Dear Mrs. Dantas
An ineradicable intellectual fashion - probably a survival of logical positivism which for long had exercised a real dictatorship on philosophy of science - reduces mathematics to "a kind of language."
It is a real pleasure for me to discover in your essay some as original as relevant objections to this commonplace.
Before we talk about your paper, I wish to reiterate my great astonishment to see so many people persist on the reduction of mathematics to "a kind of language", knowing that some rather trivial reasonings are sufficient to shake these certainties.
Language in the original sense of the term is a cultural phenomenon which for this reason (i) changes when we move from a linguistic community to another and (ii) evolves over time in an at least partly contingent manner. Mathematics, among humanity, is universal, and any evolution of mathematical knowledge - because "mathematics" and "knowledge of mathematics" do not mean the same thing - is never contingent. The emergence of a new paradigm, such as non-Euclidean geometry, always occurs so that the old paradigm becomes a particular case of the new. When new fields appear, such as probabilities at the time of Pascal, they always represent bonds of unity with already existing fields, or even enhance this unity, as it is the case of Cantor's set theory. (Here, obvious links can be established with (i) "insaturation of mathematics" and (ii) "[nature as a] space for realizations to come.") Finally, Frege already said that changing mathematical symbols does not affect the meaning of mathematical propositions - for example, the jargon of Schwartz distributions prefers "Dt" to "df / dt" and "I" to "∫" without changing anything at this level) - whereas contingent changes in the configuration of phonemes regarding a given language would make the latter ineffective on its role in communication.
In short, even at this basic level, a gap opens between mathematics and "a kind of language", but by going further, epistemological discernment inevitably leads to your starting assumption. If we consider mathematics as a kind of language, it is impossible to attempt a rational explanation of this "unreasonable effectiveness" of mathematics in physics. To try such a rational explanation, we first have agree the idea that mathematics must be something else / more than just language. Your final words mentioning the famous Galileo quote aptly summarize the problem: It is not sufficient that the great book of philosophy (here philosophy of nature) is written in mathematical terms. This book can be adequate if and only if th nature in turn "behaves mathematically", and if this is the case, the reduction of mathematics to a language like any other can not be satisfactory.
I totally agree with you that mathematics can be used as language since its allows the communication of laws of nature, but this does not explain how or why nature "behaves" in a mathematizable way. For the "unreasonable effectiveness" of mathematics in physics to be rationally explainable, there must be some deep analogy between mathematics and nature.
Regarding this analogy, your approach in terms of self-reference and insaturation is as interesting as convincing.
Nature is there because it is there, knowing that it could although not be there. On the other hand, nature is becoming, i.e. "a space for realizations to come", as you say. (If you envisage a longer study, you might perhaps reformulate the latter point in terms of block universe suggested by relativistic constraints. The block universe, certainly controversial, is not inconsistent with "a space for realizations to come", but requires reformulations. Since you are reading Jean CavaillГЁs dans le texte, you manifestly speak French; therefore I recommend you, in this specific context, the french paper of Fabien Besnard "Temps des philosophes, temps des physiciens, temps des mathГ©maticiens" http://fabien.besnard.pagesperso-orange.fr/articles/temps.pdf . Regarding mathematics, please allow me to make a little remark. When you say - with CavaillГЁs - that "Mathematics constitutes a becoming (...)", you risk claiming in spite of you - and despite CavaillГЁs - constructivist views opposing your own self-referential approach of mathematics. It is for this reason that I insisted above on the anti-constructivist differentiation between "mathematics" and "knowledge of mathematics." Our human knowledge of mathematics is obviously becoming ("en devenir" chez CavaillГЁs), but this does not conflict with the conception of mathematics existing per se, timelessly, irreducibly to anything, in short, existing only by self-reference. Certainly, in absolute terms, all approaches claiming objective existence of mathematics are metaphysical theories, nor provable nor refutable. But in my own essay, A Defense of Scientific Platonism without Metaphysical Presuppositions by Peter Martin Punin, I try to show (i) that competing theories of this ("Platonistic") approach are in turn metaphysical theories and (ii) that, compared to its (neither less, nor more metaphysical) competing theories under epistemological criteria such as simplicity, economy of assumptions, consistency and so on, approaches claiming objective existence of mathematics as edifices irreducible to anything are still the most plausible. Here let -me just mention a historical example which, in my opinion, denotes that mathematics exists before being discovered. 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 making non-euclidean geometry, and this almost a century before it was discovered by Bolyai, Lobachevsky and others. Deduce from the previous point that non-euclidean geometry existed before its discovery is perhaps metaphysical. But explaining the necessary failure of Saccheri on the bases of ultra-formalism or constructivism would lead to arguments approaching farfetchedness.
Let us therefore assume the objective existence of irreducible mathematics. From this perspective, knowledge of mathematics is insaturated, since at any time, something remains to be discovered. However, from an ontological perspective we can not speak of unsaturation of mathematics; the latter exists as such. By contrast,we can mention the "physical" insaturation of mathematics, in the sense that only a small part of mathematics is interpreted by nature. But, on the other hand, nature as "space for realizations to come" tends to invest "physically" insaturated mathematics. Finally, since our knowledge of nature is in turn insaturated, discoveries about nature can favor discoveries in mathematics.
Please, just tell me even if my interpretation of your final words is correct:
Laws of nature and mathematics are the same, except that mathematics transcends knowledge areas covered by laws of nature.
Well, I hope, that my understanding of your essay is not too far from what you mean. Anyway, this reading was actually beneficial for me. I obviously have to apologize for my English, but I am (i) not a native speaker and (ii) always under time pressure, knowing that il faut de tout pour faire un monde. And if you have the time, it would be a pleasure to know your comments on my own contribution My paper is unfortunately not animated by the admirable humor being yours. It may be a matter of personal style and probably also of generation.
With best regards
Peter Punin