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In plain language, what's the difference between two things that are 'equivalent', 'equal', 'identical', and isomorphic?

If the answer depends on the area of mathematics, then please take the question in the context of logical systems and statements.

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    $\begingroup$ You might like to add "isomorphic" to your list. $\endgroup$
    – Travis Willse
    Dec 9, 2014 at 3:05
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    $\begingroup$ @Travis Done. Thanks. $\endgroup$
    – Hal
    Dec 9, 2014 at 3:09

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Convention may vary, but the following is, I guess, how most mathematicians would use these notions. Identical and equal are very often used synonymously. However, sometimes identical is meant to say that the two things are not just equal, but actually are syntactically equal. For instance, take $x=2$. The claim that $x^2=4$ is saying that $x^2$ and $4$ are equal. The claim that $x^2=x^2$ is saying that $x^2$ is equal to $x^2$, but we also say that the left hand side and the right hand side are identical.

Equivalence is a strictly weaker notion than equality. It can be formalized in many different ways. For instance, as an equivalence relation. The identity relation is always an equivalence relation, but not the other way around. A typical way to obtain an equivalence is to suppress some properties of the objects you study, and only look at particular aspects of them. A classical example is modular arithmetic. We say that $10$ and $20$ are equivalent modulo $5$, basically saying that while $10$ and $20$ are not equal, if the only thing we care about is their divisibility by $5$, then they are the same.

Isomorphism is a specific term from category theory. Two objects are isomorphic if there exists an invertible morphism between them. Informally, two isomorphic objects are identical for the purposes of answering any question about them in their category.

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    $\begingroup$ I'm glad you mentioned that $x$ and $2$ in $x=2$ are equal, but that there might be a distinction between $x=2$ and $x^2=x^2$. There seems to be some reason to secern identity and equality At the very least we could say that no matter what system you're working with, statements such as the latter are true, but statements like the former could be false. (E.g. $2+4=7$ is false in a field where the greatest number is $5$.)Also, we might say that two triangles that are the same except for their location on a Cartesian plan are equal, but they aren't identical - I can tell one apart from the other. $\endgroup$
    – Hal
    Dec 9, 2014 at 3:28
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    $\begingroup$ In the example of the triangles, there's definitely something we aren't considering that distinguishes them (their location). So is it better to say they're equivalent? $\endgroup$
    – Hal
    Dec 9, 2014 at 3:29
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    $\begingroup$ yes, certainly. The common terminology for such triangles is that they are congruent. $\endgroup$
    – Ittay Weiss
    Dec 9, 2014 at 3:33
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    $\begingroup$ btw, there is no field in which the greatest element is $5$. You are probably thinking about $\mathbb Z_5$. In that field $2+5=2+0=2=7$ (since you need to compute modulo $5$). In any case, no finite field is an ordered field, so it is incorrect to say it's greatest element is $4$ (which is what you meant probably since $5=0$ in $\mathbb Z_5$. $\endgroup$
    – Ittay Weiss
    Dec 9, 2014 at 3:40
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    $\begingroup$ How would you characterise the following usage (assuming you agree it's acceptable)? "Consider real functions $f(x) = x^2$ and $g(x) = x^3$. Then $f$ and $g$ are not identical. Find where they are equal". It's clear that we're using one to mean $f = g$ (which is false) and another to mean $f(x) = g(x)$ (which is true for some $x$), but I don't think it's really covered by "not just equal, but actually are syntactically equal". Rather for some reason "identical" here means "actually equal", whereas "equal" means something obvious but slightly more complicated -- equal "at a value". $\endgroup$
    – Steve Jessop
    Dec 9, 2014 at 19:10
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They have different types.

"Equal" and "identical" take as input two elements of a set and return a truth value. They both mean the same thing, which is what you think they'd mean. For example, we can consider the set $\{ 1, 2, 3, \dots \}$ of natural numbers, and then $1 = 1$ is true, $1 = 2$ is false, and so forth.

"Equivalent" takes as input two elements of a set together with an equivalence relation and returns a truth value corresponding to the equivalence relation. For example, we can consider the set $\{ 1, 2, 3, \dots \}$ of natural numbers together with the equivalence relation "has the same remainder upon division by $2$," and then $1 \equiv 3$ is true, $1 \equiv 4$ is false, and so forth. The crucial point here is that an equivalence relation is extra structure on a set. It doesn't make sense to ask whether $1$ is equivalent to $3$ without specifying what equivalence relation you're talking about.

"Isomorphic" takes as input two objects in a category and returns a truth value corresponding to whether an isomorphism between the two objects exists. For example, we can consider the category of sets and functions, and then the set $\{ 1, 2 \}$ and the set $\{ 3, 4 \}$ are isomorphic because the map $1 \to 3, 2 \to 4$ is an isomorphism between them. The crucial point here is, again, a category structure is extra structure on a set (of objects). It doesn't make sense to ask whether two objects are isomorphic without specifying what category structure you're talking about.

Here is a terrible place where this distinction matters. In ZF set theory, in addition to being able to ask whether two sets are isomorphic (which means that they are in bijection with each other), it is also a meaningful question to ask whether two sets are equal. The former involves the structure of the category of sets while the latter involves the "set" of sets (not really a set, but that isn't the problem here). For example, $\{ 1, 2 \}$ and $\{ 3, 4 \}$ are particular sets in ZFC which are not the same set (because they don't contain the same elements; that's what it means for two sets in ZFC to be equal) even though they are in bijection with each other. This distinction can trip up the unwary if they aren't careful.

(My personal conviction is that you should never be allowed to ask the question of whether two bare sets are equal in the first place. It is basically never the question you actually want to ask.)

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    $\begingroup$ With ZFC, bijectivity is the same as having same cardinality, but often we care about equivalence as ordinality. If we care about "sets as trees", then bare equality is on topic. (+1: "types") $\endgroup$
    – Charles Stewart
    Dec 9, 2014 at 8:45
  • $\begingroup$ Forgive me my lack of mathematical knowledge, I am a beginner. How would you then describe the equivalence of chunks of mathematical theory or certain mathematical propositions. What does it mean to say that Zorn's lemma is equivalent to the axiom of choice, apart from the fact that they imply the same things and each other? Is this a different form of equivalence? Why are equivalent but not equal? $\endgroup$
    – Max
    Dec 3, 2020 at 23:25
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    $\begingroup$ @Maximilian: when we say that Zorn's lemma is equivalent to the axiom of choice the simplest way to say it is that they imply each other (in $ZF$). Two statements implying each other is an equivalence relation on statements so it falls under the equivalence relation case of what I wrote above. They aren't equal because they aren't literally the same statement: two statements are equal if they consist of exactly the same symbols in exactly the same order. $\endgroup$
    – Qiaochu Yuan
    Dec 3, 2020 at 23:46
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    $\begingroup$ @Maximilian: things get more complicated once you realize that "Zorn's lemma" and "the axiom of choice" don't even refer to unique statements in the first place; there are several different strings of symbols you could write down that would deserve to be called Zorn's lemma or the axiom of choice respectively. This gets into muddy waters; there's a third, largely implicit, notion of what it means for two statements to be "basically the same" which is different from either literal symbolic equality or logical equivalence and it's tricky to say much about it precisely. $\endgroup$
    – Qiaochu Yuan
    Dec 3, 2020 at 23:48
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In general it's a tricky question. For example is the natural number 2 equal to the real number 2? As sets, they are not equal. But as numbers, everyone would consider them equal. But then what's a number? As soon as you try to answer this question, you're down the rabbit hole and into category theory and philosophy.

Barry Mazur wrote a famous essay on this subject, "When is one thing equal to some other thing?" You'll find it of interest.

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

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  • $\begingroup$ any given system of real numbers contains a subsystem of the natural numbers. When you write $2$ you assume a system of real numbers to be given. Then $2$ is also a natural number, no question about it. The fact that you can construct number systems in various ways is not quite a rabbit hole. $\endgroup$
    – Ittay Weiss
    Dec 9, 2014 at 3:13
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    $\begingroup$ @IttayWeiss The natural number 2 and the real number 2 are not equal as sets. This is a philosophical problem if one regards sets as foundational. Rather than try to work through the issues I figured I'd let Mazur do a far better job than I could. Of course there's a natural injection from the naturals to the reals, but the ontological issue remains; and I thought that's what the OP was getting at. I don't have any disagreement with your point. The distinction between equality as sets and implied equality via injection is, if not a rabbit hole, at least a small warren. $\endgroup$
    – user4894
    Dec 9, 2014 at 3:17
  • $\begingroup$ this problem only pops up if you insist to work with a particular model of numbers. If instead you define the numbers systems axiomatically then there is no problem. Any construction of a number system simply proves that the axiomatization is (relatively of course) consistent, but once that is done you can throw the model to the garbage bin. Every time you talk about numbers, you simply say "let $M$ be a model of the (for instance) reals". Then the element in $M$ are, by definition, the real numbers, and they are not (necessarily) sets. In fact, you just don't care what they are. $\endgroup$
    – Ittay Weiss
    Dec 9, 2014 at 3:20
  • $\begingroup$ @IttayWeiss I'm in complete agreement, I believe Mazur makes all these points. I only wanted to draw the OP's attention to Mazur's essay and could have left it at that. But when you talk about throwing out the model and working only with the axioms, that's a structuralist viewpoint; that is, it's a particular philosophical position and not necessarily the ultimate truth of the matter. Was my rabbit hole remark objectionable? I'm in complete agreement with everything you say and not sure how to respond. $\endgroup$
    – user4894
    Dec 9, 2014 at 3:26
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    $\begingroup$ @IttayWeiss But when you say "there are no canonical models for the number systems," surely that is a very modern viewpoint and not at all the opinion of the early 20th century mathematicians who did build models of the number systems. We still teach these models to students. If you can't tell me what the number 2 is, you do have a philosophical problem. Saying that it's anything that behaves like we expect the number 2 to behave is somewhat unsatisfactory. I don't think the final word's been said on this question. $\endgroup$
    – user4894
    Dec 9, 2014 at 3:38
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Given you have a person "John Doe", then to...

  • his picture he is 'isomorphic'
  • his brother he is 'equivalent'
  • his twin brother he is 'equal'
  • himself he is 'identical'
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Equal means two entities are the same entity; equivalent means that two entities have the same EFFECT, in some sense. That is, when two things are same in some specific way, but not identical, they are said to be equivalent. (Identical is not really a math term, it is an English word used to convey the idea of exact sameness).

See Link or http://www.differencebetween.com/difference-between-equal-and-vs-equivalent/ for an explanation in words.

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