Thanks for the kind words John :)

In response to your question, I would honestly answer with; I don't know. Given the infinite nature of say, real numbers, it doesn't seem like they are physical. But in saying this, we humans with our mushy brains are finite creatures which dreamed up mathematics, hence why I drew on Landauer's work so much: He would argue mathematics must be bounded by physical constraints for this reason. Therefore, is it possible to bridge these two? I argued that if real Turing machine are an attempt to physically realise mathematics, but because of thermodynamics and the energy requirements to run the thing, you still can't realise all of mathematics i.e the halting problem. Thus, a Turing machine cannot be a viable way to bridge this Platonic divide.

For the wavefunction to evolve via the schrodinger equation, we need some potential energy V(t). Typically, we just add V(t) to our calculation without really asking where it comes from. In reality, it comes via the interaction of your system to an external environment i.e light matter interaction and the dipole moment. Now we can always include this entire interaction ad infinitum in our system, but we are now running with open arms into the church of the higher hilbert space. I'd argue that if your trying to build a Turing machine, eventually, there is going to be some interaction that you just can't as it's lost in the soup. At this point, errors are going to creep in and thermodynamics comes into the fray.

I completely understand the sympathy of your scepticism about hinging my arguments on thermodynamics since it is emergent. But if mathematics is Platonic and only in our heads, then it to must be emergent. I don't believe our brains hit some critical mass which allowed opened up a portal to the Platonic realm: For me, such an argument is necessarily on par with the claim that one can speak to God through prayer.

Again, thanks for taking the time for reading my essay!

Dear Michael,

thank you for an excellent essay, very clear and well-argued. As you have firstly noticed, there are many elements of resonance between our ideas. I particularly appreciated how you managed to draw a line between abstract mathematics and physical processes, a way of operationalising maths. It's insightful also your discussion on the limit of infinite processes due to the kick in of a sort of thermodynamica obsolescence. This led you to a great conclusion: "Deterministic

theories like a microscope, focus down on a highly specific piece of nature by closing out the rest of reality".

(Please also have a look at my reply to your post on my essay's page).

Congrats again on a very good piece of work, to which I gave full points! And good luck for the contest.

All good wishesm

Flavio

    Hi Flavio,

    Thanks for taking a look at my essay and the full points! :)

    As you pointed out in your response in your thread, there is several interesting points of overlap between our ideas.

    Cheers,

    Michael

    Dear Michael,

    Congratulations for your well written and interesting essay.

    Your conclusion : " The processing logic of error-correction must come from an external processing unit also sourcing its power from the environment. By implication, the physical realisation of indefinite self-referential logical program such as the Halting problem relies on an external network of other external logical processors to prevent errors." is really appealing to me, as this is quite analog to my idea that self-referential issues inside quantum theory can be dissolved by invoking an "external object", an observer, and thus the measurement problem is just a logical tension between physics-from-inside and physics-from-outside.

    However, I am not sure to follow you when you write that "the mathematical paradoxes uncovered by Turing and Gödel do not bear any consequence on the physical world because they can never be truly realised in physics." Or at least, they have a huge importance in meta-physics...

    All the best,

    Hippolyte

    Hi Hippolyte,

    Thanks for taking a look at my essay.

    Yes I agree that they have importance from metaphysics and mathematics but any attempt to execute them physically is impossible.

    We can do the abstract task of metaphysics but this---currently---appears to be orthogonal to physics i.e we can discuss perpetual motion machines but we can't make them, thus any machine that relies on perpetual motion machine cannot be realised either. Likewise, Halting problems and mathematical anomalies can't be made in physics.

    Now, this is not to say they can't be conceptually useful, but they are certainly not physical. Therefore any physical model that relies on them is somewhat meaningless; but this is not true for metaphysical models.

    I hope this clears up what I meant.

    Michael

    Hi Michael,

    I enjoyed your essay, you write very well, your explanations are clear, and you made very interesting points. I must agree with you, noise is ubiquitous, and it's not always a bad thing. Without it we couldn't exist. I wish you good luck with the essay!

    Cheers,

    Cristi

      Hi Michael!

      This is a fantastic essay! I really love how you talked about the physical nature of Turing machines, I think something that is often overlooked. I think the Church-Turing thesis is very deep, but also unfortunately allowed people to think about computation in such an abstract sense, that we forgot about the physical needs to perform computations! So thank you for bringing this to attention.

      Do you think that, because of the above, it is more useful to think of computation in terms of lambda calculus? I think Lamba calculus is useful since it does not make the distinction between data and programs, which is explicitly distinct in Turing's language of combustion. Actually I think this is an extremely powerful viewpoint when thinking about biology, and so it makes me wonder why there isn't more work in understanding the physical thermodynamics of Turing machines and their equivalent lambda functions. Landaur's principle helps a little, but I haven't found a nice way to see a Turing machine and calculate its potential energy from its look-up table.

      I'm curious to know what you think!

      Cheers!

      Alyssa

        Hi Michael,

        This is wonderful essay and opens the big picture on "computational machine". In my past essay, which was finally published as the book chapter, informational context and computation in terms of Maxwell's demon have already been discussed.

        In your essay, I do not understand the connection between thermodynamic context and quantum reality except for the irreversiblity. From computational viewpoint, do you think to open a new pathway to understand this connection?

        On the reference [11], Roger Penrose is not the author but was communicated with.

        Best wishes,

        Yutaka

          Hi Alyssa,

          Thank you for suggesting Lambda calculus, I think it would be a really interesting avenue to pursue! Personally I haven't put much thought into it, but as a guess, I would suppose it would be subject to similar constraints. The operations of any information processing framework can be done without an increase in entropy, however it is the encoding/decoding side which is irreversible.

          Encoding any information in a physical structure is intrinsically dissapative---provided you don't keep track of all the absurd amount entanglement being generated between subsystems. Biological/genetic systems exhibit a huge amount dissipation and heat generation which down at the smallest scale must be closely bound by Landaur's principle. Although it might not be the processing that is dissapative but the transmission reading/writing side that is generating heat.

          To be honest, I'm being very speculative but I think I will certainly dig into lambda calculus more. Thank you for the suggestion!

          Michael

          Hi Yutuka,

          Thank you for the kind words (and pointing out the reference error).

          In the Quantum reality section I was arguing that even the Church-Turing-Deutsch principle is still subject to the same irreversible constraints arising from encoding/decoding information in physical subsystems. While the underlying hardware and processing capabilities between classical and quantum computers is fundamentally different, the process of reading and writing will introduce noise in the exact same way for both. In this sense, a quantum computer is subject to same thermodynamic constraints as a classical computer and cannot realise an undecidable problem.

          I hope this clears up my intent!

          Thanks again for taking the time to look at my essay.

          Michael

          Dear Michael,

          It's very kind of you to read my essay and comment on it, thanks so much, as it has led me to your well crafted and most enjoyable essay, which I will give a very good rating.

          As a practicing physicist measuring magnetic fields I deal with noise on a daily basis. As a hobbyist radio-astronomer noise is our bread and butter. As a retired EMC practitioner noise was mostly the enemy. And finally as an armchair philosopher noise is a most important facet of arguments against quantum misinterpretation.

          I believe QM is an unreasonably effective mathematical tool of the physicist but I do not agree with the quote of Feynman "nature isn't classical, dammit, and if you want to make a simulation of nature, you'd better make it quantum mechanical" that you gave in your essay, for I truly believe that nature is classical, provided you are using the right physical ontology. I am not a nay-sayer of QM, just a free thinker who thinks it has been wrongly placed on a pedestal as the 'crown jewels'.

          I have several areas where I feel QM needs revision:

          1. I understand that there are Maxwellian explanations for the photoelectric effect, that gave birth to the notion of the photon as a particle.

          2. HUP has been extended way too far past what Heisenberg originally developed. (I have made a few comments on this in posts to other essayists)

          3. Wave/particle duality is a nonsense as presented by QM, as particles and their fields (and attendant waves) must always exist together, not seen as separate entities dependent on mode of observation.

          4. The photon is not a particle, just purely a wave in the classical aether, and as such doesn't partake in the duality mentioned above. Young's slit experiments with particles need to be interpreted carefully as their fields and their Maxwellian radiation pass through both slits even if the particle does not.

          5. The various interpretations of the collapse of the wave function truly require one to believe in magic, and put the 'observer' on a pedestal that is very misleading.

          6. I believe in a certain type of entanglement that occurs during pair production and 'photon' splitting. I believe that from the moment of birth, noise in the environment cause degradation of the entanglement known as decoherence.

          7. I do not think there is sound evidence for quantum superposition, and hence I do not think quantum computers will ever achieve their lofty claims. (See Alan Kadin's excellent essay notes regarding this).

          8. I think that Bell's tests re EPR have been following an incorrect path of interpretation (see Barry Gilbert's challenging essay on EPR).

          I realise that you work in the area of quantum optics and that I have made some big challenges. I say what I have said because I have developed a working preon theory (gimli theory) that reduces matter to a single particle, and thus gives structural insight to particle physics that is sadly lacking in the Standard Model of particle physics (a tapestry of quantum field theories, and a testament to the great mathematical ingenuity of particle physicists). The gimli model of the electron allows it to produce two distinct magnetic fields, one of which is interpreted to be the vector potential, thus explaining many electromagnetic phenomena.

          For the record, I am not a believer in the Platonic realm, as you will have picked up from my essay. Another aside, in the realm of human endeavour bitcoin mining must be one of the most dissipative activities outside of exploding nuclear weapons!

          All that aside, I agree with your overall conclusion and I certainly enjoyed reading and commenting on your fine essay.

          Good luck with your ratings, as yours is an important essay,

          Lockie Cresswell

          (btw my brother is a Prof. at your Uni)

            Dear Michal James.

            "MEMORY" is just a time-effect in our emergent reality. (so it originates from a Point in a Point of TS).

            I will read your essay, I promise, but today I am hospitalized for an operation on my colon where they will take away a tumour. I will have to stay ca 12 days before coming home.

            best regards

            Wilhelmus

              Dear Michael,

              you've provided a well-argued essay that takes the abstract notions of 'austere' mathematical Platonic domains and confronts them with the hands-on reality of the actual world. Do the issues noted by Gödel, Turing, and others survive the confrontation with a messy, noisy world of increasing entropy and finite resources?

              You answer in the negative---and in the way you frame the debate, I can only agree: there is no device that could be built that actually realizes the necessary preconditions for undecidability to apply. Even the simplest possible systems that could hope to achieve computational universality eventually fall prey to the loss of energy to the environment, and to the degradation of their signal-bearing vehicles.

              Sure, one could quibble at certain points. For instance, the famous result by Cubitt et al., demonstrating that whether a certain system has a gapped ground state is an undecidable question, essentially only applies to systems with an infinite number of degrees of freedom---and while such systems do exist in quantum field theory, arguments from e. g. the Bekenstein bound may be taken to suggest that an ultimate theory of nature ought to be finitary in this sense.

              Likewise, one can show that the question of whether a certain output port in a chain of iterated quantum measurement (Stern-Gerlach type) experiments stays dark, is undecidable. But neither of these really demonstrates an exploitable source of undecidability.

              But I don't think that the failure of concrete systems capable of universal computation in a realistic context to exist entails the inapplicability of Gödelian reasoning in a physical setting.

              Consider randomness. Famously, no computational process can produce genuine randomness---not indefinitely, at least. Any given formal system can only approximate a random number to a finite degree; the digits of the number beyond that threshold correspond to undecidable propositions. Hence, as Feynman put it, "it is impossible to represent the results of

              quantum mechanics with a classical universal device". To the extent that there is randomness, undecidability is relevant to physical reality.

              Fine, one might say. We only ever make finite observations; it's good enough to use some pseudorandomness for our purposes. But then, it turns out, one can use entangled systems to send faster-than-light signals, in conflict with special relativity; hence, all such models must be noncomputable.

              More generally, I think that even though undecidability may not apply on the level of concrete devices, this doesn't entail that it can't have no role to play in the formulation of laws themselves.

              One typical argument one encounters is that we only ever make a finite set of observations, hence, collect finite data, which can be produced by a finitary system, such as a finite state automaton. That's of course true. But such a model would, effectively, be equivalent to a computer program that just outputs that data, without significantly compressing it; hence, if would yield poor predictive power.

              But then, how could one have predictive power if, for instance, the data is due to a non-computable process? Well, any such process can be decomposed into a finite algorithm and a source of randomness---say, a string of random digits. An observer faced with (finite) observations of noncomputable data could then figure out the algorithmic part, and thus, explain their observations in terms of some algorithmic process interspersed with random events---which is, of course, exactly what we see.

              So, in the end, I think that there's a window for the applicability of undecidability to natural law that your arguments leave open---and, as is probably no surprise given my own essay, I believe that this is what actually is realized in nature.

              Still, I wish you the best of luck in the contest!

              Cheers

              Jochen

                Hi Lachlan,

                Thank you for the kind words. Yes, noise is ubiquitous and it's always a central concern as an experimentalist.

                You have a lot of big challenges with QM as you list. Personally, I would subscribe to the argument that if observed reality cannot be explained by factorising the observed statistics in probability distributions over hidden variables, then it cannot be explained by a classical theory.

                However, there is still certainly room to move since we still haven't reconciled the tension between locality and realism, nor have we brought gravity fully into the fray (or dark matter on that note), so it is hard to put QM on a complete description of reality of pedestal as you claim. For the time being I would submit that QM in its current form is incredibly useful and explains a huge chunk of observations, but is it the full picture? Certainly not.

                Anyways, thanks again for taking the time to read my essay! Good luck in the competition.

                Michael

                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