In plain language, what's the difference between two things that are 'equivalent', 'equal', and 'identical'?
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.
Solution 1:
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.
Solution 2:
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.)
Solution 3:
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
Solution 4:
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'