Essay Abstract

Abstract: From the Platonistic standpoint, mathematical edifices form an immaterial, unchanging, and eternal world, that exists independently of human thought. By extension, "scientific Platonism" says that directly mathematizable physical phenomena - in other terms, the research field of physics - are governed by entities belonging to this objectively existing mathematical world. Platonism is a metaphysical theory. But since metaphysical theories, by definition, are neither provable nor refutable, anti-Platonistic approaches cannot be less metaphysical than Platonism itself. In other words, anti-Platonism is not "more scientifical" than Platonism. All we can do is to compare Platonism and its negations under epistemological criteria such as simplicity, economy of hypotheses, or consistency with regard to their respective consequences. In this paper, I intend to show that anti-Platonism claiming in a first approximation (i) that mathematical edifices consist of meaningless signs assembled according to arbitrary rules, and (ii) that the adequacy of mathematical entities and phenomena covered by physics results from idealization of these phenomena, is based as much as Platonism on metaphysical presuppositions. Thereafter, without directly taking position, I try to launch a debate focusing on the following questions: (i) To maintain its coherence, is anti-Platonism not constrained to adopt extremely complex assumptions, difficult to defend, and not always consistent with current realities or practices of scientific knowledge? (ii) Instead of supporting anti-Platonism whatever the cost, in particular by the formulation of implausible hypotheses, would it not be more adequate to accept the idea of a mathematical world existing objectively and governing certain aspects of the material world, just as we note the existence of the material world which could also not exist?

Author Bio

Holding two french M.Phil. degrees (D.E.A), one in philosophy of physics, and another in cognitive sciences, lecturer at Pôle Universitaire Léonard de Vinci and at IPECOM, Paris, France, I am focusing on mathematical Platonism and physical reversibility/irreversibility. More precisely I argue that in a material world universally subjected to irreversibility, no organized material or partially material entity could manifest/maintain itself without the intervention of essentially immaterial, eternal, and immutable principles existing objectively.

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Dear Mr. Fisher

First allow me to thank you very much for your kind words. On the other hand I want to emphasize that according to my personal vision, philosophy of science is a confrontation of viewpoints regarding issues which do not admit definitive answer. My participation in this contest seeks above all to promote such an intellectual confrontation for the only purpose of mutual enrichment. Any reaction to my article is welcome.

Here in France, it is 7:55 p.m. and I have a stressful day behind me. But I undertake to respond tomorrow during the day to your really interesting remarks.

Warm regards,

Peter Punin

Dear Mr. Fisher,

As I promised, I will try to respond to your comments.

First, I do not believe that the physical universe is "EASY" to understand. More specifically, I think the use of the adjective "easy" is in this context threatened by a misunderstanding. Certainly contemporary physics hopes that the phenomena belonging to its research field are reducible to "SIMPLE" logical structures, i.e. structures (i) based - in a first approximation - on a minimum of assumptions being as simple as possible, and (ii) provided with clear and distinct deduction procedures. But in this context, "simple" does not mean "easy." Consider two competing physical theories Th1 and Th2 advanced about a physical phenomenon not yet understood. Until further notice, the theory Thi, i = 1,2, which is more "simple" than the other, i.e. which requires, roughly speaking, fewer assumptions while using a less complex mathematical apparatus, is more likely to approach truth that the competing theory, "less simple." It is clear that in the context of contemporary physics neither Th1 nor Th2 can, regardless of their degree of "simplicity" in the logical sense, be "easy" (to understand)

In absolute terms I fully agree with you that any entity within the universe is unique. I would add that there is a kind of uniqueness of second degree involving the knowing subject and the item this knowing subject beliefs to know. The "same" airplane is not the same" one for the pilot and the passenger, knowing that each pilot and each passenger lives each flight in his or her personal way. The uniqueness of each thing is reflected by the infinite complexity of the whole, and in these conditions, no knowledge of the universe seems possible.

Yet knowledge exists, otherwise you and me would not be able to communicate via internet. How then to explain the existence of knowledge relating to an universe inappropriate to be known. In my opinion, this question does not admit definite answer. All we can do is confronting our personal standpoints, and I suppose that this contest was organized for this purpose.

If I understand correctly, you want to say in an expressly symbolic way, that behind the manifest hypercomplexity of the universe, something unique must be hiding, and that science, instead of looking for what must be unique, focuses on what is disparate. Perhaps this can be defended. But, for my part, I consider the Platonic approach as simpler: (i) Mathematics exist independently of human thought. (ii) Some of the phenomena of the physical universe is governed by this mathematics. (iii) Humans having to some extent access to mathematics can hope to understand to a certain degree the physical phenomena mathematics govern. And history shows that physics, without having found its unity, nevertheless made great progress in this direction.

I hope I have answered your remarks, but would say for a second time, that this field, probably will forever remain an arena of controversies.

With best regards

Peter Punin

Dear Peter Punin,

You need to take a remedial course in written English. You did not understand a blessed word I wrote. Reality is easy to understand, abstract physics and abstract mathematics are impossible to understand.

Amazed,

Joe Fisher

Dear Mr. Joe Fisher

Fortunately, there are people who understand some mathematics and physics, specifically the minimum of mathematics and physics being required to develop and operate internet. If our humanity was reduced to your vision - math and physics are not understandable; "only reality can be understood"; I hope that this time I finally managed to assimilate your philosophy - internet probably would not exist and I would be deprived of the pleasure to receive your posts.

Best regards

P. Punin

    ROFL:-)

    Fisher makes the same post in everyone's forum. I will certainly make an effort to read and understand your essay.

    Best Regards,

    Gary Simpson

    Dear Mr. Punin,

    I went through your essay and I found it interesting. Some parts were too technical for me, but only because of my poor skills.

    I have a question on metaphysical presuppositions. You highlight in your essay that they are propositions that we admit because they are neither provable nor refutable. Do you believe this impossibility to refute them is absolute or is it because we are unable to create perspectives on them?

    I will give an example to explain my question: For the simplicity of my question, I will assume the earth does not turn around the sun.

    1. If one man on earth looks at the stars, they seem fixed in the sky. They could be seen as absolute light.

    2. With today technology, Hubble would look at the same star with a slight different angle and the light will be slightly different. Thanks to these two measures, today scientific are able to calculate a distance to the star. The star's light is not absolute anymore because a perspective has been created.

    Do you believe this approach could be applicable to metaphysical presuppositions? If yes, can we imagine "a distance" to presuppositions?

    I hope my question is not too exotic.

    Regards,

    Christophe

    Peter,

    This is a very elaborate and elegant work of formal logic. Unfortunately, I think that much of the FQXI readership will struggle with it :-) I think I understand your major point though. Taking the negation of an argument and determining what that implies is a fairly effective tool. The weakness of it is that even if something seems like nonsense, it can still be true. So without a formal proof the circularity that you reference remains.

    I think that there is truth, and that both mathematics and science attempt to understand it and describe it. To the extent that they are both successful, it appears that mathematics describes science.

    Best Regards and Good Luck,

    Gary Simpson

    Dear Mr Tournayre

    The issue you raise is highly interesting (i) There metaphysical and non-metaphysical propositions. Can we establish an exact delimitation between both types of propositions? (ii) Suppose that a proposition has in a given epistemological context a metaphysical status. Could this status change over time? I'll try to answer these questions, knowing that this field is controversial.

    For Kant, a metaphysical question remains metaphysical. Auguste Comte thinks that the formulation of metaphysical issues is a step in the advent of scientific thought that unmasks metaphysics as such.

    My personal opinion is this: some questions are metaphysical, while other issues has a metaphysical status related to the epistemological context of the moment.

    The category of essentially metaphysical questions include issues like "Why does the universe exist rather than not to exist?" Or "Can we affirm the objective existence of immaterial entities such as we affirms the existence of the material universe? "

    The second category allows to go back to the example in your own question.

    Auguste Comte still asserted the impossibility of what is called astrophysics today. For him, any proposal in this area was metaphysical. But his position is already inadequate. Meanwhile, Fraunhofer had (i) invented the spectroscope (ii) launched the chemical investigation of matter light emitting light.

    Anyway, before the invention of spectrocope a proposition like "a fixed star is an immaterial source of pure light" was a metaphysical neither provable nor refutable metaphysical proposition. But since the invention of the spectroscope, this proposal is refutable and no longer belongs to the category of metaphysical propositions.

    It is the same for the distance of stars. Previously, humans believed that all fixed stars were at the same distance from Earth and populated the same celestial vault. Newton himself would have designed the negation of this belief as a metaphysical assumption which did not concern science as such.

    Today, we have several methods allowing the rather accurate determination of stellar and interstellar distances. On the other hand, powerful telescopes have taught us that the universe is made up of galaxies and clusters of galaxies. Thus the proposition "The fixed stars are at the same distance from the earth" is no longer a metaphysical proposition. It is now a physical proposition empirically refuted.

    I hope I have answered your question.

    Let me thank you for reading my essay. If it includes passages requiring additional explanation, I am of course entirely at your disposal.

    I intend to read your essay next week, feeling myself concerned by links between physics and information theory, especially in the perspective of L. Brillouin and O. Costa de Beauregard.

      Message to Mr TOURNAYRE (suite)

      Dear Mr Tournayre

      It is not my style to finish a letter without a few polite words. Excuse me, I'm under time pressure.

      With best regards

      Peter Punin

      Dear Gary,

      Your remark concerning circularity is absolutely adequate.

      In this moment, I am terribly under time pressure, but I promis you to respond next week.

      Best regards and good look also to you

      Peter

      Dear Peter,

      I have just read your essay (except the technical notes; I'll go through them as time allows). An excellent text, technical, raises interesting and relevant points, and offers a nice entry to the literature (very useful for newbies in the subject like me). I need to go back to it and think about your points.

      One thing that crossed my mind, apologize if it is not related to your points. I wonder which position would Cantor place himself (or would we place him, since he no longer can defend himself), given the scheme that you delineate.

      He was, perhaps, a kind of "mixture" of idealist and realist, despite these being opposite concepts. He was quite certain of an independent "existence of mathematics", but not only "externally" to us, but (perhaps specially) "internally" to us at the same time. He created "his" transfinite numbers freely, even though such a creation was constructed using the previous definitions of mathematics, but was imposed on him by several issues concerning real numbers, which he was not satisfied. (Paradoxes were gradually understood and the edifice based on his concepts gradually refined and extended by others.) I suppose physicists would go at ease using reals without the need for such formalizations, but mathematicians cannot live a tranquil life while such issues are not clarified... (There is a tension here that is also mysterious, I think, but that's another question). So would you say that Cantor's position could only be justifiable by the need of various metaphysical presuppositions? Or would you place him in what position? Thanks.

      Best,

      Christine

      Dear Christine,

      Your comments about Cantor are absolutely relevant. In this contest, the number of characters is limited, which is understandable. If I had had more space I would have spoken Cantor and especially the position of Hilbert and Gödel regarding the epistemologic status of transfinite numbers.

      Today, I can not go further, this sunday is a hard day for me, but tomorrow, in better conditions, I will answer exhaustively.

      Best regards

      Peter

        Dear Peter,

        No need to answer quickly or feel a pressure to do so. Any brief comments or pointers to the literature in that regard will be highly appreciated, whenever time or energy allows. It's a very interesting discussion, which I am just begining to study in more detail. Your essay has already provided me a good material to make me think for a while. I hope you have a better day and your essay receives better rating and classification.

        Best,

        Christine

        Thank you for your detailed response.

        My essay is short, I wish I would have introduced my arguments in more details. In any case, I suggest you not to have high expectations. I am very far from what L. Brillouin or O. Costa de Beauregard could say on this topic.

        Regards,

        Christophe

        Dear Christine,

        I'll finally try to answer your absolutely pertinent remarks about Cantor, but the issue is complex, and it is difficult to find the right beginning.

        The double challenge is in the polysemy of both concepts of Platonism and idealism, knowing that some interpretations of these concepts can meet, and this is somewhat the case for Cantor.

        The most common interpretation of the concept of "idealism" refers to the so-called "German Idealism", initiated by Kant and pursued by Fichte, Schelling perhaps - but this is debatable - and finally Hegel. This heterogeneous movement has a common denominator: Reality as it appears to us, would be conditioned and even generated by human mind and or reason.

        In the days of Cantor, the main stream philosophy in mathematics and physics is still Kantism, although countering increasing difficulties. According to Kant, mathematics belongs to human reason as such. On the other hand, reality exists objectively, but would not be knowable per se. What we can know would be limited to "phenomenons", ie the reality as it is conditioned by the "categories of pure reason." And since mathematics belongs in turn to this pure reason, it would be only natural that all physical laws take the form of mathematical expressions.

        But besides the reference to German idealism, the term "idealism" can take an openly Platonic connotation. According to Plato - specifically according to what posterity made from Plato; but this is not necessarily a problem - the material reality is only a rough and imperfect representation of "ideas" hiding behind. These "ideas" supposed immutable, immaterial and eternal exist independently from human mind and so form the authentic reality beyond deceitful appearances. Referring to the concept of "idea", some authors characterize Platonism as "idealism."

        Cantor refuses any form of German idealism, including Kant's philosophy, speaking in this regard about "regrettable exaltation." In a letter to Paul Tannery, Cantor protests against all those who qualify himself as an "idealist" - Tannery is one of them - and characterizes his own position roughly (I quote from memory) in the following terms: "I am absolutely far from idealism as it has evolved following Kant. My own position is akin to that of Aristotle and Plato, which is a kind of realism. I am as much as realist as an idealist."

        Let us leave aside Aristotle. This would lead too far. But for the rest, I personally interpret this quote as adherence to Platonism, that some authors assimilate to idealism by reference to the concept of "idea."

        Regarding transfinite numbers, I personally do not think that Cantor has considered them a creation from him. I believe, he discovered them in spite of himself. After showing - against intuition and common sense - that all infinite and countable sets have the same cardinal, he wanted to prove that this was also true for any infinite set. In connection with his unsuccessful attempts, Cantor finally realized that the set of all subsets of natural numbers is not countable. On this basis, he proved without difficulty that (i) the set of reals has the same size (Mächtigkeit) as the set of all subsets of natural numbers, that (ii) the reals therefore are not countable, and that (iii) all infinite sets do not have the same cardinality. These three points make all the glory of Cantor, but they are diametrically opposed to its original intentions. So I think it's more discoveries. This is of course a personal vision as part of a debate that probably will never find an end. If this was not the case, it would not be a philosophical problem, and on the other hand, if we consider philosophy as a mutually respectful confrontation of views on issues admitting no definitive answer, we must accept that no one can pretend to be absolutely right.

        The Cantor approach encounters serious difficulties. You rightly say that Cantor's elucidation led to a better understanding of paradoxes. But the resolution of the paradoxes is based on what some people, including myself, qualify as ad hoc axiomatic. Cohen proved the undecidability of the continuum hypothesis. The choice axiom is to complex for an axiom; perfectly plausible but not obvious for the Platonists, it is hardly acceptable or unacceptable for most other tendencies. Research initiated by J.D. Hamkins on the basis of forcing developed by Cohen perhaps will lead to the emergence of a new paradigm: Just as the undecidability of Euclidean parallel postulate had led to the discovery of non-Euclidean geometries, the undecidability of the continuum hypothesis perhaps will lead to the consolidation of set-theorerical multiverses, existing according to Hamkins objectively in the Platonistic sense. But I am not sure that this kind of approach could be unanimously accepted.

        Anyway, you are right to say that "mathematicians cannot live a tranquil life while such issues are not clarified." It is regrettable, that some among them work on mathematical entities without asking questions regarding their epistemological status. The situation may be somewhat different for physicists. As long as the physical discoveries indirectly justify the involved mathematical structures, the ontological status of these latter is perhaps less important. But it is a great pleasure for me to see physicists - and you are among them - questioning themselves about the ontological status of mathematics. Neither "english descriptions", nor "meaningless assemblages of symbols according to arbitrary rules" possess the deductive power that confirms an initial hypothesis at the level of its consequences, or reveals the necessity of a new paradigm. An interrogation about the status of mathematics transcending undoubtedly that of "English descriptions" or "arbitrary assemblies of signs" concerns both physicists questioning the reliability of their deductive tools, and mathematicians doing better than wasting time with "meaningless systems reduced to syntax" etc.

        The elucidation of the reals by Cantor therefore does not just respond to a simple need to confront metaphysical approaches. On the other hand, when metaphysical presuppositions are involved, we must assume them as such. But this is not a sterile exercise.

        My participation in this contest primarily aims to initiate discussions that would not be possible on other media. I am really indebted to your essay I have read several times. In this way, it is now certain that my participation was not useless.

        If you want, we can continue during this discussion somewhat later with the positions of Hilbert and Gödel about Cantor.

        Best regards, good luck

        Peter

        Dear Peter,

        Thank you very much for taking the time and energy to offer again a very lucid and detailed account (considering it is a comment), and of a highly pedagogical character, which is enlightening not only with respect to my current studies but also beneficial for the readers at this section.

        My question refers to the book on Cantor by Jean-Pierre Belna, where he explicitly states the difficulty to frame Cantor's philosophical position. I believe that your points complement those on that book, but I have to think further, and go back to that book as well. I do not wish to put you any pressure to discuss Hilbert and Gödel, with respect to Cantor. But feel free to post your comments as time permits, evidently they will be valuable.

        By the way, do you know Cao's book "Conceptual Developments of 20th Century Field Theories" concerning "structural realism"? I read that book some years ago, and I should go back to it, and try to see conections with what has been discussed. It is a challenging book, but at least you see one of those few physicists seriously integrating philosophy and foundations.

        I highly appreciate your reading of my essay, and as I commented before, yours is of great value both as a critique and pedagogical introduction to the matter, interesting and relevant.

        Best wishes,

        Christine

        Dear Gary

        I try to answer your really relevant remark. My paper DOES analyze an inevitable circularity, where we just can try to do for the best.

        Platonism IS a metaphysical theory, in other words a theory, that in absolute terms can neither be proved nor disproved.

        But therefore all competing theories of Platonism are also metaphysical theories. If only one among all competing theories of Platonism was not a metaphysical theory, Platonism in turn would cease to be a metaphysical theory. It would become (i) a scientifically refutable theory, refutable by previously unknown means and (ii) a theory effectively refuted by the competing theory in question. Carnap and Reichenbach being among the main references of logical positivism, detect a lot of metaphysics within all approaches which do not belong to their movement. But Penelope Maddy representing naturalism also discovered some metaphysics in Carnap's and Reichenbach's approaches. Personally, I think that there is also a lot of metaphysics in Penelope Maddy's work. When naturalism will no longer be part of philosophical main stream thinking, other voices will probably go in the same direction.

        Instead of denouncing metaphysics detected in competing approaches, it would be more honest to admit - and to assume - that any approach of mathematical foundations is ultimately metaphysical. Or, more precisely: When anti-metaphysicians as ultra-formalists, constructivists etc. say "I operate AS IF mathematics were arbitrary signs assemblies" or "I operate AS IF mathematics were human constructions made from natural numbers" etc., there is no problem. But from the moment we say "Mathematics IS an assembly of meaningless signs." or "Mathematics IS a human construction ...", we already do ontology, so metaphysics. At this point, the only way to proceed non-dogmatically is to compare the various metaphysical systems between them. In my opinion, this comparison is not a question of sense or nonsense. The challenge consists of criteria such as economic assumptions, simplicity, consistency, etc., which is not the same thing.

        An example: Let us admit that mathematics "IS" a set of formal systems, ie a set of meaningless signs assemblages configured according to arbitrary rules. These systems certainly allow mechanical deductions from axioms being also arbitrary. Now would be very unlikely that these mechanical deductions correspond to predictions within physics. Even if we rightly accept that physics selects the only phenomena which comply to its formalizations, a correspondence between ONE single physical prediction and some "meaningless arbitrary formal system" would be an incredible coincidence. For the theory of physical phenomena formalizable by arbitrary formal systems to hold, we must assume a lot of metaphysical hypotheses formulated ad hoc. If we add that mathematics IS an arbitrary formal system, then all mathematics used by physics would inherit this lot of metaphysical hypotheses with their ad hoc formulation. I therefore raise the question, if it would not be easier to note (i) that there is objectively existing mathematics from which our knowledge is certainly imperfect, as we note the objective existence of material reality that also could not exist, and from which our knowledge is also imperfect, and (ii) that certain phenomena of material reality align objectively certain mathematical laws? Note that I'm just asking this (kinf of) question, leaving everyone answer in his way. In absolute terms, there is an inevitable circularity, and I would be the last to claim that I'm right. But I am also astonished that some people are convinced to hold the only and unique truth within manifestly ENDLESS debates, which sometimes extends from antiquity.

        Well, I hope I have answered your question. Obviously I have to apologize for my English, I am (i) not a native speaker and (ii) always under time pressure. At this moment I must write very quickly.

        I will read your essay as soon as possible, in principle next Friday, before replying on your own forum.

        Best regards

        Peter

        Hello. After seeing many essays filled with anti-Platonistic prejudices, I tried to follow your reasoning to see if it can be taken as a good reference for a defense of Platonism, but I found it both hard to follow and in some aspects disappointing. Of course the main idea is clear and worthy, that anti-Platonism is itself a metaphysical presupposition and it has many troubles. The problem is in the details. The very fact of being hard to follow is a weak point for an argument to convince people who are not fond of mathematics and are naturally tempted to skip technicities that look not so clear (indeed for the concern of anti-Platonist essays of the present contest, authors usually base their denial of mathematical realities on their natural dislike towards mathematics, and thus towards any formalized kind of exposition), and people are usually only ready to "accept conclusions by faith" in doing so when it does not conflict their basic philosophical convictions.

        I am usually not afraid of technicities when they are needed, since I am mathematician. However, technical details also deserve their own kind of clarifications, that is, they need to be appropriate. And more precisely what puzzles me with your very formal way of writing, is its strange combination with what seems to be your exclusively philosophical background, that is, you seem to have only learned the topic as "philosophy of mathematics", a branch of philosophy in the way taught by academic philosophers (and you even mention your degree to be only in the philosophy of physics, not mathematics), and not as "mathematical logic" taught by professional mathematicians.

        And the problem I found with academic philosophy and its diverse branches of "philosophy of science" such as the "philosophy of mathematics" and "philosophy of physics", is that they usually keep a poor understanding of the fields they are pretending to discuss : they poorly understand science, which does not even mean they better understand its philosophical aspects, since philosophers usually keep a fuzzy and unreliable ways of reasoning in their own field. Because they usually focus their study on traditional ideas and what other "science philosophers" wrote, texts are supposed to be about the philosophical aspects of science, and ideas that previous philosophers took note of as philosophical. In doing so, the ideas of science philosophers often remain quite outdated with respect to scientific progress, ignoring what scientists actually discovered just because it is not written "Philosophy" on the title of their works.

        I guess this explains why in your attempt to defend Platonism in mathematics, you did not even say a word about the Completeness theorem of first-order logic, which originally was Godel's PhD thesis, and which I see as a terrible (but usual) omission in the debate. You only mentioned his later Incompleteness theorem. Why ? Maybe because you only read other academic philosophers teaching about the incompleteness theorem because it turned out to be more "famous" among non-mathematicians, while they failed to notice the no less fundamental importance of the Completeness theorem, actually a cornerstone of mathematical logic (no less famous than the incompleteness theorem among specialists), for their "philosophical debates", where they seem glad to picture things in a "philosophical" style, that is a coexistence of competing "viewpoints" supposed to be antagonistic, irreducible and equally defensible, with no chance of any rational resolution. But whose viewpoints are these, seriously ? Mainly those of philosophers and a few fringe mathematicians, it seems, while the overwhelming majority of mathematicians, even specialists of mathematical logic, does not feel concerned by these "debates" in their works anyway.

        And specialists of mathematical logic have a good reason to no more show themselves in arguments between opposite "philosophies of mathematics" as expressed in terms of the old debate between Platonism, formalism and so on, because this old debate turned out to be outdated in the light of the more recent understanding of mathematical foundations. This way, the new scientific updates fall out of the radar screen of professional philosophers, who keep repeating the old debate without noticing that they are waiting in the wrong train that is not going to leave because the right train already left.

        Now to try analyzing the contents of your arguments on Platonism in mathematics :

        Your whole discussion seems contained in the century-old formulation of foundational problems as expressed by Hilbert's program, that is, the search for a formalization of mathematics, when the global picture of mathematical foundations was not well understood yet.

        You seem to express the question of Platonism in terms of the correspondence between "a given mathematical edifice E" that is not yet formalized, and a formalized version of it, Sy ; and you seem to define Platonism as the idea that E preexists before Sy, while being essentially different from it. Sorry but I find this quite a strange way to define Platonism, I would rather be tempted to see this claim as better associated with anti-Platonism than with Platonism.

        What the heck is E ? Non-formalized mathematics, you say. But precisely I don't see the sense of trying to make any logical reasoning about such a thing as "non-formalized mathematics". Because as soon as some "non-formalized mathematics" is regarded as an object of discourse, it becomes reified, analyzed... and finally turns out to be a sort of formal system itself ! Otherwise if a "non-formalized mathematics" (whatever it might be) fundamentally differs from any formalized version, then in which sense can it be considered any kind of perfect mathematics independent of human beings at all ? I do not see any fundamental structural difference between "non-formalized" mathematics and formalized mathematics, and I do not see the issue of Platonism as having anything to do with the hypothesis of such a difference. Where is the contradiction between mathematics being pure and perfect, and being defined by a perfectly rigorous formalization ? I cannot see any.

        As for me, I did not waste time "learning philosophy" but followed the more recent theories of mathematics and physics, from which I cared to both further clarify the expression of theories from a purely mathematical viewpoint, and directly identify their philosophical dimensions without the "help" of philosophers. And the picture I found this way of philosophical aspects, is quite different from what philosophers usually say. Before my work, I would have called myself a Platonist. Now the evidence I found from the (mathematical) foundations of mathematics led me to a sort of intermediate position (as was mentioned in comments about Cantor, which I did not check), close to Platonism but with an important difference : I found that the mathematical universe in itself is not fixed but has its own time, independent but similar to our time !

        You can find my ideas and findings on these issues in my web site, especially the introduction and the philosophical aspects page.

        Now this understanding of time in the foundations of mathematics also brings light by analogy, to the role of time and its irreversibility in physics. As I see you also interested both about irreversibility in physics and the interpretations of quantum physics, you may be interested with my essay A mind/mathematics dualistic foundation of physical reality where I propose a solution to these issues.

        A few more details: you wrote "A valid formalization of E by Sy via Φ presupposes the consistency and completeness of Sy". What ? Theories normally need to be consistent, not complete. Moreover, Euclidean geometry is consistent and complete, so that it is possible for a theory to be consistent and complete. The point is that Euclidean geometry is unable to express arithmetic. And for the same reason, physical theories may be considered consistent and complete as far as their physical results are concerned (not making sense of arithmetical formulas which depend on the infinity of natural numbers, as this is not something physically measurable)

        You wrote: "[Sy is] complete if no deduction of a theorem θv belonging to Sy would require the widening of Ax by other axioms". You have a strange way to define completeness. What do you mean by "deduction of a theorem ?" Normally, "theorem" means that it is deduced from the given axioms. If a something cannot be deduced then it is not a theorem. In a consistent theory, negations of theorems are not theorems, and there is no such thing as a need to still make them theorems by adding more axioms that will make the theory inconsistent.

        You wrote: "the consistency proof concerning all Sy as strong as or stronger than formal arithmetic prevents their completeness proof". The Incompleteness theorem says that any consistency proof of Sy inside Sy itself would would make Sy actually inconsistent (so that it has "proven" something false). Of course Sy is assumed to contain arithmetic, otherwise it would be hard for it to formulate and prove any claim of consistency in the first place.

        Then, theories containing arithmetic cannot be both complete and consistent. However it does not speak about "proof of consistency vs. proof of completeness" as the question whether a theory is consistent or complete (because any inconsistent theory is complete) can remain itself undecidable.