Yeah, I get the issue in general, but the quote I referenced was specifically referring to the equations from Wikipedia.
My thought is that pretty much any mathematician would not choose to use absolute value bars in a case where it would be ambiguous. For example (|a|)b(|c|) would be instead written |ac|b and |a(|b|)c| would be written |abc|. And so unless it the equations are specifically about showing properties of absolute value (in which case just use parenthesis), I don't see a case where the ambiguity would pop up in real discourse.
That changes the meaning. You're now relying on multiplication being commutative which is not true for all sets.
As I said at the end of my last comment: I don't see a case where the ambiguity would pop up in real discourse. If you are dealing with non-commutative objects you can deal with a few more parenthesis. Just like non-associative objects result in many more.
An interesting way of justifying how two things are really the same thing, that is, by acknowledging their difference.
The point is that the pointed ambiguity (the difference) can be quite significant, and these become apparent when studying these sentences in a formal logic. Hence, the significance of the ambiguity.
This (and the article under discussion) just echoes the idea that mathematical sentences and proofs are considered "rigorous" but "informal", where "formal" is the domain of logic.
In CS we talk about l-values and r-values. Only an l-value can be assigned to, but you can compare r-values for equality. So there is a solid distinction there between types of expression which is upheld in programming languages.
For example i^2 = -1 is the generally accepted definition of the imaginary unit.
See n+k patterns in Haskell for a programming exception to the rule you mention.
Definition and equality are really the same thing. A definition is just an introduction of a term/variable, and an equality giving it a value. So the only difference is if a term in the equation hasn't been introduced yet.
> The bracketing placement of symbols for absolute value, is not a matching pair, thus is ambiguous (when nested or sequential).
How was it not a matching pair? I only see two |'s in each equation so they have to pair up. What other way is there to interpret it?
> the notation for differentials “dy/dx” rapes the division notation. The differential notion has no position in math regarded as a computer language, as implied by the math philosophies of formalism and logicism.
I believe the original reasoning for notating derivatives as dy/dx is that it is a limit of a faction and so approaches a fraction of infinitesimals. with that background the division notation is simply extended to a new class of numbers.
I am not sure what the second sentence is trying to say.
Also pretty much all of the issues could be solved with gratuitous parenthesis. So we have a solution to get a formal grammar; we use the current notation because it is faster to read and write. So without any concrete proposals, this critique has very little content.