Hacker News Reader

Comment by quietbritishjim | Hacker News Reader

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.

View on HN

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

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

Unfortunately that's not uncommon in mathematical texts either!

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

You are indeed right!

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...

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).

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.

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.

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.

> - ∀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.

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.

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.

>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.

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

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").

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.

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.

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.

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 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.

> * 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."?

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".

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.

Reply on news.ycombinator.com