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Terry Tao on some desirable properties of mathematical notation

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Re: Terry Tao on some desirable properties of mathematical notation

#181

Post by quantdev »

JadeNB wrote: post #42
quantdev wrote: post #25

Earlier quoted context omitted.

I'm both a programmer and a mathematician. Mathematics is written in English (or another human language) plus added symbols. The symbols are what we refer to as the notation. It's not comparable to programming language. Here the words are equal to the symbols. Also, no better notation could help you understand most modern mathematics. There's no better choice of symbols or names that will help you understand Galois t…

> There's no better choice of symbols or names that will help you understand Galois theory. You simply need to understand high-school algebra, then group theory, then Galois theory. This probably isn't completely true—as will be clear to anyone, even a subject expert, who tries to go back and read the original papers. At least part of this is due to fads in notation—in my field, I can easily read the papers of people…

Yes, you're right. I should have said, there's no better choice of symbols that will remove the huge set of prerequisites you need to first understand. There isn't as large of a tower of abstractions to understand most codebases.

Re: Terry Tao on some desirable properties of mathematical notation

#182

Post by madcaptenor »

watersb wrote: post #82
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…

Thus: 1. Write a good narrative and then 2. Write good code Which is to say that a mathematics text is an instance of literate programming.

It's not surprising, then, that Knuth introduced it.

Re: Terry Tao on some desirable properties of mathematical notation

#183

Post by omaranto »

rytill wrote: post #79
Koshkin wrote: post #18

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

Not true at all, there are several times I've attempted to read through a textbook only to be stopped by notation because something was introduced prior to being referenced, or notation is overloaded with multiple meanings. I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me. Does no one else feel…

You are making the huge mistake of assuming that if you hadn't given up when facing difficulty with the notation, the rest would have been easy, that you would have no trouble understanding the concepts!

Don't worry: you are not alone in making that huge mistake, plenty of people do. People that don't give up when confused about notation usually quickly learn that is the concepts and relations between them that require careful thought (and also, usually, that the notation was advantages they didn't realize when at first blush they found it confusing).

Re: Terry Tao on some desirable properties of mathematical notation

#184

Post by madcaptenor »

rytill wrote: post #79
Koshkin wrote: post #18

Difficulties, if any, perceived or real, arising in connection with notation, are usually incomparably smaller than those presented with the subject itself. (Personally, I only wish mathematical notation were better integrated with software in general and programming languages in particular.)

Not true at all, there are several times I've attempted to read through a textbook only to be stopped by notation because something was introduced prior to being referenced, or notation is overloaded with multiple meanings. I consistently have run into "perceived or real" confusing mathematical notation as an impediment to learning in a way that programming languages have never, ever caused me. Does no one else feel…

I have seen math books that have "tables of notation", with a list of all the notations and where they're defined in the text. That helps. (Also, I suspect that the sort of author who would include such a thing is one that's thinking more carefully about notation.)

Re: Terry Tao on some desirable properties of mathematical notation

#185

Post by madcaptenor »

OscarCunningham wrote: post #124
tsimionescu wrote: post #117

Earlier quoted context omitted.

There are some cases where this is true (the most infamous being Roman numerals, where the notation makes even simple arithmetic difficult - XI * VI = LXVI???), but in general notation is just something you learn once and mostly remember, as long as you understand the concepts. However, I fully agree that reading a text that uses notation you are not familiar with without introducing it is almost futile. Even the sim…

Incidentally, the example of roman numeral multiplication you chose is actually fairly easy. Just break it up as (XI×V) + (XI×I). Then since X×V=L and I×V=V (Romans would have these memorised just like we memorise multiplication tables for all the decimal digits) so the first multiplication is LV and the second is XI, which add to give LXVI by simply interleaving the letters.

That example works well because you don't have to do V * V. You can do XVI * VI = (X + V + I) * (V + I) = (XV) + (XI) + (VV) + (VI) + (IV) + (II) = L + X + XXV + V + V + I = LXXXVVVI = LXXXXVI, which "simplifies" to XCVI, but it's already a good bit messier.

Re: Terry Tao on some desirable properties of mathematical notation

#186

Post by omaranto »

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…

If informal descriptions confuse you, skip them and read actual definitions instead.

I actually like informal definitions a lot. I think they serve two different purposes:

1. For beginners they usually soften the blow of a fully rigorous definition, letting them get an idea of the concept before getting it exactly.

2. For experts they can often suggest what the exact definition is faster than it would be to read a precise definition!

But if you don't get anything out of them, you can skip 'em. Definitions are more important: informal descriptions are there to help you grasp the definition faster and to help you develop an intuition for the concept being defined. If you can do those things faster from just a rigorous definition, you don't need an informal description. (But as I said, I feel informal descriptions benefit both beginners and experts, so I'd also suggest practicing reading them more to get a feeling for what kind of details people tend to omit or emphasize.)

Re: Terry Tao on some desirable properties of mathematical notation

#187

Post by olikas »

JadeNB wrote: post #43
Koshkin wrote: post #23

Earlier quoted context omitted.

Mathematicians will never agree to write dblEulerConstant instead of e .

And, issues about the length of the name aside, we shouldn't! `e` isn't a double or any other standard numeric type. It is an infinite-precision number.

I know what you mean, and a number that is not infinitely precise is an interval (i.e. a set of numbers). e is a number that is quite hard to pin on the imaginary number line, because the chances of hitting it with a pin is virtually zero. But the same argument can be made about the number 2 or 3. Those are just notations to abstract ideas, even though it is easier to formulate analogies for some of those numbers.

Now, formalising that above sentence even with symbols is quite difficult.

Re: Terry Tao on some desirable properties of mathematical notation

#188

Post by gmadsen »

andrepd wrote: post #167
gjulianm wrote: post #135

Earlier quoted context omitted.

But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, wh…

You are comparing apples to oranges. Your English definition is missing the definition of neighborhood. To actually be an accurate comparison you'd need to add ", where a neighborhood is a set that contains an open ball that contains the point." But now, you're also missing the definition of open ball...

if every definition needed to redefine definitions of its terms it would be impossible to discuss anything.

Within a certain domain, it is assumed you know basic definitions within it that can be used to talk about more complex things.

Re: Terry Tao on some desirable properties of mathematical notation

#189

Post by brmgb »

gjulianm wrote: post #135
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…

But that example is a little bit artificial, isn't it? A lot of mathematical concepts are more complex than that and sometimes symbols are not the best option. Say, for example the definition of Hausdorff space, in words and symbols: - Any two distinct points in the space have disjoint neighbourhoods. - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅. Another example would be Navier-Stokes equations, wh…

> - ∀x,y ∈ X with x ≠ y, ∃U,V ⊂ X s.t. x ∈ U, y ∈ V and U ∩ V = ∅.

Amusingly and to go in the direction of your argument, this is not a very rigorous definition of a Hausdorff space. You should introduce the topology τ associated with X and specify that U and V are open sets of τ.

Re: Terry Tao on some desirable properties of mathematical notation

#190

Post by parralelex »

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…

If you ever found yourself lost in Plato's cave, you'd be super confused as to what those black splotches on the cave wall are.
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