Leaving Cantor’s Paradise through Paul Cohen’s Golden Door by John Benavides

Dear John,

I too am flattered by your comments. I was pondering for some time questions along the lines of your essay and your essay resonated strongly with what I am thinking. Hope you will win a prize.

Interesting essay John. I appreciate in particular the emphasis you give to emergence, and emergence in computation. Also, the closing quote by Deutsch shines. I did not know it, and I am very pleased by his mentioning, among the deepest explanatory theories, (i) quantum theory, (iii) evolution theory for living organisms, and (iii) theory of computation, side by side. Surprisingly, Relativity is not in the group of four. Why do you think he excluded it?

I am worried/confused by the fact that the set P at the right of your central Figure 1 can be equated to two very different things such as (i) 'a space of boolean algebras whcih represent history propositions...' , or (ii) just a causal set modeling spacetime. I agree that 'richness can become an enemy', as you write -- and I could add 'meta-theories eventually need instantiation', or 'the devil is in the details', etc...

Finally, you write that 'The duality between discrete and continuum is just one more of the misunderstandings caused by a classical logic reasoning'. I am not sure I can retain a clear and strong argument explaining why this would be the case, after reading your essay. Does it mean that I have missed your main point?

Tommaso

You have written an interesting essay. I give Topos theory some discussion in my essay

http://fqxi.org/community/forum/topic/810

However, that is more mentioned with respect to the Zariski topology.

I am not entirely sure what the continuum hypothesis אּ_1 = 2^{אּ_0} has to do with physics. The Cohen Bernay theorem indicates this is consistent with the ZF set theory by Godel's theorem. I am somewhat familiar with these developments, but they are not entirely my area of expertise. Your paper gets a good thumbs up from me.

Cheers LC

Hello John,

I make references to Topos theory, largely with connection to the algebraic or projective varieties with Zariski topology. This is the basis of what might be called "pre-topos" theory. It is a form of pre-sheaf construction, which can be used to build a sheaf theory. One main interest is in a twistor geometry with E_6 subroup (or E_6xE_8) with a sheaf or pre-sheaf construction for twistor geometry.

There is a relationship between lightcones and Heisenberg groups within this Kleinian quotient system. I tend to see this as a precursor for the far more generalized system you present in your paper. I am also interested in the prospect for monster-moonshine structure, which is are projective varieties in 26 dimensions (eg the bosonic string) with Lorentzian structure. These projective varieties form the pre-sheaf construction for topos or grothendieck-Etale structure.

The space of lightlike geodesics is a set of invariants and then due to a stabilizer on O(n,2), so the space of lightlike curves L_n is identified with the quotient O(n,2)/P, where P is a subgroup defined the quotient between a subgroup with a Zariski topology, or a Borel subgroup, and the main group G = O(n,2). This quotient G/P is a projective algebraic variety, or flag manifold and P is a parabolic subgroup. The natural embedding of a group H - -> G composed with the projective variety G - ->G/P is an isomorphism between the H and G/P. This is then a semi-direct product G = P x| H. For the G any GL(n) the parabolic group is a subgroup of upper triangular matrices. An example of such a matrix with real valued elements is the Heisenberg group of 3x3 matrices (sorry for the inconvenient representation, but I have bad luck with these html-TeX systems)

|1 & a & b|

|0 & 1 & c|

|0 & 0 & 1|

which may be extended to n-dimensional systems to form the 2n+1 dimensional Heisenberg group H_n of n + 2 entries

|1 & a & b|

|0 & I_n & c|

|0 & 0 & 1|

where for O(n,2) the Heisenberg group is H_{2n+3}. The elements a and c are then n+2 dimensional row and column vectors of O(n,2). These are Borel groups, which emerge from the quotient space AdS_n/Γ, where the discrete group Γ is a manifestation of the Calabi-Yau 3-cycle, and which as it turns out gives an integer partition for the set of quantum states in the AdS spacetime. So both spacetime and quantum structure as we know them are emergent.

This of course can exist in more general setting, which is the type of construction you are presenting with the continuum hypothesis. I will not write that with alephs, for the Unicode representation does not work well. This suggests that the universe (the system of multi-cosmologies or multiverse) has this underlying system of topos between different structures. The system above illustrates how lightcones and Heisenberg groups emerge from the same quotient structure, where the topology indicates this is a form of topoi.

Cheers LC

John,

I find your idea that theory limits what we 'see' of Nature very interesting and also very relevant to the raging debates going on. Let me add my voice to that debate and point you to a result along the same lines in my essay.

I show that Planck's constant is a necessary boundary to our ability to 'see' the Universe because h is the 'measurement standard' that defines Kelvin temperature. The entire theoretical regiment through which we measure, observe and understand the Universe breaths existence into h. Planck's constant is the 'focal point' to the physics we have created beyond which we cannot 'see' the world. It is NOT a necessary fact of Nature!

Just the other day I posted a very short paper, "If the speed of light is constant, then light is a wave", that mathematically proves that light must be a wave.

I look forward to your comments and your support for these significant results.

Best wishes,

Constantinos

Dear John,

My approach is a bit more "conservative" I suppose. I think that this sort of topos construction may pertain on a deeper layer than what I am working on. I happen to be more focused on hypothetical connections with physics we understand currently. I would say a relationship between what I am doing and the sort of topos you advocate is similar to the relationship between pre-sheaf and sheaf theory.

Cheers LC

I promised in my reply to your post that I will read your essay and let you know what I think. Sorry for taking so long. It is a very interesting paper. I like the way you relate mathematics progess to physics progress.

I do have a feedback about your feedback to me earlier. You mentioned in your post to my essay that "... what you call zero-distance connection in my context is the fact that the each point in the order determine a universe which structure depends ...". I think there is a missunderstanding. The connectiviy mentioned in my paper was between two space-time positions that may be planck-scale apart. The link in your Figure 1 is between individual universes. They are quite different both in scale and in concept. I hope I haven't missundertood what you presented in your paper.

Thanks!

Honda

Congratulations John! I am pleased that you made it to the final round!

Now that this frenzy is over, I like to pick up our conversation concerning Planck's constant and also the short post I linked you previously regarding light.

Good luck with the panel!

Constantinos

Dear John,

Congratulations on your dedication to the competition and your much deserved top 35 placing. I have a bugging question for you, which I've also posed to all the potential prize winners btw:

Q: Coulomb's Law of electrostatics was modelled by Maxwell by mechanical means after his mathematical deductions as an added verification (thanks for that bit of info Edwin), which I highly admire. To me, this gives his equation some substance. I have a problem with the laws of gravity though, especially the mathematical representation that "every object attracts every other object equally in all directions." The 'fabric' of spacetime model of gravity doesn't lend itself to explain the law of electrostatics. Coulomb's law denotes two types of matter, one 'charged' positive and the opposite type 'charged' negative. An Archimedes screw model for the graviton can explain -both- the gravity law and the electrostatic law, whilst the 'fabric' of spacetime can't. Doesn't this by definition make the helical screw model better than than anything else that has been suggested for the mechanism of the gravity force?? Otherwise the unification of all the forces is an impossiblity imo. Do you have an opinion on my analysis at all?

Best wishes,

Alan

John,

As promised, but with some delay ;-) , here is my (very brief) comment which you asked for.

As I mentioned in my reply to your message (on my essay's page), I believe in the priority of considerations related to (object or data) representation in science over logical considerations: the latter should follow the "logic" of the chosen representation, i.e. the logic should emerge during our analysis of "data "manipulation". Mathematics so far has been dealing basically with the numeric representations (for over four thousand years), and it is quite possible that the way out of our present difficulties in physics and other sciences require a radical revision of this basic, numeric, form of representation rather than any escapades into new forms of logic that are not motivated by any new forms of object/data representation. Again, to see the nature in a new light we need to change the form of data representation, which we have never done so far.

I'm sorry if I have not been as helpful as you expected.

My best wishes to you!

Dear John Benavides,

1JB: "I think, you are misunderstanding something. You cannot separate mathematics and logic. If you are saying you have found some problems with some basic notions in mathematics it is because the logic that you are using to think about these notions do not agree with the logic that define and govern them. "

1EB: My logic is the logic used by Euclid, Galileo, and many others. Where is the logic that defines and governs the basic notions in mathematics? Galileo compellingly concluded that the relations smaller, equal to, and larger do not apply for infinite quantities. Perhaps even Goedel was not able or not willing to accept that. He imagined the cardinality of continuum much larger than 2^aleph_0. Goedel proved CH does not contradict to ZF, and Cohen proved its opposite does also not contradict ZF. Hence CH is with Popper's words not falsifiable, in Pauli's language not even wrong. My logic tells me that the notion transfinite cardinality contradicts Galileo's conclusion. In other words, Georg Cantor misguided mathematics to leave the path of clean logic.

2JB:"I read your essay and what you see as problem is because you are unconsciously denying the excluded middle principle of classical logic that allows to abstract numbers with the limit point of the extension they represent."

2EB: I did definitely not unconsciously deny the TND. Read Fraenkel 1923. He admitted that there is what he called a fourth logical possibility besides the three above mentioned: not smaller, not equal to, not larger: incomparable. Brouwer clarified: The TND only applies within the realm of rational numbers. Unfortunately, Hilbert disserviced mathematics by ousting Brower. Hilbert was disappointed that his successor Weyl rejected much of set theory. Unfortunately Weyl emigrated soon. Mathematicians before Weierstrass and Cantor did not really bother treating the irrational numbers as if they were rational and accordingly subject to trichotomy.

Why do you mean classical logics dictates the TND, and why do mean every limit point belongs to the realm of countable numbers? The continuum as defined by Peirce cannot be split in single points.

3JB: " I agree with you extensions are more fundamental, but what you are ignoring is that if you say we should not identify extensions with its points, what are you saying is that we shouldn't use classical logic to describe the continuum, ...

3EB: My effort in this contest was not in vain if mathematicians like you agree on the reinstating of the Euclidean notion "number" as an extension. I suggested to identify the number with the extension between zero and an endpoint which is a limit from inside. I did not say we should not use classical logic to describe the continuum. On the contrary, I would like to make the mathematicians aware that Cantor cheated them.

4JB: ... that kind of misconceptions is what my essay is about. "

4EB: If only you did arrive at helpful conclusions.

5JB: "On the other hand you don't need a quantum computer to justify that quantum reality is ruled by a non classical logic, you just need a half-silvered-mirror and light to construct a machine that cannot be described using classical logic."

5EB: I am just an old engineer. Nonetheless I will try and hopefully find out on what possible mistakes the so far not functioning quantum computer is based. What experiment do you refer to? If something possibly very profitable does not work, this is a strong indication of something wrong. Isn't it?

Regards,

Eckard

Dear Eckard

"My logic is the logic used by Euclid.."

.

Are you sure your logic only rigt logic?

How about non-Archimedian logic noted by Winterberg?

"Archimedes believed he could determine the value of K through a limiting process, by drawing a sequence of polygons inside a circle with an ever increasing number of sides. This "exhaustion" method though must fail if there is a smallest length. It was Planck who in a 1899 paper had shown that the fundamental constants of physics, h, G and c, give us such a small length, the Planck length

These three quantities are sufficient for the architecture of a non-Archimedean geometry for a finitistic formulation of physics. The square root in the expression for m0 gives us only the freedom to have two possible signs for m0, but nothing more.

In such a finitistic formulation one can, in an arbitrary number of space dimensions, replace differentiation operators by finite difference operators"

Dear Yuri,

Even ZFC includes the Archimedian axiom of infinity. Do we really need a special logic as to set a more or less speculative finite limit to resolution? For practical use Planck length, time and energy are obviously irrelevant.

I feel not yet in position to comment on Winterberg's interpretation of mass as a small difference between positive and negative Planck masses.

Regards,

Eckard