what is the exact meaning of the identity relation?

Question

I would like some clarification regarding the exact meaning of the identity relation. Specifically, if $a=b$, does this mean

$(1)$ $a$ is the very same object as $b$

or does it leave open the possibility that

$(2)$ $a$ and $b$ are two distinct objects having all properties in common

Note that $(1)$ above implies $a$ and $b$ have all properties in common, but the converse does not appear to be necessarily true. In other words, it seems to me there can exist two objects, not one and the same, that are exact duplicates of one another in every respect.

This feels like deep philosophical waters. Ultimately, I just want clarification on the exact meaning of the identity relation. Thanks!

Answer 1

This depends on several factors, but primarily the foundation of mathematics you're working over.

The most common framework for mathematicians to be working in these days is a theory of sets, described by a collection of axioms known as the Zermelo-Fraenkel axioms, or ZF for short. Often this is called ZFC because people also toss in the axiom of choice. I like to write ZF(C) to leave open the possibility of neglecting choice ;p

Working in ZF(C) set theory, everything is a set. All mathematical objects and concepts, be they numbers, functions, or complex structures are ultimately defined in terms of sets. There is only one notion of equality in ZF(C) set theory, which is that we write $X=Y$ for two sets $X$ and $Y$, if and only if $\forall x\in X, x\in Y$ and $\forall y\in Y, y\in X$.

For example, in David's comment, he mentions the example of $\sqrt{4}=2$. While this isn't often mentioned at the start of one's mathematics education, here $\sqrt{4}$ and $2$ are really shorthands for sets (being Cauchy sequences of rational numbers!), and the equality there is really a set equality.

So, that makes (more) precise your first definition of equality, what about the second one? Well, this also depends on several factors, such as what you mean by "distinct" and "having all of the same properties". Again, if we're working in ZF(C) then to say that two sets $X$ and $Y$ are distinct typically means that $X\neq Y$.

What does it mean to say they satisfy all of the same properties? Well, in my opinion, the most sensible interpretation is that for any well-formed statement $P(a)$ ranging over sets, $P(X)$ is true if and only if $P(Y)$ is true. Now consider the statement: "$P(a): a = X$"

Certainly $X=X$, hence $P(X)$ is true, and since "$X$ and $Y$ satisfy all the same properties" it must be that $P(Y)$ is true, ie. $X=Y$. However, this is a problem, because we're assuming that $X\neq Y$. So, in short, it's not possible to have "distinct" objects satisfying "all the same properties". In other words, if two objects satisfy all the same properties, then they are in fact equal.

Now, you might disagree with my interpretation of terms like "distinct" and "all the same properties in common", but I think I've given the most straightforward and reasonable approach to this. If you have something else in mind, then the onus is on you to provide precise and rigorous definitions of these concepts.

It is, however, important to note that there are different notions of "sameness" other than set equality. In many disciplines, mathematicians don't care about strict equality, but rather the notion of isomorphism which is when for two objects $X$ and $Y$, there are some structure-preserving maps $f:X\to Y$ and $g:Y\to X$ where $f\circ g$ is the identity map on $X$ and $g\circ f$ is the identity map on $Y$. In many fields, such as group theory and topology, isomorphism is the notion of sameness that mathematicians ultimately care about, and are only interested in those properties which are preserved by isomorphism.

There are other notions of sameness still. It's far too much to begin to explain in one post here (let me shamelessly plug this video I made about it), but there's a new-ish foundation of mathematics called Homotopy Type Theory, which has as an axiom that equivalence is equivalent to equality! (I'm somewhat imprecise here, apologies to any experts reading this)

You're correct that this ventures into very deep philosophical waters, there's much more to say about this topic! I hope this gets you started on your journey of thinking about equality and sameness. You might also find this incredible talk by Dr. Emily Riehl enlightening.

Answer 2

Standard equality $=$ in the way that we normally use it means that two items are in fact the same object.

If $a=b$ that means that $a,b$ are the same (in the model you look at).

It is possible to define new versions of equality that satisfy certain properties that we expect from equality, these are equivalence relations.

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For example, equality $\mod 2$ is an equivalence relation so we can say $$ 4\equiv 2 \mod 2 $$ because both $2,4$ satisfy divisibility by $2$. However $2\neq 4$

Answer 3

Philosophers often make the distinction between qualitative identity and quantitative identity (sometimes called numerical identity or true identity). This distinction maps to your two notions 2) ('same object') and 1) ('same properties') respectively. It is a useful distinction just to make sense of a concept like 'change' which, if you think about it, seems a little paradoxical: if something changes, it is no longer the same.... and yet, it is still the same object. I go through many changes .... and yet it is still me, not some other person. If I get a haircut, no one says "Hey, who are you? ... and where is Bram?" So, when things change, it is different .. and yet the same. How can that be? Easy: when something changes, it is no longer qualitatively the same, but it still is quantitatively (truly) the same.

Leibniz' Law states that two objects are identical if and only if they are indiscernible. Here, Leibniz clearly had the same two concepts in mind as well: by 'indiscernible' he clearly meant qualitative identity, and by 'identical' he clearly meant quantitative identity.

Now, does Leibniz' Law make sense? It seems it would be easy to consider scenarios where two objects are indiscernible (e.g. two factory-produced objects), but not identical, thus denying the claim that indiscerniable objects are identical. Then again, doesn't the fact that one object takes up a different location mean it really is indiscernible? Thus, discussion quickly turn to the question of what exactly you mean by 'property'.

Going the other way, you can point to something changing as a possible counterexample to the claim that identicals are indiscernible. However, what would something's 'true' identity consist in/of? Is there some 'essence' that remains unchanges, even as something goes through a change? What might that be? So maybe we should just work with what we can measure and observe ... and thus only work with qualitative identity. Indeed, scenarios like the Ship of Theseus, and other splitting, merging, combining scenarios might even question the very notion that there is even such a thing as 'true' identity ... indeed, even your 'identity' is really just an illusory construct of the mind.

So yes, lots of philosophy here!

However, regardless of what your stance is on Leibniz' Law, it is clear that he used 'identity' as quantitative identity, not qualitative identity. In fact, in second-order logic, Leibniz' Law is often symbolized as:

$$\forall x \forall y (x = y \leftrightarrow \forall P (P(x) \leftrightarrow P(y))$$

And so here, the $=$ is clearly meant as 'same object', not 'same properties'. Indeed, even if Leibniz' Law would be true, the concepts of 'identity' and 'indiscernibility' are different concepts.

Of course, in the end you may not really care as to which concept the $=$ exactly refers to. Certainly in mathematics, things are typically very static. I think we see some of this in how we use set identity: typically, we say that sets are identical if and only if they have the same elements. And that is not a theorem, but a definition of set-identity. Interestingly, this means that objects cannot be added to a set while remaining that set: once I add an object to a set, it is immediately a different set: so note: we didn't really add $0$ to the set of natural numbers, and we didn't really remove $1$ from the set of primes.

I guess it's a good thing things don't change too much in mathematics!

Sorry for my rambling answer ... but I guess I would lean towards treating the two notions as different concepts ... that $=$ is best seen as your 1), but that in mathematics it's ok (and much more practical) to use 2)

Answer 4

Very weirdly, all the existing answers have not mentioned the obvious fact about foundations of mathematics and even rational thinking in general.

The logic that applies to the real world as far as we know is classical FOL (first-order logic), and nowadays FOL includes equality and the standard interpretation of it, namely that "E = F" means that the expression "E" refers to exactly the same object as the expression "F" does. For example, when you write "1+2 = 3", you mean that "1+2" refers to exactly the same object as "3" does.

But in history, for whatever reason, some people did not have this conceptual understanding, and so there is such a thing called "FOL without equality". In FOL without equality, the symbol "=" is just a relation-symbol, and it is reflexive, symmetric, transitive, and is respected by every other property. The first three conditions are just the defining properties of any equivalence relation. The fourth condition means that if we have "E = F" then the truth-values of "Q(a,...,b,E,c,...,d)" and "Q(a,...,b,F,c,...,d)" are the same, for any property Q. Wikipedia has an example of an equivalence relation "≡" (modulo n) that is respected by "+" (addition). In general, for "=" to behave in a manner to deserve being called equality, it must be respected by every single property, not just one or some of the function/predicate-symbols.

If we use FOL without equality to describe a structure M, and we add axioms imposing all these four conditions, it may still be the case that there are structures with distinct elements a,b that behave identically in terms of all the function/predicate-symbols. This is a problem with FOL without equality. Today most logicians want full FOL (with equality), so that every model of an FOL axiomatization has the desirable feature that if the model thinks "a = b" for two objects a,b then they are actually the same object!

Note that the answer by Kristaps misses the real point for the reason stated by Carsten and elaborated in this answer.