Dialogue : Roger PENROSE and SATO Humitaka

Human Understanding of the "Abstract"

SH: Is the beauty of pure mathematics something a non-specialist can feel ? Or do we need to be specially trained ?

RP: That's a good question. Mathematics is a very esoteric thing. One really has to be an expert to appreciate some aspects, and even one mathematician may have trouble appreciating what another mathematician is doing in a different area. It's the sort of thing that's very hard to communicate. And certainly very difficult for people who are not mathematicians-the general public-to appreciate for its artistic values. Nevertheless, it is possible to appreciate that such things exist, to see through simple examples that they have this aesthetic quality. One doesn't need a great deal of mathematical understanding to see that. And yet there is always a great frustration in mathematics that one may find something of tremendous beauty and then try to express this to somebody else who can't really appreciate it. That can be very disappointing in a way, because it's difficult for other people to appreciate fully what these things might mean.

SH: Pure mathematics may be too specialized, but geometric symbols or geometrical beauty and simplicity remain accessible to non-specialists.

RP: True. I have always found it remarkably ironic in a way. When people ask me to explain ideas to the general public, they say, "Well, use a lot of pictures. Use geometry to get the ideas through." Yet students in mathematics sometimes have great difficulty with geometric arguments; they are much happier with the calculations. Very often, appreciating geometry is much harder for professional mathematicians. Of course, some mathematicians can do the geometry, but they're the exception. I certainly found this when I was an undergraduate: very few of us found the geometry easy; most found the geometry much more difficult.

SH: Some inspiration is needed, even to solve an exercise....

RP: It's not so automatic. One really does need to think, I suppose.

SH: With algebra, there are basic techniques, how to start, et cetera... which maybe also distances it from the general public.

RP: Yes, that's true. But people can be very different, too. In mathematics, I've found that some people will react very differently from other people. I tend to think more geometrically. But I find I'm in the minority. There are very few people who find geometry an easy way to think.

SH: Then how is it that ordinary people have no trouble in visualizing a circle, for example, which is really such a very abstract concept?

RP: You mean a real circle, not just an approximate circle that one would draw? To appreciate the notion of an abstract circle, that's a Platonic question. People have this ability to appreciate the abstract thing... that's a good question.

SH: Circle or triangle or straight line-nobody can say they don't exist, but the concepts themselves are so abstract.

RP: Very abstract. So why do ordinary people not find that difficult? Is it the appreciation that it is only an approximation? When one thinks about a circle, the ordinary person may not realize the difficulty that even the boundary of that table is not really a circle.

SH: No, it's not. But in their minds, it's very exact.

RP: But obviously one could think about these things without really knowing what the problems are. It's only a mathematician who'd start worrying what a circle really is. Or a philosopher, perhaps.

SH: Still it seems to me that many people have very abstract concepts.

RP: I sometimes get letters from people on all sorts of strange subjects. Quite recently someone wrote to me claiming that pi--was not a constant. He thought that pi could be a variable, so that it might be a function of time. So I had to write to him and say, look, pi is a mathematical number. It couldn't be anything else; there are all sorts of mathematical formulae for pi, which do not directly refer to the length of the circumference of a circle. This relates to your question about an abstract circle. You see, he thought that each different circle might have a different pi. I had to try to explain to him that even in non-Euclidean geometry, pi itself is one thing. It's not that pi is something other than the value we give it, but whether the ratio of the circumference to the diameter is actually pi or not.

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