Cartesian Coordinate System, the union of geometry with numbers #0 Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept

[Archimedes Plutonium]

The beauty of future editions of a textbook is that errors can be
removed and improvement of teaching can be improved. Already I made a
major error, in that suggesting that we do not connect points of the
graph of the function when we have holes between points, for we
certainly do connect them since the hypotenuse atop the picketfence is
a straight line segment that is connected to a "number point of the
graph". In fact, that is what the Fundamental Theorem of Calculus is
going to be in New Math. In Old Math, they thought their important
Fundamental theorem was going to be the union of the derivative as
inverse to the integral. In New Math we build the derivative as
inverse and why need to prove it when we built it as such. So the
Fundamental Theorem of Calculus in New Math concerns itself with why
and how is it that Euclidean geometry with straight lines and straight
line segments is the only geometry that can build a Calculus, while
Elliptic geometry and Hyperbolic geometry are impossible geometries to
build a Calculus therein?

Now I realized after post-page #3 yesterday (almost 1/3 done with the
book) that it is too complicated for a High School student. So this
post, which should be #4 is now numbered to be #0 so as to make simple
and inviting to High School students. This should be the first page so
that I can constantly refer to this simplified model.

Exercise on Graph Paper

Now I do not know if I can get 100 dots per line in the Usenet
sci.math newsgroup. Let me try.

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The above is a grid of 100 dots for the x-axis and 100 dots for the y-
axis and 100x100=10,000 dots altogether. I doubt the post will show it
in a square 100 dots wide and 100 dots long.

So I need the student or reader to get a graph pad. I have an
Engineering graph pad where each sheet is 35 dots wide and 50 dots
long so I need to take 6 sheets and cut and paste, or cut and tape
them together to form a large sheet that is 100 dots wide and 100 dots
long for a total of 10,000 dots altogether. So if a High School
student, it is wise to do this exercise for you want to refer to it
and to constantly make pencil drawings of functions and graphs to know
what is going on.

Now we pretend, pretend that 10 is infinity borderline, the borderline
where 10 is the last finite number and the largest finite number. So
that has a deep and powerful implication. For it means that 1/10 is
also a borderline of the small.

Now mathematics is the science of precision and that means that
mathematics can handle precision and accuracy only with finite
entities, finite numbers, finite points of geometry. Mathematics
starts deteriorating once it reaches infinity, for it loses precision
and accuracy. When mathematics is in infinity territory, it no longer
is mathematics. The planet Earth is a good analogy here. The planet
Earth has a borderline of solid matter Earth which is the surface of
Earth as a borderline, including water of oceans. Another borderline
is the atmosphere, but beyond the atmosphere we start getting into
Outer Space. Now the magnetosphere of Earth that protects Earth from
Solar rays that are harmful can be considered a borderline for Earth
but beyond that, it is hard to say that Earth has any physical
parameters. Now that is an analogy to mathematics. Mathematics is the
science of precision and can be precise about finite numbers and
finite points of geometry but once it reaches the borderline of
infinity numbers or infinity points of geometry, we no longer have
mathematics of total precision, and math begins to fall apart into
imprecision. Now in the next post, I will talk about the precise
borderline of infinity, but in this post, we pretend the number 10 is
that infinity borderline. For High School students, they can handle
the number 0, 1/10, 2/10, 3/10, .. on up to 10. And they can easily
graph functions with only these numbers in play. They can draw
picketfence structures on this large graph paper.

And as I write the next 9 pages of this 10 page textbook, I will
constantly refer to this 100 x 100 = 10,000 points of the Cartesian
Coordinate System.

--
More than 90 percent of AP's posts are missing in the Google
newsgroups author search archive from May 2012 to May 2013. Drexel
University's Math Forum has done a far better job and many of those
missing Google posts can be seen here:

http://mathforum.org/kb/profile.jspa?userID=499986

Archimedes Plutonium
http://www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies

[Archimedes Plutonium]

Alright, let us go for this 3rd edition and hopefully it will be
logically tight, clear, and mistake free. That is a lot to ask for,
but that is what I am striving for. And success should be near because
this textbook is only 10 post-pages long. (I do not know how long if
in hardcover book print).

Now the Calculus is about two operations, derivative and integral,
much like number theory is two operations of multiply with divide or
add with subtract. In Calculus these two operations of derivative and
integral are inverses of one another and the derivative is rate of
change and given the symbolism of dy/dx. The rate of change is also
called slope or tangent of the graph function at a specific point of
the graph function. The integral is area under the graph function with
the x-axis. So derivative is rate of change and integral is area.

Now Calculus exists within geometry and numbers and it exists because
of a Euclidean Geometry Cartesian Coordinate System that always puts
points of geometry as numbers of number theory and the relative
position of those points yields a fixed angle. So where add and
subtract is about numbers and multiply and divide is about numbers,
the derivative and integral are about numbers that yield fixed angles
between numbers.

In a Cartesian Coordinate System the number- 
points are so spaced and
arranged in order that this spatial 
arrangement yields an angle that
is fixed. So that if you have an 
identity function y = x, the
position of points (1,1) from (2,2) is 
always a 45 degree angle. So
Calculus of derivative and integral is 
based on this fact of
Euclidean Geometry that the coordinates are so 
spatially arranged as
to yield a fixed angle. Numbers forming fixed 
angles gives us
Calculus.

Exercise on Graph Paper
Now I do not know if I can get 100 dots per line in the Usenet
sci.math newsgroup. Let me try.

....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
....................................................................................................
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etc
etc

The above is a grid of 100 dots for the x-axis and 100 dots for the y-

axis (although I have only 10 rows for the y-axis when I should have
100 rows, and 100x100=10,000 dots altogether.

Now let me call this the 10-Grid because 10 is pretended to be the
last
integer number and each integer is divided up into 10 parts so from 0
to 1
we have .1, .2, .3, .4, .5, .6, .7, .8, .9, 1.0

Coordinate System, this 10 Grid.

--
More than 90 percent of AP's posts are missing in the Google

newsgroups author search starting May 2012. They call it indexing; I
call it censor discrimination. Whatever the case, what is needed now
is for science newsgroups like sci.physics, sci.chem, sci.bio,
sci.geo.geology, sci.med, sci.paleontology, sci.astro,
sci.physics.electromag to
be hosted by a University the same as what
Drexel
University hosts sci.math as the Math Forum. Science needs to
be in education
not in the hands of corporations chasing after the
next dollar bill.
Besides, Drexel's Math Forum can demand no fake
names, and only 5 posts per day of all posters which reduces or
eliminates most spam and hate-spew, search-engine-bombing, and front-
page-hogging. Drexel has
done a excellent, simple and fair author-
archiving of AP sci.math posts since May 2012
as seen
here:

[sperm...@yahoo.com]

All talk about the union of geometry with numbers is meaningless as mathematician dont even know what a number is- without circularity
all their definitions about numbers reduce to just this

a number is a number-circularity impredicative

thus we then dont know what a number is



mathematicians give all these proofs about numbers but they dont even know what a number is so their proofs are worthless
as without knowing what a number is they then cant even IDENTIFY what a number is

Australias lead erotic poet colin leslie dean points out Mathematicians cannot define a number with out being impredicative-ie self referential thus mathematicians dont even know what a number is- thus maths is meaningless All mathematicians can say is a number is a number ?thus they don?t know what a number is thus maths is meaningless

http://www.scribd.com/doc/40697621/Mathematics-Ends-in-Meaninglessness-ie-self-contradiction


http://www.iep.utm.edu/predicat/
http://www.iep.utm.edu/predicat/

In many approaches to the foundations of mathematics, the property N
of being a natural number is defined as follows. An object x has the
property N just in case x has every property F which is had by zero
and is inherited from any number u to its successor u+1. Or in
symbols:
Def-N N(x) ? ?F[F(0) ? ?u(F(u) ? F(u + 1)) ? F(x)]
This definition has the nice feature of entailing the principle of
mathematical induction, which says that any property F which is had by
zero and is inherited from any number u to its successor u+1 is had by
every natural number:
?F{F(0) ? ?u(F(u) ? F(u + 1)) ? ?x(N(x) ? F(x))}
However, Def-N is impredicative because it defines the property N by
generalizing over all arithmetical properties, including the one being
defined.


again impredicative definition
Let n be smallest natural number such that every natural number can be
written as the sum of at most four cubes.
again impredicative definition


http://en.wikipedia.org/wiki/Impredicativity
Concerning mathematics, an example of an impredicative definition is
the smallest number in a set, which is formally defined as: y = min(X)
if and only if for all elements x of X, y is less than or equal to x,
and y is in X.


http://en.wikipedia.org/wiki/Set-theore ... al_numbers
http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers

A consequence of Kurt Gödel's work on incompleteness is that in any effectively generated axiomatization of number theory (ie. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other effectively generated formal system cannot capture entirely what a number is.

Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.

Poincaré, amongst others (Bernays, Wittgenstein), held that any attempt to define natural number as it is endeavoured to do so above is doomed to failure by circularity. Informally, Gödel's theorem shows that a formal axiomatic definition is impossible (incompleteness), Poincaré claims that no definition, formal or informal, is possible (circularity). As such, they give two separate reasons why purported definitions of number must fail to define number. A quote from Poincaré: "The definitions of number are very numerous and of great variety, and I will not attempt to enumerate their names and their authors. We must not be surprised that there are so many. If any of them were satisfactory we should not get any new ones." A quote from Wittgenstein: "This is not a definition. This is nothing but the arithmetical calculus with frills tacked on." A quote from Bernays: "Thus in spite of the possibility of incorporating arithmetic into logistic, arithmetic constitutes the more abstract ('purer') schema; and this appears paradoxical only because of a traditional, but on closer examination unjustified view according to which logical generality is in every respect the highest generality."

[hbe...@gmail.com]

On Thursday, May 23, 2013 11:32:58 AM UTC-7, Archimedes Plutonium wrote:
> The beauty of future editions of a textbook is that errors can be
>
> removed and improvement of teaching can be improved. Already I made a
>
> major error, in

Every single thing you've done in your life; particularly every single thing

yu have said about (what you call) Mathematics.

[hbe...@gmail.com]

On Thursday, May 23, 2013 11:32:58 AM UTC-7, Archimedes Plutonium wrote:

> The beauty of future editions of a textbook is that errors can be
>
> removed and improvement of teaching can be improved.

Even better, you can print the book in nice-n-soft 2-ply, and it will

be of some use --to at least one person.

[hbe...@gmail.com]