I just came across this statement when I was lecturing a student on math and strictly speaking I used:
Assuming that the value of x equals <something>, …
One of my students just rose and asked me:
Why do we assume so much in math? Is math really built on assumptions?
I couldn’t answer him, for, as far as I know, a lot of things are just assumptions. For example:
- 1/∞ equals zero,
- √−1 is i, etc.
So would you guys mind telling whether math is built on assumptions?
To respond to the charge that “we assume so much in math”: math involves the analysis of various hypotheticals. When I say “if X then Y,” I might need to assume X hypothetically while in the process of proving the implication, but this does not require presumptuousness in any way. One may easily accept “if X then Y” and its proof without actually accepting X at all!
On the other hand, there are various configurations of axioms that can be chosen as the logical foundation of mathematics, with ZFC as the tacit standard. Outside of foundations and set theory and model theory and logic and so on, the axiom list is relatively small (compared to the whole of the theory of any mathematical context with all of the various theorems and formula) and more importantly unchanging, so this is not relevant to the charge “assuming so much in math.”
And further away from presumptuousness, many things in math (as anywhere else) are not actually “assumptions” but are conventions and definitions. I don’t “assume” an apple is a red or green fruit that grows on trees, for example, that’s just how it’s defined.