4 ms·

2 pi doesn't seem like a big deal to you because you've internalized the notation. You paid the cost a long time ago (and perhaps it was a small one for you).

by colah3 10y ago

2 pi doesn't seem like a big deal to you because you've internalized the notation. You paid the cost a long time ago (and perhaps it was a small one for you). I think it can be hard to empathize with what it's like to be a beginner. I learned about the pi definition debate a few years after I learned trigonometry -- there were a bunch of essays prior to the tauday one -- and it seemed tremendously better. I was also still in high school at that point, and I saw students struggling daily with these ideas and it seemed like part of the pi definition was the stumbling block. So, of course the pi vs 2 pi thing looks trivial to a mature mathematician: we aren't the people paying the cost.

CogitoCogito 10y ago
That argument _may_ apply to very young learners. I.e. students in high school or younger. It certainly does _not_ apply to the people the original article is talking about (i.e. researchers). Anyone who can't handle the circumference of a circle being 2 pi has no chance at understanding modern mathematical research no matter how simplified it becomes. It's not even on the radar. By the way I simply am not convinced by this 2 pi vs tau debate. Now the area of a unit circle would become tau / 2. The original problem doesn't go away it just changes. The argument can be that you write write tau more than you'd write tau / 2, but the fundamental difficulty remains. Notation in mathematics is _extremely_ important, but pi vs tau is so far down the rabbit hole of diminishing returns, that I just see the debate as a total waste of time (and that's before I consider the fact that it won't ever change due to momentum...).
- buzzybee 10y ago
  I see tau as a thing that's mostly useful to programmers because writing "2*Math.PI" (or equivalents) everywhere is a good way to bloat up a trig-intensive algorithm. I have no comment on its applicability to other uses of mathematics.
  - CogitoCogito 10y ago
    Keeping the 2 around can be annoying in math for exactly the same reasons. Also by the way, 2 * Math.PI is only one character less than Math.TAU. Even if you were to bind it locally to pi and tau, then 2 * pi is 4 characters whereas tau is 3. I mean it's a little annoying to keep track of, but it's certainly not the hardest thing programmers have to deal with. edit: I'm a bit new to hacker news and am not really sure why I'm seeing these italics... edit2: I see the issue is the asterix. Everywhere you see ' * ' imagine there are no spaces around them. I'm not sure how to "escape" an asterix...
    - nercht12 10y ago
      >> I'm not sure how to "escape" an asterix... You can't, sorry. It's automatically formatted. >> Even if you were to bind it locally to pi and tau I use things like Math.PI2 or PI2, so it's not any longer.
- jessriedel 10y ago
  The fact that you write down factors of 2 less is a weak signal of the improved notation, but it's not the point. (If you can't see that tau is conceptually a more fundamental quantity than pi, then you actually don't understand the point!) The reason to use an elementary example like pi/tau is that it's accessible to a wide audience. Of course it's nearly trivial for experts. It wouldn't make sense for the author to pick a high-level stumbling block that's only important for a tiny group of experts in one subfield.
  - CogitoCogito 10y ago
    > (If you can't see that tau is conceptually a more fundamental quantity than pi, then you actually don't understand the point!) Well then I guess I don't understand the point. That claim is essentially equivalent saying that circumference is more fundamental than area. You may accept such a strong claim, but I certainly do not.
    - jessriedel 10y ago
      It's not equivalent, but it's connected: circumference is "more fundamental" than area simply in the sense that 1 dimension is more fundamental than 2. In particular, when we build things up in higher dimensions, it's much more clearly done using a 1D primative than a 2D one, e.g., for the Fourier transform, there are powers of 2pi=Tau in any dimensions.
      - nercht12 10y ago
        >> it's much more clearly done using a 1D primative than a 2D one Or you could just tell people that pi is the edge length of half of a circle. Pi and Tau are both 1-dimensional numbers that happen to be related to these things, so the argument really is silly. The reality is, we're going to have to pay some kind of intellectual debt in learning anything. So we keep teaching Pi. Suppose everyone switched to Tau. That would make all kinds of older documents inaccessible and harder to grasp, and we would be paying intellectual debt that way.
mrob 10y ago
If you're going to change notation there are things with much bigger payoff. I'd start by eliminating implicit multiplication. This does great harm to clarity by encouraging single character variable names.
- mncharity 10y ago
  You might like the Preface to the Structure and Interpretation of Classical Mechanics [1]: "Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings that depend on context, and often even change within a given context." [1] https://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-5.html https://mitpress.mit.edu/sites/default/files/titles/content/...
mncharity 10y ago
> I think it can be hard to empathize with what it's like to be a beginner. In a similar vein, there's pi ≈ 3 (within 5% error). When we've an issue with high-school students having an infirm grasp of the very concept of volume, it seems an odd allocation of effort to have middle-school students struggle with 4/3 pi r^3, while failing to tell early primary-school students that "the volume and area of a ball are about half that of the box that wraps it".

by colah3 10y ago

CogitoCogito 10y ago

That argument _may_ apply to very young learners. I.e. students in high school or younger. It certainly does _not_ apply to the people the original article is talking about (i.e. researchers). Anyone who can't handle the circumference of a circle being 2 pi has no chance at understanding modern mathematical research no matter how simplified it becomes. It's not even on the radar. By the way I simply am not convinced by this 2 pi vs tau debate. Now the area of a unit circle would become tau / 2. The original problem doesn't go away it just changes. The argument can be that you write write tau more than you'd write tau / 2, but the fundamental difficulty remains. Notation in mathematics is _extremely_ important, but pi vs tau is so far down the rabbit hole of diminishing returns, that I just see the debate as a total waste of time (and that's before I consider the fact that it won't ever change due to momentum...).

buzzybee 10y ago
I see tau as a thing that's mostly useful to programmers because writing "2*Math.PI" (or equivalents) everywhere is a good way to bloat up a trig-intensive algorithm. I have no comment on its applicability to other uses of mathematics.
- CogitoCogito 10y ago
  Keeping the 2 around can be annoying in math for exactly the same reasons. Also by the way, 2 * Math.PI is only one character less than Math.TAU. Even if you were to bind it locally to pi and tau, then 2 * pi is 4 characters whereas tau is 3. I mean it's a little annoying to keep track of, but it's certainly not the hardest thing programmers have to deal with. edit: I'm a bit new to hacker news and am not really sure why I'm seeing these italics... edit2: I see the issue is the asterix. Everywhere you see ' * ' imagine there are no spaces around them. I'm not sure how to "escape" an asterix...
  - nercht12 10y ago
    >> I'm not sure how to "escape" an asterix... You can't, sorry. It's automatically formatted. >> Even if you were to bind it locally to pi and tau I use things like Math.PI2 or PI2, so it's not any longer.
jessriedel 10y ago
The fact that you write down factors of 2 less is a weak signal of the improved notation, but it's not the point. (If you can't see that tau is conceptually a more fundamental quantity than pi, then you actually don't understand the point!) The reason to use an elementary example like pi/tau is that it's accessible to a wide audience. Of course it's nearly trivial for experts. It wouldn't make sense for the author to pick a high-level stumbling block that's only important for a tiny group of experts in one subfield.
- CogitoCogito 10y ago
  > (If you can't see that tau is conceptually a more fundamental quantity than pi, then you actually don't understand the point!) Well then I guess I don't understand the point. That claim is essentially equivalent saying that circumference is more fundamental than area. You may accept such a strong claim, but I certainly do not.
  - jessriedel 10y ago
    It's not equivalent, but it's connected: circumference is "more fundamental" than area simply in the sense that 1 dimension is more fundamental than 2. In particular, when we build things up in higher dimensions, it's much more clearly done using a 1D primative than a 2D one, e.g., for the Fourier transform, there are powers of 2pi=Tau in any dimensions.
    - nercht12 10y ago
      >> it's much more clearly done using a 1D primative than a 2D one Or you could just tell people that pi is the edge length of half of a circle. Pi and Tau are both 1-dimensional numbers that happen to be related to these things, so the argument really is silly. The reality is, we're going to have to pay some kind of intellectual debt in learning anything. So we keep teaching Pi. Suppose everyone switched to Tau. That would make all kinds of older documents inaccessible and harder to grasp, and we would be paying intellectual debt that way.

mrob 10y ago

If you're going to change notation there are things with much bigger payoff. I'd start by eliminating implicit multiplication. This does great harm to clarity by encouraging single character variable names.

mncharity 10y ago
You might like the Preface to the Structure and Interpretation of Classical Mechanics [1]: "Classical mechanics is deceptively simple. It is surprisingly easy to get the right answer with fallacious reasoning or without real understanding. Traditional mathematical notation contributes to this problem. Symbols have ambiguous meanings that depend on context, and often even change within a given context." [1] https://mitpress.mit.edu/sites/default/files/titles/content/sicm/book-Z-H-5.html https://mitpress.mit.edu/sites/default/files/titles/content/...

mncharity 10y ago

> I think it can be hard to empathize with what it's like to be a beginner. In a similar vein, there's pi ≈ 3 (within 5% error). When we've an issue with high-school students having an infirm grasp of the very concept of volume, it seems an odd allocation of effort to have middle-school students struggle with 4/3 pi r^3, while failing to tell early primary-school students that "the volume and area of a ball are about half that of the box that wraps it".