Also it is not clear to me what is meant by "constructive mathematics", as, if it is about some non-classical logic, I could not find the sense of such a logic. My impression is that it looks like some funny toy for philosophers, and maybe a senseless formal system for the pleasure of defining extravagant formal systems without clear sense, far from genuine mathematics. What I know about, though, is Godel's constructible universe, that is something making rather clear sense to me. This is a model of set theory in the classical sense, formed by the mere "constructible sets" defined in a way that still needs to be quite elaborate to indeed form a model of ZF (where the axiom of choice turns out to be true).

SYLVAIN POIRIER

Dear Mr. Poirier

Having spent the day on the train, I just received your 3 posts.

I will reply as soon as possible, in principle this weekend.

Meanwhile, thank you for taking the time to comment my essay so thoroughly.

Just an information:

I mean by "constructive mathematics" the approach initiated by Brouwer and continued by authors from diverse backgrounds, among them Kolmogorov. "Non-classical logic" denotes HERE the so-called Brouwer-Heyting-Kolmogorov interpretation. In both cases, it is not philosophical byzantineries. For advanced computerized mathematics and/or computational approaches, constructive frameworks approaches are precious, even if in my personal opinion, their role in FOM is disputable.

If you agree, this discussion - I appreciate it - can be continued some days later.

Best regards

Peter Punin

Dear Peter,

Your comment on Gary Simpson's essay page wistfully observed that there aren't that many Platonists around these days.

I have to say that the effect on science is not that much influenced by one's Platonist convictions (or anti, but perhaps "non" might also apply). Let's instead look at this pragmatically...

Suppose you come up with a scientific theory. It will be judged on its correspondence to observations and its other utility (or generalizability, etc.) No one will ponder on whether the author of such a theory is a Platonist. The subject may still be of interest to philosophers, (neuro) psychologists, or even cognitive scientists.

Unless the actual process of conducting science is adversely impacted by one's Platonist beliefs (such as searching for answers in some metaphysical meditation, although even this might not be all bad), I cannot see why it makes a difference. This situation may be analogous to a scientist believing in God. Being religious, as I see it, has far more potential to skew what a scientist will investigate, how she might do it, and what she might believe is correct.

Your essay was well written and well-reasoned, and I see no benefit in taking up any of your arguments. I also found it interesting that you show very little underlying tendency to impose French syntax on your English prose.

En

Dear En,

I will respond as soon as possible.

Best regards

Peter

Dear Mr Poirier

Not being always on holiday despite of current certitudes, unfortunately I can not respond to all the details of your posts. But I will do my best, while thanking you for your efforts invested in your 3 posts; it is a form of interest beyond all disagreements that you express in this occasion. In my vision, agreement is not necessary and can not exist; I come back to that.

Let us start with the question of formal language. Any discussion requires a minimum of shared language. Although general philosophy does not belong to your preferences - that is your right - please pay attention to the few following mots. In freshmen handbooks about philosophy of human sciences we can read for example "that for Dilthey, sociology is un 'understandig discipline' whereas Durkheim conceives it as an 'explaining' one." To get a personal idea that about, one must know what Dilthey and Durkheim mean respectively by "understanding" and "explainig". As nobody is in the respective heads of both authors which moreover died since much time, eternal misunderstandings cannot be avoided in the also eternal debate about Dilthey and Durkheim.

The use of a minimum of universal FORMAL language, in the case of my FQXi essay just a little bit of set theory leaving beside all specific problems striking this discipline and also a little bit of model theory focusing simplified phi/psi functions, perhaps cannot eliminate all potential misunderstandings, but it is an attempt to reduce it.

When you say that "mathematicians don't need 'the help' of philosophers'", you are are absolutely right, but I am not concerned. First, it would be pretentious on my part to call myself a philosopher. I am working on philosophy, which is not the same thing. As someone working in philosophy of mathematics, I need by definition mathematicians of the present and the past. By "mathematicians" I mean women and man who have left significant traces in mathematics. Without these men and women, there would not be mathematical knowledge, and, subsequently, no philosophy of mathematics. Of course, mathematicians don't need philosophy. But it is the personal affair of each mathematician to be interested in philosophy or not. Someone are.

Similarly, I have no vocation to become involved in reserved domain struggles. Here, I would ask you politely to tolerate another digression. Admit that someone working on cultural philosophy raises the following question: "What is art?" This person has the absolute right to refer to a particular painter or sculptor working by definition in another field than philosophy, and more precisely in a field requiring artistic inspiration, artistic talent, ability of hands etc., but not necessarily philosophical thinking. In these conditions, the philosophy worker in question DOES NOT infringe upon the reserved domain of painters or sculptors, IF THIS RESERVED DOMAIN EXISTS. Personally I think that constructive dialogue is better than defending some CHASSE GARDEE. Analogously , the above mentioned painter or sculptor is free to claim that he or she can work "without help of 'philosophers'", but I also believe that there are a lot of cultural philosophy workers which don't claim that artists could not work without their help. PERHAPS it is the same about mathematics.

When teaching me that Gödel proved the consistency and completeness of first order logic, I do not see why you do it. Every freshman knows it. I would add that Gödel had originally no intention to prove the incompleteness of anything. He discovered in spite him both famous GIT while trying optimistically to prove the consistency and completeness of analysis. In this occasion he saw that already arithmetic was problematic. Analogously I know just like every freshman that we can not simultaneously prove consistency and completeness of Sy within Sy. I don't anywhere say the opposite. We can prove the consistency AND completeness of Sy1 by embedding Sy1 in a "stronger" Sy2, ie a Sy2 with more (appropriated) axioms, so that the set of axioms of Sy1 is included in the set of axioms of Sy2. I do not understand why this offends you.

Anyway, I had no other purpose than saying that in contrast to some prejudices, both GIT are not necessarily anti-Platonistic arguments.

Since you ask me what I meant by "formalized / non-formalized mathematics", I think there is a misunderstanding concerning the difference between "formal" and "formalized". Any mathematical edifice obviously is written by formal signs. Formalized mathematics is something other. The only way to avoid ontological proposals on mathematics, like "mathematics IS ..." consists until further notice on FOM approaches in the terms of "Let us do AS IF mathematics was ..." For this we can try to reconstruct mathematics AS IF it were a "system arbitrary signs" or a "system constructed on the base of N", etc.etc. For this approach, supposed phi-functions supposed bijective are particularly convenient, even if it goes back to the almost a century old approach of Hilbert, but still revisited by Hintikka and others more or less contemporaries.

Incidentally, you reproach me focus to "academical philosophy of former times." Well, I am hugely interested on Michael Detlefsen and Joel Hamkins. Detlefsen was recently visiting Professor at the Collège de France, an institution which uses to consecrate atypical thinkers. Hamkins, starting from forcing in the sense Cohen, is working on set theoretical multiverses. Considering himself a non-Platonist, Hamkins thinks that theses multiverses have an objective existence as suggested by Platonism. This makes me think that the term "Platonism" still needs further elucidation. This is an essentially philosophical question that interests me in this respect.

In order to end with something constructive: Yes, I am interested in irreversibility, a debate dating back to Boltzmann. Since this debate continues for much more than a century, I cannot belief that anyone will there find the ultimate truth. Thinking that only a confrontation of a maximum of viewpoints can lead to a better view of this problem, I will of course read your essay on irreversibility, without any a priori.

Best regards

Peter Punin

    5 days later

    Dear En,

    I see that you know the subtleties of French language. Moreover, behind the author's name appearing in the head of your essay, I guess a pleasant french pun suggesting many ideas such as chess...and of course your personal attitude concerning the present contest. Am I really à côté de la plaque?

    Regarding my poor English, I do my best in order to not "impose French syntax on (my) English prose"... I am just trying to be clear in a language which is not my mother tongue, and the result cannot be a perfect native speaker text. And especially HERE I am under time pressure; thus my English prose should be even worse.

    But let us turn to more important things.

    I totally agree that being Platonist, anti-Platonist or simply non-Platonist does not affect scientific discoveries as such. Again I would like to put a small damper. Until the late Renaissance, mathematics were regarded as something unrelated to physical reality. This had not fostered the emergence of physics in the modern sense. Finally, Kepler was led by DECLARED Platonistic motivations to glimpse intrinsic links between expressions and physical phenomena. The mathematical formulation by Kepler, of the three laws bearing his name was among the founding founding acts - and even IS THE founding act of modern physics. We can carry out similar reasoning about Galileo, although his (too) famous quotation about the "book of nature" needs nuanced analyzes.

    But in our day where it is established - at least DE FACTO - that physical laws are to be expressed mathematically, Platonism as such does not add anything to effective physical research. Hence you are absolutely right when you says that a new scientific theory is evaluated "on its correspondence to observations and its other utility (or generalizability, etc.)", an that the position of its author(s) in matter of Platonism changes nothing.

    Yes, but this contest DOES NOT FOCUS on the impact of different philosophical options on effective physical research AS SUCH. This contest is on the PHILOSOPHICALLY problematic links between mathematics and physics. You add still rightly that Platonism (or anti-Platonism or whatever) may still be of interest to philosophers, (neuro) psychologists, or even cognitive scientists. Yes, and the subject of this contest address precisely philosophically interested scientists and/or people working in/on philosophy, knowing that some among these latter qualify themselves as "philosophers". Now, philosophy is a finality per se. "Utility" of philosophy is certainly the issue of an endless debate. Personally, I think (i) that utility is not all in life (and you seem agree with me, saying that your participation is not motivated by the perspective of winning 10 000 dollars), (ii) that scientific research strico sensu, such as cognitive science, goes often back to philosophical investigations, (iii) that philosophy is interesting per se and does not need to justify its existence, and (iv) that FQXi would not have chosen this subject, if philosophy was really "useless".

    Please permit me now to say that I do not at all agree with you when you do not see any difference between Platonism and religion. Others, and among them H. Fields, say or have said the same, but although materialism declined under various forms sometimes hidden belongs to and even IS the main stream thing of our days, this vision continues to be debated. Sharing a religion presupposes faith, and faith, either you have it - that IS my case, but here this point has no importance - or you do not have it. Philosophy, comprising metaphysics, is a question RATIONAL inferences, and it is better to avoid any confusion between faith and philosophical rationality, although a scientist and/or worker in/on philosophy believing in God can be PERSONALLY glad to see that her or his scientific and/or philosophical results meet her or his faith. But this is another debate. Concerning scientific Platonism, I try to defend in my essay the following position: (i) All foundational approaches concerning mathematics and/or links between mathematics and physics are ultimately metaphysical. (ii) Platonism belongs to metaphysics as well as anti-Platonism, whatever it would be. (iii) Even cognitive approaches ultimately DO NOT escape to metaphysics: all investigations of human cognition are conditioned by human cognition. To appreciate this latter "objectively", human cognitive scientists should be able to get out from their cognition, to go "beyond" of all links between cognition and its objects, and this remains a genuine metaphysical idea. (iv) All we can do is to compare several competing metaphysical theories under EPISTEMOLOGICAL criteria such as simplicity in the logical sense of this term, complexity of primary and secondary hypotheses, internal consistency, consistency on the level of consequences and so on. On these bases, I try to show that (necessarily metaphysical)(necessarily metaphysical) Platonism is more plausible than its (necessarily metaphysical) competing theories. My approach probably and even certainly will not convince everyone, and I would be the last to pretend the contrary. Philosophy is philosophy, nor more, neither less. But I do not see how this kind of rational philosophical debate could resemble to the religious faith I share and assume, yet without confusing faith and reason.

    Well, I think, all is said.

    Like you, I do not participate to this contest to win 10 000 dollars. My motivation focus on all forms of constructive dialogue, comprising dialogue with authors defending opposite standpoints. Some authors - fortunately not all, far from it - have another vision and present their certainties as the ultimate truth and, on this occasion, do not always manage to avoid impoliteness and perhaps more. The present dialogue has been established following your initiative, and I appreciate your ability to communicate regardless of agreement or disagreement dimensions. I have read a first time your essay which is interesting because of its atypical aspects doubtlessly in connection with your pseudonym since it should be one, isn't it? Please excuse this repetition. However, precisely because of the atypical aspect of your essay, I have to read it again and to think about how to respond. But it would be done soon.

    Best regards

    Peter

      Hi Peter,

      I have not read all of your essay in detail but see that you have very clearly set out your arguments. Coming to the end I am left with two choices neither of which is wholly satisfactory, as I see it.

      I would prefer to see the idealized mathematics as a distillation from nature.A distillation of relations between and among concrete elements of reality, not existing in distilled state in an immaterial universe. I once visited the Oban single malt Whiskey distillery in Scotland. The purified, distilled alcohol is colourless. But it has had to be extracted into that state.(Its later maturation in barrels reintroduces colour.) Humans know the exact properties of such pure alcohols without requiring that such properties exist in an immaterial realm as overseers of the behaviour of the material product. Similarly a perfumier mixes pure odours to create new perfumes. The pure odours are extractions from nature. He may well have rules that control which odours are compatible but such rules do not come from an immaterial odour universe.

      The mathematics in nature is, as I 'see it', neither externally governed nor human made. Just as for the alcohol, this particular structure will behave in this way, other structures and co-existant relations between elements of reality behave in other ways; purely because of what they are ('muddy' or purified )and the environment in which they exist, in the here and Now.

      This is just food for thought regarding the possibility of a middle ground conclusion, that you may or may not have already considered. Thank you for sharing your deduction.

      Good luck in the contest. Georgina

      Dear Georgina

      Thanks for your comments. Your metaphorical approach is really interesting and I have appreciated it. But it is also double-edged. I will respond you, tomorrow as I hope, in greater detail.

      Best regards, good luck

      Peter

      Dear Georgina,

      Having carefully read your comment, I find your metaphorical reasoning very interesting. But it is also double-edged. (Concerning English expression, this opus is a catastrophe, but what can I do? I am (i) not a native speaker and (ii) terribly under time pressure.)

      You are absolutely right that a distilled alcohol does not exist in this state in nature. Yes, but for humans can distill alcohol by techniques remaining STABLE over time, the distillation ingredients belonging to nature must meet a sort of timeless and immaterial logic. Here, obviously, I have to clarify what I mean by IN THIS CONTEXT by "timeless and immaterial logic." Compare the following elementary propositions "The door is closed, so it is not open.", My sister arrives on 30th April 2015, so it does not happen to another date. "," Paul thinks that the sky is always orange, so he does not think the sky may have a different color." and so on and so on ... All these propositions meet the same elementary logical sentence "p or non-p". To say that logic including the sentence "p or non-p" is immaterial simply means that its sentences apply to materially very different phenomena, and that because of this principle only phenomena "interpreting" this logic are tangible; logic as such IS NOT. However, logic "exists" either way, and personally I do not find reasonable to say that the principles which for example "prohibit" the occurrence of simultaneously open and closed doors are man-made (or something similar). On the other hand, it seems plausible to assume that within 10 months (or 3 minutes, or 10 000 years ...), a closed door still can not be open.

      Now, if the fermentation of materially different substances such as malt infusion, grape must, other fruit juices, aqueous suspensions of potatoes or rice etc. allows alcoholic distillation allows all together under essentially equivalent conditions, this is because of chemical principles immaterial AS SUCH in contrast to the various substances to which these principles apply, knowing that these chemical principles existed before humans invented distillation and even before humans appear. Now to your second metaphor, the perfumer. Let this perfumer extract from 5 natural substances A, B, C, D, E, the corresponding essences e(A), e(B), e(C), e(D), E(e). From these essences, he or she creates a fragrance formula, for exemple F = 30% e(A) 20%e(B) 15%e(CC) 10%e(D) 25%e(E). Since this fragrance formula is "good", the perfumer takes the decision to keep it, ie to "reproduce" it in the future. In absolute terms, this idea of "reproduction in the future" is an IDEALIZATION. For reasons of complexity all fragrances effectively obtained in the future from this formula will be very close to the initial fragrance, or, in "mathematical" terms, will tend to the initial fragrance, but it will never be exactly identical to it. Certainly this infinitesimal approximation would not be possible without extraction and/or distillation, and moreover this IDEALIZATION we mentioned above has nothing to do with the fact that neither the ideal fragrance, nor its very close approximations do exist in nature.

      But now several problems DO occur.

      We are free to consider the fragrance formula "F = 30% e(A) 20%e(B) 15%e(CC) 10%e(D) 25%e(E)" as a "mathematized expression". But it is not a law of nature,and within laws of nature expressed by "real mathematics" the epistemological status of idealization changes.

      Everybody knows that the ancient Egyptians and Babylonians had obtained geometrical proto-theorems by idealization of empirical data, before the ancient Greeks discover the principle of mathematical demonstration. Consider the theorem of the angular sum of the triangle as it is formulated in elementary geometry due to the Greeks. The proof of this theorem requires directly

      Euclid's parallel postulate. Since the latter can be replaced by its both negations without undermining the coherence of the system, Euclidean geometry implies its non-euclidean extensions independently of the effective/human discovery of non-euclidean geometries. In other terms, idealization of empirical data can initially can foster the onset of mathematical discoveries. But it seems reasonable to assume that mathematical entities and their relations exist independently of their dicovery. Metaphorically speaking, mathematical entities and their relations are NOT "manufactured by intellectual distillation."

      This point can be illustrated by a more historical approach: As you know, Saccheri, around 1730, tried to prove Euclid's parallel postulate by reductio ad absurdum. He thought that replacing the parallel postulate by one of its both negations would entail a lot of inconsistencies. But these inconsistencies did not occur, could not occur. Despite himself, unwittingly, Saccheri was doing non-euclidean geometry, and this almost a century before it was discovered by Bolyai, Lobachevsky and others. Deducing from the previous lines that non-euclidean geometry existed before its discovery is perhaps "metaphysical". But explaining the necessary failure of Saccheri on the bases of anti-Platonistic approaches should require an inextricable configuration of ultra-complex hypotheses.

      It is now the good moment to recall that the expression "F = 30% e(A) 20%e(B) 15%e(CC) 10%e(D) 25%e(E)" is not a law of nature in the sense of physics. In contrast to F, a law of nature in the sense of physics permits the mathematical deduction of new laws. Two possibilities then arise: (i) The mathematically deducted law is confirmed by experience, and (ii) the mathematically deducted law is not confirmed by experience. If the case (i) occurs, all is said and done, whereas the occurrence of the case (ii) implies the necessity of a new paradigm comprising the ancient one as particular case. Until further notice, the definition of such a new paradigm is possible if and only there is ALREADY a mathematical edifice which (i) meets the new paradigm whilst including a mathematical sub-edifice corresponding to the former paradigm.

      In closing, let me clarify that from my personal perspective, nobody can definitively be right in philosophy and that this latter must remain a confrontation of standpoints for the sole purpose of mutual enrichment. Through the preceding developments, I just tried to explain why from my own standpoint, Platonism is the most plausible among all hypotheses responding to the double issue concerning the nature of (i) mathematics and (ii) the link between mathematics and phenomena belonging to the research field of physics. But probably this debate never will be closed, and other people have other standpoints.

      Thanks to make me share in turn your own viewpoint about all this.

      Best regards

      Peter

        Dear Peter,

        don't worry about your English it is brilliant. I really appreciate the time you have given to thoroughly and clearly addressing my comments. Lots of really interesting things in here, including the accidental discovery on non-euclidean geometry. That's new to me.You have done a very good job of defending your standpoint and explaining how the analogies I gave fit within it. Thank you so much.

        Best regards, Georgina

        Dear Peter,

        Thank you for writing such an elaborate comment.

        Before getting to the content of it, let me reassure you about your English. As you might imagine, you are dealing with a sophisticated English-speaking audience. They make allowances for any lapses due to the fact that English is not your first language. Btw, English is also not my first language, but I started learning it early enough to master it.

        So relax any concerns about your English.

        Despite what you (or others) might think, your view of the world is not that far removed from mine. I know that your religiosity is very sophisticated, and does not cause any trouble for science.

        I will first ask you about certain aspects of how you see Platonism. Is there a way to imagine the form (or format) of how structures (forms) can exist without people? If I could succeed at that, it would help me in understanding your view of things.

        Now I will introduce a concept that is not going to sit well with virtually anyone. Do you think that when we think we are deploying logic, that it is any different from other brain functions subject to the chemistry of the brain? Are we somehow connected to a greater "logic" of the universe? It seems to me that at all times we are subject to the "laws" of the universe, and whatever those lead to, we appear to believe that they are necessary and unavoidable. That is why logic is so compelling. Our brains are wired up so that we cannot defy "logic."

        It seems to me that we are not able to "shake off" the necessity imposed on us by physics and chemistry - that we must think in certain ways.

        I will leave it at this point until I see what you have to say about this.

        En

        P.S. It occurred to me that the above is not clear. All our thinking is "psychological." That is to say that despite our thinking that we are following some rules of "logic," we are in fact following rules of "psychology." It is inescapable. Yes, we make best efforts to follow the rules of "logic," and that is very desirable. But, in the end, if you find something to be persuasive, it is the psychology of your brain that is convinced.

        Dear Peter,

        The reason why I mentioned the Completeness theorem of first-order logic, is that I see it essential to the present discussion. Namely, I see it as showing that the truth of Platonism should not be searched for in opposition to formalism, but as equivalent to it, since it shows that the systems studied by mathematical theories are constructible from the formalism itself. So I cannot see how you came to present things as if Platonism and formalism were opposite to each other, and as if there was any interest for issues of Platonism in considering mathematical edifices in a non-formalized manner, unless you somehow failed to grasp the full meaning of the Completeness theorem.

        Dear Peter,

        Btw, why didn't your parents name you Pierre?

        I really liked your idiom "à côté de la plaque." It is just so descriptive that its meaning could hardly be captured by a simple phrase stating as much.

        It is a bit like the English idiom "kick off." It is used in business to indicate the beginning of a project that requires cooperation from many people. It may sound crude, but there isn't a comparatively efficient expression. You just cannot express the meaning of everyone ready for the run (which is where it comes from - North American football) more effectively. When I started my projects while working for business, it just would not work to say the "start." "Kick off" is just so much effective in conveying the intended meaning.

        En

        P.S. You might be interested in a person named Hugo Martin Tetrode (although there isn't a direct relationship). He independently derived his version of the Sackur-Tetrode Equation (at the ripe old age of 17).

        Folklore has it that Einstein (accompanied by another guy) tried to visit him in Holland, but he turned them away. He could not be bothered (with Einstein, of all things). There are things in this world that defy our attempts to "understand" them.

        So if you wonder why "Considering himself a non-Platonist, Hamkins thinks that [set theoretical] multiverses have an objective existence as suggested by Platonism", the reason to me is clear : the results of mathematical logic are not leaving us any significant philosophical choice what to believe in. The Completeness theorem essentially proves the existence of set-theoretic multiverses, leaving the labels "Platonism" and "non-Platonism" essentially obsolete by lack of a remaining unsettled question to disagree on.

        The only non-refuted philosophical alternatives to the belief in set theoretic multiverses are quite poor and hardly defensible : denying the existence of an actually infinite set of natural numbers, or suspecting set theory of inconsistence. I gave a philosophical argument for the consistency of ZF, inspired from the logical proof of equivalence between the axiom schemes of replacement and reflection.

        Dear Georgina

        Thanks for your kind respond. Just a little remark which seems to me essential in the context of this debate.I find it is difficult to decide whether non-euclidean geometry was discovered accidentally or not. It is certain that a "pre-programmed" discovery, and, a fortiori, a discovery "known in advance" is not a discovery. On a purely heuristic level, any discovery made by humans involves contingent factors and is at least partially determined by trial and error. But in absolute terms, I think the formulation following is more adequate: Within the framework of global geometry, Euclid's parallel postulate has in some way the structural property to be substitutable by its two possible negations. In any case, this property would have been discovered, perhaps by another trial and error approach.

        The case of Saccheri is interesting; it evokes a bit that of Columbus. The latter, hyperfocussing on his project to discover a new route to India, did not realize until the end of his life that he had discovered a new continent. At least, this is our common contemporary vision. But anyway, Saccheri corresponds to this scenario. Hyperfocussing on his project to "prove" Euclid's parallel postulate by reductio ad absurdum, desperate by the unavoidable failure of this project, Saccheri could or would not realize that he had discovered the possibility of non-euclidean geometry and even non-euclidean geometry as such. But it is commonly agreed that Sacceri was doing non-euclidean geometry without being conscious of it. For my part I think, for we can do something unwittingly, this "something" must exist objectively. Hence the case of Saccheri represents an argument advocating Platonism.

        Kind regards

        Peter

        Dear En,

        I will try to answer to your question if there is a way to imagine the form (or format) of how structures (forms) [[in a Platonistic sense]] can exist without people?

        First it should be clarified that the question "how XYZ can exist without people?" concerns not only Platonistic items but all kind of existence comprising its material aspects. Here we touch an old philosophical problem which has never found a satisfactory solution. Consider a video system filming material reality. This system reproduces reality "in its own way," including 3D 竊' 2S mapping and thereby reconfigurations of proportions, and generally some deformation of colors.

        As humans, we are outside of the circuit linking the system to reality, and vice versa. In other terms, we can check the degree of adequacy between reality and its mapping by the system. And more especially, w can know whether the system is effectively video REPRESENTING parts of reality, or for example a hologram synthesizer CREATING it.

        But regarding the links between human cognition and reality this cognition is supposed to represent, the situation is essentially different. Here, we cannot get out of the circuit, since we are unavoidably integrated to it. Of course, if someone assure us that trees are red or blue for her or him, we can reply to her or him that for a very significant majority of humans trees are green. However, we cannot know anything about elements of reality "pe se", ie reality such it exists beyond its conditioning by cognition. Perhaps our cognition deforms colors just like video systems can do it, and only cognition modalities globally shared by a significant majority of humans give us the illusion of "objectively green" trees and of a sky "objectively varying from blue to gray". Perhaps reality is created by our cognition, and only the (highly hypothetical) "fact" (??) that a significant majority of humans share very similar cognition modalities permits us to say (or to hope?) that we share a "same" reality. Here, a big problem occurs: since all philosophical and/or scientific approaches of the link between human cognition and reality are intrinsically conditioned by human cognition, there is a genuine CIRCULARITY, and, by definition, any tentative to break out of circularity is metaphysical. So a lot of philosophical/metaphysical theories are competing about this issue; some among them seem rather zany, others are more plausible. But in absolute terms they are all equivalent in this sense that they are neither provable, nor refutable. The thesis claiming that the material reality such as we perceive it could also exist WITHOUT PEOPLE is NOT the most convincing; by contrast it encounters terrible lacks of consistency.

        Briefly speaking , not only the idea of a Platonistic reality is highly problematic. No conception of reality escape to irresolvable epistemological problems. Material reality is "familiar", but sole familiarity does not mean knowledge.

        Facing material reality, all we can do is to seek the simplest way to approach it. I mean the simplest way in the philosophical sense of the term, which is not ever simple in the common sense.

        Consider a macroscopic physicist, in order to avoid supplementary problems concerning quantum epistemological considerations about "reality" taking sometime very exotic forms. This macro-physicist, before beginning her or his purely physical investigations should theoretically interrogate her/himself if the phenomenon to investigate "really exist", because - always theoretically - it would be futile to investigate a phenomenon which perhaps "does not exist." But since all consideration about the "existence of material reality" are metaphysical and thus without definitive solution, and since on the other hand these metaphorical controversies do not add anything to a purely physical investigation, the physicist mentioned above can operate AS IF the existence of the material reality would be "obvious", and although this is NOT the case, the macro-physicist's (tacit) choice to neglect all metaphysics is a legitimate way to avoid complications being absolutely useless in her/his domain.

        That is what I mean roughly speaking by "simplicity" in the philosophical sense of the term.

        Now I wonder why people working about foundations of mathematics and/or links between mathematics and physics would not adopt an equivalent attitude regarding Platonism IF this represents the simplest way to approach these issues. I insist on this IF. For the moment, the issue if Platonism represents the simplest way to approach mathematical foundations and links between mathematics and physics. But it is a manner to say that there is no reason to develop any apriorism against Platonism because of the immaterial reality it presupposes. The material world the physicist mentioned above consider tacitly as existing could also not exist. It is not materiality which could elucidate the question why the material world exist instead of not existing. Why then deny a priori

        the existence of an immaterial word existing - exactly like the material world - because it is existing, in the event that mathematics as object of DISCOVERY would the simplest way to approach FOM and related issues?

        Of course to know if it is so, me must return to your other question: How can we accede to an immaterial world au cas oテケ il existerait?

        Here I think we must avoid that what some people reproach to Paul Benacerraf. According to Benacerraf, Platonism is weakened by the "fact" (?) that for an immaterial Platonistic world of objectively existing mathematics, there is nothing equivalent to human empirical cognition. But this point is neither specially astonishing, nor an anti-Platonistic argument. Since a Platonistic world is not a material one, its investigation modalities are not necessarily equivalent to empirical cognition.

        According to my own standpoint - it is not the only one; our little Platonistic family is not family quarrel free, as this contest confirms it - the best way to decide between Platonism and its challenging approaches consists on comparison under epistemological criteria such as simplicity in the logical sense of this term, complexity of primary and secondary hypotheses, internal consistency, consistency on the level of consequences and so on.

        Here is an example how proceed in this way, and more precisely an example to consider in its historical context. As you know, Saccheri, arround 1730, tried to prove Euclid's parallel postulate by reductio ad absurdum. He thought that replacing the parallel postulate by one of its both negations would entail a lot of inconsistencies. But these inconsistencies did not occur, could not occur. Despite himself, unwittingly, Saccheri was doing non-euclidean geometry, and this almost a century before it was discovered by Bolyai, Lobachevsky and others.

        The case of Saccheri is interesting; it evokes a bit that of Columbus. The latter, hyperfocussing on his project to discover a new route to India, did not realize until the end of his life that he had discovered a new continent. At least, this is our common contemporary vision. But anyway, Saccheri corresponds to this scenario. Hyperfocussing on his project to "prove" Euclid's parallel postulate by reductio ad absurdum, desperate by the unavoidable failure of this project, Saccheri could or would not realize that he had discovered the possibility of non-euclidean geometry and even non-euclidean geometry as such. But it is commonly agreed that Sacceri was doing non-euclidean geometry without being conscious of it. For my part I think, for we can do something unwittingly, this "something" must exist objectively. Hence the case of Saccheri represents an argument "advocating" Platonism, but "advocating" of course does not represent a proof. As on the other hand ALL this is metaphysics, and the question of proof does not even arise. But we can compare Platonism to its competing approaches under the criteria mentioned above.

        Saying that the Platonistic mathematical world exist just like the material world which exist on its side instead of not existing, well, it is perhaps a heavy hypothesis, but not heavier than the hypothesis concerning the existence of the material world.

        Let us now compare the Platonistic hypothesis to yours.

        If I understand well, you are postulating that our brains belonging to the material universe are subsequently subjected to the laws governing this latter. Symmetrically our brains can accede to the laws of the material universe. This is another case, but personally I think to hold in terms of overall consistency, it requires a lot of secondary hypotheses. First, our brains certainly belong to the material universe in this sense that they are in space-time. But do they also belong to the material universe in a physical sense emphasizing specificity of physical laws? The opinions are divided about this point. First me must mention here the so-called "anthropic principle" in its weak and strong versions. For the weak version, I think there is no problem. Simplifying, we can formulate it as "Since there is an evolution beginning with inert matter and being achieved by human brain", the properties of the material universe must be compatible with this evolution. No problem, but un universe "allowing" the emergence of the human brain is only a necessary, but certainly not a sufficient condition for the effective apparition of our brains. So some people formulated the "strong anthropic principle", somewhat like ""Since there is an evolution beginning with inert matter and being achieved by human brain", the properties of the material universe must IMPLY this evolution". Well, if I said "somewhat like ...", that is because the exact formulation of the "strong anthropic principle" is controversial even among its supporters. And other people deny its validity. So this "principle" is not a principle but just a very complex and controversial hypothesis. On the other hand, each approach making links of cerebral structures and our knowledge concerning physical laws should entirely master the issue touched by the "strong anthropic principle", and this it not possible since neither this latter, neither its challenging approaches can be proved or refuted.

        A each one to see if the Platonistic hypothesis is not, epistemologically speaking, much simpler.

        Well, this discussion keeps open and must be continued. I will also do my best to answer to your further questions formulated in other posts.

        With kind regards

        A bientテエt, j'espティre

        Peter

          5 days later

          Dear Peter,

          Thank you for your detailed explanation. I still plan to explore some of your ideas further, and will post a comment here later this week.

          So don't give up on me yet.

          I wish you and your family a Happy Easter.

          En

          Dear Peter

          nice essay, though too technical for me. I couldn't seriously grasp the technical part, since this would require a thorough study from my side.

          I can tell you what is my position, and you can advise me if there is any incoherence from your standpoint:

          1) Metaphysics (onthologies) are methodologically irrelevant.

          2) Mathematics is an intimate and structural part of human language, and as such comes from experience (see e.g. George Lakoff and Rafael E Núñez (2000), Where mathematics comes from: How the embodied mind brings mathematics into being. Basic books)

          3) Language is conventional, and as such is formalized.

          4) By definition, the "laws" of physics are true everywhere and ever, otherwise their variation is regulated by another law.

          What do you think?

          Happy Easter

          Mauro

          Dear Mauro

          Thank you for your very rapid response; I'll try to take a position regarding your philosophical ideas.

          Saying that metaphysics (onthologies) are methodologically irrelevant, you are absolutely speaking right, but unfortunately we are condemned to metaphysics. Carnap, main figure of logical positivism, relegates ontological realism to metaphysics, just like idealism and other ontologies. Of course, ontology is metaphysics par excellence, but asserting "A 'is' B", for example "Mathematics 'IS' an intimate and structural part of human language", or "Mathematics 'IS' an immaterial reality" and so on, ALL this is already ontology. If assertions about the link between mathematics and physics were scientifically refutable or at least temporarily confirmable, this contest should operate on very different bases. But it is not so. All we can do is to compare competing metaphysical approaches under epistemological criteria such as simplicity, economy of hypotheses, internal consistency, consistency concerning consequences and so on.

          Regarding point 2) of your reply, I think it is much more complicated. Human language as such has no any deduction potential. It is the same for experience as such and experience related by common language. Elementary geometry had been discovered by experience and initially described by common language, but common language would not have sufficed to derive non-Euclidean geometries, whereas human experience was hindering non-Euclidean geometries to be accepted. Christine DANTAS wrote a very interesting essay non-reducible to a simple mathematical language. Having had with her very rewarding exchanges, I recommend you particularly to read her essay.

          Yes, language is conventional. It is the same for formal systems, but formal systems are not the same than FORMALIZED systems. Formal systems are arbitrary assemblages of meaningless signs following arbitrary rules on the base of arbitrary axioms. According to Hilbert, formalizing a given mathematical edifice means to operate AS IF we reconstruct this edifice like a formal system. This kind of "formalism" - I do not particularly like the term "formalism" because of its polysemy difficult to control - treating mathematics (and physics) AS IF they were ... is perhaps the only way to escape to ontological assertions like "Mathematics IS ..." or "Physics IS ..."

          Personally, I would add that we can lessen the weight of the ontology focusing on relations between mathematical entities instead of aiming these entities as such. That is why group theoretical approaches are precious, provided it is adequate. But this is the case concerning the links between mathematics and physics stricto sensu.

          Well, I hope having answered your questions.

          Best regards

          Happy Easter

          Peter

          Dear En

          Take your time, il n'y a pas le feu. But I am of course curious about the continuation of this fascinating debate.

          Happy easter

          Peter