SOLUTION OF TWO DIFFICULTIES.
206. Must not the theory which supposes the relations of bodies to be variable, put an end to all the natural sciences? Can there be science without a necessary object? and can there be a necessity which is compatible with variability?
The natural sciences have two parts: one physical, andthe other geometrical. The first supposes the data furnished by experience; the second forms its calculations relative to these data. Change the relations of external beings, and the data will be different, you will have a new experience producing a new physical science: the calculation will be the same, only new results will be obtained from the new data. The difficulty thus disappears. All the physical sciences are based on observation, all their combinations are made from data furnished by experience; therefore the physical sciences are not wholly absolute, but they have a part which is conditional. The theory of universal gravitation is developed as a body of geometrical science, but it starts from the data furnished by experience. Destroy these data and from a body of physical science it becomes a body of pure geometry. In mechanics, the problems of the composition and decomposition of forces have a physical signification, inasmuch as they presuppose the data of experience; suppress these data and there remains only a composition of lines which mean nothing when we call them forces. Therefore mechanics is only a system of geometrical applications.
207. Here another difficulty arises which is apparently more serious than the other. If the relations of bodies are not essential, but are subject to variation; if our calculations upon them are not founded upon data which are intrinsically necessary, it seems that geometry is destroyed, or limited in such a way to the ideal order, that it cannot be sure that on descending to the field of experience it will not find that false which it regarded as true, and that true which it reputed false. For example, the distances of bodies are calculated by considerations of geometry: if the relation of distance is variable, and a body may be in many places at the same time, geometry turns out false. Such a supposition is no more than the application of the foregoing theory; for, if the relations are variable, this variation may affect distance, which is only a relation. I said this difficulty was more serious than the other, because it leaves the field of experience, and attacks the order of our ideas, an order which we must hold to be indestructible, unless we wish to give up our reason. What would become of our reason if geometry were contradicted by the reality? what would become of an order of ideas in contradiction to facts? Still I repeat that the force of this difficulty is only apparent, and if analyzed will be found of no more weight than the objection which we have already answered.
A body which is a hundred yards distant from another, cannot be only one yard distant; geometry would be opposed to it. But if the relations of bodies are variable this proposition can mean nothing with respect to the reality. Therefore geometry is false. I admit the consequence; but the principle on which it is based involves a supposition contrary to my theory. If you alter or destroy the relations of bodies, you destroy distance, which is a relation, consequently you cannot have a distance of one hundred yards, nor of one yard, nor any distance at all, and if there is no distance there is no contradiction. If, then, you ask how great is the distance between them, your question is absurd; for it supposes a distance, whereas there is no distance at all.
208. This solution rests on a fundamental principle which we ought never to lose sight of. Geometrical truth is true in reality when the conditions of geometry exist in reality; if these conditions do not exist, there is no real geometry. There is nothing strange in this: in fact, the same occurs in the purely ideal order; even there, geometry rests on certain postulates, without which it is impossible. Two triangles with the same base and altitude are equivalent toeach other. This is a true proposition, but only on the supposition that there are those orders of points which are called lines, and that the lines form angles, and are united at three points. If these relations are not presupposed, the geometrical theorem has no meaning.
209. Geometry in itself, or in the purely ideal order, is founded on the principle of contradiction. The truth of this principle being absolutely necessary, that of geometry is equally so. But the principle of contradiction, like all purely ideal principles, abstracts existence, and is applied to nothing in practice, unless we suppose some fact to support it. Yes and no at the same time are impossible; but the principle determines nothing for or against either of the extremes. It only affirms that one excludes the other; if we supposeyes, it excludesno, and if we supposeno, it excludesyes; that is to say, it always needs a condition, a datum which only experience can furnish.
It is the same with geometry. All its theorems and problems refer to the ideal field within us, where there are certain conditions which lead to certain results, by virtue of the principle of contradiction: whenever the conditions exist, the results are true; but if the former fail the latter are false. Ideal sciences consider theconnectionof conclusions with principles in the order of possibility, but take no note of facts. If the connection is admitted the science is true.