And I'm not trying to be clever, just trying to avoid misconceptions, and help those interested sort things out. They abound in the "foundations of mathematics".
You won't have to delve deeply. Mathematics, and foundations of mathematics, have parted ways as scientific disciplines.
Yes of course and is the same as an intersection. Analogies is what maths is about.
And your prof is wrong. Ask him about the mathematician called
Jean Leray