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

Re: For mathematicians, = does not mean equality

#191

Post by lou1306 » Sat, Apr 14, 2018, 10:12 PM UTC

jtc1983 wrote: post #178

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 intuition…

> Further topics to explore: the BHK interpretation of intuitionistic proof and the univalence axiom in homotopy type theory.

Thanks! For the interested, Girard's book focuses on the Curry-Howard isomorphism, another great result linking computer programs (actually, the typed lambda-calculus) to mathematical proofs.

Re: For mathematicians, = does not mean equality

#192

Post by sykh » Sat, Apr 14, 2018, 10:16 PM UTC

gizmo686 wrote: post #180

sykh wrote: post #170

Earlier quoted context omitted.

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 corr…

You wrote:

This means that we have proven that x=0.

Later you have x=2i or x=-2i. Persisting with the idea that you can view x as an element in the base ring and at the same time allowing its value to change or vary indicates you don’t really understand these issues. If x is in R it is a single value. If you want to vary it you need to expand R to include x and add the approroate algebraic structure.

It’s shocking that you think x^3+4x is not a polynomial. The whole discussion I started originally was that math language, like all other human languages, is nuanced and there are lots of abuse of notations. This is ok because math is written for humans by humans. The standard interpretation of x^3+4x is that it’s a polynomial. This is not disputable.

Re: For mathematicians, = does not mean equality

#193

Post by sykh » Sat, Apr 14, 2018, 10:26 PM UTC

diffeomorphism wrote: post #171

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 equa…

I don’t know the precise definition of overloaded. I was merely claiming that the use of = in an equation, like a polynomial equation, is very much different than the use of = in something like the statement of the distributive property of the real numbers.

I’m a mathematician so it’s something my teacher said as such but something that I noticed consciously when I started doing some programming.

Re: For mathematicians, = does not mean equality

#194

Post by gizmo686 » Sat, Apr 14, 2018, 10:51 PM UTC

sykh wrote: post #190

gizmo686 wrote: post #183

Earlier quoted context omitted.

>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 numb…

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

Then you are not reading my comments.

> To be clear, outside of very particular contexts I would still call x^2-x+1=0 a polynomial equation, because it is extremely useful to talk about polynomials without invoking all of the machinery of formal polynomials.

>You don’t understand the underlying algebraic theory.

My claim is that the underlying algebraic theory is not necessary to rigorously state or solve the problems under discussion.

>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”.

My claim is that "what values make x+3 the number 5" translates directly into { x \in F | x+3=5}. I further claim that the "x+3" in this interpenetration is not a polynomial in the formal sense

I am not disputing that this is set is the same set as the pre-image of 5 under the natural map induced by x+3.

Re: For mathematicians, = does not mean equality

#195

Post by gizmo686 » Sat, Apr 14, 2018, 10:56 PM UTC

sykh wrote: post #192

gizmo686 wrote: post #180

Earlier quoted context omitted.

> 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 corr…

You wrote: This means that we have proven that x=0. Later you have x=2i or x=-2i. Persisting with the idea that you can view x as an element in the base ring and at the same time allowing its value to change or vary indicates you don’t really understand these issues. If x is in R it is a single value. If you want to vary it you need to expand R to include x and add the approroate algebraic structure. It’s shocking th…

>Later you have x=2i or x=-2i.

To be clear, by the time I got to this statement, I had changed the question (from a base of Q to C)

Further, the complete statement I was making at that point was:

If ((x^3+4x=0) AND x != 0) then ((x=2i) OR (x=-2i))

I am not allowing x to vary at all here. Suppose, for the sake of arguement, we had x=2i. It would still be true that ((x=2i) OR (x=-2i)).

> Persisting with the idea that you can view x as an element in the base ring and at the same time allowing its value to change or vary indicates you don’t really understand these issues.

Persisting with the idea that I am doing this indicates that your are not reading what I am writing.

Re: For mathematicians, = does not mean equality

#196

Post by fjsolwmv » Sat, Apr 14, 2018, 11:05 PM UTC

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!

You'd love APL

Re: For mathematicians, = does not mean equality

#197

Post by fjsolwmv » Sat, Apr 14, 2018, 11:09 PM UTC

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.

https://www.google.com/search?q=formula

Definition 1 ("equation") vs definition 2b ("recipe")

As a theoretical physicist you might find definition 2b uninteresting.

Re: For mathematicians, = does not mean equality

#198

Post by fjsolwmv » Sat, Apr 14, 2018, 11:14 PM UTC

giomasce wrote: post #123

The author itself admits in the postscript that he has embellished a bit the article, but allow me to take it at its face value: to me, it seems that the article confuses mathematics with its notation (and the same for computer science, but at this level CS is just a branch of mathematics). All the funny stuff he goes on describing follow from this confusion. When a mathematician does mathematics, they have very well…

The article and the article it is written in response to are explicitly about notation, not semantics.

Re: For mathematicians, = does not mean equality

#199

Post by fjsolwmv » Sat, Apr 14, 2018, 11:18 PM UTC

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…

> I've never heard a software developer say this about coding.

This is sarcasm, right? We say this all the time about legacy spaghetti code or excessively clever magic our languages or (closest to von Neumann's statement, perhaps) the mountain of APIs and libraries and OS we use..

Re: For mathematicians, = does not mean equality

#200

Post by fjsolwmv » Sat, Apr 14, 2018, 11:27 PM UTC

hyperpallium wrote: post #106

Geometry distinguishes between equivalence and value. An "angle" isn't its degrees, but the geometric figure (two rays or segments meeting at an end-point of each). It's the measure of the angle that is the degrees. You don't say "angles are equal" - you say they are congruent . It's their measures that are "equal". Although congruency implies measure equality, it doesn't really mean that, but that the shapes are the…

Congruent means that you have some relevant information that you are discarding (projecting away) information, such as position and rotation. It's just as correct to say equal if you've already established the relevant context / quotient space.

Two triangles on a page are congruent, because they have different position. The three corners of an eauikao trisne are congruent, because their corner+angles are equal. If you take angle to mean corner, you say congruent. If you aren't also talking so about their position, you say equal.

In math you can slice things every which way, so it's impossible to use the different words for every different concept, so you have to establish a context

Equal vs congruent (and equal vs equivalent, which are also synonymous in math) is a crutch for beginners who are over reliant on their informal intuition.