[Smaug123]

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.

[quietbritishjim]

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 equation is that the objects (speed, distance, time) are visually very distinct from the operations and relationships. In the second form, there's a bit of a word soup so you need to "manually" parse the sentence rather than letting your eyes (really the visual cortex) do that bit of the processing for you.

[gjulianm]

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, where they're much easier to understand in words than in symbols. Symbols are ok when you don't have to search too much to see what they mean and when the idea you're trying to transmit with them is relatively simple, but trying to build complicated phrases and definitions with symbols, for me, ends up being a mess.

edit: fixed hausdorff space definition, points should be distinct

[C4stor]

I find it a bit funny that the whole discussion seems to be on what definition is better. The two complements each other perfectly, and reading them actually makes both clearer for me.

Since there is in the modern world almost no use case where you're heavily space contrained (maybe the final print version of an article for some publications ? I'm not that familiar with the research world), I don't see why you'd try to choose one instead of including both where needed.

[andrepd]

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...

[gjulianm]

Doesn't that add to the point that sometimes literal definitions are better than symbolic? Either you have a symbol that says "this is a neighbourhood" or you have the symbolic definition of neighbourhood (which is missing in my symbolic definition of the space, btw, I just noticed) and then you force the reader to identify those symbols and say "oh, this is a neighbourhood". The former is the same issue as in English, and the latter adds unnecessary complexity (no people reading about Hausdorff spaces will be unfamiliar with the concept of a neighbourhood of a point).

[gmadsen]

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.

[quietbritishjim]

I think my overall point is that symbols can significantly make things simpler, not that they always do. It's very much like the analogy with diagrams that I made in that comment: making things visual can help enormously, but a bad diagram or equation can be as unclear, or worse, than some descriptive text.

Communication is hard, and unfortunately doing it well requires experience and thought rather than simplistic rules like "always use symbols rather than words" or vice-versa.

[toxik]

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

[Bootvis]

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

[nicoburns]

Unfortunately that's not uncommon in mathematical texts either!

[gjulianm]

You are indeed right!

[brmgb]

> - ∀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.

[erezsh]

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 it's actually simple division, so other academics won't find them too obvious or boorish.

Some even take it a step further and use random greek letters (without exhausting the English alphabet first ofc.), where small gamma and big gamma mean completely different and unrelated things.

[HPsquared]

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.

[swilliamsio]

>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.

[Sharlin]

I would assume that the "s" comes either from Latin spatium or German Strecke.

[abenga]

This is (velocity) = (displacement) / (time). s is used because of the Latin word _spatium_ for space. If I remember correctly, there was a difference between distance and displacement. (displacement is "net").

[quietbritishjim]

Displacement is a vector whereas distance is a scalar (similarly for velocity vs speed). Even if you restrict to 1 dimension there's a difference as displacement (and velocity) can be negative.

[Smaug123]

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.

[contravariant]

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.

[auggierose]

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.

[quietbritishjim]

* 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 isolated equation with unusual notation, I still think it's more clear to define the notation and then show the equation rather than spelling it out it words! (I'm sure you could find terrible counterexamples!) The visual benefit seems so great to me that it would be worth splitting it into two parts. A bit like a diagram or a graph can make things clearer even if it needs a bit of explanation.

[nicoburns]

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.

[throwaway744678]

> * I find it quite unusual in practice for genuinely new symbolic notation to be introduced by an author.

This sentence seems to contradict the rest of your comment; did you mean "I find it quite unusual in practice for genuinely new symbolic notation to NOT be introduced by an author."?

[quietbritishjim]

Thanks, that was ambiguous and the way you read it wasn't what I intended. I meant it was unusual for an author to use a new symbolic notation at all, without saying anything about whether they define it in those cases where they do use something new. I've edited "introduced" to "used".

[jvvw]

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!

[Smaug123]

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.

[ChrisLomont]

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.

[Smaug123]

I think you've got the same misconception as quietbritishjim above. Start stacking quantifiers, and the symbols get hairy much faster than the English does.

[ChrisLomont]

Maybe you're not used to reading quantifiers. I also find them much easier, faster, and more accurate to read, because it's from practice.

Sure you can take simple English statements, and write them with quantifiers, and claim the English is simpler. But going the other way, expressing complex items in English, is a non-starter. English is far too sloppy, whereas the quantifier version is mathematically precise.

Try converting something professional, such as Godel's incompleteness proofs, into English. Without precise quantifiers you'd quickly get lost, make mistakes, and take forever to get anywhere.

For example, look at page 17 of the proof [1], at AG(6)(a) (after the "Thus"), where there is a long statement in logic. Convert to English (near impossible, certainly not possible without something like parentheses) and tell me it's easier for a logician to read. It's not. As written it's concise and parseable without confusion for a logician.

The simple English sentence case is a tiny part of what mathematicians do.

[1] https://web.yonsei.ac.kr/bkim/goedel.pdf

[Smaug123]

Look, do you really want me to wade through sixteen pages to discover the notation first (written by someone who doesn't know about \langle and \rangle, either)? I would consider it bad argumentative form of the same water as Euler's apocryphal "does God exist" debate with Diderot.

At the very least, you will struggle to persuade me that the use of \wedge is easier to understand than the English word "and" with line breaks.

Also you've picked a specific example where the objects of study are these long strings of symbols. Of course any paper worth its salt is going to use them - they're literally the things that the paper is there to examine. It's the metamathematical statements in this context, not the quotation of the formulas under study, that I want to replace with English.

[ChrisLomont]

>do you really want me to wade ....

No - just pointing out that no matter how hard you studied that an English equivalent of such a terse expression will be a mess, vastly harder to understand.

>At the very least, you will struggle to persuade me that the use of \wedge is easier to understand than the English word "and" with line breaks.

You're making my case for me :)

The word "and" and "or" are ambiguous in common English, and have no common or even technical well-defined precedence. "Or" in English both can mean "inclusive or" or "exclusive or," yet most people simple write "or".

In math they have well defined precedence, and math has parentheses to order correctly, unlike English.

The only reason you find English easier to understand is you have used more than math symbols at a ratio that makes that true for you. It's not true for everyone, especially professionals, that use some set of notation a lot.

>Also you've picked a specific example where the objects of study are these long strings of symbols

Avoiding the point. The math does not have to self-referential to use complex expressions not amenable to writing in English. This expression is not complex because it's a meta argument. Such expressions can occur in all sorts of places.

At this point I'm sure your present these red herrings to avoid considering that your opinion is not common among professionals that can read symbols much faster and more accurately than can be done in English.

No sense in continuing.

[Smaug123]

> No sense in continuing.

On this we agree, at least!

[commandlinefan]

Hm - well, I'm not a mathematician, but I am a programmer and I know that there are many, many times when I've seen somebody try to describe an algorithm in what ends up being incomprehensible English followed by a code example that actually clears up what it was they were trying to say.