
Re: Circular reasoning on a curve
Posted:
Oct 29, 2013 11:46 PM


On Tuesday, October 29, 2013 8:07:17 PM UTC7, fom wrote: > On 10/29/2013 9:18 PM, Ross A. Finlayson wrote: > > > On Tuesday, October 29, 2013 4:36:44 PM UTC7, 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, > > >> > > >>> ordering, and even counting require some dimensional framework in order > > >> > > >>> to even think them. I am including time as a dimension for purposes of > > >> > > >>> 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. > > >> > > >> > > >> > > >> 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. > > >> > > >> > > >> > > >> 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 > > firstorder 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 selfidentical. 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, postMinkowski." > > > > > > Minkowskian geometries have a timelike 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 Minkowskispace is > > > superEuclidean 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 unidirectional). > > > > > > 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 "wellfounded" or "regular" cardinals you can change that for "ordinary" in the way of Russell and "irregular" or "nonwellfounded" or "antifoundational" cardinal, as "extraordinary", and it reads the same.
And here it is number theory.
Regards, Ross Finlayson

