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For mathematicians, = does not mean equality

jeremykun.com

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Re: For mathematicians, = does not mean equality

#181

Post by baddox »

grigjd3 wrote: post #157
baddox wrote: post #125

Earlier quoted context omitted.

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.

Can you find any reference to the idea gas law as a “formula?” As far as I can tell, equations without a single variable on the left side are referred to as simply equations, while solutions of such equations in terms of one variable are referred to as formulas. This seems to be the case for every well-known identity I can think of, like the quadratic equation/formula. Can you think of any counterexamples?

Re: For mathematicians, = does not mean equality

#182

Post by nitrogen »

nitrogen wrote: post #176
golergka wrote: post #142

Earlier quoted context omitted.

I remember starting with QBasic when I was around 7 or 8 years old, and I quickly got an idea – just put in the equations from math homework to find out the answer! Fighting through the error messages without English and trying to wrap my mind around the basic concepts of procedural programming was a world of pain.

Nice. I had fun in some junior high math competitions in part by writing TI BASIC programs on my calculator to brute force hard problems while I did the easy ones. Eventually they stopped allowing calculators altogether and focused on cheap memorized tricks instead of generalized problem solving.

I'm curious what prompted two people to disagree with this comment so quickly, with no comment.

Re: For mathematicians, = does not mean equality

#183

Post by gizmo686 »

sykh wrote: post #159
gizmo686 wrote: post #119

Earlier quoted context omitted.

>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 g…

>The polynomial (3, 1, 0, 0, 0, ....) Induces a natural map from R to R that is generally called “evaluation”.

I am well aware of this. If you look at my comments within this very chain, you will see that I made reference to polynomial evaluation, which I will call Ev(f, x). I Am not denying that the partial application given by f' = x -> f(f,x) is naturally induced by the polynomial f. Nor that this is so natural that it often makes sense to identify f with f' so that we would consider f=f', even though they are different types of objects.

>Ask a million mathematicians, “is x^2+3x a polynomial” and without hesitation they’ll say yes.

As will I, because the distinction between formal polynomials and expressions which can be naturally modeled as polynomials is so unimportant that it is almost never worth thinking about.

Put another way, how would you compute the following sets:

{ x \in C | x^3 - x = 8 }

{ x \in C | x^3 = x + 8 }

{ x \in C | log(x) = x^x }

{ x \in C | sin(x) = .7 }

{ x \in N | exists y \in N such that 5x + 3y = 1 }

Are you really claiming that these questions are ill-posed without stating them in terms of algebraic geometry?

Re: For mathematicians, = does not mean equality

#184

Post by CJefferson »

sykh wrote: post #102
comex wrote: post #94

Earlier quoted context omitted.

I feel like that example counters your claim in two different ways. First, (1, 2) only makes sense if you assume that x is first and y is second, or in other words that they correspond to X_1 and X_2 for some vector X. This is probably a reasonable assumption for x and y, but what if you have some other arbitrary choice of variables? a ρ + 2 = 0 Then the only unambiguous way to write a solution is with explicit label…

Polynomials in multiple variables always have the variables ordered. Sometimes the ordering is not important or explicitly stated but in reality they are supposed to be ordered. If we start with x y + 2 = 0 we can apply the function f(x) = x - 2 to both sides of the equation. This gives us the equation x y = -2 This is a different equation than the one we started with but these two equations have the same solution se…

Technically, people have polynomials with uncountably many variables, so you can't order them.

Re: For mathematicians, = does not mean equality

#185

Post by gaurav_v »

nitrogen wrote: post #182
nitrogen wrote: post #176

Earlier quoted context omitted.

Nice. I had fun in some junior high math competitions in part by writing TI BASIC programs on my calculator to brute force hard problems while I did the easy ones. Eventually they stopped allowing calculators altogether and focused on cheap memorized tricks instead of generalized problem solving.

I'm curious what prompted two people to disagree with this comment so quickly, with no comment.

I didn't downvote, but the reason is likely because your comment is off-topic. The parents are talking about experiences un-learning one-sided equality that they picked up programming before learning algebra; your comment is about programming a T-83 to help with schoolwork.

Re: For mathematicians, = does not mean equality

#186

Post by jcranberry »

svat wrote: post #167
jcranberry wrote: post #62

Earlier quoted context omitted.

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(…

Wow, so disagreeing with the point of this article immediately provokes a typed-out condescending sigh from you? That's worth a sigh more than anything. Furthermore, I was a mathematics person first, and got into computer science later, so I'm not a "CS person" insisting on calling it an abuse of notation.

I have two points:

Firstly, although the article does a good job making the point that there's no reason to get on programming languages' case for using formalized, ad hoc syntax, it does a terrible job of backing up it's actual title. The bottom line is the equals sign is very closely associated with the concept of RST equivalence in mathematics, to the point where a specificity such as the symbol for the isomorphism (which, ironically, he used as a counterexample) is abstracted away, because isomorphism is an RST relation. If two things are isomorphic, they are, in some sense, the same (or equal), so use of an equals sign is natural.

Secondly, if you had read the first source you posted, maybe you wouldn't have claimed programmers/CS people were the ones pedantically insisting on calling it an abuse of notation since, on page 6, Bruijin writes, "The trouble is, of course, due to abusing the equality sign =."[1] Furthermore, after defining the parameters around the use of the O-symbol, he also writes, on page 7, "It is obvious that the sign = is really the wrong sign for such relations, because it suggests symmetry, and there is no such symmetry."[1]

1. https://shreevatsa.wordpress.com/2014/03/13/big-o-notation-a...

Re: For mathematicians, = does not mean equality

#187

Post by jcranberry »

sykh wrote: post #81
jcranberry wrote: post #67

Earlier quoted context omitted.

Suppose x is an integer such that x + 3 = 1. Then, x is -2. There's no solutions here, just implications and an alternative way of defining the value of x. I think variables in equations are not meant to express the existence of variance within an equation, but a sense of context-dependency of the value of x. At least, IMO.

There is a solution. An equation is really a question. x+3 = 1 is asking the question, “what value for x makes x+3 the number 1?” The polynomial x+3 is defined for all values in R, the base ring you are working in. We are trying to find the elements of R for which x+3 is the element 1.

An equation is an instance of equating.

A solution necessitates a question, and questions associated with equations involving variables aren't restricted to: what is set of possible values which satisfy those equations?

Re: For mathematicians, = does not mean equality

#188

Post by sykh »

CJefferson wrote: post #184
sykh wrote: post #102

Earlier quoted context omitted.

Polynomials in multiple variables always have the variables ordered. Sometimes the ordering is not important or explicitly stated but in reality they are supposed to be ordered. If we start with x y + 2 = 0 we can apply the function f(x) = x - 2 to both sides of the equation. This gives us the equation x y = -2 This is a different equation than the one we started with but these two equations have the same solution se…

Technically, people have polynomials with uncountably many variables, so you can't order them.

Each polynomial has only finitely many variables in it. If you were to work with uncountably many variables I assume one would impose a well ordering on those variables. I’ve never worked with a polynomial ring with uncountably many variables. No one is going to solve an equation with uncountably many variables so I’m not sure the point you are trying to make.

Re: For mathematicians, = does not mean equality

#189

Post by sykh »

jcranberry wrote: post #187
sykh wrote: post #81

Earlier quoted context omitted.

There is a solution. An equation is really a question. x+3 = 1 is asking the question, “what value for x makes x+3 the number 1?” The polynomial x+3 is defined for all values in R, the base ring you are working in. We are trying to find the elements of R for which x+3 is the element 1.

An equation is an instance of equating. A solution necessitates a question, and questions associated with equations involving variables aren't restricted to: what is set of possible values which satisfy those equations?

You are incorrect. This is not how mathematicians view a polynomial equation like the one I used as an example. That equation does have a solution.

Re: For mathematicians, = does not mean equality

#190

Post by sykh »

gizmo686 wrote: post #183
sykh wrote: post #159

Earlier quoted context omitted.

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 g…

>The polynomial (3, 1, 0, 0, 0, ....) Induces a natural map from R to R that is generally called “evaluation”. I am well aware of this. If you look at my comments within this very chain, you will see that I made reference to polynomial evaluation, which I will call Ev(f, x). I Am not denying that the partial application given by f' = x -> f(f,x) is naturally induced by the polynomial f. Nor that this is so natural th…

I believe you are the only person trained in mathematics in the world that thinks x^2+3x=0 is not a polynomial equation.

You don’t understand the underlying algebraic theory. This is evidenced by your claim

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.

You can’t write such a statement if you really understand that “what values make x+3 the number 5” is a nicer way of conveying the question “under the natural evil map induced by x+3 what is the pre-image of 5”. The text of your I quoted is wrong.

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