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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!

RyRy the Fly Guy
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    Please give an example where this matters. Do you care if $\sqrt{4}$ really does equal $2$? – David G. Stork Jul 20 '23 at 15:32
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    I think of your first bullet as $a=b$ and your second bullet as $a \equiv b$, meaning $a$ is in the same "equivalence class" as $b$. In probability we can say random variables $X$ and $Y$ satisfy $X\equiv Y$ if $P[X=Y]=1$. – Michael Jul 20 '23 at 15:43
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    See Identity – Mauro ALLEGRANZA Jul 20 '23 at 17:21
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    The philosophical definition of identity is: to share all properties. – Mauro ALLEGRANZA Jul 20 '23 at 17:21
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    These is really a philosophical question, not a mathematical one - there are too many undefined terms. Nevertheless, I will try to give you an argument that you might find satisfying. – Alex Kruckman Jul 20 '23 at 17:48
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    Suppose $a$ and $b$ have all properties in common. Certainly $b$ has the property of being the very same object as $b$. Thus $a$ has the property of being the very same object as $b$. So (2) implies (1). Now you may object that "being the very same object as $b$" is not a valid property. But you used this property to formulate (1). I see two possible ways in which the objection might succeed. First, we could agree that (1) can't even be formulated. Second, we could make the notion of "property" precise in such a way that it uses a more restricted language than we do when stating (1). – Alex Kruckman Jul 20 '23 at 17:50
  • What does "distinct" mean in your second alternative? – JonathanZ Jul 20 '23 at 18:16
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    This question would be better suited for the Philosophy SE: https://philosophy.stackexchange.com/ – Rivers McForge Jul 20 '23 at 18:19
  • @AlexKruckman I think your argument makes sense from a mathematical perspective, but let me give an example to motivate skepticism towards 2. Consider your body at a given instant $t$. Your body at $t$ is clearly not identical to your body ten years prior, but you are the same organism as when you were a zygote, and you will be the same organism until you die. So, at $t$, your body and your organism have all and only the same properties, but they are distinct. At an instant of time both ‘your body’ and ‘your organism’ refer to the same thing, but they do not always. – PW_246 Jul 20 '23 at 18:40
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    @PW_246 I was responding to the question about whether (1) and (2) are equivalent. I did not mean to address whether either are good definitions of $=$ in a philosophical context. In response to your example, I would argue that if you want to consider objects existing across time (as you do explicitly with "your organism"), then "having all the same properties" is much more expansive than 'having all the same properties at fixed time $t$". – Alex Kruckman Jul 20 '23 at 18:50
  • @AlexKruckman I found your argument to be particularly enlightening. Thanks – RyRy the Fly Guy Jul 20 '23 at 19:22
  • @AlexKruckman yes, but if it can be violated at a specific time, the argument goes that they can’t be equivalent in a more general sense. That is, the general case seemingly should imply the specific case. – PW_246 Jul 20 '23 at 19:27
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    This philosophical issue goes a long way back: Leibniz postulated the idea of the identity of indiscernibles. For another take relating to the notion of number see Benecarraf's papers on what numbers could not be. – Rob Arthan Jul 20 '23 at 21:02
  • Pretty much every formulation of set theory has the Axiom of Extensionality, which explicitly rules out case 2. Also, the axiom is not a trivial one--removing it significantly lowers the strength of set theory. – eyeballfrog Jul 21 '23 at 12:10
  • @RiversMcForge: If you ask on Phil SE, you will be extremely unlikely to get any mathematically backed answers, nor are they welcome there. – user21820 Jul 21 '23 at 16:06
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    @eyeballfrog: Your first sentence is technically incorrect or misleading. If the base logic is FOL without equality (see my answer), even Extensionality fails to rule out case 2. You might argue, ZFC is based on logic with equality, but the point is that it is that feature of today's conventional choice of FOL that rules out case 2, rather than anything to do with set theory! – user21820 Jul 21 '23 at 16:09
  • this was a question 100% relevant to the topic of logic and set theory. hard to understand why this was closed. – RyRy the Fly Guy Jul 25 '23 at 16:39
  • @RyRytheFlyGuy: I don't know why your question was closed, but have you read my answer and understood it? Some troll on SE always stalks me and downvotes my answers, so just ignore that... – user21820 Aug 12 '23 at 10:28

4 Answers4

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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.

Mark Foskey
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Kristaps John Balodis
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    Thank you for taking the time to write this response. i gave you an upvote but will hold off on the check for now to encourage more answers. I think you make a good point that the answer depends on the mathematics we are working over. I understand there are different notions of sameness in different contexts. We can used the notions to define equivalence relations. but i thought the identity relation $a=b$ was particularly unqiue, asserting two objects $a$ and $b$ are in fact one and the same object. i will check out your video. – RyRy the Fly Guy Jul 20 '23 at 19:29
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    It all comes down to what you want to mean by "same". If you want "one and the same object" to mean that $\forall x\in a, x\in b$ and $\forall x\in b, x\in a$", then you define "$a=b$" to mean exactly that. As this is what many mathematicians want to mean by "one and the same object", that's why the Axiom of Extensionality in ZF(C) defines "$=$" as such. – Kristaps John Balodis Jul 20 '23 at 20:00
  • Also, perhaps it's worth mentioning that the concepts of "partitions" and "quotients" in set theory, allows one to translate between equivalence and literal equality. For example, we say that 0 and 6 are equivalent modulo 3. This is just an equivalence relation, however, if we form the quotient set $\mathbb{Z}/\sim$ of the integers $\mathbb{Z}$ under the equivalence $\sim$, then the equivalence class $[0]$ of 0 under this relation, is literally equal $[0]=[6]$ to the equivalence class of 6. – Kristaps John Balodis Jul 20 '23 at 20:06
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    A nice answer, but it's important to remember that set theory does not provide canonical representations of important mathematical notions like numbers. See the paper by Benecerraf referenced in my comment on the question as regards numbers.. From some points of view, the set-theoretic foundations of mathematics involves a multitude of arbitrary decisions about representations that are irrelevant to our mathematical intuitions about notions like a point in 3-space or a finite group or a banach space or ... – Rob Arthan Jul 20 '23 at 21:14
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    I think this answer misses the point because equality is not part of the axioms but of the language of first order logic in which they are defined. – Carsten S Jul 21 '23 at 08:46
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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)

Bram28
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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.

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 $2\neq 4$

wjmccann
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  • Yes, but in a mathematical world where we have to deal with notions like relations and functions, working modulo equivalence relations becomes extremely complicated. – Rob Arthan Jul 20 '23 at 21:17
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    @RobArthan The typical foundation for integers is an equivalence relation on ordered pairs of natural numbers. Similarly the typical foundation for rational numbers is an equivalence relation on ordered pairs of integers (excluding those where the second is $0$). But you can work with integers and rational numbers without having to go back to the foundational equivalence relations each time. – Henry Jul 21 '23 at 09:29
  • Yes, I have read Landau's delightful book on the foundations of analysis and understand the issues you raise. However, when you relate this to the original question, it isn't that simple: you can't just identify all isomorphic groups and expect group theory to make sense (for example). wcjmcann is suggesting you can easily do something like that, but that is an oversimplification. – Rob Arthan Jul 22 '23 at 20:34
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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.

user21820
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  • A fun fact is that it is easier to prove the semantic completeness theorem for FOL without equality, and that one can then prove it for full FOL by (unsurprisingly) taking the model (without equality) modulo the equivalence relation given by its interpretation of "=". – user21820 Jul 21 '23 at 16:01