The Basic Nature of Formulas
Recently
I was asked about whether I thought individual formulas “existed” in an
abstract sense, and as is typical I answered first and then thought about
it. Fortunately I still agree with my
answer, but as I thought more I thought I should clarify my thoughts (you can
be the judge of whether this is clear or totally confusing). My original answer was that formulas do not
exist in an abstract sense, as if just waiting for some random person to happen
upon them. However, I’d like to clarify
that at the time I was thinking only of physical (scientific) formulas and not
of mathematical ones, and my working definition of a formula is: a mathematical
description of reality that gives accurate results. Obviously this doesn’t include mathematical
formulas, which don’t describe reality but instead are a statement of some
truth.
The
reasons why I decided this was that formulas are created, and not found. Anyone with a shovel (and enough luck) can
find an archeological site, but not everyone with a calculator can find a
physical formula. No amount of work with
a calculator will show that F=d/dt(mv). Physical formulas are created by people who
take variables that experimentation (actual or thought) has shown should be
related and arranging them into some sort of plausible mathematical statement
that seems to give accurate results.
Unfortunately, they are not just discovered out of thin air, and they
are subject to constant revision, such as the lovely gammas that relativity
throws everywhere. Since our formulas
are often only simplifications (such as of the relativistic versions), it’s
difficult to support the idea that our formulas “exist” abstractly when they’re
wrong.
However,
this doesn’t preclude the idea that “ideal” true formulas exist somewhere out
there, waiting for us to have a complete description of force, momentum,
etc. Of course, all of this depends on
whether you believe that force, etc, exist abstractly or are simply a slight of
hand used by physicists to describe reality.
My answer for this is sort of a variable-by-variable analysis that I’m
not going to describe right now because you’re probably already incredibly
bored. I’m actually not so concerned
about whether these perfect formulas exist or not as I believe that even if
they do exist, humanity will never attain them.
I don’t think we can ever completely describe reality in a set of
mathematical relationships; instead I believe that there will always be another
level for us to discover. Once,
physicists thought that all that was left was to simply measure a few constants
to a precise level and they would know everything; and then relativity and
quantum mechanics destroyed those thoughts.
This pattern has occurred throughout the history of physics; as soon as
people thought it was completely figured out, something new is discovered about
reality, or as is most often the case, our understanding of reality is expanded
based on a new explanation. Rarely does
something just come from nowhere; typically physicists are working on what
seems to be a simple problem that ends up becoming more complicated. For example, you can indeed see the effects
of relativity in everyday life. Go put a
magnet on the fridge. Ta-da- Einstein
discovered relativity while trying to explain magnetism, and now it forms our
basis of understanding as to what creates the magnetic fields. Sorry about going off onto that tangent- I’ll
return to my point now. As often as
humans think that we have achieved the perfect forms of the formulas, we
discover that there’s another level that we’re missing. I see no reason for this trend to stop, and
firmly believe that there will always be another level of physics to discover
(that may or may not require additional factors in our formulas) and therefore
I don’t believe that we’ll ever reach an “complete” version of a formula. I honestly don’t know whether God has decreed
a complete set of formulas for reality.
My gut belief is that God, who created this universe, knows of all the
subtleties of reality that would create factors in the formulas and therefore
He knows “complete” formulas for force, etc, but these formulas are
unattainable for humanity.
Gee,
you’re still reading? I haven’t bored
you to sleep yet? Great, because now
comes part 2- mathematical formulas.
Since I’ve concluded that our physics formulas don’t exist abstractly
and that we can’t obtain perfect ones if they do indeed exist, does that same
hold for mathematical formulas? Well,
first of all, there are inherent differences between physical formulas, which
describe reality, and mathematical formulas, which are a statement of
truth. A mathematical formula always
arises from some incredibly long and often difficult proof, which should be
completely undeniable. This, I think, forms
an incredible chasm between physical and mathematical formulas. While physical formulas cannot ever be proven
completely true (or to be completely free from factors that are irrelevant in
given situations), mathematical formulas by definition are proven absolutely
true through a nice, rigorous proof. So,
then, the question of whether mathematical formulas exist abstractly reduces to
the question of whether truth exists abstractly. I believe that God does indeed will some
things to be true, and so truth does exist.
The question is, does this include mathematical
formulas? I do believe that mathematical
formulas are true- Green’s Theorem, etc, will always hold, because they do not
depend on humanity to take individual variables and contort them into a
relationship that is then tested- and therefore do exist in a stronger sense
than physical formulas. However, I’m not
sure how strong of a claim I want to make.
It could be that they exist based on definitions we have created and the
application of logic, which is true because God wills the laws of logic to be
true. Or, it could be that our
mathematical definitions do exist, and God actually directly wills for the
mathematical laws to be true. I’m not
sure, but I do think mathematical formulas have a stronger claim to existence
than physical laws.