Analytic Foundations of Geometry

Robert S. Wilson

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This website uses a format, which is standard for presenting mathematical material: A sequence of theorems. A theorem is a statement, which has a proof. A proof is a sequence of statements, where each statement has a reason, and each reason is a previously established statement. It could be a statement, which has been previously established in the current proof, or it could be a previously proven theorem. The theorems are stated so that they may serve as reasons in subsequent proofs. It also contains definitions, which can also serve as reasons.

 

The person who gets credit for developing this format is Euclid, who used it in his Elements (c. 300 BC). One problem with this format is that one cannot have statements and reasons going back forever. One must have a starting point. Euclid took as his starting points, in addition to his definitions, some axioms and postulates. The axioms were what he called "common notions", such as "The whole is greater than the part." The postulates were more mathematical in nature, such as "Two points determine a straight line." Now days, rather than distinguish between axioms and postulates, we generally just use the term a "axiom" to denote the statements, we are going to assume without proof, for starting points. The way it actually works, is that whenever you can show that the axioms, upon which a theory is founded, are true, then all of the propositions in the theory will be true.

 

Euclid did not do a good job with his axioms and postulates. His first theorem does not follow from them. The first theorem is to construct an equilateral triangle with a given side. Euclid’s axioms are true in the set of ordered pairs of real numbers, in the Cartesian coordinate plane, which have rational coordinates, but it is possible to prove that there is no equilateral triangle in the Cartesian coordinate plane, which has all rational coordinates.

Euclid’s definitions also leave something to be desired. In his definition of "A line is breadthless length." what is "length"? Before getting to what "breadthless" means, what is "breadth"? Some of these definitions, upon which everything rests, use undefined terms.

 

If one is going to build a towering structure of logic, like Euclid’s geometry, the worst place to have flaws is in the foundation.  Since Euclid, especially in the last several hundred years, there have been many successful attempts to devise a system of axioms which will provide a satisfactory foundation for Euclid’s geometry.

 

Euclid did not include some axioms or postulates, which he needed. Two axioms, which have been added, are the so-called ruler axiom and protractor axiom. The ruler axiom states that the distance from the endpoint of a line segment to a point on the segment sets up a one to one correspondence between the points on the segment and the set of real numbers in the interval between  0  and the length of the segment, and the protractor axiom similarly says that the arc length sets up a one to one correspondence between the points on the arc of a circle and the set of real numbers in the interval between  0  and the length of the arc.

 

Another issue was Euclid’s fifth or parallel postulate. An axiom or postulate should be a simple statement, to which everybody would agree. Where the other axioms and postulates were succinct and seemed obvious, the fifth postulate was not and struck people as being more like a theorem than an axiom or postulate.  In 1795, John Playfair (1748-1819) proposed the following substitute:

 

Given a line and a point not on the line, there is exactly one line through the given point, which, is parallel to the given line.

 

Attempts to prove that the parallel postulate from the other axioms and postulates came to a halt when, several years later, Gauss showed that there were geometries where the other axioms held where the parallel postulate did not. There are two other possibilities. In an elliptic plane, there are no lines through the given point, which are parallel to the given line, and, in a hyperbolic plane, there are more than one.  There are examples of all three types of geometric spaces.  There are many other equivalents to the parallel postulate, such as the Pythagorean Theorem, and the fact that the angles in a triangle add up to180o. In an elliptic plane, a2 + b2 > c2, and in an hyperbolic plane, a2 + b2 < c2. In an elliptic plane, there are more than 180o in a triangle, and in an hyperbolic plane, there are less than 180o. Your axiom system will have to include some equivalent of the parallel postulate in order to specify which type of plane you have.

 

Another trouble with Euclid’s approach, which is called the synthetic approach, is that while it rolls along quite nicely when you have a nice backlog of theorems, which are proven and can serve as reasons, when one is first starting off, one can find oneself rather cramped.

 

This website deals with these issues by using a different approach, which is called the analytic approach and uses algebra to establish the foundations of geometry.  Euclid has to be excused for not using this approach, because the algebra which it requires was not available in  his time. You will hear that, algebra was developed by the Arabs in the ninth century AD. Even though this is about 1100 years after Euclid, it was still not sufficient for our purposes. Algebra is arithmetic with unknowns, but where we would write

 

2x + 5

 

the Arabs of that time would write

 

Double your unknown and then add five

 

long hand in Arabic. While many people might think it would be easier to express themselves in their native tongue than to use our algebraic notation, notice that the second method uses much more ink than the first. Moreover, simple tasks such as simplifying a linear expression in one unknown, which has many terms and parenthesis sprinkled liberally throughout, would not be nearly as easy using long hand as it is using our current short hand. It wasn’t until about the end of the sixteenth century that Francois Viete developed notation along the lines of that which we currently use.

 

It was very shortly after this, in the early to middle seventeenth century, nearly two thousand years after Euclid, that Rene Descartes got credit for developing analytic geometry. In the Cartesian coordinate plane, geometric objects like lines and circles have algebraic equations.

 

In the synthetic approach, some features for which one looks in the set of axioms are completeness – are there enough axioms to prove all of the theorems you want, and independence – can any of the axioms be proved from the other axioms? Another feature, which follows from independence, is how many axioms are you using? It is not unusual to find axiomatizations, which use around a dozen or so axioms. While the more axioms you have, the more statements, which can serve as reasons in proofs, also the more unproven propositions upon which your theorems will rest. Using the analytic approach, it is possible to use the absolute minimum number of axioms: none.

 

In the analytic approach, we define a point to be an ordered pair of real numbers. The plane is the set of all ordered pairs of real numbers. A line is the set of points, which satisfies a linear equation. There are some axioms lurking around here. They are the axioms of the real number system. However, these could be taken to be the definition of real number system, and not axioms at all. They are derived from the axioms of set theory and the rules of logic. We need to use the real number system for the ruler axiom and the protractor axiom.

 

One question, which always comes up, is what are you using as your equivalent to Euclid’s parallel postulate. We are not using any axiom, which is equivalent. We are using the standard distance formula for defining the Euclidean distance between two points in Cartesian coordinate plane. This, of course follows from the Pythagorean Theorem, but we are not assuming the full Pythagorean Theorem. With the distance formula, we are only assuming the Pythagorean Theorem for right triangles where the perpendicular sides are horizontal and vertical. We will need to prove the full Pythagorean Theorem from that assumption.

 

This is not intended to eliminate synthetic approaches. There is a wonderful tradition of synthetic approaches to geometry going back thousands of years to Euclid. It is also not going to cover all of geometry. This is intended simply to get enough basic theorems proved that a student can proceed synthetically. Basically we need to get through congruent triangles, which is a very ubiquitous and powerful topic. Along the way we will prove that all of the traditional axioms of Euclidean geometry are true in the Cartesian coordinate plane

 

This will prove to be useful is in dealing with questions of completeness and independence for axiom systems. This will show that one way to demonstrate that your axiom system is adequate would be to show that you can use your axioms to coordinatize your plane into a Cartesian plane isomeric to the set of all ordered pairs of real numbers, with the distance function between such ordered pairs given by the distance formula of the Euclidean metric. Any axioms, which are not needed to accomplish this will then be seen to be not independent.

 

In California, there is a state wide prerequisite of intermediate algebra for any course, which will satisfy the General Education requirement in mathematics for the California State University system. Students will ask, "What is algebra good for?" Algebra is good for geometry. And the algebra is the most incredible algebra you have ever seen. It has been an incredible amount of fun going through it all, and I am very pleased to be able to share it with you.

 

We only need beginning algebra until we get to arc length, which is a calculus problem. While it is possible to explain to students, who are nowhere near ready for calculus, the process of taking the limit of finer and finer polygonal approximations of the length of an arc, it is a calculus problem.

 

The reader will hopefully find adequate opportunities to navagate through the website. The material is divided up into six sections or chapters. The home page for the website consists of a table of contents, which lists six sections or chapters. Each section contains a list of definitions of terms used in the section, and a list of theorems, which is the homepage for the secton. In the header for each of these pages, The "Analytic Foundations of Geometry" is a link back to the table of contents homepage for this geometry site, and anywhere you see any form of my name is a link to the homepage for the entire website. The reader may find it convenient to have a list of the theorems in a section together to survey without the space which the proofs would take up. For the proofs, in the list of theorems for the section, the theorem numbers are links to the proofs. In the proof, the theorem number is link back to the statement of the theorem in the list of theorems for the section.

 

A website is an excellent venue for such mathematical presentations. When one reads that something is true because of, say, Theorem 3.2, one may need to look up Theorem 3.2. On a web page, the reference to Theorem 3.2 is a link, and, instead of having to flip pages, one needs only click on the link, and Theorem 3.2 will pop up. If it depends on the definition of a word, the word is a link to its definition. At the end of the proof, there is a link to the next theorem.

 

Another capability of a website is the opportunities for footnotes. In the lists of theorems and the definitions, there are some numbered footnotes. The numbering of the footnotes starts anew with each page, but it really doesn't matter, because the footnote number is a link to the footnote, so one need only click on the footnote number and arrive immediately at the footnote. At the end of the footnote is a link "Return to text" which will take you to the start of the paragraph which contains the footnote. The ends of the footnotes also contain links back to the "Analytic Foundations of Geometry" table of contents homepage for this site, followed by the author's name which will take you back to the author's website.

 

The first section deals with lines. This is just algebra with first degree polynomials. Finding the point where two lines intersect is a matter of solving a system of simultaneous linear equations. We can show that two lines intersect unless they have the same slope, so we define parallel and perpendicular using slopes. Playfair’s Postulate is then the point-slope form of the equations. Things like the transitivity of parallelism are quite immediate using slope considerations.

 

One point, which comes up in developing the foundations of Geometry, is the Pasch property: if a line goes into a triangle it will come out the other side. If you bisect an angle in a triangle, how do you know that it intersects the opposite side? People often unconsciously assume this without proof, and may authorities take exception when that happens. The second section is devoted to the Pasch property. We use parametric equations of lines, which is a very powerful technique, although it actually only uses first degree linear polynomials.  We need it for refining outside approximations when taking arc length. The reason for putting it in a section as early as the second is that it is good for proving the segment addition axiom, which is in Chapter 3.

 

 Chapter 3 deals with circles. At this point we start to use quadratic equations. This is when the algebra displays its incredible intelligence. One direction of the full Pythagorean Theorem falls out as a result of a simple algebra problem. Of course the algebra is rather extensive, but if you enjoy mind spangling walls of algebra, you will enjoy this. Later, the triangle inequality will just drop out of the formula for the points of intersection of two circles.

 

After we have established the formulas for the intersection of lines and circles, and parametric equations for lines, The primary goal is to establish the criteria for congruent triangles. Once you have congruent triangles, a synthetic approach will work quite nicely. We will accomplish this by using rigid motions of the plane. Since we will be able to accomplish everything we want by using translations, reflections, and rotations, we will not bother to prove that any rigid motion can be obtained in this manner, or that any rigid motion which is not a translation , reflection, or rotation is a glide reflection, i.e. a reflection followed by a transaltion. We cover translations and reflections in Chapter 4. By this point, we have a sufficient backlog of previously proven results that a synthetic proof of the fact that a reflection is an isometry is feasible. However, there is also an algebraic proof, which compares with some of our other spectacular algebraic proofs, so we present them both. Rotations will have to wait until we get to angles, and angle measurement depends upon arc length, which we cover in Chapter 5. Chapter 6 is congruent and similar triangles. At this point, in place of a dozen or so axioms, we have eight-two theorems. I believe you will find that this will serve as a good supply of reasons for proceeding in a synthetic development, without any unproven assumptions beyond the axioms of set theory and the rules of logic.

 

Enjoy.