James, one of the things I've always liked about you, is that you do your homework before you risk a heated blowup (wish I could say the same for myself). You're right -- "radical empiricism" is a philosophy term, mainly applied to British empiricists of the 18th century whose views survive in various forms today. You write, "I keep pointing to that first theoretical error concerning the mathematical definition of mass." And I keep pointing out that we don't need a mathematical definition of mass. :-)

"With regard to incompleteness, I do not see incompletelness ever being resolved. Not necessarily for the 'stuff' reason that you gave, but, every attempt of completeness that I have seen grabs something important for free without explanation." In fact, though, a mathematically complete theory *is* an explanation that closes all logical judgments in its specified domain. Relativity is mathematically complete. Joy Christian's framework is the basis of a potentially mathematically complete theory.

"It is usually denied that that is the case, but, it sure seems clear to me that nothing leads to nothing." Have you read Lawrence Krauss's latest book?

"Anything that uses something more than its beginning nothing is obviously beginning with something." Which always leads to the question of what "something" and "nothing" mean. Is the quantum vacuum something or nothing?

"The mathematical loops that bring ends together is not representative of nothing. It is representative of extensive pre-existence." Does pre-existence exist? What about pre pre-existence?

"My opinion is that we should acknowledge both what we think we know and even more importantly that which we do not know. Anyway, while we see things differently, I take nothing back about my remark and rating for your essay. You are amazing in your ability to comprehend mathematic concepts is a manner that is broader than but is useful for application to theoretical physics."

Thanks for the high praise -- I don't feel deserving -- yet you absolutely describe the limit of my ambition. Mathematics -- indeed, all language -- does not transcend its utility. While I am saddened by the co-option of Wittgenstein by postmodern apologists, I agree in some deep sense with his aphorism, "Of that which one cannot speak, one must remain silent."

"The conversations that still take place concerning Joy's work are among the best I get to read. It doesn't matter if I personally do not accept spheres and such, especially spheres that are flat surfaces :). Thank you for the responsible role you fullfill during discussions that sometimes become 'radical' here."

You're a treasure, James. I know that topology isn't easy, but it's worth the trouble. Yes, I suppose it is a radical foundation compared to conventional views. Speaking of utility, though -- it works.

"I say that your essay is brilliant!"

As the politicians say, vote early and vote often. :-) Thank you so much for dropping in.

All best,

Tom

ROTFL! I am delighted to see the mood lightening. Joy, thanks for posting the great illustration. (Come on, Steve -- take yourself less seriously and you're bound to get more joy from life, no pun intended.)

"Tom, Do we live inside three tomatoes or am I missing the point? :)"

Now that's intersting James. Can one differentiate three flat 3-dimensional tomatoes from three flat dimensions when one is on the inside of a simply connected 9-dimensional tomato? Topology is about sufaces (manifolds).

Thinking of flatness, though -- you do live on the flat surface of a 2-sphere without boundary, do you not?

Best to all,

Tom

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  • Post #84

Tom,

Thank you for your comments. This stuff is extremely fun and interesting! But it also makes me feel so "deprived" and "underprivileged," and even "deficient" for lack of better terms. Whereas you and other are "living large," as it were, in your fancy 9-dimensional or even 11-dimensional universes, I'm stuck here in my lowly, humble, evolving 3-dimensional universe, believing that space-time is a myth perpetuated by a long-standing, faulty notion regarding the fundamental purpose and role of clocks.

These discussions often remind me of the adage that "using words to describe ideas is like using lumber to build a tree." It seems we're all just doing our best with the lumber we find available to us.

Cheers,

jcns

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  • Post #85

I was being flip, but to be serious I should have said 10 dimensional tomato, where every point is a 3-sphere.

(Yes, jcns, it's fun! Ray Munroe must be proud of us now.)

Best,

Tom

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  • Post #86

James,

I've got it now. I remember that I wanted to explain Scott Aaronson's argument in simpler and more accessible arithmetic, and then realized that the explanation grows ever more sophisticated and that I couldn't make it simpler.

So let me try again.

Remember, I claim that the answer to the first question *has* to be "yes," no matter what the question is. Here expressed in arithmetic, is why:

1 - 1/2 1/4 - 1/2 1/4 - 1/2 = 0

The important thing is that no value in the sum of these terms exceeds unity. Consistent with Scott's upper bound of probability .75 for a 2-player cooperative game, the first iteration (where the zeroth iteration is 1 - 1/2) is 1/2 1/4 = 3/4. When continued through the second and third iterations, we find 3/4 - 1/2 = 1/4 and 1/4 1/4 = 1/2. To make this easier to read:

(eqn 1) 1 - 1/2 1/4 - 1/2 1/4 - 1/2 = 0

= .5, .75, .25, .5

0 1 2 3 4 iterations

If we were to sum and average all the values of the results of these iterations -- which is what quantum mechanics does by averaging values of a run of experimental results -- we would get 1/2. This is the upper bound of a coin toss probability for sufficient length of n independent Bernoulli trials. When we revese signs for the above, we get:

(eqn 2) 1 1/2 - 1/4 1/2 - 1/4 1/2 = 2

Which is the CHSH upper bound for Bell-EPR type experiments.

What's the problem? The former case is algebraically closed; zero indicates that the computation halts. The Bell-CHSH case does not halt; it will increase monotonically to infinity. This is interpreted to mean that every experimental event not measured has a vanishing but non-zero probability of happening. The non-probabilistic result (the algebraic equation 1 summing to zero) tells us that no event is probabilistic; perfectly random tosses of a fair coin are precisely determined.

The way that quantum mechanics reconciles its probabilism (implying quantum entanglement and nonlocality), with actual physical reality, is by quantum unitarity -- the average of the set of iteration results in eqn 2 (1.5, 1.25, 1.75, 1.5) is unity, or probability 1.0 that the upper bound of any quantum pair correlation experiment obeys the upper bound of CHSH.

Now this is the simple arithmetic that Scott Aaronson, Richard Gill and many others believe without question is at the foundation of a probabilistic physical reality. Look, though, when expressed as the answers to yes-no questions (pairs of binary values):

(eqn 1) 1 - 1/2 (Yes) 1/4 (No) - 1/2 (Yes) 1/4 (No) - 1/2 (Yes) = 0

(eqn 2) 1 1/2 (No) - 1/4 (Yes) 1/2 (No) - 1/4 (Yes) 1/2 (No) = 2

If we allow the first iteration to answer "No" such that all subsequent iterative values exceed unity, we have loaded the dice in favor of an infinite dimensionless range of values that we ASSUME constitutes the sum of perfect information that we can treat by probability theory. This is reconciled to actual physical experiments (in which probability plays no actual physical role) by normalizing to quantum unitarity for any experimental result.

Joy Christian's inspired realization was that ORIENTATION of the topological initial condition determines the experimental outcome which must be ALGEBRAICALLY CLOSED as eqn 1 shows, and which thereby obviates quantum entanglement, nonlocality and probabilistic measure. This initial condition -- if one follows Joy's objective mathematical argument without imposing one's personal beliefs on it -- clearly produces E(a,b) = - a.b. Very straightforward.

I've been playing with the consequences of algebraic closure for topology, for quite a while. I decided to attach a draft of one such effort; section 4.25 relates to the present argument.

Tom

Dang it. Lost my log-in and the attachment -- forgot about the attachment size limit. Will get it on my personal page and link it later.

yes of course, templeton becomes crazy and makes irrational extrapolations, yes of course, and the lected is ...no but frankly, and what after ? a beer from belgium in a spherical 24 dimensions.

Now you are going to make your intelligent with your humor.

All is said and the team is known !!!

Universal , yes of course.

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  • Post #89

Posting the link

http://home.comcast.net/~thomasray1209/SophieGermainprimes8411.pdf

Naturally, I posted the link in the wrong place. Sorry. Look for 31 August, 0937.

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  • Post #91

Hi again jcs,

You wrote, "Whereas you and other are 'living large,' as it were, in your fancy 9-dimensional or even 11-dimensional universes, I'm stuck here in my lowly, humble, evolving 3-dimensional universe, believing that space-time is a myth perpetuated by a long-standing, faulty notion regarding the fundamental purpose and role of clocks."

Hmmm. I suppose Minkowski space could be described, in that context, as a myth. If one were disabused of the notion, though, that clocks have some fundamental purpose and role -- then the space-time model is the only one that makes sense in a relativistic universe. This is because we only infer facts of our physical existence by making measurements -- rods for length, clocks for time -- and if these were absolute, we would live in a dualistic universe; as it is, Minkowski discovered, we can only preserve physical reality for either of these notions by a continuum of the two. The fundamental role in physics is for spacetime.

Don't fear that any of us are living apart from you in a higher dimensional universe -- after all, we communicate within the same spacetime, and that's the only way, after all, in which we even know that either of us is "real" to the other. Extra dimensions are said by most string theorists to be hiding in our common space of measurement as curled up, compactified. I much prefer the explanation that Joy has offered: extra dimensions as an artifact of topological structure, not hidden at all. Living in 8-dimension space is not different, to our measurement and perceptions, than living in 4-dimension space.

"These discussions often remind me of the adage that 'using words to describe ideas is like using lumber to build a tree.' It seems we're all just doing our best with the lumber we find available to us."

Whether we used words or not, we'd still be compelled to use some formalized language. Even if we all had little computers with broadcasting stations in our heads to communicate with one another, the language would still have to have a common structure for encoding and decoding information.

All best,

Tom

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  • Post #92

Tom,

"I much prefer the explanation that Joy has offered: extra dimensions as an artifact of topological structure, not hidden at all. Living in 8-dimension space is not different, to our measurement and perceptions, than living in 4-dimension space."

Are you equating 'not hidden at all' with 'not different, to our measurement and preceptions'?

James

Hi James,

"Are you equating 'not hidden at all' with 'not different, to our measurement and preceptions'?"

Sure I am. The case is the same with Einstein's abandonment of the ether. If we find that electromagnetic waves propagate in a vacuum, the ether is superfluous to our physics, even if it exists in some sense. After all, ether may be necessary to explain some yet unknown physical phenonomenon.

Similarly, an experimental record of measurement events in 4 dimensions -- such as the quantum correlations in Joy's framework -- that are explained by an 8 dimension topological structure (and cannot be explained otherwise) contribute to our understanding physics in a rational model, without superfluous or mystical assumptions. Because Joy's model is mathematically complete, it begs the topology in which the mathematics corresponds to physical predictions 1 to 1. (As EPR-Bell requires for non-probabilistic explanation of reality.)

Tom

No, James, I don't think that is what Tom has in mind.

If I may pre-empt Tom's reply, the issue here, technically speaking, is between the "compactified" dimensions that string (et al.) theories require to bring us back to the "regular" 3+1 spacetime, versus what I am saying---which is that these dimensions are not compactrified at all. The strong quantum correlations we observe are in fact the *evidence* of these dimensions. They are all there in their naked glory, unashamed to not have been artificially compactified. And yet our measurement schema take place in the 3+1 dimensions.

But to understand the latter one must dig deep into the technical details of my (or rather Nature's) construction, well beyond the standard EPR correlation.

Joy

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  • Post #95

Joy,

"They are all there in their naked glory, unashamed to not have been artificially compactified. And yet our measurement schema take place in the 3+1 dimensions."

Thank you. I understand this. My non-expert opinion about compactification of dimensions is another hidding place allowing the theorist to offer an answer that is a guess and proceed forward with theory. I believe that I have seen this practice occur repeatedly during the development of theoretical physics. I do not presume that you, or anyone else, agree with this.

I do presume that you are equating "They are all there..." with "...not having been aritificially compactified." If this is correct, I wonder if the word 'artificially' were removed would you support the remaining statement. Obviously my question has to do with whether you say that dimensional compactification is inherently artificial or do you allow that it could be real but just not in this case?

James

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  • Post #96

Dear Joy,

Sorry 'equating was a poor choice of word'. I meant 'the opposite of'.

James

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  • Post #97

J.C.N Smith, James,

In Roger Penrose's lecture on "Twistors and quantum non locality" also available via FQXi resources, he gives an illustrated explanation of how the quaternion structure has to be projected onto the flat space-time representation. They are different representations of the same phenomenon. One of the representations seems more intuitively correct because it matches our own experience though it is not necessarily the most helpful way to think of what is occurring.

Hi James,

"My non-expert opinion about compactification of dimensions is another hiding place allowing the theorist to offer an answer that is a guess and proceed forward with theory."

Actually I agree with this, at least partly. In fact compactification is worse than a hiding place. One of the problems---a very serious and ancient one---with approaches like Kaluza-Klein theories (and their modern incarnations like sting theories) is that compactification---far from offering a physically meaningful solution to the unobservabilty of "extra" dimensions---opens up a Pandora's box of many other intractable problems.

On the other hand I do not have a problem with guessing in general and then proceeding to produce a theory. That is exactly what theoretical physics is all about, and it has produced some marvellous results for us---such as the general theory of relativity.

I am indeed contrasting "They are all there..." with "...not having been artificially compactified." I think compactification is an inherently artificial process. It arises from our attempt to fix the problem by hand rather than following the natural course of mathematics.

Joy

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  • Post #99

Tom,

I just found this message:

"I've got it now. I remember that I wanted to explain..."

I will let you know if it worked. :)

James

Joy is absolutely correct that compactification imposes an artificial structure on the space. He is also right that the idea is driven by the history of Kaluza-Klein theory -- there never was a *physical* reason to believe that the 5th dimension of K-K is smaller than a measurable quantity of spacetime; the explanation was given to reconcile the mathematics of Minkowski space in a metric tensor 4-dimension model with the fact that the 5th dimension added to the tensor accounts for the propagation of electromagnetic waves in the Minkowski spacetime field, even though we don't actually have the means to physically measure the relation between wave propagation and the propagation of the gravity field that Minkowski space and Riemannian geometry describe. Both of these wave functions are continuous to infinity.

One must never forget the rational premise that Einstein lived by: that a mathematically complete structure has to match every element of the mathematical theory to every element of the physical observation. That rock-solid premise allows us to distinguish what is a mathematical *artifact* of a physical model from what is physically real; an artifact describes the limit of a measure, as opposed to the measure of a limit. To give an example, I'm sure I am not the only one who is dismayed to see the number of arguments in this forum suggesting that the variation of the speed of light under special conditions falsifies special relativity -- that's nonsense. The constant term only describes how we *know* that rest energy of matter *can* be precisely measured in a relativistic framework. It has absolutely nothing to do with how we measure the speed of light.

By the same reasoning, Joy demonstrates how we *know* that measured quantum correlations *can* be explained in a fully relativistic, analytical model of octonionic dimensions. There is nothing mysterious or ad hoc (much less mathematically incorrect) in Joy's framework. It is no more difficult to explain the limit of 8 dimensions in our physics, than it is to explain 4.

Tom