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Terry Tao on some desirable properties of mathematical notation

mathoverflow.net

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Re: Terry Tao on some desirable properties of mathematical notation

#151

Post by HPsquared »

erezsh wrote: post #132
quietbritishjim wrote: post #116

Earlier quoted context omitted.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

That's not a genuine example. Most academic texts I've read will use something like s = d/t And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it. Some will even make up their own notation: s = d(t) And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that…

That sounds more an issue of unclear naming, than with notation itself.

Edit: Though I agree, mathematical notation lends itself to using single letters to denote objects. This can of course be problematic. I'm a fan of 'pseudocode' - somewhere between natural language and rigorous notation - where possible.

Re: Terry Tao on some desirable properties of mathematical notation

#152

Post by Smaug123 »

quietbritishjim wrote: post #116
Smaug123 wrote: post #107

Earlier quoted context omitted.

I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

I'm certainly not saying that symbols are always bad. Rather, I am attempting to argue that the parent is false in asserting that "mathematician[s]… just wish it would be written with symbols so that they could know precisely what the book is trying to say". The Hausdorff space example is a good one: it's very easy to say what a Hausdorff space is, but if you have to spell it out formally then the definition is kind of big and ugly.

Re: Terry Tao on some desirable properties of mathematical notation

#153

Post by joshuaissac »

cryptica wrote: post #142
Koshkin wrote: post #49

Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothin…

Math symbols are a minor issue for me. What confuses me the most are descriptions of mathematical concepts. For example, Wikipedia describes a 'field' like this: "In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do." It doesn't make sense to me. What does it mean if an operation 'is defined…

> What does it mean if an operation 'is defined' on a set? Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?

For a binary operation f to defined on a set, f(x,y) must exist for every x and y in the set. There is no requirement that f(x,y) itself is in the set. Adding that requirement would mean that the set is "closed" under the operation f. So if we take Z+ = {1, 2, 3, ...}, ordinary division is defined on Z+, but Z+ is not closed under division, since we can get results like 2/3 that are not in Z+. Whereas division is not defined on Z0+ = {0, 1, 2, ...} because we can get undefined results like 2/0.

However, some definitions of "binary operation" include the "closed" property, so under such definitions, division would not be considered a binary operation on Z0+.

>Does it mean that any 2 elements combined together using that operation always need to output an element which is also in the same set?

Specifically in the case of a field, yes; addition needs to be defined, closed and invertible for the set; multiplication needs to be defined, closed and invertible for the set excluding the additive identity (zero).

Re: Terry Tao on some desirable properties of mathematical notation

#154

Post by throwaway_pdp09 »

Koshkin wrote: post #49

Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothin…

> but rather looks like a crazy mess of characters and other marks in a multitude of fonts

You're projecting (in the psychological rather than set-theoretic way).

Re: Terry Tao on some desirable properties of mathematical notation

#155

Post by jimhefferon »

dhosek wrote: post #80
Koshkin wrote: post #49

Mathematical notation is great at facilitating formal manipulations. This is its critical feature, and without it we would get stuck at the level of ancient mathematics. This is the reason it was invented a few hundred years ago in the first place. That said, I find that notation is often abused in texts as a mere substitute for the normal human language which, while allowing to compress the text, does in fact nothin…

I remember, as a callow college freshman waiting in the hallway of the math department to be able to go into a classroom reading an article which talked about mathematical writing and the first thing it said was to prefer English text over mathematical symbols in numerous cases (e.g., writing "For all $x$ in the Reals" over $\forall x\in\mathbb{R}$). As someone who was highly skilled in TeX at a time when such skills…

When I teach the intro to proofs class I require that they learn LaTeX. Some students like the availability of symbols so much that they go a bit nuts. Something like this sentence: $\forall x\in\mathbb{R}$ $\exists y$ that is $>$ the number $x+1$. Sigh.

Re: Terry Tao on some desirable properties of mathematical notation

#156

Post by ChrisLomont »

Smaug123 wrote: post #107
logicslave wrote: post #98

Earlier quoted context omitted.

"does in fact nothing to help the reader better understand what is being said" For a mathematician it is the opposite, they just wish it would be written with symbols so that they could know precisely what the book is trying to say

I don't think this is true in general. It may be true for a novice, who needs all the available help to keep them rigorous (but even then, there is definitely room for reading-to-build-intuition), but symbols definitely slow you down while you translate them.

As a math PhD I disagree. I can read math notation far faster and more accurately than ambiguous English. We don’t translate symbols. If anything, when reading English we have to translate into symbols.

For example, reading 5-7, I don’t have to translate the - symbol to the word “subtract”. I know this is -2. And I don’t translate the - in that to the word “negative,” and certainly not to the word “subtract”. And it’s vastly faster to agree 5-7=-2 is correct than “5 subtract 7 equals negative 2.”

Symbols are how mathematicians think.

Re: Terry Tao on some desirable properties of mathematical notation

#157

Post by scribu »

Darkstryder wrote: post #122

Steal this idea: a Shazam of mathematical notation. In an app you would draw (or take a picture) of a mathematical symbol you don’t recognize and get a link to the appropriate Wikipedia page. My biggest pet peeve with mathematical symbols is the difficulty of looking them up when you don’t know them already. If I’m reading a text on a topic I'm unfamiliar with, I can at least google the keywords I don't know. This is…

Shapecatcher is a free webapp that allows you to draw any Unicode symbol:

https://shapecatcher.com/

Re: Terry Tao on some desirable properties of mathematical notation

#158

Post by jimhefferon »

enriquto wrote: post #137

It would be nice to have an equivalent post, but with programming languages. The fact that different programs perform an identical computation is important. For example, in Python/numpy you can write c = 0 for i in range(u.size): c = c + u[i] * v[i] or c = u.T @ v and even if the result is identical, the computation is not, the first one being orders of magnitude slower. There is no good reason for it to be so, unfor…

When Python first came out, one of its most appealing characteristics was that, in very strong contrast to Perl's There's-More-Than-One-Way, in Python there was a canonical way. That's no longer true, and it makes the language less valuable.

Re: Terry Tao on some desirable properties of mathematical notation

#159

Post by yw3410 »

wheresmycraisin wrote: post #55
foobar_ wrote: post #50

No one uses mathematical notation for practical purposes. This is just like the medival music notation which is neither practical nor what modern composers use, which is more visual in nature. Infact modernism is a rejection of medievalism. I think in the future programming will force all mathematicians to code or give out simulations. Most mathematical notation was intended to be throwaway by the original authors, t…

Programming != proofs, or in general communicating abstract mathematical ideas. Writing mathematics is nothing like writing software.

I don't think this is true (or at least not true for all programs); there's a whole discipline of software which shows how close writing software is to formal logic/inductive proofs in Agda Coq, etc.

Re: Terry Tao on some desirable properties of mathematical notation

#160

Post by swilliamsio »

erezsh wrote: post #132
quietbritishjim wrote: post #116

Earlier quoted context omitted.

Do you really think > speed = distance / time is less clear than > speed is the ratio of distance over time ? To me the first equation is genuinely easier to read for the purposes of understanding, not just for formal manipulation. For larger equations the difference is only more stark, not less. I have a maths PhD so your comment about symbols being for novices doesn't apply. I suppose the beauty of the first equati…

That's not a genuine example. Most academic texts I've read will use something like s = d/t And sometimes not even explain what the components mean, because obviously 't' stands for time. That's what all their lecturers used, so why bother explain it. Some will even make up their own notation: s = d(t) And somewhere will say "f(x) in this article describes an inverse multiplicative relation", without explaining that…

>s = d/t

Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.

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