True Geometry without that ugly contradiction #01 High School or

True Geometry without that ugly contradiction #01 High School or College 8th ed.: TRUE CALCULUS

Alright, good, well, I need for the first 10 pages to correct Geometry, to lay out what is True Geometry before the next 10 pages gives us True Calculus.
This is the 8th edition of this textbook and I mentioned the contradiction of Geometry as taught in schools across the world today, the ugly contradiction that lies at the heart of geometry. I mentioned or referred to this contradiction in all 8 editions but here I need to actually correct Geometry and make it True Geometry. Children and students rebel when they are taught something and later told that what they were taught was fakery and need to relearn the true subject. I want to avert any such rebellion by students, by actually teaching them what True Geometry is, so there is no need to say to students-- what was taught before was fakery and now you learn the real true subject.

Fortunately for me I have a High School geometry text at home titled Geometry, 2nd edition, 1987, Harold Jacobs, so that I can refer to this book for the changes that need be made.

Now let me tell the student that Euclidean Geometry was invented or discovered by the Ancient Greek mathematicians such as Pythagoras and Euclid and many others. Geometry is over 2,500 years old with the Pythagorean theorem that the sides of a right triangle are related as the square of those sides. This theorem inspired the Ancient Greeks to form geometry into a deductive science, a subject ruled by logic, where reasoning is steps of logic that reach a conclusion. The logical reasoning starts with ideas that are beyond doubt and called axioms. Axioms are givens that start the subject and are used to attain more truths. Now Euclid over 2,000 years ago had 10 axioms, five of which he called postulates and five of which he called common notions. But they all mean the same, whether we call them axioms or postulates or notions that they are taken as knowledge beyond doubt.

Now in those 2,000 years since Euclid, many flaws of his axioms were found, so that by recent times of 1899 David Hilbert had to revise much of Euclidean Geometry with not just 10 axioms but with 26 axioms and specifically with three primitive notions:

point

line

plane


and then, three primitive relationships. In Hilbert's book he adds up 3 notions, 3 relations, and 20 axioms for a total of 26 axioms.

In True Geometry, notions, relations, axioms, postulates, primitive terms, are all called axioms. We are not foolish in thinking that different names for something makes them different, for all of these items listed are axioms because axioms are taken as given knowledge beyond doubt. Hilbert, was rather being silly and illogical by calling notions and relations as different from axioms.

Now Jacobs in his book on page 51 lists 3 axioms (he calls them postulates) and on page 52 lists the 4th axiom.

Now in the history of Geometry, Euclid defined the point as "that which has no parts". And for the line, Euclid defined the line as "breadthless length". That was over 2,000 years ago and in the intervening centuries Euclid's axiom of point and line became this:

point: no length, no width, no depth

line: length, but no width, and no depth


Now in Jacobs' first axiom (postulate) he says:

Postulate 1: If there are two points, then there is exactly one line that contains them.

Jacobs then gives the next postulate as this:

Postulate 2: If there is a line, then there are at least two points on the line.

Jacobs' then says this: "There are in fact, infinitely many points on a given line. Such points are called collinear."

So now here we come to the ugly contradiction at the heart of Old Geometry.

In Jacobs book he makes a good attempt of discussing the subject of Logic, for he devotes the first 48 pages to Logic, and specifically on page 37 details what a contradiction is in the method of indirect proof.

But, here is the question before us, whether we compare Jacobs book or any other High School or College text on geometry. I have the college text of Elementary Geometry From An Advanced Standpoint by Edwin Moise, 3rd ed, 1990, and Moise makes the same contradiction that Jacobs makes.

In fact all geometry texts make the same big ugly contradiction that lies at the heart of geometry.

You cannot have a point and line in geometry without the 3rd entity of "empty space between successive points".

A point can be a entity that has no length, no width, and no depth.

A line can be an entity that has length, but no width and no depth.

You can have those two entities that Euclid to Hilbert wanted to have, but only if you admit there is a third entity that must exist alongside point and line, and it is "empty space".

If you cannot admit this third entity, then you have that massive ugly contradiction at the heart of Geometry. Because without empty space, a line composed of nothing but points (Jacobs says infinitely many points) cannot have length itself since the points that compose the line have no length. So a third new entity gives a line a length since points cannot give it length.

I call it empty space, but in actuality, it is composed of numbers that are infinity-numbers. Space is composed of points and those points are finite-number points. Between one finite number and another finite number is empty space of infinity-numbers.

So, take out a ruler and you use it to measure length or distance and you see the hash marks between centimeters having ten hash marks that mark out millimeters. You see no hash marks between two successive millimeters. What do you see between two successive millimeters?

You see empty space between two successive millimeters. What is true for the ruler is true for geometry. To have a length or distance, you need empty space between gradations of length or distance.

--
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 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