Tuesday, November 25, 2014

Simplicity, language-independence and laws

One measure of the simplicity of a proposition is the length of the shortest sentence expressing the proposition. Unfortunately, this measure is badly dependent on the choice of language. Normally, we think of the proposed law of nature

  • F=Gmm'/r2
as simpler than:
  • F=Gmm'/r2.000000000000000000000000000000000000001,
but if my language has a name "H" for the number in the exponent, then the second law is as brief as the first:
  • F=Gmm'/rH.

One common move is to employ theorems to the effect that given some assumptions, measures of simplicity using different languages are going to be asymptotically equivalent. These theorems look roughly like this: if cL is the measure of complexity with respect to language L, then cL(pn)/cM(pn) converges to 1 whenever pn is a sequence of propositions (or bit-strings or situations) such that either the numerator or the denominator goes to infinity. I.e., for sufficiently complex propositions, it doesn't matter which language we choose.

Unfortunately, one of the places we want to engage in simplicity reasoning in is with respect to choosing between different candidates for laws of nature. But it may very well turn out that the fundamental laws of physics—and maybe even a number of non-fundamental laws—are sufficiently simple that theorems about asymptotic behavior of complexity measures are of no help at all, since these theorems only tell us that for sufficiently complex cases the choice of language doesn't matter.


Heath White said...

I don't know enough theoretical stuff to properly put this into words, but...

I find it EXTREMELY difficult to believe that the simplicity of our scientific laws is a feature of our language, rather than of the laws. (Here "the laws" is supposed to pick out, er, the *thing* that is the laws. I realize that's problematic.)

The reason that g=Gmm'/d^2 is simpler than g=Gmm'/d^2.0001 is related to the fact that 2 is a "simpler" (more primitive in construction?) number than 2.0001. Something like: you can have integers without rationals, but no rationals without integers.

Also, I am attached to the teleological argument that appeals to the intelligibility of the world, and I would be reluctant to give that up, or to make intelligibility more a feature of our language than of reality. I realize, also, that what I am attached to is not really an argument.

Alexander R Pruss said...

You may be right, but it may still be the case that the way to *measure* the simplicity of a law is by looking at syntactic features of expressions of them.

Maybe laws are propositions and propositions are structured Platonic entities. It could be that the right language is one whose structure is isomorphic with the structure of the Platonic entities. (But then one could define simplicity directly in terms of the structure of the Platonic entities.)

Anonymous said...

There are more interesting and relevant measures of complexity:
cf. http://tomkow.typepad.com/tomkowcom/2013/09/the-computational-theory-of-natural-laws.html

= MJA said...

Simplicity and the Universe

Einstein reduce the Universe to his famous equation e = mc2 then sadly got lost in complexity going back the other way. Had he understood the flaw of measure he would have found it was only the speed of light that stood in his way; and then reduced his own equation even further and found the solution for unification he died searching for.

I finished his work and found truth is much more simple than thought. =