Hi Jochen,

Thanks for the feedback! You raise some very good points and I will do my best to try and answer them all.

I agree that the notions of undeciability and uncomputability can be extremely useful tools and essential for formulating theories and laws of nature. This gives us incredible power to make predictions and derive some intuitions about our reality, but at the end of the day they are still idealisations informed by continuous feedback from physics.

You wrote that 'even though undeciability may not apply to the level of concrete devices, this doesn't entail that it can't have a role to play in the formulation of laws themselves''. I agree completely, but it implies that if a law of natures themselves rely on uncomputable ideas, then they can never be verified. This was the contention of my essay; any axiomatic system that contains undeciable problems can never be verified in our physical reality.

Now this comes to your last point about finite observations. I'm not sure I follow exactly why the finite model you describe wouldn't be able to compress the input data and make predictive models? It should surely still be able to make predictions, however its predictive power would scale with the size of dataset it consumed---which would of course be tied to its energy consumption.

This comes to your last excellent point; consider an an input data set that can be partly described by some definite algorithm and the other part by random noise (a close model to what experimentalists actually observe in the lab). The noisy string comes from an external environment and we can implement all sorts of tricks to suppress this and explain as much of the randomness as we can, but it will always be there. This is not to say that we cannot learn the algorithmic side, but we can only learn it to with some small error.

But what if the algorithm your trying to answer is undeciable? Physics won't give you an answer, it will only give you an error. At every time step the error continues to compound with every time step and in the limit of infinite steps, you are guaranteed to measure an error. This is the main take home message of my essay.

You raise some very good points and I will certainly reflect on them in more! Thank you for the stimulating questions.

Michael

Dear Michael,

Thank you for steering me to your great essay. Kudos to you for asserting the fundamental inevitability of noise from the environment. You brilliantly capture the impossibility of isolating it by your statements: "The afterglow of the big bang buzzes in the background and virtual particles pop in and out of existence. No matter where you are in the universe, you cannot escape noise."

I believe that your premise, despite contradicting the conventional interpretation of physical reality, is solidly founded. The analysis in my essay concludes that the empirical facts of classical and quantum mechanics are completely and best explained by a physical reality that is contextually defined with respect to a positive ambient temperature. This leads to a physical reality in which the 2nd Law, entropy, and irreversibility are truly fundamental. The prevailing conception of a fundamentally deterministic physical reality, in which irreversibility and entropy are emergent, is an idealization based on an ambient temperature equal to absolute zero. Absolute zero can be approached, but it is physically unrealizable, and the prevailing conception is a fiction.

A positive ambient temperature effectively and objectively coarse-grains physical reality. Given a contextual reality and positive ambient temperature, you correctly assert that "asymptotic predictions of closed axiomatic systems can never exist in the physical world." You go on to provide an insightful well-written analysis of the implications on the operations of an actual Turing machine.

I have to disagree with you, however, regarding quantum computations. You state "their processing logic is entirely reversible as its evolution is governed by the Schrodinger equation." The Schrodinger equation describes a unitary and deterministic transition from a definite eigenfunction to a superposed wavefunction. However, I do not believe that a superposed wavefunction describes a physically superposed state. Superposed cats and superposed states in general do not exist. A superposed wavefunction describes the spontaneous and statistical process of transition from an initial eigenstate to one of the system's physically allowable eigenstates of higher entropy and stability. So, while a quantum computer's processing logic may be reversible, its physical calculations are not.

Best,

Harrison

Dear Michael,

This is very nice and well argued essay you have proposed here. I learnt many things, I did not know about conservative logic for example, and will for sure come back to it multiple times in the coming months.

You are perfectly correct that noise should not, and maybe cannot, be discarded. And the reference to the 3 Kelvin CMB is pot on about this.

A small issue I have with one of the theses you develop however, is that you say that because a computer ultimately relies on external resources (whatever they are for: memory, energy etc...), once this storage somehow runs out the programme will halt. This is perfectly true but I would not consider this as being the same as saying that the halting problem does not apply. If the programme is terminated before it terminates on its own then it is still a major problem and this is not, I believe, what the original Halting problem was about.

So, to me, if anything, you actually put forward, like Paul Davies does in his essay, an additional limitation to computation.

So, instead of dispelling these undecidability and incompatibility problems, I think you actually add to them by considering more realistic scenarios.

Another interesting point you mention is that mathematics can only go as far as the tools of mathematics, themselves governed by the laws of physics, enable them to go.

I would venture to object that the very laws of physics we have developed are equally prone to the same critic. So I am not sure how one can be used to undermine the other.

This reminds me of Penrose's claim that the proof of Godel's first incompleteness theorem could not be checked by a Turing machine and out of which he would conclude that our brains go beyond such idealised machines. Do you have any thoughts about this?

Many thanks again for this inspiring essay.

Best of luck for the contest.

Fabien

    Hi Michael,

    I really lied this paper. Well argued. I might note also that, in the event that someone tries a Lucasesque move (à la Minds, Machines and Gödel) of claiming that the mind can see things that computers cannot, your results apply their as well. So there is no escape into the "peace and quiet" of Plato's forms: doing this would require the same kinds of dissipative processes in the brain.

    Good luck - you should do well!

    Best

    Dean

      Hi Dean,

      Thanks for the kind words and I'm glad you enjoyed my essay!

      Cheers

      Michael

      Hi Fabien,

      Thanks for the kind words and taking the time to read my essay! I'm glad you enjoyed it and found it useful.

      You raise and excellent point that I should have clearly delineated; is the halting of the algorithm equivalent to it halting because it ran out of resources? As you point and Paul point out this is a limitation on realising axiomatic systems (like mathematics) in physical hardware. I should have sharpened this point up and distinguished more clearly between the two.

      Yes, I certainly believe that mathematics is limited by the laws of physics, however I understand this is a contentious point. There are many examples, which on face value appear to violate this. For example, it does not require an infinite amount of time to take the limit of an exponential decaying curve and state that it converges to zero. However, here I am not doing an infinite computation but rather using heuristics which are governed by some other computable result that doesn't require an infinite amount of time.

      This ties into Penrose's point, which I currently disagree with. Our brains have a huge amount of representational power which makes them extremely fluid in moving between numerous seemingly abstract tasks. But underneath there is hardware processing this. Even our brains can't check Godel's theorems, we just have enough sense to not get caught in an infinite loop until our bodies decay. I don't believe our brains have some secret sauce that is fundamentally off-limits to a Turing machine. To leave it with a quote from Sam Harris; ``there is nothing special about a computer made of meat''.

      I'm glad you enjoyed my essay. Thanks again for taking the time to read it.

      Michael

      4 days later

      Dear Michael James,

      I greatly appreciated your work and discussion. I am very glad that you are not thinking in abstract patterns.

      While the discussion lasted, I wrote an article: "Practical guidance on calculating resonant frequencies at four levels of diagnosis and inactivation of COVID-19 coronavirus", due to the high relevance of this topic. The work is based on the practical solution of problems in quantum mechanics, presented in the essay FQXi 2019-2020 "Universal quantum laws of the universe to solve the problems of unsolvability, computability and unpredictability".

      I hope that my modest results of work will provide you with information for thought.

      Warm Regards, `

      Vladimir

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