I do like Jordan Peterson. I his "12 rules for life" you will find many goodies. Let me quote just one:
"Scientific truths were made explicit a mere five hundred years ago, with the work of Francis Bacon, René Descartes and Isaac Newton. In whatever manner our forebears viewed the world prior to that, it was not through a scientific lens (any more than they could view the moon and the stars through the glass lenses of the equally recent telescope). Because we are so scientific now—and so determinedly materialistic—it is very difficult for us even to understand that other ways of seeing can and do exist. But those who existed during the distant time in which the foundational epics of our culture emerged
were much more concerned with the actions that dictated survival
(and with interpreting the world in a manner commensurate with that goal) than with anything approximating what we now understand as objective truth."
I would like to stress the word "action". Actions, processes, connections are more fundamental than "things" subjected to actions. Of course at the "beginning" where there were only actions, they were feeling lonely, so the need was generated for "things" to act upon. So "matter" was created out of the dense vortices of actions.
That is my own view of Genesis.
But now I have started with "things", even though in the Platonic world of numbers. These "things" under our scrutiny are Pythagorean triples.
We need to compensate for the lack of actions so far. How these triples were "created"? In what kind of a process? By what kind of actions? We will deal with it in the coming posts. For now we need some vocabulary.
Pythagorean triples can be primitive or not. The triple of positive integers (a,b,c) is primitive if the pair (a,b) is "coprime" or "relatively prime" if a and b have no common (natural number divisor different from 1. So (3,4,5) is a primitive triple, while (6,8,10) is not.
One easily proves that if (a,b,c) is a primitive triple, then one of (a,b) is odd and the other one is even. Like it is with (3,4). On the net, or, more seriously, in
"Elementary Theory of Numbers" by W. Sierpinski, North Holland 1988,
And then the book goes with some kind of an action:
"In order to list systematically all the primitive solutions of equation (1) we take values 2, 3, 4, ... for the number m successively and then for each of them we take those numbers n which are relatively prime to m, less than m and being even whenever m is odd. Here is the table of the first twenty primitive solutions listed according to the above-mentioned rule:"
Thanks! For me spinors are vectors of the space of an irreducible representation of the Clifford algebra.
ReplyDeleteOf course one can discuss what do we mean by irreducible representation and does it really have to be irreducible - but these subtleties do not change the essence. By definition then the spin group (contained in the Clifford algebra) knows how to act them. The spin group always covers the corresponding orthogonal group - that follows by the definition.
That is roughly the idea as far as I know.
And spinors and spin group will show up already in the next post.
Also via Clifford algebra (Tony Smith's use of it), I kind of have a loose view of spinors, but I have a loose view of lots of things. Tony had them as a row or column of the Clifford matrix algebra. Via commute vs anticommute, he had the odd grades as spinor components.
ReplyDeleteHe had other ways of getting spinor components via getting big Lie algebras from really big Clifford algebras. He also admitted to morphing spin and pin (not that I knew what that meant). I mainly just stuck to his odd grade view which was a view he didn't have originally or talk about much at the end; it was kind of his midpoint view.