Saturday, December 24, 2011
Recondite Fermat Number Space
Recondite Fermat Number Space L. Edgar Otto December 23, 2011
Well, it did snow a little and it got a little chilly so I did not go to the Festivus at the bar called the Joynt after all- it did not look like as many old friends would show up like last year, the season is more of a brown Christmas- not that I did not relish my time alone (well, I did have some time one on one flirting with a more isolated soul across the way - I mean people are much more important than theories, and I calmed down everyone after my roommate got into a fight when another friend tried to push his way into the door... and yes, Lucinda and her half wolf pet named Coyote did manage to make the way without serious incidents driving the oxen and wagon into the valley to face both the beauty and the immensity of the virgin west in Zane Gray's novel, the tension in the novels the wide new world and the foreboding of isolation in Luce's soul when we return to the frontiers.)
So with such interruptions I offer a more formal graph or question as well as some stray notes errors (but significant ones for the general theory) and all.
For the book Fearful Symmetry I almost feel intimidated or fearful to read it as it is very well dressed out despite some silly humor and repetitions- worse than that how can I show the uncanny synchronicity of it when the philosophical concepts I developed recently came before this beautiful permutation theory of Galois groups when it comes so close in time- In any case I am not more familiar with the standard terminology. In general I see it as the unification of the usual arithmetic axioms with the all important independent ones of the continuum hypothesis and axiom of choice- if in a sense these can be so unified. I disagree with some of the general comments and results of some of the laws- like for example the numbers that describe what is in an equation cannot be that different or hidden from those of the coeficients of the algebra of it or that I do not see the great (and yes thomnic) sea of infinity as something we cannot grasp a little better or maybe totally with the number and topological ideas arranged in the quasic plane (or a much better one) I mean, the book, concerned with Diophantus ideas of what is the finite in all this suggests it is a "generalization of reciprocity" of which I seem to address too but in a different way perhaps. So now I wonder again, in synchronicity just how this relates to Pitkanen's view- and I can see his posting today. Here. I am not that clear on what the objections are all around.
In the illustration of the Fermat numbers on a quasic grid (and the location of the primes as a square root of that grid, the general idea also that the concept of generations can also be generations of our idea of fields but maybe not quite in a perfect computation as expected by the application of these ideas alone. The next prime on the graph is of course 4097- 2^32 - 1 not a Mersenne prime... all these questions on which we can almost answer- and I wonder moreover if I am seeing Pitkanens view of this correctly. His is a generalization too of the ideas of reciprocity and space (presumably the ones in this book somehow goes beyond Riemann's treatment of planes or multiplanes, but sometimes simple observations of number properties are very much a result of our generalizations- Ramanujan and the idea that, well it is obvious that certain numbers all are divisible by 24 on some level and what after all is this hidden intuitive view of things that those in another faith hold as mysterious or some sort of difference of genius or method that some of the exalted old mathematicians seem thought to have?
I only left the simple graph here, I did have plans to map other of these 2^n or 2n numbers- but there is a lot of information there already to see, including if it is trivial or not- there is certainly a lot of information in these books but I do not feel I am making a hopeless chase trying to adsorb and understand it- revise to a better level what the ideas are in terms of their suggestion of uses and importance. Some things in my notes are not my own btw but come from the book as a reference as in Conways. I do not see the wide world of maths as overwhelming to explore btw.
Now Pitkanen's post as with many who use the group theories (and the issue is about symmetry- if the Monster Group is the maximum symmetry why would Conway suggest it may be surpassed if I read it right?) that make sense in what they are saying (also I see where I have used a word or concept differently when it is a standard word would it make sense if translated as a statement to the concepts those words?
I hope Pitkanen makes it a little more clear why he feels some of these issues are on the wrong track or not. We are not quite there yet either in our separate paths of theories- of course the established theorists are a little intimated and afraid also- just as with government who only allows so much faith or magic before it is suppressed for the stability and order of the whole. What is wrong with the matrix approach or that of gauge theory (even if some papers while more advanced and generalized in these concepts do not get it right enough let along not even wrong?).
Anyway I may slowly figure out these theorists positions on my own in my own dialog. I see several areas I have to clear up for myself of words I do not know quite well enough yet that the bloggers have used. A simple idea like only 2 or 6 not able to make Graeco-Latin squares seems to me all important as to how we reduce these ideas of symmetry, its breaking, and any generalization of inverses and reciprocity in numbers or space. While I do welcome an explanation of Fermat's last theorem as a matter of Elliptic algebra proof thus these them of this book, Fearful Symmetry - I feel pretty sure we should still look for more elementary proofs- the Galois theory is not a deep enough foundation as explained for the patterns in the universe. I find it interesting also that there are things like Bell's numbers and that we can use but two of six expected permutations (now I understand how Kea used this idea) but more than that I begin to see a need to look for a deeper why if it not stated or posted somewhere already but not pointed to as to the relevance or application.
As I said, this is a "Brown Christmas" not that I want the record snow of last year-
so I also put up a minimal "Festivus" tree, a reduction to but one aluminum pole in the artificial tradition, less religious and less commercial as the usual mediocre themes and so on... If we think of the general idea of field as the snow all merged at least in the light as a sea of white, if we see the 80 odd types of snowflakes but that no two were ever the same across geological time and season, then we think of a particle as a grain of salt put on the side walk- well, that is a simple picture of the space around the grain before the snow begins again where you can see the sidewalk underneath where it has melted. Is it possible that in the end we find a more elementary theory of everything to which all these complicated edifices are our powerful over excited imaginations?
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Related posts today on facebook:
L. Edgar Otto to singer songwriter LiLi Roquelin: Christmas in New York, where its at... and Happy Festivus yesterday too...
L. Edgar Otto to posted photo of festivus tree: At festivus our shortest days will grow longer... inside the snow angels is our desire to look to the future, to get some alone time before a fireplace, to hibernate without the clock, and to snuggle.
Also posted as comment to Vanessa (but do not see it today, may she get better and feel better)
L. Edgar Otto
On Festivus, December 23. 2011 I put up the minimal tree (an aluminum pole, traditionally). Good for a reductionist xmas and a brown one snug inside reading and just maybe a better time for us all next year- may your stress of the shopping and family obligations be lighter. I never imagined I would miss New York city...
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