NOTES ON A PRIORI PHYSICS

Ronald E. Merrill

A Possible Solution: The Mathematical Necessity of the Universe; or, Back to Descartes

Let's set aside what we know about physics and imagine looking at physics from an a priori position. We might make a plausibility argument (and I would stress that I claim nothing more than plausibility) as follows.

We first accept "existence exists." There is a universe, and it has some sort of "stuff" (call it "matter") in it. Now it is plausible (again I emphasize that this does not remotely approach a proof) that conservation of matter (mass-energy, if you like) should be true. For instance, if matter could be destroyed, it would be possible in principle for all of it to disappear, leaving us with no existents, ie, no universe.

If we go this far, pure mathematics can answer another question for us. Specifically, is matter continuous or made up out of particles? If matter is continuous, conservation of matter cannot hold. There is a theorem in topology which states that if a sphere is made of continuous matter, it can be sliced up and reassembled to form a smaller sphere. So we may conclude that matter is made up of particles. Now, to avoid an infinite regress (and, in the limit, a return to continuous matter) we must assume that there are some smallest or most elementary particles.

What could be the properties of elementary particles? They cannot, by their nature, have any internal structure, shape, size, etc. The only permissible properties are characteristics which affect their influence on (and by) other particles. Here "influence" may be identified with "force" as the term is used in physics. These properties must specify quantitatively how particles will mutually influence one another; that is, the properties must be representable by numbers. But there is something just a bit disquieting about assigning several numbers to a certain particle (or type of particle). Suppose the numbers are 3, 0, 83.748, 1/2, and 3. These numbers completely characterize the particle; they carry all the information there is to know about it. But the numbers (which of course are, or can be made, dimensionless) are fundamentally indistinguishable. Does it not seem plausible to require that the numbers which specify the properties of an elementary particle ought to fall into separate classes, with only one in each class?

Suppose we require this. There are, it turns out, only four kinds of numbers (more precisely, there are only four division rings): real numbers, complex numbers, quaternions, and Cayley numbers. Again we find that an argument from what seems plausible leads, through the medium of mathematics, to congruence with empirical evidence. There are only four fundamental forces.

We might identify the real number with mass and associate it with the gravitational force. Electromagnetic forces can be efficiently described in terms of quaternions, and at one time they were; vector calculus turned out to offer greater computational convenience. I suspect the weak nuclear force can be associated with the complex number (it's rather like electromagnetic force without range). Cayley numbers have the interesting property that multiplication is not associative. That is, (A X B) X C does not necessarily equaly A X (B X C). This might have something to do with the weird behavior of quarks under the strong nuclear force.

We should also note that there are only four conserved quantities: mass-energy, electrical charge, momentum, and angular momentum. This follows from the proof that "black holes have no hair". If any other quantity (eg, baryon number) were conserved, it would have to be a property of black holes; otherwise we could destroy it by dropping particles having it into a black hole.

These four conserved quantities may be identified with properties of space. It seems rather reasonable a priori to demand that space have the following properties:

extension demands charge (repulsive forces)

isotropic momentum

content angular momentum (Mach's Principle)

non-continuous mass-energy

I have started out this essay, then, with an example of the way in which certain laws of physics might be developed a priori. The idea is that given certain axioms (such as "existence exists"), plus the laws of logic, plus mathematics, physics might be completely determined. Stated another way:

There is only one possible set of laws of physics which is not self-contradictory.

What of the elementary constants such as the speed of light, Planck's Constant, and so on? Note that these can be replaced by a new set of dimensionless constants, which might just as plausibly fall out of mathematical constraints in the way that pi and e do.

This conjecture could in principle be disproved easily. One need simply design a (non-trivial) physics which is demonstrably self-consistent. So far as I know, this has never been done. Newtonian physics, for instance, cannot incorporate electromagnetic (let alone nuclear) phenomena. Furthermore, it leads to discontinuities and inconsistencies even when confined to gravitational forces.

Another Possible Solution: Merrill's Conjecture; or, the Church of God the Engineer

Suppose for a moment that it were possible to travel, or at least communicate, essentially instantaneously. One consequence would be, ultimately, a sort of homogenization of intelligence in the universe. (We see this happening on earth, as rapid transport and communication makes Europe and East Asia, once exotic locales, look more and more like American cities.) The universe would not be very interesting. On the other hand, if something like the speed of light limitation made interstellar communication absolutely impossible, intelligent species would never encounter one another. This situation too would be less interesting than the one which actually exists, in which such encounters should be limited but not impossible.

In practice, even a small change in the speed of light, or in other elementary constants, would lead to a universe in which intelligence, or even life, would be impossible. That situation would certainly be a great deal less interesting than the way things actually are.

I therefore propose the following conjecture:

We live in the most interesting of all possible universes.

That is, the laws and constants of physics are optimized with regard to making the universe interesting.

Now, what do we mean by "interesting"? Would we not say that "interesting" is an inherently subjective evaluation, which is meaningless unless in the context of "interesting to person X"? Not necessarily.

Take an analogy. Music is sound, and a piece of music can be characterized by the power spectrum of the sound. It turns out empirically that music tends to have a "1/f" power spectrum. Sounds or sequences of sounds which have a power spectrum much different from the 1/f form do not sound "musical". We may thus construct an objective, though limited, quantitative measurement of "musicalness". It is quite possible that a piano prelude by Rachmaninoff, and a song by the Beatles, could have the same amount of "musicalness", even though one person strongly prefers the Rachmaninoff and another the Beatles.

Perhaps we can do something similar with the property of being "interesting". Obviously, for instance, a text on elementary chemistry would be rather uninteresting to me, since I long ago mastered the material in it. On the other hand, a beginning student might find it fascinating. The two of us, however, both loving the field of chemistry, might agree on the intrinsic interest of the text. Indeed, a third person, who had zero personal interest in chemistry, might still agree on this intrinsic interest. It is quite possible to say, "That is a very interesting subject, though I am not interested in it myself."

How could we make this quantitative? It is known that we can quantitate complexity, in the manner developed by Kolmogorov. Now, let us measure the complexity of a set of phenomena (say, the phenomena of electromagnetism) in this way. There will be various representations (or "models" or whatever term one wishes to use) that will generate the phenomena in question completely. In the case of electromagnetic phenomena, Maxwell's Equations would constitute a representation. Then we assume that we can find a minimal representation--the representation of lowest complexity which is yet capable of completely describing the phenomena. The ratio of these two complexities may be called the "elegance" of the phenomena; it is the optimization (specifically, the maximization) of the elegance of the representation, holding the phenomena constant.

However, this is not quite satisfactory as a measure of interest. Thus a very simple set of phenomena, which can be representated in an extremely simple form, may not be as interesting as as a more complex, though slightly less elegant, set of phenomena. We might therefore define interest as:

interest = [complexity(phenom)]**2/complexity(rep)

Modelling the Issue of Interest

There is a well-known proof that there are no uninteresting numbers. (If there were, then there would be a smallest uninteresting number, and it would of course be of interest as such.) So we may find it instructive to examine the interest of real numbers.

Any real number may be "represented" (as we have used the term above) by an equation of which it is the solution. The most complex numbers on this basis would have to be the transcendental numbers, which cannot be represented by (are not the solution of) any polynomial equation. The most interesting numbers would be those transcendental numbers which are the solutions of the simplest transcendental equations.

Now it is interesting that we find that at least some of these extremely interesting numbers are also among the most important numbers. Thus pi and e are ubiquitous in mathematics.