# Is 'equality' ultimately grounded in empirical observation?

Let's say I invent a concept X in my own imaginings. The only property it has is X-ness; it is defined as 'that which is represented by X'. I have just defined that to be the case. It seems to me, now, that it must be true that `X=X`. X is the same thing as itself. I thought this was hard to argue against, but a guy on YouTube did just that; he says:

The only way that you could possibly abstract the idea x = y is because you are imagining they represent numbers. And the only way that you can imagine numbers is because you imagine they represent quantities. And the only way you can be familiar with quantities is by counting them in the real world. Thus, yes, EVEN equality is grounded ultimately in empirical observation. And you have been captivated by the illusion that it is not because you have forgotten where you learned it from.

His argument is that any algebraic or logic statement must be somehow grounded in empirical observation, but this doesn't seem correct to me. I accept that mathematical observations are grounded in empirical observation - the notion that `2+3=5`, for example. However, if I have invented a concept X, independent of the real world and in my mind only, surely the fact that 'X is the same as itself' can be said to be completely independent of any empirical observation, and must inherently be true? How could it be false?

The youtube poster appears to be conflating logical and mathematical equality. Mathematical equality is the state of being quantitatively the same, which he argues requires empirical observation to prove. Logical equality is applied to the values of propositions. For your purposes, Leibniz's law is relevant, as you appear to be more interested in identity:

The identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical (are one and the same entity) if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa.

Or more formally:

Given any x and y, x = y if, given any predicate P, P(x) if and only if P(y).

• It'd be great if we could get the Youtube poster to respond to this argument. – Pacerier Jul 10 '14 at 8:27

What you've defined is known in formal logic as a tautology, which is a statement that is rendered true merely by definition, or by virtue of you declaring it to be true. Essentially, you've defined X as being X.

By their nature, tautologies can be considered to be absolutely true. They assert a universal, unconditioned truth. So, the statement `X = X` is certainly and absolutely true because you've defined it that way.

However, the problem here quickly becomes that tautological statements do not convey any useful information whatsoever. They do not tell us anything at all about the nature of the objects involved. I can't draw any conclusions or engage in any sort of reasoning about objects of type X  because the only thing I really know about them is that they are themselves.

That is why, in philosophy, tautologies are essentially worthless. I am fond of likening them to a metaphysical "division by zero," much like the well-known mathematical "proof" that `1 = 0`:

``````   x = 0
∴ x(x - 1) = 0
∴ x - 1 = 0
∴ x = 1
∴ 1 = 0
``````

Just as when you divide by zero you are no longer dealing with numbers, when you deal with tautologies, you are no longer dealing with logical statements. You haven't proved anything at all, and in fact, your entire argument (were you to derive an actual argument from the claim in this scenario) becomes specious.

So while yes, you've essentially "proven" that the statement is "inherently true" by defining it as such, you haven't really made a logical claim. It doesn't have to be grounded in any sort of empirical observation because it's merely grounded in definitions. But you haven't found a hole in the system of logic any more than a proof that involves dividing a number by zero has found a hole in the system of mathematics.

• Has anyone proved the statement that "tautologies can be considered to be absolutely true", though? I've always been taught that this is just an axiom. I do know that there are people who work to prove axioms, but don't know how they go about doing this. – Ethel Evans Jun 7 '11 at 23:38
• @Ethel: No, that's one of the primary problems with tautologies. Under our system of logic, tautological statements are always true. There are formal, systematic proofs of it, but they're basically meaningless for the reasons I establish in my answer. – Cody Gray Jun 7 '11 at 23:40
• 1. A tautology is not merely true by definition in formal logic. A tautology comes as true for all possible combinations of truth values for the truth set involved. Also, if there exists a soundness metatheorem of the logic (as there does for classical logic, and most logics), then if a statement form has a formal proof, then it qualifies as a tautology. So, it comes as true by proof! 2. Tautologies aren't universal. They come as local with respect to the truth set. The set of tautologies of 2-valued logic, is by no means the same set as that of Lukasiewicz's or Kleene's 3-valued logic! – Doug Spoonwood Aug 29 '11 at 6:20

I would disagree with the argument of the person you are arguing with, though I don't think I could prove that X=X is always true; I accept that as an axiom. I would challenge anyone to produce a counter-example, though. There is certainly no empirical observation of a thing that is not equal to itself.

My counter-argument to the person you argue with is that concepts are not empirically observable, yet can have quantity. Therefore, empirical observation is not necessary for reasoning about quantity. Since this is one of the principles of his arguments, his argument fails.

• As a counter-example to X=X always being true, in C++, testing whether a float equals itself is one way to test for whether the float represents a number. See this Stack Overflow answer for more: stackoverflow.com/questions/570669/… – Ben Hocking Jun 8 '11 at 1:49
• @Ben: That's because numerical equality is not defined for things that are not numbers. This is exactly what my answer addresses. That's not really a counter-example, it just proves that equality is not applicable here because you've introduced a category error (like divide by zero, or not-a-number). – Cody Gray Jun 8 '11 at 8:51
• @Cody Gray: Continuing to play devil's advocate, this also applies to the statement `++i == ++i`, because the `++` modifier changes the object in the midst of equality evaluation. I can imagine the objections to such a contrived example, but the "real world" analogue in my opinion would be in a quantum mechanical formulation where observing an object changes its nature. To wax semi-philosophical: you can't step in the same river twice. I think there's a reason that reflexivity is an axiom of mathematics. – Ben Hocking Jun 8 '11 at 18:12

There are several things one could say in reply.

1.) It could be argued that your equation is in fact not an equation because it isn't about numbers, but rather an identification. In that case, Youtube guy's position seems reasonable.

2.) Let's assume identification counts as equation. "The only way that you could possibly abstract the idea x = y is because you are imagining they represent numbers": your Youtube friend would seem to disregard the fact that we can imagine mental images of physical objects or concepts as identical. I can think of two pictures of my mother at different ages and decide they depict the same person. I need no numbers to identify both persons.

3.) If we assume that it is indeed an equation, it could be argued that all you need to prove it is Artistotle's Principle of Contradiction: a proposition cannot be true and false at the same time. It is evident that `x!=x` violates this law and is hence impossible; therefore, `x=x`. It would be even easier with the Law of Identity: a thing is the same as itself.

Are these laws grounded in observation? One could argue that mankind developed these modes of thinking (for that is what they are) inspired by observations that the moon cannot both shine and not shine at the same time, and that it cannot be not the moon.

But that connection seems trivial, since we apply these laws equally to abstract things. If we should call them grounded in empirical observation, we should consider any and all aspects of human thought grounded in empirical observation as well; some Empiricists (Locke?) might hold this position. Kant would say that these laws are a-priori parts of the framework of human thought. That seems sensible.

Category theorists are fond of playing with notions of equality. This piece is interesting and insightful.

In short, whether two things are equal depends rather a lot on what purposes you're interested in.

Where x=x in the physical world, as in an assignment of a singular property that can no longer be subdivided into additional properties, then the notion of equalness can be truly asserted.

But as long as the observer (in time), can change the scope of properties a given object consists of, it can no longer be assigned equality with the other, because the potential for those properties no longer being equal exists, and thus each object must be observed once again before true equality can be re-affirmed.

• Where there is life, there are new properties of life. – Chris L. Fleshner Aug 20 '12 at 15:57

This is actually not true. An argument against it is known as Russel's Paradox. From Wikipedia

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is Russell's paradox.

Regarding conflating logical and mathematical equality: If one can reasonably discern one object to another, that object would of course have equality, if the properties of the objects are identical.

However, since time is a function of discernibility, it is asserted that identical properties of objects cannot exist in space and time at any one instance.

Regardless, all equalities, for practical purposes can be rationalized as mathematical, as an umbrella above logical equalities, depending on the accepted level of precision desired.

• Welcome to Philosophy.SE! Not sure what to make of your first two sentences - does the fact that "objects cannot exist in space and time at any one [instant]" mean that no two objects can ever be equal? What does it mean that all equalities can be rationalized as mathematical? – James Kingsbery Nov 25 '15 at 18:25
• I agree with the above comment, I think it might help if you made it a little more clear how you're answering the question. – commando Nov 26 '15 at 3:20

Your (X = X) is true but, in reality, not true forever.

Just to note, there are empirical approaches to mathematical realism, using set theory, that attempt to correlate sets of "natural kinds" with brain states or "cell assemblies," just as perceived objects also produce certain neural "cell assemblies." These are assumed to develop physically as the brain develops "object permanence" and "identity" in childhood.

Presumably these "cell assemblies" would apply to the cardinality of sets and thus to a primal equivalence (=). But equivalence is not identity. Obviously, 1 + 1 is not identical to 2, and X = X entails two X's in two nonidentical mirror positions. The = marks a potentially broken symmetry.

The equal sign (=) was created by the 16th-century Welsh mathematician Robert Recorde. His rationale was that two parallel lines (=) were as ideally alike as two nonidentical things could be. Thus the equal sign bears a lost historical relationship to Euclid's fifth postulate. And this postulate proves famously paradoxical, because it introduces a negative infinity. How can we assume the two lines "never ever" meet? What keeps them apart?

Likewise, how can we know the infinite nonidentity (=) of left-hand X and right-hand X? We can see them and perhaps categorize them into overlapping "cell assemblies." But I do not see that this accounts for the idealization of the negative "infinity" of their "equality." Here, the (=) symbol is a good example of what Hegel calls the "identity of identity and difference."

Sorry, this is a bit of mental meandering inspired by this very interesting question. Perhaps there is a partial answer in there somewhere. Basically, I agree that the purely empirical correlation cannot hold up.