Identity

"Mathematics has the completely false reputation of yielding infallible conclusions. Its infallibility is nothing but identity. Two times two is not four, but it is just two times two, and that is what we call four for short. But four is nothing new at all. And thus it goes on and on in its conclusions, except that in the higher formulas the identity fades out of sight."
page 1754
-Johann von Goethe, In J. R. Newman (ed.) The World of Mathematics, New York: Simon and Schuster


This identity is infact a core part of Mathematics. Example. How would you prove that six times six and nine times four are one and the same thing? (Johann might say that they are not. They are infact what they are: six times six and nine times four respectively. But these are just different perspectives of the same object. Read on.) In the decimal world, you can name six times six thirty-six and nine times four also, thirty six and come to the conclusion that we are talking about the same object. In the hex world, we can name the same quantity twenty four. Names don't matter, what matters is that using these identities we can keep aside perspectives to focus on absolute objects. That is the essence of Mathematics! It kinda puts us in a frame of reference where we can safely define absolute w.r.t that frame of reference.

Now, one might argue that he can name six times six fooday-foo and nine times four, looday-loo. But, then asking for fooday-foo pencils or looday-loo pencils will give you the same number of pencils. So, its better to call them thirty-six.

Until and unless counting is redefined in a revolutionary new way, six times six and nine times four will have the same identity and thats a conclusion that is not just made-up from thin air.