be unphilosophical logic? Perhaps logic is not in much need of philosophy. Mathematics, which is a species of logic, has never had the least need of philosophy in doing its work. Besides, even if logic should require subsequent remodelling in the light of philosophy, yet the unphilosophical logic with which we are obliged to set out will surely be better than no logic at all.
The object of a theory is to render something intelligible. The object of philosophy is to render everything intelligible. Philosophy thus postulates that the processes of nature are intelligible. Postulates, I say, not assumes. It may not be so; but only so far as it is so can philosophy accomplish its purpose; it is therefore committed to going upon that assumption, true or not. It is the forlorn hope. But as far as the process of nature is intelligible, so far is the process of nature identical with the process of reason; the law of being and the law of thought must be practically assumed to be one. Hence, in framing a theory of the universe, we shall do right to make use of those conceptions which are plainly essential to logic.
The two words logic and reason take their origin from two opposite views of the nature of thought. Logic, from λόγος, meaning word and reason, embodies the Greek notion that reasoning cannot be done without language. Reason, from the Latin ratio, originally meaning an account, implies that reasoning is an affair of computation, requiring, not words, but some kind of diagram, abacus, or figures. Modern formal logic, especially the logic of relatives, shows the Greek view to be substantially wrong, the Roman view substantially right. Words, though doubtless necessary to developed thought, play but a secondary role in the process; while the diagram, or icon, capable of being manipulated and experimented upon, is all-important. Diagrams have constantly been used in logic, from the time of Aristotle; and no difficult reasoning can be performed without them. Algebra has its formulae, which are a sort of diagrams. And what are these diagrams for? They are to make experiments upon. The results of these experiments are often quite surprising. Who would guess beforehand that the square of the hypotheneuse of a right-angled triangle was equal to the sum of the squares of the legs? Though involved in the axioms of geometry and the law of mind, this property is as occult as that of the magnet. When we make a mathematical experiment, it is the process of reason within us which brings out the result. When we make a chemical experiment, it is the process of nature, acting by an intelligible, and therefore rational, law, which brings about the result. All reasoning is experimentation, and all experimentation is reasoning. If this be so, the conclusion for philosophy is very important, namely, there really is no reasoning that is not of the nature of diagrammatic, or mathematical, reasoning; and therefore we must admit no conceptions which are not susceptible of being represented in diagrammatic form. Ideas too lofty to be expressed in diagrams are mere rubbish for the purposes of philosophy.
If we do not know how to express relations of virtue, honor, and love, in diagrams, those ideas do not become rubbish; any more than red, blue, and green are rubbish. But just as the relations of colors can be expressed diagrammatically, so it must be supposed that moral relations can be expressed. At any rate, until this is done, no use can be made of such conceptions in the theory of the universe.
Good reasoning is concerned with visual and muscular images. Auricular ideas are the source of most unsound thinking.
6
The Non-Euclidean Geometry Made Easy
| late Spring 1890 | Houghton Library |
We have an a priori or natural idea of space, which by some kind of evolution has come to be very closely in accord with observations. But we find in regard to our natural ideas, in general, that while they do accord in some measure with fact, they by no means do so to such a point that we can dispense with correcting them by comparison with observations.
Given a line CD and a point O. Our natural (Euclidean) notion is that
1st there is a line AB through O in the plane OCD which will not meet CD at any finite distance from O.
2nd that if any line A′B′ or A″B″ through O in the plane OCD be inclined by any finite angle, however small, to AB, it will meet CD at some finite distance from O.
Is this natural notion exactly true?
A. This is not certain.
B. We have no probable reason to believe it so.
C. We never can have positive evidence lending it any degree of likelihood. It may be disproved in the future.
D. It may be true, perhaps. But since the chance of this is as 1:∞ or
E. If there is some influence in evolution tending to adapt the mind to nature, it would probably not be completed yet. And we find other natural ideas require correction. Why not this, too? Thus, there is some reason to think this natural idea is not exact.
F. I have a theory which fits all the facts as far as I can compare them, which would explain how the natural notion came to be so closely approximate as it is, and how space came to have the properties we find it has. According to this theory, this natural notion would not be exact.
To give room for the non-Euclidean geometry, it is sufficient to admit the first of these propositions.
Either the first or the second of the two natural propositions on page 25 may be denied, giving two corresponding kinds of non-Euclidean geometry. Though neither of these is quite so easy as ordinary geometry, they can be made intelligible. For this purpose, it will suffice to consider plane geometry. The plane in which the figures lie must be regarded in perspective.
Let ABCD be this plane, which I call the natural plane,1 seen edgewise. Let S be the eye, or point of view, or centre of projection. From every point of the natural plane, rays, or straight lines proceed to the point of view, and are continued beyond it if necessary. If three points in the natural plane lie in one straight line, the rays from them through the point of view will lie in one plane. Let A′B′ be the plane of the delineation or picture seen edgewise. It cuts all the rays through S in points, and so many of these rays as lie in one plane it cuts in a straight line; for the intersection of two planes is a straight line. The points in which this plane cuts those rays are the perspective delineations of the natural points, i.e. the corresponding points in the natural plane. We extend this to cases in which the point of view is between the natural and the delineated points.
It is readily seen that the delineation of a point is a point, and that to every point in the picture corresponds a point in the natural plane. And to a straight line in the natural plane corresponds a straight line in the picture. For the first straight line and the point of view lie in one plane, and the intersection of this plane by the plane of the picture is a straight line.
All this is just as true for the non-Euclidean as for the Euclidean geometry.
But now let us consider the parts of the natural plane which lie at an infinite distance and see how they look in perspective.
First suppose the natural, Euclidean, or parabolic geometry to be true. Then all the rays through S from infinitely distant parts of the plane, themselves lie in one plane. For let SI′ be such a ray, then if SI′ be turned about S the least bit out of the plane parallel to the natural plane it will cut the latter at a finite distance.
These rays through S from the infinitely distant parts of the natural plane, since they lie in one plane, will cut the plane of the picture in a straight line, called the vanishing line of the natural plane.
The delineations of any two parallel lines will cross one another on this vanishing line.
Next,