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Comment by onedognight | Hacker News Reader

Sussman's critique of mathematical notation is hollow. The point of good notation is to hide the details while conveying the high level compactly and succinctly.

Lagrange's equations written the way he derides are a perfect example. His need to think about what spaces the symbols live in based on their context in the expression is typical when translating math to code. He should be talking about about what kind of type inference engine one needs to allow the original notation and end up with his positional notation internally.

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Actually, the notation is confusing even to professional mathmaticians. The notation used in differential geometry is the archetype of this phenomenon; it's sometimes incredibly hard to work out what the simple-looking notation means.

In fact, differential geometry has been described as the study of objects "invariant under a change of notation" playing on the lack of clarity in this field, along with the standard description of a tensor as an object "invariant under a change of coordinates".

See, for example, the introduction to this piece: http://www.math.jhu.edu/~yrubinst/RiemGeom/Thesis-pp-23-30-a...

Einstein joked about this http://books.google.com/books?id=U2mO4nUunuwC&q=%22I%20h... and Penrose have made an elaborate prank out of situation too, http://abstrusegoose.com/130.

That said, notations here really correspond to different ways of thinking and are important and shouldn't be ridden.

Maybe, but a notation arrived at by an accretion of historical accident is not the way to introduce this stuff to beginners, inertia be damned.

Edit: Notation

So raise your kids in Esperanto? Or does it really matter that much in the end?

I thought someone might try that tack. Math is completely decoupled from its representation, so you can use whatever notation you want and not lose anything. I'm not gonna guess to what extent linguistic communication is bound up with its representation, but it's certainly more than math, meaning meaning can be lost. So your analogy doesn't work. Also, since mathematical notation is a tool primarily to facilitate mechanical transformations, and only secondarily for communication, you should use whatever lets you think biggest.

An excellent point as long as you do not need to actually communicate your results to anyone. As soon as you do, it becomes precisely as bound as language is.

So raise your kids in Esperanto?

I hope not.

http://www.xibalba.demon.co.uk/jbr/ranto/

A Nobel laureate in physics who grew up in a non-English-speaking country recommends learning English as a key to physics research.

http://www.phys.uu.nl/~thooft/theorist.html

The author of that article is well known in the Esperanto world for being the author of that article... and nothing more ;)

A true expert in Esperanto, the linguist and psychologist Claude Piron, debunked that article years ago:

http://claudepiron.free.fr/articlesenanglais/why.htm

Good teachers do try to use consistent notation, but they also understand the need to communicate with the rest of the world and so end up having to teach historical accidents.

Here's a fun one: pi is wrong! http://www.math.utah.edu/~palais/pi.pdf

"At school Feynman approached mathematics in a highly unconventional way. Basically he enjoyed recreational mathematics from which he derived a large amount of pleasure. He studied a lot of mathematics in his own time including trigonometry, differential and integral calculus, and complex numbers long before he met these topics in his formal education. Realizing the importance of mathematical notation, he invented his own notation for sin, cos, tan, f (x) etc. which he thought was much better than the standard notation. However, when a friend asked him to explain a piece of mathematics, he suddenly realized that notation could not be a personal matter since one needed it to communicate."

The story is either in "Surely You're Joking, Mr. Feynman" or "What Do You Care What Other People Think?". I forget which stories are in which book. Both are great though.

     I thought my symbols were just as good, if not better, than the regular

symbols -- it doesn't make any difference what symbols you use -- but I discovered later that it does make a difference. Once when I was explaining something to another kid in high school, without thinking I started to make these symbols, and he said, "What the hell are those?" I realized then that if I'm going to talk to anybody else, I'll have to use the standard symbols, so I eventually gave up my own symbols.

-- Surely You're Joking, Mr Feynman (http://www.gorgorat.com/)

That said, I'm inclined to say that at some point it makes sense to break backwards compatibility in favour of a better system.

Sussman's critique of mathematical notation is hollow.

I don't do enough math to have a useful opinion, but it strikes me that something as simple as 1 + 1 = 2 is ambiguous (does f = x+x or x+1?) and using unambiguous notation would be tedious and useless.

Unless you're trying to turn it into a program. I sometimes think of programming as the opposite of math.

type inference

Guy Steele's ‘Fortress’ language is headed in that direction.

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