Conclusion & Appendix on Transformation Groups¶
David Hilbert — The Foundations of Geometry (Authorized Translation by E. J. Townsend, 1902)
Conclusion¶
The preceding work treats essentially of the problems of the euclidean geometry only; that is to say, it is a discussion of the questions which present themselves when we admit the validity of the axiom of parallels. It is none the less important to discuss the principles and the fundamental theorems when we disregard the axiom of parallels. We have thus excluded from our study the important question as to whether it is possible to construct a geometry in a logical manner, without introducing the notion of the plane and the straight line, by means of only points as elements, making use of the idea of groups of transformations, or employing the idea of distance. This last question has recently been the subject of considerable study, due to the fundamental and prolific works of Sophus Lie. However, for the complete elucidation of this question, it would be well to divide into several parts the axiom of Lie, that space is a numerical multiplicity. First of all, it would seem to me desirable to discuss thoroughly the hypothesis of Lie, that functions which produce transformations are not only continuous, but may also be differentiated. As to myself, it does not seem to me probable that the geometrical axioms included in the condition for the possibility of differentiation are all necessary.
In the treatment of all questions of this character, I believe the methods and the principles employed in the preceding work will be of value. As an example, let me call attention to an investigation undertaken at my suggestion by Mr. Dehn, and which has already appeared. [1] In this article, he has discussed the known theorems of Legendre concerning the sum of the angles of a triangle, in the demonstration of which that geometer made use of the idea of continuity.
The investigation of Mr. Dehn rests upon the axioms of connection, of order, and of congruence; that is to say, upon the axioms of groups I, II, IV. However, the axiom of parallels and the axiom of Archimedes are excluded. Moreover, the axioms of order are stated in a more general manner than in the present work, and in substance as follows: Among four points $A, B, C, D$ of a straight line, there are always two, for example $A, C$, which are separated from the other two and conversely. Five points $A, B, C, D, E$ upon a straight line may always be so arranged that $A, C$ shall be separated from $B, E$ and from $B, D$. Consequently, $A, D$ are always separated from $B, E$ and from $C, E$, etc. The (elliptic) geometry of Riemann, which we have not considered in the present work, is in this way not necessarily excluded.
Upon the basis of the axioms of connection, order, and congruence, that is to say, the axioms I, II, IV, we may introduce, in the well known manner, the elements called ideal,–ideal points, ideal straight lines, and ideal planes. Having done this, Mr. Dehn demonstrates the following theorem.
If, with the exception of the straight line $t$ and the points lying upon it, we regard all of the straight lines and all of the points (ideal or real) of a plane as the elements of a new geometry, we may then define a new kind of congruence so that all of the axioms of connection, order, and congruence, as well as the axiom of Euclid, shall be fulfilled. In this new geometry, the straight line $t$ takes the place of the straight line at infinity.
This euclidean geometry, superimposed upon the non-euclidean plane, may be called a pseudo-geometry and the new kind of congruence a pseudo-congruence.
By aid of the preceding theorem, we may now introduce an algebra of segments relating to the plane and depending upon the developments made in § 15, pp. 46–50. This algebra of segments permits the demonstration of the following important theorem:
If, in any triangle whatever, the sum of the angles is greater than, equal to, or less than, two right angles, then the same is true for all triangles.
The case where the sum of the angles is equal to two right angles gives the well known theorem of Legendre. However, in his demonstration, Legendre makes use of continuity.
Mr. Dehn then discusses the connection between the three different hypotheses relative to the sum of the angles and the three hypotheses relative to parallels.
He arrives in this manner at the following remarkable propositions.
Upon the hypothesis that through a given point we may draw an infinity of lines parallel to a given straight line, it does not follow, when we exclude the axiom of Archimedes, that the sum of the angles of a triangle is less than two right angles, but on the contrary, this sum may be
(a) greater than two right angles, or
(b) equal to two right angles.
In order to demonstrate part (a) of this theorem, Mr. Dehn constructs a geometry where we may draw through a point an infinity of lines parallel to a given straight line and where, moreover, all of the theorems of Riemann’s (elliptic) geometry are valid. This geometry may be called non-legendrian, for it is in contradiction with that theorem of Legendre by virtue of which the sum of the angles a triangle is never greater than two right angles. From the existence of this non-legendrian geometry, it follows at once that it is impossible to demonstrate the theorem of Legendre just mentioned without employing the axiom of Archimedes, and in fact, Legendre made use of continuity in his demonstration of this theorem.
For the demonstration of case (b), Mr. Dehn constructs a geometry where the axiom of parallels does not hold, but where, nevertheless, all of the theorems of the euclidean geometry are valid. Then, we have the sum of the angles of a triangle equal to two right angles. There exist also similar triangles, and the extremities of the perpendiculars having the same length and their bases upon a straight line all lie upon the same straight line, etc. The existence of this geometry shows that, if we disregard the axiom of Archimedes, the axiom of parallels cannot be replaced by any of the propositions which we usually regard as equivalent to it.
This new geometry may be called a semi-euclidean geometry. As in the case of the non-legendrian geometry, it is clear that the semi-euclidean geometry is at the same time a non-archimedean geometry.
Mr. Dehn finally arrives at the following surprising theorem:
Upon the hypothesis that there exists no parallel, it follows that the sum of the angles of a triangle is greater than two right angles.
This theorem shows that, with respect to the axiom of Archimedes, the two non-euclidean hypotheses concerning parallels act very differently.
We may combine the preceding results in the following table.
The sum of the angles of a triangle is |
Through a given point: No parallels to a straight line |
Through a given point: One parallel to a straight line |
Through a given point: An infinity of parallels to a straight line |
|---|---|---|---|
$> 2$ right angles |
Riemann’s (elliptic) geometry |
This case is impossible |
Non-legendrian geometry |
$< 2$ right angles |
This case is impossible |
Euclidean (parabolic) geometry |
Semi-euclidean geometry |
$= 2$ right angles |
This case is impossible |
This case is impossible |
Geometry of Lobatschewski (hyperbolic) |
However, as I have already remarked, the present work is rather a critical investigation of the principles of the euclidean geometry. In this investigation, we have taken as a guide the following fundamental principle; viz., to make the discussion of each question of such a character as to examine at the same time whether or not it is possible to answer this question by following out a previously determined method and by employing certain limited means. This fundamental rule seems to me to contain a general law and to conform to the nature of things. In fact, whenever in our mathematical investigations we encounter a problem or suspect the existence of a theorem, our reason is satisfied only when we possess a complete solution of the problem or a rigorous demonstration of the theorem, or, indeed, when we see clearly the reason of the impossibility of the success and, consequently, the necessity of failure.
Thus, in the modern mathematics, the question of the impossibility of solution of certain problems plays an important role, and the attempts made to answer such questions have often been the occasion of discovering new and fruitful fields for research. We recall in this connection the demonstration by Abel of the impossibility of solving an equation of the fifth degree by means of radicals, as also the discovery of the impossibility of demonstrating the axiom of parallels, and, finally, the theorems of Hermite and Lindeman concerning the impossibility of constructing by algebraic means the numbers $e$ and $\pi$.
This fundamental principle, which we ought to bear in mind when we come to discuss the principles underlying the impossibility of demonstrations, is intimately connected with the condition for the “purity” of methods in demonstration, which in recent times has been considered of the highest importance by many mathematicians. The foundation of this condition is nothing else than a subjective conception of the fundamental principle given above. In fact, the preceding geometrical study attempts, in general, to explain what are the axioms, hypotheses, or means, necessary to the demonstration of a truth of elementary geometry, and it only remains now for us to judge from the point of view in which we place ourselves as to what are the methods of demonstration which we should prefer.
Appendix: Foundations of Geometry Based Upon Transformation Groups¶
(Author’s Summary of the 1902 Memoir on Lie Transformation Groups) [2]
The investigations by Riemann and Helmholtz of the foundations of geometry led Lie to take up the problem of the axiomatic treatment of geometry as introductory to the study of groups. This profound mathematician introduced a system of axioms which he showed by means of his theory of transformation groups to be sufficient for the complete development of geometry. [3]
As the basis of his transformation groups, Lie made the assumption that the functions defining the group can be differentiated. Hence in Lie’s development, the question remains uninvestigated as to whether this assumption as to the differentiability of the functions in question is really unavoidable in developing the subject according to the axioms of geometry, or whether, on the other hand, it is not a consequence of the group-conception and of the remaining axioms of geometry. In consequence of his method of development, Lie has also necessitated the express statement of the axiom that the group of displacements is produced by infinitesimal transformations. These requirements, as well as essential parts of Lie’s fundamental axioms concerning the nature of the equation defining points of equal distance, can be expressed geometrically in only a very unnatural and complicated manner. Moreover, they appear only through the analytical method used by Lie and not as a necessity of the problem itself.
In what follows, I have therefore attempted to set up for plane geometry a system of axioms, depending likewise upon the conception of a group, [4] which contains only those requirements which are simple and easily seen geometrically. In particular they do not require the differentiability of the functions defining displacement. The axioms of the system which I set up are a special division of Lie’s, or, as I believe, are at once deducible from his.
My method of proof is entirely different from Lie’s method. I make use particularly of Cantor’s theory of assemblages of points and of the theorem of C. Jordan, according to which every closed continuous plane curve free from double points divides the plane into an inner and an outer region.
To be sure, in the system set up by me, particular parts are unnecessary. However, I have turned aside from the further investigation of these conditions to the simple statement of the axioms, and above all because I wish to avoid a comparatively too complicated proof, and one which is not at once geometrically evident.
In what follows I shall consider only the axioms relating to the plane, although I suppose that an analogous system of axioms for space can be set up which will make possible the construction of the geometry of space in a similar manner.
We establish the following convention, namely: We will understand by number-plane the ordinary plane having a rectangular system of co-ordinates $x, y$.
A continuous curve lying in this number-plane and being free from double points and including its end points is called a Jordan curve. If the Jordan curve is closed, the interior of the region of the number-plane bounded by it is called a Jordan region.
For the sake of easier representation and comprehension, I shall in the following investigation formulate the definition of the plane in a more restricted sense than my method of proof requires, [5] namely: I shall assume that it is possible to map [6] in a reversible, single-valued manner all of the points of our geometry at the same time upon the points lying in the finite region of the number-plane, or upon a definite partial system of the same. Hence, each point of our geometry is characterized by a definite pair of numbers $x, y$. We formulate this statement of the idea of the plane as follows:
Definitions of the Plane. The plane is a system of points which can be mapped in a reversible, single-valued manner upon the points lying in the finite region of the number-plane, or upon a certain partial system of the same. To each point $A$ of our geometry, there exists a Jordan curve in whose interior the map of $A$ lies and all of whose points likewise represent points of our geometry. This Jordan region is called the domain of the point $A$. Each Jordan region contained in a Jordan region which includes the point $A$ is likewise called a domain of $A$. If $B$ is any point in a domain of $A$, then this domain is at the same time called also a domain of $B$.
If $A$ and $B$ are any two points of our geometry, then there always exists a domain which contains at the same time both of the points $A$ and $B$.
We will define a displacement as a reversible, single-valued transformation of a plane into itself. Evidently we may distinguish two kinds of reversible, single-valued, continuous transformations of the number-plane into itself. If we take any closed Jordan curve in the number-plane and think of its being traversed in a definite sense, then by such a transformation this curve goes over into another closed Jordan curve which is also traversed in a certain sense. We shall assume in the present investigation that it is traversed in the same sense as the original Jordan curve, when we apply a transformation of the number-plane into itself, which defines a displacement. This assumption [7] necessitates the following statement of the conception of a displacement.
Definition of Displacement. A displacement is a reversible, single-valued, continuous transformation of the maps of the given points upon the number-plane into themselves in such a manner that a closed Jordan curve is traversed in the same sense after the transformation as before. A displacement by which the point $M$ remains unchanged is called a rotation [8] about the point $M$.
In accordance with the conventions setting forth the notions “plane” and “displacement,” we set up the three following axioms:
Axiom I. If two displacements are followed out one after the other, then the resulting map of the plane upon itself is again a displacement.
We say briefly:
Axiom I. The Displacements Form a Group.
Axiom II. If $A$ and $M$ are two arbitrary points distinct from each other, then by a rotation about $M$ we can always bring $A$ into an infinite number of different positions.
If in our geometry we define a true circle as the totality of those points which arise by rotating about $M$ a point different from $M$, then we can express the statement made in axiom II as follows:
Axiom II. Every True Circle Consists of an Infinite Number of Points.
As preliminary to axiom III, we make the following explanations:
Let $A$ be a definite point in our geometry and $A_1$, $A_2$, $A_3$, …any infinite system of points. With the same letters we will also denote the maps of these points upon the number-plane. About the point $A$ in the number-plane take an arbitrarily small domain $\alpha$. If then any of the map-points $A_i$ fall within the domain $\alpha$, we say that there are points $A_i$ arbitrarily near the point $A$.
Let $A$, $B$ be a definite pair of points in our geometry, and let $A_1B_1$, $A_2B_2$, $A_3B_3$, $\ldots$ be any infinite system of pairs of points. With the same letters we will denote the maps of these pairs of points upon the number-plane. Select about each of the points $A$ and $B$ in the number-plane an arbitrarily small domain $\alpha$ and $\beta$, respectively. If then there are pairs of points $A_iB_i$ such that $A_i$ falls within the domain $\alpha$ and at the same time $B_i$ falls within the domain $\beta$, we say that there are segments $A_iB_i$ lying arbitrarily near the segment $AB$.
Let $ABC$ a definite triad of points in our geometry, and let $A_1B_1C_1$, $A_2B_2C_2$, $A_3B_3C_3$, $\ldots$ be any infinite system of triads of points. With the same letters we will also denote the maps of these triads of points upon the number-plane. About each of the points $A$, $B$, $C$ in the number-plane take an arbitrarily small domain $\alpha$, $\beta$, $\gamma$, respectively. If then there are triads of points $A_iB_iC_i$ such that $A_i$ falls in the domain $\alpha$, and likewise $B_i$ in the domain $\beta$ and $C_i$ in the domain $\gamma$, then we say that there are triangles $A_iB_iC_i$ lying arbitrarily near to the triangle $ABC$.
Axiom III. If there are displacements of such a kind that triangles arbitrarily near the triangle $ABC$ can be brought arbitrarily near to the triangle $A'B'C'$, then there always exists a displacement by which the triangle $ABC$ goes over exactly into the triangle $A'B'C'$. [9]
The content of this axiom can be briefly expressed as follows:
Axiom III. The Displacements Form a Closed System.
We call special attention to the following particular cases of axiom III.
If there are rotations about a point $M$ of the kind that segments lying arbitrarily near the segment $AB$ can be brought arbitrarily near the segment $A'B'$, then there is always such a rotation about $M$ possible by which the segment $AB$ goes over exactly into the segment $A'B'$.
If there are displacements of the kind that segments arbitrarily near the segment $AB$ can be brought arbitrarily near to the segment $A'B'$, then there is always a displacement possible by which the segment $AB$ goes over exactly into the segment $A'B'$.
If there are rotations about the point $M$ of the kind that points arbitrarily near the point A can be brought arbitrarily near the point $A'$, then there is always such a rotation about $M$ possible by which $A$ goes over exactly into the point $A'$.
I now prove the following proposition:
A geometry in which axioms I–III are fulfilled is either the euclidean or the bolyai-lobatchefskian geometry.
If we wish to obtain only the euclidean geometry, it is necessary merely to make in connection with axiom I the additional statement that the groups of displacements shall possess an invariant sub-group. This additional statement takes the place of the axioms of parallels.
In what follows, I will briefly outline the general idea of my method of proof. [10] Within the domain of a certain point $M$ construct in a particular manner a certain point-configuration $kk$, and upon this configuration construct a certain point $K$. We then base our investigation upon the true circle $k$ about $M$ and passing through $K$. It may be easily shown that the true circle $k$ is an assemblage of points which is closed and in itself dense. It constitutes, therefore, a perfect assemblage of points.
The next objective point in our demonstration is to show that the true circle $k$ is a closed Jordan curve. We do this in that we first show the possibility of a cyclical arrangement of the points of the true circle $k$, from which it follows that we may map in a reversible, single-valued manner the points of $k$ upon the points of an ordinary circle. Finally, we show that this map must necessarily be a continuous one. Furthermore, it follows also that the originally constructed point-configuration $kk$ is identical with the true circle $k$. Moreover, the law holds that each true circle inside of $k$ is likewise a closed Jordan curve.
We turn now to the investigation of the group of all the displacements which by the rotation of the plane about $M$ transforms a definite true circle $k$ into itself. This group possesses the following properties: (1) Every displacement which leaves one point of $k$ undisturbed, leaves all points of $k$ undisturbed. (2) There always exists a displacement which changes any given point of $k$ into any other given point of $k$. (3) The group of displacements is a continuous one. These three properties determine completely the construction of the group of transformations of all the displacements of the true circle into itself. We set up the following proposition: The group of all the displacements of the true circle into itself, which are rotations about M, is holoedric, isomorphic with the group of ordinary rotations of the ordinary circle into itself.
Moreover, we investigate the group of displacements of all the points of our plane by a rotation about $M$. The law holds that, aside from the identity, there is no rotation of the plane about $M$ which leaves every point of the true circle undisturbed. We now see that every true circle is a Jordan curve and deduce formulæ for the transformation of that group of all the rotations. Finally, the proposition easily follows that: If any two points remain fixed by a displacement of the plane, then all points remain fixed; that is to say, the displacement is the identity. Each point of the plane may be indeed made to go over into any other point of the plane by means of a displacement.
Our further important objective point is to define the idea of the true straight line in our geometry and deduce those properties of it which are necessary in the further development of geometry. First of all, the notions “semi-rotation” and “middle of a segment” are defined. A segment has at most one middle, and, when we know the middle of one segment, then every smaller segment possesses a middle.
In order to pass judgment as to the position of the middle of a segment, we need particular propositions concerning true circles which are mutually tangent, and indeed the question depends upon the construction of two congruent circles tangent to each other externally in one and only one point. We derive also a more general proposition concerning circles which are tangent to each other internally and consequently a theorem covering the special case where the circle which is tangent internally to a second passes through the centre of that circle.
Moreover, a sufficiently small definite segment is taken as a unit segment, and from this by continued bisection and semi-rotation a system of points is constructed of the kind that to each point of this system a definite number $a$ corresponds, which is rational and has as denominator some power of $2$. By setting up a law concerning this correspondence, the points of the above system are so arranged that the above laws concerning mutually tangent circles are valid. It is now shown that the points corresponding to the numbers $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots$ converge toward the point 0. This result is generalized step by step until it is finally shown that every series of points of our system converges, so soon as the corresponding series of numbers converges.
From what has been said, the definition of the true straight line follows as a system of points which arise from two fundamental points, if we apply repeatedly a semi-rotation, take the middle point, and add to the assemblage the points of condensation of the system of points which arises. We can then prove that the true straight line is a continuous curve, possessing no double points and having with any other true straight line at most one point in common. Furthermore, it can be shown that the true straight line cuts each circle drawn about one of its points, and from this it follows that any two arbitrary points of the plane can always be joined by a true straight line. We see also that in our geometry the laws of congruence hold, by which however two triangles are proven to be congruent if they are traversed in the same sense.
With regard to the position of the systems of all the true straight lines with respect to one another, there are two cases to distinguish, according as the axiom of parallels holds, or through each point there exists two straight lines which separate the straight lines which cut the given straight line from those which do not cut it. In the first case we have the euclidean and in the second the bolyai-lobatschefskian geometry.
Footnotes