Thursday, December 29, 2011

Surreal Number Tables over Quasic Space

Surreal Number Tables over Quasic Space L. Edgar Otto December 29, 2011

I thought I would do the obvious and convert Conway' "nimber" tables of addition and multiplication and powers into the quasic grid ordering.

I wonder what it means for someone to hold that if we find a hypothesis true like Riemann's then "a large part of number theory must be redone" those that begin with something like "assuming the Riemann Hypothesis is true, then the following theorem...". Is it not the same for Turing who believed this hypothesis not true? Is it the same for Conway who feels the continuum hypothesis is false?

This is a logic problem I imagine and we do achieve things along the way- if a student of the abstract can master them or if someone is in touch with such concepts in some form intuitively, or that they have a sense of intelligibility of letters and numbers- for example an idea of base called lexicographic wherein each digit is treated as if part of a general base number. The problem is where these ideas meet in our heads and the structures of our heads as analog and computational devices over some form of physicality, physics. Thus those looped in a theory may or may not be justified in the exclusion of a different or equivalent view if that is the ultimate case or not. Or we have not reached a new intellectual cognitive level and sufficient to the day of problems thereof all research is glorious and all errors in the innocent spirit of things can be forgiven in the eyes of history.

I would ask myself after doing what seems so trivial yet suggestive in this illustrated chart (forgive all the dots of prime coordinates for now as I had to recycle my qs graph paper.) If in a generalization beyond the quasic and surreal can I imagine an empty distance as an analog to those lesser geometries or in some sense like some feel the string formalism achieves a totality of exclusion within the bounds of physical reality this is the case decidability. That we cannot prove God exists or not by some method or level is a matter of (well accepting all experienced inputs) a matter of philosophy but it may be a matter of science as well. We could hardly reason without some such intelligible, even common sense ground of these abstractions in some form on some level we can get our head around, but in doing so we should not assume we have then mastered the infinite that once so challenged us.

So much of patterns can suggest if not tell us something. I see in one of the three tables above that the quasic regions, 3 x 3 or 2 x 2 come together to arrange the zeros in the first 4 x 4 quadrant region. There are seven here (presumably the two sets of 4 which I regard as quasic generations merge at the zero or unity position. So this seems the Fano plane number or the similar idea of octonions and so on in that we may go a little further to generalize our ideas of the projective plane. Of course it is instructive to read the diagonals and rows and columns as if a matrix.

In the numerology beyond these 7 into three space of 2^8 do we not just barely get into the endless patterns to say in a coordinate we can imagine 11 dimensions? Just a suggested idea- much as the idea that there is a dynamics of multiplication between the generations one down on the same level of abstraction as per Riemann's insight on space dimensions, (is there a nimber to the nimber power chart- if not why? Is there something perhaps I am missing again that does not make the abstraction clearer to me? If missing then why can our theoreticians come so close and be so intricate and miss it?

Whatever else I can say about this quasic idea it inspires a lot of abstract thoughts, and evidently a hell of a lot of trying to express things in words- so it gives as much or more than it saves us in visualization. Something in us also wants a solid grounding and maybe the certainty of a totality, not just the quicksand of change in descent endless into the surreal infinitesimal.

But I wanted to illustrate with a picture of the quasic chess game with the ordinals and cardinals and surreal levels and physical dynamics focused and implied in such a complex (here four space compressed into three)... But I am sure you get the picture.

Oh well, I made a simplified cover illustration...
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