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Fabien Paillusson

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

Thank you for having read our essay and for the further thoughts about it.

You said "I think Kuhn was right when he proposed that we go through cycles of revolutionary and normal science.As a consequence, it appears we become stuck in an 'undecidable' trap when we are doing normal science and don't know how we could possible answer particular philosophical questions. ".

Yes, in a way, a philosophical question that cannot be answered by science could be considered as an example of what Kuhn would call an anomaly. Given our definitions of philosophical and scientific questions, we would claim is that this is tautological; of course philosophical questions cannot be answered by science.

I think Kuhn was thinking more about scientific theories failing to answer scientific questions. In this event, we might say the unanswered scientific questions must be 'recoded' as philosophical ones?

You said "The current dominant trend in much of modern science is reductionism where we try to build up a global understanding of the laws of nature by studying them independently then bolting them onto one another. Do you think the undecidable philosophical problems could also be a result of this?"

Yes reductionism is a prominent philosophical doctrine in contemporary scientific circles and it does come with its set of issues. Anderson was already pointing this out in 1972 in his famous paper "More is Different" (https://science.sciencemag.org/content/177/4047/393). Incidentally he did not formulate it in term of undecidability but it is tempting to imagine that in todays terms he may have phrased in that way. In fact some people have done it https://arxiv.org/abs/0809.0151 . This was also already mentioned in one of S. Wolfram's last scientific publication in PRL in 1985 (https://www.stephenwolfram.com/publications/academic/undecidability-intractability-theoretical-physics.pdf ). So in the end yes reductionism brings along with it some undecidable questions which could be decidable if a different doctrine was adopted.

With regards to your comment on dualism and reductionism, would you mind specifying what you meant? I would think that dualism refers to a necessary difference in kind, as philosophers call it, whereas reductionism simply focuses on the premise that a whole is made of parts and if the purported parts do not confer an alleged phenomenon then the said phenomenon is, sometime, deemed illusory.

Many thanks and best of luck in the contest.

Fabien and Matt

Dear Jason,

This was a very stimulating and well-argued essay.

If I would have any complaint, it would be that the notation appeared to me quite obscure at times, especially in the part on Godel's proof.

I would have some questions:

From LEM and LNC we can construct reductio ad absurdum arguments in logic and mathematics. Do you have any thoughts on the value of this type of argument in physics?

Could you please elaborate on what you mean by "meaning" in the introduction?

Many thanks and good luck for the contest.

Best wishes,

Fabien and Matthew

Dear Jason,

Thank you for your comment and for the positive feedback.

With regards to your comment "Should I go so far as to say it is an argument that science (and mathematics) is, at least to some extent, invented?" I would say that there is some degree (or maybe a lot) of invention but the corresponding creative freedom is bound by rules which compel the invented narrative (or its consequences) to match some agreed-upon aspects of reality (where the minimum required tends to be a form of consistency). Contemporary sciences use as matching tools essentially mathematical tools.

I will make sure to look at your essay.

Best wishes and good luck for the contest.

Fabien

Dear Flavio,

This is really an impressive and excellent essay. I enjoyed it very much.

I have not much to comment but to ask some questions with regards to where your essay could lead:

- There is an interesting trend in mathematics (I find it interesting at least) which attempts to extend concepts of standard mathematics to fuzzy sets. This is sometimes called fuzzification. I am wondering whether your essay could not be a starting point for a fuzzification of classical mechanics. Note that fuzzification has also the advantage of using fuzzy logic which has a working procedure to determine entailments and the like (this could be relevant to your notion of cause and effect).

- Your last optimistic quote and paragraph on the openness of the future does remind me of the interpretation of probability by Carl Friedrich von Weizsacker. As far as I understand his view, probability can only be about the future because the future is "open" to re-use your wordings. Are you aware of his work on the matter (like his temporal logic) and, if so, do you have particular thoughts about it?

While your essay mostly focuses on indeterminacy, the essay I have submitted focuses on undecidability and asks a similar question in substance "As physics and science ever been decidable?". If you are interested you can read it there https://fqxi.org/community/forum/topic/3477 .

Best of luck for the contest.

Fabien

Dear Markus,

This is a very insightful and very well written essay you have got there!

The structural realism you put forward coincidentally resonates with some of my recent readings on Poincaré who was also advocating for a form of structural realism well before quantum mechanics.

A naive query I would have about an ontology based on structure is that it seems to rely on a form of first order logic where predicates, and the rules they may obey, are what remains when what they can act on is forgotten. But I cannot help wonder how would that work if the predicates themselves are instantiations of models in higher order logics; it would seem to run into a form of infinite recurrence of Russian dolls structures that in some sense never stops; unless we select a given model or order of logic.

I would be interested to read your thoughts on this :) .

In case you would be interested I develop a similar view in my submitted essay where, as far as I understand your perspective, we claim that finding meaningful differentiations within a given structure (of observational phenomena for example) is in fact a defining feature of scientific practice https://fqxi.org/community/forum/topic/3477 .

Best of luck for the contest.

Fabien

Dear Mozibur,

Thank you very much for having taken the time to read our essay. I am happy to read that our thesis seems to resonate with some of your views.

With regards to your first question on Aristarchus, I must confess that I knew that he, and others before like Pythagoras, had proposed heliocentric models which somehow did not "take off" so to speak, but I do not know the exact details.

I personally believe that sense data is a very difficult argument to overcome. Now, I am not an expert in antiquity and some much more knowledgeable than I am on the matter have discussed your question here (https://hsm.stackexchange.com/questions/1979/why-didnt-aristarchus-theory-of-heliocentrism-stick) for example. I found the discussion therein particularly illuminating.

As for the influence that Eastern Philosophy has probably had on the development of quantum mechanics, I totally agree with you. Most physicists of then were versed in their classics of Western philosophy and many appeared to think that physics and Western thought had reached a stalled state and were looking for ideas in Eastern philosophy. While you mention Wigners and Schrodinger (I would have to read more about them), interestingly I would have directly mentioned Bohr (who famously designed his coat of arms with the Yin-Yang symbol at its centre and strongly advocated for a universal form of his Complementarity Principle) or Pauli (who developed a theory with Jung on the I Ching and thought that there were there principles that would enable physics to move forward).

This is not to say that the development of quantum mechanics was not also driven by experimental results, but the inconceivability of these results within the inherited Western philosophical and scientific tradition of the time compelled these people to turn Eastward to find different ways of thinking.

Best wishes,

Fabien

Andrew,

Thank you for your reply.

I suppose my 2nd question was precisely on the blatant incompatibility between the reversibility of the microphysical laws (based on Hamiltonian mechanics) and the irreversibility of macrfphysical observables such as "raining".

This is not at all an original question. It reminds me of the objection from Loschmidt to Boltzmann's H theorem relying on uncorrelated velocities where Loschmidt observed that you could in principle reverse all velocities and the entropy should decrease accordingly (since it was increasing with time for the "forward process"). Boltzmann would have allegedly replied "go on reverse the particles velocities".

What I am saying is that I am not convinced that, even in principle, events leave measurable traces of their happening at all later times.

Even a three body problem is already not reversible even when people try hard (https://www.sciencealert.com/three-black-holes-orbiting-each-other-can-t-always-go-backwards-in-time).

This is just a thought if you think that can help you improve upon your argument against this concern.

Cheers.

Fabien

Dear Jochen,

Yes I totally agree with your view here. I do believe scientific questions are important, first because they allow a given perspective to be followed through and through for a relatively long time. Second because, from them, may emerge new philosophical questions too. And third, because ultimately they strongly participate to enrich our understanding of a topic.

Reading on the history of science, scientists in the first half of the 20th century were, for the most part, all versed in literary subjects and very much so in philosophy. Later on, Carl Friedrich von Weizsäcker (whose writings I do enjoy very much) did try to bring closer philosophy and the sciences and was somehow disappointed at the lack of literary culture of the new scientists he saw emerging in his time.

That philosophy appears to be entirely separated from the sciences and possibly "dead" appears to me as a somewhat recent (post-war) phenomenon that has been further fuelled by events like the "science wars" in the 1990s and by the two-culture paradigm. Whether this schism will survive the various challenges that we have to face nowadays, which call upon moderation, humility and collaboration from all sides, we will see.

That being said FQXi does manage to bring like-minded people from philosophy and the sciences by making them interact on questions where there is still much contention or where an apparent consensus can be looked at with a more critical outlook. That is an opportunity that should not be missed indeed.

Best,

Fabien

Dear Jochen,

Thank you for your reply. That there are hierarchies of halting problems was insightful.

With regards to my initial question, I suppose I was mostly confused by the leap from the two states of coin tossing to infinitely many states at the core of any diagonalisation argument.

As I said, I am just trying to restate what is actually being stated in your proof and in particular what are the necessary assumptions for the proof to hold. That the system can be in infinitely many states appears as such an assumption.

That the system can be in infinitely many states does appear actually reasonable a priori, for we never know what "the" state of a system is before we measure it in some way.

I actually think that the coin example could be used to go beyond the two state system. In fact m1 would still be "Head is showing once the coin has landed" but the state could be something much more complicated linked to the initial setting of the experiment...or to the mechanical state of the coin.

Then I believe what your proof is saying is that:

"Given that a system can be in infinitely many states, there necessarily exist states such that the outcome of some measurements on those states is undetermined"

If that is the case, different physical theories can choose to work only with subsets of the states the system can be in (in the same way that one can decide to work with integers instead of the reals...in some sense, as long as this subclass of states forms a closed set under some dynamical rules and chosen operations, then it seems fine). Classical physics restricts the states to those that are determinable while QM embraces both kinds of states.

Is that a fair restatement of what you are saying?

Many thanks.

Best,

Fabien

Dear Israel,

I enjoyed reading your essay.

It does propose various claims with which I would tend to disagree but in any case they are well argued for.

I have some questions/ comments if I may:

-You said that "For if science is not about truth, then scientific activity becomes meaningless and in that case I should not be writing this essay". What if science is about unravelling "facets" of the truth rather than some absolute one way of looking at the world? Would that still make it meaningless?

- In your well-thought diverse examples to show that mathematics alone is not enough and ampliative principles from physics are necessary, I would more than agree with you.

But I thought that the way it was phrased was somehow unfair to the practice of mathematics. When solving an equation, an actual mathematician would ask in what space we are looking for the solutions. In the case of the degree 2 polynomial equation for the radius of a quantum dot, physics compels us to search for solutions in the set of positive real numbers. With regards to the particle in a box problem, I would argue the same. Although I totally agree with the main message, the mathematics problem that should be posed is that we are looking for a wave function psi(x) that satisfies the time-independent Schrodinger equation, hard boundary conditions at the walls and is normalised. If one chooses n=0, the last condition is not fulfilled since the wave function is identically zero everywhere. It is not that we decide to discard it for the sake of it, it just not satisfies the properties of a wave function.

- Of course we can also come back to the while discussion on relativity of motion later on :) .

I would be happy to know your thoughts on the questions/comments above.

Best,

Fabien

Dear Ernesto,

Thank you for your comments. We do hope you found some aspects insightful.

With regards to the "End of Science", with the thesis we defend in our essay, it is possible to reach a state where scientists believe that "they have got the whole picture right" and maybe what remains to be done is just getting better and better quantitative agreement between theory and observations. It is unclear whether the remaining quest driven by quantitative agreement can ever reach an end though. That being said, even if most scientists believe that the "End of science" is reached, it does not follow that philosophers agree and nor does it follow that future scientists are going to agree either. There are at least two situations that come to mind where scholars thought that "we have got the whole picture right" and yet that was not the end of it. The end of the 19th Century and the pre-Renaissance period. Copernicus is a very interesting example because, although he motivates to some extent his theory based on disagreement between observation and the Ptolemaic model, his main drive was reportedly the antiquity texts of the Pythagoras school that had been translated from Greek and Arabic (leading up to Renaissance movement) and lead him to look at the world from a unique vantage point with respect to his contemporaries.

Although this is much more complicated than what I am going to say here, one cannot dismiss the influence that Eastern philosophy has played on the development of Quantum Mechanics in the 1920s, in particular when it comes to the development of the Copenhagen school.

With regards to Turing's uncomputability, we think it plays a very practical role in many problems of physics where one wants to infer the infinite time and/or infinite size limit of a system; which is relevant to computer simulations but also to experimental works as well. Alongside Goedel's incompleteness theorem it also gives trouble for answering fundamental questions on the spectral gap. Note that for the latter example, extrapolating to infinity is not the only issue. The point is that it is undecidable to know which finite system size can actually reproduce the real system the model is trying to characterise (https://www.nature.com/news/paradox-at-the-heart-of-mathematics-makes-physics-problem-unanswerable-1.18983).

Beyond the above detailed example, what has come from Goedel's proof of his incompleteness theorem and from other examples such as the Barber's paradox etc.. is whether some self-referential questions can put physics in some problematic situations. A somewhat recent article claims that indeed QM runs into problems when we try to "apply it to itself" so to speak (https://www.nature.com/articles/s41467-018-05739-8).

Hope that answers in part some of your questions.

Best,

Fabien

Dear Israel,

Thank you for your reply.

Of course Newton's work constitutes a breakthrough of a sort that by no means we wish to diminish (and we absolutely don't in our essay).

Have you read Mach's critique of Newton's work by any chance?

Best,

Fabien

Thank you. Will have a closer look.

Dear Klass,

I found your essay truly brilliant. Combining in a clear manner Quantum mechanics, Goedelian-like results and using them for a discussion of the possibility of deterministic theories of quantum mechanics.

Since this is not my area, I must say I was quite blown away by some statements such as " fair classical coins do not exist". I am still recovering in fact and will have to look at the references you gave in your essay.

With regards to that statement, I wanted to make sure I understood what is meant here:

- Do you mean to say that a coin whose motion is determined by Newton's (or Hamilton's) equations of motion cannot eventually give rise to a fair coin toss (unless true fair randomness is put in the initial conditions)? or

- Do you mean to say that a fair coin model within classical probability theory is actually not fair?

I believe this is the former but just want to make sure.

Finally, given that the argument relies, as far as I understood, on infinite sequences, is there a finite version of it whereby, say, a membership function (for the Kolmogorov random character of a sequence) would be in between 0 and 1 for any finite N but would tend to zero when N tends to infinity?

Congratulations again on this very nice essay.

Many thanks.

Best,

Fabien

Dear Andrew,

This was a nice and thought provoking essay.

It seems naive but I really liked your definition of a fact!

I would have a couple of questions though, as it seems that some things are escaping my understanding:

- First, it is not clear to me what really constitutes a superposition. From the example you gave I do not see the difference between superposition (with possibility of interference) and simply not knowing all facts (tossing a coin). Can you please clarify on this?

- Second, at the very end you discuss the fact "it rained yesterday" and the various traces that would follow even many years after. I appreciate that this is a microscopic world view but what of the 2nd law of thermodynamics?

- Third. It is more a comment but your definition of fact and the following detailed example with particles reminded me of simulation method called Event Driven Molecular Dynamics (https://academic.oup.com/ptps/article/doi/10.1143/PTPS.178.5/1869834) which literally implements what you describe in your essay. Are you familiar with this method? Do you see any way of using it to emulate the kind of things (e.g. entanglement, superposition) in your essay?

Many thanks.

Best,

Fabien

John,

Thank you for suggesting further ideas.The GR vs QM would fit within the problem of contrastive underdetermination and we can interpret the current programme of find a theory that combines the two as the belief that there is "better" theory, in some sense, than the existing alternative of having GR and QM kind of separate. The request for unification appears to me as an aesthetic constraint.

I have read Optiks some time ago and light as being made of particles is only addressed in the last book from what I remember. I do not recall the aether argument. Would you have a link to suggest where these ideas are explored further?

Thank you.

Best,

Fabien

John,

Unless I made a mistake in my last observation I do not believe this is the case. It is not absolutely decidable on the sole observational basis. One needs to assume the background of the fixed stars to be fixed for the parallax evidence to actually make the problem decidable from astronomical observations.

If the stars in the background are not fixed, the reality of their observed motion from the Earth becomes as undecidable as that of other planets and the Sun.

So, you are right that it did support a non-geocentric view but this was not for free.

Best,

Fabien

Dear Tim,

It is a very nice and original idea you have presented in your essay. As many others have said I would probably need a few more reads to grasp all the details though.

Few questions if I may:

- If an underlying fractal geometry can give rise to quantum-like behaviour, how does classicality emerge from this picture, if it does at all?

- Would you have any toy-example with the Lorentz attractor of non-computable counterfactual?

Many thanks.

Best,

Fabien

Dear Jochen,

Thank you very much for this essay it was a real pleasure to read. Very well written and thought provoking too.

Pardon my slow-mindedness but I would like to grasp the essence of the argument for the non existence of f(n,k) i.e. what are some necessary preconditions for it to hold.

- Is the claim valid even if there are finitely many states?

- If k belongs to a finite interval of integers then I could build a finite set of experiments that would create a f(n,k) table for all possible values of 1 and -1 for each state (like when designing a truth table). In that case the mg operation must bring back an existing row since all rows would have already been exhausted by my truth table.

Many thanks for your help on this.

Best,

Fabien.