tag:blogger.com,1999:blog-8148573551417578681.post6064553337914221253..comments2019-10-19T12:46:34.585-07:00Comments on Dark Buzz: Classical and quantum theories are similarly indeterministicRogerhttp://www.blogger.com/profile/03474078324293158376noreply@blogger.comBlogger9125tag:blogger.com,1999:blog-8148573551417578681.post-74488725150080019232019-10-02T17:03:31.352-07:002019-10-02T17:03:31.352-07:00We have those same conservation laws in classical ...We have those same conservation laws in classical and quantum mechanics, so I don't see what there is to reconcile.<br /><br />Intrinsic probability is a funny notion. We don't know that there is any such thing, or that the probabilities in quantum mechanics are any more intrinsic than the ones in classical mechanics.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-22182713862555220842019-10-02T10:04:52.237-07:002019-10-02T10:04:52.237-07:00Dear CFT,
I am delighted!
...The first time I h...Dear CFT,<br /><br />I am delighted! <br /><br />...The first time I heard the idea that the universe is, fundamentally, made of numbers was in my 11th standard. I found the very idea itself very stupefying back then. Stupefying enough that I became distinctly suspicious about it right then and there. But I didn't know how to think about it. (Fortunately, our coaching class teacher back then, one Mr. Wani, an MSc in Maths, himself was inclined to think that the idea was ridiculous, and even said so!)<br /><br />...It took me a lot of time (3--4 decades), and also studies of philosophy (esp. epistemology by Ayn Rand, and commentaries by Dr. Binswanger), plus going through history of maths, hitting calculus books once again (even after being in graduate school), etc. But finally, I got it! <br /><br />The idea that if all people are annihilated, so would all knowledge, all concepts, including mathematical ones, isn't mine; I got it from Dr. Binswanger.<br /><br />Coming back to maths, personally, the most difficult part for me was getting to realize that there *is* this distinction between sizes (which can be found in the physical reality, as sizes of attributes of objects), and numbers or other mathematical concepts (which can't be). <br /><br />The second most difficult part was realizing that in counting, the idea of a *set* of $n$ objects is a purely man-made construct. Physically, $n$ objects might exist in a bowl, but their *set* doesn't. It is us who deliberately decide to mentally club *all* of them together. But, just because all these objects physically exist in the same bowl, you are not therefore obliged to take them all into a single set; you can always partition the same collection into two (or more) distinct sets. Tracing the logic back, you get the distinction between the sets suggesting the numbers 2 and 1. So, the idea that even when there is just one object in a bowl, you still have to take a *set* of this singleton (even if only implicitly) before you can define the counting operation---the idea that in counting you are basically comparing only *sets* (mentally constructed things)---the idea that concrete objects don't naturally form groups... <br /><br />It now seems ridiculously obvious, but this idea *also* was pretty hard for me to get at. But yes, finally, I got it! (And, no, I didn't find it mentioned so explicitly anywhere! Descriptions everywhere made direct references to objects but not to sets.) So, in that sense, I am happy to have got it right.<br /><br />Pedagogy does stand to benefit a lot if the nature of maths is spelt out clearly, removing from it all traces of Platonism.<br /><br />Thanks, anyway! <br /><br />Best,<br /><br />--Ajit<br />PS: Once again a very lengthy reply! Sorry!Ajit R. Jadhavhttps://ajitjadhav.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-46132457347152705322019-10-02T01:18:40.756-07:002019-10-02T01:18:40.756-07:00 But I am inclined to agree with Laplace. How to... But I am inclined to agree with Laplace. How to reconcile indeterminism or discontinuity with Noetherâ€™s theorems, laws of conservation and and the wavelike nature of underlying microprocesses. There is also a problem here for the notion of intrinsic probability in standard quantum mechanics.Dan Shanahanhttps://www.blogger.com/profile/11240342697478880476noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-61205725172948626222019-10-01T23:58:09.168-07:002019-10-01T23:58:09.168-07:00@Ajit,
At last. About time someone put the pretent...@Ajit,<br />At last. About time someone put the pretentious metaphysical Platonic nonsense of the 'math/numbers informs the universe' trope to rest. What we call Math clearly is an abstract construct that required quite some time for humans to develop it, it did not fall from the sky or spontaneously appear. Math followed numbers which followed logic which followed simple counting which followed after humanity, it did not proceed it, obviously. It also DOES NOT exist independently of mental abstraction, and never has.<br /><br /> CFTnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-27173130847250766752019-10-01T11:28:13.614-07:002019-10-01T11:28:13.614-07:00The Banach-Tarski paradox should be evidence enoug...The Banach-Tarski paradox should be evidence enough that real numbers are a flimsy model. They assume a completed infinity and that leads not just to paradox but outright contradiction. The problem I have is that there's a false dichotomy in mathematics between continuity and discreteness. Continuity is defined as degenerate (infinite) discreteness, so mathematics does not really have a distinction between parts and wholes. Zeno's paradox still stands. Joseph Mazur pointed this out in his book The Motion Paradox. Certainly, mathematics is not physics. Try telling the qubit people that.MD Coryhttps://www.blogger.com/profile/05342743632013663077noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-10537767317095510182019-09-30T22:49:34.426-07:002019-09-30T22:49:34.426-07:00Dear Roger,
The thing is: even real numbers are, ...Dear Roger,<br /><br />The thing is: even real numbers are, in the end, only numbers. <br /><br />Numbers themselves are only mathematical concepts. All mathematical objects (points, lines, surfaces, numbers, operators, functions, whatever) are just results of the methods of measurement we have invented; they are concepts of methods, not of physically existing objects. <br /><br />The entirety of mathematics is what we have devised/invented to be able to measure the actually existing sizes of the actually existing physical objects in the real world out there. So, sizes exist. But numbers don't.<br /><br />If we all get annihilated, numbers would cease to exist, but physical objects will continue to exist---with the same sizes with which they now exist. The moon's diameter will not suddenly change or become zero simply because we cease to exist---and with us also ceases all knowledge, including the bit that says that there is a number (real, rational fraction, or integer) which represents that size (that diameter) in some units.<br /><br />Thus, the very questions "Can real numbers exist?" or "Can physically existing numbers be such that they have an infinity of digits in some (decimal or other) representation?" are basically wrongly formulated. <br /><br />Numbers don't exist; sizes/magnitudes do. Numbers are just a means to measure the existing sizes according to some method of measurements (and according to some standard, and within some context).<br /><br />In this context, the real utility of the real numbers system is that they never can run out of supply, for any arbitrary demand on precision while exercising any method of calculations.<br /><br />Notice, no one has ever *defined* real numbers *as* an end-product of some *specific* method. Integers are defined as an end-product of counting; fractions via divisions, themselves as inverses of multiplications, themselves as repeated additions; some of the irrationals via certain geometric constructions and their algebraic representations, etc. <br /><br />But the real number system has not been defined as the end product of any one, specific, method of constructing numbers, period.<br /><br />The real number system (aka continuum) is a mere place-holder for saying that no matter what method of calculations or measurement you might ever devise, if its logic can also produce the other known numbers (like integers, fractions, known irrationals, etc.), then *any* other product it produces via the same logic *also* is a *number*, and this one must lie somewhere between two otherwise known numbers (i.e. in an interval of vanishing size, via limiting arguments, i.e., effectively, on the so-called real number line). <br /><br />So, real numbers are just a convenience of maths, a reassurance that we will never run out of the precision in calculations. They originally have nothing to do with the sizes of objects existing out there. If they were to have something to do with the actual sizes, they would be too restricted a set, and hence, wouldn't be useful for arbitrary calculations.<br /><br />Ditto, for the continuum of solids, fluids, or of EM. I think I wrote a PDF doc about the real number system a while ago. May be I should update it further to include points such as those noted here.<br /><br />Best,<br /><br />--Ajit<br />PS: Sorry for a long reply.<br />Ajit R. Jadhavhttps://ajitjadhav.wordpress.comnoreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-90446523546571305172019-09-30T15:03:45.426-07:002019-09-30T15:03:45.426-07:00I agree, insofar as models, much like maps, are no...I agree, insofar as models, much like maps, are not <i>perfectly</i> like the world, aka the territory, and yet, a mathematical model is ipso facto a systematic, formal development from some starting point. That's not determinism in the evolution over time sense, but consequences are determined by the rules and premises as they are given.<br />I feel more-or-less OK using anything I like in a mathematical model, including "real" numbers, but I can't see that whatever structures I use will correspond more than loosely, albeit usefuly, as of map to the territory, to the real world. I'd hate not to be allowed to use real numbers, integration, and differentiation and other formal rules in models just because they're not real. It's enough that they're useful, provided we keep some humility about us.Peter Morganhttps://www.blogger.com/profile/06075268176382429701noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-14935513370668135942019-09-30T14:19:49.158-07:002019-09-30T14:19:49.158-07:00Thanks for the link.
More and more I am taking th...Thanks for the <a href="https://arxiv.org/abs/1901.00526" rel="nofollow">link</a>.<br /><br />More and more I am taking the view that there are no real numbers in the real world. Saying that we cannot measure to arbitrary precision is true, but it suggests reality is a thing with arbitrary precision real numbers, just like that Hamiltonian model.<br /><br />I say that's wrong. A real number could encode all the world's information, plus oracles for unsolvable problems. Why would we find such a thing in nature? Once you abandon that, the determinism is meaningless.Rogerhttps://www.blogger.com/profile/03474078324293158376noreply@blogger.comtag:blogger.com,1999:blog-8148573551417578681.post-39688348361552153242019-09-30T13:44:14.822-07:002019-09-30T13:44:14.822-07:00One can argue that Classical Mechanics as an ideal...One can argue that Classical Mechanics <i>as an ideal</i> is deterministic, because in mathematical models that's OK. The quote "the alleged determinism of classical physics relies on the tacit, metaphysical assumption that there exists an actual value of every physical quantity" is unfortunate, in that it conflates the model and things modeled. In a Hamiltonian model ---as mathematics, with no metaphysics in sight--- there does exist an actual value of every function of the position and momentum variables, however I very much doubt that any such model is exactly the same as any part of the real-world, which in any case we cannot measure with infinite accuracy and precision.<br /><br />For all that, FWIW, I agree with you that in practice theories are quite different from theories in the ideal. With care, particularly with an acknowledgement that we can't measure anything precisely, so we have to introduce probability of <i>some</i> kind, an abstract classical measurement theory looks very close to the same as an abstract quantum measurement theory. I'm pretty sure I've posted the link to your blog before, but you might try arXiv:1901.00526, "Unary Classical Mechanics", which takes a Liouville state approach.Peter Morganhttps://www.blogger.com/profile/06075268176382429701noreply@blogger.com