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For mathematicians, = does not mean equality

jeremykun.com

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Re: For mathematicians, = does not mean equality

#171

Post by diffeomorphism »

sykh wrote: post #61

Think about the equation x + 3 = 1 Typically we write that the solution is "x = –2". This to me is the most abusive form of usage for "=" in mathematics. The solution to the equation is –2. The solution to the equation x = –2 is also –2. Solving the equation x = –2 is very easy. We can solve it just by looking at the equation. What we are really doing when solving an equation is transforming the original equation int…

> Typically we write that the solution is "x = –2"

That is not overloading "=" at all. It is overloading the word "solution".

What your teacher probably told you is:

"x+3=1 is equivalent to x=-2 by .... Hence, the solution of x+3=1 is the same the solution of x=-2 (which is -2 in both cases). Since the solution of x=-2 is so obvious, we also say the solution is x=-2 to mean that it is the same solution as of the equation x=-2."

Same difference as in "Q:Would you like a large coffee or a small coffee? A: Large " The correct answers should only be either phrase "A large coffee" or "a small coffee", but the answer was an adjective.

Re: For mathematicians, = does not mean equality

#172

Post by svat »

contravariant wrote: post #83
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.

Could be worse. I've seen things like f(x) = g(x) + O(h(x)) a few times.

That's not “worse”, that's the entire point of using O() notation! The beauty of O() notation is that it lets us carry out, fully rigorously, computations like

(n + O(√n))(n + O(log n))^2 = (n + O(√n)(n^2 + O(n)) = n^3 + O(n^(5/2))

without dealing with a mess of sets and quantifiers. Please take a look at some works where asymptotic expressions are dealt with proficiently; you'll understand. (de Bruijn's book https://news.ycombinator.com/item?id=16834297 for example, or at least Chapter 9 of Concrete Mathematics.) I recently worked out an example here: https://cs.stackexchange.com/a/88562/891 — replacing all O() equations with “∈” and “⊆”, though it can be done ( https://math.stackexchange.com/a/86096/205 ), is just cumbersome and only distracts from what's going on.

Re: For mathematicians, = does not mean equality

#173

Post by abrowne »

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 two black squares on Firefox beta on my Android 5.1 phone.

Re: For mathematicians, = does not mean equality

#174

Post by barrkel »

abrowne wrote: post #173
Recursing wrote: post #146

Earlier quoted context omitted.

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

And two black squares on Firefox beta on my Android 5.1 phone.

Works fine in Firefox Beta (and normal Firefox) on my Android 8 phone.

Re: For mathematicians, = does not mean equality

#175

Post by analog31 »

grosjona wrote: post #161

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 co…

My understanding was that the "d" is an operator, and there are notations in which an operator on an operand is notated by just putting the one before the other. Also, the dx corresponds to delta x in the limit definition of the integral. One reason for keeping it, is that it makes the units of measure work out if the integral involves things that have units. So, notational consistency aside, it saved my arse when doing physics problems. ;-)

(even in my pure math classes, I sometimes imagined that the variables had units, to help find mistakes).

But your point is well taken about the consistency of math notation. Math spent most of its history being scribbled by hand and read by humans. It got the job done. And it was not uncommon to invent a new notation on the fly to replace an abstraction with a single symbol. That's the precursor to the subroutine.

The need for perfect formality of notation is a new thing, brought on by the computer age. This may illustrate the point that programming is not "just math," and math is not a form of programming.

Re: For mathematicians, = does not mean equality

#176

Post by nitrogen »

golergka wrote: post #142
hyperpallium wrote: post #78

Earlier quoted context omitted.

Because I started programming before taking maths at school, I didn't properly appreciate equality for a while. Sure, algebra was fine, a(x+y)=ax+ay can go either way; but not ratios and other relationships. What helped me was was geometry, where you can see it's just a relationship. All the components move together; one part isn't priviledged as the result . e.g. you enlarge a circle. It doesn't make sense to ask wh…

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.

Re: For mathematicians, = does not mean equality

#177

Post by [deleted] »

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.

[deleted]

Re: For mathematicians, = does not mean equality

#178

Post by jtc1983 »

lou1306 wrote: post #56

Some more food for thought on the meaning of =, from Girard's "Proofs and Types" [0]: > There is a standard procedure for multiplication, which yields for the inputs 27 and 37 the result 999. What can we say about that? A first attempt is to say that we have an equality "27 x 37 = 999". This equality makes sense in the mainstream of mathematics by saying that the two sides denote the same integer [...] but it misses…

These quotes from Girard are great, as is the mention of Frege below.

Typically, the objects related by equality can be thought to have the same meaning with respect to extension and different meanings with respect to intension. Further, the difference in intension reveals something of the computational content of the extensional object being referred to.

Further topics to explore: the BHK interpretation of intuitionistic proof and the univalence axiom in homotopy type theory. Both of these topics give one some insight on the relationship between the computational content of mathematical objects and how this content pertains to the question of whether two objects are “the same.”

Finally, I did skim the article itself and found it lacking. The author seems to be aware of the fact that there are surprising, highly non-trivial properties of the (seemingly trivial) notion of equality in mathematics. And also to be aware of the fact that the use of ‘=‘ in CS contexts isn’t some sort of abuse of notation. But there seems to be very little of interest here beyond some circumstantial verification of these two general (and well-known) facts about equality in the mathematical and computational contexts.

Re: For mathematicians, = does not mean equality

#179

Post by posterboy »

raphlinus wrote: post #33
lvh wrote: post #27

Earlier quoted context omitted.

The usual notation for modular arithmetic uses three dashes, not two, to denote congruence.

I probably should have said "common" notation; it's certainly what I learned. I think you're right in that many authors prefer ≡ to emphasize that it is an equivalence rather than equality relation. Source that both are in use: https://math.stackexchange.com/questions/196081/the-right-wa...

I don't understand the schism. For example, 11=1 gives the equivalence of two different programs. Whereas 33 mod 4 and 1 are equal in Z_4 ...

It bothered me a lot when I took logics lectures that equality wasn't treated as operator or even just anything within a theory (whereas turnstyle was described, at least).

Re: For mathematicians, = does not mean equality

#180

Post by gizmo686 »

sykh wrote: post #170
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…

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…

> Saying x=0 and that x is an element base ring means that x is the element 0.

I never said that x=0. I said that x could be 0. More specifically, I stated that the statement x^3+4x=0 AND x=0 is not inconsistent.

All I have shown in is that (assuming we are working in C), the statement x^3 + x=0 implies that x \in {0, 2i, -2i}.

I suppose you could complain that I have not defined a sense in which {0, 2i, -2i} is correct while {0, 2i, -2i, 7} is incorrect, as it is still a true statement that x^3+4x=0 implies x \in {0, 2i, -2i, 7}.

However, you can easily make this intuition rigourous by saying that the question is to compute the set {x | x^3+4x=0}. Sure, this is invoking machinery not explicitly present in the statement x^3+4x=0. I will even concede that we do not make this machinery explicit when teaching highschool students. However, it is far less machinery than your approach.

I am not claiming that the algebraic approach is not rigourously sound; merely that it is not the only rigourously sound approach.

As far as I can tell, you are claiming that it is the only rigourously sound way of stateing the question.

>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.

I believe I have made this point clear, but I never claimed x is 3 different values. The claim I made was that x is a member of the set {0, 2i, -2i}

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