Mathematical Notation, Past and Future (2000) • Hacker News Board

Re: Mathematical Notation, Past and Future (2000)

#21

Post by kmill » Tue, Dec 28, 2021, 4:23 PM UTC

Qem wrote: post #15

kmill wrote: post #10

Earlier quoted context omitted.

> preventing students to think of math as lego bricks in which you can use a set of functions to build newer functions Isn't this something that mathematical notation lets you be able to see at all? (I'm also not sure what you mean here exactly.) By the way, have you heard what Euclid allegedly said to Ptolemy I when the king found the Elements to be difficult? "There is no royal road to geometry." Math is inherently…

But surely notations vary a lot in quality. Think for example in roman numerals vs. positional number systems as alternative notations to work with arithmetic. One was surely a great advance over the other. So I wonder, what elements of our modern notations are akin to roman numbers, a suboptimal solution awaiting to be replaced with something vastly better. Is all perceived complexity in math inherent to it, or at l…

> Is all perceived complexity in math inherent to it

No, of course not, but I'd argue that every notation that's in use gives practitioners that use it strictly more mathematical capability. Notations make things that previously only geniuses could comprehend become things with wider accessibility.

Consider Roman numerals. Like anything, they are suboptimal, but without them, large numbers are essentially impossible to manipulate. They gave the business class the ability to record their finances and inventory, for example, which is a remarkable achievement. It's great when better things come along like positional number systems -- people can learn the art of division in grade school because of it, rather than needing to leave that to the experts -- but we shouldn't dismiss what was replaced as merely holding us back.

Re: Mathematical Notation, Past and Future (2000)

#22

Post by kmill » Tue, Dec 28, 2021, 4:28 PM UTC

syngrog66 wrote: post #13

I love math but prefer programming type syntax to express math ideas, rather than much of traditional math notation. Much of it feels too alien and non-intuitive. Too irregular and random, and not predictable. Various styles duct taped together over centuries. The names of the various symbols are hard to remember, and dont get me started on the pain from not being on the standard PC/ASCII/Western computer keyboards.…

You might like Structure and Interpretation of Classical Mechanics by Sussman and Wisdom. It does classical mechanics using an MIT-Scheme library.

Lean/mathlib is a mix of functional programming and classical math notation. The community likes unicode quite a lot, with support by the editor (using quasi-Latex input methods in both VS Code and emacs). Here's an example from graph theory (I removed some conceptually irrelevant parts):

  theorem simple_graph.sum_degrees_eq_twice_card_edges (G : simple_graph V) [fintype V] :
    ∑ (v : V), G.degree v = 2 * G.edge_finset.card

Re: Mathematical Notation, Past and Future (2000)

#23

Post by sieste » Tue, Dec 28, 2021, 5:14 PM UTC

kmill wrote: post #20

DarylZero wrote: post #19

Earlier quoted context omitted.

Also I just want to add... mathematicians have ⋅, ×, and ∗ to express multiplication, but decided that they also needed ADJACENCY OF CHARACTERS to express multiplication, and therefore mathematics is not allowed to use variable names longer than one letter. Come the fuck on, mathematics.

It's fine, you just add a space. It's also used for function application, like "sin x". I'd say single-letter variable names are mostly due to ease of handwriting when doing calculations (or for giving a chalkboard talk), and authors tend to just use what they've already come up with when writing things up. Mathematicians seem to be very tolerant of ambiguous parsing, so that can't explain all of it.

When reading a math proof it helps to spell out what the symbols stand for, instead of saying the symbol names, for example, instead of "E equals m c squared" say "energy equals mass times speed of light squared". This also greatly helps understanding. If you can't keep track because there are too many symbols, make a little lookup table.

Re: Mathematical Notation, Past and Future (2000)

#24

Post by syngrog66 » Tue, Dec 28, 2021, 5:27 PM UTC

DarylZero wrote: post #17

syngrog66 wrote: post #13

I love math but prefer programming type syntax to express math ideas, rather than much of traditional math notation. Much of it feels too alien and non-intuitive. Too irregular and random, and not predictable. Various styles duct taped together over centuries. The names of the various symbols are hard to remember, and dont get me started on the pain from not being on the standard PC/ASCII/Western computer keyboards.…

I think having 15 different ways to express multiplication is charming.

agreed. heh. thankfully though at least 3 of them can be expressed with a normal keyboard:

ab

a * b

a X b

Re: Mathematical Notation, Past and Future (2000)

#25

Post by kmill » Tue, Dec 28, 2021, 6:34 PM UTC

kmill wrote: post #22

syngrog66 wrote: post #13

I love math but prefer programming type syntax to express math ideas, rather than much of traditional math notation. Much of it feels too alien and non-intuitive. Too irregular and random, and not predictable. Various styles duct taped together over centuries. The names of the various symbols are hard to remember, and dont get me started on the pain from not being on the standard PC/ASCII/Western computer keyboards.…

You might like Structure and Interpretation of Classical Mechanics by Sussman and Wisdom. It does classical mechanics using an MIT-Scheme library. Lean/mathlib is a mix of functional programming and classical math notation. The community likes unicode quite a lot, with support by the editor (using quasi-Latex input methods in both VS Code and emacs). Here's an example from graph theory (I removed some conceptually ir…

Speaking of which, there's a comment today about SICM in Clojure: https://news.ycombinator.com/item?id=29714267

Re: Mathematical Notation, Past and Future (2000)

#26

Post by DarylZero » Tue, Dec 28, 2021, 8:03 PM UTC

kmill wrote: post #20

DarylZero wrote: post #19

Earlier quoted context omitted.

Also I just want to add... mathematicians have ⋅, ×, and ∗ to express multiplication, but decided that they also needed ADJACENCY OF CHARACTERS to express multiplication, and therefore mathematics is not allowed to use variable names longer than one letter. Come the fuck on, mathematics.

It's fine, you just add a space. It's also used for function application, like "sin x". I'd say single-letter variable names are mostly due to ease of handwriting when doing calculations (or for giving a chalkboard talk), and authors tend to just use what they've already come up with when writing things up. Mathematicians seem to be very tolerant of ambiguous parsing, so that can't explain all of it.

You can point to "sin", but in practice math just doesn't use multi-char variable names. The trig functions are special cases. Doesn't mean you're allowed to name arbitrary vars like that.

Re: Mathematical Notation, Past and Future (2000)

#27

Post by kmill » Tue, Dec 28, 2021, 9:13 PM UTC

DarylZero wrote: post #26

kmill wrote: post #20

Earlier quoted context omitted.

It's fine, you just add a space. It's also used for function application, like "sin x". I'd say single-letter variable names are mostly due to ease of handwriting when doing calculations (or for giving a chalkboard talk), and authors tend to just use what they've already come up with when writing things up. Mathematicians seem to be very tolerant of ambiguous parsing, so that can't explain all of it.

You can point to "sin", but in practice math just doesn't use multi-char variable names. The trig functions are special cases. Doesn't mean you're allowed to name arbitrary vars like that.

What about ker, coker, im, hom, end, aut, lim, colim, Set, Top, Man, Mat, Vect, GL, SL, SO, Lie, Gr, Tor, Ext, tr, Rep, char, rank, Isom, ann, hull, Diff, ...? I'm just copying multiletter names out of my preamble.tex file here.

I get that these aren't variables, but it's really not so uncommon outside of trigonometry to have multi-letter names for things that appear in equations.