Proofs Without Figures - Mathematics Form and Function

Nevertheless, it is truly remarkable that our experiences of spatial extent and of motion through space can be organized so completely. By taking certain figures-line, plane, segment, angle, and circle-as basic ones, and by assuming certain simple facts as axioms, other geometric figures can be constructed and other geometric facts deduced from the assumed axioms. Some of the resulting facts are surprising; for example, the three medians of a triangle meet in a point, as do the three altitudes.

The resulting deductive structure of Euclidean geometry is a model of Mathematical method.

2. Proofs Without Figures

In elementary Euclidean geometry, proofs of various facts about figures are developed from the axioms, with occasional implicit use of intuitively evident properties of the figures concerned. For instance, to find the per

pendicular bisector of a segment

AB

one constructs two circles with centers

A

and

B

and both with radius

AB;

these circles obviously intersect on opposite sides of the line

A B,

^{. . .}. But what is meant by a "side" of

AB

and why must the circles intersect? (See Figure 1 .) If

all

facts are t o be deduced from axioms, then these plausible observations must also be proved from the axioms, rather than from inspection of the figure. Once this austere requirement was recognized, it turned out to be possible to add suitable new axioms to those of Euclid so that all arguments could be free of any intuitive or pictorial content. This meant that geometry was not really "about" the figures themselves, but was about certain corresponding notions defined exclusively in terms of basic notions to be taken as primitive or undefined. This austere program was systematically carried out by David Hilbert in a book,

Grundlagen der Geometrie

(B. G . Teubner, first edition 1 899, twelfth edition 1 977). His axioms were for both plane and solid geometry; we will examine only those for the plane.

He presented the axioms in the five following groups:

Group I (Incidence Axioms). These axioms involve only three primitive notions: "point", "line", and "the point P lies on the line I ". They require :

( 1 ) Two distinct points P and Q lie on one and only one line. (Except for the meticulous logical formulation, this is the familiar requirement that "two points determine a line".)

(2) There are at least two points on every line.

(3) There are three points P, Q,

R

not all on a line.

With these axioms, one can define a

triangle

(the figure formed by three non-collinear points). However, one cannot yet define "right angle".

64 I I I . Geometry

Figure 1 . Do the circles meet?

Group II (Axioms of Order). In Euclidean geometry a line does not have a selected direction, so we cannot describe the order of points P, Q on a line in terms of binary relations such as "P lies to the left of Q" or "P comes before Q" or "P

<

Q". Instead, the description is given means of a

ternary

relation

"B

lies between

A

and

C",

available when

A, B,

and

C

are all known to lie on a line. The axioms then read:

( I )

B.

This axiom provides the existence of at least one point D in this prolongation.)

(3) If

A, B,

and

C

are three distinct points on a line I, then exactly one of these three is between the other two.

With these axioms, one can define the

segment AB

for

A

ABC

and does not contain

C,

then I meets either

A C

BC

(Figure 2).

In pictures, this states that a line which crosses into the triangle over side

AB

must again come out, either through

A C

or through

CB

(or through C). From the austere, no-pictures, point of view, it was this axiom which was most sharply missing in the Euclidean formulation. This axiom will show that each triangle divides the plane into two parts, an "inside" and

2. Proofs Without Figures

65

A B

Figure 2. Pasch's axiom.

an "outside". One can also prove that any closed polygon (which does not cross itself) divides the plane into two parts, and inside and an outside

but the proof is quite involved because the polygon may be convoluted, with many reentrant angles.

With these axioms, one can also define that is meant by a ray or "half line" starting at a point

between two rays r and

s

can then be defined as the figure formed by two (distinct) rays r and

s

from the same point. In particular, this defines the (three) angles of a triangle

ABC.

B '

A '

and

C '.

This amounts to describing the addition of segments.

A '

and on a given side of the line of

66

I I I . Geometry Given this much about the two triangles, one proves that

A C

⁼⁼

A 'C '

and hence that the triangles are congruent. This is the familiar first congruence theorem (side-angle-side, or SAS) of Euclidean geometry. It is convention

ally "proved" by moving one triangle till its parts coincide with the other.

Though we regard such "motion" as lying in the practical origins of geometry, it use in a formal axiomatic proof is not acceptable ; hence this axiom. From the congruence axioms, one can define right angles.

Two lines are defined to be

parallel

1 ,

. . . , n - l (Figure

3).

(1 ) Archimedian axiom.

Given

C

on that ray from

A,

there is a natural number

n

with

C

between

A

and

Bn .

That is, multiples of

A BI

will eventually exceed the given segment

A C;

in other words,

C,

as measured by the unit

AB,

cannot be infinitely far away.

(2)

Completeness.

The system of points and lines satisfying the given rela-tions of incidence and congruence cannot be a part of a larger system of points and lines with the same relations satisfying all the same axioms.

These are the two axioms of continuity as formulated in Hilbert's book.

However, this pair of axioms can be replaced by a single axiom which concerns the division of a line I into two rays. Each point 0 on I divides I into two rays; if S is the set of all points

A

on one ray and

T

that on the other (and 0 is put in neither ray), then 0 lies between any point

A

of S and any point

B

T.

The axiom desired is essentially a converse to this:

A c

Figure 3 . The Archimedean axiom.

In document Mathematics Form and Function (Page 74-78)