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What are the differences between equality and equations and identities?

Look at the following:

  1. $x=y$, $x$ and $y$ are mathematical objects.
  2. $x+2=4$
  3. $(a+b)^2=a^2+2ab+b^2$

What are all of the differences between the three?

Edit #1: From my understanding the $1$ claims that $x$ and $y$ are the exact same object, meaning that $x$ and $y$ are simply two names for the same things.

The second one is a conditional that "asks" what $x$ make the predicate $P(x):x+2=4$ true.

The third one claims that the predicate $P(x): (a+b)^2 = a^2+2ab+b^2$ is a true statement for any $x$. I don't really know why it is different from the first one. Basically $(a+b)^2$ is the same thing as $a^2+2ab+b^2$, they are 2 different names for the same thing.

Mauro ALLEGRANZA
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mawaior
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  • Well, what are your thoughts on these? – MathematicianByMistake Jan 05 '24 at 15:01
  • Look at the edit. @MathematicianByMistake – mawaior Jan 05 '24 at 15:08
  • Your edit would serve as a valid answer I believe. The only point that merits a little more attention is that of identity. Algebraically yes, they are the same. But say for example, when stating Pythagoras' Theorem, we are equating two equal quantities but ones describing quite different things. – MathematicianByMistake Jan 05 '24 at 15:14
  • They are formulas. The difference is the "question" we ask about them: are they true for evey/some values of the variable? $x=x$ is alwYs true while $x=2$ is not. – Mauro ALLEGRANZA Jan 05 '24 at 15:57
  • I think all three mean the exact same thing: the thing we label by the string on the right is the same thing as the thing we label by the string on the left. The two differences are 1) the unstated implication as to why we are saying it. And 2) how were thing strings introduced in context in the first place and whether they are meant to be specific objects in a context or general objects or general objects in a specific context. – fleablood Jan 05 '24 at 17:59
  • all are instances of the equality relation applied to the theory of mathematics. all are equations. all happen to be identities as well. – RyRy the Fly Guy Jan 05 '24 at 20:22

2 Answers2

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Note: the equality $(a+b)^2=a^2+2ab+b^2$ is not really an equality of mathematical objects, since it is a priori not true. In any context where $ab \neq ba$, this does not hold.

soggycornflakes
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You're missing quantifiers on all of these, so all your descriptions are guesses based on assumed context. It's like going into a kitchen and asking "what is an egg?", when we could be making an omelette, or a cake, or glazing some buns.

However, independent of you examples:

  • Equations and equalities assert that two things are equal.

  • An identity asserts that two things are equal for all values that variables may take on in a specified (or implicitly specified) range.

The "exactly same object" description you've given for #1 isn't part of math as I know it. In the C++ programming language

int &a = b;

more or less does that. I can also imagine philosophy.SE saying something like that when discussing exactly what "is" means. But it doesn't seem like math, at least to me.

You also make several references to "naming" things. This also feels more like philosophy than math.

JonathanZ
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