Mathematics by Timothy Gowers

Author:Timothy Gowers
Language: eng
Format: epub
ISBN: 9781402782169
Publisher: Sterling
Published: 2010-12-07T05:00:00+00:00

High-dimensional geometry is yet another example of a concept that is best understood from an abstract point of view. Rather than worrying about the existence, or otherwise, of twenty-six-dimensional space, let us think about its properties. You might wonder how it is possible to consider the properties of something without establishing first that it exists, but this worry is easily disposed of. If you leave out the words “of something,” then the question becomes, How can one consider a set of properties without first establishing that there is something that has those properties? But this is not difficult at all. For example, one can speculate about the character a female president of the United States would be likely to have, even without any guarantee that there will ever be one.

What sort of properties might we expect of twenty-six dimensional space? The most obvious one, the property that would make it twenty-six dimensional, is that it should take twenty-six numbers to specify a point, just as it takes two numbers in two dimensions and three in three. Another is that if you take a twenty-six dimensional shape and expand it by a factor of two in every direction, then its “volume,” assuming we can make sense of it, should multiply by 226. And so on.

Such speculations would not be very interesting if it turned out that there was something logically inconsistent about the very notion of twenty-six dimensional space. To reassure ourselves about this, we would after all like to show that it exists—which it obviously couldn’t if it involved an inconsistency—but in a mathematical rather than a physical sense. What this means is that we need to define an appropriate model. It may not necessarily be a model of anything, but if it has all the properties we expect, it will show that those properties are consistent. As so often happens, though, the model we shall define turns out to be very useful.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.