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How to Read Mathematics

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Re: How to Read Mathematics

#101

Post by extra__tofu »

andrepd wrote: post #94
zitterbewegung wrote: post #3

How I read Mathematics. First try to find the paper. First use http://front.math.ucdavis.edu . If you can't find it then use google scholar. If that fails then go on the authors academic website. If that fails then see if the author is live and try to contact them. Then repeat for their grad students. If all else fails post it to r/math or HN and ask for the pdf. First you want to see if this paper is even worth read…

>First try to find the paper. First use http://front.math.ucdavis.edu . If you can't find it then use google scholar. If that fails then go on the authors academic website. If that fails then see if the author is live and try to contact them. Then repeat for their grad students. If all else fails post it to r/math or HN and ask for the pdf. You can replace all that by a trip to sci-hub ;)

I believe sci-hub has been taken down, at least temporarily. The .cc and .bz domains weren't working for me. I think it's possible to setup custom DNS servers on your machine and access that way, but I haven't played around with it.

Re: How to Read Mathematics

#102

Post by catnaroek »

internetman55 wrote: post #68
jordigh wrote: post #2

One thing that I really would like to impress upon computer people is that mathematics is not (usually) a formal computer language. Symbols and technical terms change meaning depending on the author, people write with different "accents". There's a little ambiguity and informality at times, but a whole lot less than other kinds of writing. Mathematics is (usually) written for humans, not computers. Don't attempt to r…

This is true. The CS folks are my university tended to be quite strict about writing out quantifiers for everything, while a very accomplished math researcher in our department told our class to "please write in English" instead of using those symbols, because he wouldn't be able to read them. I asked a friend why he thought this was, and he suspected that the CS department's focus on formal languages and automatic t…

I am very much in the math camp, not the computer science camp, but I enjoy annoying mathematicians by explicitly using quantifiers every now and then. Other ways to annoy mathematicians are bringing up pathological examples and remarking that what they have just said doesn't hold in trivial degenerate cases (e.g., the empty set).

Re: How to Read Mathematics

#103

Post by agentultra »

rsj_hn wrote: post #61
retox wrote: post #32

Earlier quoted context omitted.

Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? I have a deep dread and fear of numbers and mathematics in general because I don't understand them and I have never learned. Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation…

What do you mean by "writing" mathematics? The glib answer is that math is a superset of programming and there are non-computable arithmetic functions, so programming is not rich enough to describe what you need to describe when you are talking about math. I think a more intuitive answer is that in math you are generally talking about relations between infinite sets of objects, whereas in programming you are living i…

> You can ask, why is it so painful to give a formal proof of correctness of your program?

It's not!

It turns out that the dependently-typed lambda calculus is sufficient. See the research into Lean[0].

It seems we had discovered a portal called, Curry-Howard. You can throw in theorems in one side and out come proofs on the other. Programs are mathematical objects. And programming is a subset of mathematics. I like to think of it as an applied mathematics.

It just so happens that practitioners of this language don't require knowledge of the other side of the portal in order to write their proofs. They just write their proofs without any theorems, or at best, a large suite of examples.

To me writing proofs is programming.

[0] http://leanprover.github.io/

Re: How to Read Mathematics

#104

Post by vidarh »

lostcolony wrote: post #76

"Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space" "A well-written math text will be careful to use a word in one sense only" So many papers are not that careful, and thus, not so precise. There are only so many greek letters, so they end up overloaded. I can't count the number of times I've struggled over an equation, only to find out that the…

My main exposure to mathematics is via CS papers, and this is doubly true there it seems to the point where my warning bells go off if they present their solutions in equations - it's all to often a sign that they're going to hand-wave away essential details that'd be obvious were missing if they presented code or even pseudo-code.

E.g. I did my MSc. on statistical approaches to reducing error rates in OCR, and so many of the papers I reviewed for my literature review suffered from leaving out absolutely critical information when they presented equations etc. that most of the time I spent implementing a lot of the methods presented tended to be to try to reverse engineering missing information (and often that was only possible because many of the papers used one of a few well established public datasets for their experiments).

To me, for those CS papers, maths was a warning what was lying ahead was likely not precise or well defined. It's incredibly frustrating as there were generally no good reason to do so - it wasn't a matter of leaving out large bulks of complicated code. Often it could be as simply as leaving out the concrete values of given parameters.

I'm not implying maths have no place in CS papers, but the CS papers that I've seen that have used it best have tended to also present code fragments, or use maths very sparingly and coupled with painstakingly defining variables etc.

Re: How to Read Mathematics

#105

Post by agentultra »

What I most enjoy about reading mathematics is convincing myself that what I'm reading is true. I like to keep a pad of dot-graph paper and a pencil with me as I read to break down the theorems, lemmas, and expressions as I go; try derivations, etc. Martin Gardner got me into the habit as I read his articles while I was growing up.

These days I can even toy with the idea in Lean[0] or TLA+ and find different ways of formulating the results. What a time to be alive!

[0] http://leanprover.github.io/

Re: How to Read Mathematics

#106

Post by protonfish »

andrepd wrote: post #95
protonfish wrote: post #81

Earlier quoted context omitted.

But humans can't read it, nor can computers. Maybe it's time to ask if we can do better than traditional formal mathematical notation.

What do you mean by "humans can't read it".

What I mean is that mathematical notation was never developed with the intention of being easily understood. Nobody writes out a formula and then spends time making it more readable. In programming they do - it's called refactoring. How mathematicians typically polish an already correct formula is called "simplification." This usually leads to their collection of symbols being even more inscrutable. But the more mathematical proofs, formulae, and other statements are unintelligible, the more they make you appear really smart, so the natural tendency is to obfuscate mathematical statements as much as possible. And because this notation cannot be executed by a machine, there is no way to easily prove that it isn't complete gibberish.

Re: How to Read Mathematics

#107

Post by balabaster »

wimagguc wrote: post #28

Feynman’s method to understand complex problems is so simple and elegant! Surely you’re joking, mr Feynman: ”I can’t understand anything in general unless I’m carrying along in my mind a specific example and watching it go. Some people think in the beginning that I’m kind of slow and I don’t understand the problem, because I ask a lot of these “dumb” questions: “Is a cathode plus or minus? Is an an-ion this way, or t…

This is so exactly how I think...

When you're talking about complex abstract systems, I have 2 or 3 real world models going on in my head and playing out chains of consequences at the same time. When your abstract system falls down I can tell you where and what caused it despite not understanding a word of your explanation as to why it should. All those mathematical equations and models may as well be Greek to me.

Re: How to Read Mathematics

#108

Post by gizmo686 »

klibertp wrote: post #53
wiz21c wrote: post #31

FTA : >>> One can now check that the next statement is true with a certain amount of essentially mechanical, though perhaps laborious, checking. I, the author, could do it, but it would use up a large amount of space and perhaps not accomplish much, since it'd be best for you to go ahead and do the computation to clarify for yourself what's going on here. I promise that no new ideas are involved, though of course you…

> I hate that. When I program my computer, I specify all the steps to accomplish something. If it's too verbose, then I abstract it in a function. > I even add comments to make sure that a human reader easily and completely understands what I do. Yeah, honestly, any single time I read a maths-related paper I feel like I would be better off with an example, well-written, implementation in any programming language. The…

Because mathamaticians have thousands of years of history where they learned how to write math in a way that is readable to other mathamaticians.

Software engineering is a very different activity from the math that mathamaticians do. It makes sense that it has a different standard for what mathematicians do.

There are undoubtadly ways to improve mathematics, but it seems highly arrogant to say that your <100 year old field has discovered a massive improvement that mathematics has missed for centuries.

Re: How to Read Mathematics

#109

Post by BeetleB »

smhost wrote: post #6
vinchuco wrote: post #5

I disagree with 'do not read too fast': Skim all. Skim again the next day. But the third time, get pencil and paper and take notes, and effort. If it's neither uncomfortable nor exciting, you're doing it wrong.

So basically take a couple days to understand the material rather than going all in all at once?

Few math professors grok a journal paper within a day. One professor told me he strives for a page a day.

Re: How to Read Mathematics

#110

Post by neaden »

lostcolony wrote: post #76

"Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space" "A well-written math text will be careful to use a word in one sense only" So many papers are not that careful, and thus, not so precise. There are only so many greek letters, so they end up overloaded. I can't count the number of times I've struggled over an equation, only to find out that the…

Another problem is that different subfields of math use jargon a little bit differently. Normal for instance can mean so many different things.
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