For mathematicians, = does not mean equality • Hacker News Board

Re: For mathematicians, = does not mean equality

#161

Post by grosjona » Sat, Apr 14, 2018, 1:53 PM UTC

I remember thinking the same thing after reading https://www.hillelwayne.com/post/equals-as-assignment/

Math symbols and expressions are inconsistent just like regular languages. But, unlike math, other languages don't claim to be consistent.

It's not surprising that John von Neumann said "in mathematics you don't understand things. You just get used to them." - I've never heard a software developer say this about coding.

For example, I did not enjoy integrals at school because of the 'dx' at the end which means 'with respect to x' but which actually looks like a multiplication (* d * x).

I think that the reason why I never got deep into math is because the language of math is too inconsistent and has too many logical shortcuts and I can't operate in such environment.

Re: For mathematicians, = does not mean equality

#162

Post by pron » Sat, Apr 14, 2018, 2:12 PM UTC

narag wrote: post #153

pron wrote: post #150

Earlier quoted context omitted.

> Trying to make it easier to type, a bigger usability problem is created, assigning counterintuitive symbols to functionality that most people would not associate to them. How do you know? It seems no such problem has been reported, so the burden is on you to show it exists. I don't think people assume that the same token has the same meaning in different languages.

It seems no such problem has been reported Oh, please. This has been a flamewar since the 80's. Every angle you can think of has been tried before. The real question is why are we still using plain text. A lot of the difference between languages is this kind of absurd minutiae taken too seriously as if we're discussing about the reality fabric instead of mere conventions.

A flamewar is a disagreement of opinion. I am not aware of any actual problem reported as a result of using the `=` sign for things other than equality or anything suggesting it is generally counterintuitive.

Re: For mathematicians, = does not mean equality

#163

Post by acqq » Sat, Apr 14, 2018, 2:46 PM UTC

Recursing wrote: post #146

ancarda wrote: post #141

Earlier quoted context omitted.

Off-topic: how did you format that equation (K = mv^2/2) so nicely? e.g. mv is in italics. It looks really nice. Does Hacker News have built in LaTeX, or is that just very clever use of Unicode?

Seems to be just unicode in a code block. mv is 𝑚𝑣

And the verbatim lines are explained here:

https://news.ycombinator.com/formatdoc

Re: For mathematicians, = does not mean equality

#164

Post by narag » Sat, Apr 14, 2018, 2:54 PM UTC

pron wrote: post #162

narag wrote: post #153

Earlier quoted context omitted.

It seems no such problem has been reported Oh, please. This has been a flamewar since the 80's. Every angle you can think of has been tried before. The real question is why are we still using plain text. A lot of the difference between languages is this kind of absurd minutiae taken too seriously as if we're discussing about the reality fabric instead of mere conventions.

A flamewar is a disagreement of opinion. I am not aware of any actual problem reported as a result of using the `=` sign for things other than equality or anything suggesting it is generally counterintuitive.

If I were you, I would start searching articles about a more interesting question: can everybody learn to program?

When you're teaching children, this is one of the most common complaints.

Re: For mathematicians, = does not mean equality

#165

Post by jcranberry » Sat, Apr 14, 2018, 3:26 PM UTC

pron wrote: post #149

smadge wrote: post #122

I agree that “=“ as interpreted by people doing math requires context, but in most situations they are able to translate it into a “correct” or formal notion of equality. For example, translating on the fly these ad hoc notions of equality into precise notions of equality in first order logic and/or set theory. For example, f(x) = 2x + 3 Might be translate into something like, For all x in the domain of f, f(x) = 2x…

In every formal system, such as FOL/ZFC, all operators must have a precise meaning. And while it is true that most informal uses of the `=` sign in mathematics could be translated to some formal expression making use of a precisely-defined `=` sign, I think the point was that some (or most) mathematical notation is not formal, and therefore saying that `=` always means a precise notion of equality is wrong, even if i…

I agree with your interpretation of the article, but I find that the article doesnt match the title. I think the equals sign almost always represents usage of an RST equivalence of some sort, even if the formula containing it isnt an equality (with the irritating exception of Big O notation).

However, I think that using this as a justification for programming languages' use of the equals sign for assignment (whether or not it needs any justification at all) I think is a moot point because mathematics is more expository, and isnt always strictly rigorous as long as its clear that everything is well defined and correct in the end (although there is expressiveness in the formalities as well).

Re: For mathematicians, = does not mean equality

#166

Post by coldtea » Sat, Apr 14, 2018, 3:49 PM UTC

kps wrote: post #31

(I assume this was inspired by https://news.ycombinator.com/item?id=16803874 ) The use of ‘=’ for assignment in programming languages comes, not directly from mathematics, but indirectly from the use of mathematics in science and engineering. As an example, consider the formula for kinetic energy, commonly written 𝑚𝑣² 𝐾 = ─── 2 Why isn't it written 2 K = m v ², which expresses the same mathematical equality in a s…

>Why isn't it written 2K=mv², which expresses the same mathematical equality in a smaller, simpler form?

Because you want to find the value of K, which means that if it was expressed thusly, you'll still need to divide by 2 to get to the desired result.

Re: For mathematicians, = does not mean equality

#167

Post by svat » Sat, Apr 14, 2018, 3:50 PM UTC

jcranberry wrote: post #62

abhishekjha wrote: post #21

Also the asymptotic notation where f(n) = O(n^2) means a set membership operation.

IMO one of the most irritating abuses of notation that I've come across given that it requires no additional effort to use the 'is an element of' symbol instead.

Sigh, even here, on an article whose very purpose is to point out that = does not mean equality as used by mathematicians, there are people calling this an abuse of notation. It's not; it's the standard notation in asymptotics. Using set-theoretic notation is cumbersome — you have to use “∈” in things like 3n^2 + 5 = O(n^2), but “⊆” in things like (n+O(√n))^2 = n^2 + O(n√n) — and defeats much of the point of using O() notation in the first place. I've mostly seen programmers / CS people insist on calling it abuse of notation and try to pedantically and counterproductively use set-theoretic notation, when mathematicians have been freely using the = sign for more than a century, from before the birth of computers. (Not repeating my comment from earlier: https://news.ycombinator.com/item?id=16834297)

Re: For mathematicians, = does not mean equality

#168

Post by diffeomorphism » Sat, Apr 14, 2018, 3:56 PM UTC

xg15 wrote: post #24

j2kun wrote: post #18

Earlier quoted context omitted.

Fair point. I'd add that f(x) = 7 can be both equality of functions and equality of evaluations, and binding x=2 suddenly changes the meaning of the equality and the expression.

True. I don't want to dispute that there is a hell of ambiguity in using =.

I wouldn't call that ambiguity. While several shorthand notations use =, it is always clear from the context which one and only one is referring to (and if there are multiple that all of them agree).

This touches on another point that one sees much of in mathematics lectures but little in math lectures. Mathematical notation needs to be unambiguous but also facilitate communication and hence tends to be very terse. Thus when discussing addition on a finite field F_5, one usually starts defining equivalence classes [j] and an addition operator "+" and then says that [3]"+"[3]=[3+3]=[5+1]=[1] followed by a disclaimer like:

"We hence see that this notion is compatible with the previously defined one when identifying ... . Hence, if there is no possible confusion, we will simply write ..."

See also Tao's comments on rigor in mathematics: https://terrytao.wordpress.com/career-advice/theres-more-to-...

All notation can be made rigorous, if one wants to, but discussion is more concise if we let everybody do that by themselves.

Re: For mathematicians, = does not mean equality

#169

Post by dhimes » Sat, Apr 14, 2018, 4:01 PM UTC

Nullabillity wrote: post #158

dhimes wrote: post #155

Earlier quoted context omitted.

or ≡

I've only seen that used for congruence (x ≡ y f(x) = f(y), for some context-dependent f).

It used to mean "define" in ancient times.

Re: For mathematicians, = does not mean equality

#170

Post by sykh » Sat, Apr 14, 2018, 4:01 PM UTC

gizmo686 wrote: post #119

sykh wrote: post #116

Earlier quoted context omitted.

One of the points of my original comment was that when it comes to equations the = does not mean equal in the sense of stating two objects are the same. In the context of an algebraic equation to solve the = sign is really a question. It’s asking, what is the set of values that make the statement true? I know of no mathematician who thinks x^2-x+1=0 is anything other than a polynomial equation. Specifically it’s shor…

>It’s asking, what is the set of values that make the statement true? For this to be the case, there would need to be a statement in the first place. And that statement would involve the =, so you necessarily still have the = symbol representing something other than questioness. I would further say that the equation itself is still just a statement, and any "question" interpretation is based entirely on the context w…

In your example of solving x^3+4x=0 you write

..x=0 is consistent..

You don't see the problem with this? Saying x=0 and that x is an element base ring means that x is the element 0. You can't later in your problem write "...x=2i..." if you are going to persist in your view that x is an element of the base ring. What you have shown is that the variety of the ideal generated by x^3+4x is the same as the variety of the ideal generated by x, x-2i, x+2i if we are talking about C as the base ring. If the base ring is Q then a different thing is shown.

You can't logically say, in a consistent manner, that x is in R and x^3+4x = 0 and that x is 3 different values. An element of a ring is not three different elements. An element of a ring is itself. If you want to vary the object x then you need to enlarge your ring to an algebraic structure that admits x so that it behaves the way you wish to view it. Your view of what is really happening when solving an equation is not rigorously sound. The proper way to view this is in the context of algebraic geometry.