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How to Read Mathematics

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Re: How to Read Mathematics

#121

Post by lostcolony »

vidarh wrote: post #104
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 ma…

Yeah, my attempt at a master's in CS (before deciding the time would be better spent elsewhere) left me feeling largely the same. "Oh, isn't that equation elegant. Oh, wait, actually trying to implement it...what the hell, where is all the other information I need?! That's a gaussian...but where do the parameters for that gaussian come from? What is this variable? WHAT IS THIS VARIABLE?!" etc.

I had a 4.0 departmental in CS from the same school I tried to get a master's in, but yeah, the master's experience was so bad, and working through those issues so time consuming, I just said "Screw it, I can get a better return on this amount of time doing my own things". Even if it's learning the same things, but to an explicit goal rather than just 'understanding' (and an eventual test/assignment that may or may not relate to what I care about), and where I'm not constrained by an academic policy that prevents me from going to others to have them explain exactly what I need to know.

Re: How to Read Mathematics

#122

Post by nogbit »

klibertp wrote: post #97
pbhjpbhj wrote: post #85

Earlier quoted context omitted.

So you'd never say to someone "use a bubble sort" or "try a bitwise trie" you'd lay it out in full, commented code each time? When you say "I abstract it in a function" that's surely what a mathematician is doing when they say "then we have a BVP and it naturally follows ...", they're applying the function "treatAsBVP". If you don't understand that step then you go and learn that, just as someone would have to find a…

> When you say "I abstract it in a function" that's surely what a mathematician is doing In principle, yes. In practice, I can right-click on any such function and get to see its source (edit: instantly, automatically) whenever I need it. That doesn't work at all with math papers. EDIT: Furthermore, when writing about something where bubble sort is used, for the general audience, I would simply link to an explanation…

> In maths, papers are devoid of any links, the teaching materials are up to 50 years old and never updated to take advantage of new presentation techniques, authors frequently name-drop some algorithm and then forget to say what parameters they used or how exactly it connects with preceding and following paragraphs.

That there is the crux. If math articles/papers were written in a different medium then you could drill down (right click, view source) from the most abstract level down to the basics. Some kind of tool would need to exist to make this possible.

Re: How to Read Mathematics

#123

Post by dekhn »

After many years of frustration not being able to read physics papers, a helpful physics guy told me: when it's bolded, that means it's a matrix, and that other one, they didn't say so, but it's a vector, so this step is a vector matrix operation.

(that's why, today, I use numpy, where the operations are much easier to see).

Re: How to Read Mathematics

#124

Post by gizmo686 »

klibertp wrote: post #97
pbhjpbhj wrote: post #85

Earlier quoted context omitted.

So you'd never say to someone "use a bubble sort" or "try a bitwise trie" you'd lay it out in full, commented code each time? When you say "I abstract it in a function" that's surely what a mathematician is doing when they say "then we have a BVP and it naturally follows ...", they're applying the function "treatAsBVP". If you don't understand that step then you go and learn that, just as someone would have to find a…

> When you say "I abstract it in a function" that's surely what a mathematician is doing In principle, yes. In practice, I can right-click on any such function and get to see its source (edit: instantly, automatically) whenever I need it. That doesn't work at all with math papers. EDIT: Furthermore, when writing about something where bubble sort is used, for the general audience, I would simply link to an explanation…

>In maths, papers are devoid of any links

Not really. Math papers have citations; and important references to prior work are mentioned in the text. It would be a nice quality of life improvement if these were as easy to follow as clicking a link, but relative to the amount of effort you will spend within each link, it is pretty close. Assuming that the referenced paper is in an accessible location; although if it isn't and I need to track it down, I would prefer a standard citation over a broken URL with no archive.

Re: How to Read Mathematics

#125

Post by j2kun »

sundarurfriend wrote: post #57
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…

Also from the same book: "I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) – disjoint (two bal…

What kind of second-rate mathematician tries to prove a theorem without writing down examples? That's Proving Theorems 101.

Re: How to Read Mathematics

#126

Post by jloughry »

extra__tofu wrote: post #101
andrepd wrote: post #94

Earlier quoted context omitted.

>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.

It's still possible to use Sci-Hub, it's just that DNS queries for it tend to fail. Even knowing the IP address isn't enough, because although that gets you to the familiar Sci-Hub home page, search results always time out.

The trick is to find out the IP address of a DNS server or two that knows about Sci-Hub, and put those in your /etc/resolv.conf file.

Note: They have to come first in the file, before any 'compromised' DNS servers like Google's or your ISP's, that will always report 'sorry, I don't know how to find this "Sci-Hub" you speak of'.

If you're on a Windows machine, do it in Control Panel instead; dig down in Network settings, TCP/IP, Advanced, DNS to find the place to type in new DNS server IP addresses.

Do that, and Sci-Hub works fine. Twitter is a good place to find the latest working IP addresses for their DNS server. Last time I looked, they were 80.82.77.83 and 80.82.77.84.

Re: How to Read Mathematics

#127

Post by Koshkin »

gizmo686 wrote: post #108
klibertp wrote: post #53

Earlier quoted context omitted.

> 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…

> your

We should not forget that the progress that science, mathematics and technology have made in the past 100 - 150 years by far exceeds all that had been done before.

Re: How to Read Mathematics

#128

Post by Koshkin »

gizmo686 wrote: post #116
gfodor wrote: post #115

Earlier quoted context omitted.

appeal to tradition, poor argument. not arrogance to assert perhaps a field needs to update its methods.

A software engineer suggesting that math should adopt the practices of software engineering sounds a lot like someone trying to generalize their domain into another domain; and saying that they know better then the domain experts in the field they are generalizing into. Occasionally they are right. Most of the time however, the domain experts are right.

Well, it's not just about who is right and who is wrong, not in this case. Comparing industry practices with those of scientific (especially theoretical) research may not make much sense. Plus, if you consider the way software is actually developed, you will see that the kind of honesty that is required or assumed in regard to a scientist cannot be practically expected from today's software engineer. Engineering is all about compromises, while science is not.

Re: How to Read Mathematics

#129

Post by ColinWright »

j2kun wrote: post #125
sundarurfriend wrote: post #57

Earlier quoted context omitted.

Also from the same book: "I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) – disjoint (two bal…

What kind of second-rate mathematician tries to prove a theorem without writing down examples? That's Proving Theorems 101.

I have no idea why you were being down-voted, and I wonder how many of the down-voters are actually mathematicians. If they were responding to your tone they need to know that it's incredibly mild compared with what one could say.

I upvoted you, but clearly didn't undo all the downvotes.

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