The Effectiveness of Mathematics in Describing the Universe

NOVA just had a show on this question.  But besides the original question is that of what type of answer are you looking for, or what answer will you accept?  Otherwise the question is largely undefined.

There are several simple answers, if you are looking for a simple one.  I will try to discover a few.  If you want a complex or subtle answer, better ask another.

First of all, we do not yet completely know everything about the universe in a direct way.  I also think that we do not yet know the subtleties embedded in the universe.  When we think of scientists or philosophers trying to answer questions historically, we realize how the lack of knowledge rather limited them in answering questions about science and mathematics.  We are undoubtedly in the same situation, and will remain so, perhaps forever.

On the other hand, we also do not fully know or yet understand the completeness of mathematics, or all the subtleties underlying it.  Probably it is endless.  Is the Universe also endless in its subtleties and complexities or in its simplifying principles?

There are some very simple answers, which rob the question of its presumed depth.  The one I like best is the most immediate.  The universe must act according to very simple and universal rules.  Otherwise it would be very complicated, depending on position, time and all sorts of qualities.  It is of course up to physicists and other scientists, using mathematics, to discover what the basic underlying laws are.  The laws often seem very complicated, but once they are learned, explored and taught, they are masterable and in a sense simple.  An in fact, the universe is complex, but experiments and observations can be designed to minimize so called higher order corrections, and focus on one or a few properties that lead to a simple law.  As when Newton reasoned that gravity that worked on falling apples also controlled the movement of the moon around the earth, and the planets around the sun.  So if you can isolate simple circumstances, simple laws can be found.  Then simple math can be found to describe these universal laws.  By simple math I mean a few universal equations, not necessarily simple in the sense of taking time to learn or calculate.  However, physics must have enough complexity to account for both protons and neutrons, so that atomic nuclei can be built up, and chemistry can exist to make our world and life in it.  In the standard model, that requires a substructure of protons and neutrons composed of up and down quarks with three colors for each.

A deeper answer to the connection of math and nature, is the search for uniqueness in the Universe, applying a test of uniqueness.  When I started in particle physics, I worked on the ideas of Goeff Chew, that strong interaction theory would be found to be self generating by consistency.  At the start of string theory, there was a search for the unique or uniquely consistent string theory, which would explain why the universe is the way that it is.  This replaces the earlier ideas that the unique theory would somehow be the most beautiful one.  While string theory has failed this test, it will undoubtedly arise again.  In contrast to this are theories that the initial conditions determine the universe.  A more popular variant is that there is a statistical ensemble of universes, and we are one of the very rare ones, where the cosmological constant can have an incredibly small value, so that the universe lives a long enough time for us to arise and contemplate it.

As far as the adaptability of mathematics to describe the universe, it is not well known publicly, but mathematics is not a unique and absolute system.  You can build an arbitrary number of mathematical systems, depending on the initial axioms that you choose.  This was proved by Kurt Gödel.  So mathematics can be a lot more flexible in adapting to describe however the universe needs to be described.  This occurred with the discovery of curved space-time, and the need for Riemannian geometry.  It also occurred with the discovery of quantum mechanics and the math needed to describe it.  It will probably be needed to solve the description of gravity with quantum mechanics.

Did I miss the sense in which you were seeking an answer to the unreasonable agreement of mathematics with natural law?  Of course I did.  There is still an unknown truer formulation of the laws of the universe, and lots of un-invented mathematics, and a further depth of understanding of current theories.