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>and onto irrelevant metaphysical tangents. For those with some time to spare, you might enjoy: Clearing Up Mysteries -- The Original Goal http://bayes.wustl
by GregBuchholz 10y ago
>and onto irrelevant metaphysical tangents.
For those with some time to spare, you might enjoy:
Clearing Up Mysteries -- The Original Goal
http://bayes.wustl.edu/etj/articles/cmystery.pdf http://bayes.wustl.edu/etj/articles/cmystery.pdf
"Put most briefly, Einstein held that the QM formalism is incomplete and that it is the job of theoretical physics to supply the missing parts; Bohr claimed that there are no missing parts. To most, their positions seemed diametrically opposed; however, if we can understand better what Bohr was trying to say, it is possible to reconcile their positions and believe them both. Each had an important truth to tell us. But Bohr and Einstein could never understand each other because they were thinking on different levels. When Einstein says QM is incomplete, he means it in the ontological sense; when Bohr says QM is complete, he means it in the epistemological sense. Recognizing this, their statements are no longer contradictory. Indeed, Bohr's vague, puzzling sentences -- always groping for the right word, never finding it -- emerge from the fog and we see their underlying sense, if we keep in mind that Bohr's thinking is never on the ontological level traditional in physics. Always
he is discussing not Nature, but our information about Nature. But physics did not have the vocabulary for expressing ideas on that level, hence the groping."
>we understand "infinity" just fine.
I enjoyed "Meta Math" by Gregory Chaitin:
http://arxiv.org/abs/math/0404335 http://arxiv.org/abs/math/0404335
"So the set of real numbers, while natural—indeed, immediately given— geometrically, nevertheless remains quite elusive: Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light."
- eli_gottlieb 10y ago>"So the set of real numbers, while natural—indeed, immediately given— geometrically, nevertheless remains quite elusive: Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? And each of these things happens with probability one! The real line from 0 to 1 looks more and more like a Swiss cheese, more and more like a stunningly black high-mountain sky studded with pin-pricks of light." Bah! Chaitin makes so much of his discovery that there is randomness in mathematics, but doesn't ever seem to have made the shift to thinking of structures as parametric over randomness. To wit, generating random real numbers by simply flipping a countable number of coins and writing down the bits is trivial, and from that point of view: 1) An open set is just a set about which we can have finite information. 2) A closed set is just a set about which we can have complete information. 3) A compact set is just one we can quantify over and prove theorems about given only finite information. 4) A continuous function is one that has finite character: producing finite information about an element of the image only ever requires finite information about the argument. These statements are a bit rough, but hey, if I ever have a gajillion dollars I can spend my newfound spare time merging the computational view on synthetic topology, Chaitin's algorithmic information theory, and the old "Age of Stochasticity" talk to get a view of mathematics that properly accounts for its finities and infinities. http://www.stat.uchicago.edu/~lekheng/courses/191f09/mumford-AMS.pdf http://www.stat.uchicago.edu/~lekheng/courses/191f09/mumford...
- wnoise 10y agoSee also "abstract stone duality". https://ncatlab.org/nlab/show/abstract+Stone+duality https://ncatlab.org/nlab/show/abstract+Stone+duality
- jsprogrammer 10y agoIt doesn't appear that the set of rel numbers exists. If it did, someone should be able to point to it and say, "this is the set". No one can do that because the available space to present the set appears finite, but producing the set would require an infinite amount of space (assuming you need some exclusive space to represent each number). It is no surprise that the set remains elusive, it doesn't exist.
- chc 10y agoSets of numbers aren't physical things. Physical things can't be infinite, but I don't see why you think a set can't. By this thinking, not even the natural numbers can exist, since you can't point to them either.
- jsprogrammer 10y agoNatural numbers are easy to write: 1, 2, 3 You can even write a set of them: {1, 2, 3, 4} You cannot write the set of real numbers.
- chriswarbo 10y ago{1, 2, 3, 4} is also a set of real numbers. Why is it a valid existence proof of the naturals, but not the reals?
- Govannon 10y agoPossible linguistic pedantry; he used "a" set of natural numbers, but "the" set of real numbers. It's equally difficult to represent the set of natural numbers.
- jsprogrammer 10y agoIt's not pedantry, it's just what the words mean. To the question: I'm only saying that the "set of real numbers" and the "set of natural numbers" don't seem to exist. Despite numerous downvotes, no one has yet produced them here (or even a link to them).
- 77pt77 10y ago> if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even refer to it? That guy is just a constructivist. Nothing new here.