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Terry Tao on some desirable properties of mathematical notation

mathoverflow.net

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Re: Terry Tao on some desirable properties of mathematical notation

#131

Post by throwaway744678 »

quietbritishjim wrote: post #121
auggierose wrote: post #119

Earlier quoted context omitted.

The problem is when it is not quite clear what the symbolic notation stands for. With division, that tends to be less of a problem.

* I find it quite unusual in practice for genuinely new symbolic notation to be used by an author. Maybe that just reflects the fields I read about most (information theory, Bayesian modelling, harmonic analysis). * Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times. * Even if you did have an iso…

> * I find it quite unusual in practice for genuinely new symbolic notation to be introduced by an author.

This sentence seems to contradict the rest of your comment; did you mean "I find it quite unusual in practice for genuinely new symbolic notation to NOT be introduced by an author."?

Re: Terry Tao on some desirable properties of mathematical notation

#132

Post by erezsh »

quietbritishjim wrote: post #116
Smaug123 wrote: post #107

Earlier quoted context omitted.

I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

That's not a genuine example.

Most academic texts I've read will use something like

    s = d/t
And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it.

Some will even make up their own notation:

    s = d(t)
And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that it's actually simple division, so other academics won't find them too obvious or boorish.

Some even take it a step further and use random greek letters (without exhausting the English alphabet first ofc.), where small gamma and big gamma mean completely different and unrelated things.

Re: Terry Tao on some desirable properties of mathematical notation

#133

Post by quietbritishjim »

throwaway744678 wrote: post #131
quietbritishjim wrote: post #121

Earlier quoted context omitted.

* I find it quite unusual in practice for genuinely new symbolic notation to be used by an author. Maybe that just reflects the fields I read about most (information theory, Bayesian modelling, harmonic analysis). * Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times. * Even if you did have an iso…

> * I find it quite unusual in practice for genuinely new symbolic notation to be introduced by an author. This sentence seems to contradict the rest of your comment; did you mean "I find it quite unusual in practice for genuinely new symbolic notation to NOT be introduced by an author."?

Thanks, that was ambiguous and the way you read it wasn't what I intended. I meant it was unusual for an author to use a new symbolic notation at all, without saying anything about whether they define it in those cases where they do use something new. I've edited "introduced" to "used".

Re: Terry Tao on some desirable properties of mathematical notation

#134

Post by foobar_ »

jjgreen wrote: post #128
foobar_ wrote: post #65

Earlier quoted context omitted.

I'm guessing there was a physical intuition behind the theorem, if you can simulate it you will probably do something better than the proof. Now it's your turn to tell me why 1 + 1 = 2.

See Principia Mathematica , A. N. Whitehead and B. Russell, Proposition 110.643

1 + 1 = 10

Re: Terry Tao on some desirable properties of mathematical notation

#135

Post by gjulianm »

quietbritishjim wrote: post #116
Smaug123 wrote: post #107

Earlier quoted context omitted.

I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols:

- Any two distinct points in the space have disjoint neighbourhoods.

- ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅.

Another example would be Navier-Stokes equations, where they're much easier to understand in words than in symbols. Symbols are ok when you don't have to search too much to see what they mean and when the idea you're trying to transmit with them is relatively simple, but trying to build complicated phrases and definitions with symbols, for me, ends up being a mess.

edit: fixed hausdorff space definition, points should be distinct

Re: Terry Tao on some desirable properties of mathematical notation

#136

Post by foobar_ »

Smaug123 wrote: post #113
foobar_ wrote: post #72

Earlier quoted context omitted.

I guess you missed the sarcasm. If you want me to s-p-e-l-l it out - maths is convention.

(I missed the sarcasm, and honestly I still can't see it after you've pointed it out - and I'm British, supposedly a native expert!)

Wow .. an expert in sarcasm. Amazing! I have so many questions! Do you have good dental insurance ?

Re: Terry Tao on some desirable properties of mathematical notation

#137

Post by enriquto »

It would be nice to have an equivalent post, but with programming languages. The fact that different programs perform an identical computation is important. For example, in Python/numpy you can write

    c = 0
    for i in range(u.size):
        c = c + u[i] * v[i]
or

    c = u.T @ v
and even if the result is identical, the computation is not, the first one being orders of magnitude slower. There is no good reason for it to be so, unfortunately.

Re: Terry Tao on some desirable properties of mathematical notation

#138

Post by toxik »

gjulianm wrote: post #135
quietbritishjim wrote: post #116

Earlier quoted context omitted.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, wh…

Curiously, you have been ambiguous in the mathematical notation. Counterproof: let x = y.

Re: Terry Tao on some desirable properties of mathematical notation

#139

Post by foobar_ »

augustt wrote: post #68
foobar_ wrote: post #65

Earlier quoted context omitted.

I'm guessing there was a physical intuition behind the theorem, if you can simulate it you will probably do something better than the proof. Now it's your turn to tell me why 1 + 1 = 2.

Honestly what are you talking about. You can simulate for 100 years without finding a counterexample, but that doesn't make a proof. The whole point of math is to understand why things are true, not to just be satisfied that it seems true.

The way I see it ... Most mathematicians nowadays use mathematica or matlab or even python, proving my point. The notation is medieval ... and probably the only reason it survives is because of form factors of paper.

> Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.

https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

I see simulating as a part of the experiment. If the proof is wrong it wouldn't last a seconds worth of simulation. I suppose a proof in essence is a pattern or an invariant of the system ... but most proofs have really no meat to them. The notation is merely intimidating like obfuscated code.

Re: Terry Tao on some desirable properties of mathematical notation

#140

Post by nicoburns »

toxik wrote: post #138
gjulianm wrote: post #135

Earlier quoted context omitted.

But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, wh…

Curiously, you have been ambiguous in the mathematical notation. Counterproof: let x = y.

Unfortunately that's not uncommon in mathematical texts either!
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