CAN I GET HELP WITH HISTORY OF MATHEMATICAL PHILOSOPHY? (edited) (request for he

CAN I GET HELP WITH HISTORY OF MATHEMATICAL PHILOSOPHY?

(edited) (request for help) (foundations of mathematics)

Need a mathematical historian. Someone very well versed in teaching theory.

I am going to say this badly because I don’t know the correct way to ask it:

0) the classical construction of mathematics is an operational and (identity, counting and measurement) and analog one. It is the practical uniting of counting, measurement, geometry, and algebraic logic (deduction).

1) The ZFC+AC argument (the set argument) converts the practice of math from one of dimensions (space and analog) to one of sets (binary). This allows the excluded middle. It is a very artful way of solving the problem, by simply returning to the very basics of the origination of counting. But the set solution is achieved by removing scale and therefore contextual utility from the calculation, leaving us with no means of external reference for choice of precision. I see this solution as useful, but a fabrication.. a ‘trick’. Whereas, one could just say ‘precision of N’, and increase or decrease that precision as needed. (Although this approach would require tagging variables or numbers I think, or maybe prevent us from reducing ratios including real numbers.) The solution to the problem of scale and context (analog representation) by converting to binary (set membership) representation is actually very interesting one. The question is, was it knowingly made, or what this solution achieved without understanding that the problem of context and scale was solved by effectively reducing math from analog (related to the real world scale) and binary (independent of real world scale). I can’t figure it out from the literature.

2) The constructivist argument relies on a binary proof. (“Russian Recursive Mathematics”) This method disallows the excluded middle. (and double negation). It is a higher standard of proof. However, I don’t understand why we could not construct a syntax for the explicit preservation of scale (correspondence with whatever context we have in mind) and thereby retain correspondence as well as the excluded middle. (I am not sure about double negation. I haven’t thought it through yet.)

3) Computational mathematics is both operational and binary.

But why aren’t these three methods a spectrum – just like description, deduction, induction, abduction, guessing and intuitive choice? I mean, at the early end of the spectrum (0, 1) we require deduction, and at the later end of the spectrum (2,3) we require computability. The reason we have a problem with (1) and (2) is because they give upon correspondence (context). And with that we lose the use of context for determining the precision of a calculation.

Deduction in context is always easier because we have information with which to make a choice (precision). But outside of context we cannot use external information, so we must rely on a binary choice (or decidability). Deduction is a very different problem from computation.

Or, can we say then, that the foundations of mathematics have been wrongly divorced from correspondence and context by cantor through ZFC? When we could just say that binary is a universal substitute for arbitrary precision? I mean, that’s the functional equivalent of it?

I need a frame of reference within the language of mathematics to talk about this issue and I don’t know how to get to it. I don’t even know how to ask this question any better than this?

Was the solution to the foundations of math, culminating in ZFC+AC, understood as providing a solution to creating independence from the problem of correspondence and scale at the expense of ‘truth’ while retaining ‘proof’ and internal consistency?

Or stated this way: Did mathematical philosophers understand that they were divorcing ‘departmental mathematics’ from physics (cause and correspondence) and logic (truth) by adopting ZFC+AC, thereby creating a study of pure relations independent of context?

I have worked through both sides of the debate to the best of my ability.

Why can the reason that sets work – reduction to binary in order to escape the burden of retaining context – simply be stated openly? I mean, if all it does is render scale infinitely variable, then that explains why ZFC works, and all these platonic devices are necessary: they create deducibility and computability. And it’s not ‘wrong’ per se, in the sense that it doesn’t produce correct calculations independent of context, or rather, independent of SCALE and therefore independent of correspondence. But it does sort of render mathematics platonic and almost magical rather than computational and rational.

In that sense, we get to logically state WHY these methods work and when and when not they are applicable. The excluded middle is a problem of scale (analog, and correspondent values).

In the end, the set method is useful because is just SO MUCH LESS BURDENSOME, but that’s all.

But still, teaching people operational mathematics, and higher criteria of proof under constructive math, and then explicitly stating that we can move to sets at the expense of correspondence in order to obtain the ability to practice double negation and the excluded middle is not a problem, it’s a tool not a truth.

I don’t need to solve this problem for my work. But since math is the gold standard, and contains this particularly burdensome problem, if I can describe the consequences in mathematics of non-operational language leading to platonism, I can explain why non-operational language in ethics, likewise leads to platonism.