Wednesday, July 24, 2013

About naturality, fine tuning, and the recent ego catastrophe in theoretical physics

There has been a considerable amount of discussion in blogs about the recent situation in fundamental physics. Do the results from LHC mean end of theoretical physics as a predictive discipline? Should we accept multiverse? Should we give up the notion of naturalness and accept fine tuning? These are the questions.

Phil Gibbs answered affirmatively to these questions in his blog post, Lubos wrote in more skeptic tone about these topics, and Peter Woit touched these questions in his comments about talks held in "Prospects in Theoretical Physics" - a program for graduate students and postdocs in Princeton. I wrote a couple of comments about the situation to the blog of Phil Gibbs and combine them below to a more polished commentary.

To my opinion the situation in fundamental physics should be looked from a wider perspective than that given by last forty - not very successful - years of theoretical high energy physics. Physics is definitely in crisis. Multiverse scenario and the view about necessity of fine tuning are conclusions from sticking to certain basic dogmas and refusal to admit that some of them might be badly wrong. I do not believe in all these dogmas and therefore do not share these pessimistic conclusions. I tend to see the recent situation as an ego catastrophe (we failed to find the theory so that there can be is no theory) which is outcome of accepting quite too many ad hoc assumptions as final truths. The situation can be also seen as the final collapse of reductionistic view about physics.

Alternative for multiverse

I believe that standard model symmetries have fundamental meaning being selected by their very special mathematical and physical character. GUT approach denied this possibility and led theoreticians on wrong track leading to standard SUSY and eventually to M-theory landscape. Also the fact that that the observed space-time is 4-dimensional very probably contains a very important message. But also the idea about 4-dimensional space-time became old-fashioned as super string revolutions revolutions followed each other. Sociological factors played a key role in the process. The attitude that thousands of brilliant theoreticians cannot be wrong allowed the situation to develop to a catastrophe made manifest by the findings at LHC. Even in this situation we are told that we should continue to follow the leaders and now give up even the belief that theoretical physics can explain and predict - the very motivation of super string theory originally. And this only because few generations of theoretical particle physicists became victims of mass psychosis. I will not eat this cake!

Standard model symmetries and also space-time dimension would be forced by the existence of geometry for infinite-dimensional space - "world of classical worlds" (WCW) consisting of 3-surfaces defining the analog of Wheeler's superspace. WCW geometry would realize a generalization of Einstein's geometrization program to a geometrization of the entire quantum physics rather than of only the classical physics. In the case of much simpler loop spaces the mere existence of this Käahler geometry fixes it uniquely for given group G defining the loop group (the existence of Riemann connection requires infinite-dimensional Kac-Moody group as isometries as shown by Freed). Standard model symmetries fix WCW (equivalently, the imbedding space H =M4×CP2 containing space-times as 4-surfaces) and the conjecture is that the mathematical existence of WCW Kahler geometry implies the same WCW. In accordance with the vision about physics as generalized number theory, standard model symmetries would have also number theoretical interpretation in terms of classical number fields. For instance, color group would correspond to isometries of CP2 and subgroup of automorphisms of octonions.

Concerning the fine tuning of coupling parameters: I believe that fine tuning of dynamical parameters is a basic aspect of quantum evolution leading to life as we identify it, but that standard model symmetries and space-time dimension are mathematical necessities rather than outcomes of evolution in some region of multiverse. p-Adic length scale hypothesis and selection of preferred p-adic primes provides a realization of the evolution selecting preferred mass scales for elementary particles. One should therefore accept the obvious: superstring model describes physics of 2-D space-time but - as has become clear - the attempts to deduce real physics from it are doomed to fail. Nature does not love tricks. Super-conformal symmetry remains the genuine contribution of string models to physics and the natural next step is to finally generalize this symmetry to four dimensions.

Reductionism as the basic cause of the catastrophe

I have spent much time during last decades in trying to understand why we have gradually ended up with this dead end and why the professionals are not able to see that there is no way out except radical rethinking of fundamentals. The history of physics is history of bold and often wrong generalizations. The naive length scale reductionism is one of the most influental of these wrong assumptions. It has been raised to a level of dogma and together with materialistic world view more or less defines nowadays what it is to be scientific. Fractality is very natural candidate for replacing the reductionism and quantum theory strongly encourages to give up materialism but still taken as givens by particle physicists. Reductionism is indeed responsible for many far reaching and probably wrong dogmas in recent day physics.

Reductionism forces us to believe that the strange findings at RHIC and LHC about heavy ion collisions and proton heavy ion collisions are consistent with QCD although here we would have the new physics that we are so desperately searching for. This relates also to naturalness. To my opinion, the attempt to understand mass ratios of various fermion generations group theoretically is doomed to fail. If one accepts the notion of length scale hierarchy implied by fractality there is no need to extend standard model symmetries. The fact that separate B and L conservation is consistent with experimental facts provides an additional strong constraint.

Length scale reductionism also forces us to believe that biology and brain science are just complexity, consciousness is just an epiphenomenon, and free will is an illusion. Theoretical physicists lose a huge treasure trove of anomalies which could help to achieve the sought for unification. As a consequence of this isolation from experiental reality, theoretical physicists have divided into half-religious sects such as super-stringers and loop gravitists. Feynman has talked about general relativists gathering to their yearly meetings and discussing again and again the same old dead ideas. Sadly, Feynman's characterization seems to apply quite well also to Strings 2013 and Loops 2013.

Length scale reductionism guides us to search dark matter from elementary particle length scales. This direction might be completely wrong: TGD suggests generalization of quantum theory by introducing the hierarchy of effective Planck constants and in this framework dark matter as quantum coherent phases would emerge in long length scales. Ironically, already Tesla made observations, which one might be interpret as indications for the existence of something behaving much like dark matter in TGD sense. Tesla spoke of "cold electricity" not seen in ampere-meter but as a child of his time assigned with it what he called aether particles. Did Tesla discover the dark matter for more than century ago? One cannot exclude this possibility since his experiments typically used high voltages, low frequencies, and using sudden pulses resulting in switching on of electrical circuits and in this manner testing the boundaries of Maxwell's theory in long rather than short scales (as particle physics does). In this context one must mention also the strange quantum like effects of ELF radiation on vertebrate brain and the fact that cell membrane resting potential corresponds to an electric field above the dielectric breakdown in air. Tesla's vision about future technology was also surprisingly far reaching and he also saw a possible connection with the energy technology and biology: his ideas are still revolutionary. To me the example of Tesla demonstrates that the history of science is not steady linear evolution but a continual fight between mediocrits and visionaries and mediocrits quite too often win in the short run.


At 3:57 AM, Blogger Ulla said...

At 5:55 AM, Blogger Ulla said...

On Julian Barbours wikipedia page are three links.

Anderson, Edward (2004) "Geometrodynamics: Spacetime or space?" Ph.D. thesis, University of London.
-------- (2007) "On the recovery of Geometrodynamics from two different sets of first principles," Stud. Hist. Philos. Mod. Phys. 38: 15.

Baierlein, R. F., D. H. Sharp, and John A. Wheeler (1962) "Three dimensional geometry as the carrier of information about time," Phys. Rev. 126: 1864-1865. this I cannot find free on net.

At 1:19 PM, Blogger Ulla said...

“Why do some people get caught by an idea that takes over your life? I don’t know, but I do know that as long as it doesn’t drive you crazy, it is a blessing,” Barbour says gently. “When I started out on this 40 years ago, I said to my family that I know what I want to do and it will take me the rest of my life to do it—and that is the way it has worked out.” from 1982

At 5:53 AM, Blogger Ulla said...

Maximum negentropy principle

At 6:32 AM, Blogger Ulla said... here is the text

At 9:24 PM, Anonymous Matti Pitkanen said...

The basic problem of Wheeler's geometrodynamics is that basic objects are 3-dimensional. One should be able to get out 4-D space-time from this approach. Barboux is quite right in saying that time is lost in general relativity based on Wheeler's geometrodynamics. Wheeler's geometrodynamics has also profound mathematical difficulties: for instance, it is difficult to get fermions out of it and the geometry of super-space is poorly defined.

But why Nature should obey Wheeler's geometro-dynamics? Wheeler's notion of super space must be modified so that it exists mathematically and is consistent with the basic space-time symmetries (Poincare group) and geometric description of gravitation.

"World of Classical Worlds" allows to achieve this elegantly. The news is that after more than two decades this is still news!

At 9:47 PM, Anonymous Matti Pitkanen said...

Still to Ulla:

Thank you for links. They allow to clarify the relationship of TGD to other theories.

The Negentropy Principle introduced by Mahulikar brings in mind Negentropy Maximization Principle.

The abstract does not give much idea about what is involved but thermodynamics is claimed to give NP. I find it difficult to take this seriously, at least if one talks about conscious information.

Thermodynamical definition of negentropy allows to consider only entropy gradients and claim that the flow of entropy out of space-time volume corresponds to flow of information into it. But I cannot take this seriously. One should speak about absolute information rather than changes of information. With this criterion the best thermodynamics can give is that there is zero entropy (and also zero negentropy).

Negentropy maximization and entropy maximization (second law) are apparently contradictory. Conflict disappears when one realizes that number theoretic entanglement negentropy characterizes a decomposition of system to two parts: hence *two-particle* property is in question. This notion is not thermodynamical but purely quantal. Thermal entropy in turn characterizes average member of ensemble of identical systems and is *single-particle* property.

As a matter fact NMP implies second law and entropy maximisation for ensemble for ordinary matter. In dark matter sector where negentropic entanglement is possible, the situation remains unclear.

There is however very close correspondence between entanglement negentropy and thermodynamical entropy. A system having n degenerate states (corresponding to sheets of n-multifurcation of space-time sheets and h_eff=n*h) is highly entropic as single particle system: one does not know at what sheet the particle is.

By entangling two systems of this kind by nxn unit matrix one obtains negentropically entangled system stable under NMP. Living systems are this kind of systems and the multi-sheetedness provides them with high representative powers. Highly entropic systems (in sense of having large h_eff) are buildings blocks of highly negentropic systems.

One can argue that this kind entangled systems are highly critical ("in the intersection of reality and p-adicity") and unstable. Here standard measurement theory - that is basic law of nature - comes in rescue. It predicts that state function reduction leads to states which are characterized by nxn unit matrices for them so that number theoretic negentropy is positive and NMP does not force further state function reduction.

At 11:14 AM, Blogger Stephen said...

Ulla, I busted the knowledge outta jail, you can find it @

At 12:31 PM, Blogger Ulla said...

And I do think it is important to link TGD to other theories, sadly I know so little though.

About origin of life.

At 8:53 PM, Anonymous Matti Pitkanen said...

To Stephen:

Imbedding is also now needed to obtain time: to 4-manifold which solves Einstein's equations rather than fixed 8-D imbedding space M^4xCP_2.

They imbed 3-geometries as sub-manifolds of 4-manifolds which are solutions of Einstein's equations. In this manner they can imbed two 3-manifolds as time=constant cross-sections of 4-manifold and say which was earlier and also calculate temporal distance between them.

This works of 3-geometries that happen to be space-like intersections of the *same* solution of Einstein's equations but not otherwise: in TGD this works for all 3-surfaces.

Another problem is that there is infinity of space-times containing a given 3-geometry as cross section so that in practice one cannot apply the procedure. One would also like holography: more or less unique space-time as analog of Bohr orbit but this one does not obtain.

The further problem is related to the classical conservation laws since Noether's theorem based conservation laws are lost since symmetries are lost.

At 11:27 AM, Anonymous Anonymous said...

On the sidenote, here's so far best introduction to fractional 10-adic or "reversimal" arithmetics I've seen, touching also very deep issues like Euler's doubly infinite identity, and that the computation on the "reversimal" side is actually much simpler than on the decimal side:

PS: googeling "doubly infinite identity" gives quite a lot of results, which may be of interest for those developing arithmetics and algebra between p-adics and reals.

At 2:12 PM, Anonymous Anonymous said...

More good stuff, problems of theory of real numbers:

So it seems that the Original Sin of Metaphysics aka Fall of Physics creeped into natural philosophy in the theory of real numbers based on axiom of choice - which is the current standard belief, according to which most do also physics.

On pi as "metanumber".
(Finnish difference between 'numero' and 'luku' sound very important, but difficult to translate into English.)

At 6:48 PM, Anonymous Matti Pitkanen said...

Thanks to Anonymous:

The link gives excellent intro to p-adic numbers. p-Adic numbers are indeed in many respects much simpler than reals. Basic challenge is to understand the correspondence with reality! How reals and p-adics correspond to each other. Here the notion of p-adic manifold that I have discussed here comes in help.

In p-adic context pi is indeed metanumber in the sense that for finite-D extensions (and algebraic extensions in general) one can have only roots of unity: only cosines and sines of 2*pi/n but not 2*pi/n itself. One can allow infinite-D extension (not algebraic anymore) bringing in all powers of pi but somehow I do not like it all.

At 3:42 AM, Anonymous Anonymous said...

Sines and cosines etc. of current standard trigonometry based on radius and length have never made sense to me, so I very much like good professors approach to trigonometry (and its implications), which he calls 'universal hyperbolic geometry. Here's the introduction to subsequent lectures:

This approach has much to contribute to learning TGD, and perhaps TGD can also benefit from universal hyperbolic geometry and other ideas and views of this good professors, such as centrality of rationals in all number theory, etc.

One question and idea that first came to mind was proof of "metanumber" or "metarational" Euler's Identity of relation of pi, e and i in universal hyperbolic geometry instead of standard trigonometry.

At 5:07 AM, Anonymous Anonymous said...

Some discussion on 'Is there p-adic 2pi i, and where to look':

At 5:23 AM, Anonymous Anonymous said...

The deep structure of rationals:

This touches also the questions in the OP, as this lecture presents simply and beautifully the natural fractality and holonomy of the deep structure of rational numbers.

And as hinted before, the Fall of Physics can be perhaps most informatively originated to sloppy metaphysical approach to reals with the purely metaphysical axiom of choice. Trying to run before learning to walk. The intuition that "reals" can make sense only with "doubly infinite identity" with p-adics rises naturally.

At 5:32 AM, Anonymous Matti Pitkanen said...

In the link it is said that there might be a manner to define p-adic 2pi i but certainly the usual definition fails. It this definition exists, it should be also physically meaningful. Cosines and sines however exists for roots of unity and this leads automatically to finite measurement resolution for phases giving a hierarchy of phase/angle resolutions.

pi and also other transcendentals appear in the scattering amplitudes of QFTs . The assumption that they are absent from amplitudes poses very strong conditions on the p-adic counterparts of scattering amplitudes and also real scattering amplitudes.

Bringing 2pi i and its powers via infinite-dimensional non-algebraic extension of p-adics might lead to inconsistencies. exp(i2pi/n) defined as power series should be equivalent with its counterpart introduced as root of unit via algebraic extensions. All roots of unity would actually follow as a consequence of introducing 2pi i via non-algebraic extension. Is the introduction of 2pi i equivalent with the infinite-dimensional abelian extension containing all roots of unity?

And what about the numbers exp.(i 2pi *x/n) where x is p-adic integer containing infinite pinary digits and therefore infinite as real integer. Can one define x modulo n for any integer to make the formula sensible. Or could one *define* x by giving x mod n for all n?

At 9:55 AM, Anonymous Anonymous said...

Given the problems of smuggling unnecessary metaphysics into real numbers, can we be sure that those problems don't continue into complex roots of unity?

Reciprocal mapping of natural philosophy and (dynamic) number theory is very deep idea, and as such requires critical and careful cleaning of number theory from unnecessary metaphysical assumptions. One such assumption, as you seem to be saying, is the idea that either extension of rationals - real or p-adic - could be meaningfully and consistently approached and studied without the other. Our collective thinking is still very much contaminated by metaphysical reals as they are taught in standard way, so the task is not easy, as we see from the open and difficult questions challenging our best mathematical minds.

Here's one clear and well defined suggestion for working on the problem of p-adicization of pi and other transcendental metanumbers and metarationals: if Wilderberg's trigonometry is indeed more consistent and potent and beautiful than standard trigonometry, use it first to prove Euler's identity of metarational of pi, e and i, and then see what, if any, implications that proof has for p-adic and real extensions of rational numbers. Hope this sounds fun to someone with the required skills. I have strong intuition that such approach would offer much firmer ground to approach the problems of p-adicization. :)

At 11:47 AM, Anonymous Anonymous said...

To continue and clarify the basic problem of reductionism, as done in practice and hoped for it simply put: The belief that natural philosophy can be reduced to a finite set of equations, not unlike Einstein's theory reduces to finite set of Einsteins equations.

Needless to say, such wishful thinking does not pass the most basic Gödel test, but stubbornly insists on forgetting Gödel's proof about consistent mathematical structures and finite sets of axioms or other generative principles - in other words, there is no finite algorithm to generate principia mathematica or any other number theory that is also physically meaningfull.

Number theory is the language that equations and other algorithms use, not vice versa, and just like forest and language of forest is not dependent from a house in forest and language of house, but a house is dependent from the larger inclusive forest, any set of of equations or more generally an algorithm is sentence in the language of number theory, which it implicates as larger inclusive whole and from which it depends from. And thus reductionism looks only at single trees and does not see the forest, the purest math of number theory.

So, hopefully these metaphors make it even more clear than before why reductionism does not and cannot work, and why at the level of a respectful TOE of mathematical physics the only meaningful description can be the whole of number theory. And that natural philosophy aka mathematical physics requires *only* naturalization of number theory, cleaning out unnatural metaphysical assumptions from number theory.

At 2:22 AM, Anonymous Anonymous said...

Does the theory of complex reals do anything worth doing that complex rationals and rational transcendentals can't do?

If I'm not mistaken, with e^ipi you can plot each point of circle with precise number theoretical meaning just as well and even better on complex rational plane, than on complex real plane, when you treat also e and pi (etc.) as "whole (meta)numbers" also giving rational relations or rational numbers.

Here's the Euler's doubly infinite identity, from the introduction to p-adics linked above: ... + x^3 + x^2 + x + 1 + 1/x + 1/x^2 + 1/x^3 + ... = 0
Again, if I'm not mistaken, it states that e.g. p-adic pi is simply -pi. The sum of real pi and p-adic pi is zero.

At 3:07 AM, Anonymous Matti Pitkanen said...

The problem with Euler identity deriving from 1/(1+x) +1/(1-1/x) -1=0 by using formal geometric series expansions is that 1/(1-x) converges for [x|<1 only and 1//1-1/x) for |x|>1 only. For x=1 both give infinity.

In p-adic case N_p(x)<1 is necessary for converge for 1/(1-x) and N_p(x)>1 for the convergences of 1/(1-1/x).

Therefore it is difficult to make any conclusions from the identity.

At 3:08 AM, Anonymous Matti Pitkanen said...

To Anonymous:

I believe that number theory has a lot to give for physics. Number theoretical universality for scattering amplitudes is very powerful constraint. Classical number fields appear in TGD framework and allow to understand standard model symmetries. The notion of infinite prime has direct physical interpretation too.

At 3:14 AM, Anonymous Anonymous said...

Hm. Perhaps it would have been more correct to say that pi-adic pi is -pi, but not a biggie, as we know that there is deep connection between pi and primes, e.g. as stated by this formula (again, by Euler):

π/4=3/4⋅5/4⋅7/8⋅11/12⋅13/12⋯ etc

where the numerators are every prime number and the denominators are the multiple of 4 nearest that prime number.

At 4:03 AM, Anonymous Anonymous said...

Sorry for multiple comments, but there is an underlying story here that concerns the OP and future development of TDG. As Bishop Berkeley and many others, as well as basic experiments of Quantum theory, such as Wheeler's delayed choice experiment have shown, Realism as philosophical position is not rational naturalism, but a metaphysical leap of faith. And at least since Descartes, Western natural philosophy has been plagued by leap of faith into Realist metaphysics.

So, what is the implication of this problem for TGD, which does not share the realist metaphysics but strives to renaturalize physics? As Matti says, physics is a discipline, and mathematical physics is about finding natural number theory. And this is two-way process, there is constructive side in the discipline, leading all the time to more and more complex mathematical structures that becomes occult language of those who spent years and years getting indoctrinated into the occult language of the Discipline. But by consciously or subconsciously smuggling unnatural metaphysical assumptions - such as realism - into the Discipline, it can become a mere intellectual house of cards, not grounded in nature. And when it becames clear that this is the situation, the task of the Discipline is to trail back to where the unnatural assumption or assumptions originate from. To deconstruct the house of cards and to simplify, simplify, simplify.

As a child of it's era, also TGD as work in process cannot be totally free of all the unnecessary and unnatural metaphysical assumptions that have plagued the constructive side of the Discipline, which can be found also at the basic level of number theory that students are taught. Is all the jargon and the structures they refer to really - or better said naturally - required, or can there be found more simple and more beautiful and more expressible math to show and share these ideas and their naturality in the circle of dialogue. Which is the true origin of all academic disciplines originating to circle of dialogue founded by Plato in the grove of Akademeia. And Platonic dialogue is if not proof, at least working hypothesis of "anamnesis" stating that mathematical ideas are inherent in all of us and comprehensible to all of us, through skillful participation in the open dialogue circle. As long as the mathematical ideas remain natural, and not just metaphysical occult constructions.

So in the context of TGD, much has been done and much remains to be done, and the best platonic guideline for keeping it natural, keeping it simple, is to carefully look at the language and constructions, where they are too occult and cannot be shared by means of dialogue and platonic anamnesis, and to constantly to critically search for more simple and more natural expressions, going back to roots and removing unnecessary metaphysical assumptions that only add to confusion, not to clarity of natural philosophy, test of which is that it is also open to dialogue and can be communicated in dialogue not limited to any occult circle of authoritative make-believe.

At 5:47 AM, Anonymous Anonymous said...

Dear Matti, I don't see how your objection, if that is what it is, connects to what I'm trying to think and share. What conclusions do you want to derive, in the first place, and exactly from where?

The way I see now, e^ipi is the _number theoretical_ definition of a circle (or more generally can give any curvature of a cone), which emerges as given from the dawn on Greek geometry and is implied at every level of geometry since the origin. If I'm wrong, I'll be happy to be corrected. But if we see e^ipi as number theoretical definition of circle, all geometry is in a sense conclusions from that number theoretical equation.

You seem to be saying that there is some problem of dead ends of no-conclusions, which again, _may_ rise from standard trigonometry and the proof by Taylor series giving meaning to the sine, cosine etc. functions of standard trigonometry, based on notions of "length" and "angle". But we know also from experience of educators and students that standard trigonometry is far from obvious and intuitive, and starts making (some) sense only after learning to do analytic functions.

So, p-adically, if that is the realm of mathematical (etc.) cognition, as you claim, standard trignometry is not easy and may not be the "most p-adic" - most obvious and intuitive - way of doing geometry. But we can't remember and apply a more obvious and intuitive way of doing geometry by staying inside the box and thinking about just the dead ends that the standard approach leads to.

And if there is a better way to do geometry, we cannot know unless we try. As said above, now we have a good candidate for a better way of doing geometry which is not based on notions of length and angle, but perhaps more simple and general notions, and a good challenge to test this candidate is to see can we prove Euler's identity with it, and does such proof reveal something new. Could it, for example, lead to new insights about p-adics, which would be the TGD version of enlightenment, becoming more and more fully aware of the deep number theoretical structure of cognition, as part of the Socratic program of gnothi seauton?

But the main point is, we don't know if we don't try and just stay thinking inside the box (of standard trigonometry, reals, etc). There are many toys to play with, outside the box, reminding of the maxim that if revolution is not fun, don't do it. And of course a failure would also be a result.

At 6:14 AM, Anonymous Anonymous said...

For example, Wildberger uses "quadrants" and "separations" instead of length and angle, for exact definitions see the youtube lectures or the book. So, what happens if p-adic analysis and meanings are applied already at the level of basic ratios of quadrants and separations, cutting through lot of constructionism based on reals etc. and p-adicing geometry already at the most basic level of this new (old) approach?

At 9:48 AM, Blogger Ulla said... Dark mass problem (solved?). Using Newton?

At 10:07 AM, Anonymous Anonymous said...

This IMHO is a must, Logical weaknesses in modern pure math.

And as for the failure of physics, partially problems of mathematical physics originate from problems of pure math.

At 7:16 PM, Anonymous Matti Pitkanen said...

To Ulla,

the "solution" just adds additional ad hoc term to Newton's equation. Nothing is said about what happens relativity.

MOND is similar ad hoc approach to dark matter.

At 7:55 PM, Anonymous Matti Pitkanen said...

To Anonymous:

Your arguments have been very interesting. I am sorry that I have not had time to enough to participate in full.

I am not a mathematician in the technical sense and cannot take any strong views about fundamentals of mathematics: I am mathematically just an infant becoming conscious about ideas in my own primitives manner, I am not a builder of axiomatic cathedrals. I would like to call myself quantum Platonist who tries to include also mathematician into his world view by postulating that quantum states as purely mathematical objects representing objective existences and quantum jumps between them define conscious information, which is basically about these mathematical ideas but represented also as our sensory experiences and cognition and to which we usually do not assign anything mathematical. My views differ from those of Tegmark in that Tegmark does not include consciousness in this Platonia explicitly.

TGD Platonia is also subject to evolution: every quantum jump recreates the quantum state describing existing potentially conscious information about mathematics (quantum Akashic records that I have discussed in some postings). In mathematics this means endless updating of axioms: new axioms emerge by discoveries taking place in quantum jumps: these discoveries are not deductions from axioms but new axioms.

In this framework the basic goal is to understand the geography if Platonia - that is physics. The two visions behind TGD are physics as infinite-D geometry unique from its mere existence and physics as generalized number theory: generalisation means many things.

*For instance, the introduction of the notion of infinite prime as a process analogous to second quantisation applied repeatedly. Physics as a hierarchy second, third,... quantizer ordinary number theories would sound rather sexy;-)! I am not sure whether these two approaches are equivalent or whether both are needed.

*For physicists real continuum - and for me also p-adic continua - are fundamental since continuum is needed for calculus and one cannot do much physics without calculus. This leads to the vision of generalization of the notion of number by gluing real and p-adic number fields and their algebraic extensions together along rationals (and possibly also common algebraics). One would obtain a book like structure and this structure would be correlate for the sensory world and cognition. To formulate this precisely is a huge challenge: in particular, how the quantum physics in different number fields is related is a fascinating challenge and number theoretical universality meaning kind of algebraic continuation between number fields, would pose very strong constraints.

To be continued...

At 7:56 PM, Anonymous Matti Pitkanen said...

Continuation of previous comment...

For instance, the notion of plane wave fundamental for physicist as something periodic does not have p-adic counterpart: the analogs of trigonometric functions exist but are not periodic. This forces the introduction of algebraic extensions and roots of unity defining them. In this manner the notion of finite angular resolution emerges. Only the trigonometric functions of angles would exist mathematically in the extension, not angles themselves except by introducing infinite-D non-algebraic extension generated by powers of pi. Evolution would be emergence of more and more complex algebraic extensions in quantum jumps. Mathematics would evolve like living organisms.

Going from reals to p-adic means algebraization: for instance, circle is defined by its algebraic representation x^2+y^2=R^2 as set of discrete p-adic points satisfying this condition. The definition of the length of circumference of circle is very difficult since the notion of definite integral does not exist in any obvious sense in p-adic context: this reflects the fact that angle as a notion does not exist geometrically in p-adic context. Also the transfer of notions of real topology central for TGD based notion of particle to p-adic context is fundamental challenge: the notion of p-adic manifold in which p-adic manifolds has real chart maps and vice versa gives excellent hopes of solving these problems and makes finite measurement resolution - central notion in physics - a fundamental notion. For old-fashioned Platonist accepting that our knowledge is always only partial and can be also erratic is certainly something new.

I am not only a child of my time- I am a mathematical infant so that TGD certainly involves unnecessary and unnatural metaphysical assumptions about mathematics. I I have done met best to eliminate such assumptions about physics: TGD is basically a story of getting rid of un-necessary and very probably wrong assumptions which have plagued the approach to fundamental physics for last four decade: reductionistic dogma, GUT approach to particle spectrum, SUSY in standard form, belief that only 2-D string world sheets allow conformal invariance, neglect of the fact that GRT does not allow definition of standard conservation laws in Noetherian sense, etc.. I really see myself as innocent child: real mathematicians are needed to find the minimal language formulating these ideas precisely (assuming such language exists at all;-)).

At 8:16 PM, Anonymous Matti Pitkanen said...

Dear Anonymous,

I am not quite sure what you mean when you say that e^ipi is the number theoretical definition of circle?

*Angle measurement is one aspect of Riemannian geometry and in p-adic context angles are replaced with phases exp(i2pi/n) appearing in algebraic extensions of p-adics. There exist always only finite number of representable angles depending on extension. This is quantum physically natural and corresponds to quantisation of angular momentum in terms of waves exp(i2pi*phi/m).

It also conforms with the notion of finite measurement resolution. For pure p-adics with odd p would have only angles pi and 0 (p mod 4=1) and 0,pi/2, 3pi/2,2pi) (p mod 4=3). This framework also gives good hopes about the continuation of real harmonic analysis to p-adic harmonic analysis and the notion of p-adic manifold as I have proposed it also relies on it.

*Length measurement is second aspect of Riemannian geometry. 2*pi as length of unit circle would be defined by integral in real calculus and here p-adics lead to problems. One can define algebraic extensions and in finite measurement resolution as a p-adic number by replacing circle with n-gob one can define the arc length as n times the length of single segment of n-gon defined by angle 2pi/n. This expression involves square root but a finite-D extension allowing square roots takes care of this.

The basic principle in real-to-padic transition would be that p-adic structures are cognitive representations for real structure but in finite cognitive/measurement resolution. For instance, the chart map of real manifold theory becomes cognitive map representing thought about reality or its inverse representing intention transformed to real action.

This means that all real notions - notions of length and trigonometric functions and even topological notions like genus- have p-adic counterparts as discretised versions. Finite measurement resolution as basic principle is central and this is consistent with the fact that we can only perform mechanical computations using rationals and do so in finite resolution.

At 8:27 PM, Anonymous Matti Pitkanen said...

To Anonymous:

Thank you for very interesting and inspiring comments. p-Adicizaton of geometry at basic level is a very nice idea. My own view is that real and p-adic geometries appear naturally as pairs since one does not have only the real geometry any but also representation of geometers thoughts in terms of p-adic geometry! In TGD inspired theory of consciousness observer indeed becomes part of the quantum Universe.

p-Adic geometry is cognitive representation of real geometry and defined by the analog of chart map taking discretized point set of real manifold to a discreted set of p-adic counterpart of manifold (finite measurement resolution realized as number theoretical existence constraints forces this). The inverse of this chart map would represent transformation of p-adic intention to real action.

If someone would ask what we should do to help the poor pure math;-), my proposal would be this: make mathematician part of mathematics. In this framework mathematics would become an evolving organic structure developing continually new axiomatic branches inspired by irreducible moments of Eureka.

At 10:14 PM, Anonymous Matti Pitkanen said...

To Anonymous:

A little "technical" appendix. The length of circle in N-gon approximation is simply 2N*sin(pi/N) giving 2pi at the limit N--> infty. Therefore simple lengths are expressible in terms of trigonometric functions. Also in discretised versions of Schrodinger equation pi is effectively replaced with expression based on trigonometric function.

This suggests that in QFT scattering amplitudes the problematic 2*pi is replaced with the length of maximal polygon approximating unit circle. This would provide additional ingredient to the notion of algebraic universality.

At 3:04 AM, Anonymous Anonymous said...

"For physicists real continuum - and for me also p-adic continua - are fundamental since continuum is needed for calculus and one cannot do much physics without calculus."

No doubt some number theoretical structure for continuum is necessary, but can you show and prove that rational continuum is not enough and more deep, being naturally fractal etc? Wilderberg makes good case against poorly defined and messy notion of irrationals that we need to take seriously.

As for p-adic + "real" extension of rationals, logically sound and well defined structure could be perhaps most consistently and beautifully build upon what has been now called "Euler's doubly infinite identity", which rhymes at very deep level with what you are doing not only with p-adics and reals but also with ZEO. Unity of thought-like and stuff-like aspects of being and experiencing, as you say.

"I am not quite sure what you mean when you say that e^ipi is the number theoretical definition of circle?"

I'm following this explanation of Euler's identity, which I understand as circle being the metarational of well ordered e, pi and i:

Wilderberg gives this classical definition of points of circle, claiming that it's much more accurate that using sine and cosine, but the problem is self-referentiality, that the starting point can be defined only as infinity:

Number theoretical anatomy of circle seems to remain genuine problem. So, as said, the challenge of Wilderberg's approach is to prove Euler's identity with his
Rational trigonometry instead of standard trigonometry, which he considers problematic for good reasons, as far as I understand.

"The basic principle in real-to-padic transition would be that p-adic structures are cognitive representations for real structure but in finite cognitive/measurement resolution."

If we stay true to ZEO, as I believe we should, your language here is misleading - originating from blatantly wrong realist metaphysics that we have be conditioned to believe in and is difficult to get rid of. The point of Plato's cave analogy is not to stay in the cave imagining that shadows on the cave wall are poor representations of "reality out there", but to step out the cave of realist metaphysics. ZEO and Euler's DII are infinitely more deep platonic truths than any representation theory presupposing realist metaphysics. If this arises, that arises; if this ceases, that ceases.

At 3:32 AM, Anonymous Matti Pitkanen said...

To Anonymous:

Thank you for good questions.

I am not quite sure what you mean with rational continuum. What I want is differentiation and also integration. Real and p-adic continua give differentiation. If I have understood correctly, p-adic continuum is obtained by adding to rationals having periodic pinary expansion aftersome pinary digit also those which do not have periodic pinary expansion.

In the case of reals same applies since rationals allow periodic expansion. The analogy with chaos theory is obvious: pinary sequence corresponds to orbit in discrete dynamics and transcendentals correspond to non-periodic chaotic orbit.

Real (and p-adic) worlds are continua and cognitive representations of these are discrete. This would be the philosophy.

Seeing p-adics and reals as parts of bigger structure suggests that irrationals such as exp(ipi/N) are necessary if we want to speak about trigonometric functions and approximation of pi as N*sin(pi/N).

I must listen Wilderberg's talk what the idea behind Euler's identity is. Rational trigonometry brings in my mind Pythagorean triangles (I hope I am not wrong!), which are also physically very interesting.

A comment to your last point:

p-adics and reals are in rather democratic positions. p-adic-to-real transitions are also there and correspond to transformation of intentions to actions. Neuroscientist would speak about sensory representations and motor actions. They are dual and correspond to the state function reductions to the opposite boundaries of causal diamond in ZEO.
Of course, these representations are only space-time correlates, not cognition or intention.

Everything represents something else: p-adic space-time sheets to real ones and vice versa, also p-adic space-time sheets can represent other p-adic space-time sheets with different p. This view is very category theoretic: objects and morphisms.

At 2:03 AM, Anonymous Anonymous said...

Thank you for your clarification on the last point, I should have read further before commenting, but it's nice to see that these clarifications only bring out our basic agreement more clearly. :)

Wilderberg has not discussed Euler's identity of trancendentals and imaginary numbers yet, but hopefully we see and hear that in future. But as for generalisations of pythagorean etc. triangles in various basic geometries, I hope you watch and enjoy this talk, called 'Rational trigonometry, generalized triangle geometry and four-fold incenter symmetry':

Here is direct link to the youtube channel, with lot's of interesting and entertaining topics:

PS: As for most basic idea of continuum or line, how about the wild idea that we start defining it by Euler's DII, ... + x^3 + x^2 + x + 1 + 1/x + 1/x^2 + 1/x^3 + ... = 0???

At 5:21 PM, Blogger Stephen said...

My friend says approximations are mathematical masturbation that hide the most interesting bits. is this fair? what kind of argument would refute this assertion?

At 12:02 AM, Anonymous Anonymous said...

In Wilderberger's criticism of modern pure math many concepts stand as accused of being ill-defined and less than rigorously derived and in need of restructuring, starting from first principles without fuzzy areas in the chain of deduction of theorems and proofs. Among them is also the concept of 'manifold'. Could it be possible to derive and define e.g. Kähler manifolds and their central meaning in TGD directly from definitions of quadrant and spread of rational geometry and universal hyperbolic geometry?

At 1:16 AM, Anonymous Matti Pitkanen said...

To anonymous:

Generalization of riemannian and thus also of kahler geometry would require integral of square root line element. It is difficult to see how spread could generalize to an integral along curve.

In purely p-adic context line element that metric tesor defining length squared and cosine squared (1-spread) as rational for rational vectors makes sense. Also complexification and kahler form. One cannot however define kahler fluxes since the notion of form is defined as purely local notion. Local geometric notions can be defined also p-adically: in particular field equations for kahler action involving p-adic counterpart of induced matric and kaler form. Kahler action itself cannot be defined in purely p-adic context.

If one considers as basic objects pairs of real and p-adic manifolds with charts between different number fields and restricted to dicrete subsets (finite measurement resolution), situation changes. One can have global aspects of the geometry also in p-adic context as "cognitive representations". Integrals in p-adic realm would be algebraic continuations from real realm using dicretized piecewise linear path existing in rational realm. 2pi=2nsin(pi/n) in finite angle resolution is a good example.

At 10:26 AM, Anonymous Anonymous said...

"Generalization of riemannian and thus also of kahler geometry would require integral of square root line element. It is difficult to see how spread could generalize to an integral along curve."

Integration, alternative way of thinking:

Did this help you to see?

More on algebraic caluclus:

"One cannot however define kahler fluxes since the notion of form is defined as purely local notion."

Sorry, but "purely local" does not make any sense to me and I seriously doubt the validity of any such notion. Wilderberg's approach has been very refreshing so far and strengthened my conviction that there is very little in modern mathematical physics that should be taken seriously by a philosopher of nature. TGD has many genuinely good ideas and it's certainly a step in the right direction, but I remain also convinced that the problems of communicating TGD originate to the "discipline" that in many critical areas and nodes in deductive chains have been less than rigorous; both in math and physics aspects of mathematical physics.

At 11:26 AM, Anonymous Orwin O'Dowd said...

The harmonic Pythagorean is sin^2x + cos^2x = 1. Geometry works better in the symmetrical case, by logic of distinctions: distinction of isosceles triangle is by apex angle alone; of inscribed from centre of circle, gives chord. Sines and cosines arise only with symmetry-broken case, where mirror symmetry introduces negative values. Chords were the old way, like the geometry of distinctions. This is hardly known today.

Also Tesla cold electricity = cold electron discharge as in Kirlian photography, the classic evidence for natural fractals. Now thought in US to be effect of slightly radioactive potassium 40. Interestingly, potassium is the generic of the Schussler cell salts. They care called homeopathiuc, but properly alchemical, like the old chemical names - from Egyptian/pagan alchemy of soils and growth factors.

At 11:55 AM, Blogger Ulla said...
Quantum Communication Controlled by Resonance in 'Artificial Atoms'
"We have developed a new way of controlling the electrons so that the quantum state can be controlled without measurement, using resonances familiar in atomic physics, now applied to these artificial atoms,"

important for biology.

At 9:25 PM, Anonymous Matti Pitkanen said...

To Anonymous:

To me "purely local" looks like a well-defined notion. Tangent bundle and vectors in tangent space with quadratic form defined by metric defines gives something existing also in rational context but trying to define integrals over curves, surfaces, etc... leads to problems unless boundaries of objects and integrands are special. Basically talk about signed areas is about definition of differential forms and their integrals.

Quadratic counterpart of length is well defined only in Euclidian geometry and its rational restriction. For Riemannian geometry Euclidianicity applies only in local approximation.

I looked the video but I see the problems of purely p-adic integration same as before. p-Adic numbers are not well-ordered so that one cannot define orientation for integration path/volume and has no manner to say which are its end point/boundaries: boundaries do not exist in *purely* p-adic context. That *definite* integrals become very problematic in p-adic context is of course one of the basic problems of the entire modern mathematics trying to build number theoretically universal calculus. My way out is the the reduction of integrals effectively to real context by what I call canonical identification defining the coordinate chart maps between number fields. The real well-orderedness is induce to p-adic side (not globally) since canonical identification assigns to real rational two p-adic numbers by the generalisation of 1=.999999....) and one can define p-adic boundaries.

One would get over the problems by an extension of number concept by fusing number fields to a book like structure: both reals and p-adics and "cognitive representations" and their inverses as part of generalized calculus. The extension of the domain of what is assumed to be mathematically modelable to include correlates of cognition and intention would provide a solution to a deep problem of mathematics.

Real number based physics - should I call it materialistic? - is only about those aspects of existence which can be weighed - mathematically weight reduces to the real norm. Numbers have only magnitude in this approach. In number theoretic approach the notion of number generalizes and numbers have also number theoretic anatomy.

At 9:26 PM, Anonymous Matti Pitkanen said...

To Ulla:

Looks very interesting. I should find time to look at it.

At 2:51 AM, Anonymous Anonymous said...

"Quadratic counterpart of length is well defined only in Euclidian geometry and its rational restriction. For Riemannian geometry Euclidianicity applies only in local approximation."

Doctor Pitkänen, your claim is simply not true:

(Unless, of course, that you can prove that Wilderberg's algebraic generalization of geometry is fundamentally flawed. :))

As for p-adics, reals etc., I feel that the heart of the problem is the notion of completion ( Exact meaning of which is not at all clear. Even the Wiki article gives (more general?) topological alternatives that are not dependent on cauchy "reals".

To begin with, according to your approach, if I understand correctly, any metric real space (or line) is not actually "complete" without the p-adic counterpart, joined by common rational. A Western philosopher might call this relation the codependent dialectic of subject and object. And naturally, this dual notion of completeness would require careful restructuring of many basic theorems of number theory.

A question: what is the relation of notion of "completeness" - at this moment both historically and ideally - and the desired WCW restrictions? My first intuition is that number theoretical restriction of WCW might be the rational restriction???!

At 4:36 AM, Anonymous Matti Pitkanen said...

To Anonymous:

Sorry, I did not have intention to be provocative. My claim was about *generic* Riemann geometry.

The excellent talk at of Wildberger was about hyperbolic spaces- not about hyperbolic manifolds in general. If I remember correctly, 3-D hyperbolic manifolds allow a special metric allowing to represent them as coset spaces obtained from 3-D hyperbolic space by identifying points under discrete subgroups of isometries SL(2,C). Note however that this is true only for this very special highly symmetric metric used to classify topologically hyperbolic 3-manifolds. For these very special metrics of hyperbolic 3-manifolds one might be able to generalize quadrance and spread to these spaces. This is however not true for general hyperbolic manifolds, and certainly not general Riemannian geometries.

This representation of hyperbolic manifold is very interesting also from TGD point of view since effective identification of points of Lobatchevski space (t^2-x^2-y^2-z^2= a^2) under discrete isometries (physical quantitie are invariant under these isometries) gives effectively hyperbolic manifolds very abundant amongst 3-manifolds.

By the way, if one considers submanifolds in say Euclidian or Minkowski space or more generally, in H= M^4xCP_2 familiar to me, it it is possible to generalize this kind of *bi-local* (here the problem lies for the generic abstract Riemann geometry) functions as restrictions of those defined by the geometry of M^4xCP_2 so that they make sense for sub-manifolds of of H. Minkowskian distance squared is one such function.

The simplicity of H is crucial also in the definition of the notion of pairs of real and p-adic sub-manifolds of H. Real H and its p-adic counterparts form also pairs possessing a highly unique p-adic manifold structure (note however measurement resolution), and the p-adic manifold structure for -say - space-time surface can be *induced* - as everything in TGD framework. This is very important practically since the manifold structure for H leads to discretizations defined by lattices and perhaps quasilattices characterised in terms of discrete isometries and one can cope with the constraints from general coordinate invariance.

For spheres and hyperbolic spaces the distance from origin of flat Euclidian or Minkowskian imbedding space is constant and one can use the coordinates of imbedding space and define functions like quadrant and spread. Quite generally, n-point functions in quantum field theories in generic curved space are difficult to define if one wants general coordinate invariance. However, if space-times are 4-surfaces in M^4xH situation changes since imbedding space coordinates dictated by isometries provide highly unique coordinatization for arguments of n-point functions.

At 5:11 AM, Anonymous Matti Pitkanen said...

To Anonymous:

thank your for interesting questions. I am a mathematical infant but do my best to answer.

1. I think that the notion of completion is well-defined (at least for the needs of physicist) for both reals and p-adics. I have written something about infinitesimals and infinite numbers with inspiration coming from the notion of infinite prime allowing detailed number theoretic anatomy for strictly infinite numbers and their infinitesimal inverses. The connections to quantum physics are fascinating.

The generalisation of the notion of number by fusing reals and p-adics and their extensions along rationals (and perhaps common algebraics) means fusing all possible completions to single structure.

2. The basic inspiration for the fusion of real and p-adic physics comes from number theoretic universality on one hand, and from the vision that cognition has p-adic number fields as correlates. I had intention to write that without the attempt to construct quantum theory of consciousness I would not have ended with number theoretical universality. I am not actually sure about this: I ended up with p-adic physics by starting from p-adic mass calculations and this represents elementary particle physics.

3. Loop space is much simpler than WCW but already infinite-dimensional and should possess Kahler geometry. The restrictions on loop space geometry come indeed from topology although I do not understand the details.

In any case, infinite-D topology is very delicate notion and the tangent as Hilbert space is not necessary one and unique as is clear for even physicists who have worked with various Hilbert spaces defined by function basis. The outcome depends on what boundary conditions one poses, how smooth the functions allowed as elements of the Hilbert space are, etc... The condition that Riemann connection defining parallel translation does not lead out from tangent space of loop space, is satisfied only if the metric possesses maximal isometry group and is thus essentially fixed by its values at single point: helps a lot in infinite-D context. The isometries correspond to Kac-Moody group.

For WCW this forms the underlying vision. The generalisation of conformal symmetries of 2-D surfaces to those of metrically 2-dimensional light-like 3-surfaces is also central and makes space-time dimension D=4 unique

At 5:12 AM, Anonymous Matti Pitkanen said...

To Anonymous:

Your question about rationality is interesting.

1. What rationality means is the first question. At space-time level it means finite resolution effectively replacing space-time surface with a discrete set of points. If one accepts the notion about pair of real and p-adic manifolds and induces the manifold structure from H=M^4xCP_2, this means a discrete set of points of H which are rational. This set can contain very few points: Fermat's theorem says that the surface x^n+y^n=z^n contains no rational points for n>2!

At WCW level rationality would perhaps mean rational functions of preferred imbedding space coordinates appearing in the conditions defining space-time surface (or only partonic 2-surface perhaps).

The simplest WCW geometries at the bottom of evolutionary hierarchy would be expressible in terms of rational functions constructed using polynomials with rational coefficients. Rational points of WCW would correspond to 3-surfaces expressible in terms of rational functions. As a matter fact, strong form of holography suggests that everything reduces to partonic 2-surfaces and their 4-D tangent space data: this data should be expressible using rational functions.

A more general view is that one has an evolutionary hierarchy of algebraic extension meaning that 3-surfaces become increasingly complex algebraically. Rational functions with algebraic coefficients in algebraic extensions of rationals appear firsts. Maybe even transcendental functions pop up eventually! This hierarchy would represent evolution at the level of WCW. It is amusing that Neper number e is very spacial p-adically e^p is ordinary p-adic number and one has only finite-D extension p-adically (1,e,2^2,...e^(p-1). It would seem that p-adically e is an algebraic number since it satisfies the condition e^p= p-adic number.

At 10:10 AM, Anonymous Anonymous said...

Thanks, Matti.

First, about the notion of 'infinity', we should be very careful when using such notion and make clear whether we are speaking on the level of WCW and participant observer, when we can not discuss "infinity", but just "finite resolutions" or forms that seem to us "finite and then some", generated by finite algorithms. OR discussing infinity and absolute e.g. as Spinoza did, based on strong holographic principle and speaking from "God Within".

This difference and confusion between epistemical (WCW and general measurement theory) and ontological(?!) (holonomic "space" for WCW) "infinities" is present also at our notions about completeness. Your notion of evolution, as well as many basic mathematical structures, makes it clear that there is no strict boundary between the two levels referred, but a borderzone of creative play.

To me rationality means simply, being related. Aho Mitakuye Oyasin is Lakota expression meaning "For all our relations". And thus, "purely local" as linguistic and philosophical notion would mean "unrelated". Which brings to mind on the level of physics worst multiverse theories obsessively holding only to purely deterministic causality.

To return little bit closer to the main theme of this blog, Wilderberg analyses attitudes towards reals into three classes: 1) "anything goes", ie. "completeness" derived from arbitrary axiom of choice. 2) Algebraic (or more generally finite algorithmic) generation of all reals. 3) no such thing, which is his own position. E.g. from your article on p-adic manifolds it seems clear that your position is the second category. Wildergerg's objection to second category is that that it's a beautiful dream, but currently just wishful thinking or at best work in progress, as we lack, to begin with, a general theory of algorithms. And I'm sure you agree that in the Discipline there is important distinction between creative speculation and rigorous math (and mathematical physics). :)

At 2:54 PM, Anonymous Anonymous said...

Perhaps this is how a collective quantum jump rationally manifests to observers like us, observing theory of physics as self creating organism. Self correcting process, isn't that what it means, that it just keep getting better? When we learn it by heart?

At 8:48 PM, Anonymous Matti Pitkanen said...

To Anynymous:

Concerning the notion of infinity. The notion of infinite primes demonstrates that this notion has many interpretations.

a) Usually infinity is seen as something obtained as a limit. this notion is used in completion process. Also Cantorian infinite based on the notion of follower also relies on this notion.

b) Infinite prime is not a limit of anything. It is just a formula based on introduction of product of all finite primes as a mathematical object and then using just the notion of divisibility to define primeness. In p-adic sense there is nothing infinite in infinite primes.
What is interesting is that there is infinite number of infinite primes smaller than X+/-1, where X is product of all finite primes.

The amusing observation is that infinite primes at the lowest level of hierarchy can be mapped to finite rational numbers!

Second nice thing is the connection with states of second quantized supersymmetric arithmetic quantum field theory. At higher levels one can perform the map to possibly infinite variants of rationals.

On axiom of choice I cannot take stance for the simple reason that I do not have technical understanding in this area. I work basically as a physicists and classify myself as a mathematical dreamer;-).

At 9:09 PM, Anonymous Matti Pitkanen said...

To Anonymous:

Whether reals and p-adics really exist remains a question of belief: the existence of calculus however makes me as a physicists a believer on transcendentals.

A hasty attempt to formulate what I think about rational mathematics and physics. Rational mathematics represents to me what we can observe and represent physically. p-Adic and real transcendentals to what we can imagine and what realities are.

Computers use rationals. Next powerful computational step is to introduce algebraic extensions of rationals and use rational coefficients for them in computations and at the end approximate algebraics with rationals.

Algebraic extensions and completion are necessary for having matter and cognition. Without completions we would have only the rational intersection of matter and cognition (identified as life in TGD Universe!).

What is amusing that state function reduction as quantum measurement of density matrix for system and its complement makes the entanglement maximally negentropic with nxn unit matrix defining the density matrix in the final state. Number theoretic negentropy for unique prime factor gives the negative Shannon entropy.

One could very metaphorically say, that state function reductions does its best to make the Universe rational by projecting the system to what one could call rational intersection of matter and cognition (of reals and p-adics). Quantum jump however involves a decollimation at the opposite boundary of CD and loss of rationality there. The next state function reduction occurs for it. Since CD has two ends, the Universe can never make itself rational despite all these attempts!!;-)! This however means also continual re-creation, which is a good thing;-).

At 2:32 PM, Blogger Stephen said...

At 2:54 PM, Anonymous Anonymous said...

Perhaps this is how a collective quantum jump rationally manifests to observers like us, observing theory of physics as self creating organism. Self correcting process, isn't that what it means, that it just keep getting better? When we learn it by heart?

^^^^^ I like like like this comment. Enthusiastic thumbs up. In the theory of Hawkes process / point processes / compensators / etc (which extend to fermion and boson processes via permanents and determinants) there is the notion of feeding back the over and under shoot of the expectation of the future occurrence of events. So it should keep getting better, and literally learned by heart. Peace, Stephen

At 2:44 PM, Blogger Stephen said...

I think it is possible my friend thinks "all this stuff" is "mathematical masturbation" maybe because he hasn't learned the specifics of this math and is raving about math in general? even though he knows a lot of math already and is quite the established guru when it comes to technological achievement...I wish people would learn enough TGD to rationally criticize it instead of just dismissing due to bias

At 11:38 AM, Blogger Ulla said...

Ye, most doesn't understand and they think themselves that THEY are the experts. They have hard to accept this may not be the case, and if THEY don't understand it must be wrong... stupidly thought.

But critique is good, and unclear things are certailnly there. To explain is a good teacher.

Dark matter as parallell universe

At 6:24 PM, Anonymous Orwin O'Dowd said...

Gamechanger! This long-sought proof in Morse theory is actually doable by more simple and general methods of a combinatorial nature:

The point is, Peter Woit is saying SUSY may be provable in 1D but he point is 3+1D: but doen't the CPT contraint give 2D over to Charge and parity, with only the anomalies free? That's where Mattie got a focus, and with the double-take in Morse theory, the game opens again.

The hazzard: old habits of Paltonizing 3D realism, echoing through von Neumann, Andre Weil, Grontendieck, etc. But the above works oine the side of Smale/chaos, which is much more interesting...

At 9:18 PM, Anonymous Matti Pitkanen said...

It is a pity that SUSY has to a great extent reduced to an industry around very special variant of supersymmetry known as N=1 SUSY proposed as minimal generalization of standard model. This very specific option is attractive because it allows routine methods for producing papers with high technical standards. As we knowm LHC almost excludes it.

N=1 SUSY requires Majorana spinors and breaks separate conservation of lepton and baryon numbers. This should have been as a very questionable feature from the beginning. Maybe the basic problem is industrialization of science.
There are however many vaariants of SUSY if N>1 is allowed and considers models involving higher-dimensional spacetime. N--> infty limit could be interpreted in terms of superconformal algebra. For N>1 the separate conservation baryon and lepton numbers is possible to achieve: in dimension D=8 M (M^4xCP_2) one indeed has at least N=2 SUSY.

In TGD SUSY corresponds to a 4-D generalization of super-conformal symmetry with right handed neutrino taking the role of generator of least broken supersymmetry. Quite generally, fermionic oscillator operators take the role of SUSY generators so that the interpretation is different from that in the standard framework.

What is remarkable that SUSY - at least part of SUSY algebra- allows in TGD framework genuine geometrization in terms of the gamma matrices of "world of classical worlds" in this TGD framework.

The technical problem is that SUSY algebra is defined in terms of fermionic super charges which are indeed conserved by modified Dirac equation. Infinite-D isometry algebra and gamma matrix algebra combine to infinite-D superconfromal albebra. Superspace formalism does not exists and one cannot apply existing super-space methods. Here the work of mathematicians would be needed to make things explicit.

To my opinion the problems of SUSY derive from the lack of general vision (this would be regarded as "philosophy" - bad!) . As a consequence, the work has degenerated to endless and useless model building with wrong basic premises. Reminds me about similar situation for aether models before special relativity.

At 7:30 AM, Anonymous Orwin said...

That's why the chaos matters so much: not 1D but 1+; and the combinatorial approach under order relations which opens cleanly on number theory. As Peirce showed you can't algebraicize logic without order: its x<y and y<x that do the work, the operators aren't enough, they just shunt you to posets.

And so the operator-based quantum logics go nowhere either. But you can say the same for pure geometries: it practice, compactness matters, and that introduces topology, which is why Seiberg-Witten and Chern-Simmons are landmarks. Even so, as Anderson points out, the 3D theory is underdetermined.

A new physical insight is required. Here's a 2+1 view of fractionalization(!) and low energy (!) excitations opening on stringly stuff at the large N limit. Still wanted is an approach to the combinatorial and factorial logic that underlies the large N limit. that's just where I flagged your difficulty earlier.

Take these 2+1 surfaces as yoru preferred extremals/boundary conditions and it looks feasible at last.

At 7:33 AM, Anonymous Orwin said...

The link:

At 11:34 AM, Blogger Stephen said...

Maaaaaan, I studied a lot of chaos theory back in the day. Matti, what do you think about this article about topological insulators and quantum computation as it compares to your vision of quantum computers?

At 11:40 AM, Blogger Stephen said...

Orwin, you're comparison operator is funny. There’s a Zen saying: “Comparison is the root of suffering.” While this may be hard to comprehend on the esoteric levels, this insight is certainly true for girls and women who compare themselves – or are compared by others – to ideals of beauty that even the supermodels themselves don’t live up to (without intensive hair & make, wardrobe, lighting, and retouching).

funny how the male mind admiration of feminine beauty can parallel the scientific mind's admiration of theories

At 8:45 PM, Anonymous Matti Pitkanen said...

To Orwin:

In the link

emergence of supersymmetry is associated with criticality using kind of lattice model.

I have considered something analogous but for super-conformal algebras of TGD.

a) The generators of ordinary super-conformal algebras are labelled by integer n and thus contain an infinite hierarchy of subalgebras with n mod k=0, k=1,2,..... Inclusion sequences for these algebras define hierarchies of superconformal symmetry breakings.

Criticality in the infinite-D context defined by world of classical worlds (WCW) defines this kind of hierarchy and the natural proposal is that breakings of superconformal symmetry correspond to transitions to sub-algebra: only the sub-algebra would act as a genuine gauge algebra meaning that effectively the original algebra would contain only finite-number of generators creating physical states.

b) The reversals of symmetry breakings correspond to restriction to a higher-D critical manifold with higher degree of criticality. The interpretation would be in terms of quantal version of self-organized criticality. Now however the criticality would not be due to attractor property for ordinary dynamics but due to state function reduction leading to eigenspaces of density matrix that is states with nxn unit matrices as density matrix. This means that density matrix is measured in quantum jump. NMP at criticality guarantees that these states are stable and can be transformed to their p-adic counterparts defining cognitive representations. Life and intelligence emerge at quantum criticality.

An interesting question that popped up as I wrote this, is whether self-organised criticality could serve as a space-time correlate for the above quantum version of criticality made possible by quantum measurement theory and NMP.

The analogy with Thom's catastrophe theory deserves to be mentioned. In finite-D case the sub-manifolds of catastrophe manifold as defined in Thom's theory define this kind of hierarchy. For cusp one has the manifold in which the derivative of potential function with respect to behavior variable x as function of two control parameters vanishes and the tip of cusp at which also the second derivative vanishes. This kind of hierarchy becomes in the case of WCW infinite and would be characterized by sub-algebras of super-conformal algebra. Connections with the inclusion sequences of hyper finite factors of type II_1 and hierarchies of measurement resolutions suggests also themselves.

At 8:52 PM, Anonymous Matti Pitkanen said...

To Stephen:

Looks interesting. Unfortunately, I know too little about topological insulators to be able to build any associations to TGD.


At 12:12 PM, Blogger Ulla said...

Topological insulators must serve as some kind of memory function by creating 'islands' of different energetic and informational/entropic content.

About the quantum jump and the recreation of world see Poincaré's Electromagnetic Quantum. Mechanics. Enrico R. A. Giannetto
Can you see if it is something good?

At 11:42 PM, Blogger Ulla said... Information transfer as a 'chip'.

At 2:01 PM, Anonymous Orwin said...


Conrad Waddington reframed Thom's theory in terms of the Baldwin Effect to give a surface where environmental effects "steer" development, so distinguishing phenotype from genotype. Natural selection acts on the phenotype, so the space-time marker tags selection events, or points in biological space-time. A formal treatment of Waddington like this is wanting in theoretical biology today.

Whether the selection process is prior to cognition or consciousness remains an open issue, dependent on how one interprets the behaviour-environment interaction. And no-one expect such psychological issues to resolve ahead of the biology.

At 2:01 AM, Anonymous Anonymous said...

At 1:49 PM, Blogger Ulla said...
healing happen through theta waves?

At 4:39 PM, Blogger kamran shiakh said...

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At 11:55 AM, Blogger hammad sohail said...

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At 1:24 AM, Blogger hammad sohail said...

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At 5:36 AM, Anonymous Matti Pitkanen said...

Sorry about this crazy fellow. I sincerely hope that he could get professional help.


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