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?