Date: Feb 15, 2013 5:58 PM Author: ross.finlayson@gmail.com Subject: Re: Outline: A Program to establish the continuity of points in a line On Feb 2, 2:47 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>

wrote:

> On Feb 2, 1:22 pm, FredJeffries <fredjeffr...@gmail.com> wrote:

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> > On Feb 2, 1:02 pm, "Ross A. Finlayson" <ross.finlay...@gmail.com>

> > wrote:

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> > > It might be remiss to not note that of course there are a wide variety

> > > of mathematical developments over time and in history that don't

> > > necessarily have as much approbation as they should in the

> > > contemporary, with Cauchy/Dedekind/Weierstrass in analysis then to

> > > Cantor, Russell, and Zermelo and Fraenkel in axiomatic foundations as

> > > "modern". Newton's, Leibniz', and du Bois-Reymond's infinitesimals

> > > are notably absent from the one (though Leibniz' notation survives),

> > > and primary notions of Kant, Hegel, Frege, Quine, Popper the other.

> > > As well, there are modern attempts to formulate these particular

> > > notions of the integers as infinite and reals as complete that aren't

> > > the standard, in light of and in extension of the standard, for

> > > example of Aczel, Priest, Boucher, Paris and Kirby, and Bishop and

> > > Cheng.

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> > There is one outstanding difference between all of those and the

> > gibberish you post: All of them can be used to solve actual problems

> > whereas you still cannot show how to use your nonsense to do even

> > something as simple as determining the area of a triangle.

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> This could be done in this program in this manner, establishing:

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> 1) the integer lattice points

> 2) area bounded by integer lattice points (here 4-many, the unit

> square)

> 3) rationals (here 1/2 particularly for symmetrical complements, then

> generally)

> 4) the triangle (or rather tri-lateral) halving the unit square via

> symmetry

> 5) its area then generally

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> This has unit hyper-volume of the unit n-cube.

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> Fred, the area of the triangle is determined by its sides.

>

"Does Euclid tell us how to measure the size of a triangle?"

In Proposition 41, Euclid tells us that a triangle is half the size of

a parallelogram which has the same base and is within the same

parallels as the triangle. Since a rectangle is a special kind of

parallelogram, we may also infer that a triangle is half the size of

the rectangle having the same base with it and which is within the

same parallels. This proposition gives us the area of the triangle,

provided we know the area of the rectangle in question. Does Euclid,

in Book I, tell us what the area of the rectangle is? Why do you

suppose that he does not, but rather reserves the topic for Book VI,

after he has developed the theory of ratio and proportions?

Why does Euclid not simply say, as we learn in high school, that "the

area of a triangle is equal to one-half of the base times the

altitude"?

-- Adler, Wolff, "Foundations of Science and Mathematics", 1960

Here there is the typical disconnect between magnitude and multitude,

that area is defined and teleological, where instead with the spiral

space-filling curve, there's a notion that the areas are constructed,

then via the properties of the square, that area (in the square) is

established, final cause as first principle.

Regards,

Ross Finlayson