10.2 Axiomatic Systems

One of the primary goals of secondary mathematics education is to develop ways of thinking that extend beyond mathematics. One of these ways of thinking is the ability to use logical arguments to prove mathematical statements using only the most basic of assumptions and how the changing of assumptions can drastically change later outcomes.

An axiomatic system consists of certain undefined terms and a list of axioms or postulates concerning these undefined terms. One can then build a mathematical theory by proving propositions, lemmas, theorems, and corollaries using only the axioms, postulates, or previously proven statements. In the process of building the theory, additional definitions are often developed to aid in the precision of language.

In chapters 2, 3, and 4, we worked through an axiomatic system based on the ZFC axioms of set theory to construct the various number systems used in K-12 mathematics. We now turn our attention to axiomatic systems used to study Geometry.

10.2.1 Euclid’s Common Notions and Postulates

One of the earliest axiomatic systems was developed in Ancient Greece by Euclid, around 300 B.C., based upon the work of many previous philosophers and mathematicians (Heath, 1908a). Euclid’s system begins with five common notions that are independent from geometry, followed by five postulates of geometry (Heath, 1908a, pp. 154–155). Both the common notions and postulates would be considered axioms in today’s language.

Common Notions

The Common Notions are ideas that Euclid determines are generally well accepted.

Things which are equal to the same thing are also equal to one another.

If equals be added to equals, the wholes are equal.

If equals be subtracted from equals, the remainders are equal.

Things which coincide with one another are equal to one another.

The whole is greater than the part.

We can note that these common notions have a strong connection to the concept of equivalence that we studied in Chapter 3 and the basics of set theory in Chapter 2. In this regard, we can consider these common notions to be the precursor to our modern ZFC axioms, though not as rigorous.

Postulates

To draw a straight line from any point to any point.

To produce a finite straight line continuously in a straight line.

To describe a circle with any centre and distance.

That all right angles are equal to one another.

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.

Many mathematicians worked to prove that the fifth postulate was a consequence of the prior four postulates. However, we now know that this postulate is independent of the others and with replacing this postulate with a variation one can derive spherical geometry or hyperbolic geometry.

10.2.2 Hilbert’s Axioms

David Hilbert was one of the preeminent mathematicians of the late 19th and early 20th centuries and was a leader in the movement to formalize mathematics as a response to the discovery of inconsistencies and paradoxes in the logical system of mathematics being built upon the set theory of George Boole. One of his first endeavors in this direction was to create a set of axioms that would not contain the problems and inconsistencies of Euclid’s Elements (Hilbert, 1910).

Hilbert divided his axioms into five categories (Hilbert, 1910, pp. 3–26).

I. Axioms of Connection

Two distinct points $A$ and $B$ always completely determine a straight line $a$. We write $AB=a$ or $BA=a$.

Any two distict points of a straight line completely determine that line; that is, if $AB=a$ and $AC=a$, where $B\neq C$, then is also $BC=a$.

Three points, $A$, $B$, $C$ not situated in the same straight line always completely determine a plane $\alpha$. We write $ABC = \alpha$. We employ also the expressions: $A$, $B$, $C$, “lie in” $a$; $A$, $B$, $C$ “are points of” $\alpha$, etc.

Any three points $A$, $B$, $C$ of a plane $\alpha$, which do not lie in the same straight line, completely determine that plane.

If two points $A$, $B$ of a straight line $a$ lie in a plane $\alpha$ then every point of (the line) $a$ lies in (the plane) $\alpha$. In this case we say: “The straight line $a$ lies in the plane $\alpha$,” etc.

If two planes $\alpha$, $\beta$ have a point $A$ in common, then they have at least a second point $B$ in common.

Upon every straight line there exist at least two points, in every plane at least three points not lying in the same straight line, and in space there exist at least four points not lying in a plane.

Axioms of Order

If $A$, $B$, $C$ are points of a straight line and $B$ lies between $A$ and $C$, then $B$ lies also between $C$ and $A$.

If $A$ and $C$ are two points on a straight line, then there exists at least one point $B$ lying between $A$ and $C$ and at least one point $D$ so situated that $C$ lies between $A$ and $D$.

Of any three points situated on a straight line, there is always one and only one which lies between the other two.

Any four points $A$, $B$, $C$, $D$ of a straight line can always be so arranged that $B$ shall lie between $A$ and $C$ and also between $A$ and $D$, and,furthermore, that $C$ shall lie between $A$ and $D$ and also between $B$ and $D$.

Let $A$, $B$, $C$ be three points not lying in the same straight line and let $a$ be a straight line lying in the plane $ABC$ and not passing through any of the points $A$, $B$, $C$. Then if the straight line $a$ passes through a point of the segment $AB$, it will also pass through either a point of the segment $BC$ or a point of the segment $AC$.

Axiom of Parallels

In a plane $\alpha$ there can be drawn through any point $A$, lying outside of a straight line $a$, one and only one straight line which does not intersect the line $a$. This straight line is called the parallel to $a$ through the given point $A$.

Axioms of Congruence

If $A$, $B$ are two points on a straight line $a$, and if $A'$ is a point upon the same or another straight line $a'$, then, upon a given side of $A'$ on the straight line $a'$, we can always find one and only one point $B'$ so that the segment $AB$ (or $BA$) is congruent to the segment $A'B'$. We indicate this relation by writing \[AB \equiv A'B'.\] Every segment is congruent to itself; that is, we always have \[AB\equiv AB.\]

If a segment $AB$ is congruent to the segment $A'B'$ and also to the segment $A''B''$, then the segment $A'B'$ is congruent to the segment $A''B''$; that is, if $AB \equiv A'B'$ and $AB\equiv A''B''$, then $A'B' \equiv A''B''$.

Let $AB$ and $BC$ be two segments of a straight line $a$ which have no points in common aside from the point $B$, and, furthermore, let $A'B'$ and $B'C'$ be two segments of the same or of another straight line $a'$ having, likewise, no point other than $B'$ in common. Then, if $AB\equiv A'B'$ and $BC \equiv B'C'$, we have $AC \equiv A'C'$.

Let an angle $(h,k)$ be given in the plane $\alpha$ and let a straight line $a'$ be given in a plane $\alpha'$. Suppose also that, in the plane $\alpha'$, a definite side of the staight line $a'$ be assigned. Denote by $h'$ a half-ray of the straight line $a'$ emanating from a point $O'$ of this line. Then in the plane $\alpha'$ there is one and only one half-ray $k'$ such that the angle $(h,k)$, or $(k,h)$, is congruent to the angle $(h',k')$ and at the same time all interior points of the angle $(h',k')$ lie upon the given side of $\alpha'$. We express this relation by means of the notation \[\angle (h,k) \equiv \angle (h',k').\] Every angle is congruent to itself; that is, \[\angle (h,k) \equiv \angle (h,k)\quad \mbox{ or } \quad \angle (h,k) \equiv \angle (k,h).\]

If the angle $(h,k)$ is congruent to the angle $(h',k')$ and to the angle $(h'',k'')$, then the angle $(h',k')$ is congruent to the angle $(h'',k'')$; that is to say, if $\angle (h,k) \equiv \angle (h',k')$ and $\angle (h,k) \equiv \angle (h'',k'')$, then $\angle (h',k') \equiv \angle (h'',k'')$.

If, in the two triangles $ABC$ and $A'B'C'$, the congruences \[AB \equiv A'B', \quad AC \equiv A'C', \quad \angle BAC \equiv \angle B'A'C'\] hold, then the congruences \[\angle ABC \equiv \angle A'B'C' \quad \mbox{ and } \quad \angle ACB \equiv A'C'B'\] also hold.

V. Axiom of Continuity

Let $A_1$ be any point upon a straight line between the arbitrarily chosen points $A$ and $B$. Take the points $A_2, A_3, A_4, $ so that $A_1$ lies between $A$ and $A_2$, $A_2$ between $A_1$ and $A_3$, $A_3$ between $A_2$ and $A_4$, etc. Moreover, let the segments \[AA_1, A_1A_2, A_2A_3, A_3A_4, \ldots\] be equal to one another. Then, among this series of points, there always exists a certain point $A_n$ such that $B$ lies between $A$ and $A_n$.

10.2.3 School Mathematics Study Group Axioms

Following World War II, a group of mathematicians and mathematics educators called the School Mathematics Study Group (SMSG) developed a curriculum for K-12 education with funding from the National Science Foundation. In order to simplify the axiomatic systems of Euclid and Hilbert, they created a system of 22 postulates that is equivalent to those of Euclid and Hilbert. Their primary goal was to not focus on minimality, but on accessibility, by allowing some of the postulates to be consequences of prior postulates. This allowed the curriculum to instead focus on certain steps in the process of building a consistent geometric system.

The School Mathematics Study Group (1960) axiomatic system has three primary undefined terms of point, line, and plane. From these terms we have the following postulates.

(Line Uniqueness) Given any two distinct points there is exactly one line that contains them.

(Distance Postulate) To every pair of distinct points there corresponds a unique positive number. This number is called the distance between the two points.

(Ruler Postulate) The points of a line can be placed in a correspondence with the real numbers such that:

To every point of the line there corresponds exactly one real number.

To every real number there corresponds exactly one point of the line.

The distance between two distinct points is the absolute value of the difference of the corresponding real numbers.

(Ruler Placement Postulate) Given two points $P$ and $Q$ of a line, the coordinate system can be chosen in such a way that the coordinate of $P$ is zero and the coordinate of $Q$ is positive.

(Existence of Points) Every plane contains at least three non-collinear points. Space contains at least four non-coplanar points.

(Points on a Line Lie in a Plane) If two points lie in a plane, then the line containing these points lies in the same plane.

(Plane Uniqueness) Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane.

(Plane Intersection) If two planes intersect, then that intersection is a line.

(Plane Separation Postulate) Given a line and a plane containing it, the points of the plane that do not lie on the line form two sets such that:

each of the sets is convex;

if $P$ is in one set and $Q$ is in the other, then segment $\overline{PQ}$ intersects the line.

(Space Separation Postulate) The points of space that do not lie in a given plane form two sets such that:

Each of the sets is convex.

If $P$ is in one set and $Q$ is in the other, then segment $\overline{PQ}$ intersects the plane.

(Angle Measurement Postulate) To every angle there corresponds a real number between $0^\circ$ and $180^\circ$.

(Angle Construction Postulate) Let $\overrightarrow{AB}$ be a ray on the edge of the half-plane $H$. For every $r$ between $0$ and $180$, there is exactly one $\overrightarrow{AP}$ with $P$ in $H$ such that $m\angle PAB = r$.

(Angle Addition Postulate) If $D$ is a point in the interior of $\angle BAC$, then $m\angle BAC = m\angle BAD + m\angle DAC$.

(Supplement Postulate) If two angles form a linear pair, then they are supplementary.

(SAS Postulate) Given a one-to-one correspondence between two triangles (or between a triangle and itself). If two sides and the included angle of the first triangle are congruent to the corresponding parts of the second triangle, then the correspondence is a congruence.

(Parallel Postulate) Through a given external point there is at most one line parallel to a given line.

(Area of Polygonal Region) To every polygonal region there corresponds a unique positive real number called the area.

(Area of Congruent Triangles) If two triangles are congruent, then the triangular regions have the same area.

(Summation of Areas of Regions) Suppose that the region $R$ is the union of two regions $R_1$ and $R_2$. If $R_1$ and $R_2$ intersect at most in a finite number of segments and points, then the area of $R$ is the sum of the areas of $R_1$ and $R_2$.

(Area of a Rectangle) The area of a rectangle is the product of the length of its base and the length of its altitude.

(Volume of Rectangular Parallelepiped) The volume of a rectangular parallelepiped is equal to the product of the length of its altitude and the area of its base.

(Cavalieri’s Principle) Given two solids and a plane. If for every plane that intersects the solids and is parallel to the given plane such that the two intersections determine regions that have the same area, then the two solids have the same volume.

10.2.4 Discussion

While Euclid has significantly fewer axioms, it also has some holes in the logical consequences particularly in the proof of the SAS triangle congruence. It also makes some assumptions regarding properties of the real number system. In particular, it assumes completeness of the real numbers. Hilbert deals with the real number system with the Axiom of Continuity and the SMSG system uses properties of the real numbers built up in parallel with the Distance Postulate and the Ruler Postulate.

While the goal of the formalists to develop a foundation for all of mathematics from a single set of axioms can never be achieved (see the Gödel Incompleteness Theorems), the goal of using axiomatic methods for mathematics education in order to help students to reason with logic and to justify their arguments is attainable.

10.2.5 Exercises

Compare the parallel postulates/axioms between the different systems. How are they similar and different?
Discuss the similarities of the Common Notions of Euclid with algebra of sets from Chapter 2.
The introduction to High School: Geometry in the Common Core state,

During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs. Later in college some students develop Euclidean and other geometries carefully from a small set of axioms.
1. Based on this statement, what is the role of studying geometry from an axiomatic perspective in secondary mathematics?
2. Which students develop Eucliden and other geometries carefully from a small set of axioms? Does this impact the implication discussed in the previous question?

References

Heath, T. L. (1908a). The thirteen books of Euclid’s elements: Translated from the text of Heiberg with introduction and commentary: Vols. 1, Introduction and Books I and II. University Press.

Hilbert, D. (1910). The foundations of geometry (Second). The Open Court Publishing Company.

School Mathematics Study Group. (1960). Mathematics for high school: geometry. Yale University.