replacing Picketfence model with 2 pure triangle model Re: Maxwell Equations as axioms over all of physics and math #9 Textbook 2nd ed. : TRUE CALCULUS; without the phony limit concept
May 25, 2013 3:42 AM
On May 25, 12:43 am, Archimedes Plutonium <plutonium.archime...@gmail.com> wrote: > Alright, I am learning more new things, for in this 2nd edition I have > an alternative to the picketfence model. I have the pure and straight > rectangle model and the pure and straight triangle. In the rectangle > model we fill the dx of 10^-603 width and the height is y itself. In > the pure triangle we have a right triangle on the leftside of the > point of the graph and the same triangle on the rightside with its > hypotenuse in the reverse direction as pictured like this: > > /| > / | > / __| > > unioned with this triangle > > |\ > | \ > |__\ > > is the same area as the rectangle model of the point on the function > graph. > > The problem, though, is that the angle of the hypotenuse does not like > like the slope or tangent to the point of that function graph. So I > need to see if that hypotenuse is related to the slope or tangent or > derivative at that specific point. If it is, then, clearly we see how > derivative is the inverse of integral, because both have the same area > and the triangle hypotenuse would be the derivative. So instead of > rectangles forming the integral we can take two triangles. So > hopefully I can work this out in the 3rd edition which I plan to start > in the next day or so. > > Alright, this is the 10th page of the 2nd edition and the last page. I > want to devote the last page to showing how all this math is begot > from the Maxwell Equations. > Now on this last page I want to show how Calculus of its empty space > between successive numbers is derived from the Maxwell Equations as > the ultimate axiom set over all of mathematics. The Maxwell Equations > derives the Peano axioms and the Hilbert axioms. But I want to show > that the Maxwell Equations do not allow for the Reals to be a > continuum of points in geometry but rather, much like the integers, > where there is a empty space between successive integers. > The Reals that compose the x-axis of 1st quadrant are these: > > 0, 1*10^-603, 2*10^-603, 3*10^-603, 4*10^-603, 5*10^-603, > 6*10^-603 . . on up to 10^603 > > Pictorially the Reals of the x-axis looks like this > ...................> > and not like this > ____________> > > So in the Maxwell Equations we simply have to ask, is there anything > in physics that is a continuum or is everything atomized with empty > space in between? Is everything quantized with empty space in > between? > > I believe the answer lies with the Gauss law of electricity, commonly > known as the Coulomb law. The negative electric charge attracts the > positive electric charge, yet with all that attraction they still must > be separated by empty space. If there was a continuum of matter in > physics, then the electron would be stuck to the proton. The very > meaning of quantum mechanics is discreteness, not a continuum. > Discreteness means having holes or empty space between two particles > interacting of the Maxwell Equations. > So if physics has no material continuum, why should a minor subset of > physics-- mathematics have continuums. If Physics does not have > something, then mathematics surely does not have it. > > Now I end with reminders for the 3rd edition: > > REMINDERS: > (1) First page talk about why Calculus exists as an operator of > derivative versus integral much the same way of add subtract or of > multiply divide because 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. > > (2) Somewhere I should find out if the picketfence model is the very > best, for it maybe the case that a rectangle model versus a pure > triangle model may be better use of the empty space of 10^-603 between > successive Reals (number points). The picketfence model is good, but > it never dawned on me until now that there is likely a better model > even yet-- pure rectangle versus two pure triangles. My glitch is to > get the hypotenuse related to the derivative. If I can solve that > glitch, I have a crystal clear understanding of the derivative, > integral and why they are inverses. > > (3) I am really excited about that new method of arriving at the > infinity borderline of Floor-pi*10^603 via Calculus. The first number > which allows a half circle function to be replaced by a 10^1206 > derivatives of tiny straight line segments and still be a truncated > regular polyhedra, is when pi has those 603 digits rightward of the > decimal point. The derivative of half circles of any number smaller > than Floor-pi*10^603 does not form a circle. And is that not what > Calculus is all about in the first place-- taking curves and finding > Euclidean straight line segments as derivative and area. Calculus is > the interpretation of curved lines into straight line segments. So, > onwards to 3rd edition. >
I am not going to count this as the 11th page of a 10 page textbook, but rather as a reply. I found out tonight that these hypotenuse of right triangles of points on the graph of a function are related to the slope or derivative of the function at that point.
So in my previous graph of the function y=x^2 in 10 grid:
at x=.3, y=.09 at x=.4, y =.16 ?at x=.5, y =.25 Now each of those intervals of .1 width has 2 pure triangles as these two
/| / | ?/ __| unioned with this triangle
|\ | \ |__\
So in the interval between .3 and .4 of a dx of .1 sits two triangles where their hypotenuse cross one another and intersect at a point and the same is true of the next dx =.1 interval of two triangles intersecting and if we draw a line between the two intersections we end up with the derivative. Sort of reminds me of the projective geometry Desargues theorem.
But I need to confirm all of this.
The importance of this is that the picketfence model gets thrown out and replaced by the pure 2 triangles aside each point of the graph of a function and the 2 triangles determine the derivative and the integral and it is easy to see how the derivative is the inverse of integral.
-- 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: