These are two questions I have been asked lately and I
feel they are sort of related to each other, so I will attempt to address them
in a single post. I didn’t really have a
well thought out response when one of these discussions was started, but I feel
I have formulated a good idea of what’s actually going on. My professor says my view of math is a lot
like that of G.H. Hardy, so I apologize with anyone that heavily disagrees with
this and sees things in a different light.
I will try my best to explain why I think my view on mathematics is
quite accurate as it interrelates to other subjects and why it is both invention
and discovery at the same time.
First off: Does math need science?
I firmly say no and most mathematicians will heartily
agree with this assertion. Does science
need math? I say yes and again many will
agree. It’s this interplay between the
disciplines that generates some confusion on the nature of math. Non-mathematicians, for the most part, seem
to view math as an extremely powerful tool for explaining observed phenomena in
our world. I completely agree with this
view and I honestly can’t see things being done any other way. However, does this necessity work the other way? Certainly, mathematics’ level of abstraction
does not require our world, our universe, or anything else physical in nature
to define its world.
Most of the tactics used in this discussion deals with
the nature of how science proves things versus how mathematics proves things,
but I will go with a different tact, because I think it is far more
interesting.
In science, scientists spend time doing experiments and
gathering data. Sometimes this data will
lead scientists to generalize the data with a mathematical equation. We see this in physics all the time with
things like F = ma. But does F=ma have
anything to say about any mathematical truths?
No, not really. Please note, I am
not saying that scientists can’t prove things in mathematics, this is patently
untrue. Physicists have developed
wonderful things in mathematics, but they were not doing physics, they were
doing math and these are different techniques.
Here’s the real crux of the problem, scientists do not
get to choose their world. Reality is
what it is whether they want it to be that or not. They do experiments to see what they can find
out about the universe. Sometimes this
requires the creation of a mathematical model that helps explain a
phenomenon. This is surely invention on
the part of the scientists and they are using rules of mathematics as a tool to
delve deeper into this creation they have made.
However, any truths they claim to find about reality by doing math will obligate
them to do an experiment to verify it.
If the experiment does not confirm the mathematics, then their
mathematical model is simply not the right one.
That is all, so they move on and choose a better one. This is how science tends to trudge along.
Second: Is mathematics invention or discovery?
It is both!
Mathematics is entirely different. In mathematics we choose our world and we
have no choice over the results.
Mathematics always begins with laying out definitions; this is the piece
that is pure creation. However, we then
see what logical implications the definitions behoove us to make; this is why
it feels like discovery when a new Theorem is “discovered”.
For example, if we set up an ordered algebraic field for
the Real Numbers, then we further come up with definitions for surpemums and
infimums… the Mean Value Theorem will result!
As long as all of the conditions exist this theorem is true and it isn’t
just true it is an absolute truth. You
are guaranteed this as a result of the definitions.
Science clearly never works this way. They always have “here is the universe, here
is the earth in the universe” etc, these are the givens they must work with and now
it is up to them to construct mathematics to help explain what they
observe. The other wrench in this is
that there is also some percent error, so things are never absolutely
true. They may be very close, with
99.995% certainty, but there is always a degree of uncertainty. Furthermore generally accepted ideas in
science can be overturned if some other fact presents itself, or some
experiment shows it to be false.
Mathematics does not need to worry about this. Once a theorem is proven there is nothing
that can ever come along to show the theorem is false.
I am not trying to downplay science. Science is very practical and there really is
no other realistic way to study the world around us. Mathematics is a very powerful concept, but
you pay a price for that power. You are
no longer bound by reality. If
mathematics were bound by reality then we would never be able to talk about
things like right triangles. There would
always be some atom partially out of place throwing off that proper 90 degree
requirement. Mathematics talks about a
purely mental idealization of things.
This idealization lets us talk about things that may never exist in
reality and that’s all good fun, but it really is just head games.
Hopefully this has made sense and if it hasn’t, I may
spend time revising it later.
