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Infinite tetration and superroot of infinitesimal
#61
Sorry Ivars for my joke concerning the SR.
I completely agree with you. You see, ... I was at the other side. It was a sarcastic joke of mine (ask Henryk, who was surprised in a similar situation, concerning other issues). But, sometimes, life is a kind of Moebius strip and it is difficult to go to "the other side". We shall discuss of that, later, but maybe not in this Forum.
I don't know the precise Andrew's interpretation of that formula, by which I suppose that you (both) mean: h(e^(Pi/2)) = h(4.810477381..) = {-i, +i} . Plus and minus "i", I suppose.

I think about that and, then, I tell you.

GFR
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#62
Ivars Wrote:1) What would be its orientation relative to I ? ( most likely 90 degrees )
If your "imaginary infinitesimal" is equivalent to , then it would be parallel, but that depends on which interpretation you use for infinitesimals. Since infinitesimals could be added to any real number, you could consider the hyper-real field to be 2D, or 1D, its up to you, there is no right or wrong way. I think its useless to think about things like this, since you can visualize it however you want.
Ivars Wrote:2) In which space we would see this orientation?
*sigh*, in whatever space you want.
Ivars Wrote:3) If multiplying by I means rotation by 90 degress in complex plane, would multiplying by dI mean infinitesimal rotation in what plane?
This is only in the standard interpretation of complex numbers, you could construct a skew coordinate system for visualizing complex number, in which case the amount of rotation would depend on each point.
Ivars Wrote:4) What is self root of dI? Or what tetration would result in dI?
Whatever it is, the conversion of this number to the reals will be zero. Since any possible solution would be "zero" in the reals, the distinction of different scales in hyper-reals wouldn't make any difference. Even if it is a number like , the distinction of scales only holds for hyper-real numbers, not real numbers, which would make this value convert to zero. So basically, anyone who doesn't use hyper-reals would say its just 0.

Andrew Robbins
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#63
Dear Ivars!

Actually, I think that we should refer to a nice heart-like plot, element of a series made by Gottfried and re-visited by Andrews, where h, indicated as t = b^t is shown for b real and t complex, if I recall it correctly. The plot shows, in point t = + i (coordinates 0, +i, in the complex "t" plane, equivalent to complex number t = 0 + i*t) that the corresponding "base" parameter is b = 4. 810477381.. . Let call "r" this specific individual real number and try and solve the equation:
r ^ x = x, with, as we said, r = e^(Pi/2) = 4. 810477381..
Obviously, since we have: e^((Pi/2)*i) = i , but also e^((Pi/2)*(-i)) = -i , and since r is a real number > 1 (and therefore we must also have: r^(+oo )= +oo), the solutions of that equation are: x = {+oo, -i, +i}.

In fact, we have:
r ^ (-i) = -i
r ^ (+i) = +i
r ^ (+oo) = +oo

In fact, the Gottfried heart-like plot for real bases b > e^(1/e) shows an intersection with the imaginary axis (Real t = 0) in two symmetrical points, where the base parameter must be, in my opinion, the same: b = r.

In all the three above-mentioned solution cases, y = r ^ y ( or t = r ^ t) is equivalent to y = r # +oo, the infinite tetrate of r. We could then write that, in y = b # x , we might have:
y = b # +oo = -i )
y = b # +oo = +i ) for b = r
y = b # +oo = +oo )

The third line can be well accepted by everybody. For the first two lines there is a big problem of understanding (an infinite tetration, for b > e^(1/e), can be imaginary?). This is the ... terrific fact that is attracting all our attention. A mysterious (just to quote Henryk) link is connecting the infinity with the imaginary unit, as a condequence of ... tetration. I presume that this is also what is really bothering you! I shall try to post a note for explaining what I really mean, but I don't expect to be supported by everybody. We shall see!

To understand all this is very difficult. Nevertheless, please, let us try to avoid to differentiate constants!
(; -> )>>>

GFR
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#64
So does it really diverge to real infinity (even if it can be accepted by everybody)?

Ivars
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#65
I think that 3^3^3^3...^3 [infinite times, priority to the right] indeed diverges to real infinity. Let us try to start the calculations. Then, we immediately see if it doesn't. The surprise is the appearance of also two unitary imaginary "heights". Good grief !!!!!

GFR
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#66
If we miss one dimension (in my opinion, along hyperdimensions created or causing tetration of (e^pi/2) = -I,) we have no way to calculate properly h(3), because You will never get to infinity by direct calculation. It is just an axiom, that divergent series reach REAL infinity-another symbol. Axiom needed and well grounded, but still possible to improve or change.

Euler. Borel etc . summations, just a simple Symbolic or formal geometric series application to infinite series gives results which seems not to be sums, but are true since they lead to correct results obtained via usage of them. And there has to be logic, not magic, why it is so, but initial definitions have to made less definite.

I think You know that in non-standard analysis things are treated differently, the results are the same. But non-standard does not go far enough. So it does not harm anyone to have new axioms. What harms is that only 1000 people in the world may be understand string theory, tetration etc. -theories become more complex, which is because we are chasing reality not trying to create it. But that is not what knowledge is about- to have more complex theories- or it should not be about that. Knowledge has to be simple, inputs may wary.



I can not promise stopping differentiating "constants" like i, pi, because they are just symbols and any symbols can be combined with another symbol ( e.g. differentiation) and attributed a different symbol as an outcome. The less definitions You have in the beginning, the more chance there to reach something new, and not necesserily changing true results, but just changing dogmas. If you start with axioms and definitions which are never to be challenged, You will always end within the logical limits they impose. Which is all what dogma is about.

The true constants, like 1/2 , 1/3 etc are not differentiable.
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#67
GFR Wrote:I think that 3^3^3^3...^3 [infinite times, priority to the right] indeed diverges to real infinity. Let us try to start the calculations. Then, we immediately see if it doesn't. The surprise is the appearance of also two unitary imaginary "heights". Good grief !!!!!

GFR


If we add missing, or "spin" dimension I mentioned before, it would be just 2 opposite rotations, having seemingly? the same (or infinitesimally different) projection on real plane XY.
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#68
Hi Ivars!

Do you really mean that Pi, i and e are just symbols an that 1/2 and 1/3 are true constants? How can we detect if a constant is ... true? If I say, for example, that the measure of an angle is alpha = Pi, shall we immediately understand that it is not true ? I think that I am missing something.

My old sarcastic reference to the SR was caused by the fact that I read a paper of very famous Soviet mathematician, former president of an important Academy, who said that a consistent mathematical theory of complex numbers is impossible, because they are just symbolical useless contradictory ... approximations. The example he gave was the fact that in the physical reality (SR...?) they only appear as complex conjugate entities. He also added that it was impossible to distinguish between +i and -i.

Well, ... I go and drink a cup of coffee. Then, ... we shall see.

GFR
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#69
Hi GFR!

It is on a gut feeling level +mathematics must explain processes possible . I think that 1/2 is a true dynamic constant, while i is continuously variable and pi most likely discrete (but may be not).
I think +i and -i are different, but the difference is dynamic, sorry to say that, so that there is space where they mean rotation in diffferent directions and are not interchangeable.

The fact that angle can truly equal pi is by itself amazing since pi is a limit of discrete approximation of circle via polygons, from both sides, geometrically - in plane. if You try to find out pi by measuring the lenght of circle in 3D You end up with the fact that pi definitely may vary from 1/2 pi to 3/2 pi depending on the angle in 3D between the measurable and measuring disc, size of measuring circle etc. When the measuring circle gets infinitely small, than value of pi is achieved, but the cost is that You do not know anymore the angle at which You measure since all give the same result. When you know the angle, measuring disc is of finite sizes, and pi varies depending on that angle.

See the very interesting link, please: http://cvt.com.sapo.pt/tp/tp.htm

That thinking I have does not come from mathematics, its more like .... coming from thinking that things should be simpler, but probably, little unorthodox.

Any dynamic balance could divide any 2 processes in 1/2 - even if we now nothing about the quantity involved nor about the scale it is happening in. e.g if we would like to weigh the whole Universe and assume it is infinitely divisible , than only way would be to balance one half of it against another (do not ask me how) . Despite the fact that Universe might be infinite, might consist of things we can not define, we could imagine that such balance can be achieved. Then , if we take half we have balanced,we again can balance it ...etc. Zeno paradox. In that sense we can balance anything in any scale with the same and always achieve 1/2, if that what we balance is infinitely divisible. So 1/2 in each (hyperdimensional) scale is , means the same, so it is a true constant. If quantum world is not infinitely divisible, we can not do it if we do not add subquantum part- but that is nothing special since as we can not directly measure it, only via interactions with quantum world, we may add it as well- for a sake of symbol.

I was wandering why complex plane and real axis are looked upon as something separeate- in my opinion, real axis has imaginary rotating ( so that left is not right) extension perpendicular to it and every function just crosses real axis on it rotating way via infinite hyperdimenions . So basically You have kind of a cylinder, but each projection of functions on real axis may contain 3 parts- 2 rotation in opposite directions in that hyperspace , 1 truly real - like 3 parts of h odd, h even and x^1/x of h(x) when x< e^-e. That is probably the only place where they separate so obvously, the other being h(x) where x> e^(1/e).
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#70
Ivars Wrote:I think that 1/2 is a true dynamic constant, while i is continuously variable and pi most likely discrete (but may be not).
What?
Ivars Wrote:If You try to find out pi by measuring the lenght of circle in 3D You end up with the fact that pi definitely may vary from 1/2 pi to 3/2 pi depending on the angle in 3D between the measurable and measuring disc, size of measuring circle etc.
.......
See the very interesting link, please: http://cvt.com.sapo.pt/tp/tp.htm
Oh, ... I saw the CVT simulations. The problem there is to achieve a CVT (Continuous Variable Transmission). We should not confuse, in that example, the angular velocity of the "gear" with the measurement of the circumference length. One of the two circles has the double diameter of the other one, in both cases being Pi*d, where d is the diameter. If the gear rotates 1, 2, or 3 times, depending on the inclination of the planet gear with the plane of the ring, the results of the measurements are the same. See the conclusions on the same Web page. However, thanks! It is interesting. But, it doesn't mean that Pi is ... variable!

But this is Engineering and not Mathematics, don't confuse the two "worlds". In the Eng. world, for instance, we will not be ever able to know "exactly" number Pi, because we shall need an "infinite computer" for storing all its decimal figures. We shall know it approximately only. But this doesn't mean that engineers think that the decimal figures of Pi are random. Of course not! It is an axample of ... deterministic process, which can univocally produced by a special kind of Turing machine. In the Math. world, Pi is a very precise constant with all the characteristics of a transcendent number. It cannot be representes as a fraction between two whole numbers, but who cares. No mathematician would dare to say that it is variabe, or discrete (?!?). We must use serial developments, with "acceptable" rests. But this is (the mathematical) life!

Ivars Wrote:I was wandering why complex plane and real axis are looked upon as something separeate ... in my opinion, real axis ... (is a) ... kind of a cylinder ... (containing) ... 3 parts- 2 rotation in opposite directions in that hyperspace , 1 truly real - like 3 parts of h odd, h even and x^1/x of h(x) when x< e^-e.

The first part of your reasoning (orthogonality between the representation of the real and imaginary part of a complex number)depends on a convention adopted in the theory of complex numbers. The two parts are uncorrelated. So, ... orthogonal (perpendicular). But it is only a convention. The second part, I think, is influenced by the physical theory of collapsed dimensions supposed existing around the the traditional three (or ... four) classical physical dimensions.

Nope, nay, no-no-no!! The problem of the "yellow zone" or "transition area" can be analyzed, mathematically (don't mix up math with engineering and/or physics), as follows.

(a) We defined, for any base b, y = b # 1 = b-tetra-1 = b;

(b) We then concluded that we must have:
y = b # 0 = b-tetra-0 = 1, and:
y = b # (-1) = b-tetra-(-1) = 0
and, in conclusion, points (0,1) and (-1, 0) must belong to the tetrational smooth function, if any.

(c ) We also know that the values of y = b # x are determined by the log/exp relations from x = -2 to x -> +oo, for all integer x.

(d) Now, the problem is to ask to ourselves if we have or not a "line" interlinking the two (0,1) and (-1, 0) points. Or, else, if the behaviour of y = b # x between the two (0,1) and (-1, 0) points is "dusty". In the first case, we must also have "lines" interlinking any two adjacent discrete points and, therefore, we are authorized to study this global "line" and see if it is smooth and analytic. In the second case, we cannot do that and we would be reduced to use something like "fuzzy mathematics" or probability theory.

(e) In the first hypothesis (continuous line), the curves of the real values passing through all the odd or the even points defined by y = b # n are just "envelopes" of the actual almost sinusoidal real line oscillating around a mid-value. The upper and lower envelopes may very well be continuous, but the almost periodicity of the "evelopped lines" describing y = b # x will be always 2 (odd/even alternations). No discontinuous jumps between max and min y are detectable. "Tetratio non facit saltus".

(e, .... sorry, I mean: f) In the second hypothesis (dusty distribution), we have not only sudden jumps between any dx variations but also fuzzy point distributions, which would suggest us to give up and do other things.

I am for hypothesis (e). "I mean the first one, the second (e) is (f)".
[Corrected on 2008-01-28, at 14h26, CET]

GFR
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