Comparison of Physics and Maths



Comparison of Physics and Maths

There have recently been some good programs on BBC TV about Physics and
Maths, this led me to speculate on the relationship between Physics and
Maths, I would be interested to get any feedback about whether I'm on the
right track?

(BBC science output is usually dumbed down rubbish, apart from Sky at Night,
but the programs 'Dangerous Knowledge' and 'Atom' have been very good).

We usually understand Physics as the study of the structure of the world and
the way that things behave and interact in it. Mathematics is a more
abstract system which is studied for its own sake but is also needed to
encode our physics laws.

Often we can think of mathematics as a toolbox, where we have a choice tools
for a given job, for example if we want to combine rotations in 3
dimensions we have a choice of using quaternion algebra or matrix algebra.
Both of which will give the required result.

I get the impression that modern physics is blurring this distinction in
that Physics theories tend to be purely mathematical these days, even the
inventor/discoverer does not understand them intuitively and some theories
like Noether's theorem seem to bridge Maths and Physics.

However maths and physics have a different philosophy; a different was of
assessing theories. For a mathematical system to be accepted, it only has
to be self-consistent, it does not have to apply to any practical problem.
For a physics theory to be accepted it needs to represent the real world,
it doesn't matter how beautiful the theory is, if it does not give the
correct results, its no good. So, for example, a photon may be represented
as a wave in some circumstances and a particle in other circumstances but
provided we get the correct results this is more important than
consistency.

Its ironic then, given the importance of constancy to mathematics, that many
mathematical subjects have contradictions at their heart:

* Calculus is based around the idea of infinity, a concept that is
inherently contradictory (Georg Cantor and Kurt Gödel both tried, without
success to find a way to eliminate these contradictions).
* Logic has shown that any system has things that are un provable and we
can't know what they are (Kurt Gödel and Alan Turing).

In Physics, things may appear to be inconsistent due to the limits of our
knowledge and intuition and the apparent strangeness of quantum theory but
there aren't fundamental inconsistencies at the heart of the subject. For
example, physics does not have infinities, either infinitely large or
infinitesimally small. Dimensions may be very large but finite between the
limits of the big bang and the big crunch. At the very small scale, things
can't get infinitesimally small, at the scale of atoms things start to
get 'fuzzy' so we don't have an infinite amount of structure.

But what about calculus? This was invented to solve physical problems but it
is based on the concept of infinity and therefore potentially inconsistent.
For instance, velocity is the limit of distance divided by time as these
quantities become infinitesimally small, this will give the precise value
of the velocity at a given point. However the Heisenberg uncertainty
principle says that there is a limit to what we can know about the velocity
and position of a particle so again we are saved from the need to take the
final step of making the quantities infinitely small.

I'm not saying calculus isn't useful in physics, its a useful fiction, but
we don't have to make the quantities infinitely small, just very, very
small.

So where else does infinity occur in physics? What about quantum
entanglement, does this allow particles to communicate instantaneously? Are
we protected from the consequence of this at the large scale by quantum
decoherence?

more on this page:
http://www.euclideanspace.com/physics/principles/physicsAndMaths/

Martin
.