Equals VS Define: Why = is not the same as :=

I assume that you have all seen the “equals” sign \(=\) before, as in \(1 + 1 = 2\). You have probably also seen usages such as Compute \(f(x)\) for \(x = 5\). These are actually two very distinct uses of \(=\).

In \(1 + 1 = 2\), the equals sign expresses an equality. An equality may or may not hold, and a statement about equality may be true or false. While \(1 + 1 = 2\) is in general a true statement when talking about numbers, there can be cases when it is false. For example, in Boolean matrix multiplication we cannot have values larger than 1 and hence \(1 + 1 = 1\), which means that \(1 + 1 = 2\) would be false in that case.

In our \(x = 5\) example, on the other hand, we are not saying that \(x\) is equal to \(5\), we are defining \(x\) to be \(5\). There is no way for this to be true or false, you cannot tell me that I am wrong to define \(x\) as \(5\). I could have just as well defined it to be \(7\), or \(-35.3\), and there is nothing you could have done to stop me. There is no way for you to show that \(x = 5\) is false.

In linguistic terms, the equals sign in \(1 + 1 = 2\) expresses a proposition, whereas in \(x = 5\) it expresses the speech act of establishing in the discourse that the name \(x\) refers to the number \(5\). If you have some experience with programming languages, then you can think of this as the difference between a truth condition and a variable assignment. Mathematicians care so much about this difference that they use two distinct symbols. Equalities are expressed with \(=\) as usual, but the speech act meaning of \(=\) is denoted by \(\mathrel{\mathop:}=\) instead. Hence a mathematician would not write “Suppose \(x = 5\)” but rather “Suppose \(x \mathrel{\mathop:}=5\)”.

Long story short: Don’t get confused when you see \(\mathrel{\mathop:}=\) instead of \(=\), it simply means that you are looking at a definition/variable assignment rather than an equality.