fundamental mathematical objects—added knowledge of its structure is
of paramount importance, just as added knowledge of the structure of
fundamental objects such as viruses or stars is of paramount importance
in their respective sciences. From the real-world standpoint, physical real-
ity uses both discrete structures (in quantum mechanics) and the con-
tinuum (elsewhere). We have not yet discerned the ultimate nature of
reality—possibly a greater knowledge of the continuum would enable us
to make strides in that direction.
Additionally, computations made using the assumptions of the contin-
uum are often much simpler. If the continuum is abandoned, there are
no circles—just a bunch of disconnected dots equidistant from the center.
One would not walk around a circular pond, traversing a distance of two
times pi times the radius of the pond, but would walk in a sequence of
straight line segments from dot to adjacent dot. The computation of the
length of such a path would be arduous—and would turn out to equal
2 r to an impressive number of decimal places. The circle is a continuum
idealization that does not exist in the real world—but the practical value
of the circle and the simplifying computations it entails are far too valua-
ble to be summarily abandoned.
Finally, the quest for models that satisfy different systems of axioms of-
ten has surprising consequences for our understanding of the real world.
Attempts to derive models in which Euclid’s parallel postulate was not
satisfied led to the development of hyperbolic geometry, which was incor-
porated in Einstein’s theory of relativity, the most accurate theory we have
on the large-scale structure and behavior of the Universe. As Nikolai
Ivanovich Lobachevsky put it, “There is no branch of mathematics, how-
ever abstract, which may not some day be applied to phenomena of the real
world.”^11
NOTES
- This quote is from Plato’s Theaetetus, section 152a. More on Protagoras can be
found at http://en.wikipedia.org/ wiki/ Protagoras. Even though Wikipedia is
user-edited, my experience has been that it’s accurate when dealing with math-
ematics, physics, and the histories thereof—possibly because no one has any dog
in the race, possibly because there isn’t even a dog race with regard to matters
such as this. - This quote is so famous that most sources just reference Einstein! The vast ma-
jority of its occurrences seem to be from math teachers who, like myself, wish to
put their students at ease. Many people think that Einstein was a mathematician
rather than a physicist, but his only mathematical contribution of which I am
aware is the “Einstein summation convention,” which is essentially a notation—
like inventing the plus sign to denote addition.
26 How Math Explains the World