Re: randomness in math and physics. Putting aside ignorance-randomness and pseudo-randomness, as well as the issue of whether "randomness" given its multiple employments is all that coherent a concept, we're still left with the specter of irreducible randomness. Mathematically, per the algorithmic information approach, that's represented as numerical information which can't be algorithmically compressed, full-stop, although in the view of many in the field it can be expressed as a Chaitin Omega in which form it has even been "computed" by means of some really smart hacks to 64 digits. Anyway, a concern that arises from your essay, at least for me, is whether that stuff is actually in the same category as the irreducible physical randomness held as an article of faith by many physicists -- represented for example by radioactive decay and the informational output of beam-splitters.

(I know for a fact that Greg Chaitin and Anton Zeilinger perceive an identity. They worked together formulating the Quantum Omega Number. They also share enemies.)

Finally, a question: in your opinion is there any possibility that either or both of the two irreducible randomnesses -- the one of physics and that of mathematics and AIT -- might represent (1.) actual features of nature, if not indeed (2.) an identical feature of nature?

    Good question. Those ideas of randomness seem completely different to me. Yes, both are unpredictable in some sense. But no one is going to claim that radioactive decay follows the Chaitin Omega.

    The math concept depends on infinitely many decimals. Physical observations only have finitely many decimals. How can one be the other?

    We seem to have the free will to make random choices. Others would dispute that.

    The Scarani paper you link to in your refs mentions Antoine Suarez. The two have done a lot of physics work together over the years and remain affiliated with Nicolas Gisin's hugely prestigious Geneva group. They're respected physicists and Suarez is undoubtedly the go-to guy on Quantum Free Will. Watch out for Catholic theology under the asphalt but once warned you can steer around it unless you want the full-on experience.

    .0000001000000100000110001000011010001111110010111011101000010000 are 64 digits of an Omega, actually has Chaitin's imprimatur. Why couldn't the zeros and ones be the two possible directions taken when 64 lined-up photons encounter the interferometer? Some claim it's possible. The math is complicated and controversial.

    I'm not a Platonist, a Tegmarkian. But even empirically, Aristotley, Froglike, just maybe ...

    16 days later

    Dear Roger Schlafly

    Your clear and nice essay came to me in the right moment to know in detail references, which are contrary to my way of thinking. When I am reading your section of Free Will, I can accurately see the trends of thinking in this area. However, my model is not yet indirectly commented in your essay. Namely, in my essay I defend panpsychism, where decisions of the basic units of consciousness are random according to outside observer. ''In basic units'' means in every qubit, so it is also outside of the brain. Otherwise, the ''free will'' in the brain is more correlated, but there deviation from randomness has not yet been measured officially. (Otherwise, it is claimed by Sylvain Poirier, who knows for some parapsyhological measurements.) If you dislike panpsychism, you should to know that it is advocated also by Tononi and Koch.

    And I claim also that QM is incomplete because consciousness is not explained and QG is not yet known.

    My old essay about this topic is. As a detail I have added this year that consciousness does not exist without Free Will. So my theory is distinct from Tononi's one.

    For the rest, I argue (in the essay) that axioms are not the essence of the of math, but the germ of formation of math is physics. So I defend naturalism, so, similarly just as Smolin in this contest, but as distinction from him, I believe that the basic math of QG is simple, so I believe in reductionism. What about you?

    I also claim, tham infinity in physics do not exist. You claim for both options. Why?

    p.s. By the way about pi: I think That its number structure is not random, Because the rule for its calculation is simple. What do you think?

    Best Regards

    Janko Kokosar

      I agree that QM does not explain consciousness, and I doubt that anything else will anytime soon.

      I do not exactly say that physics has infinities. Some people say so. Mathematics has infinities, and physics uses math a lot, so it is often useful for physicists to use infinities.

      Dear Roger Schlafly

      As correction, we disagree about quantum consciousness. But, I agree if QM is assumed not to explain consciousness, ''I doubt that anything else will anytime soon. ''

      But, I like positivism, but at physics is a problem of selection of words and sentences, where a little changed sentence tell some clearly distinct things. Thus, because of some details, our opinions are distinct. Thus, I am positivist, but not 100%. (My phrase in this contest.)

      In prolonged version of essay (ref [1]) you can see my positivisic explanation of free will: ''Let us suppose that Turing experiment gives distinct answers of a human versus computer. (Otherwise free will does not exist.) If we respect non-quantum physics, then explanation of free will needs new physics. But a quantum computer always gives distinct answers than a human, thus free will does not need new physics.''

      Best Regards

      Janko Kokosar

      6 days later

      Dear Roger,

      I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

      All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

      Joe Fisher

      7 days later

      **************************

      rujing_tang at yahoo com

      **************************

      Roger, I'd like to get your idea on my thoughts on simplicity.

      Your view of probability requires some time to digest but appears to be along the line of what I'm thinking.

      Thanks,

      RJ

      Simplicity is that a relative few theories and mathematical models can explain a number of phenomena. While complexity is the opposite where there seems to be an unending need to invent new theories. By this definition, physics and astronomy are in the former camp and social science and biology belong to the latter.

      Why is the universe is even understandable? This itself is hard to understand according to Einstein. I propose a line of reasoning here. Simplicity is a result of long term evolution in a close system. The resulting equilibrium gives rise to simplicity. The infinite possibilities of any member of the system have been largely reduced to a highly confined options. Most of the possibilities are prohibited due to forces that have long been cancelled out during the long evolution. Because of this simplicity, there appears to be causal effect. In other words, causal effect is a direct product of simplicity. Take our universe as an example, the universe is in equilibrium by and large. Only a handful forces remain. Because there are relatively few forces and laws, the universe appears to be orderly and thereby allows mathematics to even exist and work. Mathematics owes its existence to the equilibrium of the universe. Equilibrium brings orderliness and slowness to change. Just imagine, if one puts one stone by another stone, and because the stones decay so fast, by the end of this action of moving them together, one counts zero stone. The law of addition will be forever different from what we know today. In this sense, math and physics have 'this worldliness' feature, and is a localized knowledge to this universe at this phase of equilibrium. It could be vastly different in other possible states of the universe or other universes.

      A corollary is that the rules governing a simple universe is discoverable and free of controversy. The simplistic nature of the rules make it hard to miss the mark, so to speak. Once the framework of the rules is tested true over numerous times, what's left is refining of the details. Contradiction between the rules and the reality should be worked away over the time. This is good news to scientists because it solves the age old anxiety of finding all theories are invalid one morning.

      One notable exception to the simplicity in universe is the complexity in bio-sphere. Because the bio-sphere is inherently expansive and interactive, we cannot reduce the theories to a few laws and mathematics models. The bio-sphere is NOT an equilibrium system. Therefore it is very hard to apply causal effect to explain human society for instance. It is very hard to generalize theories or apply mathematics in bio-sphere or human society, as we are able to in cosmology.

      Humans' brain is wired to understand simple things and not complex things. We seek patterns and generalize. This skill helps tremendously in our evolutionary past. For instance, our eyes are adept in figuring out patterns like straight lines. Our eyes are especially good at spotting moving object in a static background. The predisposition to seek simplicity gave humans survival advantage in its evolutionary history. We appreciate simplicity over complexity. Humans process limited computational power. It is most efficient to apply the limited resource to a fast algorithm. The design principal of the fast algorithm is simplicity. There is an aesthetic side from human eyes for simplicity, whether be a new physics theory or a design of a gadget. The propensity of seeking simplicity is a very human specific trait, and has nothing to do with the reality whether the world is simple or complex.

      The coincidence of the simplicity of the universe and human's preference of simplicity is fortunate and fruitful. Specifically in the math and physics the coincidence yielded amazing results. There is no reason to doubt that more amazing discoveries will surface in the future. However, a grain of salt must be added so that we are conscious that there is less mysterious processes or agent involved in the coincidence. This article hopefully explained why. In fact, if we are blindly led by our pursuit of simplicity, we might fall into traps of naturally complex traps. For instance, any attempt to gain simplistic insight into a complex system is not a good idea. Our brain comes into my mind as an example of complex system. There are so many facets to this simple object that no one can claim a brain can be modeled with a finite number of rules.

      However, the simplicity on the surface for the natural world might be just a disguise of a chaotic and unpredictable reality. The equilibrium masks over much of the chaos and most noises or complex nuisances cancel each other out. What's left is the poetic simplicity skin. Underneath the skin, things might not be that smooth, or elegant or simple after all. It is a possibility. We probably are able to see some hints as we get more refined data, better models and more powerful observation tools.

      6 days later

      Roger,

      You approach the subject in a nicely different way to your previous essays and to others. Your prose is as good as ever and I still agree both with your viewpoint, and with 'vive la difference'! You're still not quite alone in suggesting maths can't faithfully model everything, though I take a slightly subtler line saying (and showing with an important case) that it often 'DOESN'T do so. Not adopting the absolute 'can't' is to me like saying it's complexity is so beyond all conceivable computational possibility that it's as good as impossible for humanity. I don't like absolutes, and that case the trust we put in present maths is as equally misplaced as in the 'absolute' case. The difference needs recognizing.

      I'm glad I made it to your essay which I think has been well undervalued. The reminder of the philosophical variants was also useful. I hope you'll also make it to my essay as I'd value your thoughts and comments.

      Best of luck in the last few days.

      Peter

        Thanks. I enjoyed your essay also. Yes, there hvae to be others who doubt that math can faithfully model everything, but most seem to assume that it can. Smolin says that it cannot, but he is saying something different, and I do not agree with him.

        a year later

        The points about positivism are ignored largely because people can't resist going beyond what they can't really talk about. It's the product of a childish human nature. Probabilities and classical particles are simply an interpretation of quantum mechanics. The wave function is a mathematical fiction that encodes our uncertainty. It cannot be taken too literally.

        In terms of your points about randomness, I'm not sure how one can even suggest causality or pretend one even produces a coherent model without assuming randomness is simply about ignorance. The only alternative is mysticism. I don't think people appreciate that chaos IS random and it doesn't just look random. The problem is that people have abused a term that is simply associated with various mathematical distributions. The reason why the term "essential randomness" fails to have any meaning is that one runs into an undefined wall when tracing the argument backwards. In other words, no explanation is not an explanation. To the extent you can't explain something, you don't have a theory at all. It's impossible to prove that anything is "essentially random."

        Your point about uncountable numbers were addressed by Stanford mathematician Solomon Feferman and he showed that Cantorian mathematics had no application in the physical sciences. However, the proofs are not agreed to be sound but are debated. A mathematician can't even provide one example of a real number, so they are even fanciful from a mathematical standpoint. Additionally, even countable sets are debated by ultrafinitists, since they are associated with the completed infinite set of natural numbers. The main paradoxes introduced by Cantor, Godel, Turing, etc.. were the result of completed infinity (self-contradictory) and "diagonalizing" (Poincaire's impredicativity and paradoxical self-reference). It's really quite trivial and has no bearing on real mathematical truth or decidability properly defined. Incompleteness is a very poorly understood result and amounts to a parlor trick of no real consequence. It's an extremely easy result but the simpler proofs have not been seen by most people. It says nothing about the undecidibility of other problems (even major mathematicians fail to understand this).

        Mathematics is seen to be separate from science but it cannot also be free from the law of non-contradiction. Additionally, I would argue that when mathematics leaves physical intuition too far behind, it isn't very productive. Mathematics is certainly only a map but it has to be a map of something. For instance, ZFC is said to establish contemporary mathematics but that explanation is completely backward. Math is largely discovered and later described in terms of axioms. Then we have the fatal blow to this entire "disease," as Poincare described it: axioms are sensitive in the same manner as chaotic initial conditions. There are concrete problems where different axioms produce different answers, such as coloring problems. We have to know all their applications ahead of time and so they only work like fitting a curve to the past. It's like trying to pull yourself up by your own bootstraps. ZFC axioms are certainly not obvious but in fact quite arbitrary and one may even find a different starting point for the same system. I don't buy the idea that math is set theory.

        What I'm saying is that these disagreements point to poor logic and not equally valid ways at looking at the world in different contexts. I would not be so kind to science or mathematics. Both need to study logic.

        Write a Reply...