#02 True Geometry without that ugly contradiction; High School or

We usually think of algebra as the science of manipulation of numbers. And we think of geometry as the science of space and the shape of figures in space. That is the commonsense view of algebra and geometry. In physics, algebra and geometry are duals of one another where both are needed just as the right hand is the dual of the left hand so that both hands work together. In physics it is called electricity and magnetism duality, in mathematics, it is algebra of numbers and geometry of space and figures.

So in Old Math, of its Old Geometry that has a huge gigantic contradiction at the heart of geometry, where you have lines composed of only points and possession of length yet points have no length.

The resolution is simple and easy enough, for we are missing an entity for which a line composed of points of no length can have length. That missing entity is "empty space between each successive point".

So how do we arrive at what this empty-space is?

We get it from the fact that a point can be either a finite point or an infinity-point.

Imagine for simplicity that infinity starts after 10 and that 10 is the largest finite number. So that the number 10.001 is an infinity number, as well as the number 11 or 12. Well, if 10 is the last finite number then 1/10 or 0.1 is the smallest nonzero finite number. So that between the number 0 and 0.1 are no longer any finite number but all sorts of infinity-numbers. For example the number 0.0000003 or the number 0.012345 are two infinity-numbers.

So the solution to the contradiction in Fake Geometry, is to find the borderline of large infinity, such as the number 10^603 which is 1 with 603 zeroes after it. And once it is taken that 10^603 is large infinity, the inverse of that number is 10^-603
or 1 divided by 10^603. The number 10^-603 is the empty space between all finite numbers and their successor finite number.

To give an example that High School students understand. The counting numbers are 0, 1, 2, 3, 4, 5, etc etc. They are all separated from their immediate successor by a unit length or distance of 1. So that a hole of 1 is between 0 and 1 and between 1 and 2 and between 2 and 3, etc etc.

So a line composed of 0, 1, 2, 3, 4, to some large counting number would have length because of the 1 unit of empty space between successive points (successive counting numbers).

So, Old Fake Geometry had a contradiction at the heart of its system where its axiom of line contradicted its axiom of point, unless a new entity of empty space is introduced as a new axiom. And the introduction of empty space is the large infinity which we call macroinfinity, so that if 10 were the last and largest finite number then 1/10 is the smallest nonzero finite number and it is called microinfinity.

So if 10 was macroinfinity border, then 1/10 is microinfinity border, and in such a system, all the numbers that mathematics and geometry can use are the numbers 0, .1, .2, .3, .4, .5, .6 on up to 9.8, 9.9 and finally 10. Every number in between those are part of microinfinity numbers and make up empty-space.

We cannot use infinity-numbers in mathematics because math is the science of precision and precision is finite-numbers. A number in Old Math of 0.33333.... or 0.99999...... are infinity numbers and not part of mathematics as the science of precision. Infinity numbers do exist, but unuseable for mathematics. They play an important role in mathematics for they compose Space we call "empty space".

In physics, when we have infinity numbers in our equations we get rid of them in a process called normalization.

In mathematics, we recognize infinity numbers are plentiful and abundant just as empty-space is plentiful, but we cannot use them in mathematics.

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Drexel's Math Forum has done an excellent search engine for author posts as seen he
http://mathforum.org/kb/profile.jspa?userID=499986

Now, the only decent search for AP posts on Google Newsgroups, is a search for for it brings up posts that are mostly authored by me and it brings up only about 250 posts. Whereas Drexel brings up nearly 8,000 AP posts. Old Google under Advanced Search
could bring up 20,000 of my authored posts but Google is deteriorating in quality search of old posts to the science newsgroups.

So the only search engine today doing author searches is Drexel.

All the other types of Google searches of AP are just top heavy in hate-spam posts due to search-engine-bombing practices by thousands of hatemongers who have nothing constructive to do in their lives but attack other people.

Funny, and I have to laugh here, for in the recent news of spying on Europe by USA, such a big fuss, whereas I want that sort of full recording of all of my posts to Usenet. I am hoping the spy agencies of the USA has recorded every last one of the posts that AP has posted to the Usenet and please keep up that spy activity. I have kept a personal archive and figure that in these 20 years of posting of say 5 posts per day makes 365 x 5 = 1825 times 20 = 36500. So roughly I have about 36,500 posts to Usenet in these past 20 years. I hope many computers and servers have all those 36,500 posts safely stored and please feel free to display them. Whereas Google is working in the exact opposite, by trying to hide or destroy them, or bias the listing of those author posts.

Archimedes Plutonium