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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

#141

Post by rm445 »

JadeNB wrote: post #3

I found this post a shame. (The post itself, not putting it here; I love seeing math posts on HN, and automatically upvote. Bringing hackers and mathematicians together is highly worthwhile for both.) Usually Tao's posts are so insightful, and crystallise some idea so perfectly that it feels like I was just on the cusp of discovering it myself—a rare talent, and hard to cultivate since it goes against the ego. In thi…

As an engineer, not a mathematician, I'm glad that good mathematicians care about good notation. The relatively-elementary maths that I learnt didn't always have good notation, the tradition seems suited to chalk and pen i.e. complex glyphs are easy, but perhaps because with hand writing the size and position of elements can be ambiguous, too much notation is overloaded and re-used. Even simple stuff like an exponent of -1 meaning inverse function, it's hardly unusual for it to be mixed up with numerical exponents.

One bugbear is that mathematical writing leaks into engineering science. I wouldn't ask professional mathematicians to start caring about units or change their style while communicating amongst themselves, but in my view textbooks ought to define notation before they use it, and clearly define the units used in all expressions.

Re: Terry Tao on some desirable properties of mathematical notation

#142

Post by cryptica »

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' 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? But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size (have an infinite number of elements) wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together, the property which requires that the result also be present in the same set could not be met because the result would be greater than the highest element in that set...

The problem is that if you start with a highly abstracted math concept and you dig through all the links and definitions of sub-concepts, they all have huge gaps like this... So when you try to combine all the definitions together to make sense of that original highly abstracted concept, you end up with tens or hundreds of possible interpretations. But in fact, Math should only have 1 interpretation for each concept so this is a very bad situation to be in.

I think math definitions should be more elaborate and repetitive if necessary. They should not try to sound terse and clever. They should not assume that the reader can fill in the gaps. The most rational readers will not be able to fill in the gaps because rational people know the dangers of making assumptions.

Re: Terry Tao on some desirable properties of mathematical notation

#143

Post by nicoburns »

quietbritishjim wrote: post #121
auggierose wrote: post #119

Earlier quoted context omitted.

The problem is when it is not quite clear what the symbolic notation stands for. With division, that tends to be less of a problem.

* I find it quite unusual in practice for genuinely new symbolic notation to be used by an author. Maybe that just reflects the fields I read about most (information theory, Bayesian modelling, harmonic analysis). * Usually you don't come across a journal article or even blog post with a single isolated equation. So any new or unusual notation can be explained once and reused many times. * Even if you did have an iso…

It doesn't need to be genuinely new to be confusing. It just needs to be unfamiliar to the reader. At the very least, one should point the reader to a resource where they can read about the notation.

Re: Terry Tao on some desirable properties of mathematical notation

#144

Post by Bootvis »

toxik wrote: post #138
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…

Curiously, you have been ambiguous in the mathematical notation. Counterproof: let x = y.

IMO It should be "Any two distinct points in..." in English as well. Precisions is hard!

Re: Terry Tao on some desirable properties of mathematical notation

#145

Post by contravariant »

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…

The first one isn't really symbolic though. So really you should compare:

- speed is the derivative of position with respect to time.

and

- Let x(t) be the position of an object at time t then its speed v(t) is:

   v(t) = (dx/dt)(t)
Also note that most of the time I'm just putting the symbols after the word explaining what it means, while this does allow me to use the symbolic notation for differentiation it doesn't really make the first part any shorter. Also it would be a mistake to introduce speed by just 1 specific formula (even if I didn't specify the types of the object involved) since speed is a far more general concept.

Re: Terry Tao on some desirable properties of mathematical notation

#146

Post by gjulianm »

toxik wrote: post #138
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…

Curiously, you have been ambiguous in the mathematical notation. Counterproof: let x = y.

You are indeed right!

Re: Terry Tao on some desirable properties of mathematical notation

#147

Post by lapinot »

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…

You're quoting the introduction of the article, which is notoriously a fuzzy abstract in all the wikipedia articles about mathematical concepts. Let's quote the actual (textual) formal definition (sec. 1.1) as it would stand in a textbook:

> Formally, a field is a set F together with two binary operations on F called addition and multiplication. A binary operation on F is a mapping F × F → F, that is, a correspondence that associates with each ordered pair of elements of F a uniquely determined element of F. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Similarly, the result of the multiplication of a and b is called the product of a and b, and is denoted ab or a ⋅ b. These operations are required to satisfy the following properties, referred to as field axioms. In these axioms, a, b, and c are arbitrary elements of the field F. [...]

Still, i don't know how to say it in another way but you probably don't have much experience in mathematics, even the first quote is arguably quite accurate.

> 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?

Yes, unless told otherwise an operation is an internal binary operation, it's really the most common form. When it is not the output is notable enough to be specified.

> But if that was the case then "behave as the corresponding operations on rational and real numbers do" would mean that the fields would always need to be of infinite size wouldn't it? Because if the field had a limited number of elements and you added the last two (highest) elements together [...]

No, the important point here is your use of "highest". A set by default only has equality (and mappings, in and out) but no order relationship. So the most conservative interpretation of "behave as the corresponding operations on rational" would be to only include stuff that can be written using the 4 operations and equality, not ordering.

---

Maybe i'm biaised by the fact that i know what a field is, but still, this particular intro is also how i would present a field: give the most common example and say which operations it has. It sure can create false intuitions like yours about the size, but this will always be the case when we use non-normalized language.

Re: Terry Tao on some desirable properties of mathematical notation

#148

Post by wavegeek »

I like his point about lack of ambiguity. Nothing makes me want to punch an author in the head (without, to be clear, any possibility I would actually do it) like lazily creating an ambiguous notation, which is supposed to be "clear from context", but rarely is. As for example the Einstein summation convention which is to be ignored "when clear from context".

I would add

1. Clearly telegraphing notations. Not hiding them in the middle of long paragraphs or even, and yes I have seen this a few times, defining essential notation in an optional exercise.

2. Having a glossary of notations, so people don't have to remember every single notation and to read every word of the book sequentially.

3. Not creating low value notations that may be used only once and then, possibly forgotten. I have read books with > 1 new notation or definition per page, mostly forgotten thereafter but some random subset needed later, and you are not to know which.

Re: Terry Tao on some desirable properties of mathematical notation

#149

Post by sannee »

dopu wrote: post #91

Is it just me, or does probability theory in general have fairly terrible notation? Ambiguity between random variables and their distributions because of them simply being distinguished by being upper-case or lower-case, writing likelihood functions alternatively with an L() or p(), and using p() (with different arguments) to refer to different probability distributions. Perhaps I'm just having such a difficult time…

Glad I am not the only one finding this extremely frustrating. At least mathematicians tend to be much more explicit than engineers. Unfortunately not every topic which uses probability has a textbook written by a mathematician available.

Re: Terry Tao on some desirable properties of mathematical notation

#150

Post by Smaug123 »

jvvw wrote: post #126
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.

I've got a PhD in maths and can translate symbols into concepts in my brain much quicker than words into concepts in general. Also writing symbolically forces a rigour on the writer. I've been trying to read some semi-mathematical stuff written by scientists but non-mathematicians recently and it is painful trying to figure out what they really mean!

Sure, but the problem is that it was written by non-mathematicians, not that they were not using symbols. That's kind of what I was trying to say: symbols help you be rigorous, but they slow you down, and often the complete text-on-the-page doesn't actually need all the rigour that's forced on you by the symbols.
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