The Fourth Geometry

Among these explicit axioms there is one which seems to me to deserve some attention, because when we abandon it we can construct a fourth geometry as coherent as those of Euclid, Lobatschewsky, and Riemann.

To prove that we can always draw a perpendicular at a point A to a straight line AB, we consider a straight line AC movable about the point A, and initially identical with the fixed straight line AB. We then can make it turn about the point A until it lies in AB produced.

Thus we assume two propositions—first, that such a rotation is possible, and then that it may continue until the two lines lie the one in the other produced. If the first point is conceded and the second rejected, we are led to a series of theorems even stranger than those of Lobatschewsky and Riemann, but equally free from contradiction.

I shall give only one of these theorems, and I shall not choose the least remarkable of them. A real straight line may be perpendicular to itself.

Lie’s Theorem

The number of axioms implicitly introduced into classical proofs is greater than necessary, and it would be interesting to reduce them to a minimum.

If this reduction is possible—if the number of necessary axioms and that of imaginable geometries is not infinite?

This is answered by a theorem from Sophus Lie. Suppose the following premisses are admitted:

1. space has n dimensions
2. the movement of an invariable figure is possible
3. p conditions are necessary to determine the position of this figure in space.

The number of geometries compatible with these premisses will be limited. I may even add that if n is given, a superior limit can be assigned to p.

If, therefore, the possibility of the movement is granted, we can only invent a finite and even a rather restricted number of three- dimensional geometries.

Riemann’s Geometrie

However, this result seems contradicted by Riemann, for that scientist constructs an infinite number of geometries, and that to which his name is usually attached is only a particular case of them.

All depends, he says, on the manner in which the length of a curve is defined. Now, there is an infinite number of ways of defining this length, and each of them may be the starting-point of a new geometry.

That is perfectly true. But most of these definitions are incompatible with the movement of a variable figure in Lie’s theorem. These geometries of Riemann, can never be purely analytical. They would not lend themselves to proofs analogous to those of Euclid.

On the Nature of Axioms

Most mathematicians re- gard Lobatschewsky’s geometry as a mere logical curiosity. Some of them have, however, gone further. If several geometries are possible, they say, is it certain that our geometry is the one that is true?

Experiment teaches us that the sum of a triangle’s angles is equal to 2 right angles. But this is because the triangles we deal with are too small.

According to Lobatschewsky, the difference is proportional to the area of the triangle.

Will not this become sensible when we operate on much larger triangles, and when our measurements become more accurate?

Euclid’s geometry would thus be a provisory geometry.

What is the nature of geometrical axioms?

Are they synthetic à priori intuitions, as Kant affirmed?

They would then be imposed upon us with such a force that we could not conceive of the contrary proposition, nor could we build upon it a theoretical edifice.

There would be no non-Euclidean geometry.

To convince ourselves of this, let us take a true synthetic à priori intuition—the following, for instance, which played an important part in the first chapter:

If a theorem is true for the number 1, and if it has been proved that it is true of n + 1, provided it is true of n, it will be true for all positive integers. Let us next try to get rid of this, and while rejecting this proposition let us construct a false arithmetic analogous to non-Euclidean geometry.

We shall not be able to do it. We shall be even tempted at the outset to look upon these intuitions as analytical.

Besides, to take up again our fiction of animals without thickness, we can scarcely admit that these beings, if their minds are like ours, would adopt the Euclidean geometry, which would be contradicted by all their experience.

Should we conclude that the axioms of geometry are experimental truths?

But we do not make experiments on ideal lines or ideal circles; we can only make them on material objects. On what, therefore, would experiments serving as a foundation for geometry be based?

The answer is easy. We have seen above that we constantly reason as if the geometrical figures behaved like solids.

What geometry would borrow from experiment would be therefore the properties of these bodies.

The properties of light and its propagation in a straight line have also given rise to some of the propositions of geometry, and in particular to those of projective geometry, so that from that point of view one would be tempted to say that metrical geometry is the study of solids, and projective geometry that of light.

But a difficulty remains, and is unsurmountable.

If geometry were an experimental science, it would not be an exact science. It would be subjected to continual revision.

Nay, it would from that day forth be proved to be erroneous, for we know that no rigorously invariable solid exists.

The geometrical axioms are therefore neither synthetic à priori intuitions nor experimental facts. They are conventions. Our choice among all possible conventions is guided by experimental facts; but it remains free, and is only limited by the necessity of avoiding every contradiction, and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate.

In other words, the axioms of geometry (I do not speak of those of arithmetic) are only definitions in disguise. What, then, are we to think of the question: Is Euclidean geometry true?

It has no meaning. We might as well ask if the metric system is true, and if the old weights and measures are false; if Cartesian co-ordinates are true and polar co-ordinates false.

One geometry cannot be more true than another; it can only be more convenient. Now, Euclidean geometry is, and will remain, the most convenient:

1st, because it is the simplest, and it is not so only because of our mental habits or because of the kind of direct intuition that we have of Euclidean space; it is the simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree;

2nd, because it sufficiently agrees with the properties of natural solids, those bodies which we can compare and measure by means of our senses.