It all starts with Aristotle. He said that there needs to be some sort of deductive system \u2014 that you can only prove new things by basing them on things you already know and are certain of, going back to certain \u201cprimitive statements,\u201d or axioms.<\/p>\n

So then the question is: What are those basic things that you know to be true? For a very long time, people just said, well, a line is a line, a circle is a circle; there are a few things that are simple and obvious, and those should be the assumptions we start from.<\/p>\n

That perspective has lasted forever. It\u2019s still around today to a large extent. But the Euclidean axiomatic system that developed \u2014 \u201ca line is a line\u201d \u2014 had its problems. There were these paradoxes discovered by Bertrand Russell based on the notion of a set. Moreover, one could play word games with the mathematical language, creating problematic statements like \u201cthis statement is false\u201d (if it\u2019s true, then it\u2019s false; if it\u2019s false, then it\u2019s true) that indicated there were problems with the axiomatic system.<\/p>\n

So Russell and Alfred Whitehead tried to create a new system of doing math that could avoid all these problems. But it was ludicrously complicated, and it was hard to believe that these were the right primitives to start from. Nobody was comfortable with it. Something like proving 2 + 2 = 4 took a vast amount of space from the starting point. What\u2019s the point of such a system?<\/p>\n

Then David Hilbert came along and had this amazing idea: that maybe we shouldn\u2019t be telling anyone what\u2019s the right thing to start with at all. Instead, anything that works \u2014 a starting point that\u2019s simple, coherent and consistent \u2014 is worth exploring. You can\u2019t deduce two things from your axioms that contradict each other, and you should be able to describe most of mathematics in terms of the selected axioms. But you shouldn\u2019t a priori say what they are.<\/p>\n