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Comment by mortdeus | Hacker News Reader

because in math it means assignment. y = mx+b

That is an assignment. There is also a math symbol that specifically designates equivalence.

The concept of assignment only makes sense in a context with a time axis. Computation has a time axis (location of the instruction pointer) but math does not.

Nope. Logical axioms are required logically, but not associated with time. Comprehension is associated with time, so I can understand how you might arrive where you did.

Sorry but I don't understand how you're using the word 'comprehension' here. I agree that logical axioms aren't associated with time (this is what I mean when I say that math doesn't require an idea of time).

It should be noted that in mathematics equivalence and equality are not the same thing.

Letting ~ represent an equivalence relation on a set X:

Equivalence relations have specific properties:

  x ~ x holds for all x in X
  x ~ y => y ~ x for all x, y in X
  x ~ y /\ y ~ z => x ~ z, for all x, y, z in X

However, they do not have to be equal, merely equivalent by whatever our relationship requires (though they may be equal).

A common example that I would expect most CS students to have encountered is the idea of "congruent modulo n".

We would say that `x ~ y (mod n)` iff `(x mod n) = (y mod n)`.

So over the whole of the integers we can see that every odd number is equivalent/congruent to each other over 2. But we would not say that they are equal to each other.

On the other hand, we would say that 1/4 is equal to 2/8 (or `1/4 = 2/8`).

Equality doesn't exist in mathematics.

http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf

The punchline:

> In appropriate deference to the manifold ways an object can be presented to us, objects need only be given up to unique isomorphism, this being an enlightened view of what it means for one thing to be equal to some other thing.

I sympathize with this view; I did some simple programming in C-like languages before I learned algebra, and it's sometimes hard to not see = this way.

But if = in math notation denoted "assignment", what could ≠ mean?

Maybe lhs = rhs is the set of assignments implied by a postcondition. So x = 5 implies a single assignment, but x = ±5 could mean two possible assignments. Similarly, x ≠ 5 would mean all the possible assignments but x = 5.

Equality is symmetric. If lhs = rhs then rhs = lhs. x ≠ 5 means exactly that, x is anything that is not 5. So if you are talking about the natural numbers, x could be any element of the set {0, 1, 2, 3, 4, 6, 7, ...}. If x = ±5 then x could be any element of the set {-5, 5}.

No it is not. Have you ever studied logic? Equality is a relation. Assignment does not make much sense if you don't elaborate what you're trying to say. For example, we can assign 1 := S(0), 2 := S(S(0))... in the metatheory so that we can meaningfully use 1, 2 ... in the theory. It is also sometimes denoted with =_{def} and some books use = for this type of assignment and \dot{=} (= with dot on top) for the equality relation inside the theory. Either way, you're not right.

‘=‘ does not mean assignment in math.

Your example

  y = mx + b

could equivalently be stated as

  y - b = mx

What does it mean to assign something to ‘y - b’?

Another example,

  2 = x

means exactly the same thing as

  x = 2

What does it mean in your semantics to assign x to 2?

No. It does not.

it’s not an assignment it’s expression of a relationship, and you can rewrite it in many equivalent ways. You can’t coherently rewrite an assignment that way.

it's actually a relation (ie equality) between two expressions. it is a logical assertion, not an action.

You assign with flowery words. For example "Let X be the blah blah blah blah blah ..."

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