Laws and Orders - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: Laws and Orders ( /showthread.php?tid=148) |

RE: Laws and Orders - Ivars - 05/20/2008
A pity GFR is away.. Speaking about "imagination" and what does imaginary powers and iterations mean, we can try the following analogy to get some idea what does I "count". We know that Cantor extended enumerability to infinite numbers, so why not try with imaginary? So let us assume we have I of something. Let as say we have a set with I elements, or size I. Now lets us ask ourselves a question: How many subsets does this set have? And let us answer ( or postulate) that as usual, the number of subsets is 2^I. The solution for fixed points to such idea I=2^I would require that pi/2= ln2 which I do not have anything against, since pi/2 = 90 degrees only in spaces where such angles are possible. There could be spaces with chaotic character, disconnected, where straight lines and trajectories does not exist at all, so pi also does not have a meaning. But angle does. This idea allows(?) to apply binomial coefficients to set with size I when looking for possible varieties of subsets. The question is, if the analoque of n is I, what is the analogue of k? Is it some fraction of I? For example, if we take I/2 and ask how many subsets with size I/2 can happen within set with size I, we have I!/((I/2)!(I-I/2)!) = I!/ (I/2)! *(I/2)! Now of course i have never in my life seen factorials of Imaginary Unit. If we look at Gamma function, we see that Gamma(I) = -0,15495, Gamma(I/2) = -0,39928, so that -0,39928*-0,39928=0,159424 -0,15495/0,159424 = -0,97193 hmm I would have loved it better if the number of subsets would have turned out to be -1. Perhaps gamma function is not the right one to generalize factorials to imaginary numbers. Or binomial coefficients are not, more likely. Or the subsets in set with size I does not follow binomial distribution, because set size I is not discrete, obviosly. Another try may be Poisson distribution as closest to binomial. Somehow understanding about what "imagination" is has to be reached above pure formal placement of symbols. Also, if we stick to integer values of n in I/n (for reasons unkonown) , we could require that all subset sizes together correspond to Cantors idea about size of transfinite cardinal sets: Sum (I/n) = 2^I That would be like Imaginary Harmonic series. I wonder if there has been any attempts to give value to them in some summation. Ivars RE: Laws and Orders - Ivars - 05/25/2008
I have given counting by imaginary and negative numbers a little bit of a "philosophical" thought. This is the first crude outcome: From the point of view of given process where we have something- let us name them Things: -Positive numbers count Things that exist -Negative numbers count Things that does not exist, but can exist -Positive Imaginary numbers count Things that neither exist, nor not exist- oscillatory existance- and they count the moments when they exist -Negative imaginary numbers count the moments when these oscillatory Things that could both exist and not, does not exist. Now obviously space of all negative numbers include all Things that does not exist but could exist- so quite naturally that Imaginary unit has a restriction on it- once it turns from oscillatiory Thing into one that does not exist, but could, it has to become a negative number- for some reason it is happening as multiplication: I*I=-I*-I= -1 The non-existant Things can be obtained either from existing part of oscillatiry unit, or non-existant- any one of them is not enough to create existing Thing. You need both, as can be seen in next paragraph. If oscillatory Thing has to become existant, it can not avoid becoming one of those counted by positive numbers as there are no others, since there are no others: In mathematics that requires participation of both of its 2 states Hence -I*I=1 . The question is what that would mean concerning negative and imaginary hyperoperations and iterations? Operation seems not to be a Thing, but something working on Things. Iteration is counting how much (how many times) Operation is working on Things. Functions are usually defined via Operations (Is it) , so they represent combination of Operations. Ivars |