Recently rereading David Deutsch’s Beginning of Infinity and the discussion of Kenneth Arrow’s Impossibility Theorem in the chapter “Choices” got me thinking of some implications for AI (subject of a future post) and also brought to mind an older work from my undergrad days involving another of the Impossibility Theorems involving the characteristics of simply-defined systems.
In the theoretical branch of my Computer Science studies we covered Gödel’s incompleteness theorems and I was blown away. The idea of proving the impossibility of what seemed an intuitively possible goal was a revelation (not saying much here as it was a mind-blowing revelation to the whole field of metamathematics back when Gödel published his work almost 100 years ago).
To complete the equivalent of a minor in Philosophy I jumped at the opportunity to involve my two majors, Physics and Computer Science, to bring together all three of my serious academic interests in an exploration titled “Gödel, Science, and Reality”.
My idea was to take a leap to a hypothetical far future state of our knowledge of Physics where it is complete, and to further speculate what it might mean to apply Gödel’s theorems to the mathematical system representing these laws. I won’t rehash everything from the paper (fully embedded at the end below), but my abstract was:
Gödel’s incompleteness theorem applied to mathematics proves properties on a metamathematical level. In this paper an argument is given for applying Gödel’s theorem to physics to prove hidden properties of reality on a metaphysical level. It is argued that Gödel’s theorem applied to physics implies the existence of portions of reality outside of observable reality, although the ontological status of such portions of reality is not necessarily reconciled by such an endeavor.
While I disagree with my younger self on the overall conclusion (more below), I made some points in the Conclusion section that I think are pretty good:
It would […] be interesting to track the development of completeness and consistency as moving forces throughout the development of science. As noted earlier, it appears that science worries first about completeness and then about consistency. This is probably due to the fact that inconsistency can be resolved by making an educated choice, while incompleteness leaves one more in the dark. That is, inconsistency can be resolved by consciously choosing one result over another while incompleteness can only be addressed by mysticism. As an example, when solving for the properties of a physical system a scientist chooses whether to use the laws of quantum mechanics or the laws of classical mechanics. Under most limits the inconsistency between the two can be resolved by choosing to work with one instead of the other.
In the paper I made a hand-waving argued that if (stress on “if”) the laws of Physics ever achieved completeness and consistency over all observed phenomenon that due to the mathematical foundation for those laws and application of Gödel’s theorem it would imply the scientific laws would also “prove” facts outside of our observation.
What I said then was:
Gödels theorem demands that there are portions of this science system which are unobservable and beyond what the idealist may refer to as reality. In some way Gödel’s theorem here lets us know that there is more to the reality than what is observable.
But with more solid epistemology under my belt rooted in the thinking of Popper and Deutsch I disagree with the conclusion of my younger self.
I suspect that inconsistency in our body of Physics knowledge, taken as a whole, may be inescapable. Gödel’s theorems may not apply to Physics taken as a whole, but if they did it could be how they manifest.
As we establish good explanations for our theories we strive for completeness and consistency, at least within a bounded domain. For example, the Theory of General Relativity tackles the domain of gravity, explaining its effects as created by the curvature of a continuous space-time. Quantum Mechanics tackles the domain of quantized behavior (such as energy levels) at the sub-atomic level. Both of these strive for completeness and consistency within a domain, although they are inconsistent with each other (a condition which the field of “Quantum Gravity” has been trying to reconcile for decades).
Emergent phenomena are perhaps another source of inconsistency in our knowledge. For example, the second law of Thermodynamics implies an inconsistency in time-reversibility compared to the laws of Classical Mechanics involved in calculating multi-body interactions. While the laws of Classical Mechanics are time-reversible, the second law of Thermodynamics (entropy) is not.
The original paper of my younger self was hand-wavy and this blog post remains so, but at least for me, if anything Gödel reinforces for me that we’re at the beginning of an infinity of progress in our growth of knowledge of how everything works; searching for completeness and squeezing out inconsistency, but never fully getting there.
Full view of my younger self trying to figure things out here:
Richard Shea's Blog 