of ‘roof’?”

We go back to the example. I get plenty more linear algebra next year at university, but however abstractly they dress it up, in my head it’s just the same old intersecting planes.

I’ve seen the words ‘Fredholm Alternative’ somewhere else, though, too, on one of my unmotivated afternoons in the library pulling books off the shelves. Ah yes, googling around it must have been Booss and Bleecker’s Index Theory, which aims to drag even applied mathematicians up to the heights:

Index Theory with Applications to Mathematics and Physics describes, explains, and explores the Index Theorem of Atiyah and Singer, one of the truly great accomplishments of twentieth-century mathematics whose influence continues to grow, fifty years after its discovery. The Index Theorem has given birth to many mathematical research areas and exposed profound connections between analysis, geometry, topology, algebra, and mathematical physics. Hardly any topic of modern mathematics stands independent of its influence.

And there’s the Fredholm Alternative in Chapter 2, one of the steps on the path.

Maybe I’m not just digging out random crap from the textbook. It looks like I accidentally found one of the Old Ways of mathematics, linking my A Level classes with some great confluence up in the stratosphere. With like a million steps above me still, but sometimes it’s nice just to know you’re on the path.

tastes in the head

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[I don’t really know where I’m going with this post, but my brain seems to be fixated on writing it. It never comes out right, so this time I’m just going to push it out the door in whatever confused form I can manage, and then hopefully my brain will shut up.]

Tastes in the head is some idiosyncratic piece of mental furniture I have cluttering up the place. I’m not even sure where the best place to look is if I want to find a better vocabulary for this sort of thing (phenomenology? meditation practice? do I have to drag my way through Heidegger or something? ugh). I first got it from Empson’s Seven Kinds of Ambiguity:

“… what the poet has conveyed is no assembly of grammatical meanings, capable of analysis, but a ‘mood’, an ‘atmosphere’, a ‘personality’, an attitude to life, an undifferentiated mode of being. Probably it is in this way, as a sort of taste in the head, that one remembers one’s own past experiences, including the experience of reading a particular poet.”

He obviously liked the phrase, because he uses it again a bit later to snark about the Romantic poets:

“They admired the poetry of previous generations, very rightly, for the taste it left in the head, and, failing to realize that the process of putting such a taste into a reader’s head involves a great deal of work which does not feel like a taste in the head while it is being done, attempting, therefore, to conceive a taste in the head and put it straight on to their paper, they produced tastes in the head which were in fact blurred, complacent, and unpleasing.”

I don’t know if I’m that close to what he meant, but I interpret his ‘tastes in the head’ as that kind of preverbal emotional tone ideas have – once you have some description in language you can manipulate that and catch some of the meaning, but there is also some qualitative essence that is harder to get at. I’m not going to give any examples now, because 1. that ‘preverbal’ bit makes it really hard, 2. the ones I tried were really distracting and ended up taking over the post, but I will quote a couple of sources further along that hopefully clarify it a bit.

This is a bit of an aside, but I’m really not aiming for clear structure in this one: while I’m talking about the the New Critics, there’s an interesting link here to the idea of the ‘objective correlative’ popularised by T.S. Eliot. I often see this used to mean just, like, imagery in literature that helps to convey the emotional tone of the piece, but iirc he originally used it to make a much stronger, wrong-but-interesting claim: that each image in literature maps to a distinct ‘taste in the head’ – a specific nonverbal emotional tone – that is ‘objective’, i.e. the same for each person (provided they have cultivated the Correct level of literary sensitivity, is the disclaimer at this point.)

This would be amazing if true: we could generate a load of poetic imagery, discover exactly what impossible-to-convey-with-normal-language emotional tone it mapped to, and then produce a giant lookup table that people could use to reliably convey the background texture of their thoughts, and then nobody would ever misunderstand each other ever again. E.g., taking the latest SSC post as an example, Scott digs up the image that exactly corresponds to lived-experience-sympathy-for-the-plight-of-overworked-junior-doctors (it’s probably some kind of whale), sticks it in the post, and we all understand it on an intuitive level and none of us ever need to argue about it again. This sounds like the ultimate rationalist project, but unfortunately we’re going to fail given the obvious problem that the same image provokes very different responses in different people. (And even in the same person at different times.) It’s actually hard for me to believe that this idea was even on the table, but, well, behaviourism was popular at one point, and this is a good deal less reductionist.

Normally I approach this stuff through my endless droning on about the role of intuition in maths. In that case I sometimes also think of the feeling as ‘falling in love with the gears’. This comes from the intro to Seymour Papert’s Mindstorms (pdf) where he talks about his early fascination with cars, and how playing with gears helped his early intuition for maths:

“I believe that working with differentials did more for my mathematical development than anything I was taught in elementary school. Gears, serving as models, carried many otherwise abstract ideas into my head. I clearly remember two examples from school math. I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x + 4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend.

“Assimilating equations to gears certainly is a powerful way to bring old knowledge to bear on a new object. But it does more as well. I am sure that such assimilations helped to endow mathematics, for me, with a positive affective tone that can be traced back to my infantile experiences with cars.

“A modern-day Montessori might propose, if convinced by my story, to create a gear set for children. Thus every child might have the experience I had. But to hope for this would be to miss the essence of the story. I fell in love with the gears.”

That ‘positive affective tone’ is what I mean by ‘taste in the head’, and for me learning maths is all about the process of finding that kind of tone in new areas:

As an example I’ve been going on about a lot, for me at the moment differential geometry is strongly associated with a positive tone, and abstract algebra most definitely isn’t. (This hasn’t always been true, so hopefully one day I can like both!) Lie groups provide one natural bridge, as an algebraic object that is also a differentiable manifold. Using that I can start to see a path to more “unpleasantly algebraic” components, like, I don’t know, classifying Lie algebras or something, and hopefully later I can move further along it.

So this has mostly just been a lot of quoting various links. I was going to try and make some rambling ill-defined claims at this point, but I don’t think I’m going to be able to manage that this time so I’ll just wave vaguely at one of them.

I must be unusual in that one part of Less Wrong that I actually like is all the stuff about ‘akrasia’ and reducing procrastination – I don’t get a whole lot of practical use out of this, but the vocabulary of ‘ugh fields’ works very well at communicating the pre-verbal feel of what’s going on. Some of Alicorn’s ‘Living Luminously’ posts are even closer to what I’m interested in, including a much more developed idea of ‘hacking yourself’ to like things (like my Lie group example, but she seems to manage a greater level of control.)

When I read this stuff (also the parts about cognitive biases, Kahneman’s System One, etc., but there I haven’t read so much of the LW content), LW seems really good at taking all this preverbal substructure seriously. And then suddenly, bam!, I’ll be reading something else and it’s all about fitting everything into some incredibly restrictive language-based formal structure. I don’t know, I am possibly just missing the part of my brain that can get anything out of philosophical discussions of ethics, but this part of LW in particular gives me this strong feeling of “how the hell have have you reduced this gigantic mess of tastes in the head to a clean conceptual system, which in certain versions is clean enough to actually take values in the real numbers? This is worse than the T. S. Eliot Emotional Lookup Table! And more importantly, what the hell have you left out by doing this?”

I don’t know, sometimes formal systematizing does work, really well. I would be very surprised if this is one of those times.

OK, I’m going to finish here, and push it out the door like I said I would. Hopefully this has got some of what I wanted out of my head.


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Actually programming is doing one good thing for me, by forcing me to engage with a more algorithmic/symbolic mode of thinking that I’ve kind of ignored as much as I can in the last few years. I find it frustrating a lot of the time, partly because it’s not particularly how my brain works but also because normally the structural component is the the main thing, there is no there there. (Or what’s there is incredibly intrinsically uninteresting to me, like parsing some file or whatever).

At the other end of the spectrum is differential geometry, which I have this kind of doomed love for despite not being especially good at it. I love it because the questions are so tangible – ‘how does this surface curve?’ – and the particular methods you use are correspondingly less important if you have the right intuition for the tangible problem. I mean they are still important, there are definitely more and less elegant ways of doing things, but structure is at least somewhat downplayed compared with the actual thing you want to know about, which is how this surface curves.

I mean I found a differential geometry book by Serge Lang in the library once, I don’t think I dreamt it, and it was a proper Bourbaki-style algebraist’s version of differential geometry. No pictures and everything was done with some kind of quadratic form iirc. After that I was kind of convinced that you can build differential geometry out of anything you have to hand.

Examples first

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I finally got nerd-sniped by the Arbital thing, so here are some rambly thoughts.

I’m definitely intrigued by the idea, because there are a lot of topics in maths that can be understood at very different levels. It’s true that there is often a level of understanding below which you have very little hope (e.g. @nostalgebraist‘s example of reading the integration by parts page without calculus). But often there are many tiers of understanding above the first. E.g. Thurston talking about different concepts of the derivative:


So I can see some potential here. However I’m not at all taken with the current implementation. The biggest dud for me is the clunky, prescriptive questionnaire interface over the top of it. I’m normally pretty good at identifying whether I can follow an explanation once I can actually see it, the main advantage I can see to the site is having many such explanations in the same place for easy access. I don’t want to be tediously clicking through branching pathways, like some Choose Your Own Adventure book where every adventure is just Bayes’s theorem again.

To my mind that part of Arbital’s just plain bad, but another aspect of the site that I don’t like may improve with time. At the moment it’s heavily curated, giving it a very homogeneous textbook feel. It sounds like the idea is to allow users with enough karma to contribute themselves, and then it may become more diverse.

I think the thing I would like is more *styles* of explanation rather than particularly different *levels*. For example, here’s the earlier part of Thurston’s list [this is such a wonderful paper, if for any misguided reason you are reading all this rubbish you should just go and read that instead :)]:


These are very different ways of thinking of a derivative, but I wouldn’t say that they are at different levels. I think different ones will appeal to different people, and your ideal starting point will vary depending on that. (Eventually you need to learn the others, of course, but initial motivation is important. For me (1) and (4) feel the most natural, and motivate me to learn the incredibly necessary (2), and even to deal with the tedium of (3)). I guess my ideal maths-explanation site would have a variety of explanations at each level.

[At which point, is it even worth trying to collect all this disparate stuff on one site? I honestly don’t know.]

My final bloody obvious objection is that politically they should definitely not have gone for Bayes’s theorem as a nice uncontroversial starting example instead of basically any other topic in mathematics, but, well, Yudkowsky and doing the politically sensible thing rarely go together.

Still, after all that grumbling I do appreciate any attempts at providing better explanations for mathematical concepts online. I find this stuff really interesting for some reason, and the idea I personally like to think about is an approach I call ‘examples first’, after these two blog posts by Timothy Gowers. (The second one has an absolutely epic comment thread – reading that and following the links has taught me more maths than any single course I ever took at university.)

I always like to learn by following concrete worked examples. This may just be a personal preference, but it sounds from the blog post that it’s pretty common. In my case, out of the Arbital explanations the one I’d personally choose was the beginner-level one (so much for that questionnaire). I would always rather learn by doing problems about socks in a drawer than read an explanation in terms of some abstract variables A and B. If I’m learning maths from a textbook I always start by looking at the pictures and reading any waffly chunks of text, then look at the examples and exercises. I only grudgingly read the theory bit when I’m really stuck. 

I guess what I would like is something like a repository of worked examples, where you search for a topic and then get a bunch of problems to try. Wikipedia generally ends up with a formalism-heavy approach, whereas I would always prefer to look at some specific function, or a matrix with actual numbers in it or something.

stupid bat and ball

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The Cognitive Reflection Test came up in the SSC Superforecasters review. I’ve seen it a couple of times before, and it always interests me:

  1. A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?
  2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
  3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

I always have the same reaction, and I don’t know if it’s common or I’m just the lone idiot with this problem. The ‘obvious wrong answers’ for 2. and 3. are completely unappealing to me (I had to look up 3. to check what the obvious answer was supposed to be). Obviously the machine-widget ratio hasn’t changed, and obviously exponential growth works like exponential growth.

When I see 1., however, I always think ‘oh it’s that bastard bat and ball question again, I know the correct answer but cannot see it’. And I have to stare at it for a minute or so to work it out, slowed down dramatically by the fact that Obvious Wrong Answer is jumping up and down trying to distract me.

I did a maths degree. I have a physics phd. This is not a hard question. Why does this happen?

I know I have a very intuition-heavy style of learning and doing maths. For the second two I have very strong cached intuitions that they map to, whereas I’m really lacking that for the first one for some reason. I mean, I can visualise a line 110 units long, and move another 100-unit line along it until there’s equal space at each end, but it’s not some natural thought for me.

Now, apparently:

The CRT was designed to assess a specific cognitive ability. It assesses individuals’ ability to suppress an intuitive and spontaneous (“system 1”) wrong answer in favor of a reflective and deliberative (“system 2”) right answer.

Yeah so that definitely isn’t getting tested for me. My System 2 hates maths and has no intention of putting in any effort on this test, but luckily System 1 has internalised the ‘intuitive and spontaneous’ answer for two of the questions for me. I will fail the first question unless my equally strong ‘the answer can’t be that obvious’ intuition fires, but that one makes me seriously worried about my answer to 3. as well.

My inability to internalise the bat and ball thing might be a quirk of my brain, but I’m sceptical of this test in general. It’s extremely vulnerable to having the right cached ideas.

free books and a rant

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Today I discovered that Springer has put a load of maths and physics textbooks online for free – here’s a helpful list someone made.

I thought it would be just duds from the back catalogue but they’ve put up loads of well-known texts at the undergrad and graduate level – particularly good for maths but some of the physics ones are also worth a look.

… and now for a pointless rant.

Why do so many maths textbooks insist on having this fucking boring introductory chapter that tells you a million preliminaries in incredibly terse prose? More symbols than words if you can possibly manage it? Like this:

I’m using Sachs and Wu’s General Relativity for Mathematicians as an example as that is the pdf I have open at the moment, but it’s not unusually bad, I could use anything really. And judging from the Preface this book is actually going to be pretty opinionated, with a distinctive writing style:

Many people believe that current physics and mathematics are, on balance, contributing usefully to the survival of mankind in a state of dignity. We disagree. But should humans survive, gazing at stars on a clear night will remain one of the things that make existence nontrivial.

That suggests a book that could be fun to read. Then it’s straight into exciting pages like this:

Who wants to read this stuff when they’ve just picked up a new book? It’s incredibly boring and does nothing to help me decide whether I’ll get anything out of the rest of the book. Why not, say, a basic example that illustrates something of what they want to cover? Or something interesting about the history of the subject? Or just a general overview of what’s coming up?

I’m sure there’s a reason I’m not getting, there usually is.

Do they want it to be self-contained? Well it’s still not self-contained, are they planning to teach me to count as well? It’s not the only book in the world anyway, surely I could just look at another book?

Do they want fix notation? That sounds a lot more reasonable, but surely they could just introduce the concepts in the context that they’re going to be used so that you actually remember them, with maybe a glossary of notation at the end?

I don’t know.

boring Saturday morning maths ramblings

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So yesterday I read about DragonBox, a game for learning algebra. I haven’t played it myself yet, just seen some videos, but it looks like you learn the rules by manipulating a bunch of pictures on a touch screen, and only later see the usual symbols and numbers.

Anyway I sent it to my sister, and it sounds like my 7-year-old nephew was happily playing it most of the evening. So that’s nice.

I read about it on Hacker News, and today there are a load of comments there probably duplicating a lot of the content of this one. I can’t be bothered to read them all right now, I’m in a writing mood more than a reading mood.

What interests me about this game is its complete focus on teaching the algorithmic component of algebra – cancelling factors, ‘throwing things over the equals sign’, multiplying both sides, whatever. Algebra as game mechanics. And that’s likely to be a good thing, as games are generally somewhat more fun than algebra classes. Learning the rules by playing around and seeing what happens is likely to be more successful than being told explicitly what the rules are, and being afraid to experiment too much in case you ‘get it wrong’.

Of course, the downside is that it’s completely divorced from a conceptual understanding of what you’re doing, and how it relates to the maths you already know. I’ve always been kind of annoyed at how late algebra is taught in schools, and how its separated so distinctly from arithmetic. I feel like a good start on the way to algebra is made right at the beginning of school, where you get those worksheets with the boxes to fill in:

1 + 3 = [ ]
2 + 4 = [ ]

and then for a bit of variety you might get

2 + [ ] = 5

later in the sheet. No big deal. Then somehow later in school the first two examples become ‘arithmetic’ and the third is some abstruse topic called ‘algebra’.

I feel like if it was introduced alongside arithmetic it might be easier to take in. E.g. when you first learn multiplication:

2 * 3 = [ ]

then why not also learn

2 * [ ] = 6 ?

It seems unfair to save all this stuff up for a few years and then intimidate you with the likes of

2 * [ ] + 3 = 7,

under the threatening new title of ‘algebra’, along with an array of confusing new algorithmic techniques for ‘solving’ an equation.

I’m not trying to criticise DragonBox. I think it’s a great idea. I guess what I’m wondering is what DragonBox’s twin looks like. The game that teaches conceptual understanding of algebra divorced from algorithmic understanding, with the same emphasis on playing around and not worrying too much about whether you’re doing the right thing. E.g. in the equation above you could just try some numbers and find that 2 works, or notice that 4 + 3 is 7 and work backwards, or anything else that helps. It would be nice to be able to use a bunch of examples like these to work towards finding a general algorithm for solving the equation, but one you’re using because it makes intuitive sense to you rather than because some teacher or some game told you to do it.


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Currently flicking through Seymour Papert’s brilliant ‘Mindstorms: Children, Computers and Powerful Ideas’. I’ve read it before but the whole book is popping with insights and I should probably do this many more times. Plus maybe this time it’ll actually teach me how to juggle too?

In the epilogue on mathematical thinking, he discusses his experiences getting people “with little mathematical knowledge” (enough to rearrange an equation, though) to work towards a proof that the square root of 2 is irrational. In the process, he gives two standard proofs, the second of which I’d never seen before:

Of these I shall contrast two which differ along a dimension one might call ‘gestalt versus atomistic’ or ‘aha-single-flash-insight versus step-by-step reasoning’.

They both start with the usual proof by contradiction: let sqrt(2)= p/q, a fraction expressed in its lowest terms. This is then rearranged to get

p^2 = 2 q^2,

and then you can go down the route of “well p must be even, so p=2r, so q^2=2r^2 WHY IS q EVEN TOO WHEN WE CANCELLED THE TWOS AT THE START?” Which is pretty compelling stuff.

But I really like the second proof, what he calls the ‘flash’ version:

Think of p as a product of its prime factors, e.g. 6=23. Then p^2 will have an even number of each prime factor, e.g. 36=2233. But then our equation p^2=2q^2 is saying that an even set of prime factors equals another even set multiplied by a 2 on its own, which makes no sense at all. Done.

Papert makes the point that if you have the right idea (decomposition into prime factors) ‘pre-loaded’ into your head, the equation is directly seen as absurd just by looking at it. In fact it now surprises me that it didn’t look wrong before! The first, more algorithmic proof gets you there, but without the insight flash.

Anyway, this is a lot of set-up just to say that reading this made me understand more clearly that that dopamine hit of insight is a large part of what I’m attracted to maths for. For me, much of the purpose of studying maths is to pre-load a pile of the necessary structure into my head to enable these insights to occur. Differential geometry is great for this, and this is probably why I like learning it so much! In the process I’m willing to put up with a load of chains of logic (which I am annoyingly poor at) but I’m always really hoping that they will be the means to an aha-single-flash insight.

OK time to go practice juggling (= ‘throwing all the balls on the floor’) again.

‘two types of mathematician’ linkdump

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This is great, and something I thought was missing from the SSC discussion (pity I’m too slow to ever get round to commenting). I only really got a tumblr to read some stuff, but I am fascinated by this topic so here is some related crap…

I think there’s a lot to this ‘different ways of understanding maths’ idea, sometimes it seems that you can pretty much give a mathematician a pen and they will start writing an essay on two types of mathematician. The clusters seem to be roughly ‘algebra/problem-solving/analysis/logic/precision’ vs. geometry/theorising/synthesis/intuition/hand-waving’ but there is plenty of variation.

I keep meaning to collect together a set of all of these I can find, so this has motivated me to make a first attempt:

“It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. 

The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.”

  • Gian-Carlo Rota made a division into ‘problem solvers and theorizers’ (in ‘Indiscrete Thoughts’, excerpt here)
  • Timothy Gowers makes a very similar division in his ‘Two Cultures of Mathematics’ (discussion and link to pdf here)
  • Freeman Dyson calls his groups ‘Birds and Frogs’ (this one’s more physics-focussed)
  • Vladimir Arnold turns the whole thing into a massive ideological war in his wonderful rant ‘On Teaching Mathematics’
  • Michael Atiyah makes the distinction in ‘What is Geometry?’:

Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words “insight” versus “rigour” and both play an essential role in real mathematical problems.

There’s also his famous quote:

Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’

  • Finally, I think something similar is at the heart of William Thurston’s debate with Jaffe and Quinn over the necessity of rigour in mathematics – see Thurston’s ‘On proof and progress in mathematics’. There is also a wonderful list of ways of understanding the concept of a derivative in Section 2.

OK hopefully I know how to write a post now! Hope this is interesting to someone.