With numbers, I don’t understand the question. I have no idea what it may mean to say that beings are number, or numbers are beings. I am strongly inclined to think that that sentences like these pop out from oddities of language, they make little or no sense to me, I’d need some kind of explanation to begin talking about it.

I think it is a very serious mistake to think that logic is about foundations, or that it helps a lot with the conduct of political arguments or even most of science. One of the problem is an embarrassment of riches. I can construct an n value logic - so what? What does this show? I can also construct boring old propositional logic - so what? What does that show? Some of these things capture our intuitions about things, which is to say they partially mirror or explicate the way natural language or political and social cognition work. But they suceed or fail to the extent that they explain, we chose the ones that seem adequate.

2. I think logic is all about foundations. Logic is the study of how arguments can ground themselves, comming up with precise definitions of what counts as truth, evidence, etc.. I can understand language games and such, but its a rather problematic gesture when applied to logic. I think here, we might be in danger of abandoning logic altogether when we believe that foundations don’t matter. Maybe that’s what you want… in that case, fine.
Historically, the formalists lost. Frege’s entire project was abandoned after Russell’s letter on the “Groundlaws for Arithmetic,” I don’t see how Frege can continue to be a viable option. But, here I might be guilty of not reading correctly. On this point, I anticipate correction.

3. The relation between proof theory and phenomenology has nothing to do with mental states. It rather has to do with experience and the possibility of experience. More on this later.

It’s kinda funny. I come from phenomenology background, but mostly, I think that is a pointless project now, largely for the reasons Searle and Austen gave. I think that this sort of project, actually the entire concern with ‘proof theory’ that is really abot proving something in some transcendental sense, rather than proof-in-a-system, well, that’s just silly. What makes you think logic would have a ‘foundation’? No doubt, actual mathematical work can be described phenomenologically in some way or another, and perhaps it would be shown that there are pre-formal acts of induction and mentalizations of various objects - who knows - but that’s sort of besides the point, since the conditions for something in mathematic being valid inference or true are not the possession or otherwise of these mental states. What are they? Well, I think the formal account is precisely the answer to this question. What else is there to be said after that? Do I really gather a better understanding of what makes two and two four if I look into the phenomenologically revealed processes of addition? I am afraid I am firmly with Frege on that these days.

The principle of sufficient reason is there a reason for it? Yes. itself. Leibniz has a 1714 discourse called “Principles of Nature and Grace, based on Reason” Its a nice little elucidation of what he means.
Reason is grounded in the investigation of substance if you agree with Leibniz (and Spinoza) but there is of course wide contestation over this. Indeed, one of Kant’s remarkable contributions to this problem of logic’s grounding (detailed in the 3 critiques and more) is that the whole question involves us in transcendental argumentation. Davidson recently revived this debate in the analytic tradition. But I think Graham Priest wrote the definitive survey book on the modern treatment of this issue: “Beyond the limits of thought” There’s also a nice workbook on the nuts and bolts of “pluralistic” logic by Priest: “An introduction to Non-Classical logic.” In the latter he details multi-valued logic which historically extends all the way from the Scholastic Buddhist tradition to modern computation theory.
Sorry, but I think this Graham Priest guy is really great.
But I think most take not the “P of sufficient reason” to be the primary axiom, but the principle of non-contradiction.

Actually, in my view, one of the fascinating movements in this controversy over axioms is to be found in the work of LEJ Brouwer. He develops a proof theory (for mathematical objects) independently of formal logic. There are few guys working on carrying out his projects. Basically the view is that demonstrations have nothing to do with formal systems (although they can be subsequently formalized) but rather rely on phenomenological sources for their “ground.” In this way, the whole question of formal axiomatics is out the window.
Here I return to Leibniz. There are many ways of viewing what the early moderns were doing. David Lewis famously takes inspiration from the Theodicy for his “modal” discourse which Kripke’s formalization largely shares. But theres something here that misses the point. I think its misleading to think of what the early moderns were doing as being anywhere concerned with formalization; its an anachronistic reading. Rather, I think we do better to focus on concepts like substance, being, cause and interaction -rather than axioms and entailment.

ps. Since I come from a largely analytic institution, I have found that it has sometimes been effective to show that when some interlocutors dismiss our conversations as “fashionable European nonsense” that one merely needs to show them that such an understanding comes out of ignorance rather than insight. Some elementary demonstrations can accomplish that task effectively.

Thanks for the heads up. I’ve got it sussed by now. He works in the IT industry and so has a lot of screentime and few social skills. I’m deleting him as he posts, and not responding anymore. I figure it takes me less time to click delete than it takes him to write, even with all the monosyllabic words, so in terms of balance sheet of time spent I win. And winning rules.

I also figure I’m doing the rest of humanity a service by sucking up bits of this guy’s time now and then and deleting what is produced. It’s a shame I can’t do that in one fell swoop, but my finger is not on the big red button. Yet. Someday, I hope.

Tzuchien, thanks for the support, but don’t lower yourself to his level. Besides, feeding them just encourages them. My troll detecting magics are weak, but since the magi at Long Sunday and elsewhere have tipped me off now I know.

take care,
Nate