Godel\'s Incompleteness Theorems and Platonic Metaphysics by Aleksandar Mikovic

Aleksandar Mikovic

Essay Abstract

We argue by using Godel's incompletness theorems in logic that platonism is the best metaphysics for science. This is based on the fact that a natural law in a platonic metaphysics represents a timeless order in the motion of matter, while a natural law in a materialistic metaphysics can be only defined as a temporary order which appears at random in the chaotic motion of matter. Although a logical possibility, one can argue that this type of metaphysics is highly implausible. Given that mathematics fits naturally within platonism, we conclude that a platonic metaphysics is more preferable than a materialistic metaphysics.

Author Bio

I am a professor of mathematics at Lusofona University in Lisbon and I do research in quantum gravity, string theory and related mathematics. I obtained my PhD in physics in 1990 from University of Maryland and I was a postdoc at Queen Mary College and at Imperial College in London. I published over 60 papers in quantum gravity, string theory and related mathematics.

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Colin Walker

Dear Aleksandar,

Cosmology is where philosophy meets science, where metaphysics meets physics. I expect you will agree that a universe undergoing accelerating expansion is difficult to reconcile from the point of view of platonic metaphysics.

However, there does exist a plausible platonic cosmology. My essay discusses a new interpretation of supernova data in terms of photon energy loss. Walther Nernst around 1930 found that the energy lost from a photon due to the cosmological redshift would be Hh (Hubble times Planck) each cycle, independent of wavelength. Nernst-MacMillan cosmology proposes recycling between energy and matter so that the universe is eternal and isotropic in time, timeless in the platonic sense.

Would you agree that present-day cosmology clashes with platonic metaphysics?

Best wishes,

Colin

Aleksandar Mikovic

Dear Colin,

Time evolution can be incorporated in a platonic metaphysics by considering space-time trajectories, i.e. histories. As far as the idea of passing of time (change in time) is concerned, one can also incorporate it into a platonic metaphysics, which I discussed in my paper "Temporal Platonic Metaphysics", arxiv:0903.1800.

John Hodge

I think the idea of science is that mankind can understand natural law.

Lev Burov

Dear Aleksandar,

In your overview you mention of materialism, "Although a logical possibility, one can argue that this scenario is highly implausible." However in our paper we provide an exact proof that chaosogenesis by anthropic principle is a logical impossibility based on discoverability and precision of the laws of nature. Perhaps you would find it interesting, as it supports your hypothesis.

http://fqxi.org/community/forum/topic/2353

Lev

Sylvain Poirier

Dear Aleksandar,

I have similar ideas to yours, that is a metaphysics combining a mathematical Platonism with a non-mathematical reality made of consciousness with the flow of time. I presented this in my essay, showing how it provides what I see as the most natural and coherent interpretation of quantum physics.

However I have a different view about the role of the Incompleteness theorem. First, I consider that we should not forget the completeness theorem of first-order logic that provides a strong case for mathematical Platonism. Second, I see the Incompleteness theorem as one manifestation among others of an intrinsic time of mathematical reality, but that does not affect the laws of physics of our universe, and is unrelated to our time.

Why do you say that "any TOE has to include the arithmetic" ? Euclidean geometry doesn't, and, as I explained in my essay, I see the laws of physics as avoiding the dependence towards the actual infinity of natural numbers with its undecidabilities.

You describe the second incompleteness theorem as "a consistent theory T cannot have a statement referring to the consistency of T". This is not what it says. What it says, is that any consistent theory able to express arithmetic and thus describe itself (if its axioms are algorithmically defined), will also be able to express its own consistency as a formula, but will be unable to prove the truth of this formula. See my exposition of the incompleteness theorem for more details.

Aleksandar Mikovic

Dear Sylvain,

Thank you for your comments. My idea of time in a platonic metaphysics is that it serves to distinguish between an abstract mathematical universe and a real one, i.e. a real universe is a mathematical universe in time.

By definition, a TOE has to be able to explain arithmetics, as well as geometry, so it must contain the axioms for arithmetics.

I agree with your formulation of the 2nd theorem. I wanted to say basically the same, but I have chosen a too short sentence.

Peter Punin

Dear Dr Micovic,

Being myself a convinced Platonist convaincu, I read your essay several times and really with great pleasure.

I totally agree with you that anti-Platonistic approaches - materialist approaches and other - are as metaphysical as Platonism. In my own essay, I insist further on this point, since this latter does not seem obvious to everyone. Many people continue to think that only the consideration of material reality can be "scientific" whereas any hypothesis on the objective existence of immaterial entities, such as mathematical entities would be "exactly the kind of metaphysics" that scientific thought should ignore. I'm still totally agree with you that all we can do is comparing all metaphysics involved in the debate about mathematical foundations and links between mathematics and physics, in order to determine BEST metaphysics in the scientific sense of the term "best". Here, the evaluation of competing metaphysical theories must respond to epistemological criteria such as economy of hypotheses, logical simplicity, consistency of the theory particularly with regard to its consequences and so on. Under these criteria, Platonism is indeed the best metaphysical approach of mathematical foundations and the link between mathematics and physics. Certainly Platonism requires heavy hypotheses, but the hypotheses required by competing theories being not less heavy, attain by contrast a clearly higher degree of complexity. Platonism encounters serious problems, but the problems of anti-Platonistic approaches are much more serious.

It is absolutely plausible to establish a link between the Platonistic timeless conception of a mathematical world and the concept of block universe initially suggested by relativistic constraints. But you notice rightly, that there is here a risk of tension between the classical block universe and the idea of time flow; an idea which, despite its problematic side per se, can not simply be swept away. Being myself on a paper involving a bloc universe - it is to reconcile timelessness of idealized physical laws and de facto irreversibility, knowing that against "intuition", "common sens", "evidence" and so on, irreversibility IS NOT the direct negation of reversibility; this point is briefly touched in my own essay - I will read attentively your publication on arXiv about a Cauchy 3D hypersurface splitting of bloc universe. Personally, I envisage a superposition of a metric space with +++ - signature to an underlying non-metrical 3D topological space with a fibrous structure. The business looks complicated and I hope your arXiv publication which I unfortunately ignored until this moment, will be helpful.

Regarding both GIT, I think - by relying on Gödel himself - that they do not necessarily pose a problem to scientific Platonism. I published a paper on this issue in the magazine Al Mukhatabat; electronic versions are available on academia.edu (https://www.academia.edu/5262261/Syst%C3%A8mes_formels_et_syst%C3%A8mes_formalis%C3%A9s) or philfree (http://www.philfree.org/ecrire/?exec=article&id_article=545) But it is in French. Here, I try just a short, sometimes more metaphorical statement.

You notice rightly that GIT-undecidabilities can be avoided iff the concerned system Sy1 is imbedded in a "stronger system" Sy2. But within Sy2 we have once again this same undecidability, so that Sy2 must be embedded in Sy3, and so on ad infinitum. Now arises the following question: Can a système Sy(inf) possessing an "infinity of appropriate axioms" be sheltered from GIT-undecidabilities? This question is in turn undecidable. Its only formulation would never be admitted unanimously. A fortiori, one can neither prove nor refute the validity of Both responses "yes" and "no".

Gödel believes that the correct answer is "yes". It is a metaphysical belief and Gödel does not say otherwise. But the answer"no" est in turn a metaphysical belief. So we need to compare the two options in terms of gobal consistency. Gödel, starting from his hypothesis of an undecidability free system Sy (inf), thinks, that incompleteness and undecidability arise from the fact that all human knowledge of Sy (inf) is necessarily partial, so not only incomplete but also distorted. Both GIT would be only the consequence of this fundamental problem.

Now we can carry similar reasoning regarding "material world". Certainly, the main stream philosophy of the moment claims that the material universe does not know infinity.

Well, let us admit that there is only a finite number of phenomena, and that the number of cases where these phenomena occur, is also finite. But nothing prevents us to consider at least potentially the possibility of an infinite number of combinations of these phenomena. And even if some among us really do not want to admit this, we are still all in agreement that human knowledge masters just only a small part of the aspects of the material universe. This should not prevent us to postulate the existence of the material universe in all its details. Note in passing that this postulate is in turn metaphysical, and that the other postulate concerning the existence of the Gödelian undecidability free Sy (inf) is neither more, nor less metaphysical.

But anyway, our knowledge of mathematics is necessarily incomplete, so distorted. Its the same regarding our knowledge of the material universe. Both types of incompleteness can then join. On the other hand, the exploration of the mathematical universe promotes a better understanding of the material universe and vice versa.

It seems possible to construct on these bases a Platonistic epistemology being more consistent than its competing theories requiring hypotheses much heavier and much more complex.

In this regard I recommend you the excellent essay of Christine C. Dantas, as I recommend her yours. http://fqxi.org/community/forum/topic/2373

It would of course be a great pleasure for me to know your opinion on my own essay addressing Platonism in a different way. http://fqxi.org/community/forum/topic/2356

Best regards and good luck

Peter Punin

Aleksandar Mikovic

Dear Dr Punin,

I share completely your convictions about the platonic metaphysics, and after looking at your essay, I see that the basic motivation is the same as in my essay, i.e. to show that the platonic metaphysics is the best metaphysics for science and mathematics. My further motivation was to convince my physics colleauges that materialism is a belief, and that it does not follow from science. Since we can not experimentally, nor mathematically, confirm or refute a given metaphysics, then the only thing left is to compare different metaphysics and see which one is more plausible. My main idea was to use the definition and a meaning of a natural law in a given metaphysics, and to compare. In this respect, a platonic metaphysics only makes sense for me and the Godel theorems reinforce this belief.

The opponents of platonism use the problem of learning to reject it. However, I believe that only within platonism one can obtain a sensible epistemology, and my previous essay was an attempt to demonstrate that this is possible if the idea of time flow is introduced. Also the time-flow idea helps to distinguish between an abstract and a real universe. I will read the technical part of your essay, and let you know my opinion. And the Dantas essay, as well.

Best regards,

Aleksandar Mikovic

Sylvain Poirier

In reply to : " Now arises the following question: Can a system Sy(inf) possessing an "infinity of appropriate axioms" be sheltered from GIT-undecidabilities? This question is in turn undecidable."

No it's not, except in a trivial way, which is that the assumption is ambiguous (admits several possibilities). Because there is not only one possible "infinity of appropriate axioms" :)

2 main cases can be considered. Any algorithmically definable set of axioms will still be subject to GIT-undecidabilities. In particular, we can consider algorithmic systems obtained by going up the hierarchy of stronger and stronger systems, each claiming the consistency of previous ones, up to any algorithmically definable ordinal, but this runs into troubles at undefinable ordinals. In fact, set theory already implicitly contains this procedure at a very high level corresponding to the whole hierarchy of ordinals that its model contains. On the other hand, we can abstractly accept as an axiomatic system the countable set of "all arithmetical truths", to which GIT-undecidabilities no more apply, because this axiomatic system is not algorithmically definable.

Maybe you meant : "the infinity of all axioms which are appropriate". The problem is that there is no mathematical definition for "appropriate", any more than for "the smallest uninteresting number". So this makes an undefined claim, which I do not see very meaningful. It seems Godel hoped that mathematicians would find more and more appropriate axioms that would potentially solve all problems. I think he was dreaming.

Peter Punin

Dear Aleksandar

It is really rare to find such a point of agreement with another approach, knowing that in philosophy of science, there are for n people at least n 1 opinions.

On the one hand, materialism is a metaphysical belief just like Platonism. On the other hand, the notion of "best metaphysics", against misleading appearances and false evidence can be approached scientifically: In scientific terms, the "best theory" in a given area is the theory requiring among all competing approaches less and logically simpler hypotheses than the other theories to ensure (i) internal consistency and (ii) coherence in its consequences. Evaluate competing metaphysical theories - knowing that all foundational approaches of mathematics and/or physics are ultimately metaphysical - under these epistemological criteria allows defending Platonism being certainly metaphysical without metaphysical presuppositions.

It is very interesting when you say "that only within platonism one can obtain a sensible epistemology". Yes, there is a material universe, and within this latter, certain empirically detectable phenomena manifest themselves not anyhow, but in a predictable manner, regardless of the macro-deterministic, statistical in the sens of semi-macro atomic/molecular statistical mechanics or quantum-statistical status of the corresponding predictions. Since these phenomena don't evolve anyhow, there must be immaterial principles making that it is so. And since these predictions can be expressed by mathematics, there must be an intrinsic collusion between the principles and phenomena in question.

Time flow as link between a Platonistic universe and limited human mind unable to know this universe in its totality, but able to learn progressively about it, seems obvious. It is absolutely plausible to postulate the existence of the material universe in its totality, even if it would be vain to hope that human mind could one day know all of the material universe. Thus the traditional anti-Platonism argument dealing with our necessarily partial knowledge of the Platonistic world does not seem admissible. Concerning time flow, or more precisely the human impression that there is time flow, there are obvious links between partial knowledge currently increasing and increasing entropy; L. Brillouin and O. Costa de Beauregard had already worked thereupon in terms of information theory. But a consistent approach of all this continues to encountermany pitfalls, and especially since all this should be expressed on the basis of a non-classical block universe.There, there is still a lot of work for you and for me, but it certain that this kind of research has a future.

Best regards

Peter

Sylvain Poirier

I just wrote explanations about the conflict of philosophies in this contest, especially the Platonist vs Anti-Platonist opposition, and a list of best essays I could find, especially so as to better help the Platonists find each other. Any comment ?

Aleksandar Mikovic

Dear Sylvain,

I liked your analysis of the essays, and I have the following comments:

- I would classify myself as a dualist: mind (consciousness) and matter exist independently in the platonic realm of ideas.

- I agree with the division into platonist and anti-platonists, but the relation with the interpretation of Quantum Mechanics is less clear. For example, within the dualism (both mind and matter exist independently) one can have various interpretations of QM: de-Broglie-Bohm interpretation, the many-worlds, collapse due to Quantum Gravity, etc.

- I agree that many of the highest rated essays have this status because of the popularity of the authors and also due to the philosophical convictions of the readers. But the greatest value of this contest is the exchange of the ideas.

Best regards,

Aleksandar

Sylvain Poirier

Dear Aleksandar,

Did you study the interpretations problem carefully or do you just report vague rumors about it ? As I wrote in my report, it appears to me that the de-Broglie-Bohm interpretation is a purely materialistic interpretation, only supported by materialists, and motivated as a desperate attempt to save materialism in the face of its implausibility shown by quantum mechanics. I do not see there any trace of a fundamental realm of consciousness, and even if Bohm himself seemed to mention such an idea, the logical connection between the idea of consciousness and the effective content of his interpretation is actually non-existent as far as I could see (his hidden variables may be called an "implicate order" but this does not mean any form of consciousness).

The Many-worlds interpretation does not recognize consciousness as fundamental either ; it only develops a relativity of viewpoints, in a development that claims to be equally applicable to any sophisticated physical system (with memory) in guise of observer. Or did you see anyone presenting it otherwise ? I admit that Mikhail Borisovich Menskij has a view combining the many-worlds with a fundamental role of consciousness, however I do not consider his view very philosophically coherent and I don't think many other scientists are ever likely to follow it.

I know that the "collapse due to Quantum Gravity" came famous as proposed by Penrose but I do not see it very clear either, since as far as I know it remains quite vague without any precise conception of this quantum gravity. If the collapse comes from quantum gravity then it has concretely nothing to do with consciousness, and while I did not look at his ideas in details, I would be surprised if he could effectively justify the claim of a connection between quantum gravity and consciousness. Could he ? Already as I mentioned there, even the magnitudes of quantities do not seem to fit (perceptions by consciousness involve too light physical processes for quantum gravity to be involved).

On the other hand, I think my specific presentation of the mind makes collapse interpretation makes things much more coherent, in themselves and with the proper meaning of the idea of consciousness as fundamental. So I look forward for your more detailed review of this issue.

Aleksandar Mikovic

Dear Sylvain,

I have studied the problem of measurement in QM quite thoroughly, and the standard approaches try to solve the problem without introducing the concept of consciousness. The reason is that consciousness is a complex and poorly understood concept, so that one tries to explain the measurement problem by using first only simpler concepts. It may happen that consciousness plays a role, but by the principle of Occam's razor, one first considers simpler explanations. Also I do not see why a measurement interpretation of QM which is not based on consciousness implies that consciousness does not exist or that it implies a materialistic metaphysics.

Best regards,

Aleksandar

Jonathan Khanlian

Hi Aleksander,

I obviously realize that physics could be viewed in a materialistic way, but I didn't realize there was a view that metaphysics could be materialistic? Is that the view that any theory of the physical world is encoded in physical neurons in someone's mind? In other words, would that mean that from this point of view, the an object in the material system (i.e. a brain made of material) was organizing its material neuron firings in such a way as to represent some notion about the material in the universe, of which it was a part of? I feel like this isn't what you meant, but this is the only thing I could imagine materialistic metaphysics to be.

If the universe is finite do you think a TOE is possible in theory? Can you think of a way that Godel's work might still apply if the universe were known to be of finite size at any given time?

I'd love to talk about Godel some more. And let's not forget Turing either! Please check out my Digital Physics essay if you get a chance.

Thanks,

Jon

Aleksandar Mikovic

Dear Jon,

Yes, you understood it correctly, a materialistic metaphysics is what you described in your example of a human brain. However, the whole point of my essay was to demonstrate that a materialistic metaphysics is a very untenable metaphysics, because it implies that there are no laws in nature at the fundamental level. I also tried to argue that the only tenable metaphysics is a Platonic metaphysics, which includes both mathematical and non-mathematical ideas and the idea of passage of time.

The Godel theorems imply that a TOE does not exist, where TOE means a mathematical theory containing a finite number of equations and symbols, i.e. a theory which can be written on a finite piece of paper. This result does not depend on the size of the universe, so that even if the universe is finite in size, a TOE still does not exist. However, we do not need a TOE. It is sufficient to have an approximate TOE, i.e. a theory which describes a sufficiently large number of physical phenomena, and this is something which is possible.

Regards,

Aleksandar

Joe Fisher

Dear Aleksandar,

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

Luca Valeri

Hi Aleksandar,

According the rating it seems the community does not like your platonic metaphysics.

I like the idea of a temporal platonic metaphysics. According to von Weizsäcker Platos philosophy contains two movements. The ascending from the cave to the sun, from the shadow to the light, from the phenomena to the invariant structures, eidos. From the subjective view to a objective one. This is a step most physicist do.

The second movement is the descending from the sun to cave. From eidos, the invariant structures to the phenomena. From timelessness to temporality, from possibility to factuality. This philosophy was never finished by Plato. And not finished by most physicist. As I understand your essay it is this you try to do with your temporal platonic physics.

In my essay I argue, that even a phenomenological description needs the eidos to be understandable. And that the time is a precondition for the possibility of physics, mathematics and even logic. If this is true, the time structure itself cannot be completely be described in precise concepts.

In my essay mathematics can be constructed as imagination of possible future acts. Insofar only a constructive math can be developed in that framework. And no incompleteness in the sense of Gödel appears. However I like to stress, that if we view the history of physics as a succession of closed theories a la Heisenberg, where the newer theory contains the older one as limiting case, one could argue that from the view of a TOE, whether it could be the limit of another closed theory and which are its own limit in the phenomena that it explains.

Best regards,

Luca

Aleksandar Mikovic

Dear Luca,

Thank you for your remarks, it is very helpful for me when I hear other people's views on platonism. I agree with you that platonic metaphysics has not evolved much from the original Plato's ideas, but the developements in modern theoretical physics have given a new impetus. My main inspiration was the Mathematical Universe paper by Max Tegmark. The MU was generalized in my first FQxXi essay, where I introduced the concepts of temporal change and the concept of non-finitary ideas, so that one obtains a more realistic metaphysics in the Platonic realm of ideas. In my second FQXi essay I wanted to argue that a platonic metaphysics is much more plausible than a materialistic metaphysics, and Godel's theorems play a fundamental role in that. Given that many of my colleauges are materialists, then my views are not that popular.

There are not many scientists or philosophers who tried to develop a platonic metaphysics. I am familiar with the attempts made by R. Penrose and M. Tegmark, although I heard that Godel himself had some kind of platonic metaphysics. I did not know that Weizsacker also was a platonist, so I will look into his work and the related works, including yours.

Best regards,

Aleksandar

Luca Valeri

Hi Aleksandar

I would not call von Weizsäcker a platonist. But he surely knows Plato very good and thinks, that the main philosophical questions have already been posed in Platos work. In his book 'Time and Knowledge' he referenced Plato in more then 100 pages. Kant only 70.

My view (and this became much clearer to me in the discussions in the forum) is more that objective knowledge must be mathematical. In my essay I try to avoid discussions about ontology and reality as much as possible.

I hope you'll read my essay and leave a comment on it, although I had too little time to write it and develop a clear path of reasoning.

Best regards

Luca