Definition of Equals

Question

DISCLAIMER: This is a first time Math.SE post from a 30-something who is only now learning math. I have read the Rules and this may not satisfy the "Questions with too many answers" criteria or perhaps this is just too philosophical? ... BUT Mathematical semantics is one of the key things I struggle with in Maths and its almost impossible to google this:

While refreshing my limited knowledge of symmetry I came across this on themathpage.com:

It is not possible to give an explicit definition of the word "equals," or its symbol = . Those rules however are an implicit definition. The meaning of "equals" implies those three rules.

I would have thought equals would be easily defined as identity. If a = b then a is identical to b?

Is not the case in all maths? or is this irrelevant as the page got it wrong.

Answer 1

This is a good mathematical question.

Defining equality by saying it's just identity is not a definition, but rather a reformulation of one undefined term for another undefined term.

We can't define each and everything we use in terms of simpler things since that will result in an infinite descent of concepts. Thus, at some point in the rigorous treatment of anything, we must encounter a term that we simply say is too fundamental to be defined in any simpler terms. At that point we simply adopt the axiomatic approach. Instead of defining what something is, we clearly specify how it behaves. After all, it is the behaviour of things that interests us and not so much what they are composed of. For instance, if there was another chemical compound that exhibited the exact same behaviour as $H_2O$ does, then you would use it just like you use water. You don't care about the atoms composing the molecule, only on the behaviour of the molecule (as long as you are interested in using the chemical for, e.g., drinking).

So, what are the important properties of equality? Well, everything should be equal to itself, so $x=x$ is an axiom. If a bzorkq is equal to a kawataninga then, without caring at all what these things are, we must have that a kawataninga is equal to a bzorkq. Thus we include the axiom $x=y$ implies $y=x$. Finally, I leave it to you to justify the final axiom $x=y$ together with $y=z$ implies $x=z$.

That is what equality is. Equality is not an absolute term and we don't define what it is. Only how it behaves. When two people speak of equality, they implicitly agree that the equality relation they both refer to is the same relation.

Answer 2

Consder the inequality as opposed to the equality. If I take an inequality and multiply it by -1, the sign "flips" over. $-1(x<y)\rightarrow-x>-y$ But, if we start with an equality, $-1(x=y) \rightarrow -x=-y$. Did the equal sign flip over? How would you know? An equality may be seen as a reversible inequality. Specifically, if $x>y$ and $x<y$, then x and y don't exist, but if $x\geq y$ and $x\leq y$, then x=y.

Answer 3

I somewhat disagree with the text you quoted from that webpage. Defining what equals "means" just depends on the kind of objects you are considering. Maybe you take for granted that you know when two natural numbers (i.e., non-negative integers) are equal. Then two integers $m$ and $n$ are equal if they have the same sign and the natural numbers $|m|$ and $|n|$ are equal. Two rational numbers $a/b$ and $c/d$ are equal if $ad = bc$. Real numbers are a bit trickier, but you can define their equality using sequences of rational numbers.

There are lots of other types of mathematical objects besides numbers, too. For instance, two sets $A$ and $B$ are equal if they contain exactly the same elements. Two functions $f$ and $g$ are equal if they have the same domain and take on the same value at every point, i.e. $f(x) = g(x)$ for all $x$. Two (formal) polynomials are equal if all their coefficients are equal.

In the early 1900s, an attempt was made to axiomatize mathematics, meaning people tried to write down a "minimal" list of tacit assumptions upon which the rest of mathematics could be built. One theory that emerged is called ZFC, and is accepted by many as a "foundation for mathematics" to varying degrees. In this theory, the set is taken as a "primitive notion," i.e., the thing that we leave undefined. Amazingly, it is possible to define all mathematical objects in terms of sets. Thus, if you accept ZFC, all equality comes down to an equality of sets, which as I wrote above, just comes down to checking that two sets have the same elements.