> 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.
* 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.
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."?
Most academic texts I've read will use something like
s = d/t
Some will even make up their own notation:
s = d(t)
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.
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.
Slightly worse than that. That equation is always written as v = s/t. v represents velocity and s represents distance, for some reason.
- 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
Within a certain domain, it is assumed you know basic definitions within it that can be used to talk about more complex things.
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.
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.
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.
- 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)
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.
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.
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.
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.
On this we agree, at least!