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Laws and Orders - GFR - 04/23/2008

Dear Friends!

I reiterate how glad I am for having the possibility to exchange views with all of you at this "fronteerland" of Mathematics. Nevertheless, I am more and more feeling the need of putting some order in my brain, concerning the various laws that we choose for describing the hyperoperation hierarchy. Perhaps it is only a personal problem, for which I badly need your kind assistance.

Before submitting these few lines to the Ockham's (or Occam's) Razor, which can sometimes be also a boomerang, I should like to share with you my concern, starting from the basic right/wrong stipulations. In fact, in general, I don't think it is possible to say that an assessment such as x = x + 1 is wrong, if we don't perfectly define the context and the limits of the debate.

As a matter of fact, and with all my apologies for writing such basic "Banalitaeten", we could have the following different situations:

a) - Expression x = x+1 is an equality, i.e. a statement stipulating that the two members of the algebraic equality have identical value, despite the values assumed by the letter symbols. In this case, obviously, x = x+1 is indeed absolutely wrong (come on ... for any x's ... ??!!??). But this (... sorry, Henryk, nothing ... personal), I presume, is also the case of an expression such as x = e^x, where y = e^x and y = x have not even one real common point (intersection) for any real x. Therefore, also in this second example, the expression would be wrong, if it is an equality. An example of a well known correct (right) equality is for instance (x+a)^2 = x^2 + 2ax + a^2.

b) - Expression x = x+1 is an equation, i.e. a statement that can be transformed into an equality for some (at least one) real or complex values assumed by variables, defined as the "unknown variables". If x = x+1 is an equation, then it is easy to verify that it is satisfied by both "non standard" solutions x = {-oo, +oo}. In fact, y = x and y = x+1 can be represented in a cartesian plot by two parallel lines, "crossing" themselves in one of the improper points of the plane, with the mentioned infinite coordinates. In this situation, we cannot say that x = x+1 is wrong, but that, since it implies that it must be x-1 = x+1, it can be true only for infinite values of x. The case of x = a^x is even more evident, because we know that this equation, with a as parameter, is studied for defining the infinite towers and has a double solution x = - W(-ln a)ln a, with W representing the two real branches of the Lambert function. However, in both the mentioned cases, it is not appropriate to say if an equation is right or wrong, but if it has or not any solution. An old good example of equation is x^2 +2ax + c = 0. The problem, in the case of x = x+1, is that, while we can easily accept that x-1 = x = x+1 (with x: plus/minus infinite), we should forbid to move the infinity from one to the other member of the equality, otherwise we shall get: x-x = 1 (!!!???!!!) which I have some problems to accept. The situation would be different if we consider x = a+x, with x as a parameter and a as the unknown variable, to be determined. The solution of the equation would then be a = 0. Traffic problems, I suppose. ... !

c) - Expression x = x+1 is a programming instruction (e.g. in Basic or in Fortran). In this case it is neither right nor wrong, but it has just to be executed (take x, add 1, put it in x). I goes without saying, but it goes much better by saying it! It is not a synthetical a priory judgement (V. I. Kant) but a simple a posteriori decision (Befehl ist Befehl...!). Probably, it is not even mathematics. But, this is life! With my pocket calcolator, I should better write it as: x+1 "store" x. Much better.

d) - Expression x = x+1 is a recursive statement. This would mean that x = 1+x = 1+(1+x) = 1+(1+(1+x)) = 1+(1+(1+(1+x))) = ... and so on. Actually, by trying not to create more than necessary confusion, we might think that it could mean that 1+x = x = 1*(+oo) = +oo. Or, better, we could say that: IF x = x+1, THEN x = +oo. A very famous equivalent situation is found in expression x = e^x, intended as a recoursive statement. In fact it would mean that: x = e^x = e^(e^x) = e^(e^(e^x)) = ... and so on. In other words: IF x = e^x, THEN x = e#(+oo). This result is similar to what described in situation "b", with the only difference that here the negative infinity has no apparent meaning.

Well, in our discussions, we should try to avoid crossing frontiers, without showing the ... passport. I mean that we are not authorized to start a speach under situation "d" and then stop if we find a strange equation, which is not ... an equality. Something like: "Ops, sorry! I found x = x+1 and I stop because it is clearly wrong". Probably you should (I mean: ... stop), but please, let's discuss a little bit before. We are walking in a wild territory!! Think of a honest man who, put in front of an instruction like x = x+1, would say: "I don't do this kind of things!"

Concerning the laws and orders to be applied in the study of the hyperoperations (I am convinced to have launched this name, associated to the G. hierarchy, together with KAR, but I don't insist) I should like to observe that we don't only have the mother law (ML), but also something else. I shall indicate by "s" the hyperoperation rank and by "r" the number of iterations, whenever appropriate.

GML - The Grand-mother Law, considered by all people approaching this matter for the first time, as a first definition of the hierarchy, after applying the "priority to the right" traffic rules to the hyperops operators. It sounds like:
a[s]<r>a = a[s+1](r+1).

ML - The Mother Law, discovered (or ... assumed) by most of thew the Researchers, during a deeper hyperops stydy. This can be written as follows:
a[s+1](x+1) = a[s](a[s+1]x).

DL - The Daughter Law (this is a ... new one !! Haha! Well, not really!). In fact, the recursive application of the hyperops operator gives, as a consequence of situation "b":
a[s]x = x = a[s+1]oo.
Its left-inverse operations can give:
x = x/[s]x = x/[s+1]oo,
which means:
[s]-self-hyperroot (x) = [s+1]-oo-hyperroot (x), e.g.:
self-rt (x) = oo-srt (x) = x^(1/x), and, at another rank:
x/x = oo-rt (x) = 1;
x-x = x/oo = 0 (ops, sorry, see "d" ... special case).

On the other hand (what do you mean by: "which one|!?!"), if both GML and ML are together valid (Andydude, please forgive me!), we should have (by putting r = x):
a[s]<r>a = a[s](a[s+1]r), implemented by:
(s=1, r=0) a = a+(a*0) = a
(s=1, r=1) a+a = a+(a*1) = 2a
(s=1, r=2) a+(a+a) = a+(a*2) = 3a.

On the third (... ???) hand, we have:
(s=0, r=0) a[0]<0>a = a°(a+0) = a°a ... how strange ... !!
(s=0, r=1) a°a = a°(a+1) = a+2 (particularly for GML)
(s=0, r=2) a°(a°a) = a°(a+2) = a+3.

What is pointing out is that, for s=0 and r=0, we have:
a°a = a+2, for GML
a°a = a+1 for ML
a°a = a[0]<0>a, for GML and ML together valid.
The provisional conclusion is, in my opinion, that the zero level iteration of the zeration operator is singular. I mean that a[0]<0> is not a neutral operator.

So:
a[s]<0>a = a , for any s>0 is right, but:
a[0]<0>a = a is wrong (it is a wrong equality).

I am very tired and I go drinking a "makkiato". I should write it correctly as "macchiato", but I fear that Gottfried will pronouce it as Mc Shadow, who, together with Mc Roney, are Irish people and neither have anything to do with Italy or with coffee (particularly the second one).

For to-day "ich habe fertig" (Trapp-@-Tony).

GFR


RE: Laws and Orders - andydude - 04/23/2008

Wow, that was a very interesting post!

@GFR
I recognize that both the GML and ML as fundamental properties of hyperoperations. I also recognize that, although less intuitive, the Balanced Mother Law (BML) is very successful at producing the usual hyper1, hyper2, hyper3 as we know them. If you are responding to my comment in that thread, then I was only talking about BML, not the other two. I think so far we have established that the GML-hyper-0 is "zeration" as you define it, ML-hyper-0 is "succession" and BML-hyper-0 is the empty set.

On the subject of fuzzy feelings, I think you could play around with DL a little, because, as Kouznetsov's methods remind us, there is more than one way to get to infinity. You can go + or - or +i or -i, or even some other direction, and hopefully, if things are beautiful, then you should get to some fixed point of the previous hyperoperation. I personally think the sphere is the best way to visualize this. It is interesting to point out that many different fixed points can be obtained by changing the sign/direction of the infinity. For example, but which give the fixed points oo and 0 of multiples of a.

This makes me wonder if having a function such as would enumerate all fixed points of a[n-1]?

Andrew Robbins


RE: Laws and Orders - GFR - 04/23/2008

Excellent idea. I mean: the sphere. Smile Really! I plaid long time aroud that idea, trying to figure where to place the hypothetical trans-infinite ... "Delta Numbers". Sad

I shall read and study Kouznetsov, ... very carefully. Thank you for your comment.

About a possible a[n-1]x, I am very doubtful and concerned, for n=0. Concerned, in case of validity of GML. We may very well admit:
a[-1]a = a°2 .... peculiar, but ... tolerable ....

Very doubtful, in case of ML. In fact, we should have:
a[n](x+1) = a[n-1](a[n]x), an then:
a[0](x+1) = a[-1](a[0]x), i.e.:
a°(x+1) = a[-1](a°x), or better (or ... worse):
a°x = a[-1](a°(x-1))
What, the Hell (!), would this mean ... !?! Since I figure "zeration" and similar other objects representable by linear plots (if you see what I mean) I can't figure ... "minusation". If zeration is something like "enumeration" or "counting", what could mean an action such as "minusation". ... Terrific !! Do you think that DL would help? Let me try to think about that. Mmmmmm! Smile

Gianfranco Romerio

PS: I have some new ideas about the "yellow zone" and my old ... wrong approximation. But, Andrew (ops ... sorry!), this is another story. I shall prepare a comment to your (... I suppose) old thread.


RE: Laws and Orders - Ivars - 04/24/2008

hi GFR

Euler said that minus numbers are like borrowed money- it is not there, but when real money appears, you have to give it away as You had a minus balance. Minus numbers eat away/cancel positive numbers but are not "visible" before a positive number comes near. They have a potential, so to say, but not presence.

Also, when acting on themselves, sometimes minus numbers pop-out from nowhere into positive domain.

Something similar may relate to negative operations.

They are not visible as long as positive operation does not wander into their neighborouhood. But they sit there, waitingSmile. They are there.

I like that you have thought about 0 -tion being kind of enumeration. Should be so, or close.

Ivars


RE: Laws and Orders - GFR - 04/24/2008

Hi Ivars!

Sounds like ... ghost-hunting Wink ! Or, in physics, .... "negative" energy, or particles having "imaginary" mass (tachyons). Or, in mathematics, imaginary straight-lines being perpendicular to ... themselves. Or, always in math, varieties with an infinite number of dimensions (Hilbert?). Or, very modern subject indeed, physical bodies with a "fractal extension" (Mandelbrot?), like the length of the coast of Brittany, which is not ... a length, otherwise it will be ... un-determinate Sad .

GFR


RE: Laws and Orders - Ivars - 04/24/2008

GFR Wrote:Hi Ivars!

Sounds like ... ghost-hunting Wink ! Or, in physics, .... "negative" energy, or particles having "imaginary" mass (tachyons). Or, in mathematics, imaginary straight-lines being perpendicular to ... themselves. Or, always in math, varieties with an infinite number of dimensions (Hilbert?). Or, very modern subject indeed, physical bodies with a "fractal extension" (Mandelbrot?), like the length of the coast of Brittany, which is not ... a length, otherwise it will be ... un-determinate Sad .

GFR

Yes, "minusation" . What about "+- imagination", then? Nice word, anyway. May be we can get "+-imagination" by equalling multiplication and "addition" to "zeration"?

That would lead to "multiplication" becoming equal to "minusation" and that would only be possible if somehow inside multiplication we had "+-imagination"???
Or actually, instead of multiplication.

Ivars


RE: Laws and Orders - GFR - 04/24/2008

Ivars Wrote:Yes, "minusation" . What about "+- imagination", then? Nice word, anyway. May be we can get "+-imagination" by equalling multiplication and "addition" to "zeration"?
That would lead to "multiplication" becoming equal to "minusation" and that would only be possible if somehow inside multiplication we had "+-imagination"???
How nice indeed .... "imagination" would be. Do you mean something like a[i]x ??

In fact, we should have:
y = a[4]x ..... tetration
y = a[3]x ..... exponentiation
y = a[2]x ..... multiplication
y = a[1]x ..... addition
y = a[0]x ..... zeration (GML or ML versions)
y = a[-1]x .... minusation (!!!!!!!)

And, to this, our imagination would add ... "imagination", as:
y = a[i]x ...... imagination. Sad

Why not! However, unfortunately, we cannot make any paraller reasoning. Rank s=i (with my traditional "s" symbol) has to be defined and its properties ... carefully described. But the idea is nice. It was also rapidly mentioned (by KAR and myself), as a remote possibility, in a paper submitted to the WRI Forum. However, the "imagination" name is yours! For the moment, it is only "fantasiation". But, ... who knows?!

Before that, we have to investigate the possiblity of defining fractional ranks of the hyperops hierarchy, if any, starting from the rational half-ranks. Something like:
y = a[1/2]x .... halfation, between zeration and addition
y = a[3/2]x .... sesquition ("sesqui" means: 1 + 1/2), between addition and mutiplication
y = a[5/2]x .... sestration ("sester" means: 1 + 1 +1/2), between multiplication and exponentiation, and .... so on .... !?!

At the level between addition and multiplication (s=1.5), at the "sesquition level", we have a very well known "mean", the Gauss Mean also called the Arithmetic-Geometric Mean. So, perhaps, the corresponding operation (with its mid-way hierarchical chain) could maybe be found. One of these days !!! What would then be the impact of GML, ML, BML and DL? Interesting!!

By the way, do we have any documented proposal for a fractional or an imaginary extensions of the Ackermann function?

GFR


RE: Laws and Orders - Ivars - 04/25/2008

GFR Wrote:In fact, we should have:
y = a[4]x ..... tetration
y = a[3]x ..... exponentiation
y = a[2]x ..... multiplication
y = a[1]x ..... addition
y = a[0]x ..... zeration (GML or ML versions)
y = a[-1]x .... minusation (!!!!!!!)

And, to this, our imagination would add ... "imagination", as:
y = a[i]x ...... imagination. Sad

GFR

Yes, a[i]x is "imagination".

I think for interpetation of what minusation may mean, we can look at + zeration and - zeration in a limit, so to say, what does approaching zeration ( if that is enumeration, ordering) from both sides might mean. May be "minusation" is just subtraction since negative numbers appeared because of the need to substract positively enumerated (zerated) objects in the WRONG order, so to say- the biggest from smallest. If there were no subtraction, where would not be negative numbers, if I am right.

So perhaps +zeration is ordering in right order, and -zeration is ordering in wrong order.

Then "imagination" may be coming from ordering without order, so to say. Chaotic enumeration if that means anything.

Excuse me if I am too much in a fantasy landSmile

Ivars


RE: Laws and Orders - GFR - 04/25/2008

Aim high! There is a lot of space Wink !!!!

GFR


RE: Laws and Orders - Ivars - 04/25/2008

GFR Wrote:Aim high! There is a lot of space Wink !!!!

GFR

Ok, before Henryk catches meWink

Let us assume there is some right order in enumeration, kind of natural. Let us assume it goes from small to big (do not ask me what is small and big-perhaps numbers, so we enumerate numbers , e.g real numbers with natural ( I know its not possible, just as an example) ) . Then if we subtract in the "right " order, the small things from the big, so to say we do not destroy the order of enumeration we had used, we have still positive "numbers" as a result.

Now if we destroy the order of enumeration to exactly opposite, big to small, we have negative enumeration. When we change order in our positive enumeration we had and subtract big from small, we also end in negative teritory. So far this matches.

Now if we look what more we can do with perhaps more than 2 objects, we can reorder them in more then 2 ways - right and wrong, but also in a mixed ways so that outcome is not exactly opposite to right enumeration, but something mixed. Now perhaps (?) this mixed enumeration has to be somewhere in between rigth and wrong enumeration, perhaps it is fractional operation depending on the balance of the result.

Now there has to be an enumeration that makes it impossible to see have we mixed up things to the right or wrong direction, we can magnify the enumerated sequences, which may be infinite, as much as we want and still not able to decide is it right or wrong. That could be kind of "imagination".

If we apply this to a single number,base a, to be able to explain what such operation does - that requires some subtler structure to exist in a number which defines what will be result of "imagination" or "minusation"....


Ivars