Theories in mathematics never do battle in this fashion, and they are
never resolved on the basis of statistical evidence. As with a great prob-
lem such as the truth or falsehood of the continuum hypothesis, the
resolution of the problem adds something new to mathematics. It is true
that theories fall in and out of favor with the mathematical community,
and it is also true that theories are sometimes supplanted by more all-
encompassing theories. If there are competing explanations for phe-
nomena in the real world, mathematics may provide some of the tools
needed to resolve the dispute, but without experiments and measure-
ment these tools are essentially useless.
The last chapter in this section concerns which mathematical model
best describes the small-scale structure of our universe—discrete struc-
tures or the continuum. Both theories, from a mathematical standpoint,
are equally valid—but when it came to description of the universe, there
could be only one winner.
NOTES
- See http:// en .wikipedia .org/ wiki/ Auguste _Comte. As I have mentioned, Wiki-
pedia biographies are generally reliable, and often very well documented. - See http:// en .wikipedia .org/ wiki/ Simon _Newcomb.
- See http:// sciencepolicy .colorado .edu/ zine/ archives/ 31/ editorial .html. A quick
Google search also f inds that this quote is attributed to Mark Twain, who said lots
of very clever things, and Yogi Berra, who said lots of things like this, and as a
result gets a lot of credit for things like this which he may, or may not, have said. - According to Seligman, the precise problem was to determine for which values
of n there exists a bilinear map (the multiplication) of �n�n��n such that
ab0 if and only if either a 0 or b 0. If you’re not familiar with the notation,
�n is the set of all n-dimensional vectors whose components are real num-
bers. Bilinear maps are generalizations of the distributive law in both varia-
bles—(a b)cacbc and a(b c)abac. Additionally, because a and b are
vectors, a bilinear map must satisfy (ra)br(ab) and a(rb)r(ab) for any real
number r. - This is a terrific opportunity to plug two immensely enjoyable best sellers by
Brian Greene, The Elegant Universe (W. W. Norton 1999) and The Fabric of the
Cosmos (Alfred A. Knopf, 2004). Despite what reviewers may say, these wonder-
ful books are tough sledding—deep ideas never admit easy explanations, and
both string theory and loop quantum gravity are incredibly deep ideas. Nonethe-
less, Greene does an excellent job with string theory in the first book, but since
he is a believer in string theory, loop quantum gravity is given relatively short
shrift. To be fair, loop quantum gravity is unquestionably a minority position in
the physics community—but the right of a minority to become a majority is no-
where more religiously observed than in physics. - Topology is the study of the properties of geometric figures or solids that are not
changed by deformations such as stretching or bending. The classic example is
Reality Checks 39