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Terry Tao on some desirable properties of mathematical notation

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Re: Terry Tao on some desirable properties of mathematical notation

#191

Post by lordgrenville »

enriquto wrote: post #137

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, unfor…

I was with you until your last line, which I categorically disagree with. These two alternatives perfectly map to two different ways of thinking about the vector c:

- As a regular array (for someone with Python experience but no knowledge of linear algebra). In this case you can just loop over it as with any iterable.

- As a vector, with the associated mathematical properties. In this case you can operate on the entire vector, which is much faster; this happens to be because of implementation details (using C structures instead of Python lists, parallelisation, whatever), but is also just highly intuitive.

I'd argue that this is exactly what Tao is saying, about how different notations suit different contexts, and that allowing both methods in no way violates the Zen of Python (in reference to jimhefferon's comment).

Re: Terry Tao on some desirable properties of mathematical notation

#192

Post by enriquto »

lordgrenville wrote: post #191
enriquto wrote: post #137

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, unfor…

I was with you until your last line, which I categorically disagree with. These two alternatives perfectly map to two different ways of thinking about the vector c: - As a regular array (for someone with Python experience but no knowledge of linear algebra). In this case you can just loop over it as with any iterable. - As a vector, with the associated mathematical properties. In this case you can operate on the enti…

My problem is a practical one, not philosophical. I would expect that the computer operations in both cases are identical and thus the performance exactly the same. It is 2020 and optimizing compilers exist, and even JITs. The first code is just a notation for the second one. The fact that the first code is extremely slow (thing about iterating over all the pixels of a video sequence) is utterly disheartening. Of course, in that particular case you can say "just use the vectorized version", but in practice not all the computations that you need to do can be expressed in that form. If you try to iterate, in python, over all the pexels of a realtime video using integer indices you are in for a world of pain; it is just not possible and this is a major limitation of the expressivity of the language.

Re: Terry Tao on some desirable properties of mathematical notation

#193

Post by parralelex »

foobar_ wrote: post #139
augustt wrote: post #68

Earlier quoted context omitted.

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/~m…

"Mathematics is a part of physics."

It's kind of amazing how quickly that article got so wrong. Math isn't a subset of physics. Physics is the estranged brother of Math, always needing to borrow some money from him or else the power goes out or he can't afford food or some other sob story.

Re: Terry Tao on some desirable properties of mathematical notation

#194

Post by lordgrenville »

enriquto wrote: post #192
lordgrenville wrote: post #191

Earlier quoted context omitted.

I was with you until your last line, which I categorically disagree with. These two alternatives perfectly map to two different ways of thinking about the vector c: - As a regular array (for someone with Python experience but no knowledge of linear algebra). In this case you can just loop over it as with any iterable. - As a vector, with the associated mathematical properties. In this case you can operate on the enti…

My problem is a practical one, not philosophical. I would expect that the computer operations in both cases are identical and thus the performance exactly the same. It is 2020 and optimizing compilers exist, and even JITs. The first code is just a notation for the second one. The fact that the first code is extremely slow (thing about iterating over all the pixels of a video sequence) is utterly disheartening. Of cou…

It isn't just notation, though. It's a different way of operating. For example, the following adaptation of your code:

    c = 0
    for i in range(u.size):
        if u[i]
cannot be vectorised (ignoring the off-by-one error for simplicity's sake).

Saying that the Python interpreter should reinterpret the iterative code as if it were vectorised isn't increasing the expressivity, it's reducing it, by overriding the user's intentions.

Re: Terry Tao on some desirable properties of mathematical notation

#195

Post by gspr »

foobar_ wrote: post #139
augustt wrote: post #68

Earlier quoted context omitted.

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/~m…

> The way I see it ... Most mathematicians nowadays use mathematica or matlab or even python, proving my point.

Yes. But most of us don't use those to prove anything; rather, a lot of us use it to implement computations based on those proofs (and do some exploratory "could this possibly be tru?" kind of work). Useful tools, for sure, but not something that remotely proofs your point. Most mathematicians also eat bread. That does not mean that math is a baked good.

> The notation is medieval ...

It is not. Read Gauß or Euler from the 18th and 19th century, and the notation is nothing like modern mathematical notation. I can't even imagine what medieval mathematics notation looks like!

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

That is indeed the opinion of Arnold, a giant of mathematics. An opinion that, I daresay, does not reflect the majority opinion on mathematics.

> I see simulating as a part of the experiment.

Sure. Simulating is a valuable experimental tool to many mathematicians (where available; of course it isn't always).

> If the proof is wrong it wouldn't last a seconds worth of simulation.

At face value this statement betrays how little you know about this matter. There can very well be errors in proofs that cannot be uncovered without thousands of years of simulation, if at all.

Now, even interpreting your statement in the best possible light, namely something along the light of "simulation can often uncover mistakes in proofs", I would say: fine, but what about the converse?

> but most proofs have really no meat to them. The notation is merely intimidating like obfuscated code.

Are you insane? Take something that is patently "useful" and patently "real world", like the fundamental theorem of calculus. Meatless?

Re: Terry Tao on some desirable properties of mathematical notation

#196

Post by gspr »

foobar_ wrote: post #65
wheresmycraisin wrote: post #62

Earlier quoted context omitted.

Ok, then convince me. Write 'software' of, say, the proof of the the dominated convergence theorem or something else reasonably advanced and let's compare it to the proof in conventional math notation.

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.

> Now it's your turn to tell me why 1 + 1 = 2.

By construction.

Re: Terry Tao on some desirable properties of mathematical notation

#197

Post by foobar_ »

gspr wrote: post #195
foobar_ wrote: post #139

Earlier quoted context omitted.

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/~m…

> The way I see it ... Most mathematicians nowadays use mathematica or matlab or even python, proving my point. Yes. But most of us don't use those to prove anything; rather, a lot of us use it to implement computations based on those proofs (and do some exploratory "could this possibly be tru?" kind of work). Useful tools, for sure, but not something that remotely proofs your point. Most mathematicians also eat brea…

I'm not insane ... you are just the type of person who will defend roman numerals. Maybe you just have OCD.

1. Socrates is mortal

2. Mortals die

3. Socrates dies

Deduction is really like amazing. Holy shit we really proved something spectacular here. I guess you would be really impressed if I used tau and sigma and defined death with vietnamese alphabet.

Almost the entirety of calculus was derived from problems related to physics. Volumes were calculated for doing engineering. Mathematics != Thinking. The last time I checked both logic and critical thinking were branches of philosophy.

All good mathematicians are physicists or engineers. Heck some even learnt maths on their own. All mediocre mathematicians write textbooks and hide behind notations. Come to think of it they remind me of OO programmers in their utter arrogant mediocrity. Most abstract mathematics is like the definition of protocols/interfaces and other platonic garbage. I suppose this debate will never end. Plato vs Aristotle, Deduction vs Induction, Analytic vs Synthetic ....

Re: Terry Tao on some desirable properties of mathematical notation

#198

Post by enriquto »

lordgrenville wrote: post #194
enriquto wrote: post #192

Earlier quoted context omitted.

My problem is a practical one, not philosophical. I would expect that the computer operations in both cases are identical and thus the performance exactly the same. It is 2020 and optimizing compilers exist, and even JITs. The first code is just a notation for the second one. The fact that the first code is extremely slow (thing about iterating over all the pixels of a video sequence) is utterly disheartening. Of cou…

It isn't just notation, though. It's a different way of operating. For example, the following adaptation of your code: c = 0 for i in range(u.size): if u[i] cannot be vectorised (ignoring the off-by-one error for simplicity's sake). Saying that the Python interpreter should reinterpret the iterative code as if it were vectorised isn't increasing the expressivity, it's reducing it, by overriding the user's intentions.

My point exactly! If I need to implement your algorithm in Python, why am I condemned to be hopelessly slow? Or worse, condemned to find a "trick" that vectorizes this code while it becomes unreadable?

Re: Terry Tao on some desirable properties of mathematical notation

#199

Post by ChrisLomont »

Smaug123 wrote: post #162
ChrisLomont wrote: post #156

Earlier quoted context omitted.

As a math PhD I disagree. I can read math notation far faster and more accurately than ambiguous English. We don’t translate symbols. If anything, when reading English we have to translate into symbols. For example, reading 5-7, I don’t have to translate the - symbol to the word “subtract”. I know this is -2. And I don’t translate the - in that to the word “negative,” and certainly not to the word “subtract”. And it’…

I think you've got the same misconception as quietbritishjim above. Start stacking quantifiers, and the symbols get hairy much faster than the English does.

Maybe you're not used to reading quantifiers. I also find them much easier, faster, and more accurate to read, because it's from practice.

Sure you can take simple English statements, and write them with quantifiers, and claim the English is simpler. But going the other way, expressing complex items in English, is a non-starter. English is far too sloppy, whereas the quantifier version is mathematically precise.

Try converting something professional, such as Godel's incompleteness proofs, into English. Without precise quantifiers you'd quickly get lost, make mistakes, and take forever to get anywhere.

For example, look at page 17 of the proof [1], at AG(6)(a) (after the "Thus"), where there is a long statement in logic. Convert to English (near impossible, certainly not possible without something like parentheses) and tell me it's easier for a logician to read. It's not. As written it's concise and parseable without confusion for a logician.

The simple English sentence case is a tiny part of what mathematicians do.

[1] https://web.yonsei.ac.kr/bkim/goedel.pdf

Re: Terry Tao on some desirable properties of mathematical notation

#200

Post by an_android »

foobar_ wrote: post #197
gspr wrote: post #195

Earlier quoted context omitted.

> The way I see it ... Most mathematicians nowadays use mathematica or matlab or even python, proving my point. Yes. But most of us don't use those to prove anything; rather, a lot of us use it to implement computations based on those proofs (and do some exploratory "could this possibly be tru?" kind of work). Useful tools, for sure, but not something that remotely proofs your point. Most mathematicians also eat brea…

I'm not insane ... you are just the type of person who will defend roman numerals. Maybe you just have OCD. 1. Socrates is mortal 2. Mortals die 3. Socrates dies Deduction is really like amazing. Holy shit we really proved something spectacular here. I guess you would be really impressed if I used tau and sigma and defined death with vietnamese alphabet. Almost the entirety of calculus was derived from problems relat…

Don't use phrases as "Maybe you just have OCD". This is offensive and trivializes the problems those with OCD face. OCD is a serious disorder and your use of that phrase illustrates your lack of mental maturity.

Further, that phrase is bigoted. What you are implying is that someone with OCD is "lesser" or "other" as you are using the phrase to discount the person you are talking with. Hence it is bigoted.

In fact, it is obvious that you have no idea what you are talking about. Mathematics is not "just notation" in the same way software engineering is not "just programming language syntax", music is not just "notes on a piece of paper", and literature is not just "grammar rules".

If you cannot see that, I suggest you read more and expand your view of the world. Don't hurl insults at others.

If you want a more concrete example, show that the sum:

1 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ... = pi^2/6

That is, first define what it means to take a sum of an infinite number of terms, prove that your definition is consistent with a sum of a finite number of terms, and then show that the sum is exactly pi^2/6. Showing that they agree to 100 billion decimal places is not enough. You need to show they are exact.

When you are done with that, find an exact closed form for the sum:

1 + (1/2)^3 + (1/3)^3 + (1/4)^3 + ...

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