M Theory Lesson 278
MUBs tell us to focus on the finite fields $F_{q}$, where $q = p^{n}$ is a prime power. For a fixed $p$, the inverse limit of these fields is the ring of $p$-adic integers. This gadget has the annoying property of being uncountable, and is responsible for the beautiful fractals that naturally describe embeddings of $p$-adic numbers in the complex plane.
From a logos perspective, the axioms for fields are rather messy, and they should not be considered in the context of ordinary sets. Set theory doesn't even know the difference between the continuum cardinality and other choices, so why do we use it to inspire definitions of categories? Actually, category theorists have thought about this for a long time, and there are many kinds of category capable of all the important things that sets are capable of, but which aren't at all like the usual category of sets.
M theorists need to learn more about these alternatives. For example, today David Corfield brings our attention to the concept of pretopos. If one delves a little into this idea, the rationals and finite fields start to look even more remote from the (not uniquely defined) reals than they do in a topos!
From a logos perspective, the axioms for fields are rather messy, and they should not be considered in the context of ordinary sets. Set theory doesn't even know the difference between the continuum cardinality and other choices, so why do we use it to inspire definitions of categories? Actually, category theorists have thought about this for a long time, and there are many kinds of category capable of all the important things that sets are capable of, but which aren't at all like the usual category of sets.
M theorists need to learn more about these alternatives. For example, today David Corfield brings our attention to the concept of pretopos. If one delves a little into this idea, the rationals and finite fields start to look even more remote from the (not uniquely defined) reals than they do in a topos!
posted by Kea | 1:31 AM
1 Comments:
Amusing to see that p-adic fractals have been finally found by mathematicians. I studied the continuous mapping x= sum_nx_np^n--->sum_n x_np^(-n) from p-adics to reals for 15 years ago and made plots also about 2-dimensional maps for complex extensions of p-adics for p=2,5,7.
Illustrations can be found here http://tgd.wippiespace.com/public_html/figu.html.
My original belief was that the inverse map (not completely unique) would allow to map real space-time surfaces to their p-adic counterparts and vice versa. The idea failed because the map did not respect isometries. The map however makes
sense for p-adic variants of probabilities in p-adic thermodynamics.
Maybe I must wait one decade more to before mathematicians realize that p-adics and reals can be glued to together along common rationals (and also selected algebraics) to form a larger structure and that the notion of manifold generalizes in this manner.
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