On Mar 27, 8:06 am, "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote: > On Mar 26, 10:06 pm, fom <fomJ...@nyms.net> wrote: > > > > > > > > > > > On 3/26/2013 9:23 PM, Ross A. Finlayson wrote: > > > <snip> > > > > And every mathematician knows that what is verified in the applied > > > makes a lot of sense. > > > This knowledge of which you speak may not actually involve > > verification in any empirical sense. > > > Where mathematics is applied, there is a pretense of > > measurement and calculation. If a new method of analyzing > > and manipulating empirical data arises, the general > > interest turns to identifying the theoretical relationships > > with the established mathematics. > > > In "Critique of Pure Reason" Kant makes some remarks > > that may be paraphrased as: > > > That which is intelligible is made sensible. > > > That which is sensible is made intelligible. > > > In the Kantian framework, this symmetry expresses the > > relationship between the theoretical coherence provided > > by an intelligible logic and the a priori relations provided > > by a sensible mathematics. > > > These words are being used in relation to a strict jargon > > and are not to be taken flippantly as justifying either > > side of the debate with WM. They certainly *do not apply* > > to his theory of willfully constructed monotonic inclusive > > crayon marks. I just wanted to give you a slightly different > > view of what you had been expressing for your own > > consideration. > > The character of the reals (as it were, in abstract algebra character > has a meaning) and the nature of the continuous and discrete and the > bounary or lack thereof between continuous and discrete is of direct > relevance to continuum analysis, which has direct relevance to physics > and processes in reality. The value of mathematics about the > continuous and discrete is more than the aesthetic or knowledge- > driven, it is of import to utilitarian needs. > > Then, where the character of the reals for real analysis is that > countable additivity is used to extract results basically of delta- > epsilonics (there exists e < d for any finite d), when there has been > developed this mathematical notion of uncountability, users of > mathematics look to it to offer methods for the improvement and > general development of physics and processes in reality. And, they > have looked long and hard, and _not found uses of uncountability as > mathematics that as falsifiable in experiment support the scientific > method_. > > Of course this in itself doesn't mean uncountability isn't mathematics > or is an inconsistent mathematics (for what axioms define, regardless > their truth). However, where the axioms of mathematics are the > foundation for the axioms of physics, but one of their primary > consequences has no import to the foundation for physics as verifiable > scientifically, those axioms of mathematics aren't so justified as > axioms of physics (not that they're shown un-justified without an > alternative scientific theory, but they're _not_ shown justified.) > > Then, where there are simple counterexamples, in real analysis, to > uncountability, here as EF and the sweep principle as non-anti- > diagonalizable and the rationals as dense as least discrete in the > continuous, then not only should the conscientious mathematician > deliberate as to how and why that is so, but the conscientious > physicist can design experiment about it, for consistent mathematics > with the infinite, which is of primary concern to cosmologists, for > whom modern mathematics has yet to offer a foundation, in theory. > > So, fom, a conscientious logician should be a formalist 24/7/365 - a > constant formalist - and formality in physics is of the scientific > method. That's not to say intuition doesn't have a place in > formality, in fact the opposite: from the universe of mathematical > possibilities (and possibilities of mathematics), the formalist is the > rigorist. >
Two paths are distinct at least one node, but there are only countably many nodes.
Any node is a copy of the root of the tree, for the tree of all infinite sequences or the tree of rational sequences (those ending with a finite repeating pattern). Any node in the tree of all infinite sequences exists in the tree of rational sequences, and there are countably many nodes.
And, the breadth-first sweep of the tree doesn't see uncountability (of the nodes, of the tree) thus result.
While countable the rationals are still surely larger in almost all regards than the naturals. Basically that the rationals are countable is an example of that: the second order, reduces to first order, for any N x N <-> N.
Well I'd wonder more if the uncountable was applied anywhere: please details applications of the uncountable so it would be more clear it doesn't just put off more work in the foundations as concrete.
Then, what would be remarkable, and compelling, for the foundations of real analysis, is to better work up what is where: the continuous of the continuum, and the discrete of its individua: meet and part, not just that they don't.
Because, modern mathematics is yet rather mute on that, and the continuum is additively countable: for what it's worth.