In the beginning there was Logos. Christian Bibles tend to translate this to the “Word”, or maybe the “Verb”. But Wiktionary tell us λόγος also means “speech, oration, discourse, quote, story, study, ratio, calculation, reason”. It further descends from the verbal phrase λέγω — I say or think. Insofar as logic is (quite literally) that which pertain to logos, it ultimately refers to that which can be said or thought. Our philosophical influences teach that, much like anything else in the Mechanosphere, the Said or the Thought are both operated in two movements — one of selection and one of consolidation. This is how the same word comes to mean both “oration” and “calculation”.
Logic is therefore twofold: a matter of production, and a matter of arrangement. This is so even in the 20th-century Tarskian program. The better logicians since then, such as Jean-Yves Girard, have pressed the issue of formatting. But the chrematistic promise of mathematics is to find invariants that hold for any representation within an equivalence class (e.g. up to isomorphisms or order-preserving transformations). The wages of abstraction are emancipation from the format. For example: 19th-century economists had to speak in bushels of this versus bushels of that, whereas modern mathematical economists can speak in measure spaces — therefore obtaining study, calculation and even oratory without quotes and ratios. In the Tarskian program speech and discourse should also vanish, subsumed under the mightier and mightier program of reason. There’s no reason why this shouldn’t work, but also no reason why it should. The proof is in the pudding; the pudding is in the eye of the beholder; the metaphor is a metaphor for the metaphor.
A philosophy of mathematics only emerges when philosophers fixate on mathematics as an object of philosophical inquiry. On its own, mathematics is fine; a philosopher who interviews a mathematician might come away with some kind of unstable “strategic platonism” or as well-gatekept but frank social construction — but these are philosophical, not mathematical issues. This is in stark distinction with science, which imports (in the precise sense that computer programmers are used to) a black-boxed metaphysics of materialist stabilism.
Whenever science has tried to give a scientific veneer to this imported metaphysics, it has looked childish: the agitating material cause of gravity is christened “the God particle” because, well, it causes stuff to clump together. (It takes a philosopher to imagine a world that is motion first — where differential equations are not solved anymore than “x=x” is in the stabilist’s gut) Mathematics, on the other hand, embarrasses itself only when it forgets the μάθημα — its royal knowledgeness — and tries to prove itself in an underlying true arrangement of symbols (the logic that would underwrite it). A delicious paradox — its deliciousness being that it’s no paradox at all: mathematics, which is “socially constructed”, exists anyway, like the natural course of a river. Meanwhile science might have so far scraped the tip of a rock and our theoparticles maybe special cases of special cases, contrived like epicycles and possibly very unlike the science of eventual receivers of the Voyager Golden Record.
This is why our message to the stars needs always to begin with Bach. It tells eventual interlocutors that we’re aware of the μάθημα, that it has emerged in some form in our midst. But also in the general case — there’s no harder or fraught with possibilities for error and self-delusion alternative to bringing our μάθημα to enemies or strangers of any kind. Take the arena of politics: how long until we stop pretending to have facts and figures and a tentative science of governance? Unless much of the general populace is evil (and actually, even then), there’s a raw (not royal: there is only one mathematics) knowledgeness to their points of view. What is it? Jazz musicians are interested in Chopin, photographers in Rothko, algebraic topologists in algebraic geometry. Surely enough, and exactly like in mathematics, not every raw insight has μάθημα to it — this is the point of true proofs in mathematics, to establish true theorems with true corollaries. Accessing the knowledgeness in situations requires the painful and frustrating act of thinking. But that’s precisely God’s one weird trick to transcendence and maybe even general axiology.