How to Read Mathematics • Hacker News Board

Re: How to Read Mathematics

#131

Post by threatofrain » Wed, Dec 13, 2017, 6:58 PM UTC

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…

Cross-pollination is one of the low-hanging ways one can contribute to a discipline, and all professions are very eager to have people bring in ideas from other domains.

Even if a domain is over 1000 years old, some domains have only existed for about 150-200, which means that before, the world did not have the benefit of those lenses. Now that we have them, it is only natural to mix and match to see if easy insights come out.

Is there not also a risk of arrogance to demanding that another field understand everything about yours before attempting to share knowledge? It suggests that the time of day is so limited, and you have business of such importance, that you will not hear ideas below a threshold. That seems like a fine strategy for accruing quality up to a point, but it can run into local maxima more easily.

Re: How to Read Mathematics

#132

Post by killjoywashere » Wed, Dec 13, 2017, 7:04 PM UTC

There's a contrast between the comments here about Feynman, and the author's example of birthdays. The birthday data is totally random, and the author's assert this is the sort of thing mathematicians enjoy. It strikes me that Wolfram does this too, he'll demo something by saying "let's get some data" and then generate a big pile of random numbers.

I usually space out at that point. I understand the utility of it, but I find it worthless in most cases. I'd rather understand a concrete example well, then check my understanding by trying the random case, then try to think of some clever or experience based edge case or counterexample.

I think Feynman, especially later in life, probably had the gift of jumping straight to step 3: he understood that his most valuable contribution could often be simply expanding the space of possible examples. And random doesn't really get you the whole space, and certainly doesn't get you the interesting parts of the space.

Re: How to Read Mathematics

#133

Post by bzbarsky » Wed, Dec 13, 2017, 7:42 PM UTC

remus wrote: post #44

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…

> 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 (the same equation) anywhere in the world. It's not as bad as that! You're very unlikely to hit this sort of issue until you're reading research papers (unless you decide you're very interested in mathematical history). Pretty much all the material youll come across up to…

There are at least three different commonly used notations I'm aware of for division.... And that's not counting the long division process, for which I'm aware of at least two different notations (usage varies by country).

Re: How to Read Mathematics

#134

Post by CamperBob2 » Wed, Dec 13, 2017, 8:44 PM UTC

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 .hk at the moment.

Re: How to Read Mathematics

#135

Post by Koshkin » Wed, Dec 13, 2017, 8:58 PM UTC

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.

Right. The very first step in trying to prove a theorem is to try and disprove it (by looking for a counter-example).

Re: How to Read Mathematics

#136

Post by nicklaf » Thu, Dec 14, 2017, 1:25 AM UTC

nicklaf wrote: post #15

Iv wrote: post #4

Earlier quoted context omitted.

Also a very good advice I had received a long time ago but that I had dismissed until recently: some articles are just badly written. The author sometimes forget to mention what B actually is, changes the meaning of the multiplication operator implicitly, assumes the formula given is inside an integral.... I could go on. A maths article is an article. A lot are imperfect, quite a few are bad. Some times it is not the…

I remember reading this exact piece of advice, but have been trying to remember where. Would anybody be able to provide the blog post I believe we both read, which says almost exactly what Iv wrote? I could have sworn Michael Nielson wrote it, but I was unable to produce it last time I searched. So I am following OP's advice and am asking here. :P

I may have found the post by Michael Nielson I was looking for... on Hacker News!

https://news.ycombinator.com/item?id=666615

In particular, this piece of advice he gives still rings true to my ears:

Often, when struggling with a book or paper, it's not you that's the problem, it's the author. Finding another source can quickly clear stuff up.

Re: How to Read Mathematics

#137

Post by jwdunne » Thu, Dec 14, 2017, 8:09 AM UTC

Koshkin wrote: post #135

j2kun wrote: post #125

Earlier quoted context omitted.

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

Right. The very first step in trying to prove a theorem is to try and disprove it (by looking for a counter-example).

Which are a form of example. You don't store those counter examples in your head - you write them down. If they don't turn out to be counter examples, what are they?

Re: How to Read Mathematics

#138

Post by klibertp » Thu, Dec 14, 2017, 9:28 AM UTC

gizmo686 wrote: post #124

klibertp wrote: post #97

Earlier quoted context omitted.

> 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…

Clickable links are but one of the "quality of life" improvements paper authors could provide - but don't. It's not like there are no tools (like nogbit suggests); for example, Jupyter Notebooks are a document format which can incorporate various kinds of content, from latex typesetting to interactive charts to executable code. The problem is that such tools are simply ignored by most of the paper authors. I do understand some of the reasons for this ((over)reliance on journals for instance), but that doesn't change the fact that it makes for much worse experience for the readers than it could be.

Re: How to Read Mathematics

#139

Post by klibertp » Thu, Dec 14, 2017, 9:49 AM UTC

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…

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

No, actually, they learned to write math in a way which is a trade-off between readability and ease of writing (which is perfectly ok). The problem is exactly with thousands of years of history: the notation which is convenient to write on a clay tablet or papyrus scroll is not necessarily the best notation to use in an interactive document format displayed on computer screen.

Fiction writers have as long a history as mathematicians (or much longer) but they don't insist on writing in Latin or Middle English. It is arrogant to think that the notation and practices of contemporary mathematicians are ideal and impossible to improve in any way.

Lastly, writing for other mathematicians is a problem in its own right. There is a difference between conducting your day to day work, where you can write your equations on clay tablets for all I care and writing an article with a purpose of explaining your work to others. Most of the math papers completely ignore this and instead of honestly explaining the matter, authors focus on promoting themselves as great scientists.

There is a lot that can be done to improve understandability (I used "readability" for this earlier, but understandability seems to fit better), to reduce the time needed to read and understand math papers. It would benefit everyone, yet almost nobody does it. There are reasons for this, I'm sure, but it still makes everyone worse off.

Re: How to Read Mathematics

#140

Post by klibertp » Thu, Dec 14, 2017, 9:54 AM UTC

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.

As I said in another comment[1], I'm not even talking about software engineering, just a good practices of everyday programming.

Could you please tell me why do you think that the name `x` is better than `element` (when talking about some set), the name `X` is better than `training_set`, and the `∑` is better than `sum`?

[1] Here: https://news.ycombinator.com/item?id=15914024