[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.