Ninth Lecture

It is remarkable how – forgive the philistine image – enlightened minds today sometimes approach judgment by going at it from the wrong end, as if someone were to put a horse's bridle on by the tail instead of by the head. But such a bridling is immediately accepted if the person concerned is officially authorized. Anyone with a sense for the living in thinking, feeling and willing could have suffered real torments for many years from the way much thinking is done in the present. I still remember hearing my first lecture in Vienna on elliptic function theory – forgive the word, but it depends on the mind of the person what I want to express, and not on whether one or the other understands what I am using now. So I heard lectures by Professor Leo Königsberger, who was already famous at the time. He was so famous that after being appointed a professor he could write to the government right away to request to be appointed a court councillor, not just a professor. So when I attended his first lecture, he came to the question: What about numbers? People assume positive and negative numbers. Positive numbers correspond to the money I have, negative numbers to the money I don't have, the money I owe. But there are other numbers. Now mathematicians use a line with an O in the middle to denote positive and negative numbers: plus 1, plus 2; minus 1, minus 2. And then the famous Gauss added a new line of numbers so that you can fill the plane with different types of numbers. I don't want to talk about the justification of this number level, but Leo Königsberger began his lecture on elliptic functions by saying, “It could now be that someone would say today that one could just as well take numbers perpendicular to this plane.” When I, as a very young badger of sixteen or seventeen, learned about the story of the plane of numbers, I immediately raised an objection: I said that then one could also think of space as being filled with numbers. The teacher kindly reassured me by saying, “Well, just wait until the next few centuries!” — which, of course, made a great impression on me, the young badger. Now I heard Leo Königsberger in Vienna address the same question. He said, “Let's assume there are these three types of numbers, not just the numbers that lie in the plane of the two lines, but the numbers that lie in the third dimension. We hypothetically assume that such numbers exist, and I would multiply such a number by another number. Now I will show you that when you multiply them, the product can sometimes be zero. But since that can never be the case, there can be no such number. Well, you see, having to listen to this is torture. I don't want to talk about whether the whole story is right or not, but if you accept one thing and don't accept the other, and claim that because the product is zero, there can be no such number, then having to listen to this is torture, because of course the correct thing is that if you have two numbers that equal zero, you have to assume that zero can be created by multiplication, not the opposite; that is the most obvious thing. But whether these judgments live in mathematics, whether these judgments live in political notes, for example in Mr. Wilson's notes, they always lead back to the same forms of thought. But if these forms of judgment live in those judgments that want to be effective for the fate of humanity, then an error in judgment means something quite different from an error in a merely limited scientific speculation, as it is in many respects the teaching of Leo Königsberger.