Dear Anshu,
Dear Tejinder,
First let me state that I absolutely agree with your conception of a mutually beneficial constructive discussion. The subject of this contest is the stake of a long debate that continues for many many decades and probably will never find an end. Proclaiming to possess the ultimate truth in this domain seems rather pretentious. For my part, I participate in this contest with the sole purpose of exchanging ideas and I am happy to see that is the same for you. Further this kind of discussion is a good feedback; some of your remarks make me realize that I have to find better formulations.
In response to your post, I'm not going to proceed in the same order, but start with the problem that seems the most fundamental.
To my own formulation "An ineradicable prejudice alleges that hypotheses reserving the notion of objective existence to the only material world would be "more scientific" than other hypotheses (...)(...)", you respond "Wouldn't this make all of science into metaphysics?". For my part, I think, YES, perhaps all science but certainly all epistemology IS undermined by metaphysics. Saying "mathematics IS ... ..." or "physics IS ... ... " already belongs to ontology, and ontology is metaphysics par excellence. To escape primarily to metaphysics, all we can do is to say "Let us do AS IF mathematics were ... ..." or "Let us do AS IF physics were ... ..." This was Hilbert's choice. Hilbert never said that "mathematics "is" an assemblage of meaningless signs etc.etc." For essentially proof theoretical reasons, Hilbert tried - in vain, as we know from Gödel - to establish one-to-one relations between such "formal systems", ie effectively meaningless/arbitrary systems, and GIVEN mathematics being meaningful. Now, characterized anti-Platonism REDUCES mathematics to these formal systems, ie arbitrary assemblages of meaningless signs. Of course you do not share this position, since your are not anti-Platonists but non-Platonists.
Anyway, in the context of my own approach, the AS IF strategy diverges from the approach of Hilbert. Starting with the idea that any epistemology is ultimately metaphysical, I try to determine which AS IF position about mathematics and / or physics is the most consistent one, the "cheapest" in hypotheses and so on. Here I think that among all competing metaphysics, Platonism, although not being problem free, is still the best choice. This opinion is not shared by everyone, but I'll try to explain my standpoint. Let us consider reality in the usual sense: oceans, continents, humans, animals, plants and so on. Ontologically, so metaphysically speaking, a lot of competing theories had been formulated about the status of reality. Berkeley for example thinks that the elements of reality exists if and only if they are perceived empirically. In the vision of Berkeley - a mixture of empiricism and idealism - I would exist for you and you would exist for me if we were communicating by telephone, since audition is a form of perception. But since we communicate on a forum, our respective perceptions concern only our texts, but not ourselves. So you communicate with me, and I communicate with you, but neither I exist for you, nor you exist for me.
This is of course a caricature of philosophical idealism, but in absolute terms, such a caricature is neither provable, nor refutable, and it is exactly the same for all ontological approaches of reality. We must add that no ontological conception is entirely satisfactory, including theories being much more reasonable than Berkeley's. On the other hand - I have to repeat it - no one can get out of the mental representation he or she has of material reality. Since we cannot escape to metaphysics, all we can do is to chose the - (episte)logically speaking simplest approach of reality, ie to assume simply that reality is objectively existing. Concerning the communication process between you and me, the best "ontology" is to admit that we all exist objectively. If now someone - nobody among us, probably - hypothesized that I only exist in your brains, and vice versa, she or he would face similar difficulties to those of Berkeley.
It is in this sense that in my opinion the hypothesis of an objective existence of mathematics is the best one, or more precisely the less problematic one.
Consider now the hypothesis that mathematics are an emanation of human brains. If this hypothesis is true, the mathematical research would do better to turn to brains, there where mathematics could be found. This had been said somewhere on this support. Well, perhaps such an hypothesis explains how Saccheri could make non-Euclidean geometry a century before its discovery, and without realizing it. (Further little details are in my post concerning your essay.) Perhaps non-Euclidean geometry was already in the brain of Saccheri although he ignored it. But now several question arise: Since mathematics are in our brains, why these latter do not discover spontaneously all mathematics being in them? Why mathematical discovery in fact aligns itself on dynamics we meet in any areas which, like geology, is not in our brains? On the other hand, human brains result from biological evolution. Now, can we be sure that mathematics result in turn from biological evolution? Well, let us admit that it is so. But in this case we may collide with the following point: some parts of mathematics are interpreted by physical laws already valid before the beginning of any biological evolution on our planet, as it is confirmed by astrophysical observation.
Of course, it IS possible to give consistent answers to these issues, but this kind of consistency must be paid by exponentially complex hypotheses.
Concerning the role of experience in mathematical discovery, I agree that that experience can be a first priming. Some very elementary theorems of plane geometry initially had been discovered by empirical means. Yes, but general relativity was not discovered empirically. It came out as Einstein's intellectual reaction to some unsatisfactory aspects of special relativity. On the other hand, general relativity could not have been formulated without Riemannian manifolds previously available.
And finally, I don't think that " perhaps future (more advanced) brains will be able to communicate with the Platonistic world. " Actual brains already can do it. But the communication modalities are not the same for the material world and the Platonistic world. For the the material world it is in first approximation perception, and for the Platonistic world.
Hoping that all this serves initialization for an exchange of ideas, even after this contest
Best regards
Peter
