ROBERT FRENCH

IS E U C L I D E A N G E O M E T R Y A N A L Y T I C ?

(Received 11 April, 1985)

Kant claimed that Euclidean geometry could be known to be true a p r i o r i ,

even though he also claimed that it does not consist just of logical truths, e.g. is synthetic. 1 In this paper I will challenge Kant's claim here that Euclidean

geometry is synthetic, and will argue instead that if its subject matter is inter-

preted as being restructed to a certain class of spaces, "flat spaces", then it is analytic. This claim may at first sound rather paradoxical, since it might seem that the synthetic character of Euclidean geometry has already been

amply demonstrated, both by the failure of attempts to derive Euclid's Parallel Postulate from his other postulates, and the demonstrated logical consistency of geometries denying the truth of this postulate, e.g. the hyper-

bolic geometry of Lobachevski and Bolyai, and the elliptic geometry of Rie- mann. However, I think that it is noteworthy that these so-called "non-

Euclidean" geometries have only been modeled by spaces which are curved with respect to higher dimensional spaces in which they are embedded, such

as a sphere (where antipodal points are identified, so that two points will not

lie on more than one line) for Riemann's elliptic geometry, and a pseudosphere

(a saddle-shaped space) for Lobachevski and Bolyai's hyperbolic geometry. The question which I wish to examine in this paper then is whether a homo-

geneous space which is not curved with respect to a higher dimensional space, e.g. a "fiat space", can also possess a non-Euclidean geometry. I will argue that it cannot, and hence that if the subject matter of Euclidean geometry is taken to be such spaces, it is analytic.

It has often been held that the various definitions of geometrical concepts given by Euclid in the E l e m e n t s - such as that a straight line is a line which lies evenly with the points on itself, o r that a point is that which has no

parts - are empty, in the sense that no integral use is made of them in any of Euclid's proofs. While this may be so, the question arises as to whether certain of these primitive concepts, notably the concepts of "straight" and "flat", can be given enough content so that the geometrical system making

Philosophical Studies 49 (1986) 213-217. �9 1986 by D. Reidel Publishing Company

214 R O B E R T F R E N C H

use of them will become analytic. The claim then would be that in spaces possessing "non-Euclidean" metric structures, the "straight lines" in these spaces are not really "straight", and the "planes" are not really "flat". They will instead be interpreted as being merely geodesics or minimal surfaces, e.g. the straightest and flattest lines and surfaces existing witllin the space, but

nevertheless not truly straight or flat. For example, the straightest line possible within the confines of a sphere is a great circle, but from a three

dimensional point of view such a line is not straight, since a true straight

line will cut through the interior of the sphere, and thus cannot exist on the

surface of the sphere itself. Similar remarks can be made with respect to the geodesics of a pseudosphere.

How then can the concept of "flatness" be rationally reconstructed so that a Euclidean metric structure logically follows from the assumption that a space be "flat", or in other words, how can this assumption be made to serve as a substitute supposition for Euclid's Parallel Postulate? It turns out that two rather different types of definitions of "flatness" have been given histori- cally, one referring to an external property of a space, e.g. its geometrical properties with respect to a higher dimensional space in which it is embedded;

and one referring to an internal property of a spac e, e.g. just the geometrical properties contained within the given space itself. The external definition is

that a surface is flat if it is not curved with respect to any higher dimensional spaces in which it is embedded, or equivalently ttiat it will contain geodesics

from the higher dimensional point of view. The internal definition is that a surface is flat if it possesses the minimum area for a given perimeter, which was phrased by Proclus in terms of the space being "stretched to the utmost".2 While, as I shall later argue, the denotations of these properties, e.g. the types of spaces possessing them, may either be identical or be logically related by being subsets of each other, the properties themselves are nevertheless at least

connotatively distinct. Thus, there are in fact at least three connotatively distinct propert ies

involved here; the possession of zero external curvature, the possession of a minimal area for a given perimeter, and the possession of a Euclidean metric structure. Before examining the logical relations among the referents of these connotatively distinct properties, I should point out that modern differential geometry defines some of these properties, specifically those of "minimal surface" and "zero curvature", in a different way than I do. In particular, in modern differential geometry a minimal surface is defined in terms of the

IS E U C L I D E A N G E O M E T R Y A N A L Y T I C ? 215

asymptotic lines on a space forming an orthogonal net, which is obviously

equivalent to the possession of a Euclidean metric structure since the sums of the angles of squares in the space will then be 3600. 3 The curvature of a space

is then defined in terms of systematic deviations away from a Euclidean

metric. However, I think that my alternative reconstructions of these terms

are at least as intuitive; e.g. to define a minimal surface in terms of the posses-

sion of the least area for a given perimeter, and to define zero curvature in

terms of the external property of a space that it isn't curved with respect to

any highe r dimensional spaces. I might also point out that the distinction just

noted between a surface possessing zero external curvature and one possessing

the minimal area for a given perimeter is precisely analogous to an ambiguity

in "geodesic" (a space possessing one fewer dimension) between being the

straightest line connecting two points and its constituting the shortest distance

between the two points.

What then are the logical relationships among the referents of these three

connotatively distinct properties? I will first show that the denotations o f

spaces o f zero curvature (as I have reconstructed the term) coincide with the

denotations o f spaces of minimal area (again as I have reconstructed it), and

then show that such a space must possess a Euclidean metric. The converse of this last relationship does not hold though, as is shown by the example of a

cylinder, which possesses a Euclidean metric, but which is not flat in either of the two senses which I have distinguished. I will first use a recursive argu-

ment to show that spaces of zero external curvature must also possess minimal

areas, and will then take note of a theorem from the calculus o f variations to show that spaces of minimal area must possess Euclidean metrics.

My recursive argument showing that a space of zero external curvature

must possess a minimal area is as follows. I will begin by assuming the exist-

ence of a two dimensional space of zero external curvature, although it

should be noted that the argument could also be extended to cover spaces o f

higher dimensionality. It can be recalled that Euclid defined a plane as being

a surface which lies evenly with the straight lines on itself, and this definition

contains the gist of a possible recursive argument since it defines a particular two dimensional space in terms of properties of a space possessing one fewer dimension, namely a straight line. Also, this property clearly follows from the

external definition of a plane of its not being curved with respect to any higher

dimensional spaces, since if it was not comprised of even straight lines, it

would be so curved. It can next be noted that Euclid's definition o f a straight

216 R O B E R T F R E N C H

line as a line which lies evenly with the points on itself, also involves a recur- sion to a property of a space possessing one fewer dimension than the particu- lar space being defined, in this case a property of points. Again, this property

clearly follows from the external definition of a straight line as not being curved with respect to any higher dimensional spaces, since if it was not comprised of even points, it would be so curved.

But how can we get from the property of straight lines that they are un-

curved to their constituting the shortest distances between points? A reductio ad absurdum proof is useful here. It can be noted that if even points did not

constitute the shortest distance between two points, then the triangle in- equality would not hold, since then there would be a shorter way between the points which was not straight. It can also be pointed out that the triangle

inequality was proven by Euclid without the use of the Parallel Postulate, and thus since it is logically independent of the latter's truth, it is legitimate to

use it in this proof. It is now an easy step to show that a space comprised of straight lines,

in the sense of constituting shortest distances, must also be a space of minimal area (and the argument can be extended to higher dimensional spaces), by noting that if the geodesics (in the sense of shortest lines) contained on a surface of minimal area were not also the shortest possible lines between points on it, then the surface would not really be minimal, since then a surface of even less area could be constructed using those lines which really were the shortest possible. This completes the proof that surfaces possessing

zero external curvature must also possess the least area for a given perimeter. I shall now turn to my analysis of why these surfaces must also possess a

Euclidean metric structure. There is a theorem in the calculus of variations that if a surface is min-

imal in the sense of possessing the minimum area for a given perimeter, then it must also be minimal in the sense of possessing a Euclidean metric. This theorem closely parallels another theorem for spaces of one fewer definition, geodesics, holding that geodesics, in the sense of constituting the shortest distances between points, will also be geodesics in the sense of being the straightest lines to connect them. The converse of this theorem, that geodesics in the sense of straightest lines will also be the shortest ones, does not always hold, at least uniquely, as is shown by the example of the antipodes of a sphere, which can be connected by an infinite number of equidistant geodesics. When actual straight lines are involved here though, and not just geodesics,

IS E U C L I D E A N G E O M E T R Y A N A L Y T I C ? 217

this converse relationship does hold, as I proved earlier, and as is also proven

in the calculus of variations. 4 Both theorems, those connecting the two senses of minimal surfaces and

those connecting the two senses of geodesics, involve finding solutions, called "extremals", of the relevant Euler-Lagrange equation

af d a f (1) ay dx Ty ' = 0

for the particular variational problem. A geodesic (in both senses) will coincide with the extremal for the one dimensional problem and a minimal surface

(in the sense of a surface possessing a Euclidean metric) will be the extremal of the two dimensional problem, s This completes my proof that surfaces

possessing zero external curvature will posess Euclidean metrics. It would seem then that the question of whether Euclidean geometry is

analytic or not depends upon what is taken as being its subject matter; e.g. spaces in general or just certain spaces. I f this subject matter is taken to

just be flat spaces, e.g. spaces which are not curved with respect to any higher dimensional spaces, then my claim is that Euclidean geometry is analytic. That is, I claim that it follows from the very concept of a flat space that it must possess a Euclidean metric structure, one possible route for such a deduction having been given in this paper. Of course various supplementary

assumptions were made in that deduction, such as the theorems from the

calculus of variations, however these belong to the subject matter of pure mathematics. While there may still be some unanswered questions regarding

the status of the theorems there, they are now generally considered to be

tautologous.

NOTES

1 Immanuel Kant, Prolegomena to Any Future Metaphysics, translated by Lewis Beck (Bobbs-Merrill, Indianapolis, 1950), Part One. 2 See the discussion of Sir Thomas Heath in Euclid's Elements, Vol. i (Dover Publica- tions, New York, 1956), pp. 171-176. 3 Dirk Struik, Lectures on Classical Differential Geometry (Addison Wesley, Reading, MA, 1961), p. 182. 4 Gilbert Bliss, Calculusof Variations (Open Court, La Salle, IL, 1925), pp. 21, 22. 5 Struik, pp. 142, 183.

Department o f Philosophy, Augustana College, Rock lsland, IL 61201, U.S.A.