The word "is", typically refers to the logical equals operator (written as "="). Let A = Math, and let B = Language, the phrase "Math is Language" is therefore "A = B". My claim is that "B != A", which implies "A != B" because "!=" is reflexive. "A != B" then contradicts the claim "A = B". In words: because "language is not math", therefore it cannot be true that "math is language".

This is not number theory, not set theory, just logic. I don't see how the properties of Abelian groups applies. I do suspect that non-abelian groups is a case where "¬(A ↦ B) ⟹ ¬(B ↦ A)" is false, and would be a contadiction to my counter claim if "¬(A ↦ B) ⟹ ¬(B ↦ A)" was my counter-claim. No, my counter claim is simply that because "B != A" therefore "A != B"

The statement "¬(A ↦ B) ⟹ ¬(B ↦ A)" is quite different, and I'm not 100% sure whether you are mixing set and logic notations? [0][1] It does seem you are bringing in number theory concepts, or we are misunderstanding one another?

I assume "↦" is the set mapping operator and "⟹" is logical implies, and "¬" is logical not. "¬(A ↦ B) ⟹ ¬(B ↦ A)" could potentially be phrased as: "If the element A cannot be mapped to B, then the element B cannot be mapped to A". I would agree that statement is not 'generally' true. Could you please clarify so we are not talking past each other.

[0] Per google: The symbol "↦" is a mapping symbol, typically used to indicate a function or relationship where one element maps to another. It's not a standard logical operator in the same way that symbols like &&, ||, or ¬ are. Instead, it represents a directional relationship between sets or elements, often seen in set theory and mathematical notations

[1] https://simple.wikipedia.org/wiki/Implication_(logic)

As I said elsewhere (here for others), I'm not going to play this game of *willful misinterpretation.* Your very words do not hold up to the same bar your are attempting to hold mine to. I know you are upset you got caught in a lie, but fuck around and find out. ¯\_(ツ)_/¯ Domain experts can recognize other domain experts pretty easily

  > No, my counter claim is simply that because "B != A" therefore "A != B"

We all know that this is not always true. It may be true in certain cases, in certain fields (pun intended), but you know that this is such a basic logical fallacy that it is taught to children.

Here's a counterexample so we can lay this to rest.

  Let "A" be "a square"
  Let "B" be "a rectangle"

  B != A -> "A rectangle is not a square" (True)
  A != B -> "A square is not a rectangle" (False)

Stop cosplaying, stop trolling, stop acting in bad faith.

Did you even look at my name? There's 2 people that should come to mind. Certainly any logician would notice.

The equality relationship is symmetric, "For every a and b, if a = b, then b = a" [1]. This holds for the negation as well.

Or, "Properties of Equality... The [equality] relation must also be symmetric. If two terms refer to the same thing, it does not matter which one we write first in an equation. ∀x.∀y.(x=y ⇒ y=x)" [2]

The 'is' relationship in logic is understood to be equality. The 'is a' relationship in logic is subset. Colloquially, the word "is" can be either one though.

I notice in your counter example you swapped the "is" relationship with "is a". Keeping the "is" relationship: "A rectangle is not square" (true generally, but false for specific cases for rectangles). That is really the distinction, the sometimes true vs always true.

Let's stick to the precise definition of "is" to mean a logical equality from here on please, and be precise when we use "is" vs "is a" and never infer "is" to actually mean "is a".

So, with "Math is a language" vs "Math is language". Which do you mean?

[1] https://en.wikipedia.org/wiki/Equality_(mathematics)

[2] http://logic.stanford.edu/intrologic/extras/equality.html

The proposition was "A = B" (logical equals, A and B are entities)

The simple refutation was: because B != A, therefore A != B

That WAS the math exam!! No group or set theory needed. It's simple and this is not interesting.

> That's only in your head. Inventing claims so that you can pretend other people are wrong isn't a good move.

I'm not the only one that interprets a statement as "Math is Language" to be of the form "A = B" where "math" is A, "language" is B, and "is" is the equals operator, see: https://news.ycombinator.com/item?id=43874322

Seems kinda simple.. You're engaging in a almost pure personal attacks. Care to address the substance of how the 3 word long statement is not of the form "A = B", but is of some other different form? In which case, perhaps you can guide the conversation for why it is an interesting statement or not? If you want to change all the givens and use your own definitions, please provide those. Without common ground, this remains uninteresting.

> I'm not the only one that interprets a statement as "Math is Language" to be of the form "A = B" where "math" is A, "language" is B, and "is" is the equals operator, see: https://news.ycombinator.com/item?id=43874322

This appears to be a link that contains zero support for your sentence. Neither the comment you linked nor the response below it features such an interpretation.

> Seems kinda simple.. You're engaging in a almost pure personal attacks. Care to address the substance of how the 3 word long statement is not of the form "A = B", but is of some other different form?

You're in luck! I've already provided that material, and you responded to it. For further discussion, you'd need to have a better working understanding of English.

Very cool how the last sentence you leave off with always has to be a personal attack. It's breaking the rules of the discussion, irrelevant. Stop trying to win points. Let's focus on the substance.

> This appears to be a link that contains zero support for your sentence. Neither the comment you linked nor the response below it features such an interpretation.

It's interesting, because there is a disagreement about whether there is a paradox or not. There is a paradox if you take my perspective (that an equals relationship is being expressed), but none if an implication relationship is assumed.

These two sentences:

- "By that same logic you could also say that language is math"

- "Not quite, but the inverse is true."

The "not quite" says that "language is math" is not true. So we have "A = B", but "B != A", which is a paradox. OTOH it's not a paradox if what is actually being said is "A => B" but "B !=> A"

> You're in luck! I've already provided that material, and you responded to it.

Hello bad faith! I hope you are doing well today. Let's end the conversation here.