polynomial. Division, however, is not an allowed operation—just as some
integers divided by other integers are not integers (such as 5 divided by 3),
some polynomials divided by other polynomials are not polynomials.
The study of algebras proceeds by adding other hypotheses. Algebras
beget Banach algebras, which beget commutative Banach algebras, which
beget commutative semi-simple Banach algebras—each additional adjec-
tive representing an additional hypothesis (or hypotheses). Physics doesn’t
seem to follow this scheme—the axioms of a theory are constantly sub-
ject to review. In fact, the Standard Model is not so much the deductions
as it is the Model itself—the deductions available from the hypotheses are
generally used not as a way to build a better refrigerator, but as a check
upon the validity of the Model.
The Limitations of Physics
It is generally within the last century that physics has come to grips with
its own limitations. Although the Standard Model talks about particles
and forces, one of the more modern ideas in physics is that information is
just as much of a fundamental concept. In particular, much of what we
have discovered concerning the limitations of physics can be classified in
terms of information.
Some of these limitations occur because the information we need is
simply not accessible, if indeed it exists at all. We cannot know what hap-
pened before the big bang—if indeed anything did—because informa-
tion travels no faster than the speed of light. Nor can we know what lies
over the hill—if there is a portion of the universe that is farther away
from us in light-years than the time since the big bang, and if that portion
is receding from us faster than the speed of light, no information from
this portion will ever reach us.
Some limitations are imposed because there is a limit to the accuracy of
the information concerning it. Heisenberg’s famous uncertainty princi-
ple tells us that the more accurately we are able to ascertain the position of
a particle, the less accurately we can know its momentum (or, as is more
commonly thought, its velocity). The consequences of the uncertainty
principle and other aspects of quantum mechanics, which will comprise
a significant portion of the next chapter, are among the most eye-opening
and counterintuitive results in the history of human knowledge. This
limitation also hampers our ability to predict—negating Laplace’s famous
statement concerning omniscience. We could say that the universe pre-
vents us from knowing how things will be by concealing from us how
things are.
Reality Checks 37