Popular Science Monthly/Volume 68/January 1906/The Foundations of Geometry


an historical sketch and a simple example



GEOMETRY as a logical system took its first definite form in the mind of Euclid (about 330-275 B.C.); and since the edifice constructed by the grandfather of geometry has justly retained the admiration of all succeeding students, one can perhaps exhibit the modern researches on the same subject in no better way than by contrasting them with some of Euclid's fundamental statements. The propositions which Euclid placed at the foundation of his work have come to us classified under three heads: definitions, postulates, axioms. As examples of the first we may quote (from Todhunter's edition).

1. A point is that which has no parts, or which has no magnitude.
2. A line is length without breadth.
3. The extremities of a line are points.
4. A straight line is that which lies evenly between its extreme points.
5. A superficies is that which has only length and breadth.
7. A plane superficies is that in which, any two points being taken, the straight line between them lies wholly in that superficies.
15. A circle is a plane figure contained by one line, which is called the circumference and is such that all lines drawn from a certain point within the figure to the circumference are equal to one another:
16. And this point is called the center of the circle.

It is evident that in the first of these statements, if 'point' is defined, 'magnitude' or 'parts' is not; in the second, if 'line' is defined, 'length' and 'breadth' are not; and so on. A partial list of the terms undefined in the above definitions would include magnitude, length, breadth, extremities, lie in, lie evenly, equal to. It is in fact a commonplace among teachers and schoolboys that to any one who did not already know what the terms meant, these definitions would be entirely meaningless. Another way of stating the same proposition, and the way upon which modern mathematicians insist, is that in every process of definition there must be at least one term undefined. A thing which is not defined in terms of other things we may call an element.

It is also to be observed that in the above list of undefined terms there are at least two classes to be distinguished. The first four terms are nouns and correspond to the notion element. The last three are verbs, are conjunctive of elements, and correspond to the notion relation. You observe that no formal definition is here made of the words element and relation. I simply try to call up a distinction which I suppose to exist in the reader's mind.

The postulates and axioms of Euclid are so little to be distinguished from each other that in various editions some of the postulates are put among the axioms. The axioms (common notions) were regarded by Euclid's editors and the world at large, if not by Euclid himself, as a list of fundamental truths without granting which no reasoning process is possible. It was nearly as great a heresy in the middle ages to deny Euclid's axioms as to contradict the Bible. Without emphasizing further the historical fact that the axioms were regarded as necessary a priori truth, nor the fact that this belief is now largely outgrown, I wish to call attention to a mathematically more important feature. If the axioms are necessarily true, and if they are to be used in proving all things else, they themselves are not capable of demonstration. For mathematical purposes, the axioms are a set of unproved propositions. Out of Euclid's definitions and axioms we therefore select for emphasis the presence of

1. Undefined terms relations.
2. Undemonstrated propositions.

The postulates of Euclid are as follows. Let it be granted,

1. That a straight line may be drawn from any one point to any other point.
2. That a terminated straight line may be produced to any length in a straight line.
3. And that a circle may be described from any center, at any distance from that center.

His axioms state:

1. Things which are equal to the same thing are equal to one another.
2. If equals be added to equals the wholes are equal.
3. If equals be taken from equals the remainders are equal.
4. If equals be added to unequals the wholes are unequal.
5. If equals be taken from unequals the remainders are unequal.
6. Things which are double or the same thing are equal to one another.
7. Things which are halves of the same thing are equal to one another.
8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.
9. The whole is greater than its part.
10. Two straight lines can not enclose a space.
11. All right angles are equal to one another.
12. If a straight line meet two straight lines, so as to make the two interior angles on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side on which are the angles which are less than two right angles.

Modern objections to these axioms are to the effect that most of them are too general to be true, that 2, 3, 4, 9, for example, are not valid in every case where we use the term equality; that the axioms are insufficient in that Euclid uses assumptions not explicitly stated, etc. But our present interest in looking for such faults is not great.

Of all the axioms and postulates, the last is by far the most remarkable and important historically. One is led from internal evidence to believe that Euclid introduced it only after failing to make his proofs without its aid. It is not used before proposition 29, not even in proposition 27 which states that if one line falls on two others

Fig. 1. Fig. 2.

so as to make the 'alternate interior angles' (A and A') equal, then the lines are parallel, i. e., do not meet. In proving the converse statement (29), however, he found it necessary to assume that if the sum of the two angles A' and B is less than two right angles the lines will meet when produced far enough. This assumption is axiom 12.

It is perhaps worth while to add that the parallel axiom of which we are speaking may also be stated in the form: 'Through a point, A, in a plane, α, not more than one line can be drawn which does not intersect a line, a, lying in α but not itself passing through A.' The thirty-second proposition, to the effect that an exterior angle of a triangle is equal to the sum of the opposite interior angles, may also be used in place of axiom 12.

The twelfth axiom of Euclid was a stumbling block to many philosophers and mathematicians. While they were ready to grant that they would not be able to reason logically without the other axioms, this one seemed somehow less evident and less fundamental. The natural first attempt was to construct a proof for the axiom so as to give it place as a theorem. Many so-called demonstrations have been offered even up to the present day, but none that have withstood examination. At last, however, the thought came, "what if this axiom were not true? What would become of geometry if axiom 12 were replaced by a new axiom directly in contradiction with it?" It was found that by reasoning based on the reverse of axiom 12 one could involve himself in no contradiction, that, on the contrary, there resulted a new edifice of science which, while different from the old and containing many a strange proposition, yet never denied itself nor violated any of the principles of logic.

These results were obtained first by an Italian Jesuit priest named Saccheri and timidly published in 1733. His work, however, has been known to the modern world only very recently. The non-Euclidean geometry was rediscovered by a Russian, Lobatchewsky (1826), and a Hungarian, Bolyai (1832), though their work also remained unknown to the world at large till 1866 when it fell under the notice of the German mathematician Baltzer. The investigation of the parallel axiom has been continued by Riemann, Beltrami, Helmholtz, Sophus Lie, Cayley, Klein, until it may fairly be said that, ten years ago, this twelfth axiom of Euclid which had at first seemed such a stumbling-block was better understood than any other of his definitions and axioms.

The next attempt after Euclid's to consider geometry as a whole from a purely synthetic point of view was made by a German, Moritz Pasch. His theory, delivered first in a course of lectures in 1873-4, was published in a book called 'Neuere Geometrie' in 1882.

The advance of Pasch beyond Euclid consists essentially in the clear perception of the notions undefined element and unproved proposition. In other words, he tries to state sharply just what concepts he leaves undefined and does reduce the number of these much below that of the elementary concepts employed by Euclid. He distinguishes between his definitions and axioms. He aims to include in his axioms every assumption that he makes.

His undefined elements are 'point,' 'linear segment,' 'plane surface.' These, according to the axioms, have relations such that a point may be in a segment or a surface, a linear segment may be between two points (called its end-points). There is also introduced a relation called congruence (geometrical equality) of figures which corresponds to the Euclidean idea of superposition. We will quote only a few of Pasch's axioms, since they can not signify much apart from the propositions developed out of them.

Fig. 3. 1. Between two points there is always one and only one linear segment.
2. In every linear segment there is a point.
3. If a point C lies in a segment AB, then the point A does not lie in the segment BC.
4. If a point C lies in a segment AB, then so do all the points of the segment AC.
5. If a point C lies in the segment AB, then no point can lie in AB which does not lie in AC or CB.

Out of these assumptions about the relations between points and line segments, together with three other axioms, Pasch deduces the usual propositions about the order in which points lie on a line; the complete line and order itself being defined in terms of the elements and relations mentioned above. He then introduces the plane surface by means of some further axioms, among which are:

1. Every three points are in a plane surface.
2. If a line segment lies between two points of a plane surface there is a plane surface in which lie all points of the given plane surface and also all points of the line segment.
4. If A, B, C are three non-collinear points of a plane surface and any segment DE of the surface has a point in common with one of the segments AB, BC, CA, the line DE has a point in common with one of the other segments or one of the points A, B, C.

This fourth axiom of Pasch is the one that is generally regarded as having required the greatest insight and is most often associated with his name. Fig. 4. A very great improvement over the work of Pasch was made by the Italian mathematician, Peano, who published in 1889 his 'I Principii di Geometria.' The undefined terms of Peano are the elements point and segment and the relations lie on and congruent to. The plane segment of Pasch is defined as a certain set of points.

In Italy, at this time, there was beginning a great revival of interest, largely due to the influence of Peano, in the purely logical aspects of mathematics. This has resulted in a large number of investigations not only of the foundations of geometry, but of mathematics in general. The results are mainly expressed in terms of symbolic logic and proceed a long way toward solving the problem to obtain the smallest number of undefined symbols and unproved propositions that will suffice to build up geometry. Besides Peano one needs to mention chiefly Pieri, who has investigated projective geometry and also the possibility of basing elementary geometry on the concepts, point and motion. Standing aside from the pasigraphical school of Peano, there is Veronese, who has done pioneer work in connection with the axioms of continuity.

In Germany the chief figure at present is D. Hilbert, whose book on 'Foundations of Geometry' (1899) has been translated into several languages, including English. Hilbert's work is the first systematic study that has received widespread attention, and he has therefore been credited with originating a great many ideas that are really due to the Italians. Hilbert's chief contribution to the foundations of geometry is his study of the axioms needed for the proof of particular theorems which he has collected in the latest edition of his book.

The above historical account has no doubt many important sins of omission besides those due to its brevity. But for the purpose of grasping the type of thought involved in these researches further general remarks would probably be less useful than a simple example.

In the academic year 1890-1 Professor C. Segre gave a course of lectures at the University of Turin in which he studied the analytic geometry of n dimensions. A point of n-dimensional space he defined as usual to be a set of n+1 homogeneous coordinates (x1, x2, . . . xn+1), a line as a set of points satisfying a set of linear equations, etc. To his students, however, he proposed the following problem:

To define a space Sn not by means of coordinates, but by a series of properties such that the representation with coordinates can be deduced as a consequence.

In other words, he asked for a set of axioms for n-dimensional space. The problem was taken up by one of the students, Gino Fano, now a professor at Turin, and the results published in the Giornale di Mathematiche. I wish to reproduce one of the many interesting constructions that Fano obtained and to illustrate by means of it certain concepts that have grown up since then.

Let us take the case of n-dimensional geometry where n=2 and proceed for a time as if to build the projective geometry of the plane. Let our undefined elements be called points and let us speak of certain undefined classes or sets of points which shall be called lines. Every one will recognize as valid of projective geometry the following propositions which are our axioms—our unproved propositions.

1. If A and B are points there is one line which contains them both; and
2. There is not more than one such line.
3. Any two lines have in common at least one point.
4. Not all the points are on the same line.

If we stop at this point and try to see how much we can prove on the strength of our assumptions, we are confronted at once by the fact that we can not prove the existence of even a single point. This must therefore be assumed by a further axiom. The assumption of one point is not enough either, but if we assume that there are two points, it follows from 4 that there must be at least three. There need not, however, be more than three, for if we suppose that the points referred to are A, B, C, and that the line AB consists merely of the points A and B, the line BC of the points B and C, and the line CA of the points C and A, then on rereading 1, 2, 3, 4 it is evident that they are all satisfied. Hence in order to get ahead we must assume:

5. In each line there are at least three points.

But this does not postulate the existence of even a single point till we add

6. There is at least one line.

We are now in a position to develop considerably more theory. By 6 and 4 and 1 there must be at least two lines which by 3 meet in a point A. Hence there must be four points at least, (B, C, D, E) which do not lie in the same line. For if D were in the line BC, by 2, the lines AB and AD would be the same, which is contrary to hypothesis.

A set of four points, such as A, B, C, D, of which no three are collinear, when taken together with the lines (called the sides) joining the six pairs of points, AB, BC, CD, DA, AC, BD, is called a complete quadrangle. In the diagram below, the vertices of a complete quadrangle are 0, 1, 4, 6. The three additional points 2, 3, 5, in which the sides of the quadrangle intersect, are called the diagonal points.

We have shown our axioms sufficient to establish the existence of a complete quadrangle; are they sufficient to prove the ordinary properties of such a figure? They are not. Axioms 1-6 do not decide whether the three diagonal points, 2, 3, 5, are or are not collinear. In the ordinary geometry, those points are non-collinear and form what is called the diagonal triangle. If, however, we suppose that they are collinear (one may assist one's imagination by means of the dotted line) then on rereading our six postulates they will all be found verified. In order to obtain the usual geometry it is necessary to assume as an axiom that the diagonal points of a complete quadrangle are non-collinear.Fig. 5. What we have just done is a simple case of an 'independence proof.' We have proved that the proposition that the diagonal points of a complete quadrangle are not collinear, is independent of propositions 1, . . . 6, that is, it is not a logical consequence of them. Similarly, the non-Euclidean geometry is an independence proof for Euclid's axiom 12. The ideal of students of foundations of geometry is a system of axioms every one of which is independent of all the rest. To attain this ideal it is necessary to construct for each axiom an example in which it is untrue while all the rest are verified.

After seeing the bizarre construction that this process gives rise to, one is tempted to raise the question, how can we be sure that the complete system which we use applies uniquely to the space of our intuition or experience and not also to one of these mathematical dreams? In answering this question we define what is meant by a categorical system of axioms.

Returning to our complete quadrangle with collinear diagonal points and observing the numbers placed at its vertices, we may arrange in the same column the numbers of the points that appear together in the same line.

0 1 2 3 4 5 6
1 2 3 4 5 6 0
3 4 5 6 0 1 2

The array thus obtained is known as a 'triadic system' in the seven digits 0, 1, ··· 6.

If the undefined element, 'point' of our axioms is any one of the digits 0 ··· 6 and 'line' is a column, then the six axioms incompletely describe the triadic system. They describe it completely if we add:

7. There are not more than three points on a line.

I say they describe it completely because we have proved that the axioms are satisfied by seven points arranged as in the triadic system, while from 7 it follows that no other arrangement or number of points is in harmony with the axioms. There is only one kind of thing which satisfies all the axioms 1 ··· 7. In other terms, any two systems of objects (for example, the points of the diagram on p. 28, and the triadic system on p. 29) that satisfy axioms 1 ··· 8 are reciprocally in a one-to-one correspondence which preserves all the relations of the kind specified in the axioms.

This is what is meant by a categorical system of axioms. Thus in geometry, a categorical system is capable of distinguishing Euclidean space from all essentially different constructions of the mind—and this in spite of the fact that the fundamental elements of geometry are never defined in the ordinary sense of the term definition.

If we have before us a categorical system of axioms, every proposition which can be stated in terms of our fundamental (undefined) symbols either is or is not true of the system of objects satisfying the axioms. In this sense it either is a consequence of the axioms or is in contradiction with them. But if it is a consequence of the axioms, can it be derived from them by a syllogistic process? Perhaps not.