tag:blogger.com,1999:blog-10614348.post5769193135147482787..comments2023-03-18T02:32:07.872-07:00Comments on TGD diary: Classical number fields and associativity and commutativity as fundamental law of physicsMatti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger37125tag:blogger.com,1999:blog-10614348.post-75396342126346960882015-03-29T12:25:22.129-07:002015-03-29T12:25:22.129-07:00Dedekind cuts
http://en.m.wikipedia.org/wiki/Dede...Dedekind cuts<br /><br />http://en.m.wikipedia.org/wiki/Dedekind_cut<br /><br /><br />Farey sequences / series , has relations to Riemann hypothesis too<br /><br />--StephenAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-21127535006371019342015-03-27T12:02:22.795-07:002015-03-27T12:02:22.795-07:00One way to think of EDII is that both sides conver...One way to think of EDII is that both sides converge to half of negative one. The symmetry of multiplication and division is deeper than this or that entangled property of this or that number system.<br /><br />The simplest level of number theory is base one, and generating natural numbers in base one does not differentiate between multiplication and division, there is no either-or choice between inclusive whole 1 that gets divided and atomistic 1 that gets multiplied. Both movements generate identically endless strings of 1's, the platonic 'hen kai agathon'. This unity gets hidden and forgotten when we invent no-one (aka "zero") and start playing with base 2 and must invent rules to place 1 and 0 in a partially coherent way (e.g. boolean rules, peano axioms, etc.), and are faced with constant barrage of number theoretical choices similar to and including "is it a wave or particle", "which property of particle is entangled in this Bell measurement", etc. In the entangled state of base '0, 1' aka 2-dimensional base the choice of not choosing is no longer present. <br /><br />This is what the that of EDII brought me now to contemplate, and you may consider this an empirical example of an untrained mathematical cognition having an observation event, an 'anamnesis'. What I really can't tell now if this was a movement of heff in the context or limit of base '1, 0', and if so, from 1 to 0 or from 0 to 1... :)<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-82176976075948484422015-03-27T04:01:55.434-07:002015-03-27T04:01:55.434-07:00So sorry, don't know why I thought of and call...So sorry, don't know why I thought of and called nominator exponents for real side decimal extensions as "negative". Probably because of I was also thinking of some kind of zero state ontology relation of Euler's doubly infinite identity (EDII), and I associated "determinant" p-adic side as 'positive' and "nominator" real side as 'negative'. <br /><br />I believe that in Euler's mind EDII is/was intimately tied with his identity and proof concerning pi, e, i and 1 and 0. I inteprete EDII as number theoretical presentation of the maxim "as above, so below", and in that sense metamathematical equation of "if this arises, that arises". Or, inner spaces of nominator power series dividing one and "outer" inclusive spaces of determinant power series. This Russian doll relation between p-adic and real feels to me more significant than book metaphor.<br /><br />As said, the (hindu-arabic) number fields for decimal (or any base of natural number) extensions for reals and also p-adics are homomorphic with field of all natural numbers, and the differences or meaningful relations are said to different notions of "distance". This is how - the space in which - we have been taught to think about numbers. <br /><br />The notion of distance, is however, very problematic and baffles me, as it is inherently tied with notion of 'length', and as long as we do and think math inside some dimensional(orthogonal) coordinate system, notion of length is valid only in context of 1D-line, notion of area is valid only in context of 2D-plane, notion of volume is valid only in context of 3D-space, etc. This level or rigor (...mortis ;D) is IMHO highly non-trivial in order not to commit to category errors at most basic level, when questioning cognition and numbers in philosophically sincere way. <br /><br />As dimensionality is inherent already at the level of exponential multiplying or dividing a number by itself, the 1D-notion of "distance" giving different meanings to (natural, real, p-adic etc.) homomorphic number fields, which we think of as 2D planes, needs to be reconsidered. I believe Norman is on the right track with quadrance as the most basic level of mathematical aka coordinated measuring for observers like us, and with the algebraic generalizations of quadrance into higher dimensions. This would mean in some sense letting go of karmic ties (investing more and more energy to self-deception) to what we have been authoritatively taught to believe and letting heff jump to a higher level of number theoretical self-comprehension. Greek 'a-letheia' means letting go of active ignorance of feeding energy to attachments and letting world come true by itself... in the baby steps of Tao... ;)<br /><br /><br /><br /><br /><br /><br /><br /> Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-34795802395138133072015-03-26T18:49:01.875-07:002015-03-26T18:49:01.875-07:00I meant all reals. One can of course have also ...<br />I meant all reals. One can of course have also Cauchy sequences which converge to rational.<br />What to my opinion matters is the convergence point of CSs. Not sequences as such: one identifies reals as equivalence class of CSs. <br /><br />I am not a specialist in technical details of functional analysis, Sobolev, etc… I am more interested in the notion number itself from the point of view of physics and cognition. The technique of rigorization (I cannot avoid association with rigor mortis;-) is the task of mathematicians. <br /><br />I did not understand your argument about CSs of negative powers. They are very special case about CS and the limit of power is usually 0, 1, or infinity depending on whether initial point is smaller, equal or larger than 1. For negative x one has limits 0,-infty or cycle hopping between -1 and +1.<br /> <br /><br />Euler's double identify does not make sense to me except formally and I do not see it as a gateway to new math. Either (or both) of the two series involved does not converge for reals. It fails to converge also for p-adics. If x has norm smaller than 1 then 1/x has norm larger than one. If norm is 1 for both, then both series fail to converge.<br /><br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-62634995596750456082015-03-26T13:21:24.158-07:002015-03-26T13:21:24.158-07:00I assume in the above by "(definition of) rea...I assume in the above by "(definition of) real number" you mean: irrational number. :)<br /><br />Cauchy sequenses of just negative exponents is metric space because "the continued fraction expansion of an irrational number defines a homeomorphism from the space of irrationals to the space of all sequences of positive integers". So as model for 'continuous field' Cauchy sequenses of negative powers is not more interesting than field of natural numbers. <br /><br />As you know, the whole of Cauchy space that includes also p-adic positive exponents is much more interesting, and here's some interesting discussion on Euler's doubly infinite identity: <br />http://math.stackexchange.com/questions/669078/eulers-doubly-infinite-geometric-series<br /><br />The second answer relating the identity to Sobolev-like space with Hilbert-space norm goes over my head, but hopefully not yours. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-20134789850504138702015-03-26T04:52:33.372-07:002015-03-26T04:52:33.372-07:00As far as real numbers are considered they are equ...As far as real numbers are considered they are equivalence classes of Cauchy sequences converging to the real number. All information about individual Cauchy sequences disappears in the definition of real number.<br /><br /><br /> One can also consider the sequence of differences of two subsequent points in the sequence and the sequences could be regarded as a real space in obvious inner product if one allows arbitrarily large differences and all kinds of wanderings before the converge to the real number.<br /><br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-12896515965965443132015-03-26T02:32:25.631-07:002015-03-26T02:32:25.631-07:00Yes, I also noticed infinite primes are not depend...Yes, I also noticed infinite primes are not dependent from notion of real line, or even infinite sets of set theory, but can be classified as algebraic type.<br /><br />The book metaphor for Cauchy number space brings to mind sea of white pages, where there can be found and written letters and words of symbolically meaningful interval approximations for algebraic relations. On the other hand, book metaphor does not reveal the "Russian doll" structure of Cauchy exponentiality and the zero state of Euler's Doubly Infinite Identity. Finnish expression for the essence of exponent 'kertoa itsellään' can be translated both as 'multiply by itself' and 'narrate by itself'. :)<br /><br />Intuitively the exponential sea of all possible Cauchy sequenses has fractal dimensional structure, but my technical skills are not enough to tell if it can be assigned exact Hausdorff dimension value and if so, what. (http://en.wikipedia.org/wiki/Hausdorff_dimension)<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-56521581497142554612015-03-25T07:43:11.468-07:002015-03-25T07:43:11.468-07:00Just a short comment about Cauchy sequences. Infi...<br />Just a short comment about Cauchy sequences. Infinite primes are purely algebraic notion as also the infinitude of units obtained as ratios of infinite rationals. No limits are involved as for Cauchy sequences. In p-adic topology their norm is unity for any finite prime and also for smaller infinite primes. Infinity is is in the eye of topologist. <br /><br />This notion of infinity is also different from that of Cantor which also relies on imagine infinite limits obtained by adding 1 again and again: n-->n+1. Now one has explicit formulas for infinite primes: no limiting procedures. <br /><br /><br />The notion generalises to algebraic extensions of rationals allowing also the notion of primeness. <br /><br />My view is that no number field alone is enough: all are needed. Reals and all p-adic number fields are like pages of book glued along common rationals. Algebraic extensions give rise to more<br />pages so that book becomes rather thick. This structure is the natural starting point for physics which would describe also correlates of cognition and intention.<br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-26819514007973162872015-03-25T04:15:16.906-07:002015-03-25T04:15:16.906-07:00As pure mathematician, Norman has problem with the...As pure mathematician, Norman has problem with the notion of infinite sets, and set theory as whole, and hence would not probably accept infinite primes either. Applied math of engineers can be more relaxed, but there is a category mistake if we identify approximations of applied math with rigorous algebraic definitions. As reals are defined, or rather not defined, (what do YOU mean be definition of reals?!) there is a serious logical problem of drawing ontological conclusions ("is") from epistemic limitations of applied math ("should"). <br /><br />Yes, we can and do postulate infinite fields of cauchy sequenses, create such imaginary number spaces with our minds (cf. "eulerian" in the video clip above), but their arithmetics cease to be algebraic as there is in some sense too _much_ information (cf. 1/3 and 0,333...), at least on the "real" side of comma. There is a kind of uncertainty principle _created_ by postulation of cauchy sequences on the "real" side, which would not necessarily emerge in purely algebraic approach. <br /><br />When you say that "every point of real axis becomes infinite-D space", I'm reminded again that the space of cauchy sequenses, called "line" for no good reason, is _said_ to contain inherently multidimensional algebraic roots ("irrationals"), and mostly infinite sequenses that by some theorem are considered transcendentals. Giving up the invalid notion of category mistakes called "real _line_" as 1D-continuum does not mean abandoning the sea of cauchy sequenses and using that number space for what it is good for, it means that multidimensional algebraic relations such as SQR2 etc. are not _identified_ with its approximates in the cauchy sea. The "lagrangian" of an algebraic relation and the "eulerian" observation space of cauchy sea are not identified, but understood for their mutual purposes. <br /><br />I agree with "ability to imagine within confines of logic". Notion of Real _line_ just does not fit the criteria of logic, anymore than square circle. Nothing logically solid can be constructed from an object that has not been logically derived, but by adding inherently multidimensional algebraic relations etc. to rational line, and calling the creation 'line'.<br /><br />I believe you had good and solid motivations for searching "quantum math", and I hope this "disagreement" nourishes that interest. :)Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-65152523840422733912015-03-24T20:24:53.559-07:002015-03-24T20:24:53.559-07:00I see the entire spectrum of number fields from fi...<br />I see the entire spectrum of number fields from finite fields serving as convenient approximation tools to rationals and their algebraic extensions, to reals, p-adic numbers, complex number, quaternions, and octonions as physically very natural structure of structure. To cut off everything after finite fields would destroy the whole TGD so that I am a little bit worried about such attempts;-).<br /><br />There is a lot to destroy by keeping only finite fields. One can go even beyond reals as they are defined. One an introduce an infinite hierarchy of infinite primes/integers/rationals and form a hierarchy of ratios of these number behaving like real units. This implies that every point of real axis becomes infinite-D space which could represent entire Universe: algebraic holography or Brahman= Atman mathematically. <br /><br />This all is of course something totally outside sensory perception and simple book keeping mathematics. What is however amazing that the hierarchy in question has very "physical" structure: supersymmetric arithmetic quantum theory second quantized again and again. Even more amazing, infinite primes analogous to bound states- something which is the Achilles heel of QFTs - emerge naturally so that infinite primes could pave the way to the construction of quantum theories and also TGD. Many-sheeted space-time with its hierarchical structure could directly correspond to it. <br /><br />All this I would lose besides TGD if I would accept the view of Norman about numbers. My view is that one should not take the limitations of human thinking as criterion for what is possible mathematically. We have also the ability to imagine within confines of logic. For me this is the spiritual view as opposed to the "practical" view accepting only discrete structures, a view which -rather ironically - does not work in practice.Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-55330943042804811462015-03-24T10:16:01.085-07:002015-03-24T10:16:01.085-07:00It's good to disagree! Notions of 'continu...It's good to disagree! Notions of 'continuity', continuum' and 'field' and even 'number' are not that clear and require careful philosophical thinking. When we talk about "continuous number fields" of classical physics, are we talking about 'numbers' or just practical approximations (intervals) in some limited form of (all possible) cauchy-sequenses? And how do these approximate intervals of classical fields relate to the operators of quantum fields? Also, these approximate intervals of cauchy sequenses have a strong flavor of 'ontological uncertainty' about them. Or do you mean by "continuous number field" something else than 'all possible cauchy sequenses'? <br /><br />PS: at the end of this lecture Norman comes close to an idea that is close to TDG: (classical) physical finite fields based on some high prime: https://www.youtube.com/watch?v=Y3-wqjV6z5E&list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf&index=24Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-90109135941287623402015-03-22T19:26:52.593-07:002015-03-22T19:26:52.593-07:00I remember that we debated about this and I can on...I remember that we debated about this and I can only say that I disagree. Rationals and their algebraic extensions emerge at the level of cognition and measurement where one always has finite measurement/cognitive resolution. The intersection of cognitive and sensory worlds. At the level of geometry describing perceived world continuous number fields are indispensable for practical description and I think they really exist. <br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-36109512423739291502015-03-22T12:55:33.727-07:002015-03-22T12:55:33.727-07:00Remember Pythagoras Theorem and how it gave birth ...Remember Pythagoras Theorem and how it gave birth to relations that are not ratios of whole numbers, and do not fit the 1D-continuum of rational number _line_? Pythagoras theorem does not say anything about 1D-lengths (they don't exist for it) but 2D _areas_: quadrance1 + quadrance1 = quadrance3 (diagonal of isosceles square). <br /><br />The claim that rational _line_ is not continuous is derived from fundamental category error of claiming that SQR2 is a "gap" missing from rational number line, even though there is nothing that can be reduced to 1D- length about fundamentally 2D object like diagonal of square. The information content of inherently 2D area cannot be contained by 1D number line, if we hold to the ideals of rigour and honesty. <br /><br />More: https://www.youtube.com/watch?v=REeaT2mWj6Y<br /><br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-14794853168280450282015-03-22T05:41:48.421-07:002015-03-22T05:41:48.421-07:00I DO believe I am a God :). Continuity of rationa...I DO believe I am a God :). Continuity of rational number field is not only easy to prove, it is intuitively clear. The ability to believe otherwise is proof of limitless ability of self-deception. The idea that irrationals are "gaps" in relation to perfectly dense rational line is based on false expectation that irrationals can fit 1-dimensional line like natural numbers and rationals. A well known physics blogger once imagined a Hilbertian or n-dimensional number theory space where also irrationals find their natural and exact place.<br />Again, let's remember that the ability to THINK about observables like continuous line requires +1-dimensional "observation space" aka "mind", and let not get confused with definition before we know what we are doing when we think math.<br /><br />Transcendental metanumbers are also very much loved and admired as very special (meta)rationals. Thanks for the Kähler hint for generalization of i. <br /><br />I'm sure we can agree that thinking is a kind of fluid. Here's excellent clip on lagrangian vs. eulerian approaches to fluids like thinking etc: https://www.youtube.com/watch?v=zUaD-GMARrA<br /><br />Lagrangian might get the girl, or punished for obsessive stalking, but eulerian has better change of gnothi seauton.<br /><br /><br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-79952203889587285222015-03-22T04:55:39.122-07:002015-03-22T04:55:39.122-07:00To believe on continuous number fields (both reals...To believe on continuous number fields (both reals and various p-adic number fields and the various algebraic extensions of the latter) is to me like believing in God. <br /><br />There is no way to prove that transcendentals exist (the term is certainly not an accent!) but without the assumption about their existence physics would become an aesthetic nightmare unless you happen to be a practical person loving to do numerics. <br /><br /> Imaginary unit is a represent ion of reflection or rotation by pi, as you wish. In Kahler geometry its action as purely algebraic entity is geometrized/ tensorized so that it does not look anymore so mystic. Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-23737454489393783752015-03-22T04:50:14.133-07:002015-03-22T04:50:14.133-07:00Re Einstein's followers, GRT and relativistic ...Re Einstein's followers, GRT and relativistic field theory(?), could this article on Noether's second(!) theorem and Weyl bear helpful meaning? http://www3.nd.edu/~kbrading/Research/WhichSymmetryStudiesJuly01.pdf Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-78980488396524141092015-03-22T03:11:03.269-07:002015-03-22T03:11:03.269-07:00"Top of the hill" metaphor associates wi..."Top of the hill" metaphor associates with 'Law of diminishing returns', well known for gardeners, and ideal state of growth factors. In that sense quantum jump to higher degree of planck scale makes... sense ;). <br /><br />"Simplest" or deepest mathematical generalization I can think of in this regard is the extra dimensional curve route from 1 to -1 in Euler's Identity, and imaginary-axis (rational axis = 0) of complex plane (by which I mean gaussian rationals, as I'm highly skeptical of "real" numbers). The poetic "coincidence" of square root of -1 being called "imaginary" is remarkable, letting go of karmic rational (or "real", if you insist) valuation and letting imagination flow freely.<br /><br />I just found Weyl's little book on Symmetry, which begins with reference to argument about philosophical or theological debate about left and right between Leibnitz (relativism) and Descarters (reifying objectivism). What kind of symmetry break, if any kind, is the "identity" of square root 1, and extra dimensionality of square root of -1? Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-88467097163369309422015-03-19T08:22:35.636-07:002015-03-19T08:22:35.636-07:00To Anonymous: Seeing observer as active part of th...<br />To Anonymous: Seeing observer as active part of the physical system rather than objective outsider is indeed the whole point of TGD view. I have worked hardly to understand what I might possibly mean with criticality;-). I find it sometimes difficult to understand what I am saying;-). <br /><br />Criticality is intuitively clear notion: you are at saddle or top of the hill where big things can happen with very small perturbation. Returning back to criticality is what biosystems are able to achieve: homeostasis might be seen as this ability to not fall down where dead system would fall immediately. <br /><br />My recent big surprise was that the increase Planck constant seems to happen spontaneously and generates potentially conscious information. Living systems are able to return to smaller h_eff: for instance, magnetic flux tubes connecting reacting biomolecules shorten because of reduction of h_eff and make possible reaction products to find each other. Maybe getting rid of Karma's cycle means letting go and allow the h_eff to increase;-).Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-86213698296208198202015-03-19T04:16:23.804-07:002015-03-19T04:16:23.804-07:00Searching for 'quantum criticality (QC)' I...Searching for 'quantum criticality (QC)' I found this explanation: http://www.physics.rutgers.edu/qcritical/frontier3.htm<br /><br />The notion of 'emergent matter' from QC superposition strongly suggests that dynamic participatory consciousness theory is involved in these observation events, in the form of mathematical imagination. But maybe you could expand on what you mean by 'quantum criticality', are you thinking about some kind of generalization that can be communicated, fits with other pieces of puzzle? Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-44305009937384726232015-03-19T04:01:10.022-07:002015-03-19T04:01:10.022-07:00"A circle is a circle is a circle" appro..."A circle is a circle is a circle" approach presupposes observation independent objectivist metaphysics, which any theory of consciousness can't do, while asking how - where and when etc - thought-observation events of mathematical objects happen. Physicists have age old bad habit of thinking "objectively", what interests in TGD is the challenge to consistently integrate consciousness at all levels of theory-forming. That means also letting go of the role of passive observation (in Greek: theory) and accepting dynamic participation in the drama (Greek: fronesis, sometimes translated as 'practical reason') Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-30821905270369266162015-03-18T23:04:55.593-07:002015-03-18T23:04:55.593-07:00"Seeing from higher dimension" certainly...<br />"Seeing from higher dimension" certainly gives information about geometric object.But it also creates information. Circle is just circle. When its imbedded to 3-D space it can be knotted in arbitrarily complex manner. Same about space-times as surfaces: without imbedding one would have no standard model interactions, just gravitation! Individual without social environment is much less than individual with it! Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-46042748044799297752015-03-18T23:00:49.893-07:002015-03-18T23:00:49.893-07:00To Anomymous: "Square of CP2 radius fixed by ...To Anomymous: "Square of CP2 radius fixed by quantum criticality". I am at all not sure whether I have said this;-). I talk always about ratios only: it is in the spine of every theoretical physicist! The values of dimensional quantities depend on units chosen! I might have said that the ratio of hbarG/R^2 which is dimensionless is be fixed by quantum criticality;-).<br /><br />Imbedding space could indeed have infinite-dimensionality in number theoretic sense: these dimensions would be totally outside of physical measurement since they would correspond to number theoretical anatomy of numbers due to existence of infinite number of real units realised as ratios of infinite primes (and integers). As real numbers all these copies of real number would be identical. Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-43105107734444757572015-03-18T22:54:19.661-07:002015-03-18T22:54:19.661-07:00To Anonymous: "Quantum theory is the square r...To Anonymous: "Quantum theory is the square root of thermodynamics" is what I meant. This is not at all self-evident to me and implies generalisation of the notion of S-matrix replacing it with a collection of M-matrices between positive and negative energy parts of zero energy states. They can be seen "complex" square roots of densities matrices (real diagonal square root of density matrix multiplied by unitary S-matrix). M-matrices would organise into unitary U-matrix between zero energy states. <br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-68175328188067221992015-03-18T21:04:11.088-07:002015-03-18T21:04:11.088-07:00PS: the statement that "thermodynamics is the...PS: the statement that "thermodynamics is the square root of general measurement theory" is of course fully expected and now self-evident implication of generalized notion of +1D "quadrancy" requirement to be able to view any and all mathematical objects. Kiitos ja anteeksi <3Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-39864440462049614112015-03-18T20:50:24.849-07:002015-03-18T20:50:24.849-07:00To make the main point as clearly as I can, all ma...To make the main point as clearly as I can, all mathematical objects are "internal" objects by definition, inside Platonia of mathematical imagination aka "God's Eye". Hence, any mathematical object can be meaningful and fully comprehended only from the imaginary higher dimension of the object. The dynamics of combining empirical "here-and-now" participatory nature of "material" experiencing and holistic God's Eye of mathematical imagination in communicable way is the main inspiration and challenge of TGD, from what I have gathered so far. If fully accepted and rigidly applied, this approach has of course dynamic implications for any general measurement theory.<br />Anonymousnoreply@blogger.com