How to Read Mathematics • Hacker News Board

Re: How to Read Mathematics

#111

Post by jcranberry » Wed, Dec 13, 2017, 4:04 PM UTC

protonfish wrote: post #106

andrepd wrote: post #95

Earlier quoted context omitted.

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 math…

Most math books construct narratives and have points of view on the mathematics they present. Math texts in particular are constantly "refactored" in order to build around a them, or core idea if present.

There is a big difference between code and math in that good code is written with the intent that it be understood as quickly and easily as possible in order to not stymie productivity. Non-referential math texts are written not just to teach you the theorems, but a mathematical perspective, such that the mathematics within becomes intuitive through that perspective.

Its both wildly ignorant and insulting to suggest mathematicians only portray their theorems as minimally is possible for no other purpose than mental masturbation.

Re: How to Read Mathematics

#112

Post by Koshkin » Wed, Dec 13, 2017, 4:10 PM UTC

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…

> Math is about symbolic relations

Tell this to anybody who is familiar with, say, algebraic topology. This is a very narrow view of mathematics. Mathematics is much, much deeper than that. Even the pure algebra is no longer about "symbolic relations".

In fact, today the correct view of mathematics would be much closer to that of theoretical physics: both study one or the other form of reality, using pretty much the same method.

Re: How to Read Mathematics

#113

Post by gizmo686 » Wed, Dec 13, 2017, 4:24 PM UTC

protonfish wrote: post #106

andrepd wrote: post #95

Earlier quoted context omitted.

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 math…

"To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." -Brahmagupta, 628AD

I would say we have come a long way.

Mathamaticians absolutly do spend effort on making their work readable. However, this readability is general not within the equations themselves, but rather in the prose around the equations and in how the proof is presented. Of course, skill levels vary in this, and most mathematicians only ever write for other mathematicians, so that is the audience they have practice with (and, if you read a math paper, likely the intended audience).

Also, generally the "equation" is not what mathamaticians are even trying to explain because it is vastly simpler then what they actually worked on, which is the proof.

For example, suppose quadratic equations were actually really difficult, and a mathamtician finally figured out how to solve them. She would probably say something along the lines of:

"A quadratic equation has solutions x=(-b +- sqrt(b^2 - 4ac))/2a.

[Entire paper talking about how to complete the square]"

The entire point of the equation is to be simple to write down and use. It is not intended to be understood.

Sometimes they say an equation without explanation. That is either bad writing, or knowing the audience. Ideally, every equation would come with either an explanation or reference; but if I am writing a research paper, I am probably not going to cite every fact that can be found in an undergrad calculus textbook.

Re: How to Read Mathematics

#114

Post by Koshkin » Wed, Dec 13, 2017, 4:26 PM UTC

protonfish wrote: post #106

andrepd wrote: post #95

Earlier quoted context omitted.

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 math…

In the actual practice of learning and using mathematics, mathematical notation is often much, much more readable (and potentially much less ambiguous) than any text in a natural language that would attempt to convey the same meaning as a mathematical formula.

What must be noted though is that historically the language of mathematical formulas, unlike more "human-readable" text, has been designed to serve several distinct purposes, and conveying the meaning was originally not the most important one; rather, the language of formulas serves the purpose similar to that of programming languages of today, which is to let one to efficiently and, to a large extent, mechanically perform - and thus radically simplify - 1) calculations; 2) logical reasoning; 3) transformations that lead to discovery of new facts.

Re: How to Read Mathematics

#115

Post by gfodor » Wed, Dec 13, 2017, 4:33 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…

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

Re: How to Read Mathematics

#116

Post by gizmo686 » Wed, Dec 13, 2017, 4:40 PM UTC

gfodor wrote: post #115

gizmo686 wrote: post #108

Earlier quoted context omitted.

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…

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.

Re: How to Read Mathematics

#117

Post by kalid » Wed, Dec 13, 2017, 4:44 PM UTC

klibertp wrote: post #56

kalid wrote: post #17

Earlier quoted context omitted.

Shameless plug, I write about math on BetterExplained.com For each symbol, I try to make an analogy. For "e", for example, I have the notion of "continuous growth". The formal definition is this: https://betterexplained.com/ColorizedMath/content/img/E_(mat... "The base for continuous growth is the unit quantity earning unit interest for unit time." Once you see the role of each part of the definition, the idea snaps…

Just wanted to thank you for a great job you do on BetterExplained. I was able to understand and put to use some concepts only thanks to your articles, most recently it was Fourier transform. I'd be much better at math if people writing educational materials realized that there are more effective ways to teach (and to learn) than what they had experienced. Thank you!

I appreciate it, thank you! I have a similar wish, hoping more people would share math in the way that truly helps them.

Re: How to Read Mathematics

#118

Post by x14 » Wed, Dec 13, 2017, 4:52 PM UTC

Just my perspective, but mathematics seems like badly-written software to me. Shitty single-letter variable names with all the code in main() without any separation of concerns.

Re: How to Read Mathematics

#119

Post by platz » Wed, Dec 13, 2017, 4:54 PM UTC

x14 wrote: post #118

Just my perspective, but mathematics seems like badly-written software to me. Shitty single-letter variable names with all the code in main() without any separation of concerns.

try reading some proofs in coq to see how badly software does in making mathematics readable.

nevertheless, i agree for things like white papers the overloading of identifiers is frustrating.

Re: How to Read Mathematics

#120

Post by kurtmueller42 » Wed, Dec 13, 2017, 4:56 PM UTC

klibertp wrote: post #74

ryanmonroe wrote: post #60

Earlier quoted context omitted.

> When I program my computer, I specify all the steps to accomplish something. You certainly do not specify all the steps your computer must take to run the program because that would make even the simplest program a nightmare to read. In the same way, specifying all the steps a reader must take would make math papers completely unreadable. There's a line which should be chosen wisely, but it's not as simple as "just…

You forgot this: > If it's too verbose, then I abstract it in a function. Yeah, not everything has to be included in your source and you do rely on abstractions when writing code. On the other hand, these abstractions are always there, and it "easily follows" that you can take a peek at them at any time, up to and including disassembling a binary executable. You can't do this with a maths paper - you can't just go to…

>You can't do this with a maths paper - you can't just go to Github to check the source of a library, you have to "reimplement", again and again, pieces of the paper that an author decided to omit.

Analogous to Github would be the entire mathematics ecosystem - it is not unreasonable to expect your reader to read another referenced work rather than reproduce it yourself.

I think a better analogy than Github for work just assumed to be worked through by the reader, would be uncommented sections of code. Despite what anyone says about best practices, there will always be plenty of uncommented sections of code in any serious codebase, although the hope is that they would all be fairly trivial (and this is the same hope with mathematics papers).

I would expect anyone reading my code to understand that if I loop through an array of integers, add them to a sum, and then divide by the length of the array, I am taking an average, even if I would be better served using a method that describes this properly.

I would expect anyone reading my code to intuitively understand what I am doing if I build a reverse lookup map to some other map data structure without me commenting every little line and type to explain the idea behind the construction.

Similarly, it is not always unreasonable for mathematicians to assume their reader can step between lines using elementary techniques. If you want someone to perfectly walk through every single assumption in a proof with zero ambiguity, you have arrived at Principia Mathematica.

Certainly some mathematicians are worse than others, but this is no different than the world of software. I cannot begin the number the times I have stepped into some old enterprise code and immediately wanted to scrub my eyes after viewing some thousands-of-lines method with few to no comments and code branching in every direction for hundreds of lines at a time.