What is the difference between identity and equivalence?

Question

While the problems of identity seem to have more heft in philosophy, I am actually more interested in the meaning of "equivalence" as symbolized in the (=) sign and, in guilt by association,with commodity exchange.

Obviously, 1+1=2 is not an identity, nor is 1=1, since the left 1 and right 1 are spatially distinct. Leibniz's identity of indiscernibles appears to have no place in mathematics... and perhaps not in any "conceivable" reality. Do concepts of "space" and "time" preclude identity and constitute the basis of binary "equivalence"? Is equivalence simply synonymous with "substitution" inside some framework?

Anyway, I am interested to know if equivalence (=), as distinct from identity, is well defined in philosophy, logic, or mathematics...or if there are interesting discussions of this paradoxical and prevalent concept.

Answer 1

Classically, there is no difference. Identity is a two place predicate which has the value true if its arguments are numerically identical and false otherwise. One can write such a predicate as Identical(x,y) or one can write it as x=y. The latter is just 'syntactic sugar' that makes sentences easier to read, but it expresses the same thing. This is the way identity is employed in logic, in particular in predicate logic, and by extension in mathematics.

To say 1=1 does not express identity because there is a spatial difference between the left and right is confusing the thing with the symbol that denotes it. The number one is identical with itself. Of course a serious issue arises over how symbols denote things, and there is much debate over this. Frege held that names are a kind of abbreviated definite description, but this is only one of many accounts of the meaning of names.

There are plenty of contexts (intensional contexts) in which identity relations appear to violate the indiscernibility of identicals. For example, "Mary knows that George Orwell wrote 1984" might be true, while "Mary knows that Eric Blair wrote 1984" might be false (even though George Orwell = Eric Blair). In general what this means is that while a given identity relation might correspond to a purely extensional equivalence in a particular logic, there might be a more expressive logic in which that identity relation is not extensional.

Frege, Quine, Geach and Dummett are all worth studying on this subject. The SEP article on identity is a helpful introduction.

Answer 2

Equivalence is a fundamental concept of mathematics, possibly even the most elementary fundamental concept. Taking equivalence relation is used for abstraction:

One has a set ob objects and wants to group them into classes, with all members of each class having the same property. All members of one class are equivalent concerning the given property, but they are not equal.

A simple example is the class of even numbers and the class of odd numbers. Each integer falls in exactly one class. Now one forms the set FF_2 with these two classes as its elements. Denote by "0" the class "Even" and by "1" the class "Odd". Then you can derive addition and multiplication for the two elements of FF_2 from the corresponding operations on the set of integers. E.g., one defines 1 + 1 := 0 in FF_2, because "odd" + "odd" is "even".

Hence one passes from the infinite set of integers to the finite set FF_2 by introducing an equivalence relation. And this equivalence relation respects addition and multiplication.

The above construction means to consider equivalent two integers if and only they give the same rest when dividing by n = 2. The same construction can be achieved for division by arbitrary non-zero n. The result are the restclasses modulo n, i.e. the sets with elements 0,1,2,...,n-1. Also here addition and multiplication are well-defined.

Answer 3

Being a programmer I might look at this slightly differently than others.

I treat identical and equal as different things, mostly because some programming languages do that.

In most cases both things will be identical and equal but in some they might only be equal, for example:

1 and 1 are identical and equal (same value, same type).

1 (number) and 1 (text) are equal but not identical (same value but they are of different type. If you are not a programmer you will probably think "hey how is that even possible that a number is not in fact a number?" but things can have different "states" (let's call it like that).

So I see it as equal being same and identical being exactly the same.

Answer 4

If like is like like, then is unlike like unlike or different? And if different then Being cannot be one, an argument put by Socrates to Parmenides.

But this isn't how equality is theorised in mathematics.

If you're interested in the notion of how mathematical equality is thought through, then you might find it worth looking at how equality is theorised in category theory; there's a blog entry here, by Baez on his essay on 'concepts of sameness' which is exactly on this.

A box is like itself - identity; and in this sense, here, says nothing; but what say we, if we see a box is not just a box by itself, but is so positioned in space? Were it positioned in space somewhat differently - say I shoved it to the left a little; would it be the same box?

Obviously, yes.

But this just reduces to what went before; for I took it out of space to say so - in a way, in a sense; but not actually so - for I did not; saying so, to illustrate.

So to take a box in relation to space; in terms of its relation, we see that it is different, unequal; and yet the same.

X=X

and

X!=X

So, somewhat like the Heraclitian thesis of the Unity of Opposites; dismissed by Aristotle - but not - quite - so, for he was dismissing platitudes; which are utterances empty by repetition, or by being carried aloft like a banner over being.

Or, as in the first verse of the Dao:

These Two emerge together

But differ in Nature

The Unity is said to be the Mystery

Or, in the last verse of Borges Ars Poetica:

Tambien es como el rio interminable

Que pasa y queda y es cristal de un mismo

Heraclito inconstante, que es el mismo

Y es otro, como el rio interminable

And so to time, for mathematical operations even when they are on time, never exemplify time: if A is A now, and later is B; if change is a continua, then A and B are the same in a sense; but if A is really different to B, and so properly distinct, when or how can change occur? It's a question of becoming and being: I can say, becoming is, and that being is; but that does not entail their ontological reduction; for Aristotle, at least in one sense, being is a limit of becoming.

Answer 5

In computer science, equivalence is notably different than identity, so much so that computer languages often provide both as separate syntactic units.

I have found in many environments, equivalence and identity are treated as different. Equivalence always requires some metric with which equivalence is defined ("Shut up and just pick an apple, Johnny. They're both equally good!"). Identity is typically treated as more of an intrinsic characteristic.

The most common place I find the two words treated differently in philosophy is in scenarios handling identity in the presence of cloning. In these situations, it is easy to generate two equivalent clones, but it is less immediately clear whether there is an identity relationship to be had or not. Related, the concept of the Ship of Theseus is a major question in the philosophy of identity. It is always clear to everyone that the ship after repairs is equivalent to the ship before repairs, but the debate is whether it is identical or not.

I do find that people often blur the line between equivalence and identity in situations where there is such an obvious equivalence relationship that it gets "promoted" to an identity relationship. This is easy to see in the phrase "1+1=2" which is most literally phrased "one plus one equals two," but is often vocalized as "one plus one is two."