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RECENT POSTS IN THIS TOPIC

ABDELWAHED BANNOURI: on 4/22/15 at 17:45pm UTC, wrote Dear Laurence . your essay is very interesting, indeed i consider it an...

Lawrence Crowell: on 4/20/15 at 22:37pm UTC, wrote Dear Laurence, Thank you for taking interest in my essay. The idea is...

Laurence Hitterdale: on 4/20/15 at 17:07pm UTC, wrote Dear Lawrence, Many points in your especially comprehensive essay are...

Lawrence Crowell: on 4/20/15 at 0:50am UTC, wrote There are some questions concerning foundations of mathematics. I am not a...

Roger Schlafly: on 4/19/15 at 21:11pm UTC, wrote You give the impression that there is something wrong with the foundations...

Lawrence Crowell: on 4/18/15 at 20:28pm UTC, wrote Thank you for that. I voted for your essay a month ago or so. I don't...

Alma Ionescu: on 4/18/15 at 15:40pm UTC, wrote I just realized I didn't rate your essay so I am fixing that now. As you...

Alma Ionescu: on 4/18/15 at 15:38pm UTC, wrote Dear Lc, I know what you mean by notions that are not entirely comforting...


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FQXi FORUM
November 19, 2019

CATEGORY: Trick or Truth Essay Contest (2015) [back]
TOPIC: Mathematical Physics as Topological Numbers, Types and Quanta by Lawrence B Crowell [refresh]
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Author Lawrence B Crowell wrote on Jan. 27, 2015 @ 21:55 GMT
Essay Abstract

The development of mathematics used in physics is most likely to become concerned with finite elements that are measurable. This means that topology and the computation of topological charges and indices, quantum numbers, and connection to logical switching theory are likely to supplant concerns of geometry, metrics and infinitesimal structure of manifolds. This is examined, with a possible counter direction to this as well with super-Turing machines and second order $\lambda$-calculus. Mathematics and its deeper foundations may share a similar nature with physics in regards to quantum information.

Author Bio

Doctoral work at Purdue. Worked on orbital navigation and currently work on IT and programming. I think it is likely there is some subtle, and in some ways simple, physical principle that is not understood, or some current principle that is an obstruction. It is likely our inability to work quantum physics and gravity into a coherent whole is likely to be solved through new postulates or physical axioms, or the removal of current ones.

Download Essay PDF File

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John C Hodge wrote on Jan. 30, 2015 @ 19:58 GMT
Your question (``Does this mean that older forms of mathematics will disappear?’’) suggests math has an evolution or selection--of--the--fittest history in human discovery.

Science has precipitated out of philosophy to be that part of human knowledge that predicts observations. Does science extend as far as metaphysics? Likewise math as we know it today has become the usefulness of...

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Author Lawrence B Crowell replied on Jan. 30, 2015 @ 21:12 GMT
I did not have much time to go into this in my essay but intuitive arithmetic is sort of logarithmic. This appears to be the case for many indigenous cultures. The Taylor series for the log function is to first order near x = 1 is log(x) = x – 1. Hence for numbers small the log function is linear. So for small number people tend to count more or less normally. Depending upon the numbers people might actually count to 10, maybe more, or sometimes only to 3. Beyond that numbers sort of attenuate. So if people in some culture might only count to 10, if show them 50 objects they might see 15 or 20 as the “between” number. It is a form of Gamow’s “ One, Two, Three …, Infinity.”

Mathematics is a subject far vaster than physics. There are systems of mathematics I keep learning about all the time. I just learned about albelian grebes, which is a form of Sheaf theory. Some mathematical systems find their way into applications in the real world, others do not --- or they have not yet found such application. Physics has extended its reach into mathematics, but the mathematics employed is generally used to answer rather similar problems. These problems are about symmetries and conservation laws, counting degrees of freedom, finding entropy and so forth. Mathematics by comparison is a subject that gives the practitioner far greater freedom to explore and develop systems, but is conversely very constrained by logic.

There is probably some selection process that dictates that certain areas of mathematics will become richly developed if it describes the physical world, while a system that does not will tend to languish in little read issues of journals gathering dust.

Cheers LC

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Sophia Magnusdottir wrote on Jan. 31, 2015 @ 15:59 GMT
Hi Lawrence,

This is a very interesting essay that touches on many good points, each of which would deserve its own essay! The question of the continuum or the infinitesimally small keeps coming back to me, for various reasons. I read an interview with Tegmark some months ago in which he is asking for the reality of infinity, which is basically the same question. As you say though, I suspect it is a question that is ultimately not possible to answer. (Pragmatist would disapprove of my thinking about it ;) )

-- Sophie

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Author Lawrence B Crowell wrote on Jan. 31, 2015 @ 23:20 GMT
In type theory there is no infinity, but types that are analogous to sets are unbounded in general. There is no upper bound to the size the types, but the ordinality of them must be constructed. This tends to preclude infinity, and certainly the continuum infinity or cardinality C > 2^{aleph_0} is not included.

I see there being a sort of two fold system. Standard mathematics might be thought of as the “soul,” or a “ghost,” and mathematics that is restrained by concerns of Kolmogoroff complexity, types and so forth as the “body.” It may not be possible to express all numbers between 10^{10^{10}^{10}}} and 10^{10^{10}^{10^{10}}}}, but this just means the body is not able to construct or contain the information space necessary to do so, but this still leaves room for the “soul.” Mathematicians are then free to “pick their poison,” where a pure mathematician may prefer to stay with the standard approaches to math, while a more practical minded analyst might prefer to stick with the “body.”

Cheers LC

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Gary D. Simpson wrote on Feb. 2, 2015 @ 23:18 GMT
Lawrence,

This is quite a piece of work. I agree with Sophia. There is enough subject material here for several essays of more than ten pages. But I also see why you have taken the path you have. You begin with the origin of mathematics ... likely something as mundane as counting cattle (or livestock in general) and show its practical development to produce distances and angles and algebra.

Then things start to move pretty fast. You move from calculus to more advanced topics in very short order with the objective of presenting some of the challenges and contradictions facing mathematics today. You also illustrate how mathematics and physics have developed together. You mention all the familiar names and then move towards possible resolutions.

All in all, I will say well done. You have challenged the reader without over whelming the reader. That is a delicate balance.

Best Regards and Good Luck,

Gary Simpson

PS - I've been following the discussions in Dr Klingman's forum. I know very little about Bell and entanglement, but it seems to me that if he can make a testable prediction then what he is proposing is science. Perhaps he is right or perhaps he is wrong. A properly designed experiment is the only thing that can give the answer.

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Author Lawrence B Crowell wrote on Feb. 2, 2015 @ 23:52 GMT
Klingman writes essays on how quantum mechanics is ultimately a local theory, and he keeps coming up with what Feynman calls towards the end of this recording “ball bearings on springs.” I have criticized his essays on other occasions. The problem is that FQXI is attended by many who write these essays and most who write to the blog and who have these ideas. There is a long history of this, and it in part swirls around a physicist named joy Christian. JC is, or was, an FQXI fellow and has hidden variable ideas. While few FQXI fellows are hidden variable mavens who think quantum physics is a local theory, the blog posters and many of the essay writers are. As a result Klingman’s essay gets lots of attention and lots of votes on the high side, even if the core idea is hogwash.

You might notice that the blog is populated mostly by complete cranks. It used to not be quite like this, and I think the problem is the lack of minimal moderation has lead to this situation. These essays really should be put to some level of review, even if somewhat cursory, to weed out the crankiest of the cranky ones.

Klingman’s claim is rather easy to verify. The spin of a system is measured to have discrete projections along the axis of measurement, and for a spin ½ system there are only two such projections: UP and DOWN (or + and -). This would imply there is some consistent trend away from the discrete spin splitting in the Stern-Gerlach experiment. People have made thousands of such measurements and so far “no cigar” for hidden variables.

Mathematics is entering into a multiplicity of maths. In fact there is a subject called proof theory, which I am not versed in, and my understanding is that there does not appear to be a consistent system for proofs. Instead it appears there exists a multiple draft system of math-proof systems.

Cheers LC

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Author Lawrence B Crowell replied on Feb. 2, 2015 @ 23:54 GMT
I forgot the URL for the talk by Feynman.

https://www.youtube.com/watch?v=_sAfUpGmnm4

LC

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Eckard Blumschein wrote on Feb. 3, 2015 @ 05:35 GMT
Lawrence,

Hopefully "joy Christian" is just one more of your typos, not a deliberate humiliation. I don't intend defending JC and Klingman. I merely dislike your wording "weed out the crankiest of the cranky ones", "hogwash" and "“no cigar”".

I admit having sometimes problems to understand what you means, for instance "It is likely ... [something] ... ...".

Nonetheless I intend reading your essay because it seems to address key questions.

Regards,

Eckard

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Author Lawrence B Crowell replied on Feb. 3, 2015 @ 15:58 GMT
A lot of that stuff is “cranky.” I will give both JC and Klingman the benefit that they did some manner of work in their theories. This does not make them right. The matter of quantum nonlocality and that nature is so utterly cockeyed and crazy one almost can’t believe it, as Feynman puts it, is very well settled. Since the 1980s experimental demonstrations of nonlocality have become a cornerstone of physics. Even further this is emerging as “quantum technology,” where nonlocal properties permit encryption of information and the teleportation of quantum states nonlocally.

We are really long past the time where we should be fretting over this stuff. I say that maybe those who don’t like this must either “get over it,” or as Feynman puts it, “go somewhere else.” We can’t really leave this universe, but a person who really can’t stand quantum mechanics should just get out of physics entirely, or at least physics that connects with nature on the atomic-molecular scale and smaller. One can work on orbital dynamics or many forms of engineering without ever seeing a single ħ, and the quantum world is safely outside your domain.

There are a number of serious cranks that show up on the blog posts. The worst is Valev, who insists that relativity is all washed up. The problem with people like this is that they simply refuse to really learn and hold fast to ideas that they have that are simply wrong. Five years ago and beyond the FQXI blog was not this bad, but it has become an example of “bad money chasing out good.” The great majority of blog posts are cranky.

LC

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Eckard Blumschein replied on Feb. 3, 2015 @ 22:37 GMT
LC,

In order for you to understand what I meant, I will more completely quote the beginning of the last sentence in your bio:"It is likely our inability to work quantum physics and gravity into a coherent whole is likely to be solved ...". Likely is likely?

Being a German, I did not immediately understand a lot of your slang expressions, for instance "washed up" in the sense of k.o. or "chasing out good".

I too disagree with Pentcho Valev, however not entirely and not without serious arguments.

Doctors like us should not use mere humiliation instead of convincing arguments. Don't you blush for deliberately writing joy C.?

You wrote "standard mathematics has become increasingly shaken on its core". Unfortunately I hoped in vain for a detailed sober foundational criticism of this core.

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Author Lawrence B Crowell replied on Feb. 3, 2015 @ 23:51 GMT
I didn't deliberately remove the capital J from Joy's name. It is just a typo error. Anyway, these blog posts are not formal communications.

The business of axiomatic set theory is something I am familiar with on a semi-formal sense. It is not central to my work or interests. I also made a decision to keep this essay as informal as possible. It seems the more formal these are the less attention they get.

LC

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Member Tim Maudlin wrote on Feb. 4, 2015 @ 17:39 GMT
Dear Lawrence,

Just a quick note. There are, of course, many interpretive difficulties with quantum theory. But one that has been settled is that no theory that regards the wave function as purely epistemic can recover the predictions of quantum theory. The proof is by Pusey, Barrett and Rudolph. So your suggestion that one regard the wave function as "not real" cannot be reconciled with the prediction of quantum theory, in the sense outlined by PBR. In addition, the two-slit interference phenomena already testify to the reality of the wave function in the sense that something is physically sensitive to the state of both slits on every run of the experiment, so something is, in this sense "spread out" between the two slits. Every clearly stated interpretation of quantum theory takes the wavefucntion as physically real, not epistemic.

Regards,

Tim Maudlin

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Author Lawrence B Crowell replied on Feb. 4, 2015 @ 19:11 GMT
I remember the paper by PBR, maybe about 3 years ago, which I believe is this paper http://arxiv.org/abs/1111.3328 . This was found lacking, or it had problems. It has been a while since then, so I don’t recall what the issue was. I don’t study these matters that closely, preferring string/M-theory stuff. As I see it wave functions are not real; at best we can only say they are complex, or quaternionic. What is meant by reality with quantum waves is either very strange or nonexistent. I think that if quantum waves have some sort of reality, or ontology, then we have to admit there is no objective reality at all.

LC

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Sylvain Poirier wrote on Feb. 5, 2015 @ 23:29 GMT
As I am much involved in the foundations of mathematics, I noted some inaccuracies in your essay on this topic:

"Bertrand Russell asked what would happen if you have a set of sets that does not include itself"

A consequence of the axiom of regularity of ZF is that no set is ever member of itself. And as in this theory all objects are sets, every set is a set of sets that does not...

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Author Lawrence B Crowell replied on Feb. 6, 2015 @ 02:36 GMT
The one problem that I had with this is first off that there is a severe word or symbol limitation and secondly I have to admit I am not a student of ZF set theory. I discussed these set theory parts in a pretty cursory manner, and I may have in doing this in a truncated manner have done some damage. In cutting back the size of this essay I cut mostly from the discussion on Goedel’s theorem and set theory

My point about Russell paradox is that if S = {x|x \notin x} then S \in S implies that S \notin S. So one does the opposite and one gets S = {x|x \in x} then S \notin S implies that S \in S, and one can then think of having to fix this with lists of lists and lists of lists of lists and so forth. Turing's proof is a sort of Cantor diagonal proof, and this does have a "physical" meaning that a universal Turing machine is not able to enumerate all Turing machines.

I really discussed these matters within little more than a page, and so the discussion is pretty thin. I also presented the ideas in more of a physical sense. The issue of Kolmogoroff entropy or complexity is meant to illustrate how not all numbers are computable. I stand by my statement about the complexity of numbers. Also Godel’s theorem does indicate that Peano’s number theory is incomplete, and so something funny does happen with N+1

As a rule with these essays it is best to keep the mathematical pedantry to a minimum. It works best to give readers more of an intuitive sense of these matters than it is to lay down layers of mathematical symbolism.

http://www.cambridge.org/us/academic/subjects/mathematics/lo
gic-categories-and-sets/descriptive-set-theory-polish-group-
actions

LC

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Sylvain Poirier replied on Mar. 21, 2015 @ 22:28 GMT
Hi Lawrence. Thank you for your reply to Georgina. Sorry I forgot to reply to you earlier.

As I said it is not true that "a universal Turing machine is not able to enumerate all Turing machines". Turing machines can be automatically enumerated, but what is not possible is to find a general algorithm always correctly able to prove for any other algorithm, whether or not it will ever...

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Author Lawrence B Crowell replied on Mar. 23, 2015 @ 20:06 GMT
Some of my discussions were meant to illustrate something of the divide between computational mathematics and pure mathematics. With Peano arithmetic we know that Goedel's theorem indicates that something is not complete, even though much of it involves N ---> N + 1. There are then numbers, such as some between 10^{10^{10^{10}}} and 10^{10^{10^{10^{10}}}} that have no description. There is a Berry paradox or self-referential form of incompleteness, based on the complexity or unnamable property of such numbers between these two, in not being able to describe numbers.

I had a limited amount of space to describe this, and maybe I did not do the best at it. I tried to explain some of these ideas in physical terms without getting into depth on set theory or logic. It is also best I have found that keeping these essays on a level accessible to general readers to be a good strategy.

LC

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Florin Moldoveanu wrote on Feb. 7, 2015 @ 04:21 GMT
Hi Lawrence,

Nice essay. I am intrigued by the homotopy type theory. Do you understand its basic points and motivation? On a side note PBR is actually correct (originally I thought I found a loophole there but there is none).

Cheers,

Florin

PS: I think you may appreciate this link: https://www.youtube.com/watch?v=WabHm1QWVCA

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Author Lawrence B Crowell replied on Feb. 7, 2015 @ 22:46 GMT
Hi Florin,

I am in one sense disposed to Wildberger when it comes to mathematics that is computed or that has a physical meaning. There seems to my mind there are two notions of mathematics with respect to infinity and the continuum. The pure notion, which is the standard approach to mathematics, is in some ways Platonic. I am not particularly anti-Platonic, but the problem I see with...

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Florin Moldoveanu replied on Feb. 7, 2015 @ 23:08 GMT
Hi Lawrence,

If you want to write a guest blog post at my blog about your FQXi entry to advertise it and boost the penetration of your ideas you can do so at any time. In the last few months I got caught in so many "clerical" activities that I had to put on hold learning new things. I'll participate in the FQXi grant contest but I don't know if the ghost of Joy Christian will kill my entry. If I have extra time I might write an essay for the FQXi context but I know I don't have the time necessary to invest to win. If I'll do it it will be only for advertisement purposes.

Cheers, Florin

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Author Lawrence B Crowell replied on Feb. 8, 2015 @ 16:55 GMT
I might take you up on your offer. I probably will not write about the FQXi essay topic, but on a subset of it involved with the Bott periodicity and the large N SU(N) for the structure of an event horizon.

I think this has some bearing on PR boxes. Quantum gravity requires that the field theory be nonlocal. Standard QFT has local field amplitudes with Wightman causality conditions, such as equal time commutators. Nonlocality occurs with the expectations of the operators over the Fock basis which gives quantum waves. Quantum gravity I think involves further violations of inequalities, which PR boxes or nonsignalling conditions are maybe capable of working with. Maybe there are bounds beyond the Tsirelson bound?

If you want I can send to you an article I wrote but have yet to submit for publication that addresses some of this.

Cheers LC

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basudeba mishra wrote on Feb. 8, 2015 @ 18:01 GMT
Dear Sir,

Ancient Indian texts describe in detail about number theory including what is a number, what is zero, what is infinity, what are negative numbers and irrational numbers, the difference between one and many, why one is the first number, why two follows one, why three follows two, why four follows three, why zero comes after nine, why the number system repeats thereafter, why these...

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Author Lawrence B Crowell replied on Feb. 9, 2015 @ 00:31 GMT
I think you are trying to think of things according to some intrinsic "things as they are" sort of perspective. The problem is that we are in the universe, which means we must operate within it, and are thus faced with limits to observability. I think in some ways that is more of a mystical way of knowing than a scientific one.

The lack of observability or knowledge is a sort of topological obstruction. Such obstructions can in their own way tell us things.

Cheers LC

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basudeba mishra replied on Feb. 9, 2015 @ 08:02 GMT
Dear Sir,

We fail to see how in the universe “things as they are” are not relevant. Science is all about explaining “things as they are”. Then how can these be mystical? Regarding “limits to observability”, we have said in our post: “Our instruments for perception/measurement have limited capacity both in content and time. Thus, what we measure depicts a temporal state of a limited aspect over limited period. The problem comes when we generalize our limited information. We impose our ignorance or inability to know on the Universe and call it fuzzy”. If we assume “lack of observability or knowledge is a sort of topological obstruction” and things they tell us, that certainly will be mystical.

We are not here for scoring points, but seeking to understand Nature. Hence kindly do not take these comments otherwise.

Regards,

basudeba

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Author Lawrence B Crowell replied on Feb. 9, 2015 @ 17:27 GMT
Those limitations are types of obstructions that tell us something entirely new. It also means there is a massive reduction in the number of fundamental degrees of freedom in physics. That is the valuable lesson. For those who bemoan the loss of absolute objective knowledge about the world, such as was lost with quantum mechanics, I can only say that such people have sympathies.

We are no longer in a situation where physics tells us in some intrinsic fashion what the exact nature of systems are. Physics tells us what is observable or measurable; physics does not give a complete "God's eye view" of what nature is. Certainly one problem is that we are in the universe, and we have to us the physical systems in the universe to measure physics. We are not able to make a pure observation that does not disturb a system.

LC

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Author Lawrence B Crowell wrote on Feb. 11, 2015 @ 15:34 GMT
Complex numbers employed in physics have consequences. One is unitarity that is important in quantum physics, another is holomorphy that is important in gauge theory. Quaternions, which are hypercomplex numbers, can be used to derive Maxwell's equations. So these things are useful in making calculations.

LC

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Anonymous wrote on Feb. 11, 2015 @ 17:37 GMT
Complex numbers employed in physics have consequences. One is unitarity that is important in quantum physics, another is holomorphy that is important in gauge theory. Quaternions, which are hypercomplex numbers, can be used to derive Maxwell's equations. So these things are useful in making calculations.

LC

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Harlan Swyers wrote on Feb. 15, 2015 @ 15:22 GMT
Lawrence,

Apologies for taking so long to respond. I likewise have returned the favor and rated your essay with a 10.

I right honorable essay.

Absolutely fascinating discussion about the numbers we can never count. The idea that we can define spaces we can never explore does make one feel small in a seemingly much larger universe. Is it ultimately the human condition that we must accept that there are things we can never know? What are the true limits of knowledge? When do we know we cannot go further?

On the other side, it is somewhat refreshing to know that there is always places we can go that have not been explored. The question I have is how far will we get?

Cheers!

Harlan

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Author Lawrence B Crowell replied on Feb. 15, 2015 @ 16:24 GMT
Garrison Keillor has his “Guy Noir,” who “On the tenth floor of the Atlas building still seeks answers to life’s persistent questions.” If you have ever listened to his “Prairie Home Companion” you know this well. There are persistent questions, such as “Does God exist,” that will probably never be conclusively answered.

The problem is that we transition from physics to...

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Sujatha Jagannathan wrote on Feb. 16, 2015 @ 07:46 GMT
Your conscious effort to subjectively narrate the developmental science is well-written.

Sincerely,

Miss. Sujatha Jagannathan

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Author Lawrence B Crowell replied on Feb. 16, 2015 @ 16:35 GMT
Thanks for the words of encouragement. I tried to write the description of Goedel's theorem and related matters in a physical sense, and I wonder if I fell far from the mark on that.

I see that you are in the contest as well. I will try to get to your essay as soon as possible. I have been a bit unable to read many of these the last couple of weeks.

Cheers LC

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Richard Lewis wrote on Feb. 19, 2015 @ 16:09 GMT
Hello Laurence,

I enjoyed your essay which covered many mathematical topics of great interest. I was intrigued by your diagram: 'Topological winding numbers in the two slit experiment'.

Taking the viewpoint that a photon is a real physical wave that passes through both slits of the interference apparatus it is hard to imagine that a path looping back through the slits is a real possibility.

Regards

Richard

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Author Lawrence B Crowell replied on Feb. 19, 2015 @ 16:28 GMT
Richard,

Thanks for the response and the question. This is something other people have wondered about. There is a small probability the photon will loop around the slit! This is predicted by the Feynman path integral. Feynman said the purpose of the path integral is to derive quantum properties from all possible paths corresponding to amplitdes. A particle can leave a source and reach a target with some probability that it looped around Mars. Of course the amplitdue for this looping is very small, but it is there. Again Feynman once said that most or all of quantum mechanics can be studied with the two-slit experiment. This means that with the actual experiment there should be some very small effect due to the looping of a particle around the topological obstruction that is the slit. This is something that at first seems utterly impossible until you think more closely about it.

Cheers LC

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Akinbo Ojo wrote on Feb. 21, 2015 @ 19:33 GMT
Dear Lawrence,

Your essay taught me certain things that I didn’t know. I liked the historical way the essay was written starting slowly and rising to a crescendo.

I would have wanted included in the essay concise definitions for what geometrically you mean by 'continuous' and 'discrete', in the light of Euclid's definitions of a what a line is.

In contrast to the assertion, "Geometry is then not fundamental", which I disagree with, I leave you a quote from Galileo to ponder:

“He who attempts natural philosophy without geometry is lost”

- Galileo Galilei, Dialogo, Opere 7 299 (Edizione nazionale, Florence, 1890-1909).

A very informative essay worth keeping and reading more than once.

Regards,

Akinbo

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Author Lawrence B Crowell replied on Feb. 22, 2015 @ 01:02 GMT
Geometry is important, but I think it is built up from quantum entanglement. Of course there are a number of papers in this contest which purport that Bell’s theorem, entanglement and the quantum physics of nonlocal correlations is wrong. One is wise to not bite on this. Quantum mechanics is one of the most experimentally tested areas of physics. So far all of the strange consequences of quantum physics hold up, this is even if they appear to be so utterly bizarre and upside down.

Quantum states entangle with a black hole. The observer on the outside easily loses information on the exact black hole horizon state her part of the EPR pair is entangled with. This is a form of entropy; the entropy illustrates the lack of knowledge. The horizon area of the black hole is then a direct measure of entropy, where any quantum bit in our outside world entangled with the black hole is entangled with a Planck unit of area on the stretched horizon. It is difficult to localize that of course. If one tries to find which region of the horizon your EPR part of the pair is entangled with the Heisenberg microscope argument tells us the other part of the EPR pair will be sent into a huge uncertainty in position. Hence one is not able to recover this information.

The horizon of a black hole is then built up from entanglement. Further, the null boundaries of spacetime contain the holographic information of the entire region. We then have the physics of space or spacetime really being built up from entanglement of quantum states. Geometry, at least geometry used to model physics, is then an emergent property.

LC

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James Lee Hoover wrote on Feb. 24, 2015 @ 06:42 GMT
Lawrence,

Much to ponder here, Lawrence. The Penrose triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. My connections I don't feel have a Platonic flavor, as you mention, only an attempt to mimic the mind in mathematical models for understanding natural connections between the quantum and the classical worlds. I see a functional relationship between mind, physics and math, making possible giant strides in physics and other sciences.Quantum entanglement has been found to have a role in classical phenomena such as navigation of birds, turning new pages in quantum biology.

Great essay.

Jim

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Author Lawrence B Crowell replied on Feb. 24, 2015 @ 20:02 GMT
As I remember Penrose and his triangle in “Road to Reality” had consciousness or mind with physical reality and mathematics as a triality. The I recall that he had mathematics as the foundation of physics, and physics the foundation of mind, and to complete the cycle mind gave conscious recognition to mathematics.

Platonism is not something I take that strongly. I will put on the “hat of Platonism” when it suits me, and at other times I will not. I discuss largely the aspect of mathematics that I think has “meat,” while a lot of mathematics involving infinity and infinitesimals is what I call “soul.” I am not out to deny the “soul,” but I do think the “meat” has a more direct connection with physical reality.

LC

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Philip Gibbs wrote on Feb. 26, 2015 @ 16:20 GMT
Lawrence, this is possibly your most readable essay yet in these contests, but you have still managed to maintain a high level of novel mathematical ideas. I liked the historical introduction that puts the relationship between Mathematics and physics in perspective.

The HoTT ideas are very interesting and they mesh well with my own ideas of higher category theory as a system of multiple quantisation so it is good to see this presented.

You should do well.

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Author Lawrence B Crowell replied on Feb. 27, 2015 @ 15:07 GMT
Thanks for the positive word. I wrote the bulk of this essay up in a single day. I spent another day correcting it. I tried to keep the mathematics somewhat "physical," in that the discussion on incompleteness and numbers is oriented towards what one might actually encounter in computing these things.

I read your essay early on, and as with most essays I have read I have yet to assign a score to it. So far you are running at the top. I will probably have to re-read yours. I have been a bit busy and unable to attend to this contest that much.

Cheers LC

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Michael Rios wrote on Feb. 28, 2015 @ 22:40 GMT
Lawrence

Nice work. I see you mentioned entanglement of quantum black holes. In light of twistor-scattering amplitudes, I suspect there is an operad structure, to be defined for all such quantum black holes. In the complex case, the operad is already defined (by Loday), and can be viewed as a chain of punctured Riemann spheres. This has an interpretation in terms of associahedra and binary tree diagrams. In essence, once can tile a moduli space with associahedra. The vertices of the associahedra correspond to interactions that contribute to the relevant n-object scattering amplitudes.

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Author Lawrence B Crowell replied on Mar. 1, 2015 @ 01:45 GMT
The connection with the Bott periodicity of large N, or SU(N), entanglements with associahedra is with the some sort of projection of this thing. The Stasheff polytope K_5 has 14 vertices, and this might be seen as composed of two copies of 7 elements mapped to the Fano plane, with each of these “sevens” associated with a projective point “∞”, or as associated with eight elements by the Hopf fibration. So the associahedra might be some sort of “double cover” with the Bott periodicity on A&D type Lie groups.

I think to construct such operads one has to work more from the ground up. Spacetime as a monoid, groupoid or so called magma might lead to the sort of spacetime algebra that leads to the sort of category theory. The set of possible operations would then form these “trees” and are associated with certain homotopies and categories. My essay discusses this in an informal sense, and homotopy as a route to category theory with monad/monoid structure.

I can discuss this more tomorrow or later. I see that you have an essay as well, which I will try to get to in the coming days. These are coming in faster than I can read them.

Cheers LC

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Michael Rios replied on Mar. 1, 2015 @ 22:46 GMT
The construction of spacetime, from say, category theory would be elegant. Through the lens of noncommutative geometry, one comes very close to the spirit of category theory, as classic geometrical points are replaced by pure states of a C*-algebra. If one takes the lessons learned from D-branes seriously, gauge symmetries arise from configurations of branes, and natural brane coordinates are noncommutative. In fact, the brane coordinates arise from a noncommutative C*-algebra. Let's take the case of N coincident branes, which have a U(N) gauge symmetry as long as the branes are coincident. The N D-brane positions in spacetime are packaged in a matrix, say X, in the adjoint representation of the unbroken U(N) group. Upon diagonalizing X, the N eigenvalues give the classical spacetime positions of the N D-branes, corresponding to the ground state of the system.

In category theory language, the noncommutative C*-algebra of NxN complex matrices Mat(N,C) can be interpreted as the algebra of noncommutative functions over a finite point space, with N objects. The elements of the noncommutative C*-algebra serve as morphisms over these N objects. Therefore, we can define a category C with N objects in Obj(C) and Hom(C) consisting of Mat(N,C) morphisms.

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Author Lawrence B Crowell replied on Mar. 4, 2015 @ 13:38 GMT
Michael,

I have been a bit tied up with a number of things. Your discussion about D-branes and the NxN matrix of their symmetry in U(N) (SU(N)) or SO(N) is close to what I have been working on. The Bott periodicity of these matrix systems gives an 8-fold structure. This 8-fold system has a connection of E8. I am interested in 4-qubit entanglements of 8-qubit systems that are E8. The structure of four manifolds involves a construction with Plucker coordinates and the E8 Cartan matrix. This seems to imply, though I have not seen it in the literature, that for 8 qubits there is not the same SLOCC system based on the Kostant=Sekiguchi theorem. However, I suspect that the structure of 4-spaces might hold the key for something analogous to KS theorem and the structure of 2-3 (GHZ) entanglements that are constructed from G_{abcd}. If the universe has this sort of discrete structure via computation, then it makes some sense to say the universe is in some ways a "machine" that functions by mathematics.

Cheers LC

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Torsten Asselmeyer-Maluga wrote on Mar. 1, 2015 @ 21:56 GMT
Dear Lawrence,

very good essay, I have to go more deeply into the details to post a substantial comment. Also thanks for the emails (I will answer soon).

In the meantime I also wrote an essay which appears now. Her is the link to

my essay.

Best Torsten

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Christian Corda wrote on Mar. 7, 2015 @ 10:00 GMT
Hi LC,

Once again, you made an excellent work with a very profound Essay. Here are my comments:

1) This should have been an excellent Essay also in previous contests "It From Bit or Bit From It?" and "Is Reality Digital or Analog? ".

2) My recent work on black holes, that you know, goes in the same direction that "topology and the computation of topological charges and indices, quantum numbers, and connection to logical switching theory are likely to supplant concerns of geometry, metrics and infinitesimal structure of manifolds."

3) On the other hand, I am not sure that such a statement goes in one specific direction. I try to clarify: you claim that

i) "This means that the fundamental description of reality is not with space, spacetime or anything geometric. Geometry or metric space is something which is a measure of entanglement of quantum bits with black holes and the inability to follow the entanglement phase. Geometry is then not fundamental."

But you also claim that:

ii) "Spacetime is built up from entanglements".

Thus, in my opinion, statements can be inverted. One could claim that "Entanglement of quantum bits is something which is a measure of geometry or metric space". In other words, this could be a sort of duality and/or complementarity of the fundamental description of reality. Geometry and entanglement could be two different aspects of the fundamental description of reality, but we could be unable to decide which one is the most fundamental.

In any case, I stress again that you wrote an intriguing Essay deserving the top score that I am going to give you.

I wish you best luck in the Contest.

Cheers,

Ch.

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Author Lawrence B Crowell replied on Mar. 7, 2015 @ 14:10 GMT
Dear Christian,

Event horizons are a way that a quantum state or EPR pair of qubits can be entangled with a black hole. The entanglement is coarse grained, for one does not know which of the qubits on the horizon one's EPR state in the pair you are entangled with. Any attempt to find out runs into limitations of the Heisenberg microscope argument. So horizons are an ensemble or Bayesian set of priors for quantum states. This is sort of the meaning of how geometry is built up from entanglements. The converse has some element to it, in that entanglements are coset structures of geometric elements.

I will look at your new entry and score in the next few days. I have been on travel lately and time is a bit tight.

Cheers LC

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Mark A. Thomas wrote on Mar. 8, 2015 @ 19:56 GMT
Hello Lawrence,

As a layperson I am not going to pretend to understand these new approaches and the maths being utilised but in your conclusions you state interestingly,

"Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent whole-

ness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos...

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Author Lawrence B Crowell replied on Mar. 9, 2015 @ 04:29 GMT
Mark,

There are some indications that this might be happening. With high energy physics the LHC and tests of the standard model, supersymmetry and maybe some hints of exotic physics are probably the last of the sort of direct tests physics has enjoyed or wanted. The future may see theories tested in increasingly oblique or indirect ways. I hope that progress can be had this way through the 21st century.

I see the prospect that physics becomes purely a mental game a bit disturbing. If we end up in the future where we are unable to make any test, no matter how indirect, of our physical theories we will be in a problem. In some ways it will be the end of physics as a science. Science does depend upon experimentation.

Theoretically the heirarchy problem is solved with supersymmetry. It is just an open question of whether the LHC with 13TeV beam can find evidence of supersymmetry.

Cheers LC

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Torsten Asselmeyer-Maluga wrote on Mar. 9, 2015 @ 15:21 GMT
Dear Lawrence,

Now I had time to read yiour essay. I agree with a comment above: it is one of your best. Here are my own comments:

1. You spoke about Bott periodicity but SU(N) has a 2-periodicity. It is the SO(N) group which admits an eight-fold periodicity (with integer coefficients, it is 4-periodic like the symplectic group for rational coefficients).

2. Your double slit experiment is very interesting. You view it from the topological point of view. Maybe one should remark that this approach wa already done by Berry and others using geometric phases.

3. you discussed it that HOTT will overcome the continuum approach. But homotopy needs a continuous family of maps (the deformation). It is central point in the approach and many results using implicitely the continuum (like Cerf theory, Whiteheads theorem etc.)

4. I also don't understand why you want to change from continuum to discrete. I showed in a previous essay that a smooth manifold contains only finitely many information (from topology). Furthermore, the dynamics in quantm mechanics (or field theory) is smooth (and continuous). Only the spectrum of the operators is discrete.

5. HOTT is a good approach but this proposal don't change the logic. Therefore Gödel works. Fro your approach, you need model theory (including forcing) to go over it. I remembered on a approach of Landsman to quantum mechanics using this approach. But my friend (and co-worker) Jerzy is the real expert.

I like your part explaining the Turing machine (and the relation to the Entscheidungsproblem)

Very good work

Torsten

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Author Lawrence B Crowell replied on Mar. 10, 2015 @ 02:36 GMT
In the end there is a bit of a duality here, or a dialectic of sorts. I think that what is measured in physics is discrete. We measure certain observables that have finite values, and quantum physics in particular bears this out pretty seriously. The continuum aspects to physics is pretty much a mathematical issue. Experimental data does not have any reference to infinitesimals or infinities. ...

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Torsten Asselmeyer-Maluga replied on Mar. 10, 2015 @ 15:41 GMT
Lawrence,

Ok I see the point. Of course the outcome of experiments is not a real number but as you also point out, one has problems to confirm the discrete structure of spacetime.

I see one reason in the underlying topological nature of physics. You also discussed it in your essay. I will illustrate it in a an example:

If two curves intersect then we measure the number of intersections (a discrete number, gauge or diffeomorphism invariant) but in most cases we are not interested in the coordinates of the intersection. Even sometimes we have problem to determine the coordinate system.

I see the measurement values in physics in this fashion. But then one has a dichotomy between discrete (number of intersections) and continuous. The measured values are in principle discrete but you need the continuum to express the probabilites of quantum mechanics.

I don't see any contradiction in this picture. Of course you will never measure that spacetime has a continuum structure but you can measure a discrete structure. And as you correctly point out: every experiment failed up to now.

In principle I agree with you very much. In particular I like your body-and-soul picture

Best

Torsten

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Author Lawrence B Crowell replied on Mar. 12, 2015 @ 13:07 GMT
In the subject of gauge theory a central aspect is the interaction form. These types of continuous homotopy or homotopy-like constructions involve curves that can be adjusted in certain ways so that an index is invariant or constant.

I am back home, but of course I have a lot of things to attend to here. I will try to expand on things in the future. The whole subject involves orbit spaces, or quotents of groups or spaces. The subject of four-manifolds is centered around the moduli, a 5-dim space that in a hyperbolic setting can be the AdS_5. Of course the hyperbolic setting is not Hausdorff and there are other problems. However, this is a form of orbit space that is mapped to the quantum SLOCC types of theory.

Cheers LC

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Michel Planat wrote on Mar. 12, 2015 @ 15:47 GMT
Dear Lawrence,

You start with Goedel "no mathematical system can ever prove all possible atements as theorems about itself" and you propose HOTT (homotopy and type theory together) which of course fits the great categorization process at work in mathematics. I found a very recent preprint of Yuri Manin pointing the same direction http://xxx.lanl.gov/pdf/1501.00897.pdf

"Information is physical" and you seem to suggest that "mathematics is physical", and both are quantum (in your conclusion). I like your approach and thank you for a very original and readable text with non-trivial concepts.

This year, I am exploring the most discrete and anomalous/sporadic object ever found. I hope you can comment on it.

Best.

Michel

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Author Lawrence B Crowell replied on Mar. 13, 2015 @ 00:24 GMT
This is in line with motives, categories and fundamental quantities as discrete elements from homotopy or varieties. This is as you say in line with category theory. In fact I think that ultimately the fundamental observables in the universe are topological categories, similar to Etale or Grothendieke theory.

I see there being in a sense what I call the “body” of mathematics, which are those aspects of mathematics that can be, at least in principle, solved on a computer, and the “soul,” which is the continuum mathematics of infinitesimals and infinities. My essay concentrates on the body, and not so much on the soul. I think for physical science the body is more directly associated with what is observed in the universe.

The “body-soul” duality I tend to advocate is something one can “wear” as needed. I might by virtue of some argument want to invoke a mathematical objectivity of sets, continuous spaces and even to the point of Platonism. At other times I may put this entirely aside. In my essay I largely put this aside.

Garrison Keillor has a feature on his show “Prairie Home Companion” called Guy Noir with the opening line, “On a dark night in a city that knows how to keep its secrets, one man seeks answers to life’s persistent questions; Guy Noir, Private Eye.” That is about my sense of the question about the relationship between physics and mathematics. We may never know for sure. Further, the universe may have a kernel of structure, symmetry and order to it that appears in a fractal-like form at different scales, but where nature also has this inherently chaotic or disordered nature to it as well, which I think is distilled down to the stochastic nature of quantum measurement.

I will try to get to your essay in the near future. I just got back from some travelling.

Cheers LC

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Michel Planat wrote on Mar. 15, 2015 @ 09:50 GMT
Dear Lawrence,

Every scientist has his own way and velocity in going through the wonderful secrets of nature. At FQXi you already wrote many excellent essays like "Discrete time and Kleinian structures in Duality Between Spacetime and Particle Physics". I wonder if you already looked seriously at the concept of an orbifold? I see that it plays a role in the VOA associated to some sporadic groups. I also found http://arxiv.org/abs/math/0505431 for your topic of this year.

I appreciate much the impetus you gave to my essay. After my first participation I learned how it works and don't take care to much of the lazzy inappropriate votes. You received from me the best endorsement. The goal is a continuing friendly discussion about the topics of mutual interest.

Best.

Michel

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Author Lawrence B Crowell wrote on Mar. 15, 2015 @ 20:22 GMT
Dear Michel,

Of course I am aware of orbifolds with respect to superstring theory. The vertex operator algebra with partition function p(q) =tr q^N = Π_{N}1/(1 - q^n) is related to the Dedekind eta function. The trace results in the power [p(q)]^{24} In this there is a module or subalgebra of SL(2,Z), eg S(Z) ⊂SL(2,C), that forms a set of operators S(z)∂_z. This module or subgroup...

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Michel Planat replied on Mar. 15, 2015 @ 20:49 GMT
Dear Lawrence,

I am really impressed by your knowkedge of so many things related to string theory. I propose that we start a collaboration because we have so many things to share and we are also quite complementary. I was very enthousiastic in writing the essay because new relations between several parts of maths and physics was taking place as by magic and also thanks to the computer. This is unreasonable in some sense!

My best wishes,

Michel

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Author Lawrence B Crowell replied on Mar. 16, 2015 @ 16:59 GMT
Dear Michel,

That might be interesting. I have been pondering how it might be that Γ^+_0(5) is related to the tiling and permutation of links on AdS_5. The quotient SO(4,2)/SO(4,1) = AdS_5 is not an entanglement group, at least not as I know, but this might have some relationship to entanglement. This might be through the Γ^+_0(5). Particularly if this is related to Langlands in some way.

Cheers LC

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Jonathan J. Dickau wrote on Mar. 17, 2015 @ 03:53 GMT
I want to give your paper some time Lawrence..

But I want you to know that your essay is on my radar of important papers to read for detail (and I have skimmed it), while the contest is still underway. I see that you mention Bott-periodicity, which is a topic I would have touched on in my essay - had I allowed myself adequate time. My entry this year is briefer than I intended, because I did not.

I was happy to see that you mentioned the HOTT program, which I also find to be interesting and relevant. I especially like that their pursuit of univalent foundations is geometrically constructive, but it is tied to a rigorous analytic proof checking engine. I find this usage of constructivist Math as program code particularly elegant.

More later,

Jonathan

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Author Lawrence B Crowell replied on Mar. 17, 2015 @ 12:37 GMT
Jonathan,

I am working my way through reading these essays. I will try to get to yours before too long.

The HOTT program does put mathematical foundations closer to algorithmic structures. It might be a way to address what I call the "body" of mathematics, which is that part of mathematics that is reduced to a computation. This can be computed in some way on a computer. The part of mathematics that involves infinitesimals and set theoretic infinities are what might be called the "soul." I don't deny the existence of this per se, but I don't think it has a direct connection to physics.

I am working right now to find out how Bott periodicity applies with exceptional groups. The intention is to find a way that nilpotent sets can be mapped to max compact subsets as with the Kostant-Sekiguchi theorem.

Cheers LC

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Pentcho Valev wrote on Mar. 17, 2015 @ 09:35 GMT
In your Bio you wrote: "I think it is likely there is some subtle, and in some ways simple, physical principle that is not understood, or some current principle that is an obstruction."

Einstein's constant-speed-of-light postulate is an obstruction. In a paper published in Science Miles Padgett showed that the speed of light (in a vacuum) is not a constant:

"The speed of light is a limit, not a constant - that's what researchers in Glasgow, Scotland, say. A group of them just proved that light can be slowed down, permanently."

Pentcho Valev

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Author Lawrence B Crowell replied on Mar. 17, 2015 @ 12:41 GMT
Pentcho,

You spend a lot of time thumping this theme. Sadly, mostly this is just a demonstration that you don't know what you are talking about. I have no intention of getting into an argument over this, any more than I intend to argue for evolution to a committed creationist or global warming to a climate denialist.

The speed of light is different in media, and some exotic media have been developed that can trap light. This does not falsify relativity.

LC

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Pentcho Valev replied on Mar. 17, 2015 @ 13:50 GMT
"The speed of light is different in media, and some exotic media have been developed that can trap light. This does not falsify relativity."

They slowed down light IN A VACUUM:

"Physicists manage to slow down light inside vacuum (...) ...even now the light is no longer in the mask, it's just the propagating in free space – the speed is still slow. (...) "This finding shows unambiguously that the propagation of light can be slowed below the commonly accepted figure of 299,792,458 metres per second, even when travelling in air or vacuum," co-author Romero explains in the University of Glasgow press release."

Pentcho Valev

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Author Lawrence B Crowell replied on Mar. 17, 2015 @ 15:52 GMT
This does not have a bearing on relativity, but is a quantum effect. One might say that the action of this mask that slows down photons can persist with a photon in much the same way as with the Wheeler Delayed Choice Experiment.

LC

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William T. Parsons wrote on Mar. 18, 2015 @ 20:17 GMT
Hi LC--

I loved your essay. You covered an immense amount of ground--and did so in a cogent yet concise manner. Congratulations!

I now turn to discuss some comments that you made in response to my essay. You raised the issues of super-Turing machines and the physics of super-tasking. I am not an expert on either. However, I have looked at several examples of physical super-tasking (e.g., carrying out an infinite number of physical operations within a finite time period). I did so because super-tasking appeared to be one place where physics might really need the concept of "physical infinity". As you know from my essay, I call into question the necessity and desirability of relying upon physical infinity.

In fact, for me, super-tasking was the "tipping point" against physical infinity. In every example I looked at, I found that either: (a) the super-tasking scenario was unphysical and could not work realistically (e.g., because of friction, chaos, cannot propagate a signal faster than c, etc.); or (b) the underlying physics was so murky that I couldn't tell whether the scenario was physically realistic or not. I place super-tasking via Malament-Hogarth spacetimes in the latter category. Regarding super-tasking via M-H spacetimes, I strongly recommend Earman's book, "Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes". His Chapter 4 includes an excellent review of M-H super-tasking.

Best regards,

Bill.

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Lawrence B. Crowell replied on Mar. 19, 2015 @ 00:32 GMT
I am glad you liked my essay. I threw in the subject the MH spacetimes and supertasking because that seems to be something that needs to be considered for a number of reasons. I think that non-eternal black holes can’t be supertasking machines. The black hole decays by Hawking radiation and disappears before i^∞, so there is no continuous stream of infinite amount of information that can approach an observer as they approach r^-. However, this probably means that NP-complete problems can be quickly solved for the internal observer and the exponential time is replaced with ~ r – r_- near r_-. This may mean that the NP-complete problem of compactifying all CY manifolds is computed by black holes. I do agree that it may be unlikely that superTuring computing is possible in a way that the output can be read by an exterior observer.

It is possible still that black holes are MH machines, even if they are finite in duration. This might be the case if black hole singularities are all the same thing. It could well be that black holes are all connected in a single quantum state that defines the singularity, and in a multiverse setting it could be that this is a great MH machine. The universe might then has underlying it a supertasking computer that is the ultimate quantum error correction code. I can go into this in detail if you want, though I will avoid that for now. Supertasking process in this setting is then associated with what were called shadow states. Shadow states are an old idea going back to the 1970s with S-matrix bootstrap physics. These are states which have T-matrix realizations, but they have no Born interpretation as associated with observables. The output of the MH spacetime machine can’t be read!

Cheers LC

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William T. Parsons replied on Mar. 23, 2015 @ 17:12 GMT
Hi LC--

I think that we are in agreement on the issue of super-tasking via M-H spacetimes. It's amazing the kinds of things that show up in our discussion threads! Thanks for taking the time to set out your position on this issue.

Best regards,

Bill.

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Author Lawrence B Crowell replied on Mar. 23, 2015 @ 20:10 GMT
There is not likely to be any way that supercomputing machines such as from MH spacetimes will produce readable output. This does not mean it is absent, but it may simply not involve quantum information that is directly read.

LC

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Georgina Woodward wrote on Mar. 21, 2015 @ 00:28 GMT
Hi Lawrence,

Nice historical introduction. Interesting new maths. Most enjoyed philosophical concerns, which is more "down my street". Good that your essay is getting noticed. Good Luck, Georgina

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Steven P Sax wrote on Mar. 24, 2015 @ 07:25 GMT
Hi Lawrence,

Your essay is a real wealth of knowledge, and I thoroughly enjoyed reading it. I liked your analysis of Godel and relating it to the cardinality of the continuum not being decidable, and your approach to the limits of computability and the Berry paradox. Your relation of topology to computation is very fascinating as is your subsequent in depth perspective through holographic principles. "What is fundamental are topological quantum numbers, such as those here associated with the two slit experiment or black hole horizon units of area." I'm going to think about this foundationally, and see how I can relate it to other foundational concepts, including my self-referential operators. Finally your discussion on continuous mathematics and attempting non-computable problems is interesting. I'm very intrigued by this as you know and my essay explores transcending Turing machines as well. We of course are in agreement that the physical aspects are quite important.

Thanks again very much your comments on my essay and the dialogue we had on undecidability; see also there the thread (above yours) where I tie this back to incompleteness and the undoing of Hilbert's Einscheidungsproblem. Also note Gentzen's proof of consistency for Peano axioms using transfinite induction, which affirms some of the concepts in your paper.

Thanks again, it's a great contribution to this essay topic and I rated it very highly. Please also take a moment to rate mine, especially now that we've been through them both and share a number of topics. Best, Steve

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Author Lawrence B Crowell replied on Mar. 24, 2015 @ 10:47 GMT
I am glad that you enjoyed my essay. The Einscheidungsproblem of Hilbert turned out to have this strange impact on mathematics that Hilbert never imagined at the time. On the other hand I have read that Goedel discussed with Einstien on how he was fairly unhappy that his result seemed not to have practical impact on mathematics. However, in some ways that may now be the case. The formulation of mathematical physics might involve recognition of these matters.

Your recognition that a quantum system in a superposition of two states in a qubit has undecidable nature is interesting. I think a quantum system in a superposition of states could reflect a Goedelian undecidable situation in some problem involving einselection, or maybe even deeper with problems with quantum error correction codes (QECC) in black holes. It discuss hypercomputing in my essay, and this could involve some aspect of how QECC in black holes and the erasure of quantum bits that accumulate. This may be an undecidable problem, and hypercomputing might indicate something that is concealed from observability.

I will try to look up Gentzen's proof of consistency for Peano axioms. I thought I had scored your essay earlier, but I had not, so I just now scored it.

Cheers LC

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Cristinel Stoica wrote on Apr. 1, 2015 @ 13:50 GMT
Dear Lawrence,

I finally got to read your essay, and I loved it! As usual, you make excellent and deep connection between various things, connections that allow us to see relevant subtleties. You made interesting connections between computation, quantum theory, homotopy, black holes, and proved that HOTT may be very well the way to the next stage of physics.

Best wishes,

Cristi Stoica

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Author Lawrence B Crowell replied on Apr. 1, 2015 @ 21:26 GMT
Christinel,

Thanks for the positive assessment of my paper. I gave your paper a pretty high score a few weeks ago. I did this while I was on travel and I don't think I had time to write a post on your blog page. I will try to write a comment, which will probably require rereading your paper.

There is a paper by Schreiber on directly applying HOTT to physics. This is a difficult...

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Cristinel Stoica replied on Apr. 2, 2015 @ 06:49 GMT
Dear Lawrence,

Thank you for the links, and for the explanations.

Best wishes,

Cristi

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Joe Fisher wrote on Apr. 1, 2015 @ 18:32 GMT
Dear Dr. Crowell,

I thought that your engrossing essay was exceptionally well written and I do hope that it fares well in the competition.

I think Newton was wrong about abstract gravity; Einstein was wrong about abstract space/time, and Hawking was wrong about the explosive capability of NOTHING.

All I ask is that you give my essay WHY THE REAL UNIVERSE IS NOT MATHEMATICAL a fair reading and that you allow me to answer any objections you may leave in my comment box about it.

Joe Fisher

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Author Lawrence B Crowell replied on Apr. 1, 2015 @ 22:21 GMT
I will take a look at it in the near future.

LC

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Member Marc Séguin wrote on Apr. 5, 2015 @ 19:53 GMT
Dear Laurence,

As always, you submitted a challenging and thought provoking essay, and I am glad it is doing very well so far in the ratings. I particularly enjoyed your discussion of numbers too big for a Turing machine in our universe to count to.

In your conclusion, you write:

"Chaitan has advanced ideas that mathematics is not something that exists in any sort of coherent wholeness. It is more a sort of archipelago of logically consistent systems that sit in an ocean of chaos. [...] Possibly the quantum vacuum is similar. It may be a tangle of self-referential quantum bits, where some sets of these exist in logical coherent forms. These zones of logical coherence might form a type of universe. These logical coherent forms are then accidents similar to Chaitan's philosophy of mathematics. It is very difficult to understand how this could be scienti fically demonstrated, yet maybe regularities in physics described by mathematics exist for no reason at all."

As you found out when your read my essay, this is pretty much how I see our universe in relation to the Maxiverse that results from the Mathematical Universe Hypothesis. Our universe exists for no specific reason, because all possible universes do, and the regularities that we observe between our physics and known mathematics is simply a necessary condition for the existence of self-aware substructures.

All the best,

Marc

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Author Lawrence B Crowell replied on Apr. 6, 2015 @ 00:35 GMT
Marc,

Thanks for the encouraging word here, and the upward boost.

This touches in many ways on the issue of unification of physics. In particular this concerns the plethora of unification schemes. I tend to think that they may all, or most of them, are correct. They may have some probability assignment, but they all may well manifest themselves. There may be cosmologies with very different particles and interactions than what exists in this observable cosmology. I base this on part with what I am working on, which seems to be deriving a type of landscape of string/M-theory. That is of course a good thing that I can recover something known.

I think the many worlds account of QM has some connection to the string landscape. If the spatial surface of this cosmology is infinite there are an infinite number of us sufficiently far out there. In a cosmology that is infinite, vast distance and lack of causal connection may imply quantum entanglement. This in particular would apply across the particle horizon.

Cheers LC

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Akinbo Ojo wrote on Apr. 8, 2015 @ 14:21 GMT
Dear Lawrence,

I am giving your essay a second read. In addition to my earlier comments above, I wish to take you up on a part of your essay. You said, "...in point set topology there are an infinite number of points between any two points on the real number line with a finite distance between them. This means if they exist in some meaning according to computation there must be a machine that performs any calculation of points separated by any tiny finite set of intervals segmenting the distance between these points"

If we may interrogate this, I wish to ask:

1 - Can distance be what separates two points, when distance itself is constituted of points? Or are there some distances constituted of points and other types of distances not so constituted and not having points as their extremities of their extension or segments thereof.

2 - What is an interval made of? Is it spatial or temporal?

3 - How many intervals, if such exist can be on a real number line? I ask because of the 'finite' set of intervals in the quote above.

4 - How can a real number line with an infinite number of points be divided, if points cannot be divided into parts and there is always a point at the incidence of cutting?

5 - Finally, talking about "existing in some meaning", are points eternally existing objects or can they perish? If the Universe can perish, will points outlive it? If there was a Big bang Universe creation from Nothing, did points precede it?

Regards,

Akinbo

*If you don't mind you may drop me a note on my forum so I get email notice. That is if you are inclined to discuss the above.

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Akinbo Ojo replied on Apr. 9, 2015 @ 12:13 GMT
Thanks Lawrence for dropping your comments at my forum. Appreciated.

If you have the time, you may wish to volunteer direct opinion to the 5 questions I attempted to raise here.

Regards,

Akinbo

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Author Lawrence B Crowell replied on Apr. 10, 2015 @ 01:23 GMT
I can answer some of these. An interval in relativity is the measure of a clock on a frame bundle and on a certain path. It is the integration of the path length.

The matter of infinitesimals, a length or displacement along a certain direction that is arbitrarily small, has been a subject of debate and research for a long time. This matter has only been somewhat resolved with the so called Robinson numbers, which have underlying it set theoretic concerns of forcing and the continuum. I am not a great expert on this subject, so I can really only make mention of this in a short post like this. In the end it only works, as I understand, within a certain continuum model. The underpinnings of calculus and questions surrounding the Dedekind cut do not seem to be derived according to a complete axiomatic system. However, with a few basic ideas you can develop a lot of calculus.

Mathematics in the objective or in some ways the Platonic perspective does not perish with the heat death or end of the universe. If mathematics is nothing more than a pattern system derived from the natural world then in that model it might perish. I am not terribly committed to either perspective. There are troubles with either viewpoint.

Garrison Keillor has his “Guy Noir,” who “On the tenth floor of the Atlas building still seeks answers to life’s persistent questions.” If you have ever listened to his “Prairie Home Companion” you know this well. There are persistent questions, such as “Does God exist,” that will probably never be conclusively answered.

LC

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Thomas Howard Ray wrote on Apr. 11, 2015 @ 18:14 GMT
Hi Lawrence,

If it hasn't been brought up yet -- I want to make sure that we get the spelling of Gregory Chaitin's name right. He's among my favorite mathematicians/computer scientists, so the typo jumps out at me.

There's no getting around the issue of how we differ in our views of foundations. I do not think classical physics is either finished, or emergent from conventional quantum theory -- in fact, I think it is the other way around. Although it is commonly believed , as you say, that "The classical picture of the universe is a continuum of flows [3] ..." this is not true. Continuous functions as described by differential equations or topological methods do not support a physical continuum of space independent of Minkowski spacetime, because space has no physical reality independent of time.

I think this is easier to see by critical study of Perelman's solution to the Thurston geometrization conjecture -- all singularities on S^3 are extinguished in finite time by continuation (via surgery) of the Ricci flow, on the half open interval [0, oo). This is the mathematical advantage that any simply connected 4-dimensional world -- including Minkowski space-time -- has over a multiply connected space of random functions in 3 dimensions (or in fact, Hilbert space of any dimensionality).

Nevertheless -- my highest score goes to your essay, for setting up the issues in thoughtful and highly readable terms, even though I couldn't be more opposed to the notion that "Spacetime is built up from entanglements [13]" Classical orientation entanglement explains the phenomenon just as well, when a time parameter (such as that of Hess-Philipp) is included in the dynamics.

I hope you get a chance to check out my essay as well.

All best wishes,

Tom

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Author Lawrence B Crowell replied on Apr. 11, 2015 @ 23:59 GMT
Thanks for the positive vote or score.

Before 1900 it was commonly thought the universe was a continuum, and the idea of atoms was under attack, as this was thought to not conform to the continuum picture of reality. Of course Planck assumed that energy occurred in discrete steps, and Planck and Bohr assumed discrete values of angular momentum as well to model the atom. Quantum physics does have continuum structure, such as the dynamics of the wave function or the system of paths in a Feynman path integral. However, these no longer have the sort of ontology that continuum structures have in classical physics. The existential aspects of the quantum wave function is not longer ontological, and recently it is being found that the epistemological foundation of the quantum wave is not satisfactory either.

How classical physics emerges is tough to understand. How an einselected basis occurs so that a particular eigenvalues corresponds to a measurement or is associated with a classical value is not solved. The paper by Sax proposes that Goedel’s incompleteness theorem plays a role. I had some discussions with him on this on his essay blog page. This is curiously important with D-branes, for these are classical or macroscopic structures. While they are ultimately made of strings, or are similar to Fermi surfaces of electrons or condensates of quantum states, they are nonetheless classical and important for foundations.

Sorry about the Chaitan for Chaitin. That is a regrettable typo. I don’t remember if I read your paper or not. I will try to take a look at it soon.

Cheers LC

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Eckard Blumschein replied on Apr. 13, 2015 @ 05:56 GMT
"I don’t remember if I read your paper or not.=

Isn't this one more unnecessary insult?

I recall joy Christian and Einstein's anus mirabilis.

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Alexey/Lev Burov wrote on Apr. 12, 2015 @ 03:51 GMT
Dear Lawrence,

An assumption at the end of your article

"Mathematics and physics have this curious relationship to each other for purely stochastic or accidental reasons; there ultimately is no reason for this"

provokes me to note that this possibility is refuted in our essay on the scientific ground.

Best regards,

Alexey Burov.

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Author Lawrence B Crowell replied on Apr. 12, 2015 @ 12:20 GMT
Dear Alexy and Lev.

Your paper is well argued. I will admit to being very agnostic about these sorts of ideas. In particular I am very agnostic about Tegmark’s hypothesis, which seems not mathematically provable, nor scientifically testable. Even string theory is only at best indirectly testable, but Tegmark’s Mathematical Universe Hypothesis seems impossible to test.

A...

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Alexey/Lev Burov replied on Apr. 12, 2015 @ 18:15 GMT
Dear Lawrence, I am following your good idea to double the comment.

*****

Dear Lawrence,

Thank you so much for your generous compliments to our essay. As you see, we are showing there how Tegmark's MUH is refuted on the scientific ground. Yes, it goes against the dominating opinion of the community of cosmologists (and your own), that the full-blown MUH is unfalsifiable, but our refutation looks very solid for me.

About your 'couple of points'. First, your distinction of WAP and SAP fully agree with the conventional one, as I may judge. It isn't clear to me what point you were trying to make about them. Second, we use the word "chaos" in its ancient meaning, as we stress it when this word is used the first time, pointing there to Platonic philosophy. This meaning sometimes is expressed by such words as "nothingness" or "nothing". This formless entity, chaos/nothingness, is a source of pure accidental, random, causeless factors. It has little to do with the mathematical concept of "dynamical chaos" you mention, which assumes certain mathematical forms already given.

Your ideas about "the soul of mathematics" sound very interesting to me, and I would very much wish to discuss them with you in much more detail than this specific place and occasion allows. You know how to find my email. Please be assured that I would highly value communication with you on these and other questions.

All the best,

Alexey.

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Author Lawrence B Crowell replied on Apr. 13, 2015 @ 10:13 GMT
The meaning of chaos might in that sense also mean void or nothing. Of course the trick is that there can exist vacua at different energy. Since this is quantum mechanical there can be quantum transitions or tunneling between two vacua.

Your essay did propose an alternative to the MUH. My main point with untestable is empirical. My main issue with MUH has been that it tries to "prove too much." The idea is a sort of monism, which tries to reject duality between forms and substance, but in the end it runs into difficulty I think.

Cheers LC

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RJ Tang wrote on Apr. 12, 2015 @ 17:40 GMT
There is a profound principal in the universe that says there is no central entity or notion anywhere, and that everything has no special significance than any other things in physics terms. This principal dispelled ‘earth-centric’ idea and later the Newtonian absolute time-space concept. It is a universally accepted principal in modern science. If math and physics are intertwined inextricably...

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Author Lawrence B Crowell replied on Apr. 13, 2015 @ 10:36 GMT
I will have to respond in detail later. It is the morning before heading off to work. In many ways physics is coming around to the idea that information is at the foundation of what is important with respect to phenomenology.

LC

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RJ Tang wrote on Apr. 12, 2015 @ 17:55 GMT
I think the world around us gives us all type of CLUEs to our intellectual pursuit. Great thinkers are great observers of hints and clues. Ultimately the physical world (not the bio-sphere or human society) is far simpler than anyone can think of, because of the equilibrium status of the universe. Or read differently the static nature of the universe. The universe is a dead body waiting for eons to be dissected. Complexity is largely suppressed because of the cancelling effect of forces, as well the elimination of complexity by evolution of a closed system. The complexity is decreasing as a function of increasing approximation to an equilibrium.

The trick is to change our human centric mind set and think out our own box. I pointed the natural number conundrum. It is an example of homo sapiens obsession. New type of understanding is needed and can be achieved by shedding unconscious constraints posed upon ourselves.

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Author Lawrence B Crowell replied on Apr. 13, 2015 @ 10:25 GMT
Actually equilibrium in general relativity is not well defined. Suppose you have a black hole that has the same horizon temperature T ~ 1/M as the cosmic background temperature. If the black hole emits quanta it becomes hotter and the probability that it will then emit more photons to the universe increases. Conversely, if the black hole absorbs a photon from the background universe it becomes colder which makes it more probable that it will absorb more photons than emit them. This is a bit odd with respect to semi-classical or quantum physics in spacetime.

LC

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RJ Tang wrote on Apr. 12, 2015 @ 18:55 GMT
Math is all about mapping from one artificial domain to another artificial domain. The mapping works well but does not give rise to the legit or such mappings.

I propose that quantum mechanics needs an entirely new type of math. The current approach to quantum mechanics is bewildering and confusing. The fundamental rules of math need change to adapt to the quantum mechanics world.

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Author Lawrence B Crowell replied on Apr. 13, 2015 @ 10:28 GMT
There have been some ideas along these lines. There was axiomatic quantum field theory, but this never really seemed to catch on. There is also quantum logic, which has lead to some interesting developments. There is though Bohr's statement that quantum physics is best described by a system that permits discussion in ordinary language.

LC

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adel sadeq wrote on Apr. 12, 2015 @ 22:53 GMT
Hi Lawrence,

I have read your essay and many of your posts and appreciate what you are trying to do. But please take a look at my essay if you haven't already and give it some effort since you are a good programmer, it should not take much of your time. You will see that physics can be done based on the most elementary mathematics known(addition, comparison ...etc). So fundamentally no need for exotic math or to worry about infinity, compute-able ... etc. The system can be put on more formal level(fundamentally it is a geometric probability problem), but through simulation the origin of the design of reality is so clear and makes a lot of sense. Then MUH is established(confirmed) with minimum effort, we start by postulating it and end by confirming it. Our reality's existence is the proof of the Platonic realm of mathematics.

Essay

Thanks and good luck

P.S. But please read some of the first post in my thread about running the programs if you like to delve in them more.

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Author Lawrence B Crowell replied on Apr. 13, 2015 @ 10:31 GMT
Dear Adel,

I suddenly see that I got a fair number of posts and have risen considerably up now to the top. It is interesting to respond to two people, Alexey and Lev Burov above, who refute the MUH, and then within the same half hour discuss somebody who embraces it. Of course I will have to read your essay first. I will try to get to that as soon as I can.

Cheers LC

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Jonathan J. Dickau wrote on Apr. 13, 2015 @ 05:35 GMT
Congratulations Lawrence on your high standing..

I'm just finishing reading your excellent paper, and I'll have some comments after I catch some sleep. This is one of your best essays so far, and it appears that high marks are well deserved.

All the Best,

Jonathan

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Author Lawrence B Crowell replied on Apr. 13, 2015 @ 10:33 GMT
Jonathan,

Thanks for the positive assessment. Indeed, I seem to have popped up considerably in the last day or two. I will have to take a look at your essay as well to refresh my memory on it. I can't recall if I scored it as yet.

Cheers LC

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Jonathan J. Dickau wrote on Apr. 14, 2015 @ 03:40 GMT
Excellent job Lawrence,

This essay is well-written and presents a Tour de Force of interesting Maths relevant to Physics. You have managed to work in a lot of topics that are very interesting to me, and about which I have much to learn, such as Homotopy Type Theory. The HoTT program is especially interesting to me, as it has a constructive geometric basis on the one hand, and a rigorous analytic procedure on the other. I also like that you wove in the Bott periodicity, which I was trying to find a way to fit into my own essay, because it is one of those invariant structures that one seems to bump into - as though it was there before you found it.

Being a constructivist, I think that perhaps numbers and counting are not the first Maths to arise, however. Having a Set of objects requires preexisting elements of geometric topology, so that objects with surfaces and containers to hold them are well-defined. Also, it is seen in young children that a sense of greater and lesser quantity is a kind of numeracy that exists apart from counting itself, and develops sooner. I would think that just as ontogeny recapitulates phylogeny, for developing organisms; so individual patterns of learning are reflected in the development of cultures. Perhaps counting is merely the earliest form of mathematical reckoning that could be written down.

I think you come out on the side of the formalists and logicists in the Brouwer Hilbert debate, while I am firmly in the intuitionist camp - and while this is sometimes termed anti-realist, I believe it is more realistic yo imagine that everything should be constructable for Physics. But at least you mention that there is a debate about this among mathematicians, which some might miss otherwise. It was a great effort overall, and you get high marks from me.

Regards,

Jonathan

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Author Lawrence B Crowell replied on Apr. 14, 2015 @ 13:52 GMT
Jonathan 

I am rather agnostic on any of these ideas about mathematical foundations. I don't hold to any of them to much degree. For one thing these things are a bit removed from physical theory, which I am more interested in than pure mathematics. The other reason is there seems to be no way we can decide whether one is better than the other. In some sense maybe it is best to consider them as metaphysical tools that can be used or not depending on the situation.

The homotopy and Bott periodicity involves my observation that groups involved with quantum information appear to have this period 8 structure to their topology. This seems to extend into the exceptional and sporadic groups as well. This means the quantum bits associated with a black hole event horizon have a type of degeneracy. This is the main reason why I think it is possible that this homotopy based mathematics with a correspondence to quantum bits might form the foundations of mathematical physics through this century. It would be curious to see what mathematical physics looks like in 75 years.

Really I don't pretend to know the relationship between physics and mathematics. It is a completely mystery really. It may just come down to an instrumentalist argument that because physical science involves measuring things according to numbers that the subject must necessarily involve mathematical consistency.

LC

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Jonathan J. Dickau replied on Apr. 14, 2015 @ 14:26 GMT
Thanks for the thoughtful reply Lawrence..

In his (non-contest) essay 'A View of Mathematics', Alain Connes speaks about Math as a single corpus, almost like a biological organism, that makes it hard to separate the parts or say what came first. I am of the opinion, however, that there must be some set of most elementary rudiments, from which that entire body of knowledge flows. Perhaps this does not mean it can be constructed deterministically, but there have to be some bones to hang the meat on somewhere.

You might find interesting Andrei Rodin's book 'Axiomatic Methods and Category Theory' which relates strongly to the HoTT program, and is available as a download from arXiv. But I think it is wonderful that mathematicians are pursuing a better understanding of the underpinnings of Math, to expose its underlying simplicity, or inherent congruency, as well as probing the complexities and the details of that knowledge.

All the Best,

Jonathan

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Author Lawrence B Crowell replied on Apr. 15, 2015 @ 14:51 GMT
The Rodin book looks to be a very long read. It could have some interesting insights into things. There are connections through groupoids to category theory and Grothendieck type of theory and cohomology.

It is hard to know what the totality of mathematics is. It could be infinite in extent, which of course makes it difficult to know how this applies directly to physics. That would be difficult with respect to the Tegmark MUH conjecture. The one thing that is apparent is how many areas of mathematics are mapped into each other according to functors and categories.

Cheers LC

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Michel Planat wrote on Apr. 14, 2015 @ 15:42 GMT
Dear Lawrence,

Following your last post, this is the type of application we can discuss. Until now, I focused on dessins due to their relationship to quantum geometries and contextuality as in my [12] and [17], now I mentioned in the essay the link to most sporadic groups, there are plenty of other applications, some have to be discovered. Cheers.

Michel

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Alma Ionescu wrote on Apr. 16, 2015 @ 19:59 GMT
Dear Lawrence,

I just finished reading your essay and let me tell you that it was one of the most original in this competition. I think that's justly reflected in your current position in the top.

What makes this paper special is your choice to treat computability instead of more vague questions. Surely this position is footed on more solid ground as it aims at describing potentially fruitful directions rather then simply focusing on the quirky side of the universe which brings forth coincidences and such. I found particularly striking your topological treatment of the wavefunction collapse due to a measurement needle state and I want to ask if you develop this treatment anywhere. I saw you are referencing a not yet published paper of yours on the topology of states on relativistic horizons, which is probably more to do with the equivalence you are drawing between a horizon and an N-slit (?). Anyway I'd like to get a better understanding of your work. I couldn't find it at arxiv - are your papers online somewhere where I can read them? I mean somewhere not behind a pay wall, since I have no affiliation and couldn't really afford it :)

Thank you for an engaging read! Should you have time to take a look at my essay, your comments are much appreciated.

Warm regards,

Alma

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Author Lawrence B Crowell wrote on Apr. 16, 2015 @ 21:50 GMT
Dear Alma,

Thanks for the positive assessment of my essay. Fortunately a number of people seem to share your opinion. It has been near the top since the beginning, and I have been in #1 and 2 spot for nearly a week.

I see there being a sort of two fold system. Standard mathematics might be thought of as the “soul,” or a “ghost,” and mathematics that is restrained by concerns of Kolmogoroff complexity, types and so forth as the “body.” It may not be possible to express all numbers between 10^{10^{10}^{10}}} and 10^{10^{10}^{10^{10}}}}, but this just means the body is not able to construct or contain the information space necessary to do so, but this still leaves room for the “soul.” Mathematicians are then free to “pick their poison,” where a pure mathematician may prefer to stay with the standard approaches to math, while a more practical minded analyst might prefer to stick with the “body.”

I don’t particularly get into the argument over whether the soul of mathematics exists or not. This involves things such as infinities, infinitesimal or even finite numbers that can’t ever be computed. I am agnostic on the idea of there being a Platonic realm of ideals. The idea seems in one sense compelling, but it also seems to lead to some mystical notions that are not entirely comforting.

Cheers LC

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Alma Ionescu replied on Apr. 18, 2015 @ 15:38 GMT
Dear Lc,

I know what you mean by notions that are not entirely comforting and I appreciate that :) I am not a platonist myself because it feels - to me at least - a bit useless; I am more of an utilitarian. Math is what math is. Thank you for answering my comment and thank you even more for finding the time to read and give your thoughts on my essay. Wish you best of luck in the competition and I hope to see more of your ideas as they make a very good read.

Cheers,

Alma

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Alma Ionescu replied on Apr. 18, 2015 @ 15:40 GMT
I just realized I didn't rate your essay so I am fixing that now. As you have a lot of votes, I hope mine is enough to make a difference.

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Author Lawrence B Crowell replied on Apr. 18, 2015 @ 20:28 GMT
Thank you for that. I voted for your essay a month ago or so. I don't remember the exact score I gave it. It was probably a 6 to 8 score.

Cheers LC

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Roger Schlafly wrote on Apr. 19, 2015 @ 21:11 GMT
You give the impression that there is something wrong with the foundations of math, just because different formulations and axiomatizations are possible. But these possibilities make very little difference to the great majority of math.

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Author Lawrence B Crowell replied on Apr. 20, 2015 @ 00:50 GMT
There are some questions concerning foundations of mathematics. I am not a great expert on this, but it does seem that as mathematical physics develops that it will embrace concepts that are not as tied to many aspects of point set topology with infinitesimals and the rest.

I don't say there is something wrong with the foundations, and it appears that we are increasingly in a time where there are several such foundations. These things seem in some ways to be model dependent, with different proof methods and the rest.

LC

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Laurence Hitterdale wrote on Apr. 20, 2015 @ 17:07 GMT
Dear Lawrence,

Many points in your especially comprehensive essay are worthy of comment, but I find particularly intriguing the idea mentioned at the end. This is the suggestion that mathematical reality and physical existence have the same unusual organization. It might be that in both of them we find islands of order set amidst vast and encompassing chaos. If this is so, then perhaps, as you say, there might be no reason for this similarity between mathematics and physics. However, I think we would try to find some deeper reasons, though I am not sure how we would go about that.

Thanks and best wishes,

Laurence Hitterdale

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Author Lawrence B Crowell replied on Apr. 20, 2015 @ 22:37 GMT
Dear Laurence,

Thank you for taking interest in my essay. The idea is that the quantum vacuum as a set of qubits, say (0, 1) set to a|0> = 0, has a phase structure based on how qubits are transformed into each other. We normally think of the vacuum as invariant under a certain symmetry group, but underneath that it could just be a vast self-referential loop, where there are “accidents” that occur where the vacuum has a symmetrical structure. This means zones exist where there are dynamical structures, where symmetries are aspects of division algebras.

These self-referential qubits, or loops of them, form a strange basis for the universe, or multi-verse, that can’t be derived or computed. We can’t then know what is not computable. It is similar to Chaitin’s halting probability; we can know there are incomputable symbol strings in a set of them of length N, but we can’t compute with certainty which are not computable (Turing’s halting problem), we can’t compute the number of them that are computable or not computable, or the probability for any of them to be incomputable or nonhalting. We are then faced with a bit of a conundrum; this would be a theory that tells us that this state of affairs exists, but we can’t compute much of anything with it.

If physics and cosmology reaches this state of knowledge it might be the end of these foundations. The end of scientific foundations might occur this way, though I suspect we have quite a ways to go before progress in physical foundations stops at this point.

Cheers LC

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ABDELWAHED BANNOURI wrote on Apr. 22, 2015 @ 17:45 GMT
Dear Laurence .

your essay is very interesting, indeed i consider it an extention to mine.

Sincerly yours

Bannouri

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