It means that the mathematical object to the left of the sign is the same object as the one on the right of the sign. Of course there are cases where modifiers are attached. For instance, let’s write the absolute value of x as |x|. Then we can say that x = |x| whenever x is a nonnegative number. And it’s true, but only because of the modifier “whenever x is a nonnegative number”.
The answer to this question really depends on what you mean by a real number. If you’re content with the notion that there are these things out there called real numbers and you’re not too worried about what they are, then you pretty much have to take x = y as an undefined predicate which satisfies the normal properties of equality: x = x; x = y implies y = x; x = y and y = z implies x = z.
If you’re really concerned about what real numbers are, you’re dealing with constructions. There are two standard constructions of R from Q, and in both cases every real number ends up being a set. Fortunately, = is defined for sets–for any two sets X and Y, X = Y if and only if every element of X is an element of Y and vice versa.
The fraction 1/2 is not identical to the fraction 2/4, yet they are considered to be equal.
A detailed answer would take us pretty far into formal logic, so let me sketch it out. Basically, we mathematicians are lazy. Rather than prove the same basic thing over and over and over again, we’d like to prove it once and be able to refer to that proof as often as possible.
So what we do is take a set of properties that many classes of objects have and abstract them. We can then take those properties as axioms, prove theorems from them, and be confident that those theorems are true for any class of objects that satisfy the axioms.
I learned that the axioms are that x = x, and that if x = y, you can rewrite all the occurences of x in a true statement as y and come back with another true statement. The other properties of an equivalence relation follow from those two axioms.
There are other sets of axioms that are equivalent to what I presented, but this is fairly simple.
Yes, but proving it only for sets is hardly general enough.
These are the axiom schemata from which you can deduce statements using equality, correct?
But I think the OP was asking about the semantics of equality. What does “=” really mean?
Also, saying that equality is an equivalence relation is not enough, since it is clearly not the only equivalence relation. What distinguishes it from other equivalence relations?
And sometimes mathematicians consider objects to be equal because it is convenient. Suppose we construct Q as the field of fractions of Z. So Q is a set of equivalence classes on Z x Z*. We usually identify Z with the its image under the monomorphism mapping n to the equivalence class of (n, 1), so that we can consider Z to be a subset of Q. But do you really consider the 0 in Z to be equal to the 0 in Q?
I’m not a big math guy but I took logic and I think it’s a lot of fun. I posted the metamath link in GD. It’s pretty interesting that they were able to, in an equal system, prove that x=x just from the other axioms. But their computation-oriented system makes things like that a lot easier to do.
Anyway, `=", identity, has three basic properties, and it is a two-place predicate. In linguistics that makes it an infix, like um edumacation =). It takes an argument on the left, and an argument on the right. Here’s those three properties:
[ul]
[li]The symmetry of identity: from a=b, b=a[/li][li]The transitivity of identity: a=b, b=c, a=c[/li][li]The reflexivity of identity: a=a[/li][/ul]
So the statement `x=y’ basically can only be true if x actually is y. And here’s the proof showing there is something that exists such that x=y: at least one individual exists.
I think I can see the outline of the point you’re making here, but I’d appreciate it if you could clarify it a bit. One might argue in opposition to the above that 2/4 is identical to 1/2 because it is simply 1/2 * (2/2), where (2/2) is clearly the multiplicative identity. Are you arguing that this argument is not valid in general because of its reliance on the properties of multiplication over the set of rational numbers?
x = y can mean anything you like, and there no reason why the symbol has to be used for a reflexive relation. So no, it doesn’t follow. I can’t think of an example because no one actually does that.
Right. Rational numbers are equivalence classes of fractions (in the standard construction) and so you can’t argue about fractions using the properties of rational numbers.
In most areas of mathematics where “=” is used, it’s defined as an equivalent relation, which makes the proof of “x = x” trivial. The proof consists of “Axiom: x=x”. It is possible to use some different set of axioms, which that metamath site apparently does. They’ve apparently chosen some set of axioms which does not include “x = x”, but the other axioms they’ve chosen are sufficient to prove it. Without spending time to explore exactly what all of their axioms are, I can’t say why they chose the particular set they did, but it’s certainly possible.
Polerius: Most “interesting” statements about Mathematical formulas deal with their semantics (meaning) and not their syntax.
E.g. here is a similar example that is syntactic:
“1+1” contains the plus sign. True.
“2” contains the plus sign. False.
Not all syntax tests are so easy:
“131071+1” contains a prime number.
“131073-1” contains a prime number.
But it is quite clear that applying what is basically a syntax check in an issue involving equality is fudging things to say the least.
In Theoretical Computer Science, this is a key issue. E.g., all non-trivial statements about the input/output of computer programs (semantics) is undecidable. But many questions about their syntax are decidable. (I.e., I’ve had to try to drill this point into a lot of student’s heads over the years, with little success.)
While we’re on the subject, and I hope this isn’t irrelevant - what is the difference between “equivalence” and “identity”?
From what I recall, an “identity” is a statement that is true for all values of the variable - informally, something that is “trivially true”.
I’m referring to the concept denoted by a triple “=” sign, which I don’t know how to generate in a browser-neutral way.
Thus, x^2 <triple equals sign> x * x
Isn’t it also valid to write x ^ 2 = x * x using a regular ‘=’ though?
Is “identity” really just a hint to other mathematicians that “this is an equation but I’m going to express it as an identity to show you that I consider it obvious” ?