Subject: An Invitation to Smooth Infinitesimal Analysis
Date: Wed, 03 Mar 2004 05:11:30 +0200 From: Dimi Chakalov <dimi@chakalov.net> To: John L Bell <jbell@uwo.ca> Dear Professor Bell, Let me tell you about my efforts to understand the instant at which the infinitesimal will (or would it, really?) become a "point", http://members.aon.at/chakalov/Wagh.html#NB In your "An Invitation to Smooth Infinitesimal Analysis", pp. 1011, http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf you wrote: "On comparing these we see that, if we take e as the "infinitesimal unit" for measuring infinitesimal timelike distances, then ie serves as the "imaginary infinitesimal unit" for measuring infinitesimal spacelike distances." Perhaps the reason why the transition into "point" above cannot be *actually* achieved stems from the impossibility of eliminating the intrinsic difference between timelike and spacelike regions: spacetime theory in SIA forces one to use imaginary units. If it were possible to actually shrink the infinitesimal into a bare mathematical point, then the three kinds of distances would be indistinguishable, which is forbidden. Maybe Mother Nature uses two kinds of time, so that she can 'have her cake and eat it', http://members.aon.at/chakalov/Pestov.html#continuum Does that ring some bells? Regards, Dimi Chakalov
Note: I didn't dare to say that, the way I see it, the crucial issue of the emergence of time and space should be considered on some pregeometrical and premanifold "level" of He Who Does Not Play Dice. Otherwise you just can't get a spacetime. No way. Also, if we ponder on the tacit presumptions in the old story about the infinitesimal, there is a brand new kind of infinity here, which I call 'logical infinity'. The tacit presumption in the chasing of the "point" is that it can indeed be reached 'down the road', but if this same "point" is quietly residing inside the same 'terribly small interval' which is chasing the "point", then the final instant at which the finite interval could catch the "point" is impossible on logical grounds. It is not possible to catch something if the latter is already residing inside you. You will end up with infinite series of steps: logical infinity. Here the "point" (in the global mode of spacetime) is inside the shrinking, but always finite, spacetime interval (local mode of spacetime). This is a purely metaphysical resolution of the paradoxes of The Beginning and the 'vacuum cleaner'. Hence He Who Does Not Play Dice is residing inside the instant 'now' of the cosmological time arrow. Always been there, always will. Once created [John 1:1], The Universe (insidetheuniverse & its current Holon state) has a dual age: finite in the local mode of spacetime, some 13.7 billion years "after" The Beginning, and infinite (or rather indecisive) in the global mode of spacetime. We cannot actually catch the "point" inside us, for any arbitrary large, but finite, cosmological time interval measured in the local mode of spacetime. Briefly, logical infinity. Who said it's simple to find the circumference of a circle, after Archimedes? Once I hear from John Bell, I will be happy to elaborate. I'm sure he can cast this story in the context of Synthetic Differential Geometry, which I believe is the obvious choice for addressing the ultimate task of his colleague Chris Isham: from regions to "points". More at GR17 in Dublin. Praise the Lord, and pass the ammunition!
