On Tue, Sep 11, 2012 at 11:27 AM, Paul Tanner <firstname.lastname@example.org> wrote:
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> This doesn't hold up, since almost all irrationals are not computable > numbers, according to the standard definition of computable numbers. >
Uh oh, sounds like a security breach. We've gone from a strictly professed segregationist stance, putting a fence between Q and R, and now some of the irrationals are getting through, as definitely computed. Pi to billions of digits. Sure, more digits could be crunched but to call what's probably a record-holder for computing time devoted thereto, "non-computable" is just daft.
Phi is one of my own favorites, given its importance in STEM, where the 5-fold symmetry of the virus-icosahedron is like a trademark. A continued fraction. All 1s. So simple, so computable.
The isomorphism of multiplication with addition, through log / exp notation, is another indicator of where we might go, in terms of relating these two operations.
Even though the slide rule is mostly a relic, the ability to encode a large number as powers and do convolutions, much like a Fourier Transformation, is a sign of deep and meaningful connections.
Yet you seem concerned to leave addition far behind, in kindergarten, with all those "misleading teachers" that Devlin likes to berate.
I don't think you necessarily want to bully these teachers. You just want them to be more connected to the world you live in, where irrationals abound in different species, and are proud of their differences from the rationals.
There's a Dr. Seuss novel we could read -- more a short story.
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>> We also have many examples of systems in which a binary >> multiplication operation vis-a-vis set objects has no representation >> in the sense of scaling. >> > > Exactly. >