I am traveling the next three weeks and things will go very slowly on this blog.
In case you missed it, you might enjoy two pieces I recently wrote for NOVA: Are Singularities Real? and Are Space and Time discrete or continuous? There should be a third one appearing later this month (which will also be the last because it seems they're scraping this column). And then I wrote an article for Quanta Magazine String Theory Meets Loop Quantum Gravity, to which you find some background material here and here. Finally you might find this article in The Independent amusing: Stephen Hawking publishes paper on black holes that could get him 'a Nobel prize after all', in which I'm quoted as the voice of reason.
Your article in Quanta was really nice. Thanks.
ReplyDeleteWe feel your potential presence even in your absence. Aharonov–Bohm effect!
ReplyDeleteFrom the Discrete or Continuous article: "Though the [Zeno] paradoxes have now been solved ..."
ReplyDeleteI read a book on the history of this which was supposed to convince me that Zeno was wrong, but it failed to do so. It seemed to me that mathematicians (cleverly) defined their way out of the problem. I would state Zeno's claim as follows (using the Arrow version of the paradox): in order to move by some distance through continuous space, a moving object must traverse to its end a sequence of points which has no end. (E.g. for a distance of length L, the points L/2, 3L/4, 7L/8, etc.)
Mathematicians reposed this as the problem of summing an infinite number of terms (e.g., L/2+L/4+L/8, ...). They (reasonably) defined the sum as the (epsilon-delta) limit of the sequence of partial sums, and then could easily prove the sum was L. This would not have surprised Zeno. He knew the length was L, by definition. His question, it seems to me, was this: is nature a mathematician? Does it use the epsilon-delta limit process? Or does it have to actually sum the series? I don't see how the mathematicians provided an answer to this. (Maybe someone can explain it here.)
Jim,
ReplyDeleteI don't know what you mean - it's a convergent series, the limit doesn't depend on the ordering. I'm not a historian and don't know if Zeno would have been surprised. But the point is that back then people didn't know how to show the series converges. Thus it was a paradox until someone could resolve it.
Just take a look at the entry "Zeno's paradoxes" in the Wikipedia, and you'll find that Jim has a point and your view of this issue is quite superficial. How can you write about subjects in your articles that you do not even know to a Wikipedia level?!
ReplyDeleteMaurice,
ReplyDeleteAh, so you looked at a Wikipedia entry and since it's vaguer than what I wrote, certainly I must be the one who misunderstood something. Interesting argument. Now please tell me exactly what mistake you think I made.
Jim,
ReplyDeleteSince you asked for a reference, I can recommend Amit Hagar's book "Discrete or Continuous?: The Quest for Fundamental Length in Modern Physics", which I reviewed here. He discusses Zeno's paradoxa in length. Unfortunately the book is quite pricey, maybe you can find it in some library or get it used. Best,
B.
Thanks for the response and reference, and I am sorry for provoking another controversy. The answer may be that there is a spectrum of reasoning ability among humans (and other animals) and from where I am in it I will never reach your understanding, as chimpanzees will never understand lightning.
ReplyDeleteHowever, my issue does not depend on the order of the terms. I listed them in the order that "nature" sees them as that seemed most appropriate. I agree that as Zeno started with a length L, and chopped it into ever smaller pieces, they should all sum back into L regardless of which order they are summed in. Still, it seems impossible to actually perform that summing process over an infinite number of terms. Mathematicians say, well we can prove the answer is L without actually having to add all the pieces up physically. My question, and I think part of Zeno's, is, but isn't nature constrained to use physical processes (summing) rather than mathematical proofs? (Maybe not.)
P.S. I can buy the book used for about $92 but I already read another book (different author) which had the same objective, and it didn't work, so I think the money would be better spent in a donation to this blog.
ReplyDeleteJim,
ReplyDeleteThat's not so controversial... Sorry in case my reply was harsher than intended. Unfortunately I have found that philosophers sometimes like to insist that problems which have long been solved haven't been solved, and I was afraid that's what you were after.
About the question how to 'actually sum an infinite amount of terms', I am not sure what you mean by that. The reason one uses the epsilon-delta proofs is that you don't actually have to sum them to get the answer.
Having said that, nobody knows of course why nature obeys mathematical laws or proofs and how nature 'does' the computing. I'm not even sure whether that question makes sense. On that issue I can only say that it works. Maybe that addresses your question?
Best,
B.
Dear Sabine,
ReplyDeleteYour Nova article on singularities was intriguing. Like most physicists, I am scared of infinities; I am sure nature is not fond of them either. As you implied, the gap between math and reality contain clusters of debris that occasionally inconvenience any possible smooth commuting between the destinations of math and reality. As a proposition, we will figure out what black holes contain once we sole the bigger problems of dark matter and dark energy.
The Unreasonable Effectiveness of Mathematics in the Natural Sciences 1960, Eugene Wigner, doi:10.1002/cpa.3160130102. Physical reality does not calculate. Beauty and rigor do not assure empirical validity. Macroeconomics is a disaster, Karl Marx to Milton Friedman, Pol Pot to Augusto Pinochet. Uniting GR and QM; SUSY, dark matter, and baryogenesis are intractable because Official Truth excludes ugly experiments falsifying beautiful paradigms.
ReplyDeletehttps://www.youtube.com/watch?v=w-I6XTVZXww
https://www.youtube.com/watch?v=E-d9mgo8FGk
https://www.youtube.com/watch?v=y3nwktTiSMk
https://www.youtube.com/watch?v=PCu_BNNI5x4
http://goodmath.scientopia.org/2014/01/17/bad-math-from-the-bad-astronomer/
and now the fun part
Thanks again, and I saw nothing harsh in your responses to me. One last beat of the horse: I agree nature's mysteries are not obliged to be fathomable by me, but both myself and Zeno would consider that there is one less mystery to fathom if space-time is discrete instead of continuous. And I think all our actual calculations, from abacuses to long-division to computers are done using discrete math (finite numbers of digits). (I started my career using a slide rule, but I could only read a few digits on it.)
ReplyDeleteHere's another view on math vs. reality: D. Abbott in an IEEE paper (PDF)
"The Reasonable Ineffectiveness of Mathematics"
http://www.eleceng.adelaide.edu.au/personal/dabbott/publications/PIE_abbott2013.pdf
Sabine, Jim,
ReplyDeleteThe Wiki entry on Zeno's paradox is not vague. It features its standard resolution that Sabine repeats (its Ref. 4), but it also discusses the philosophical limitations of this resolution. See this nice paper: http://arxiv.org/abs/physics/0505042, by a respected physicist, that takes a standpoint that seems similar to the one of Jim. See in particular the last paragraph of section 3 and the first paragraph of section 6. If a high-quality article addresses a philosophical problem I expect that the author either knows its literature well or asks philosophers about it.
You have probably seen this article, in particular its last paragraph, but if not:
ReplyDeletehttps://physics.aps.org/articles/v8/123
Rickyjames,
ReplyDeleteYes, I know this and similar statements. I don't find it very enlightening. To me the question is can you construct a model, does it explain something that other models can't explain.
Hi Sabine, (and Rickyjames),
ReplyDeleteyou can construct a model based on Cramer's transactionnal interpretation of quantum mechanics (TIQM) where the known particles are composites of currents going up and down the time. It seems to explain the elementary particles mass spectrum from two geometrical constants (norms) and a few integral resonances.
TIQM explains non-locality in a very simple manner.
Best,
J.