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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

#51

Post by Koshkin »

dilap wrote: post #48
Koshkin wrote: post #18

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

> with the subject itself that's the thing though, you can never grapple with the subject itself, only representations thereof. this is a mix of feelings/images/movements inside your head & mechanical manipulations of the notation; furthermore, the notation itself influences our internal model/feeling of the subject.

I am not saying that with enough dedication one couldn’t screw up the notation completely.

Re: Terry Tao on some desirable properties of mathematical notation

#52

Post by nl »

chongli wrote: post #44
nl wrote: post #35

Earlier quoted context omitted.

I'm a power user (been programming for 30 years) and I refuse to use vim keybindings. I have too many things to remember, and there are plenty of options that give me equal power but don't demand that investment from me. Math isn't like that - I do more math than the average person and I hate having to decipher the annotation. The number of times I've tried to work out if this sigma is the same as the sigma in this o…

Mathematics is more general and applicable than Vim, I'll give you that. I wasn't the one who brought in the Vim analogy, however. It seems like your complaints are with academic papers, not with mathematical notation specifically. This is a problem that is universal to academic papers. Try reading a critical theory paper, for example, and you'll find it's extremely dense with critical theory jargon the authors don't…

To be honest I think my (and most people on this thread) real complaint isn't with notation per-say (as in Euclidean vs Hilbert notation etc, as discussed by Tao).

It's really about the lazy habits of many who work at the intersection of math and computer science, and use maths to express themselves without defining things.

> The other problem with academic papers is that the authors generally don't care about reproducibility, consistency, clarity, pedagogy, or even intelligibility. They're optimizing for quantity of papers published, not quality. As long as their expert peer reviewers understand and give the green light for publication, that's good enough.

Having worked around this field, I think this (common) perception of publishing doesn't quite capture what is happening. Peer review isn't anything like code review, and software engineers (separate from computer scientists and mathematicians) think that it is.

Re: Terry Tao on some desirable properties of mathematical notation

#53

Post by xelxebar »

pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

The frustration is inarguably real, but...

Programming language syntax sucks because none of it maps to things that non-programmers know. Every time a new token is introduced, whether it be an ascii symbol or a word it's one more mapping your brain has to create to remember it. And on top of this you have to remember how these actually execute too. Yes I said execute - most programming concepts have so many implementations it's crazy. Even basic conditionals can't escape this. There are several tokens for if-statements (if, guard, case, (^:), etc.) and countless common implentations (gcc, ghc, guile, J, and whatever actual processors end up doing).

Deep, complex things are hard, my friend. The pain is real; the payoff is totally worth it though.

Re: Terry Tao on some desirable properties of mathematical notation

#54

Post by Koshkin »

kdmccormick wrote: post #37
Koshkin wrote: post #18

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

That's a strong assertion. You're implying that difficulties with the subject itself rarely have anything to do with difficulty communicating through some notation. I'm not disagreeing, but I'm curious how you would back up that assertion.

It’s an experimental fact. (I guess you have to trust me on this one.)

Re: Terry Tao on some desirable properties of mathematical notation

#55

Post by wheresmycraisin »

foobar_ wrote: post #50

No one uses mathematical notation for practical purposes. This is just like the medival music notation which is neither practical nor what modern composers use, which is more visual in nature. Infact modernism is a rejection of medievalism. I think in the future programming will force all mathematicians to code or give out simulations. Most mathematical notation was intended to be throwaway by the original authors, t…

Programming != proofs, or in general communicating abstract mathematical ideas. Writing mathematics is nothing like writing software.

Re: Terry Tao on some desirable properties of mathematical notation

#56

Post by cheez »

Koshkin wrote: post #49

Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothin…

Looks like we read the same books.

Re: Terry Tao on some desirable properties of mathematical notation

#57

Post by aspaceman »

zamfi wrote: post #17
pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

In addition, the same notation is often overloaded in new domains, where the association with the prior domain is metaphorical, or even simply allegorical. As a novice programmer, it took years to get out of the habit of using non-descriptive single-letter variable and function names. It’s a terrible habit, and makes code incomprehensible — and programmers know it. Math, as taught to kids, is full of single-letter va…

> Math, as taught to kids, is full of single-letter variable names (or worse, as you point out, using non-Roman letters and other symbols) that are devoid of any hints to the uninitiated of what their meaning might be.

Any time a new symbol is used in a text, it will typically be introduced. The phrase "Let A be the subgraph composed of G" for example defines A in relation to G. Typically, the author will also define their notation at the start of a paper with: "A graph G is defined as the set of edges E and vertices V such that...". To be frank, I always see math papers and communications written in this style.

More often than not, I think folks just say "SINGLE VARIABLE NAMES BAD. MATH HAVE SINGLE VARIABLE NAMES". It's really dense. In any given math paper, the author typically defines any such variables. And if those definitions don't make sense, it's because the underlying mathematical objects are beyond you, and you need further background.

Re: Terry Tao on some desirable properties of mathematical notation

#58

Post by wtallis »

tmpz22 wrote: post #16
pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

I dropped out of college at 19 and attempted to return at 21. My first math course back was discrete math and my teacher was a grad student who very clearly had no interest in teaching and was only there in order to subsidize his "real work". Keep in mind this is a large public university charging $40k/year. Going into his office hours was like going to another country, because his only method to explain math was rea…

In many university mathematics curriculums, discrete math is the course used to transition students from the computation-oriented mindset (instilled in them by high school and early undergraduate courses up through calculus) to the more abstract and proof-oriented mindset. Thus, teaching you to use that "really really advanced level math notation" is often one of the primary goals of the course, even if it seems like unnecessary overkill at the time. If you refuse to learn it, you're setting yourself up for failure in any future math course that uses that notation as the starting point for building new concepts and abstractions.

Re: Terry Tao on some desirable properties of mathematical notation

#59

Post by foobar_ »

wheresmycraisin wrote: post #55
foobar_ wrote: post #50

No one uses mathematical notation for practical purposes. This is just like the medival music notation which is neither practical nor what modern composers use, which is more visual in nature. Infact modernism is a rejection of medievalism. I think in the future programming will force all mathematicians to code or give out simulations. Most mathematical notation was intended to be throwaway by the original authors, t…

Programming != proofs, or in general communicating abstract mathematical ideas. Writing mathematics is nothing like writing software.

What I am trying to convey is writing software is better than writing maths, just like medieval music notation vs modern notation. Programming is better than proving because most proofs are mere tautologies or artificial constraints. This is why theorem provers in code rely on term rewriting.

A triangle has a sum of 180 ? Well how about if you push the triangle inside out. In code you can easily run a more complex simulation which gives you all possible values of the sum ... which is why ascertaining useful facts like 180 ad-nausea is boring at best. In fact most mathematics if it can't be simulated can't exist.

Re: Terry Tao on some desirable properties of mathematical notation

#60

Post by Koshkin »

chongli wrote: post #28
pubby wrote: post #9

Imagine I give you a list of words and ask you to remember them. 5 minutes later, I ask you to give me those words in reverse order. Not too hard, right? Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Suddenly the task becomes much more confusing. You aren't remembering a small handful of objects and ideas, but instead trying to juggle the individual syllables in your head…

Now imagine if those words I gave you were in Vietnamese, or some language you don't speak. Math notation sucks because none of it maps to things non-mathematicians know. I'm struggling to see how someone could conclude the latter statement from the former. Why is it reasonable for Vietnamese to be unintelligible to non-Vietnamese speakers, yet unreasonable for mathematics to be unintelligible to non-mathematicians?

I would argue that in fact most of mathematical notation happens to be intelligible to non-mathematicians (engineers, scientists, etc.)
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