tag:blogger.com,1999:blog-10614348.post7063066104540777932..comments2024-01-22T11:26:37.599-08:00Comments on TGD diary: Breakthroughs in the number theoretic vision about TGDMatti Pitkänenhttp://www.blogger.com/profile/13512912323574611883noreply@blogger.comBlogger40125tag:blogger.com,1999:blog-10614348.post-82446662167963244762015-05-14T19:19:37.413-07:002015-05-14T19:19:37.413-07:00
Continuing….
b) What is the computer like struc...<br /><br />Continuing….<br /><br />b) What is the computer like structure now? Turing computer is 1-D time-like line. This quantum computer is superposition of 4-D space-time surfaces with the basic computational operations located along it as partonic 2-surfaces defining the algebraic operations and connected by fermion lines representing signals. Very similar to ordinary computer.<br /><br /><br />c) One should understand the quantum counterparts for the basic rules of manipulation. x,/,+, and - are the most familiar example.<br /><br />*The basic rules correspond physically to generalized Feynman/twistor diagrams representing sequences of algebraic manipulations in the Yangian of super-symplectic algebra. Sequences correspond now to collections of partonic 2-surfaces defining<br />vertices of generalized twistor diagrams.<br /><br />*3- vertices correspond to product and co-product represented as stringy Noether charges. Geometrically the vertex - analog of algebraic operation - is a partonic 2-surface at with incoming and outgoing light-like 3-surfaces meet - like vertex of Feynman diagram. There is also co-product vertex not encountered in simple algebraic systems, it is time reversed variant of vertex. Fusion instead of annihilation.<br /><br />*There is a huge symmetry as in ordinary computations too. All computation sequences connecting same collections A and B of objects produce the same scattering amplitude. This generalises the duality symmetry of hadronic string models. This is really gigantic simplification and the results in twistor program suggest that something similar is obtained there. This implication was so gigantic that I gave up the idea for years.<br /><br />d) One should understand the analogs for the mathematical axioms. What are the fundamental rules of manipulation? <br /><br />*The classical computation/deduction would obey deterministic rules at vertices. The quantal formulation cannot be deterministic for the simple reason that one has quantum non-determinism (weak form of NMP allowing also good and evil) . The quantum rules obey the format that God used when communicating with Adam and Eve: do anything else but do not the break conservation laws. Classical rules would list all the allowed possibilities and this leads to difficulties as Goedel demonstrated. I think that chess players follow the anti-axiomatics.<br /><br />I have the feeling that anti-axiomatics could give a more powerful approach to computation and deduction and allow a new manner to approach to the problematics of axiomatisations. Note however that the infinite hierarchy of mostly infinite integers could make possible a generalisation of Godel numbering for statements/computations.<br /><br />e) The laws of physics take care that the anti-axioms are obeyed. Quite concretely:<br /><br />*Preferred extremal property of Kaehler action and Kaeler-Dirac action plus conservation laws for charges associated with super-symplectic and other generalised conformal symmetries would define the rules not broken in vertices.<br /><br />*At the fermion lines connecting the vertices the propagator would be determined by the boundary part of Kahler-Dirac action. K-D equation for spinors and consistency consistency conditions from Kahler action (strong form of holography) would dictate what happens to fermionic oscillator operators defining the analog of quantum Boolean algebra as super-symplectic algebra.<br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-45333456659480894342015-05-14T19:17:39.736-07:002015-05-14T19:17:39.736-07:00Some comments about quantum Boolean thinking and...Some comments about quantum Boolean thinking and computation as I see it to happen at fundamental level.<br /><br />a) One should understand how Boolean statements A-->B are represented. Or more generally, how a computation like procedure leading from a collection A of math objects collection B of math objects takes place. Recall that in computations the objects coming in and out are bit sequences. Now one have computation like process. --> is expected to correspond to the arrow of time.<br /><br />If fermionic oscllator operators generate Boolean basis, zero energy ontology is necessity to realize rules as rules connecting statements realized as bit sequences. Positive energy ontology would allow only statements. Collection A is at the lower passive boundary of CD and B at the upper active one. As a matter fact, it is a quantum superpositions of Bs, which is there! In the quantum jump selecting single B at the active boundary, A is replaced with a superposition of A:s: self dies and re-incarnates and generates negentropy. Q-computation halts.<br /><br />That both a and b cannot be known precisely is a quantal limitation to what can be known: philosopher would talk about epistemology here. The different pairs (a,b) in superposition over b:s are analogous to different implications of a. Thinker is doomed to always live in a quantum cognitive dust and never be quite sure of.<br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-647383664392653152015-05-14T18:19:26.981-07:002015-05-14T18:19:26.981-07:00
To Anonymous: You should calm down and stop talki...<br />To Anonymous: You should calm down and stop talking total nonsense. <br /><br /> You are unable to realize that things can exist although we cannot know perfectly what they are. What we can know is that real number is in some segment, we can narrow down this segment endlessly but never know exactly.<br /><br />But we have also something else than mere numerical computation: we have the conscious intelligence. It cannot be computerised or axiomatised but and most importantly, it is able to discover new truths. <br /><br /> In mathematics communication requires also learning: just watching some videos and becoming a fan of some character making strong nonsense claims is not enough. Also mathematical intuition is something very difficult to teach: some of us have it, others do not. <br /><br />Just as some people are able to compose marvellous music. It seems that we must just accept this. I am not musically especially talented but I enjoy the music of the great composers and experience the miracle again and again: I do not declare war against this music. <br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-19816535991368192212015-05-14T14:24:38.548-07:002015-05-14T14:24:38.548-07:00for example, ur little toy problem: The irrational...for example, ur little toy problem: The irrational numbers, with the metric defined by , where is the first index for which the continued fraction expansions of a and b differ (this is a complete metric space). <br /><br />the iterated map that gives rise to the continted fraction expansion of a real number .. well, it's related to the riemann zeta function, see the Wikipedia page<br />Edit<br /><br />Continued fractions also play a role in the study of dynamical systems, where they tie together the Farey fractions which are seen in the Mandelbrot set with Minkowski's question mark function and the modular group Gamma.<br /><br />The backwards shift operator for continued fractions is the map h(x) = 1/x − ⌊1/x⌋ called the Gauss map, which lops off digits of a continued fraction expansion: h([0; a1, a2, a3, …]) = [0; a2, a3, …]. The transfer operator of this map is called the Gauss–Kuzmin–Wirsing operator. The distribution of the digits in continued fractions is given by the zero'th eigenvector of this operator, and is called the Gauss–Kuzmin distribution.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-55626364773013022622015-05-14T14:17:14.883-07:002015-05-14T14:17:14.883-07:00anonymous, who is this we you speak of? get some b...anonymous, who is this we you speak of? get some books and stop watching videos. do some analysis. read up on Baire spaces if u are so caught up on notions of continuity<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-87401680942989933762015-05-14T06:19:19.324-07:002015-05-14T06:19:19.324-07:00Matti, read again. Your latest comment has very li...Matti, read again. Your latest comment has very little to do with what has been said and meant.<br /><br />Again:<br />The authorities (e.g. wiki) keep on saying that real numbers follow the basic rules of arithmetics. Obviously that claim is not true. <br /><br />The definition of 'real number' refers to infinite process ("least upper bound"), not to finite computable segment. Finite segments by definition are NOT "real numbers", they are something else. Some say "approximations", but also an approximation is NOT a real number. It is an approximation.<br /><br />If we want to keep math communicable, we must respect definitions and do our best to define as clearly as we can. The notion of "real number" is as it is usually used, horribly vague and poorly defined. <br /><br />That is of course a big if, and communication is not necessarily priority. The word "sin" has been mentioned in context of incommunicado.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-60697827632625691402015-05-14T05:44:47.097-07:002015-05-14T05:44:47.097-07:00As for relation of Boolean operators V and its ver...As for relation of Boolean operators V and its vertical inverse, and human cognition, propositional logic is far from universal; some natural languages behave closer to propositional logic, some not in the slightest.<br /><br />Leveled horizontal operators of ordinality "<" and ">" (less-more) are much more naturally universal in human cognition, I'm not aware of natural language without more-less relation, which is also naturally hyperfinite process closely related to whole-part relation. The arrows giving directions are also more-less relations: go more in the direction the arrow is pointing, less in the opposite direction. These operators predate all other written language and needless to say, propositional logic. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-6806975284129236602015-05-14T05:29:14.421-07:002015-05-14T05:29:14.421-07:00
This is my last comment in this fruitless discuss...<br />This is my last comment in this fruitless discussion. I have done pedagogical efforts in order to clarify the basics but in vain. I however strongly encourage to continue serious studies of basics before declaring a war against modern mathematics and physics. <br /><br />I have tried to explain that finite calculational accuracy is the point: it is not possible to calculate real number exactly in finite time and no-one has been claiming anything like that. The idea of giving up all the mathematics since Newton is just just because we cannot calculate with infinite precision is complete idiotism. <br /><br />And I am still unable to see what is wrong with Cauchy sequences: here I tried to concretise them in terms of decimal representation in order to give the idea what they are about but it seems that it did not help. <br /><br />The generalisation of real numbers rather than refusing to admit their existence, is the correct direction to proceed and I have been working with this problem with strong physical motivations. Fusion of reals and p-adics to adelic structures also at space-time level, hierarchy of infinite primes defining infinite hierarchy of second quantisation for an arithmetic quantum field theory, even construction of arithmetics of Hilbert spaces, replacement of real point with infinitely structured point realizing number theoretic Brahman = Atman/algebraic holography. These are interesting lines to proceed rather than a return to cave. <br /><br />Strange that someone blames me for blindly believing academic authorities;-). Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-60704959068239154582015-05-14T05:08:36.642-07:002015-05-14T05:08:36.642-07:00I don't know what all will be lost if we hones...I don't know what all will be lost if we honestly admit that "real numbers" do not behave arithmetically, at least in the boolean sense, and though many say that "real numbers satisfy the usual rules of arithmetic", obviously they don't. Any child can see that emperor has no clothes in that respect. <br /><br />Even though reals don't, AFAIK the p-adic side does satisfy the usual rules of arithmetic, at least in some areas. Worth a more careful look. Cauchy intervals within intervals is perfectly OK and very rich and interesting structure, and repeating patterns of rationals is amazing and beautiful thing worth deeper study, e.g. how do lengths of repeating patterns behave in various bases, on both sides of rationals? When repeating patterns are plotted inside Cauchy intervals, I see a wave forms at very basic level of number theory.<br /><br />In OP Matti does see the light, saying that mathematical structures follow from number theory itself, trying to deduce from "physics" does not work. <br /><br />So here is relatively simple question: what is the _minimum_ of number theory you need to observe quantum observables? I'm very much in doubt that e.g. "canonical identification" is needed (but rather, confuses and messes things up).<br /><br />I'm not IT, but even I know that computers don't do real numbers or any other infinite processes. Floating points, lazy algorithms, etc. get the job done. Finite accuracy works in boolean way, but no, we can't say that "finite accuracy" strings are "real numbers".<br /><br />If we insist that problem of mathematical continuum "has been solved with "least upper bound" completion of algebraic (e.g. roots) and algorithmic (pi, e) realside infinite processes", there is a cost: the solution is not boolean, rules of arithmetic don't work regardless of how much some people pretend that they work and push the problems under the mattress and out of text books. It's not about politics, it's just math.<br /><br />There are other options, we can admit that the problem of mathematical continuum remains unsolved, or poorly understood and formulated, and keep on thinking and questioning, instead of blindly believing the academic authorities that say that real numbers follow the basic rules of arithmetics. Eppur si muove.<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-48668009155991456202015-05-13T19:43:06.189-07:002015-05-13T19:43:06.189-07:00Correction to the last paragraph: "prime powe...Correction to the last paragraph: "prime powers such that" should read "powers of primes of extension such that"Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-46572581231392652222015-05-13T19:35:28.287-07:002015-05-13T19:35:28.287-07:00I hope that the importance of the notion of finite...<br />I hope that the importance of the notion of finite accuracy became clear. It certainly does not look like a beautiful notion in its recent formulation.<br /><br />Finite accuracy is the counterpart for finite measurement resolution/cognitive resolution and is anotion, which is often not considered explicitly in math text books. It is fundamental in physics but the problem is how to formulate it elegantly.<br /><br /><br />It is also encountered in in the adelic vision based on strong form of holography. One can in principle deduce scattering amplitudes in an algebraic extension of rationals (this for the parameters such as momenta appearing in them). One can algebraically continue this expression to all number fields.<br /><br />But what if one cannot calculate the amplitudes exactly in the algebraic extension? There is no problem in real topology using ordinary continuity. But continuation to p-adic topologies is difficult since even a smallest change in rational number in real sense can mean very big change in p-adic sense. It seems that one cannot avoid canonical identification or some of its variants if one wants to assign to a real amplitude a p-adic amplitude in continuous manner.<br /><br />Finite accuracy is also a deep physical concept: fermions at string world sheets are Boolean cognitive representation of space-time geometry. But in finite accuracy representing 4-D object using data assignable to a collection of 2-D objects rather than 0-dimensional objects (points) as in the usual naive discretization, which is not consistent with symmetries. Discrete set of points is replaced with discrete collection of 2-surfaces labelled by parameters in algebraic extension of rationals. The larger the density of strings, the better the representation. This is strong form or holography is implied by strong form of general coordinate invariance: a completely unexpected connection to Einstein's great principle.<br /><br />This leads also to an elegant realization of number theoretical universality and hierarchy of inclusions of hyper-finite factors as a realization of finite measurement resolution. Also evolution as increase of complexity of algebraic extension of rationals pops up labelled by integers n =h_eff/h, which are products of ramified primes characterizing the sub-algebra of super-symplectic algebra acting as gauge conformal gauge symmetries. Effective Planck constant has purely number theoretic meaning as a measure for the complexity of algebraic extension!<br /><br />Ramification is also number theoretic correlate of quantum criticality! Rational prime decomposes to product of prime powers such that some of them are higher than first powers: analog for multiple root in polynomial - criticality! For me this looks amazingly beautiful.Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-18268344832486783042015-05-13T19:09:59.068-07:002015-05-13T19:09:59.068-07:00I do not want to use my time to ponder whether the...<br /><br />I do not want to use my time to ponder whether there is some conspiracy of power greedy mathematicians and physicists against civilised world. I just want to make clear some elementary things about Cauchy sequences in hope that they remove the feeling of black magic.<br /><br />a) Cauchy sequences are used by everyone who can sum, subtract, multiply and divide decimal numbers. These are special kind of Cauchy sequences in which n:th term is the number in approximation using n decamical digits. One can use also binary or binary and much more general sequences. <br /><br />These particular sequences are however convenient since all arithmetic operations are for rational numbers. <br /><br />b) In numerics on introduces decimal/pinary/... cutoff. This makes sense if the functions are continuous and the operations for them respect continuity. <br /><br />c) If one wants to formulate this axiomatically one can say that one works in the category of continuous functions. Absolutely no crime against mankind is involved. Everything is finite and numbers of operations are finite but approximate with an error that can be estimated. Computers<br />use routinely binary Cauchy sequences with a success. <br /><br />c) One could of course throw away real numbers as a conspiracy against mankind and decides to use only rationals (I do not know whether algebraic numbers are also doomed to to be part of conspiracy) . This leads to difficulties. <br /><br />One must define differential equations etc as difference equations by specifying the size of different: single equation would be replaced by infinite number of them- one for each accuracy. Calculus and most of what has been achieved since Newton would be lost since no-one wants to write Newton's mechanics or Maxwell's theory or quantum field theory using difference equations: it would incredibly clumsy.<br /><br />There would no exponent function, no Gaussian, no pi, no special functions. Things become in practice impossible. Most of number theory is lost: forget Riemann Zeta, forget p-adic numbers, ... Analytic calculations absolutely central in all science would become impossible. <br /><br />Reals represent the transcendent, spirituality, going beyond what we can represent by counting with fingers. Recent science is deeply spiritual and in very concrete manner but the materialistic dogma prevents us from seeing this. <br /><br /><br /><br /><br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-77898325394809998422015-05-13T17:57:19.635-07:002015-05-13T17:57:19.635-07:00See https://app.box.com/files/0/f/0/1/f_3007347386...See https://app.box.com/files/0/f/0/1/f_30073473869 for some interesting ways that numbers 7 and 24 just pop out of some rather elementary integrals<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-79206396829705176622015-05-13T17:15:22.475-07:002015-05-13T17:15:22.475-07:00https://statistics.stanford.edu/sites/default/file...https://statistics.stanford.edu/sites/default/files/2001-01.pdf is also very interesting "Unitary correlations and the Feijer kernel" is very interesting. <br /><br />your statement of "randomly picked real numbers in base 2" is ill-posed , are you just trying to reinvent some some "floating-point" representation ?<br /><br />I'm saying its something more "weird" like a qubit.<br /><br />nowhere in the thing u described do I see any sort of room for time, much less deterministic or nondeterministic notions of system states.<br /><br />Fuzzy you say? nonsense, classical mechanics chaos, u take a Poincaire section of the flow and each orbit punctures that section at a particular point, repeat this many times (by dynamical system evaluation of integrals etc with whatever conditions) and you end up with a process whose "output" can take on a discrete number of values relative to the reference measure, well, basically, one can easily prove things such as Cantor, Levy 'dust' etc and go into fractal dimension which takes on any real value, so your argument is really just wildly stabbing in some direction or another, trying to talk big it seems... <br /><br />this boolean aspect of any set whatsoever requires the concept of indicator function, I(A)=1 if A in omega, 0 if its not in omega. integral. do you speak it? apparently not.<br /><br />you sound like an IT guy... are you an IT guy? ;-)<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-18450227435357020452015-05-13T16:46:56.215-07:002015-05-13T16:46:56.215-07:00Stephen, I have checked the link and now rechecked...Stephen, I have checked the link and now rechecked, and found this gem, under Proposition 5.1:<br /><br />"Remark. From the equality, the infinite sum of squares converges to an _almost surely_ finite limit."<br /><br />I humbly suggest that linguistic expression "almost surely" is pretty sure tell that the math in question has moved from the confines of boolean values or "boolean cognition" to somewhere else. ;)<br /><br />Or maybe you can show that randomly picked real numbers in base 2, let's say 0,000... and 0,000... , without assuming that they are rationals, do really sum up in boolean terms, ie. the sum is either a number beginning with 0 or 1, but not both or neither or something even more weird like "qubit"? <br /><br />If you can't, we can't honestly say that "boolean cognition" is behind claimed mathematical structures such as "real number line", and that the proof theory which is used to postulate such structures is a boolean proof theory. And same goes for set theory.<br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-10273506710681801392015-05-13T13:50:36.805-07:002015-05-13T13:50:36.805-07:00see http://www.encyclopediaofmath.org/index.php/Al...see http://www.encyclopediaofmath.org/index.php/Algebra_of_sets for instance, its also called a σ-field σ-algebra, they can be unconditional, or conditional, upon all sorts of other spaces<br /><br />go back and read the link I posted<br /><br />Theorem 4.1<br /><br />and quit babbling your wordy nonsense, anonymous<br /><br />random matrix theory alone cannot do it, the primes must enter in some way, and this spectral signature is universal for any and all unitary processes, apparently, if "U" know how to look , amirite M8? :)<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-24965589301024226422015-05-13T11:21:39.717-07:002015-05-13T11:21:39.717-07:00Now that we have hopefully left the dogmatic trenc...Now that we have hopefully left the dogmatic trenches or warfare and politics and are taking our baby steps, the structure of "real line" (-field) is as such an interesting object of study. In binary the sum of two points on real line (almost all of which are non-algebraic) is at least in most cases not:<br /><br />0<br />1<br />both 0 and 1<br />neither 0 nor 1<br /><br />but avoids all these extreme positions. ;)<br /><br />So the notion of qubit seems now inherently related with adding elements of non-Boolean "completion" of rational line.<br /><br /><br />Also, it would seem that the bigger the number base, the smaller the room for vague. What would be the situation with base of Biggest Mersenne Known? Is there some kind or structural relation with "modern interpretation" of consistent histories and the questions it claims to allow and exclude?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-23894518817301407622015-05-13T11:20:47.837-07:002015-05-13T11:20:47.837-07:00anonymous, you are talking out of your ass, stop w...anonymous, you are talking out of your ass, stop wasting Matti's timeAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-61492006648522830492015-05-13T10:47:15.152-07:002015-05-13T10:47:15.152-07:00PS: IFF mathematics and theoretical physics claim ...PS: IFF mathematics and theoretical physics claim to rule and conquer and control either openly or by implication, of course I revel, as any honest self-loving man would. :)<br /><br />Thusly experiencing does not reduce to nor is limited by mathematics and theoretical physics, not even TGD. 2D representation of music is not same as picking a guitar in your lap and playing music that never was and never will be.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-26304954548017578432015-05-13T09:53:33.198-07:002015-05-13T09:53:33.198-07:00No, it's not about politics, the question is v...No, it's not about politics, the question is very simple. Are "real numbers" numbers in boolean sense or combinatorical noise?<br /><br />Present real numbers in the form of hindu-arabic cauchy sequenses in base 2. Pick a pair of such real numbers and add them up. Do you get a discrete result that starts with either one or zero?<br /><br />AFAIK, no, and if not otherwise proven, hence "real numbers" cannot be said to be numbers in boolean sense. <br /><br />And as real numbers cannot be said to be definable in boolean sense, that goes also for real complex plain, complex manifolds etc. <br /><br />I don't know what the hell those things are, but they are certainly not "boolean thought and cognition", presumably meaning numbers that can be expressed as either 1 or 0.<br /><br />You can twist and dance around and play your politics and war games as much as you want - and it is sorry to see you try so hard not to admit what is so obvious -, but it's just math. This is just math, and if we choose to play boolean game, we play it by boolean rules, otherwise we would cheat. <br /><br />Also in math you need to learn to walk before you can run. When you try to run before you have learned to take baby steps, you make a glorious short dash and then end up with your face in mud. And IMHO that summarizes the state of contemporary academic mathematics.<br /><br /><br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-78610584996900196372015-05-13T09:28:39.522-07:002015-05-13T09:28:39.522-07:00Some comments about axiomatics. This is of a tech...Some comments about axiomatics. This is of a technical tool for mathematicians. The unavoidable<br />bureaucracy, one might say. <br /><br />Theoretical physicist who is really constructing a theory is working hardly to find minimal number of basic assumptions, which might be true. Trying to find a set of assumptions which forms a coherent whole, is internally consistent, and predicts as much as possible. <br /><br /> This a process of trial and error and there is no point if declaring wars against mathematics or any other branch of science. This activity could not be farther from mechanical deduction from a fixed set of axioms, which requires just algorithms and in principle can be carried out by computer.<br /><br />Theoretical physics has indeed led to powerful insight about mathematics: consider only Witten's work. Recently Nima Arkani Hamed and colleagues (mathematicians) have done similar job. This is by working as visionary: mathematicians take care of details when the dust has settled: this can take centuries. <br /><br />Theoretician of course hopes that some day this structure can be axiomatized and even average professor can use the rules to calculate. Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-16706835926115572322015-05-13T09:17:49.368-07:002015-05-13T09:17:49.368-07:00
I think it is time to stop the discussion since y...<br />I think it is time to stop the discussion since you are seem to be in rebel against mathematics and theoretical physics : windmills would be less dangerous enemy. People who argue that entire branches of science are totally wrong are usually called crackpots: I do not like that word because it is so much misused. <br /><br />I have met many people who have declared war against some branch of eel-established science: one was logician who had the fix ide that special relativity contains logical errors: he had ended up with this conclusion by interpreting special relativity in Newtonian framework. I tried to explain but in vain.<br /><br />You seems to misunderstand the idea of Cauchy sequence in a manner which remains for me black magic. <br /><br />*Cauchy sequences provide a formulation of continuity: I fail to see how you manage to assign to them Boolean interpretation. <br /><br />*You talk also about Cauchy numbers: by looking at Wikipedia you see that it is a dimensionless parameter used in hydrodynamics. I honesty admit that I fail to see the connection to the notion of continuity. <br /><br />*Also the addition of Cauchy sequences is to my best knowledge completely irrelevant for the notion of limit. <br /><br /><br /><br /><br /> <br /><br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-78379005180711584892015-05-13T07:48:55.892-07:002015-05-13T07:48:55.892-07:00The only real argument presented above is "us...The only real argument presented above is "useful", which is purely emotional and rhetoric argument. "Useful" is exactly what is meant by "black magic", in contrast to the magic and beauty of rigorous mathematical deduction. Argument from authority is to refer what others think and do, instead of establishing a well defined theory of real numbers based on foundational axioms and boolean chain of deductions and proves. If you want to play the game of boolean mathematices, play it honestly and don't cheat at every corner.<br /><br />In the boolean context, it seemingly takes a complex self-deception to lose sight of the simple fact that there is indeed significant arithmetic symmetry break. As the basic carry rules of basic arithmetics state, Cauchy numbers in form ...nnn,p + ...nnn,p make arithmetic sense in the boolean context (ie, they can be added), but irrational Cauchy numbers n,nnn... + n,nnn... do not add up but remain vague and non-boolean. <br /><br />Unless, of course, you can prove that irrational Cauchy sequenses do add up in finite life time calculation in discrete non-vague manner and can be given a boolean value. Go ahead, give it try. Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-10614348.post-51291212609910048842015-05-13T06:22:11.954-07:002015-05-13T06:22:11.954-07:00
It is a pity that your comments are getting incre...<br />It is a pity that your comments are getting increasingly emotional and rhetoric. Mathematics probably looks black magic for anyone, who does not understand it. There is a lot of mathematics which looks black magic to me, but by working hardly I can get rid of this expression. I do<br />not want to blame mathematics for my own limitations.<br /><br /> I comment those parts of your comment, which have some content.<br /><br />*As mathematical structures Boolean algebras extend without difficulty to continuous case: consider set theoretic realisation. p-Adics are completely well-defined notion and 2-adic number can be seen as infinite sequence if binary digits. <br /><br />What is important that pinay digits are ordered: the higher the digit, the lower its significance. This is the basic idea of metric topology and makes it possible to work with continuum. Mathematics without continuum reduces to mere combinatorics. <br /><br />*Completions of rationals to various number fields<br />are a standard mathematical concept and extremely successful: if someone wants to believe that this notion is mathematically flawed, he can do so but certainly remains a lonely believer.<br /><br />In a any case, mathematicians discovered for centuries about that the notions of nearness, limit, continuity, Cauchy sequence and smoothness are extremely useful notions and allow to conclude the outcome of infinite processes. Very useful notions also for a philosopher of mathematics and highly recommended;-) <br /><br />*Conscious logical thought- at least that part of which is representable physically - is discrete. Discreteness is one of the basic aspects of cognition - I formulate this in terms of cognitive resolution implying in turn finite measurement resolution. <br /><br />*We should be be careful to not project the limitations of our cognition to the physical and mathematical realities. Materialists do this: they try to identify consciousness and physical reality and end up to a dead end. <br /><br />Matpitka@luukku.comhttp://tgdtheory.fi/noreply@blogger.comtag:blogger.com,1999:blog-10614348.post-63272536039900421832015-05-13T05:27:15.219-07:002015-05-13T05:27:15.219-07:00Then please, do your best to discuss the content. ...Then please, do your best to discuss the content. Even if the only content is your confusion. :)<br /><br />"Infinite process" by definition means that the process (e.g. algorithm) continues ad infinitum, does not get finitely completed. I hope this clarifies what is illogical about "completions of infinite processes" in terms of binary logic. Based on this we can conclude that boolean bivalent either-or valuing applies strictly only to finite phenomena, not to of infinite processes and approximations e.g. by some processes of limitations. Hence, any area of mathematics that deals with infinite processes in any way is not 'boolean' in the strict sense. Boolean thought and cognition can apply only to finite, bivalent processes that involve a boolean identity. <br /><br /><br />"Axioms" are today used with variety of meanings. What was criticized was "ad hoc" axioms used to postulate what a mathematical physicist _wants_ to please himself with when rigorous deductive logic otherwise fails to produce the object of desire, not the foundational level axioms e.g. in Euclidean sense. Mathematics is a field of applied magic, and such use of ad hoc axioms is black magic.<br /><br />"Our observations about things are fuzzy, not things." This is purely a statement of personal metaphysical belief, faith in "things" having "inherent ontology". That statement has nothing to do with logic, math and science. <br /><br />Notions of 'length', 'area', 'volume', 'position', 'momentum' etc. are strictly speaking neither 'observations' nor 'things', but in this context just abstract mathematical notions, which in the mathematics we are used to do not behave in boolean way.Anonymousnoreply@blogger.com