Peter Jones wrote: ↑Fri Mar 19, 2021 11:12 am
Simon Adams wrote: ↑Thu Mar 18, 2021 12:22 am
I notice there are a couple of interesting papers on this subject
here. Not your work by any chance?
If only folks realised how completely they are being misled by the philosophy department.
Excellent!:
https://philpapers.org/archive/JONTCE.pdf
Some comments, if I may.
We have every right to define the continuum for mathematics as we currently do, and if our idea is paradoxical then it is only a problem when we investigate the foundations of analysis.
Well not every right, at least from the point of view of intuitionist philosophy of mathematics, which requires 1) constructibility of mathematical languages and 2) linguistics constructions are in some sort of harmony with intuitive/idealist ontology of mathematics. Arbitrary language games of formalism can have heuristic value, but for foundational thinking of pure mathematics, nope.
Danzig gets very close to the answer:
The notion of equal-greater-less precedes the number concept. We learn to compare before we lean to evaluate. Arithmetic does not begin with numbers; it begins with criteria.
Equal is not on same foundational level as relational operators < >, which precede the notion of equality. Already the theory of Surreal Numbers, at least Knuth's presentation of it, makes this very explicit. If A is neither more nor less than B, then A = B.
Bergson's notion of duration gives another hint:
a duration is neither unity nor multiplicity. And so does undecidability of Halting problem.
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Re Weyl's claim, an interlude:
The fact that single points in a true continuum “cannot be exhibited” arises,
This is very qualified argument, and there is an intuitive way to exhibit a single point to the eye of the mind. Let's imagine continuous flat plane, and a straight line on the plane. Let's make a cut in the line. Taking a flat-lander point of view at the other end of the cut, we can see a point where the other part of the line ends. If the plane is not flat, or if other lines are allowed and they are not parallel, the point disappears as we can see only a line. The point does not exist and exhibit itself independently from the plane and the line and the cut (cf. "transition"), and here Bohm's notions of explicate and implicate orders can be useful. Defining an explicate view to see a point, implicates lots of structure, but nothing beyond imagination in this case. In this case, the relation of implicate and explicate orders supports Whiteheadian planar instead of pointy approach to geometry and mathematics.
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Brouwer's First Act of Intuitionism, namely '
two-ity', is a relation of codependent arising, according to [ur=
https://cosmosandhistory.org/index.php/ ... ile/30/59l]this article[/url]. On the other hand Brouwer considered relational operators "negative", apparently because they are
something else than a discrete quantity. As an empirical science instead of formal speculation, intuitionism is evolutionary, also on the M@L level, where other intuitionists expanded the idealism of intuitionism after Brouwer's solipsism.
To bridge the "deep chasm" that Weyl speaks of, foundational mathematics needs to simply ask: is there a way to mathematically describe continuum, so that also discrete phenomena can be coherently derived and constructed from the foundation? Yes there is. Let's redefine relational operators as asubjective verbs denoting continuous processes. E.g. dynamic tetralemma of increases - decreases:
1) increases <
2) decreases >
3) both increases and decreases <>
4) neither increases nor decreases ><
A short for speaking such class of verbs is more-less. Form <> corresponds with notions of 'open interval' and Bergson duration. Negation/halting of process in the 4th lemma allows to define equivalence relations >=< and closed intervals aka finite sets >[]<. As we know, already finite set allows to construct discrete numbers in the style of von Neumann, but there can be also much better and interesting ways to bridge that "gap", for example the basic number-antinumber scheme of e.g. integers, which Relop language of relational operators allows to write as palindromes. Yet another possibility is to generate Stern-Brocot trees in Relop language.