Wednesday, April 08, 2009

On Mathematics as a Universal Language

I occasionally have ideas for science fiction stories, some of which I eventually decide are not worth the effort of developing, some of which continue to percolate, and some of which would be neat but probably are beyond my abilities to do. An example that falls into the third category. I once had the idea of a story in which we suddenly receive a rather extensive communication from an extraordinarily distant alien culture. It looks fairly mathematical, so humanity's mathematicians get to work trying to figure out what is going on it. And never, ever do, because while they can identify mathematical patterns in bits and pieces of the communication, they never have any way of testing whether those are the actually intended mathematical patterns, and never manage to fit any of the bits and pieces together into a whole. Doing it properly would require rather more advanced mathematics than I have: you'd want to show mathematicians throwing their best and brightest ideas at something (perhaps talking to reporters about it, in order to mediate the advanced mathematics to the reader). A variant of the idea would be for human beings to develop over several generations an extraordinarily advanced and complicated form of mathematics that finally makes sense of the communication and later to learn that it was completely and entirely wrong as an interpretation; the mathematics, while advanced, was simpler and very different, and the aliens had at one point just made a silly addition mistake or typo.

People often say that mathematics is a universal language. There's an important truth there, but we should always keep in mind that the phrase is a play on an ambiguity. Obviously, the notations in which we write mathematics are not universal, and certainly the terms attached to the notations, and in terms of which we explain them, are not universal. Moreover, things that seem natural to one culture (through long practice) may seem to be merely complicated work-arounds to others. If you discovered a Sung text on the celestial element algebra and knew no Chinese, figuring out that it is an algebra would be very difficult. Even if you knew Chinese it might be difficult; the Chinese certainly knew Chinese, and by the sixteenth century they seem no longer to have any idea what is going on in the celestial element texts, although they continued to copy them. And it is much more difficult than one might think to work out all the quirks of a notation even if you have a fair idea about what it means.

The play on words involved in saying that mathematics is a universal language lies in this: mathematics is universal, and mathematics is a language; but it is not universal as a language, nor is it a language insofar as it is universal. The underlying principles, the things discussed, are universal; quantities and structures of various kinds and the logic, so to speak, of how they can relate to each other. But we human beings do not have immediate intellectual access to these things, so we build up to an intellectual understanding of them by efforts of the imagination -- cognitive processes leading to expression in talking, writing, and drawing. Even mathematical geniuses do this; they just need less effort to get the same result. And we are very dependent on this imaginative medium. It is difficult, if you have not already gotten used to the idea, to think of zero as a number, or to recognize that you can have a geometry of things you can't draw in the sand or sculpt in clay, or to do algebra entirely geometrically, or to do geometry entirely algebraically. Because mathematics is universal, if you come to a difficult mathematical text for the first time, you can in principle rediscover all the underlying principles by yourself, and then use that understanding to help work out what the text means. But, depending on what it is, that may require all your life, or, indeed, generations of lives, centuries or millenia of lives. To shorten that, someone has to explain it to you in a not-so-universal language.

No comments:

Post a Comment

Please understand that this weblog runs on a third-party comment system, not on Blogger's comment system. If you have come by way of a mobile device and can see this message, you may have landed on the Blogger comment page, or the third party commenting system has not yet completely loaded; your comments will only be shown on this page and not on the page most people will see, and it is much more likely that your comment will be missed.