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 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.
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
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