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Topic: Circular reasoning on a curve
Replies: 14   Last Post: Nov 11, 2013 12:40 PM

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ross.finlayson@gmail.com

Posts: 1,221
Registered: 2/15/09
Re: Circular reasoning on a curve
Posted: Oct 29, 2013 11:46 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Tuesday, October 29, 2013 8:07:17 PM UTC-7, fom wrote:
> On 10/29/2013 9:18 PM, Ross A. Finlayson wrote:
>

> > On Tuesday, October 29, 2013 4:36:44 PM UTC-7, fom wrote:
>
> >> On 10/29/2013 5:34 PM, Hetware wrote:
>
> >>
>
> >>> On 10/29/2013 5:20 AM, Peter Percival wrote:
>
> >>
>
> >>>> Hetware wrote:
>
> >>
>
> >>>>> On 10/28/2013 9:57 PM, fom wrote:
>
> >>
>
> >>>>
>
> >>
>
> >>>>>> But, your geometry without Descartes' innovation has no numbers.
>
> >>
>
> >>>>>
>
> >>
>
> >>>>> Then how do I get numbers from it?
>
> >>
>
> >>>>
>
> >>
>
> >>>> In one or other of the appendices to Hilbert's 'Foundations of geometry'
>
> >>
>
> >>>> there is an account of how various axioms of geometry imply various
>
> >>
>
> >>>> properties of numbers.
>
> >>
>
> >>>>
>
> >>
>
> >>>
>
> >>
>
> >>> I'm not familiar with that discussion.
>
> >>
>
> >>>
>
> >>
>
> >>> I believe it is impossible to completely separate our geometric
>
> >>
>
> >>> reasoning from analytical reasoning. The very notions of betweenness,
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> >>
>
> >>> ordering, and even counting require some dimensional framework in order
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> >>
>
> >>> to even think them. I am including time as a dimension for purposes of
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> >>
>
> >>> this discussion.
>
> >>
>
> >>>
>
> >>
>
> >>> I find it tantalizing that our analytical mathematics can produce a
>
> >>
>
> >>> number such as pi without any direct reference to any kind of geometric
>
> >>
>
> >>> object. There is some kind of supremacy of Euclidean over other
>
> >>
>
> >>> geometries. For example, the geometry of general relativity is locally
>
> >>
>
> >>> Minkowskian, and at the low velocities of daily experience, Galilean.
>
> >>
>
> >>>
>
> >>
>
> >>> That is the environment in which our mind/brain evolved.
>
> >>
>
> >>
>
> >>
>
> >> Apparently, there are certain philosophers who uphold this view.
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> >>
>
> >>
>
> >>
>
> >> For my part, I find the rejection of geometry which occurred in the
>
> >>
>
> >> late nineteenth and early twentieth century confusing. When the
>
> >>
>
> >> disputes over set theory and the paradoxes settled, Hilbert, Brouwer,
>
> >>
>
> >> and others turned to "intuitive", "a priori" notions. At the heart
>
> >>
>
> >> of these ideas is what should be called "inscriptional identity".
>
> >>
>
> >>
>
> >>
>
> >> This kind of identity is what Wittgenstein had in mind when arguing
>
> >>
>
> >> for the "eliminability of identity" from mathematical discourse.
>
> >>
>
> >>
>
> >>
>
> >> This kind of identity is what grounds the notion of number as
>
> >>
>
> >> monotone inclusive sequences of strokes:
>
> >>
>
> >>
>
> >>
>
> >>
>
> >>
>
> >> |
>
> >>
>
> >>
>
> >>
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> >> ||
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> >>
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> >>
>
> >>
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> >> |||
>
> >>
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> >>
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> >>
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> >> ||||
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> >>
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> >>
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> >>
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> >>
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> >>
>
> >> These ideas, however, being grounded in inscriptional identity
>
> >>
>
> >> presume rigid motions. Geometrically, this is Euclidean geometry.
>
> >>
>
> >>
>
> >>
>
> >> Sometimes I think that all of these intelligent people have
>
> >>
>
> >> argued themselves stupid.
>
> >>
>
> >>
>
> >>
>
> >> Sometimes one must sit back and reconcile the arguments.
>
> >
>
> >
>
> >
>
> >
>
> >
>
> > Curve is line from zero.
>
> >
>
> > "For my part, I find the rejection of geometry which
>
> > occurred in the late nineteenth and early twentieth
>
> > century confusing. When the disputes over set theory and
>
> > the paradoxes settled, Hilbert, Brouwer, and others turned
>
> > to "intuitive", "a priori" notions. At the heart of these
>
> > ideas is what should be called "inscriptional identity".
>
> >
>
> > This kind of identity is what Wittgenstein had in mind
>
> > when arguing for the "eliminability of identity" from
>
> > mathematical discourse.
>
> >
>
> > This kind of identity is what grounds the notion of number
>
> > as monotone inclusive sequences of strokes: ..."
>
> >
>
> > "These ideas, however, being grounded in inscriptional
>
> > identity presume rigid motions. Geometrically, this is
>
> > Euclidean geometry. "
>
> >
>
> > What is this inscriptional identity?
>
> >
>
>
>
> Wittgenstein simply asserted that every different
>
> shaped symbol represented a different object.
>
>
>
> In terms of the constructive mathematics of Markov,
>
> one it given alphabets of symbols having different
>
> shapes and the mathematical language is constructed
>
> from these alphabets.
>
>
>
> In particular, natural numbers are concatentions
>
> of a single symbol alphabet. At the next stage --
>
> that is, the alphabet of expressions formed from
>
> that alphabet -- the expressions are inscriptionally
>
> distinct,
>
>
>
> ~( || = ||| )
>
>
>
> so that the system of expressions is based on
>
> inscriptional identity.
>
>
>
> Inscriptional identity is at the heart of
>
> first-order logic with identity in the metalinguistic
>
> stipulation
>
>
>
> x=x
>
>
>
> for any term x of the language. This is the ground
>
> for what Carnap calls syntactic identity. With
>
> regard to the ontological interpretation of that
>
> expression, objects are self-identical. Consequently,
>
> if, under semantic intepretation, two variables are
>
> mapped to the same object, then
>
>
>
> x=y
>
>
>
> is true. This conception makes no sense without
>
> enforcing inscriptional identity at the syntactic
>
> level. One could not possibly have
>
>
>
> ( x = y ) /\ ~( x = x )
>
>
>
> in a system in which substitution is the syntactic
>
> application of identity statements.
>
>
>
>
>
>
>
>
>

> > Viete is good for your algebra. For example he discovered
>
> > the relations between sums of squares and cubes. Mitch or
>
> > fom here has him as basically inventing algebra (number
>
> > theory).
>
> >
>
>
>
> Well, not number theory. But, his innovations permitted
>
> uniform treatment of arithmetic monads and geometric
>
> magnitudes via algebraic notation.
>
>
>

> > "I'm trying to recall the distinction twixt hyperbolic
>
> > geometry and hyperbola geometry, which is the geometry of
>
> > special relativity, post-Minkowski."
>
> >
>
> > Minkowskian geometries have a time-like dimension as then
>
> > they have (or generally as Minkowski) three space
>
> > dimensions. In the geometric algebra, it's not always the
>
> > algebraic geometry. For Minkowski it is (the geometric
>
> > algebra). With the hyperbolic and the hyperbola in
>
> > geometry, of the space, as it is the Minkowski space, the
>
> > hyperbola (sp.) and conditions on the geometry for example
>
> > of Euclid's parallel postulate where Minkowski-space is
>
> > super-Euclidean space, of geometry, those are as to
>
> > parallel transport and the instanton.
>
> >
>
> > The curve is the line through its points, the vector is
>
> > the point. Consider where the point is the curve and vice
>
> > versa. Where the vector is not the point, where the curve
>
> > goes through the points in general parallel transport, the
>
> > curve is the point. The vector is also the point as it
>
> > generally and usually is, with simply maintaining
>
> > estimation effects locally. Then, where that is transport
>
> > of effect beyond the frame, the curve is the vector
>
> > (parallel transport is uni-directional).
>
> >
>
> > Regards, Ross Finlayson
>
> >


Not, number theory?

Turn each thread into a mathematical proof with sections
Give generous proofs

I'm happy that wherever I write "well-founded" or
"regular" cardinals you can change that for "ordinary" in
the way of Russell and "irregular" or "non-well-founded"
or "anti-foundational" cardinal, as "extraordinary", and
it reads the same.

And here it is number theory.

Regards, Ross Finlayson



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