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For mathematicians, = does not mean equality

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Re: For mathematicians, = does not mean equality

#151

Post by filmor »

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…

If the expression is supposed to be the definition and not just a derived equality, at least Mathematicians (and some Physicists) would use := instead, just as Niklaus Wirth tried to establish with Pascal.

Re: For mathematicians, = does not mean equality

#152

Post by bhnmmhmd »

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…

I'm always asking "why haven't scientists come up with more symbols yet?"

I also ask "why haven't more symbols been introduced to our keyboards? Just !@#$%&*..."

It's strange, esp. when you realize coming up with a new symbol that everyone uses is easier these days than it was 4 centuries ago!

Re: For mathematicians, = does not mean equality

#153

Post by narag »

pron wrote: post #150
narag wrote: post #145

Earlier quoted context omitted.

You're trying too hard. The problem with "=" is usability. If you read the previous article about "why does = mean assignment" you'll find the true reason: Since assignment is about twice as frequent as equality testing in typical programs, it’s appropriate that the operator be half as long. Of course, that's wrong. Trying to make it easier to type, a bigger usability problem is created, assigning counterintuitive sy…

> 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.

Re: For mathematicians, = does not mean equality

#154

Post by swiley »

bhnmmhmd wrote: post #152
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…

I'm always asking "why haven't scientists come up with more symbols yet?" I also ask "why haven't more symbols been introduced to our keyboards? Just !@#$%&*..." It's strange, esp. when you realize coming up with a new symbol that everyone uses is easier these days than it was 4 centuries ago!

There are many strange symbols in math already that can be difficult to type without something like LaTeX. I feel like something pronounceable is usually better unless you have a sufficiently general idea.

Re: For mathematicians, = does not mean equality

#155

Post by dhimes »

filmor wrote: post #151
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…

If the expression is supposed to be the definition and not just a derived equality, at least Mathematicians (and some Physicists) would use := instead, just as Niklaus Wirth tried to establish with Pascal.

or ≡

Re: For mathematicians, = does not mean equality

#157

Post by grigjd3 »

baddox wrote: post #125
dnautics wrote: post #121

Earlier quoted context omitted.

PV=nRT

While I don’t think the terminology is explicitly standardized, I think most people in the relevant fields would call that statement of the ideal gas law an equation but not a formula , the latter being a special case of the former.

As a trained theoretical physicist, I am entirely unaware of the distinction you are trying to draw here.

Re: For mathematicians, = does not mean equality

#158

Post by Nullabillity »

dhimes wrote: post #155
filmor wrote: post #151

Earlier quoted context omitted.

If the expression is supposed to be the definition and not just a derived equality, at least Mathematicians (and some Physicists) would use := instead, just as Niklaus Wirth tried to establish with Pascal.

or ≡

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

Re: For mathematicians, = does not mean equality

#159

Post by sykh »

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…

Why? Suppose I don't know what R[x] or R(x) is. We certainly don't teach highschoolers what either of those are, and they seem to be able to do "algebra" just fine.

We don’t teach high schoolers what is really going on. We mask what is really going on because making it all precise is not effective or helpful at this stage of development. We teach rules to manipulate equations. We don’t use the language of algebraic geometry because that is too complicated. We don’t say to them the variety of the ideal generated by x+2 is the same as the ideal generated 4x+8 are the same so that x=-2 is equivalent to 3x+1=-x-7.

However, we are insisting that the LHS is a polynomial (again, in the formal sense). This means that there is no variable. The "x" in the LHS is literally a value. It makes no sense to ask what values of (0,1,0,0,...) make that equation true. The fact that we give (0,1,0,...) a standard name of x does not suddenly make the question sensical. Nor does the fact x is often used to represent variables.

You are clearly not an algebraist. The polynomial

(3, 1, 0, 0, 0, ....)

Induces a natural map from R to R that is generally called “evaluation”. It maps the number 5 to 8 for instance. We abuse notation and say to begininnng students, “replace x with 5”. We dumb things down. Instead of asking for the pre-image of 8 under this natural map we ask for what values of x do we get a value of 8. The shorthand way of writing this is to say solve the equation:

x+3=8

That you don’t know this is disconcerting since you’ve obscuoulsy had more than an elementary mathematical education. You are confusing the simplistic view of what is taught in basic courses with what is really going on.

Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes. Because in standard uasage of that expression it is a polynomial. That’s the default interpretation.

Read the first chapter of any beginning algebraic geometry textbook. x^2+3x-1=0 is an algebraic variety. This is the standard interpretation of that equation.

Re: For mathematicians, = does not mean equality

#160

Post by webbles »

segmondy wrote: post #97
howling wrote: post #91

All these discussions of "=" are missing the point. The notation "x = x + 1" is awful because at lhs x denotes a reference to an integer while at rhs x denotes the value hold by the reference. If you know C, it is similar to the difference between an integer pointer *x and an integer x. As an illustration, here are two programs that are doing the same thing, one in C and one in Haskell. #include int main() { int x =…

We don't have such problem in Prolog. In prolog X can never be equal to X+1. X will always be X. Once X has a value, it never changes. One of the thing that gives most procedural programmers a headache.

You have a to be a retard to get confused by variables.
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