Mathematical Logic

Abstract

In Why Is There Philosophy of Mathematics at All? [8], Ian Hacking writes: Yet although most members of our species have some capacity for geometrical and numerical concepts, very few human beings have much capacity for doing or even understanding mathematics. This is often held to be the consequence of bad education, but although education can surely help, there is no evidence that vast disparity of talent, or even interest in, mathematics, is a result of bad pedagogy. . . A paradox: we are the mathematical animal. A few of us have made astonishing mathematical discoveries, a few more of us can understand them. Mathematical applications have turned out to be a key to unlock and discipline nature to conform to some of our wishes. But the subject repels most human beings. It all rings true to anyone who has ever taught the subject. When I teach undergraduates, I sometimes say, “Math does not make any sense, right?” to which I hear in unison “Right.” My own school experience has not been much different. Even though I was considered “good at math,” it only meant that I could follow instructions and do homework to a satisfying result. Only occasionally I’d have moments I was proud of. In my first high school year, I had a mathematics professor about whom legends were told. Everyone knew that in order to survive in his class, one had to understand. At any time you could be called to the blackboard to be asked a penetrating question. We witnessed humiliating moments when someone’s ignorance was ruthlessly revealed. Once, when the topic was square roots of real numbers, I was called and asked: – Does every positive number have a square root? – Yes. – What is the square root of four? – Two. – What is the square root of two? – The square root of two.