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\title{Axiomatic Systems for Geometry\footnote{From
textit{Post-Euclidean Geometry: Class Notes and Workbook}, UpClose Printing
\& Copies, Champaign, IL 1995, 2004}
}
\author{George Francis\footnote{Prof. George K. Francis, Mathematics
Department, University of
Illinois, 1409 W. Green St., Urbana, IL, 61801. (C) 2010 Board of Trustees.}
}
\date{composed 6jan10, adapted 27jan15}
\begin{document}
\maketitle
\section{Basic Concepts}
\margin{Adaptation to the current course is in the margins.}
An axiomatic system contains a set of {\it primitives} and {\it axioms}. The
primitives are object names, but the objects they name are left
undefined. This is why the primitives are also called {\it undefined
terms}. The axioms are sentences that make assertions about the
primitives. Such assertions are not provided with any justification,
they are neither true nor false. However, every subsequent assertion
about the primitives, called a {\it theorem}, must be a rigorously logical
consequence of the axioms and previously proved theorems. There are
also formal {\it definitions} in an axiomatic system, but these serve only
to simplify things. They establish new object names for complex
combinations of primitives and previously defined terms. This kind of
definition does not impart any `meaning', not yet, anyway.
\margin{See Lesson A6 for an elaboration.}
If, however, a definite meaning is assigned to the primitives of the
axiomatic system, called an {\it interpretation}, then the theorems
become meaningful assertions which might be true or false. If for a
given interpretation all the axioms are true, then everything asserted
by the theorems also becomes true. Such an interpretation is called
a {\it model} for the axiomatic system.
In common speech, `model' is often used to mean an example of a class
of things. In geometry, a model of an axiomatic system is an
interpretation of its primitives for which its axioms are true. Since
a contradiction can never be true, an axiom system in which a
contradiction can be logically deduced from the axioms has no model.
Such an axiom system is called {\it inconsistent}. On the other hand,
if an abstract axiom system does have a model, then it must be
consistent because each axiom is true, each theorem is a logical
consequence of the axioms, and hence it is true, and a contradiction
cannot be true.
\margin{Hvidsten, in his textbook ``Geometry with Geometry Explorer",
McGrawHill, 2004, devotes a large part of Chapter 1 giving examples for
these concepts using finite geometries. Earlier editions of
MA402 treats this subject as well. In this edition of the course, we
regrettably dispense with it in the interest of having more time on
other topics.}
Finally, an axiom system might have more than one model. If two models
of the same axiom system can be shown to be structurally equivalent,
they are said to be {\it isomorphic}. If all models of an axiom system
are isomorphic then the axiom system is said to be {\it categorical}.
Thus for a categorical axiom system one may speak of {\it the} model;
the one and only interpretation in which its theorems are all true.
All of these qualities: truth, logical necessity, consistency,
uniqueness were tacitly believed to be the hallmark of classical
Euclidean geometry. At the start of the 19th century, a scant 200
years ago, philosophers and theologians, physicists and mathematicians
were all persuaded that Euclidean geometry was absolutely the one and
only way to think about space, and therefore it was the job of
geometers to develop their science in such a way as to demonstrate
this necessary truth. By the end of the century, this belief had
been thoroughly discredited and abandoned by all mathematicians.\footnote{
Curiously, it persists even today among some irresponsible, but
influential amateurs. In her column, ``Ask Marilyn" in
Parade Magazine, November 21, 1993, the world's most intelligent
woman [sic] rejects Hyperbolic Geometry on nonsensical grounds, see
\url{http://en.wikipedia.org/wiki/Marilyn_vos_Savant}.}
The main theme of our course concerns the evolution of this
idea, and its replacement by the much richer, far more illuminating,
post-Euclidean geometry of today. It is about a method of thought,
called the {\it axiomatic method.} Although at one time this method
may have developed merely from a practical need to verify the rules
obtained from the careful observation of physical experiments, this
changed with the Greek philosophers. The axiomatic method has formed
the basis of geometry, and later all
of mathematics, for nearly twenty-five hundred years. It survived a
crisis with the birth of non-Euclidean geometry, and remains today one
of the most distinguished achievements of the human mind.
As we noted earlier, the transition of geometry from \textit{inductive
inference} to \textit{deductive reasoning} resulted in the development of
axiomatic systems. Next, we look at four axiom systems for
Euclidean geometry, and close by constructing a model for one of them.
\section{Euclid's Postulates:}
Earlier, we referred to the basic assumptions as `axioms'.
Euclid divided these assumptions into two categories ---
postulates and axioms. The assumptions that were directly related
to geometry, he called {\it postulates}. Those more related to
common sense and logic he called \textit{axioms}. Although modern geometry no
longer makes this distinction, we shall continue the ancient custom and
refer to axioms for geometry also as postulates.
Here is a paraphrase\footnote{Thomas L. Heath, ``The Thirteen Books of
Euclid's Elements'', Cambridge, 1908.}
of the way Euclid expressed himself.
Let the following be postulated:
\margin{MA402 lessons A2 and E1 ff}
\begin{description}
\item[Postulate 1:] To draw a straight line from any point to any point.
\item[Postulate 2:] To produce a finite straight line continuously in a straight line.
\item[Postulate 3:] To describe a circle with any center and distance.
\item[Postulate 4:] That all right angles are equal to one another.
\item[Postulate 5:] That, if a straight line falling on two
straight lines makes
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 the angles are less than two right angles.
\end{description}
Note that the wording suggests construction.
Euclid assumed things that he felt were too obvious to justify further.
This caused his axiomatic system to be logically incomplete.
Consequently, other axiomatic systems were devised in an attempt
to fill in the gaps. We shall consider three of these, due to David
Hilbert, (1899),
George Birkhoff, (1932) and the School Mathematics Study Group (SMSG),
a committee that began the reform of high school geometry in the 1960s.
\newpage
\section{Hilbert's Postulates:}
\margin{The following sections may be skimmed with profit now, but
will not be easy to follow until additional lessons on Euclid's
Elements and the his Parallel Postulate have been studies.}
In the late 19th century began the critical examinations into the
foundations of geometry. It was around this time that
David Hilbert(1862 - 1943) introduced his axiomatic system.
The primitives in Hilbert's system are the sets of {\it points},
{\it lines}, and {\it planes} and relations, such as
\begin{tabular}{ll}
incidence: & as in `a point $A$ is on line $\ell$' \\
order:& as in `$C$ lies between points $A$ and $B$'\\
congruence:& as in ` line segments $AB \cong A'B'$' \\
\end{tabular}
An example of a formal definition would be that of a {\it line segment}
$AB$ as the set of points $C$ between $A$ and $B$.
He partitioned his axioms into five groups; axioms of connection,order,
parallels, congruence and continuity.\footnote{cf. Wallace and West,
``Roads to Geometry", Pearson 2003, Chapter 2 for a more detailed
discussion of Hilbert's axioms.}
Hilbert's axiom system is important for the following two
reasons. It is generally recognized as a flawless version of what
Euclid had in mind to begin with. It is purely geometrical, in that
nothing is postulated concerning numbers and arithmetic. Indeed,
it is possible to model formal arithmetic inside Hilbert's axiomatic
system.
We wish to show how Euclidean geometry can be modelled inside
arithmetic.\footnote{The historical significance of these two exercises
in building models of formal systems is the irrefutable demonstration
that geometry and arithmetic are equi-consistent. That means, if you
believe the one to be without contradiction, then you are obliged to
accept the
other also, and vice-versa. Hilbert's program for a proof that one, and hence
both of them are consistent came to naught with G\"odel's Theorem. According
to this theorem, any formal system sufficiently rich to include arithmetic,
for example Euclidean geometry based on Hilbert's axioms, contains true but
unprovable theorems.} For this purpose, we want the shortest possible list
of primitives and postulates, for then, we have less to check. Birkhoff
meets this requirement.
\section{Birkhoff's postulates}
\margin{These concepts are treated in greate detail in the module on
Cartesian Geometry, Lesson C1. And also in Hvidsten \textit{op. cit.}
Chapter 3}
The primitives here are the set of {\it points}, a system of subsets of
points called {\it lines}, and two real-valued functions, `distance' and
`angle'. That is, for any pair of points, the distance $d(A,B)$ is a
positive real number. For any ordered triple of points $A,Q,B$,
the real number $m\ang AQB$ is well defined\footnote{To
distinguish the figure $\ang AQB$, which we call an `angle', the number
$m\ang AQB$ is called the {\it angular measure} of the angle. Moreover,
two real numbers that differ by a multiple of $2\pi$ measure the same
angle.} modulo $2\pi$.
\begin{description}
\item[Euclid's Postulate:]
A pair of points is contained by one and only one line.
\item[Ruler Postulate:]
For each line there is a 1:1 correspondence between its points and the
real numbers, in such a way that if $A$ corresponds to the real number
$t_A$ and $B$ corresponds to $t_B$ then \[ d(A,B) = |t_B - t_A| \].
\item[Protractor Postulate:]
For each point $Q$, there is a 1:1 correspondence of
its rays\footnote{Note that once we can apply
a ruler to a line, we can identify one of the two half-lines, or rays, at
a point $Q$ as those points $P$ on the line for which $t_P > t_Q$.}
and the real numbers\footnote{We might call these the {\it circular
numbers} because they lie on the {\it
number circle}, just as one speaks of the real numbers lying on
{\it number line}.} modulo $2\pi$, in such
a way that if ray $r$ corresponds to the circular number $\omega_r$ and ray
$s$ to $\omega_s$ then \[m\ang RQS = \omega_s - \omega_r (mod 2\pi) \]
where $R$ is a second point on $r$ and $S$ on $s$ .
\item[simSAS Postulate:]
If $m\ang PQR = m\ang P'Q'R'$ and $d(PQ):d(P'Q')=d(QR):d(Q'R')=k$ then
the other four angles are pairwise equal, and the remaining side pairs have
the same ratio.
\end{description}
One says that such triangles are {\it similar}, $\tri PQR \sim \tri P'Q'R'$
with {\it scaling factor} $k$. Of course, for $k=1$,
$\tri PQR \cong \tri P'Q'R'$ .
\section{The SMSG Postulates}
There are 22 of these,\footnote{Cf. Appendix of Wallace and West,
{\it op.cit.} } and they combine the flavor of Hilbert and Birkhoff.
With Birkhoff, rulers and protractors are postulated, under the valid
impression that children already know how to deal with real numbers by
the time they study geometry. There are many postulates so that proofs
of interesting theorems can be constructed without the tedium of proving
hundreds of lemmas first. Of course, unlike Birkhoff's foursome, the
SMSG postulates are redundant, in that some postulates can be logically
derived from others. The pedagogical wisdom and usefulness of the SMSG
axiom system is a matter of some debate among educators.
\section{A Cartesian Model of Euclidean Geometry}
\margin{Hvidsten 3.6. Class lesson C2 }
We next give an example of an axiomatic system and a model for it.
For this purpose we choose a very familiar area of mathematics in
which to interpret the primitives and to test the truth of the axioms.
We all know analytic plane geometry from high school, also known as
{\it Cartesian geometry}. Birkhoff's
four postulates for Euclidean geometry appear compact enough for us
not to lose our way.
We interpret the {\it points} $A,B,C ...$ as ordered pairs, $(x,y)$, of real
numbers. {\it Lines} shall be solution sets to
linear equations of the form
$ax+by+c=0$. A point $(p,q)$ is {\it incident} to the line
$ax+by+c=0$ if it
satisfies the equation, i.e. if $ap+bq+c=0$ is true. Remember that
the distance between two points and the angle measure are also
primitives and need an interpretation. We shall do that later.
With just this much we can already attempt to verify the first
postulate which asserts the existence and uniqueness of a line through
two given points. You could do this yourself by deriving the formula for
the line through two points $(x_0,y_0), (x_1,y_1)$ in any of the many
ways you learned to do this in high school.
Here we do this by solving
this system of two linear equations for the as yet unknown
parameters $a,b,c$ :
\[
\begin{array}{clc}
ax_0 + by_0 + c &=& 0 \\
ax_1 + by_1 + c& =& 0 \\
a(x_1 - x_0) + b(y_1 - y_0)& =& 0 \\
\end{array}
\]
The third equation
eliminates $c$ for the moment; we can recover it as soon as we know $a,b$,
for example thus:
\[
c = -ax_0 - by_0.
\]
One plausible choice for a,b would be
\[
a = -(y_1-y_0),\, b = (x_1-x_0)
\]
because it fits the third equation and yields
\[
c = x_0y_1 - x_1y_0 = \left| \begin{array}{cc}
x_0 & y_0 \\
x_1 & y_1
\end{array}
\right|
\]
While this shows that both points lie on some line, it does not
demonstrate the uniqueness of this line. Indeed, our interpretation is
incomplete. If we really want the first postulate to hold we must
agree that the same line may have more than one equation, provided the
same set of points is the solution set for each. We therefore amend
our interpretation of a line by stressing that
\[ \begin{array}{c}
a_0x + b_0y + c_0 = 0 \\
a_1x + b_1y + c_1 = 0
\end{array}\]
define the same line provided the parameters are proportional:
\[
a_0:a_1 = b_0:b_1 = c_0:c_1.
\]
The distance function $d(A,B)$ Birkhoff had in mind is, of course, the
Euclidean distance as derived from the Pythagorean theorem:
For $A = (x_0,y_0), B = (x_1, y_1),$
\[
d(A,B) = \sqrt{(x_1-x_0)^2 + (y_1-y_0)^2}
\]
We are now ready to verify Birkhoff's ruler postulate in a
particularly useful fashion. First we give our two arbitrary points
more mnemonic names:
\(
Q = (x_0,y_0), I = (x_1,y_1).
\)
Now there is a canonical way of labelling
all other points on the line $QI$ with real numbers $t$ in several useful
ways as follows:
\[
\left[ \begin{array}{c} x_t \\ y_t \end{array} \right]
\left[ \begin{array}{c} x_0 \\ y_0 \end{array} \right]
(1-t) + t
\left[ \begin{array}{c} x_1 \\ y_1 \end{array} \right]
\]
In vector notation this might be written as
\[
P_t = Q(1-t) + tI = Q + t(I-Q).
\]
Notice that $P_0=Q$ and $P_1=I$,
and that the points on the segment $QI$
are given by the set
\[ \{P_t | 0 < t < 1 \}.\]
There is still something to prove here, namely that the Euclidean
distance is in fact measured by our ruler. Once again we were too
hasty in ruling lines. For the Euclidean distance
\[
d(Q,I) = \sqrt{(x_1-x_0)^2+(y_1-y_0)^2}
\]
which need not equal the parametric distance, which is 1. We may,
however, rescale our ruler by a unit
\( u=d(Q,I)\), to yield another parameter, \( s= tu\), for which
the points $P_s$ on the line are given by \(P_s = Q + sU\), where
$U$ is the unit vector, \( U= (I-Q)/d(I,Q)\), in the direction
if $I$ from $Q$ . This way, $I$ is the correct distance, $d(I,Q)$,
away from $Q$ on this ruler for the line.
For the remaining pair of Birkhoff's postulates we need a protractor,
i.e. a device for measuring angles. The simplest way of doing this in
our model is to recall the definition of the dot product of two
vectors and interpret:
\[
m\ang AQB = \arccos(\frac{A-Q}{d(A,Q)} \cdot \frac{B-Q}{d(B,Q)}).
\]
Birkhoff's axiom system achieved its remarkable economy by postulating
what turns out to be the quintessential property of Euclidean
geometry. What distinguishes it from non-Euclidean geometry are
the properties of {\it geometric similarity}. Two shapes are similar if they
differ only in scale. Birkhoff postulates that two triangles with a
similar corner, are wholly similar. By a corner we mean, of course, a
vertex and its adjacent edges. If the proportionality factor is 1,
then this postulate says that two triangles are congruent as soon as
they have one congruent corner.
We shall verify the simSAS postulate, which makes an assertion about two
triangles, by carefully measuring one triangle. Just as today we
exchange goods by means of their price, instead of bartering items for
each other, so modern geometry compares shapes by comparing their
measurements.
Given a triangle $\tri ABC$, vital statistics consists of six numbers, the
three angles and sides,
\[
\begin{array}c
\alpha = m\ang A\\
\beta = m\ang B \\
\gamma = m\ang C \\
a = d(B,C) \\
b = d(C,A) \\
c = d(A,B).
\end{array}
\]
The law of cosines, which generalizes the Pythagorean theorem to
arbitrary triangles by resolving the square of a side in terms of the
opposite corner:
\[
c^2 = a^2 + b^2 - 2ab \cos \gamma.
\]
allows us to measure $c$ in terms of the measures of two sides and
the included angle.
\section{Problems}
These problems are assigned for submission elsewhere. As you study this
lesson and as you solve these problems, enter their solutions into your Journal for
future reference.
{\bf Problem 1:}
Above we have given the parametric equations for a line.
Recall how to eliminate the paramater $t$ to obtain the
non-parametric equation of a line you may be more
familiar with.
{\bf Problem 2:}
To demonstrate that you understood our verification of Birkhoff's
Ruler Axiom, see if you can do likewise for his Protractor Axiom.
In other words, you first define the angle measure of every angle
as some circular number between $0$ and $2\pi$. Recall that
was done in terms of a unit circle centered at the vertex of
the angle. The angle measure is the length of the arc between
the two points the legs of the angle cross the circle.
Next, show that parametrization of this unit circle by the cosine
and sine of a real number establishes a bijection to the
circular reals of the angle from the from the positive x-axis.
There remains to prove that any angle situated at that origin
is measured by the difference of the parameters.
As you complete this non-trivial exercise, you will discover
that you need to use a well known trig identity.
{\bf Problem 3:}
To finish the proof that Birkhoff's simSAS axiom holds in Cartesian
geometry, you should first
use vector algebra and the definition of the dot product to verify the
law of cosines. Hint: Multiply out
\[
C^2 = (B-A)^2 = ((B-C)-(A-C))^2.
\]
Thus, knowing $a,b$ and $\gamma$, we calculate $c$. If $a$ and $b$ are
stretched, or shrunk by the same factor, so is $c$, provided $\gamma$
remains the same.
{\bf Problem 4:}
Continuing.
Thus, knowing $a,b$ and $\gamma$, we calculate $c$. If $a$ and $b$ are
stretched, or shrunk by the same factor, so is $c$, provided $\gamma$
remains the same.
Apply the law of cosines to the other two sides to calculate
$\alpha$ and $\beta$ as functions of $a,b$ and $\gamma$.
{\bf Problem 5:}
A generalization of Euclid's proof of the Pythagorean theorem
leads to another proof of the law of cosines. Label an
arbitrary acute triangle in the standard way. Construct squares on
two of its sides, say $b$ and $c$. Extending the altitudes
from $C$ and $B$ partitions the squares into rectangles
\[ \begin{array}{c}
b^2 = b b_1 + b b_2 \\
c^2 = c c_1 + c c_2 \\
\end{array}\]
Euclid's argument (do it!) proves that $c c_1 = b b_2 $
Now drop the third altitude from $A$. Of course (can you
prove this?) it passes through the same point where the
first two altitudes intersected,\footnote{This point is
called the {\it orthocenter} of the triangle.} and
partitions the third square into two rectangles.
\[ \begin{array}{rl}
a a_2 =& b b_2 = a b \cos C \\
c^2 =& c c_1 + c c_2 \\
=& b b_2 + a a_1 \\
=&( b^2 - b b_1) + (a^2 - a a_2) \\
=& b^2 + a^2 - 2 a b \cos C \\
\end{array}\]
So, we can measure the rectangle, summarize our
inferences and come up with the law of cosines.
Work your way through this argument, and supply details
which you had use to understand it, or had to look up
elsewhere.
{\bf Problem 6:}
Generalize the above argument to work also for an
obtuse triangle. Hint: Sometimes you need to
add instead of subtract and vice-versa.
\end{document}