Thursday, February 02, 2012

One more good reason for p-adic cognition

One can present several justifications for why p-adic numbers are natural correlates of cognition and why p-adic topology is tailor-made for computation. One possible justification derives from the ultrametricity of p-adic norm stating that the p-adic norm is never larger than the maximum of the norms of summands.

If one forms functions of real arguments, a cutoff in decimal or more general expansion of arguments introduces a cumulating error, and in principle one must perform calculation assuming that the number of digits for the arguments of function is higher than the number digits required by the cutoff, and drop the surplus digits at the end of the calculations.

In p-adic case the situation is different. The sum for the errors resulting from cutoffs is never p-adically larger than the largest individual error so that there is no cumulation of errors , and therefore no need for surplus pinary digits for the arguments of the function. In practical computations this need not have great significance unless they involve very many steps but in cognitive processing the situation might be different.

26 Comments:

At 3:02 PM, Blogger ThePeSla said...

Matti,

In formulas involve the natural log we have a limit of the accuracy in the computation of the tables and level of error. In a sense this is not a simple idea of limit.

In the ideal or absolute case I see your point on the utility of this for to ground consciousness as it say relates to QED from on view. We in the low dimensions and harmonics can sometimes go beyond the bounds that confines the primes and ignore this in the higher reaches and deeper accuracies.

For me, errors, noise, so on... is the general idea of meaning and information being conjugate (more of one less than the other) as the practical use by Shannon in signal transmission over noise and the entropy of natural communications (initial letters and so on). But I am a bit more radical than that in that from a higher view of things information (data) and meaning (purpose) can be interchanged.

The PeSla

 
At 1:36 AM, Blogger Ulla said...

The working memory is quite limited, about 7 things at each moment, collaborate with memory (different kinds) that relax these momentums in form of different kinds of numbers?

What IS the working memory? Is it kept up by noise?

 
At 3:02 PM, Blogger hamed said...

.Dear Matti,

I want to know evolution of concepts from classical physics to Geometrodynamics of Wheeler (Superspace) and then to TGD (WCW). So I studied correspondences between particle dynamic and Geometrodynamics at the book of gravitation by Wheeler:
1-Dynamic entity of Particle dynamics is particle and of Geometrodynamics is space ( 3-dimensional space).
2-Descriptors of momentary configuration of first is x,t(event) and of second is 3-Geometry.
3-Classical history of first is x=x(t) and second is 4-Geometry.
4-at first every point on world line gives a momentary configuration of particle but at second every space like slice through 4-Geometry gives a momentary configuration of space.

Where does WCW similar with superspace on subjects that listed above and where does is different?
does ADM formalism any position in WCW like superspace?
Maybe if wheeler had been equipped his superspace with Kahler geometry (and he had your Extraordinary ability of arguments;-)) he could like TGD assigns to each 3-surface a space-time. so space time should not mean in superspace without kahler structure and the space time is not fundamental in this view.
As wheeler wrote: “There is no space time, there is no time, there is no before, there is no after. The question of what happens 'next' is without meaning”
For more clarification I saw at The Labyrinth of Time by Lockwood: “Every such cosmological model can be foliated in infinity of different ways; and each foliation will consist of a sequence of 3-geometries. (Wheeler refers to this freedom in the choice of a foliation as many-fingered time.)”
I mean wheeler speak about some non-determinism of time? Then the time of superspace is more like subjective time(nondeterministic) of TGD than geometric time(deterministic) of it?

Apologies for the bombardment of questions at each time:-)

 
At 3:48 PM, Anonymous Orwin said...

Ulla, on the latest experimental evidence, STM is a complex, systemic capacity involving visual and frontal cortices.

http://www.mpg.de/5000193/Insights_adaptive_learning?filter_order=L
Max-Planck-Gesellschaft - New mechanistic insights into adaptive learning

It is not, as asserted by Bernard Baars, a central or superordinate register, so Matti's entanglement metaphor is descriptively apt. From what I have said previously (
The Integrative Mind
http://vixra.org/abs/1110.0046
) and in critique of Baars (Critical Foundations of the Contextual Theory of Mind. http://cogprints.org/7661/
), you have a kinetic melody (Merleau-Ponty) playing over visual loci, which is a form of psychical space-time.

It is very tiresome that what capacity we have for modeling such processes is appropriated for preaching multiverse dogma as timeless verity (Wheeler > Hammed's problem).

 
At 8:25 PM, Anonymous matpitka@luukku.com said...

Dear Hamed,

It is fun to discuss and answer good questions!

Here a brief comparison of WSP and WCW.

a) In WSP 3- geometry is basic entity. In WCW sub-manifold 3-geometry.

b) There is no time or space-time in WSP context as Wheeler emphasizes. The theory should assign in a natural manner to given 3-geometry a four-geometry to achieve this.

In WCW context the definition of WCW Kaehler function assigns to a given 3-surface a 4-D space-time surface as preferred extremal of Kaehler action. This makes classical physics as exact part of quantum theory and even that of WCW geometry. This new view about Bohr orbits as something very real instead of being a fiction produced by stationary phase approximation is the new philosophical view point.

The notion of 3-surface generalizes and eventually one ends up with zero energy ontology and strong form of general coordinate invariance implying effective 2-dimensionality: partonic 2-surfaces and their 4-D tangent space data and CDs within CDs basically due to the breaking of standard form of determinism.

On might argue that in strictly mathematical sense space-time becomes also in TGD redundant locally (only locally due to the failure of strict determinism implying the need for CDs within CDs). Space-time is however necessary for the interpretation of quantum measurements which always assign to quantum events classical space-time correlates such as frequencies. People talking about emergence forget often all about quantum measurement theory. In any case, imbedding space and time are there and very relevant for the interpretation of the theory!!

c) WSP the classical history of 3-surface is a questionable concept since topology change is not natural in this context. This view about 4-surface is very "Newtonian" and allows only X^3xR type space-time topologies.

In WCW with ZEO one gives up completely the Newtonian view: positive and negative energy parts of zero energy states as 3-surfaces at the two boundaries of CD can have totally different topologies. Particle creation in topological sense becomes possible and there is space-time topology is analogous to that of Feynman diagram. Wheeler's many fingers would in this context become external lines of particle reaction. At vertices incoming lines are join like lines of Feynman diagram so that space-time surface is not anymore even 4-manifold. 3-surfaces however are. Note that in string string world sheets are 2-manifolds but their time=constant sections are not 1-manifolds always.



d) During first year of TGD I tried ADM formalism but realized that canonical quantization fails: the connection between time derivatives of imbedding space coordinates and canonical momentum densities associated with action even in case of YM action is many-to-one and extremely non-linear so that one cannot do nothing in practice. Standard quantization recipes would also yield horribly nonlinear poorly defined functional differential equation (p->id/dx recipe generalized) so that a direct generalization from the tiny hydrogen atom to the entire infinite Universe fails. Perhaps theoreticians were too ambitious at this time;-)

For long time my heuristic guideline was that the non-locality of K\"ahler action for preferred extremal as functional of 3-surface is the manner to get rid of local divergences. Reduction to Chern-Simons terms however brings 3-D locality back but almost-topological character allows to avoid divergences now.

 
At 8:31 PM, Anonymous matpitka@luukku.com said...

Continuation to Hamed:

I have played now and then with the question whether Kaehler geometry could be defined in terms of preferred extremals of curvature scalar. What these extremals could be? How could one bring complex and Kaehler structures to the space of 3-geometries.

In the case of loop spaces Kahler geometry is unique and there are excellent reasons to expect that in higher dimensions even the target space containing 3-surfaces is more or less unique just from the existence of the WCW geometry. What is the situation in WSP: could it be that space-time dimension is fixed?

In

http://tgdtheory.com/public_html/articles/egtgd.pdf

I consider analog for Wheeler's approach in TGD based on Einstein Maxwell action.

a) Almost topological QFT property would mean that metric appears extremely implicitly in the theory. It requires that the solutions of field equations involves are such that the action in reduces to 3-D Chern-Simons term. The conformal invariance of Einstein- Maxwell equation indeed imply that action reduces to boundary terms.

b) Weak form of electric magnetic duality seems to make the 3-D contributions to action Chern-Simons terms.

c) One must allow also space-time regions of Euclidian signature and CP_2 as solution of field equation. Cosmological constant determining CP_2 size scale (CP_2 corresponds to non-vanishing cosmological constant) is needed in Euclidian regions. Average cosmological constant could be quite small and reflect the fraction of Euclidian space-time volume.

d) This would force a new view about black hole interiors as Euclidian regions of space-time. CP_2 was actually originally discovered as gravitational instanton.


e) This is very near to TGD: Euclidian regions would carry electroweak and color quantum numbers and Minkowskian regions classical quantum numbers. Could imbeddability to M^4xCP_2 follow as a consistency condition?

 
At 8:37 PM, Anonymous matpitka@luukku.com said...

Still a comment to Hamed about the loss of time. I hope that I do not repeat myself too much.


Wheeler is quite right in what he says about absence of time in WSP. This is what general coordinate invariance implies also in the 4-D view about general relativity.

In WCW approach based on zero energy ontology the position of space-time surface /CD in imbedding space brings in geometric time and one can speak about the positions of partonic 3-surfaces.

Basically I however think that "before" and "after" as we usually understand them apply to subjective time, not to geometric time. One must identify subjective time as something different from geometric time and this leads to quantum theories of consciousness and things get even messier;-).

In quantum gravity based on Schrodinger equation in WSP time could appear only as the analog of time parameter in Hamiltonian unitary evolution. The basic equations of ADM however lead to coordinate conditions saying that physical states are analogous to states with zero energy. Easy to guess since Noether charges associated with diffeomorphisms of space-time vanish identically.

Again one encounters the basic motivation of TGD: how to simultaneously realize Poincare symmetries and geometrize gravitation. All the difficulties of GRT force to make the same question: what about sub-manifold gravity.

Perhaps the strongest objection that I have invented hitherto against sub-manifold gravity relates to the superposition of fields and is circumvented if it is replaced with superposition of effects of induced fields: simultaneous topological condensation to several space-tme sheets.


In path integral formalism one makes a hypothesis that Scroedinger amplitude as a functional of 3-geometry is sum over all space-time surfaces having 3-surface as boundary. Space-time would as a emerge as the analog of point particle orbit. Classical space-times would correspond to extrema of Einstein's action. There are technical problems due to the non-existence of path integral and one must make path integral functional integral by making Euclidianization. A connection with the time of special relativity is lost and achieved only in perturbative treatment of General Relativity in path integral formalism around Minkowski space-time.

Spin 1/2 particles is a second problem of Wheeler's geometrodynamics. In fact, spinor structure does not even exist in generic space-time. CP_2 is by the way one example of this.

 
At 1:37 AM, Anonymous Orwin said...

Matti,

In Celebration of Winds of Change

*Relativity 'freezes' due to Bianchi identity and must be construed as sympletic manifold (Levi-Civita): a succession of 'instants' on 2-surfaces interrupted by radiation events.

*You can split past/future as Whitehead did by assuming manifold constituted by material points or loci, which entails 2 forms of inner product:
Whitehead, A treatise on universal algebra, with applications. http://ebooks.library.cornell.edu/cgi/t/text/text-idx?c=math;cc=math;view=toc;subview=short;idno=01950001

*In Leibniz' doctrine and the young Kant this is the nature of physical space, but for the mature Kant we only know our idea of space. Von Uexkull (bio-semiotics) showed subjective space constructed from 'lokalzeichen', indicative of relative positions. So why impose this on physical theory?

*With charges internal to sympletic neighbourhoods, relativity is always already polarized, and we find the charges balance over all. But polarizability (Slater) has FOUR dimensions of length (and two frequencies, per mass) so this potential can always give rise to new radiation events and re-constitute the relativistic situation.

*With finite speed of nerve impulses, all cognized intervention in reality must be pre-cognized, so the organism must be able to draw on physical data indicating future of process in order to survive (subject to radiation load & mutations!!).

*And now short-term memory shows the mind 'acting back' in subjective time, as required for Bohr's reflection on experimental outcomes! This is not hallucination, as imagined by Meynert and Freud. To offer a better account of hallucination and epilepsy is a break-through, for which I will always remember that dream-discourse which entangled us all.

 
At 6:42 AM, Blogger Ulla said...

This should be obvious:

By Nielsen et al 2006. Quantum computers hold great promise for solving interesting computational problems, but it remains a challenge to find efficient quantum circuits that can perform these complicated tasks. Here we show that finding optimal quantum circuits is essentially equivalent to finding the shortest path between two points in a certain curved geometry. By recasting the problem of finding quantum circuits as a geometric problem, we open up the possibility of using the mathematical techniques of Riemannian geometry to suggest new quantum algorithms or to prove limitations on the power of quantum computers.
Quantum computers run by gravity? That was cool. Angular momentum, pi. This is also the Riemannian curvature. This curvature acts on many paths in many objects, larger (GR) and smaller (SR) and quantum lattice, at one moment.
http://www.sciencemag.org/content/311/5764/1133.abstract

 
At 2:16 PM, Blogger hamed said...

.Thank you very much Matti for responding.
Just a little question: why does any map from a given 3-manifold X3 to M4*cp2 defines a surface?

 
At 2:21 PM, Anonymous Anonymous said...

Matti:

For your consideration:

---------------------------

A team of physicists detected an energy beam coming through the top of the Bosnian Pyramid of the Sun. The radius of the beam is 4.5 meters with a frequency of 28 kHz. The beam is continuous and its strength grows as it moves up and away from the pyramid. This phenomenon contradicts the known laws of physics and technology. This is the first proof of non-herzian technology on the Planet. It seems that the pyramid builders created a perpetual motion machine a long time ago and this “energy machine” is still working.

In the underground labyrinth, in 2010, we discovered three chambers and a small blue lake. Energy screening shows that the ionization level is 43 times higher than the average concentration outside which makes the underground chambers into “healing rooms”.

Further electromagnetic detection in 2011 confirmed that levels of negative radiation through the Hartman, Curry and Schneider grids are equal to zero in the tunnels. There was no technical radiation (from power lines and/or other technology) found in the tunnels and no cosmic radioactivity. Ceramic sculptures are positioned over the underground water flows and the negative energy is transformed into positive. All of these experiments point to the underground labyrinth as one of the most secure underground constructions in the world and this makes it an ideal place for the body’s rejuvenation and regeneration.

Regards.

 
At 12:02 AM, Anonymous matpitka@luukku.com said...

To Hamed:

I am not sure whether I got the gist of the question. This is the answer to the question as I understood it and perhaps totally trivial to you.

The image of continuous map of X^3 o H is image of 3-D geometric object and therefore 3-D surface.

Concrete example: h^k= f^k(x^1,x^2,x^3), for k=1,...,8 (H-coordinates) when x^i run over their value range you get point set of H parameterized by three coordinates so that it is clearly 3-dimensional.

When H and X^3 have p-adic coordinates situation is same: the image of X^3 is parameterized by three coordinates and in this sense 3-D.

Another basic manner to represent surfaces is as intersections of surfaces. The conditions H_m(h^k)=0, m=1,...5 , for independent functions would give intersection of 5 7-D surfaces which is generically a 3-D surface if non-trivial. In algebraic geometry this is the manner to represent surfaces.

In this representation the expressions for induced metric and gauge potentials are not so easy to deduce as using direct imbedding. One must in principle solve the imbedding space coordinates in term of three of three of these functions locally.

 
At 8:40 AM, Blogger hamed said...

Oh right!, Thank you Matti, i had a stupid mistake.

 
At 11:48 AM, Anonymous ◘Fractality◘ said...

Matti:

When I came across what I posted above I immediately thought of "TGD and Fringe Physics"

Could that beam be a negative energy phase conjugate wave?

Regards.

 
At 1:20 PM, Blogger Ulla said...

http://video.ias.edu/stream&ref=758 Nima

 
At 8:55 PM, Anonymous matpitka@luukku.com said...

To fractality:

Impossible to say anything without a detailed documentary.

In laser physics phase conjugate waves are everyday thing.

The interpretation as negative energy etc... is new but also consistent with Feynman graph picture allowing to talk about antiparticles as something propagating in reverse time direction with negative energy.

The sole thing differentiating it from what standard physics predicts is the reversed arrow of geometric time. An experimentally established fact also but something which one should not speak aloud since theoretical understanding is so poor;-). One could imagine all kinds of science fictive things based on phase conjugate waves. Maybe some scifi writer has done anything along these lines!

 
At 8:56 PM, Anonymous matpitka@luukku.com said...

Thank you Ulla. Seems to be a new. Nima is always inspiring.

 
At 10:41 PM, Anonymous ◘Fractality◘ said...

Matti:

Thanks for your insights.

Have you heard of the strange sounds around the world?

http://www.youtube.com/watch?v=GZ2ZcmMxehk

I do hope you aren't bothered by my "off-topic" posts.

Regards.

 
At 11:09 PM, Blogger Ulla said...

The fight on FB is like Hell has come loose. No reason at all. Gives a glimpse of what you have gone through, and suddenly Lubos seems like a kind and actually reasonable person :)

Keith Ashman, what does the name tell you? You can send it to Gabor, if you want.

 
At 5:54 AM, Blogger Ulla said...

I found a man that thinks along the same paths as you. He unify algebraic and geometric PDEs. AGOSTINO PRÁSTARO http://www.dmmm.uniroma1.it/~agostino.prastaro/HOMEPAGEPRAS.htm

http://www.dmmm.uniroma1.it/~agostino.prastaro/PUBLICATIONS.HTM

These results have opened a new sector in Algebraic Topology, that we can formally define the Algebraic Topology of PDE's. The characterization of global solutions, made by means of integral bordism groups, has allowed to obtain applications in some important PDE's of the Riemannian Geometry and Mathematical Physics, as Ricci-flow equation, Navier-Stokes equation and quantum Yang-Mills equations. Quantum super Yang-Mills equations, are discussed in the framework of quantum supermanifolds, obtaining a new approach to unify the four fundamental forces, (gravitational, electromagnetic, weak-nuclear, strong-nuclear), in an unique geometric structure at the quantum level. The geometric theory of PDE's, built in the category of quantum supermanifolds, gives us a dynamic theory to describe quantum phenomena also at very high energy levels, where quantum-gravity becomes dominant. Quantum black holes are interpreted as solutions of quantum super Yang-Mills equations with quantum- (super)gravity in action.

 
At 10:15 AM, Blogger Ulla said...

It seems they have worked with working memory in Helsinki :)

Palva et al. Neuronal synchrony reveals working memory networks and predicts individual memory capacity. Proceedings of the National Academy of Sciences, 2010; DOI: 10.1073/pnas.0913113107

http://www.sciencedaily.com/releases/2010/04/100413105704.htm

Orwins idea about a feedforward-feedback energetic mechanism creating synchrony seems plausible?

Visual working memory (VWM) is used to maintain sensory information for cognitive operations,... based on sustained neuronal activity in a complex cortical network of frontal, parietal, occipital, and temporal areas. The neuronal mechanisms that coordinate this distributed processing to sustain coherent mental images and the mechanisms that set the behavioral capacity limit have remained unknown. We mapped the anatomical and dynamic structures of network synchrony ... Interareal phase synchrony was sustained and stable during the VWM retention period among frontoparietal and visual areas in α- (10–13 Hz), β- (18–24 Hz), and γ- (30–40 Hz) frequency bands. Furthermore, synchrony was strengthened with increasing memory load among the frontoparietal regions known to underlie executive and attentional functions during memory maintenance. On the other hand, the subjects’ individual behavioral VWM capacity was predicted by synchrony in a network in which the intraparietal sulcus was the most central hub. These data suggest that interareal phase synchrony in the α-, β-, and γ-frequency bands among frontoparietal and visual regions could be a systems level mechanism for coordinating and regulating the maintenance of neuronal object representations in VWM.

Cognition = working memory (partly?)?

Individual capacity is highly correlated with intelligence measures and reduced in neuropsychiatric diseases. Simultaneous recordings from monkey parietal and frontal cortex revealed that visual capacity limitations occurred immediately upon stimulus encoding and in a bottom-up manner. Capacity limitations were found to reflect a dual model of working memory. The left and right halves of visual space had independent capacities and thus are discrete resources. within each hemifield, neural information about successfully remembered objects was reduced by adding further objects, indicating that resources are shared. http://www.pnas.org/content/108/27/11252.abstract

network topology and physical distance of synchronization ...
greater cognitive effort caused emergence of a more globally efficient, less clustered, and less modular network configuration, with more long-distance synchronization between brain regions. This pattern of task-related workspace configuration was more salient in the β-band (16–32 Hz) and γ-band (32–63 Hz) networks, compared with both lower (α-band; 8–16 Hz) and higher (high γ-band; 63–125 Hz) frequency intervals. Workspace configuration of β-band networks was also greater in faster performing participants ...
demonstrate how cognitive effort breaks modularity to make human brain functional networks transiently adopt a more efficient but less economical configuration. http://www.jneurosci.org/content/31/22/8259.full

Arousal? attention? wakenness? Is this part of negentropy maximation? At least not entropy maximation (homeostasy?)? Then comes naturally the question of emotion and p-adic relations. Seen also in Piaget? Negative emotions as panic and fear are stronger, creates a higher 'arousal' (an ad hoc word) together with memories (consciousness)? p-adics create the size of the global network? Its difference to hbar size of the Big Book remained unclear to me before. But there are also differencies left-right, so these are the Big Book or hbar then? You did get the Jill Bolte Taylor book?

 
At 10:20 AM, Blogger Ulla said...

Does p-adics and hbar work together in creating synchrony?

 
At 4:04 AM, Anonymous matpitka@luukku.com said...

Hierarchy of Planck constants and p-adics are correlates but I am not sure how. Naive guesses fail. These hierarchies are separate.

I am now working with the implications of Quantum Arithmetics. It is more and more clear that QA is the royal road to physics as generalized number theory. A beautiful unification of real and various p-adic physics emerges when number theoretic universality is realized in terms of the notion of adele generalized to Quantum Adele: QA again;-). Simple recipe: anywhere where you meat real number replace it with quantum adele!


The notion of adele is central in the Langlands program in which the goal is to understand Absolute Galois Group, a mythic objects defined as Galois group for the algebraic numbers as extension of rationals. AGG is a dragon, which has destroyed many brave and brilliant mathematical knights;-).

The needed generalization is a replacement of adeles with quantum adeles. This brings in an excellent candidate for an isomorph of AGG commutator group as Galois for the quantum adeles interpreted as extension of adeles. For ordinary adeles one has only the Galois for maximal Abelian extension of rational realized in terms of ordinary adeles.

This leads to very beautiful re-interpretation of the basic conjectures of quantum TGD and a lot of new awe inspiring insights. Takes week or two to polish it. I think I have done it;-).

 
At 11:19 AM, Blogger Ulla said...

Ah, the year of the Dragon :) You sound so enthusiastic, what is coming from this? Done it :-)
But the diamond is screaming here :(

 
At 11:57 AM, Anonymous ◘Fractality◘ said...

Matti:

When one says "I'm entering my prime."

Evolving to the next level of p-adic prime? ;)

Regards.

 
At 12:38 PM, Blogger Ulla said...

This seems to be quite interesting. p-adics controlling both cognition and affect? http://dbem.ws/FeelingFuture.pdf

What is the difference between precognition and cognition? Only about time? Libet? Remember also the Tamara Davis paper about future impact on us.

There is a workshop on this paper http://www.consciousness.arizona.edu/2012WorkshopKlein.htm

see also http://www.consciousness.arizona.edu/

http://arxiv.org/abs/1201.6624

 

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