Dear Anshu,
Dear Tejinder,
You are anti-Platonists (or rather non-Platonists), and I am a Platonist aware of all difficulties this option represents. But similar difficulties do undermine ALL options that could be discussed within the framework of this contest. Personally I see (i) this contest as a philosophical one and (ii) philosophy as the choice to focus on issues that do not allow one single answer. Of course sometime an initially philosophical question finally gets a definitive answer, but in this case it ceases to be a philosophical question. Anyway, regarding foundational research about mathematics and/or links between mathematics and physics, we must "empirically" recognize that for n people debating about, there are at least n + 1 opinions. Under these conditions, the BEST we could do is to try a mutually benefit full constructive dialogue, and this primarily with colleagues defending "opposite standpoints." You agree probably that such a constructive dialogue should be for everyone the main motivation to participate in this contest.
The calm, circumspect, and courteous background tone of your well written, highly interesting essay expressing a wide deepened culture about history of mathematics and physics make me think that you are seeking in turn dialogue.
Before going further, I would state - even repeating myself - that it has been an enormous pleasure for me to read your really well written elegant essay with its beautiful symmetry between sections I and II. You are absolutely right to emphasize that mathematics and (the interminable stammering of) physics initially knew essentially separate paths, and that the manifest collusion between physics and (a relatively modest part of) mathematics represents at least at first glance a mystery to elucidate; otherwise the subject of this contest would not have any reason to be proposed.
Well, and now let us discuss some essential points of your essay, with the sole aim to exchange ideas.
You are wondering how are we ever going to be able to scientifically prove Platonism. Here, my position as a Platonist - I expose it at length in my own essay - is clear. Platonism IS metaphysics and still metaphysics and nothing but metaphysics. Yes, but all competing theories of Platonism are as well metaphysical as Platonism they are trying to "refute". If such a refutation was possible, the competing theory in question would transform Platonism in a no longer metaphysical but scientifically refutable approach being effectively refuted, just like it can happen to other entirely scientific approaches. At first sight it seems certainly strange to postulate that "there is a mathematical universe, which we somehow grasp in an extra-sensory manner". But in my own essay, I quote a reference text of R. Carnap where this categorical anti-metaphysical main representative of logical positivism relegates material realism to metaphysics, making no difference with other ontological theories like idealism. Of course, we receive information on material reality by our sensory channels before cognition processes this information. But it seems to me that the problem is just THERE: To evaluate (i) objectively and (ii) non-metaphysically the degree of adequation between the image we have of reality following its cognitive processing and reality "as such" (??), we should transmute us in pure immaterial spirits able to go out/beyond of our cognition. The fact that all our information about material reality run through our sensory channels, whereas there is nothing equivalent concerning our access to an immaterial mathematical world being objectively given, this only fact does not refute Platonism. First, nothing allows us to predict that humans will "never" discover brain channels able to receiving directly immaterial information. At present we do not know anything about that issue - at least to my knowledge - but anyway, scientists who by definition do not have the vocation to utter metaphysical and /or arbitrary propositions should not play the prophets of what "will or will not be discovered." I would add that the (too) famous "Benacerraf argument against Platonism" - "we have good knowledge about cognition dedicated to the empirical reality, whereas there is no equivalent for an immaterial world" - has not at all "definitely refuted Platonism." Some authors like Penelope Maddy even do not think that Benacerraf would advance an anti-Platonist evidence. Anyway, discussions about "Benacerraf's argument" are continuing for 43 years, and nobody sees the end.
On the other hand, the material reality which exists - let us admit that it exists - could also not exist. A scientific is generally not interested in the issue "why exist the material reality, instead of not existing." Well, but this issue equivalent to this other: "why should an immaterial objective mathematical world NOT exist instead of existing?" Everywhere we encounter metaphysics, much more metaphysics then we would admit it spontaneously. In other terms: If we deny the existence of material world, we must assume an infinite number of epistemological problems. But we must also ask - without any a priori - if it is not the same for Platonism, and more specifically, if the denial of Platonism does not cause in turn a jolly good number of epistemological problems regarding for example the links between mathematics and physics. In my own essay, I try to raise such problems, leaving it to the reader to appreciate these problems in her/his personal manner.
In your essay you say that "in their work physicists and mathematicians generally prefer to ignore or 'forget' the brain (...)." Personally, I don't think (i) that it is a question of ignoring or forgetting and (ii) that this point is not a detail. Let me use a metaphor. Imagine you are resolving a mathematical or physical problem, using a computer to facilitate your task. Doing this, you don't "ignore or forget" your computer and its internal functioning. But in this context, the computer is accessory. Perhaps and even probably it is an unavoidable mediator between your work and the given object of your work, may it be mathematics or physical reality. This certainly does not hinder you to be interested about hardware oriented computer science computer science. However, making mathematics or physics as such, you tacitly are bracketing all issues related to computer science or hardware. Now it is clear that your computer and more generally each information processing system must be equipped so that it can carry out its task. Hence the system is supposed to share the logic (and other factors) of (i) the task to be effectuated and (ii) the object that this task concerns. But this does not necessarily mean that information treating system "GENERATE" or "CREATE" the object of its task. Moreover, in the case of a computer or other artificial information processing systems, we can KNOW that the system does not "generate" or "create" the object of its task, since we are outside the circuit linking the system to the object of its task. But it is not the same for our brains, or, more generally, for our cognition.
Of course, to be able to make mathematics or physics, our brains must be equipped for. Concerning mathematics, it is often said that elementary mathematical structures, principally counting, are innate and do pre-exist in infants to all effectively effectuated operations. Given the complexity of the issue, I am not sure that this point can be categorically asserted, but it is not so important. Not only it is obvious that our brains are disposed for mathematics - otherwise there would be no mathematics, at least no mathematics effectively written on paper - but we can still do some interesting overlaps between the theories of "constructivist" philosophy (Brouwer, Heything ea) reducing any form of mathematics effectively constructable to "intuition of counting", and, on the other hand, modern computational mathematics. Regarding physical reality (in a very vast sense), it is in turn clear that our cognition is able not only to register it, but also to investigate it. Otherwise there would be neither physics, nor other sciences.
But now complications arise.
To respond - in cognitive terms - to the following issue "How to explain that (a part of) physical phenomena correspond to (a part of) mathematics being (at least potentially) pre-programmed in / by the human brain?", we need very heavy hypotheses. In a cognitive perspective, if we assume that mathematics is at least potentially rooted in the brain, we must also assume that our brains organize the physical reality in the way that it aligns with given part of mathematics. This lead to a kind of Kantian or neo-Kantian philosophy which inevitably will be the subject of endless controversies. But regardless the option we would adopt in this area, the fundamental problem remains the same: unlike the information processing system mentioned above, where humans being outside the circuit linking the system to the reality can compare the treated information to its original, human cognition CAN NOT "get out from itself" in order to check the degree of adequacy between the reality as such and the reality as it is conditioned by cognition. In other terms, there is CIRCULARITY, and each tentative to break an objectively given circularity leads to metaphysics.
Nevertheless it is not a deadlock, but the solution can not be classical.
The position I defend in my own essay is the following: (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 a necessarily metaphysical Platonism is more plausible than its (also necessarily metaphysical) competing theories. Further details are in my own essay. Here, I try to summarize it as follows: Platonism is based on assumptions neither provable nor refutable. But it's the same for anti-Platonism. On the other hand, anti-Platonism must additionally assume some forms of circularity that Platonism can - at least in purely logical terms - avoid.
I would be glad to know your counter-propositions about my vision I tried to summarize here above and also to know the criticism that you would not fail to address to my own contribution. It would be precious to of further reformulations on the foundations of a broader vision. Ultimately, this the raison d'être of the present contest.
With best regards
Peter
