Wednesday, August 24, 2011

How to understand transcendental numbers in terms of infinite integers?

Santeri Satama made in my blog a very interesting question about transcendental numbers. The reformulation of Santeri's question could be "How can one know that given number defined as a limit of rational number is genuinely algebraic or transcendental?". I answered to the question and since it inspired a long sequence of speculations during my morning walk on the sands of Tullinniemi I decided to expand my hasty answer to a blog posting.

The basic outcome was the proposal that by bringing TGD based view about infinity based on infinite primes, integers, and rationals one could regard transcendental numbers as algebraic numbers by allowing genuinely infinite numbers in their definition.

  1. In the definition of any transcendental as a limit of algebraic number (root of a polynomial and rational in special case) in which integer n approaches infinity one can replace n with any infinite integer. The transcendental would be an algebraic number in this generalized sense. Among other things this might allow polynomials with degree given by infinite integer if they have finite number of terms. Also mathematics would be generalized number theory, not only physics!

  2. Each infinite integer would give a different variant of the transcendental: these variants would have different number theoretic anatomies but with respect to real norm they would be identical.

  3. This would extend further the generalization of number concept obtained by allowing all infinite rationals which reduce to units in real sense and would further enrich the infinitely rich number theoretic anatomy of real point and also of space-time point. Space-time point would be the Platonia. One could call this number theoretic Brahman=Atman identity or algebraic holography.

1. How can one know that the real number is transcendental?

The difficulty of telling whether given real number defined as a limit of algebraic number boils down to the fact that there is no numerical method for telling whether this kind of number is rational, algebraic, or transcendental. This limitation of numerics would be also a restriction of cognition if p-adic view about it is correct. One can ask several questions. What about infinite-P p-adic numbers: if they make sense could it be possible to cognize also transcendentally? What can we conclude from the very fact that we cognize transcendentals? Transcendentality can be proven for some transcendentals such as π. How this is possible? What distinguishes "knowably transcendentals" like π and e from those, which are able to hide their real number theoretic identity?

  1. Certainly for "knowably transcendentals" there must exist some process revealing their transcendental character. How π and e are proven to be transcendental? What in our mathematical cognition makes this possible? First of all one starts from the definitions of these numbers. e can be defined as the limit of the rational number (1+1/n)n and 2π could be defined as the limit for the length of the circumference of a regular n-side polygon and is a limit of an algebraic number since Pythagoras law is involved in calculating the length of the side. The process of proving "knowable transcendentality" would be a demonstration that these numbers cannot be solutions of any polynomial equation.

  2. Squaring of circle is not possible because π is transcendental. When I search Wikipedia for squaring of circle I find a link to Weierstrass theorem allowing to prove that π and e are transcendentals. In the formulation of Baker this theorem states the following: If α1,...,αn are distinct algebraic numbers then the numbers eα1,...,eαn are linearly independent over algebraic numbers and therefore transcendentals. One says that the extension Q(eα1,...,eαn) of rationals has transcendence degree n over Q. This is something extremely deep and unfortunately I do not know what is the gist of the proof. In any case the proof defines a procedure of demonstrating "knowable transcendentality" for these numbers. The number of these transcendentals is huge but countable and therefore vanishingly small as compared to the uncountable cardinality of all transcendentals.

  3. This theorem allows to prove that π and e are transcendentals. Suppose on the contrary that π is algebraic number. Then also iπ would be algebraic and the previous theorem would imply that e=-1 is transcendental. This is of course a contradiction. Theorem also implies that e is transcendental. But how do we know that e=-1 holds true? Euler deduced this from the connection between exponential and trigonometric functions understood in terms of complex analysis and related number theory. Clearly, rational functions and exponential function and its inverse -logarithm- continued to complex plane are crucial for defining e and π and proving also e=-1. Exponent function and logarithm appear everywhere in mathematics: in group theory for instance. All these considerations suggest that "knowably transcendental" is a very special mathematical property and deserves a careful analysis.

2. Exponentiation and formation of set of subsets as transcendence

What is so special in exponentiation? Why it sends algebraic numbers to "knowably transcendentals". One could try to understand this in terms of exponentiation which for natural numbers has also an interpretation in terms of power set just as product has interpretation in terms of Cartesian product.

  1. In Cantor's approach to the notion of infinite ordinals exponentiation is involved besides sum and product. All three binary operations - sum, product, exponent are expressed set theoretically. Product and sum are "algebraic" operations. Exponentiation is "non-algebraic" binary operation defined in terms of power set (set of subsets). For m and n definining the cardinalities of sets X and Y, mn defines the cardinality of the set YX defining the number of functions assigning to each point of Y a point of X. When X is two-element set (bits 0 and 1) the power set is just the set of all subsets of Y which bit 1 assigned to the subset and 0 with its complement. If X has more than two elements one can speak of decompositions of Y to subsets colored with different colors- one color for each point of X.

  2. The formation of the power set (or of its analog for the number of colors larger than 2) means going to the next level of abstraction: considering instead of set the set of subsets or studying the set of functions from the set. In the case of Boolean algebras this means formation of statements about statements. This could be regarded as the set theoretic view about transcendence.

  3. What is interesting that 2-adic integers would label the elements of the power set of integers (all possible subsets would be allowed, for finite subsets one would obtain just natural numbers) and p-adic numbers the elements in the set formed by coloring integers with p colors. One could thus say that p-adic numbers correspond naturally to the notion of cognition based on power sets and their finite field generalizations.

  4. But can one naively transcend the set theoretic exponent function for natural numbers to that defined in complex plane? Could the "knowably transcendental" property of numbers like e and π reduce to the transcendence in this set theoretic sense? It is difficult to tell since this notion of power applies only to integers m,n rather than to a pair of transcendentals e,π. Concretization of e in terms of sets seems impossible: it is very difficult to imagine what sets with cardinality e and π could be.

3. Infinite primes and transcendence

TGD suggests also a different identification of transcendence not expressible as formation of a power set or its generalizations.

  1. The notion of infinite primes replaces the set theoretic notion of infinity with purely number theoretic one.

    1. The mathematical motivation could be the need to avoid problems like Russell's antinomy. In Cantorian world a given ordinal is identified as the ordered set of all ordinals smaller than it and the set of all ordinals would define an ordinal larger than every ordinal and at the same time member of all ordinals.

    2. The physical motivation for infinite primes is that their construction corresponds to a repeated second quantization of an arithmetic supersymmetric quantum field theory such that the many particle states of the previous level become elementary particles of the new level. At the lowest level finite primes label fermionic and bosonic states. Besides free many-particle states also bound states are obtained and correspond at the first level of the hierarchy to genuinely algebraic roots of irreducible polynomials.

    3. The allowance of infinite rationals which as real numbers reduce to real units implies that the points of real axes have infinitely rich number theoretical anatomy. Space-time point would become the Platonia. One could speak of number theoretic Brahman=Atman identity or algebraic holography. The great vision is that the World of Classical Worlds has a mathematical representation in terms of the number theoretical anatomy of space-time point.

  2. Transcendence in purely number theoretic sense could mean a transition to a higher level in the hierarchy of infinite primes. The scale of new infinity defined as the product of all prime at the previous level of hierarchy would be infinitely larger than the previous one. Quantization would correspond to abstraction and transcendence.

This idea inspires some questions.

  1. Could infinite integers allow the reduction of transcendentals to algebraic numbers when understood in general enough sense. Could real algebraic numbers be reduced to infinite rationals with finite real values (for complex algebraic numbers this is certainly not the case)? If so, then all real numbers would be rationals identified as ratios of possibly infinite integers and having finite value as real numbers? This turns out to be too strong a statement. The statement that all real numbers can be represented as finite or infinite algebraic numbers looks however sensible and would reduce mathematics to generalized number theory by reducing limiting procedure involved with the transition from rationals to reals to algebraic transcendence. This applies also to p-adic numbers.

  2. p-Adic cognition for finite values of prime p does not explain why we have the notions of π and e and more generally, that of transcendental number. Could the replacement of finite-p p-adic number fields with infinite-P p-adic number fields allow us to understand our own mathematical cognition? Could the infinite-P p-adic number fields or at least integers and corresponding space-time sheets make possible mathematical cognition able to deduce analytic formulas in which transcendentals and transcendental functions appear making it possible to leave the extremely restricted realm of numerics and enter the realm of mathematics? Lie group theory would represent a basic example of this transcendental aspect of cognition. Maybe this framework might allow to understand why we can have the notion of transcendental number!

4. Identification of real transcendentals as infinite algebraic numbers with finite value as real numbers

The following observations suggests that it could be possible to reduce transcendentals to generalized algebraic numbers in the framework provided by infinite primes. This would mean that not only physics but also mathematics (or at least "physical mathematics") could be seen as generalized number theory.

  1. In the definition of any transcendental as an n→ ∞ limit of algebraic number (root of a polynomial and rational in special case), one can replace n with any infinite integer if n appears as an argument of a function having well defined value at this limit. If n appears as the number of summands or factors of product, the replacement does not make sense. For instance, an algebraic number could be defined as a limit of Taylor series by solving the polynomial equation defining it. The replacement of the upper limit of the series with infinite integer does not however make sense. Only transcendentals (and possibly also some algebraic numbers) allowing a representation as n→ ∞ limit with n appearing as argument of expression involving a finite number of terms can have representation as infinite algebraic number. The rule would be simple.

    Transcendentals or algebraic numbers allowing an identification as infinite algebraic number must correspond to a term of a sequence with a fixed number of terms rather than sum of series or infinite product.

  2. Each infinite integer gives a different variant of the transcendental: these variants would have different number theoretic anatomies but with respect to the real norm they would be identical.

  3. The heuristic guess is that any genuine algebraic number has an expression as Taylor series obtained by writing the solution of the polynomial equation as Tarylor expansion. If so, algebraic numbers must be introduced in the standard manner and do not allow a representation as infinite rationals. Only transcendentals would allow a representation as infinite rationals or infinite algebraic numbers. The infinite variety of representation in terms of infinite integers would enormously expand the number theoretical anatomy of the real point. Do all transcendentals allow an expression containing a finite number of terms and N appearing as argument? Or is this the defining property of only "knowably transcendentals"?

One can consider some examples to illustrate the situation.

  1. The transcendental π could be defined as πN=-iN(eiπ/N-1), where eiπ/N is N:th root of unity for infinite integer N and as a real number real unit. In real sense the limit however gives π. There are of course very many definitions of π as limits of algebraic numbers and each gives rise to infinite variety of number theoretic anatomies of π.

  2. One can also consider the roots exp(i2π n/N) of the algebraic equation xN=1 for infinite integer N. One might define the roots as limits of Taylor series for the exponent function but it does not make sense to define the limit when the cutoff for the Taylor series approaches some infinite integer. These roots would have similar multiplicative structure as finite roots of unity with pn:th roots with p running over primes defining the generating roots. The presence of Nth roots of unity f or infinite N would further enrich the infinitely rich number theoretic anatomy of real point and therefore of space-time points.

  3. There would be infinite variety of Neper numbers identified as eN=(1+1/N)N, N any infinite integer. Their number theoretic anatomies would be different but as real numbers they would be identical.

To conclude, the talk about infinite primes might sound weird in the ears of a layman but mathematicians do not lose their peace of mind when they here the word "infinity". The notion of infinity is relative. For instance, infinite integers are completely finite in p-adic sense. One can also imagine completely "real-worldish" realizations for infinite integers (say as states of repeatedly second quantized arithmetic quantum field theory and this realization might provide completely new insights about how to undestand bound states in ordinary QFT).

For details and background see the chapter Physics as Generalized Number Theory: Infinite Primes of "Physics as Generalized Number Theory" or to the article How infinite primes relate to other views about mathematical infinity?.


Santeri Satama said...

Thanks Matti,

I can't follow your language very far, but in the simplest terms that I can understand, known transcentental such as π is a *transcendental rational*, expressing the ratio between "straight" or "linear" euclidean and "curved" or "circular" non-euclidean geometrics - with einsteinian relativity and c and all that jazz. The transcendental ratio between "infinite" open ended number line and "finite" ouroboros snake eating it's own tail.

In this sense, transcendentals do not exist just as subset of reals or completion of rationals on the one dimensional real line, they are by definition also Something Else. Known transcendentals as ratio of two knowns regarded from a higher dimension (e.g. from the higher dimension subsuming both euclidean and non-euclidean geometry and algebra), unknown transcendentals as ratios between known and unknown or two unknowns (platonic) forms.

This view about π as a higher dimensionsional number theoretic rational - as it was originally and has allways been - strongly suggests that it radiates not only to real - and complex - fields but also to the p- or P-adic fields, allready at this state of mathematical cognition. Well, not only "strongly suggests" but proves by itself as a known transcendental. :)

Now to the question, what happens to the completeness axiom (, the notion of continuity and especially the Archimedean property ( of real line with this approach to transcendentals and infine integrals and primes?


Second, my obsession with RH and the question "why real part ½ and not something else?" made me grab a quitar and play some flageolets. Octave in the middle and "second octave" quantization at 3/4 (and/or 1/4) of the string (finite string from -1 to 1). Mapping infinite primes to transcendental ratios sounds full of promise of platonic harmonics, in the eerie tunes of flageolets. said...

The modification replaces real point with an infinite-dimensional space of equivalent real points. In real analysis real norm matters so that the modification is totally invisible in everyday business. Nothing happens to the basic axioms and Earth continues along its orbit as before.

If one takes seriously the algebraic holography and TGD the replacement of the space-time points with incredibly infinite-dimensional structure would represent world of classical worlds - in TGD sense- at single space-time point. This is of course a rough formulation.

The idea that mathematical point might not be something totally structureless has been long in the air, since the times of Grothendienck. This would be give explicit realization for this idea.

The understanding of transcendentals as higher level algebraics is attractive but does not have any implications for ordinary calculus. The implications would be visible only at the level of quantum physics if the vision about physics as generalized number theory is accepted. The identification of infinite integers as quantum states -or something comparable- might fix the physics completely and be extremely predictive hypothesis!

Santeri Satama said...

Here's the proof of Lindemann-Weierstrass Theorem.

What struck me first was that the proof is very sensitive to division by either p-1 or p!. said...

Thank you. Unfortunately the formulas of the document are invisible ("Missing argument for superscript") so that one can make only guesses about the auxialiary polynomials f used (the formula after "Let p be a (sufficiently large) prime..".

Santeri Satama said...

Nagging, aren't they - the missing arguments for what constitues a sufficiently large prime... :)

This blogpost contains a nice 3D-visualization of Eulers formula:

"Our jewel"-post links to wikipedia article about unit circle (, stating e.g.:

"Every compact Lie group G of dimension > 0 has a subgroup isomorphic to the circle group. That means that, thinking in terms of symmetry, a compact symmetry group acting continuously can be expected to have one-parameter circle subgroups acting; the consequences in physical systems are seen for example at rotational invariance, and spontaneous symmetry breaking."

And leads to the other theme of the day, standard model and the Higg's mechanism.

Nice to notice the very close interconnectness of all the recent issues in your blog. :)

And to keep on nagging at you, putting all these ideas together in a form with minimum jargon and approachable to layman - teachers job - would be very nice, and very likely also very rewarding for you. Much less frustrating than wating and hoping of getting attention from those at the top of the academic hierarchy, and even more importantly, helpfull to clarify your own thoughts. In real terms, one understands only as much as one can teach to somebody else. said...

I do not have too high thoughts about outcomes of popularization: impression about understanding is different from understanding. It is known that hypnotic suggestion can make a person to have an impression about what a non-entity like perfectly round square is like!

Popularization is of course important- many scientists becomes rich in this manner;-). I must however say that I am unable to sit down and begin to write popular essays about a given topic: it is extremely boring.

Apart from reactions to things like LHC all that I am able to do is completely spontaneous. These crazy ideas come and demand to get documented and I am too benevolent and too crazy to say "No". I am not a real finnish academic scientist (J. Karjalainen expresses my feelings in his song: "Mä en oo oikea cowboy";-)) having detailed research plans for next five years telling what will be discovered at what minute. I am not fully predictable and therefore I fully deserve my fate;-).

Ulla said...

"impression about understanding is different from understanding" is another way to say - a lonely millionair? All I have to do alone?

Could this understanding get condensed or breathed from the air? No struggle, no fight?

Or where shall we begin the understanding? Is it at all worth trying?

Can one take responsibility for another persons understanding?

Why the hell, Matti? Need I say more? COOPERATION to a small degree is what is needed to write a book. Fate? Rejection? Fate, what about my fate? Can you even think that what I have told you is true? Do you even want a book? One more year, no more. One year left, Matti. Then I will write it without cooperation. I think my offer is fair.

Also a composer need some cooperation, in form of an orchestra!

Spontanity and unpredictability is accepted, but I have hard to think it would be in so high degree :) More like digging into the Earth for a while, I guess. It is your human right. Eagerness... I like it much :)

I fully recognize the problem, and there are lots of questions that will never get an answer, but there are also lots of questions that has got an answer. Enough to say the worldview has changed.

What Santeri brought up here I think is very important. Yesterday I looked for Lubos post on pathological math. The borders and boundaries, constraints...

Maybe these "pathologies" also can tell us why there is matter in the Universe? Why not everything was annihilated?

Santeri Satama said...

"I must however say that I am unable to sit down and begin to write popular essays about a given topic: it is extremely boring."

Dear Matti,

I fully understand your feeling and the thrill you get from what you are used to do. The practical idea I have for learning-teaching experience is quite different: casual and informal dialogue on a certain discussion forum with mostly young open-minded and curious people, many with 'Great Experiences' and friendly and supportive atmosphere. Something not unsimilar to what was practiced in the original Academy of Plato.

Not lecturing from ex cathedra, but participating in a learning experience for the benefit of all participants.

Santeri Satama said...

Ulla's question 'what is the matter with universe' brought to my mind something that amuses me to no end.

Someone on the discussion forum I mentioned was wondering about the halo-effect and quoted this explanation:

"I don't think any one has anything to worry about. I work as a Physics research fellow, we believe this is a real effect to do with electron motion within established neural pathways inside the brain. You are seeing the true nature of sub atomic partial behavior. Stand and stair at the halo (Don't worry it wont damage your eyes) you should see that it consists of light and dark bands spreading out perpendicular from the light source, if you look at the halo closely, near the light source you may even see rings around the light. Unfortunately mankind’s understanding of sub atomic behavior still boils down to statistics. Currently we have no solid physical description of what you are seeing, but it can be considered akin to the ***famous double slip experiment***. If you would like to no more google search '' and watch the Feynman Lectures, its a real eye opener to the people unexposed to modern physics."

Yes indeed, The Famous Quantum-Freudian Double sliP Experiment!!! Teeheehee :D :D :D said...

I like informal dialogues with some wine. And no powerpoints.

My belief really is that when time is ripe, new big idea goes through irresistibly. What I can do is to make all that I can before that. My thinking is heavily mathematical and the only channel for these idea to reach audience are the brains which have enough mathematical skills to understand what I say and continue to work independently. Remember what was written in the portal to the academy of Plato;-).

Writing of popular books does not help much and also this requires high specialization to what one is writing about. The popular books about string models worth of mentioning are written by string theorists themselves.

Beethoven's music has brought joy to the life of millions of people. This is certainly not due to popularization or to co-operative skills of Ludvig! He was regarded as genuinely crazy at his time.

Santeri Satama said...

"Remember what was written in the portal to the academy of Plato;-)"

I do :^), and in return you should remember what Plato said and showed about anamnesis and even more, what you yourself say about whole Platonia present in each spacetime poing. Do you yourself take your theory seriously enough to be happy and open about it and in it?

And I would like to remind also that my own nearly total lack of mathematical skills has not been a real problem, au contraire my purely intuitive and associative approach with dare devil poetic license has been more than once quite fruitfull spark for your creative ideas towards math, physics and DNA as generalized number theory.

And as Ulla points out, often cases too much exposure and conditioning into rigid classical math thinking patterns can be harder to delearn from than for a fresh and open mind to re-member the Platonia already present there in all its richness.

Ulla said...

I have said quite terrible things to this guy, but nothing helps. He knows best, and of course it IS his theory, and he is the one that decides. But if I take only a few issues, biology most... it would be a wonderful story. I AM a teacher.

But, as Matti points out, it would need an eye of a physicist too, but nobody wants to look at TGD, so it must be the author himself. I am no physicist, but I try to learn, sic!

Sorry, Matti, for everything. "You talk like nobody else", I know.

Look at the Rossi article, a new comment :) said...

Ulla and Santeri:

I appreciate your willingness to popularize TGD but I know the best what to do. I must to do just what the others cannot: to develop TGD further to make it more understandable and to abstract the essentials.

Average colleague is what Kea would call "dude" and I can well understand what Kea means with this word but to my colleagues I must be able communicate my message;-). My luck is that there are also youngsters who have not yet reached the "dude" stage.

Delearning is important. I am doing this all the time. But to delearn one must first learn;-).

Social interaction -not as small talk but genuine communication- is decisive element and it is boring to repeat what had been said about web revolution. In any case, without web I would probably have started to write popular books long ago;-). Any question - such as he question about transcendentals by Santeri, email discussion, browsing of Wikipedia, listening a lecture of Nima, or reading article of Witten (exercise in art of abstracting the essentials from a sea of non-understood)- can stimulate a train of thoughts if one has inspiring context as background. This "job" of mine is not for a hermite;-).

Ulla said...

My comment about a blind and deaf Matti was meant only as a wake-up signal, not a delearn-signal. You was so strongly bound to your 'train-of-thoughts'. And I know you can be 'deaf' then.

You still does not convince me. So we have different opinions. Let it be so. Today you prooved maybe you are right.

The breakthrough was achieved by physicists Andrew Cleland and John Martinis from the University of California at Santa Barbara. Their machine consisted of a tiny metal paddle made of semiconductor material just visible to the naked eye. By supercooling the device to just above absolute zero (minus 273C), then raising its energy by a “single quantum”, they made it vibrate by getting thicker and thinner at a frequency of some 6 billion times a second, producing a detectable electric current. They even managed to get it to vibrate in two energy states at once, both a lot and a little – a phenomenon allowed only by the rules of quantum mechanics. said...

Dear Ulla,

I have no no obsession or need to convince anyone: TGD can do this itself. I am just doing my job as a mouthpiece helping intelligent brains meet some very interesting ideas.

I can rather objectively say that TGD is consistent with what we know about physics and also explains the anomalies that I know: there are really many of them. Mathematical theory of TGD is extremely beautiful and no doubt can revolutionize mathematics. At the same time it integrates nicely with the frontiers of modern mathematics. My passion has been to understand so that I have been lucky;-).

Experimentalists are gradually convincing also the main stream theorists that there is no separation to classical and quantum realms. For two decades ago it was still completely crackpottish to claim that macroscopic quantum effects might be real and the crux of what it is to be living.

There is however a rather remarkable group who seem to have missed the train and not even realized that this has happened: theoretical particle physicists expressing themselves in blogs;-). And of course my respected colleagues in Finland;-). We must just wait them patiently.

Ulla said...

I am quite convinced, not in everything of course, it was about the book, dumbo.

Look here what I found:

"David Potter's excellent discussion of electron diffraction (EW, April) brings out the central paradox of wave-particle duality in electronics far better than any previous article. It is well researched and entertaining. I was unaware of many of the original problems of suppression, "... Born and friends conducted a satiric seminar, making fun of de Broglie's idea... Schroedinger's ... wave mechanics alarmed his wife... she feared for his sanity." Heisenberg did not help by deriving the 'uncertainty' principle using a bogus treatment of an optical gamma ray microscope, which is impossible... Sir Karl Popper ... and his acolytes created the quagmire by defining a scientific discovery as a speculative theory or a 'falsifiable proposition.' Archimedes proofs of the criteria for buoyancy... would be denied by Popper's idea, as well as every mathematical theorem."

From 'good old days'. Also Galileo was a crackpot :) There are always besserwissers.

Now I have got THE IDEA, and would need to discuss it (not here). Still bad situation? A bottle of wine isn't enough? I want no phone calls :)

Thanks Santeri :)

Ulla said...
gravity and pi

remember gravity also includes the centrifugal force! This is why this number is a little bigger?

I think this is a good way to explain TGD physics - maybe a sixth path into it? said...

Dear Ulla,

as a nasty physics dude I like to kill crackpot ideas;-).

g=about pi^2 is example of the numerical co-incidences having no deeper physical content. g is dimensional quantity, acceleratation with dimension m^2/s. m and s are purely accidental choices of units from the point of view of fundamental physics. When one uses inch instead of meter situation changes totally.

Matti Pitkänen said...

Opps, g has dimensions m/s^2, not m^2/s

Ulla said...

the second is also defined from Earth circunference linked to meter (the length of a seconds pendulum was never actually adopted as the definition of the meter) - gives a shorter meter than we use.

g is dependent on radius. As we change our system of units, the numerical value of g changes with them. So moon has smaller and Jupiter much bigger g. Note Earth gravitaionally is not a sphere.

definition of the meter as one ten-millionth of the distance from the North Pole to the equator through Paris.

Pi^2 is very close to ten
g=9.869604401... including centrifugal force? Remember how important number 10 is in TGD.

I wonder if arbitrary choises are BETTER than geometric?

Matti Pitkänen said...

Again the problem is that g is not dimensionless number. If number is to have a number theoretic meaning it must be dimensionless. It cannot represent dimensionful quantity. Fine structure constant is the standard example about dimensionless number.

Ulla said...

No, g is a result.

calculated from the mass, the polar and equatorial radii, and the angular velocity of the Earth.
g=GM/r^2 - (2pi/86400s)^2 cos(45,5degrees) x r_equator = 9,807 m/s^2 = 1kg or 35.30394 (km/h)/s

mass is seen in the radius? Compare the newly found diamond planet. What g would it have?

Compare Schwartzchild's metric, which is a solution of Einstein's equations for the metric around a spherical massive object.

If the gravitational potential energy of the photon is exactly equal to the photon energy then
Note that this condition is independent of the frequency, and for a given mass M establishes a critical radius. said...

Of course. The value of g has been known since Newton.

My whole point is that g as acceleration is not dimensionless number. Its value is not g= 9.869.. but 9.869... m/s^2. You get something quite different when your replace meter as a unit with inch for instance. or second with millisecond or one and thousand other choices.

Santeri Satama said...


In your opinion, does every space-point having all of Platonia as inner structure make them all the same? Or what about localization, antrhopocentric, gaia-centric, etc. math.

Or basically, what is your counterargument against solipsism? If you need one?

Jason said...

OMG, what is wrong with you people? said...

To Santeri,

the numbers in the space of all units in real sense (but not number theoretically) would appear as multipliers of real numbers. This space would contain all infinite rationals reducing to unit in real sense and also algebraics with the same property. This space would be the same for all reals- or let us be careful- for generalized reals.

One can take Neper number as example about transcdendental represented as rational in the set of infinite integers. Neper number is defined as real number using the formula

e= (1+1/n)^n, n goes to infinity.

This is the magnitude of e in the proposed interpretation.

e itself corresponds to infinite number of numbers

e_N= (1+1/N)^N,

where N is ANY infinite integer and there are many of them;-). It is infinite RATIONAL with real magnitude e.

This result echoes in number theoretical manner to what one obtains in the case of surreals where given number has infinite number of numbers of same magnitude obtained by adding to it infinitesimal.

The ratio of two representatives e_N1 and e_N2 defines real unit as ratio of two infinite rationals.

This infinitude of representations of reals as algebraics has no practical implications for everyday mathematics. The number theoretic anatomy of octonionic infinite rationals might have deep physical interpretation and I have indeed proposed this in terms of standad model quantum numbers. If so then even these e_N:s might have some physical interpretation. We must patiently wait for some millenia;-)

What is to me very satisfying that it would be possible to understand transcendentals as generalized rationals if p-adic number fields are generalized to allow infinite primes. We have the notions of e and pi and transcendental. These notions must have correlates in the "real world". Infinite-P p-adic numbers would explain why we have these notions: finite-p p-adic cognition definitely cannot explain them. The future mathematics must explain also mathematician. Mathematicians cannot remain an outsider anymore;-). Physicists realized this for century ago but have not been able to get much out of this realization;-). said...

To Santeri:

Solipsism is the danger of idealistic world views in which only the subjective existence is accepted. Only my thoughts exists or can be proven to exist so that I am left to where Descartes ended up (but is even Descartes just a creation of my thoughts....?;-)).

Materialism is the mirror image of idealism. The mirror image of the above would seem to be that it is not possible to prove from experiment or logically that I do exist at all! I however directly experience my existence so that there must be something wrong with materialism!

These academic looking problems are problems of monistic philosophies. The philosophy in TGD Universe is however tripartistic. Geometric existence defining classical worlds, quantum existence at state space level defining what physicist might call objective realities, and subjective existence as quantum jumps between quantum states. The third aspect is what is really new as compared to the existing views that I have encountered. It seems that it is really difficult to realize that consciousness could be in the change although basic facts about perception demonstrate this.

There are no problems with solipsims. There is no fear about cosmic loneliness since the hierarchy of selves is basic prediction. The world is full of all kinds of Gods and semigods, sinners and saints. Selves can fuse and communicate - both classically and quantally. Selves can re-create entire sub-universes again and again and make them better place to live just by staying conscious.

And the very act of state function reduction decomposes the world to interior and outside and gives direct perception of the external world and solipsistic feelings evaporate. Say goodbye to solipsistic loneliness and enter TGD Universe full of social activitites;-)!

Ulla said...

Big News!!!
Say goodbye to solipsistic loneliness and enter TGD Universe full of social activitites;-)!

The socialism shows rather odd features, may I say.

Yesterday I made my Journey. Forgiveness, the Past is no longer active. said...

I thought that I made made a fatal typo leading to socialism association: I thought that I had said "socialize" while meaning "sosialise" (the small talk like communications in fashion to day when people build in Facebook illusions that they call social networks). There was no such typo however so that I can solipsistically congratulate myself.

Ulla said...

The interior world (self) and the exterior world communicate through Ego (illusion of self, as understood from others, parents, culture, moral etc.) and they are both needed (perceptions, measurements), but the external communication can be substituted for by 'as-if'-loops (Damasio) but this substitution is never as forceful as 'real', and they hardly give the same critique either.

I think the Ego is one of the most misunderstood features. The Ego is illusionary for good and for bad, but it is needed. It makes the 'jumps' (the force behind), seldom the 'self' do that (no free will), for social humans at least. Self is free? You used the analogy free-jail, but you can be in jail by free will too.

I must laugh. Maybe I should congrat too :)

Ulla said...