[ExI] The "Unreasonable" Effectiveness of Mathematics in the Natural Sciences

[Lee Corbin]

Just finished a very long conversation with a friend
in which I well-articulated (at least I thought so)
the long sought-after answer to Wigner's famous query
about why our universe exhibits math at every turn.

(Apologies to those who have already read less
extensive missives from me about the same idea.)

So what is the real connection between mathematics
and our physical universe, that makes the use of 
mathematics so uncannily effective?

Here is a conceptual development, kind of analogical
to a historical development, but of course without any
real reference to *time*.

In the beginning (or, as you may say, at the conceptual
foundation) we have the physical universe. Then there
are two layers upon it that solve the mystery.

First, observe that the universe is under many peculiar
constraints. For example, throughout our infinite universe
(or at least tremendously vast universe) particles do lie
in certain patterns in the vast regions of almost empty
space. Separated perhaps by many meters from any
other matter, small groups of particles may be found
which have some definite number of constituents. In
particular, I have in mind those patches which contain
a precise number of atoms or molecules.

Let us focus on those groups that consist of precisely
17 particles, or to be specific, atoms. Now these 17
particles, physically separate from adjoining groups,
may exhibit countless possible patterns. In some places
those 17 particles form themselves into various letters
of our English alphabet (very crudely), or may form
certain geometrical figures. But one pattern that you
will *never* see, indeed a pattern that the universe is
incapable of generating, is for the 17 particles to be
formed into two rows with an equal number of particles
in each row. This is a *constraint* on the ways that
the patterns may form up in space.

We have a general name for that particular constraint,
and it is "17 is not an even number". My claim is that
all of mathematics merely consists of our recognizing
certain preexisting constraints upon what is physically
possible in the universe.

So at level one in a conceptual hierarchy is the physical
universe, at level two a set of constraints somehow
imposed on the universe, and at level three our language
to describe those constraints, namely, mathematics.

Quite similar to David Deutsch's rather astonishing
and hardly predictable discussion of Virtual Reality
in "The Fabric of Reality", some of the constraints
exhibited by the universe may simply amount to what
certain machines are able to do. Of course, even
Turing machines exhibit "evenness" when a device
(to be pictured as moving left and right along a tape)
cannot possibly take a certain number of steps and
return to its starting location without taking precisely
an even number of steps. (Again, we describe the
this preexisting constraint on the motion of the device
by using our concepts of "even" and "odd".)

Certainly many other mathematical theorems reduce
to limitations upon what a Turning machine may do,
even if no simpler set of constraints (missing patterns)
can be physically recognized by us at this time.

We may, for another example, write down a number
of axioms for some system (e.g. group theory or 
geometry), and specify rules that allow deductions,
that is, formal rules that allow new statements to
be added to the axioms that are derivations from
those axioms or from earlier such derived statements.
Often it is the case that certain constraints arise here
too, as in the specific instance that no derivations
will be found that allow two separate points to each
bisect a segment (the segment bisecting point is unique). 

To sum up, the answer to Wigner's question concerning
the "unreasonable effectiveness of mathematics", as in
http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html,
is that mathematics cannot be thought of as in any way
distinct or separate from the physical universe, but only
our way of describing the universe's own preexisting
constraints.

Lee