British Journal for the Philosophy of Science, 20 (1969), pp.1-11 Printed in Great Britain

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Euclides ab omni naevo vindicatus

by J. R. LUCAS

Received 1 October 1968

I have not yet learnt how to do maths symbols in HMTL: please excuse the badly placed limits on the sums and integrals

It is often thought that the discovery of non-Euclidean geometries has discredited Kant (Paton (1936)); and some Kantians have maintained with equal fervour that nothing of the sort has occurred because what Kant really said, properly interpreted, is quite consistent with there being non-Euclidean geometries (Martin (1951), Nelson (1906), Meinecke (1906), Natorp (I92I), Becker (1927))---indeed Kant himself envisaged this possibility.1

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `two-dimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it,2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense.

The second use of the word is that of the physicist. The word `line', for example, may be taken to be exemplified by the path of a light-ray, or the path of a freely moving particle, or a geodesic (the shortest distance between two points). Under such interpretation, the axioms or some of the theorems will state synthetic propositions which can be put to an empirical test. It then becomes an empirical question whether a particular set of geometrical axioms under a particular interpretation is true or not; and, as is fairly well known, if we interpret straight lines as being the paths of light rays, space turns out to be not Euclidean but Riemannian.

It is important to stress the rôle of interpretation in the physicists'

1. `Gedanken von der wahren Schrätzung der lebendigen kraefte,' Werke T, 24; Critique of Pure Reason B 268.

2. Ibid.

J. R. Lucas

concept of geometry. It is only under a given interpretation that a set of axioms becomes physically determinate, and capable of being tested and found to be true or false. A physicist can always avoid having to reject a set of axioms as false by refusing the interpretation under which they turn out to be false. There may be another interpretation, equally acceptable, under which the axioms turn out to be true. Whether an interpretation is acceptable or not is more a matter for the physicists' judgement than a question to be decided definitely by empirical test. In general, it is possible to secure an interpretation under which a certain set of axioms-say the Euclidean axioms-will come out true, but this interpretation may be purchased at the price of having to have a more complicated physical theory than otherwise. If I interpret straight line as `path of freely moving particle' and then posit various gravitational and other forces to bring it about that a great many apparently free moving particles are not really moving freely, then I shall be able to maintain that space is really Euclidean, but that physics is a good deal more complicated than Einstein made out. Poincaré(I902, 1905) was prepared to pay such a price. He thought that non-Euclidean Geometries were so inconvenient, and that it was so important to have the geometry on which physics is based Euclidean, that he was prepared to forgo all the simplicity and elegance of the General Theory of Relativity in order to secure for physics a fundamental Euclidean basis.

Poincaré's belief in the fundamental pre-eminence of Euclidean Geometry is a view which is often, and with some show of justice, taken to be the natural modern analogue of Kant's. This is a third sense of the word `space', in which it is claimed that it is a necessary condition, perhaps a subjective condition depending on the nature of the human mind, that the space we actually think of should be a Euclidean space. Space in this sense is neither the pure mathematician's construct nor the remote object of the sophisticated physicist's discovery, but the space of our ordinary experience, given to us, and given to us under a standard interpretation, in all our visual and tactile experience. Some connection is claimed between some central aspects of experience and the Euclidean nature of the space in which this experience is given us, that makes it unthinkable that space in this third sense should not be Euclidean.

It seems to me that Kant did maintain that space in this third sense was necessarily Euclidean, but the issue is clouded by the fact that often when he talks of space, he is using the word not in this but in a fourth sense, a sense in which it makes no sense to ask whether space is Euclidean or not. `In the beginning', Newton rewrote Genesis 1:1, `God created atoms and the void.' The void is that in which atoms may or may not occupy a place,

 

Euclides ab omni naevo vindicatus 3

 

and moving through which may occupy different places at different times. Newton's concept of atoms and the void was an idealisation of our ordinary concept of things, each one occupying at a given time a certain place. The concept of space in this fourth sense provides room for things to exist in. Every point in space is a possibility of existence, in the sense that a thing may, or may not, exist at that point at a given moment; may, or may not, occupy that position. More fundamentally we may argue that some concept of space is a necessary condition, first of our being able to say that two things are qualitatively identical but numerically distinct, and secondly of our being able to say that a thing has changed while still remaining the same thing. It is not our task here to unravel the various arguments that Kant and other philosophers have used in order to demonstrate that Space is a necessary concomitant of the other fundamental categories of Time, Substance, Change or Motion. All we need to point out is that the Space that these arguments are concerned with is more primitive than the space about which it can be asked whether it is Euclidean or not. Space in this fourth sense needs to have certain topological properties-continuity, connectedness etc.-but not any metrical properties, and therefore cannot be inconsistent with there being spaces, in some other sense, that are non-Euclidean. It is because Kant was chiefly concerned with space in this fourth sense, that commentators have claimed that the discovery of non-Euclidean geometries has not in the least discredited his views.

Nevertheless some unease remains; Kant did sometimes use space in the third sense, and the discovery of non-Euclidean geometries does therefore seem to discredit the view that space in this sense must necessarily be Euclidean. It is this view that I want to rehabilitate, though not in Kant's terms. I shall attempt to vindicate the special status of Euclidean geometry and show why it holds a pre-eminent place in our affections.

Euclid was perhaps unfortunate in that his genius led him to hit upon the parallel postulate as his fundamental axiom and not some equivalent proposition. It was natural enough when geometry was concerned with laying out the boundaries of fields along the Nile to regard parallel boundaries as particularly important. But other concepts are equally adequate and even more important. John Wallis (i693), one of the first mathematicians in modern times to consider Euclid's fifth postulate, showed that it could be replaced by an axiom saying that, given a figure, another figure is possible which is similar to the given one and of any size whatsoever; and Gerolamo Saccheri (I733) pointed out that it is enough simply to postulate that there exist two unequal triangles with equal angles.

We thus see that it is a necessary condition of our being able to apply

4 J. R. Lucas

the concept `same shape though different size' that our geometry should be Euclidean. We might almost say that it was a condition of our having the concept of shape at all-for in a Projective Geometry, which contains no concept of size, similarity of geometrical configuration would be of too general application to be a reasonable analogue of our concept of shape; while in Elliptic and Hyperbolic Geometry, although there is a concept of size as well as one analogous to our concept of shape, since the two cannot vary independently, it would be unlikely, or at least difficult, for the two to be distinguished in the way we distinguish them. Euclidean Geometry, if not an absolutely necessary condition for the existence of the concept of shape, is the only ecological environment in which that concept can flourish and prosper. Thus the price of abandoning Euclidean Geometry would be the loss of an important respect in which things can be similar to or dissimilar from one another. We should still of course be able to classify things by colour, by chemical composition, by weight or by specific heat: but we should no longer be able to classify by shape, and this would be awkward; not only would our concepts of area and volume become cumbersome if squares and cubes could not be fitted together to form larger squares or cubes or subdivided into smaller constituent squares or cubes, but it would be conceptually impossible to have scale models, diagrams, maps or blue-prints; which would be a pity.

Euclid, in 1.4 `proves' the congruence of two triangles having two sides and the included angle equal by the method of superimposition; supposing triangle ABC be superimposed on to (epharmozenou epi ton) triangle DEF. He has been much criticised for this: superimposition, it is said, is not a proper geometrical method; we cannot properly derive proposition I.4 from the axioms by this method-indeed, we cannot properly derive it at all, and proposition 1.4 should not be stated as a theorem but as an axiom. In modern systems it is given as one of the axioms of congruence. The method Of 1.4 although outside the canon of geometrical methods, has some intuitive appeal, and does in truth reveal an alternative approach to geometry, an approach in which we consider what things we can do to geometrical figures without destroying their geometrical properties superimpose, move, turn, turn over, add more lines to, construct circles round. This "operational" method was extruded from the canon of geometrical propriety by Plato, who thought it ridiculous to talk of doing things (prattein)in geometry (Republic VII. 527a), not so much because of his discovery of the axiomatic method as a consequence of his allegiance to the theory of forms. Knowledge could only be of what was unchanging (tou aei ontos). Aristotle, rejecting the metaphysics that made it plausible, retained the view that geometry must be entirely

Euclides ab omni naevo vindicatus 5

static; `mathematics is a theoretical science concerned with things that are stationary (menonta) but not separable' (Metaph. 1064a30, see also 989b32): and mathematicians have had a bad conscience ever since about the operational metaphors they have continually found themselves using. It is only in the last century that a theory of operators has been developed. Felix Klein suggested in his Erlanger Program that it would be fruitful to consider for each geometry what groups of operators would leave the geometrical properties of figures unchanged. Helmholtz was on the track of the same idea, which was tidied up mathematically by Lie (1890) and now offers an alternative approach to geometry, quite as rigorous as the traditional axiomatic approach and entirely respectable, in which we consider various groups of operators operating upon the members of a given set.

Euclidean Geometry then emerges as the geometry in which the three operators of displacement, rotation, and reflection are possible without alteration of geometrical properties. These operations can be defined algebraically as transformations of the general form

xi = Sigma yijxj+ai

j

where the matrix (yij) is an `orthogonal' matrix, that is, has its inverse equal to its transpose (and therefore its determinant equal to 1). In effect, this transformation rotates, or turns, a figure through an angle (the effect of the orthogonal matrix), possibly reflects it into its mirror image (if the determinant is equal to -1), and translates, or displaces, it a distance ai in direction i.

This algebraic transformation appeals to the algebraist as being the simplest and most fundamental transformation he can deal with. We can also see, independently of algebra, that the three operations involved are pre-eminently important. Reflection (given any axis of reflection) is essentially a discrete operation: rotation and displacement are continuous operations. The group whose operator is reflection is the simplest of all non-trivial discrete groups: for if we reflect and then reflect again we are back where we started: that is, the essential structure of the group is given by

R2 = I

where `R ' stands for the operation of reflection and `I ' stands for identity. It is evident that this group has the fewest possible basic elements and the simplest possible structure, except for the group whose sole member is the identity operator, which is trivial. The operations of rotation and displacement are both continuous: they differ in that if we rotate far enough we come back to where we started, whereas we can displace

6 J. R. Lucas

further and further without ever coming back. Rotation is a cyclic continuous group, displacement a serial continuous group, and these again are the simplest and most fundamental kinds of continuous group. We can define Projective, Affine and Elliptic Geometry (but not Hyperbolic Geometry at all easily) in terms of other groups, but these other groups are less simple and fundamental than the one based on reflection, rotation and displacement. And therefore Euclidean Geometry, which is invariant under the group based on these three operators, is, from a purely formal point of view, pre-eminent.

The appeal of the theory of groups is not, however, purely formal. We see things reflected in mirrors: we see things from different sides and turn them round; and we both ourselves move, and move other things. If we did not pick out properties that were invariant under reflection, rotation and displacement, we should be unable to recognise as the same what we see in a mirror and what we see when we look direct, what we see from one side and what we see from the other, and what we see from afar off and what we see from nearby. And if we did not pick out properties that were invariant under rotation and displacement, we could not form the concept of a material object, something we can push around without affecting its properties.

The first argument is somewhat Kantian, although interestingly opposed to recent interpretations of what Kant actually said. Ewing (1938) and Strawson (1966) have attempted to save Kant's account of geometry by maintaining that it is a priori true at least of phenomenal geometry---the geometry of our visual experience---that it is Euclidean. But this is just what the geometry of appearances is not. Let the reader look up at the four corners of the ceiling of his room, and judge what the apparent angle at each corner is; that is, at what angle the two lines where the walls meet the ceiling appear to him to intersect each other. If the reader imagines himself sketching each corner in turn, he will soon convince himself that all the angles are more than right angles, some considerably so. And yet the ceiling appears to be a quadrilateral. From which it would seem that the geometry of appearances is non-Euclidean, with the angles of a quadrilateral adding up to more than 360'. And so it is; but it does not worry us, because we never think of it, hardly ever notice it. It is quite difficult to elicit from a man the answer that the angle appears to be more than a right angle. Asked simply what the angle seems to be, he will say, immediately and simply, `A right angle, of course'. For the geometry of appearances is, almost, untalkable about. If we are talking to each other, we are necessarily occupying different positions, from which things characteristically appear differently. In particular,

 

 

Euclides ab omni naevo vindicatus 7

 

apparent angles, except those placed symmetrically between us, will appear different. Therefore we cannot refer to apparent angles and apparent shapes, except by artifice and subsidiarily. We must talk not about the elliptical appearances of the penny, whose eccentricity is different for speaker and for hearer, but about the round reality, which is equally circular for both. We have to talk about the real shape, not the apparent shape, meaning by `the real shape' that which is invariant as between all likely speakers and hearers; invariant, that is, under transformations of the Euclidean group.

Shape must be a primary quality. It is different with colour. The colour of objects---apart from a few such as those made of shot silk---does not vary with the point from which they are viewed. The only external circumstance which affects colour is illumination, and this varies characteristically (for stone-age man, at least) only slowly with time. In the course of any one conversation, the illumination will be the same for both throughout. Therefore both speaker and hearer can talk about apparent colours. It was not necessary (until the invention of artificial lighting) to make much distinction between colours as they appeared to be and colours as they really were. Colours could afford to be secondary qualities, in a way in which shapes could not.

So great is the pressure that the necessities of communication exert on our minds that not only do we have to talk about real shapes rather than apparent shapes, but we see them. Psychologists have discovered `The Phenomenal Regression to the Real Object' (Thouless (I93I), Gibson (1950), Zangwill (1950), Woodworth (1963)); even when we try to concentrate on apparent angles or apparent shapes, our eyes see them more as they really are. Although the penny looks elliptical, if we are asked to choose from a selection of ellipses of varying degrees of eccentricity the one that most closely matches the apparent shape of the penny, we choose an ellipse which is less elliptical and more round than the retinal image of the penny is. Even when we try not to, we re-interpret the visual stimuli as seeming to be something more invariant than they actually are. We cannot be phenomenalists even when we try, but are naive realists at heart, and cannot help attending to those features that are invariable from place to place and person to person rather than those that are variable and subjective. The psychologists bear Kant out: the objectivity of the world is, in part at least, imposed by us, in that we choose to notice just those features that are objective-that is, invariant.

Not only as spectators do we need to be Euclidean, but even more so as agents. We could not be agents if we were floating among amorphous clouds; and in fact if we are to conceive of a stable world in which we can

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make more or less permanent alterations, we shall want it to preserve Euclidean invariances. Helmholtz (i876) was attempting to make this point in his axiom of Free Mobility: but his expression of it was open to the objection of Land (1877) and later of Russell that in basing Euclidean Geometry on the notion of rigidity, he was basing Geometry on Mechanics and putting the cart before the horse. `What is meant by the non-rigidity of a body?' asks Russell (1897) and answers `We mean, simply, that it has changed its shape. But this involves the possibility of comparison with its former shape, in other words, of measurement. In order, therefore, that there may be any question of rigidity or non-rigidity, the measurement of spatial magnitudes must be already possible. It follows that measurement cannot, without a vicious circle, be itself derived from experience of rigid bodies.' The objection is a fair one, against any attempt to build up a theory of space in terms of rigid measuring rods. And since Russell wrote The Foundations of Geometry, we have learned to be even more chary of assumptions about rigidity. But Euclidean Geometry need not be seen as the consequence of there actually being perfectly rigid bodies but as the precondition of its being conceptually possible for bodies to be more or less rigid. Measuring rods may or may not be rigid in the event: but there would be no sense in even thinking of using them unless we thought within a geometry in which there could be objects whose geometrical properties were unchanged by rotation and displacement. With rigidity, as with shape, Euclidean Geometry provides the only ecological environment in which it is a viable concept.

I have tried to show how Euclidean Geometry is a condition of our experience as passive spectators, as active-or at least mobile-spectators, and as agents. It may be felt that this somewhat Kantian enterprise is still unsatisfactory-perhaps contaminated, in some obscure way, by psychologism. I therefore turn to an entirely different justification of Euclidean Geometry, which requires no Copernican revolution, and which would appeal as much to a Platonic Deity contemplating the Forms as to any sublunary philosophers conscious of the fact that they have eyes and hands.

The culmination of Euclid's first book of Elements is the proof of Pythagoras' Theorem in 1.47. His proof is not easy. He was not able to use the much simpler proof, depending on similar triangles, because he did not have an adequate theory of proportion, which we, thanks to Cantor and Dedekind, do have. Correspondingly, the reverse chain of argument from Pythagoras' Theorem to the axiom of parallels is somewhat cumbersome, but if the reader will take on trust that the Wallis-Saccheri axiom `There exist two unequal triangles with equal angles' is equivalent to the axiom of

 

Euclides ab omni naevo vindicatus 9

 

parallels, then the following proof should convince him that so also is Pythagoras' proposition. Suppose we are given Pythagoras' proposition as an axiom, and all the axioms of Euclidean Geometry except the axiom of parallels. Let ABC and ABD be two isosceles right-angle triangles, with right angles at B. Then CBD is a straight line, and / ACB = ADB = / CAB. By Pythagoras AC2 = AB2 +BC2; AD2 = AB2 +BD2 AC2+AD2 =BC2+BD2 + 2AB2 =CD2 ; therefore /CAD is a right angle; therefore triangleCAD and triangleABC are equiangular, but of different sizes; which is Saccheri's postulate, and from which the ordinary axiom of parallels can be proved.

 

 

{figure omitted}

We thus can regard Pythagoras' proposition as the distinctive feature of Euclidean geometry, instead of the axiom of parallels. It seems a much more fundamental one. It connects the concept of distance with that of a right angle---orthogonality---and it does so in the simplest possible way. If we have any metrical space of more than one dimension we are faced with the problem of how to combine measures in different dimensions: if a place is three miles East of us and four miles North, what distance is to be assigned to the direct route? A straight addition rule (i.e. one which would give the answer `seven') would be tantamount to a reduction to only one dimension of measurement. A `squares' rule is the next simplest, and fulfils all the conditions we require of any rule for combining measures. In particular, it has the merit (which it shares with the formulae of the fourth, sixth and eighth degree) of obliterating distinctions of sign-three miles East and four South will be five miles away just the same as three miles East and four North-which suits the essentially non-negative nature of the concept of distance. Other rules could be adopted, might even be forced on us: but if we have the chance of adopting the Pythagorean rule, no further justification is needed. Mathematicians investigating differential geometries, which are not Euclidean, take care to posit nonetheless that they are `Locally Euclidean', that is to say that

ds2 = d1x2+d2x2 + . . . + dxn2,

The concession, though indeed on a small scale, could hardly be larger.

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More fundamentally, we could defend the Pythagorean rule as being the simplest case of Parseval's Theorem. Parseval's Theorem is concerned with the Fourier expansion of functions (satisfying certain conditions of boundedness or measureability) in terms of cosine and sine functions. The Fourier expansion is

 

infinity infinity

f(x) = 1/2ao + Sigma ancos nx + Sigma bnsin nx

n=1 n=1

 

an and bn are known as the Fourier coefficients, and are given by the equations [omitted] and are thus independent of x, though determined, of course by f.

Parseval's Theorem then states

infinity

1 over pi integral from 0 to 2pi (f(x))2dx = 1/2ao2 + Sigma (an2 + b, 2 )

n=1

provided that in the interval [0,2pi] f(x) is measurable and (f(x))2 is integrable.

The Fourier expansion shows how a function f(x) may be plotted in a `phase-space' in terms of its Fourier coefficients---its `co-ordinates' and a set of fundamental functions---`the axes' of the space. It will be a `space' with a denumerably infinite number of dimensions, corresponding to the basic functions cos nx and sin nx (n = 0, 1, 2 .... ), and every function will be represented by a set of values for a0,a1,b1,a2,b2, ... And then Parseval's Theorem shows that the integral of the square of the function, which we might regard as the square of the vector representing the function in phase-space, i.e. the square of the distance from the origin to the point (ao, a1, b1, a2, b2, . . . ) is (barring slight terminological difficulties with the first term) the sum of the squares of the co-ordinates. It is a result entirely uncontaminated by geometrical intuition. The notion of space is an entirely abstract one of independent parameters; the basic trigonometrical functions can be defined exponentially; measure theory is purely analytical. So that a mathematician who was so pure as never to descend into geometry, and who had never heard of Euclid or Pythagoras, would still want to have a Pythagorean rule in Hilbert space, and still pay his respects to Euclidean orthodoxy.

Merton College Oxford

 

 

Euclides ab omni naevo vindicatus 11

 

 

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BECKER, 0. (1927) Mathematische Existenz. Jahrbuch philosoph. phaenomenol. Forsch.8, 737.

EWING, A. C. (1938) A Short Commentary on Kant's Critique of Pure Reason. London. 42-7.

GIBSON, J. J. (I950) The Perception of the Visual World. Cambridge. Mass. 169-72.

HELMHOLTZ, H. (1876) The origin and meaning of geometrical axioms. Mind 1, 30I-302i.

LAND, J.P.N. (1877) Kant's space and modern mathematics. Mind 2, 38-46.

LIE, Sophus (1890) Über die Grundlagen der Geometrie. Leipziger Berichte.

MARTIN, G. (1951) Kant's Metaphysics and Theory of Science. Trans. P. G. Lucas, Manchester, 1955, 209-232. ch. 1, esp. p. 18.

MEINECKE, W. (1906) Die Bedeutung der nichteuklidischen Geometrie. Kantstudien 11,

NATORP, P. (1922) Die logischen Grundlagen der exacten Wissenschaften, 2nd edn. Leipzig, pp. 309 f.

NELSON, L. (1906) Benierkungen uber die nichteuklidische Geometrie und den Ursprung der geometrischen Gewissheit. Abh. d. Friesschen Schule, N.F.1.

PATON, H. J. (1936) Kant's Metaphysic of Experience. London. Vol. i, ch. vii, 5 and 8, pp.155-7, 160-3.

POINCARÉ, H. (1902) La Science et I'Hypothese. Paris. Chs. iii, iv, v, PP. 49-I09, esp. pp. 66-7, 79-8o, 83.

POINCARÉ, H. (1905) La Valeur de la Science. Paris. Ch. iii, pp. 59-95.

RUSSELL, B. (1897) The Foundations of Geometry. Cambridge. 70.

SACCHERI, Gerolamo (I733) Euclides ab Omni Naevo Vindicatus. Milan. Trans. George Bruce Halstead, Chicago, 1920

STRAWSON, P.F. (1966) The Bounds of Sense. London. Part V, esp. z.

THOULESS, R. H. (1931) Br. J. Psychol. 21 and 22.

WALLIS, J. (1693) De Postulato Quinto. Opera Mathematica, Oxford. Vol. ii, pp. 669-78.

WOODWORTH, R.S. (1963) Experimental Psychology. London. PP. 486-7.

ZANGWILL, 0.L. (1950) Introduction to Modern Psychology, London. pp. 30-34.