#### II

We may take for example the science of geometry which, like other mathematical sciences, has been regarded as “rational”. It has been commonly alleged that over and above the truths stated in geometrical theorems there are certain “first principles” of the science which in themselves are unprovable, and that the whole science follows from these principles. Leibniz has given classical expression to this position in the statement (New Essays on the Understanding, Bk. IV, Ch. I) that “it is not the figures which make the proof with geometers…It is the

― 7 ―

universal propositions, i.e., the definitions, the axioms, and the theorems already proved, which make the reasoning, and would maintain it even if there were no figure”. The proof of the last-mentioned theorems, of course, comes under the same general statement, and so we find that the whole science depends on axioms and definitions, supposedly identical propositions, i.e., propositions which cannot be significantly denied or conceived to be false. This position Leibniz expresses by saying that these propositions follow from “the principle of contradiction”, which therefore has embodied in it the whole of geometry and of rational science. It empowers us to reject all propositions which “involve a contradiction” and to affirm their contradictories — which neglects the fact that if a “proposition” were unintelligible, we should not know what its contradictory was.

The attempt is, in fact, to derive geometry from the notion of incompatibility or of the difference between truth and falsity. But obviously this notion could not provide us with the notion of a triangle or any other matter that geometry treats of. In order to find out that having interior angles together greater or less than two right angles is incompatible with triangularity, we require to have the specific things, triangles, before our minds. Apart from observation we could make no assertion of incompatibility whatever. To say, for example, that black is incompatible with white “because it is black” or “because it is not white” is, in either case, to presume the very thing to be proved. In demonstrating the analytic or necessary character of the proposition we surreptitiously introduce the synthetic relation, the fact. “If triangles were not X”, says the rationalist, “they would not be triangles”. Why? we ask. The only possible answer is “Because triangles *are* X”. The fact is required, and the “principle” adds nothing to it. We agree that in a sense the figures do not make the proof; men had known triangles long before they had raised the question of the sum of their angles. But without the figures there would be no proof, because there would be nothing to talk about. No more need be said to demonstrate the falsity of the view that geometry follows from axioms and definitions.

It is curious that, all the while that geometrical truths were regarded as having an ideal or rational character owing to their derivation from pure identities, application of them was made to physical phenomena. Yet, if they had not been synthetic, if they had not conveyed information which it was quite possible not to have about things of certain types, they could not have been applied at all. It is no answer to this argument to say that all that was required, in relation to the physical facts, was something approximately correct, something good enough for practical purposes. This is to say that geometrical truths could be treated as physical hypotheses; which would have been impossible unless there were definite points of contact between the geometrical and the physical. We could not say “Let us suppose this object to be triangular”, if triangularity were a “rational" entity and the object a “natural” one; and we could not go on to say “The object must then have certain other

― 8 ―

properties, and these do not differ greatly from the properties we observe it to have”, unless we could make a direct comparison between the two sets of properties. Even in *supposing* that a physical object has geometrical properties, we imply that there is no difference of order between physical and geometrical objects, that physical objects do fall within the field of geometry. Thus our geometrical hypotheses, or our hypothetical geometry, might actually be falsified by physical facts. If any such contradiction arose, the conclusion would not be that physical facts had failed to come up to geometrical requirements; it would be that our geometry had to be revised. The logic of application is simply the logic of syllogism; and if a geometrical theorem and a physical observation together imply the contradictory of a physical observation, we are as much entitled to question the theorem as to reject the observations. And if careful observation continues to give us the same results, we are bound to deny the theorem. This position will only appear arbitrary and out of harmony with our actual scientific procedure, to one who does not realise that our geometrical theorems are themselves the results of careful observation. But since, whether the conclusion be false or not, a theorem and a fact can together imply nothing unless they have a common term, we are bound to say that the fields of geometry and of physics are not cut off from one another, and that the two sciences are on the same empirical level. This conclusion will apply, however far “rational physics” may be carried. At some point there must be contact between “truths of fact” and “truths of reason”; as is sufficiently established by the fact that we know them both. And that which is capable of implying a fact is equally capable of being falsified by a fact.

It is on the basis of the view that geometry is hypothetical and so, by a curious perversion of the meaning of the term “hypothetical”, unaffected by fact, that the various “geometries” have been set up. Thus we are told that we can obtain different geometries according as we assert or deny Euclid's “axiom of parallels”. Now no doubt different consequences will follow from the two contradictories, but it is our business to seek for errors in these sets of consequences, in order that we may determine whether the “axiom” is true or not, i.e., whether as a matter of fact two intersecting straight lines can or cannot both be parallel to a third straight line in their plane. It is certainly a merit to have seen that Euclid's axiom can be denied, but what is then demanded will be either a testing of it by its consequences, or a deduction of it (or its contradictory) from propositions which we find to be true, or a direct statement that we find *it* to be true (or false). Instead of this, other propositions have been retained as axioms, and it is made a matter of choice whether we accept the proposition on parallels or not; and thus we have the various “geometries” and “spaces”. And this position is even combined with the admission that the geometries which reject the axiom have to define the straight line differently; which is really an admission that Euclid's proposition is true, and incidentally one which could not

― 9 ―

be made unless there were straight lines which answer to Euclid's description.

Bertrand Russell, in his Foundations of Geometry, does not make this admission; but he only avoids it by bringing in (p. 173) a reference to spherical space, in which, while “in general” it is true that there can be only one straight line between two points, in the case of antipodal points this is not so. Since the distance between such pairs bears a special relation to the constitution of the space in which they are, “it is intelligible that, for such special points, the axiom breaks down, and an infinite number of straight lines are possible between them; but unless we had started with assuming the general validity of the axiom, we could never have reached a position in which antipodal points could have been known to be peculiar”. Russell appears to use the word “general” in some private and personal sense. The natural conclusion would seem, however, to be that since the axiom is *not* generally valid, we have not reached a position in which antipodal points are known to be peculiar, and so no exception to Euclid's axiom has been discovered. It is also noteworthy that “unless we have started with assuming” Euclid, there would not have been terms in which to describe the “non-Euclidean” geometries and spaces. It appears to be the case that such geometries are only Euclidean geometry with different terminologies. As to the disputed proposition on parallels, it can be proved by assuming that the sum of the interior angles of a triangle is equal to two right angles. As it is employed in Euclid to prove the latter proposition, we are faced with circularity of reasoning. But the proposition on the angles of a triangle can be independently proved, if we assume that direction and difference of direction (angle) mean the same at different points; failing which there can be no question of the *sum* of such angles.

Waiving this point, however, we shall find it profitable to consider Russell's general argument. There are, he says (pp. 200, 201), certain “*a priori* axioms” which are “necessarily true of any form of externality”; but this leaves some of Euclid's “axioms”, including the proposition on parallels and that two straight lines can never enclose a space, to be “regarded as empirical laws, derived from the investigation and measurement of our actual space, and true only, as far as [the two mentioned] are concerned, within the limits set by errors of observation”. In other words, it is only by observation that we can determine whether our actual space is Euclidean or non-Euclidean. (Russell admits that we have an actual space; no doubt to save the possibility of physical applications. Nowadays this is not considered necessary, and we have the utterly illogical theory of relativity according to which nothing is “actual” and “is” has no meaning.) But there are still the *a priori* axioms (axioms proper) which are not empirical laws but are necessarily true of any form of externality. In considering this position we have to ask how, except by observation of actual externality, we discover what is true of it or what is “deducible from the fact that a science of spatial magnitude is possible (p. 175); how this deduction proceeds, so as to enable us to

― 10 ―

distinguish what is true of experienced space and what is necessary to any form of externality; how, in fact, we can distinguish in space those characters which make it external from its other characters. Russell wishes to show that there are, or may be, forms of externality which, having certain characters of the form which we have observed, do not have others. But in order to show this he must point to forms which do not have the latter characters. If space is the only form of externality that we know, then all the forms that we know have all the characters of space. In order to distinguish characters which are essential to externality from others which are accidental, we shall have to say that in the case of the former we can “see the connection” and as regards the latter see that there is no connection. In other words, necessary connections between some of its attributes and necessary disconnections between others are among the characters of space. To justify this conclusion it would have to be said that we had grasped by a single act of thought *all* the characters of externality in general and of our actual space in particular. Such a position ignores the possibility of discovery and the nature of deduction.

The question is, then, in what way the view that “all forms of externality are X, Y, Z” but “some possible forms of externality are not A, B, C”, i.e., are not Euclidean, can be supported. X, Y, Z, the *a priori* axioms, are supposed to follow from “analysis” of externality. But this analysis can only proceed by simply finding certain characteristics of externality. If analysis were taken to show the necessity of these characteristics, then this necessity in turn would be a characteristic which was simply found. In short, Russell's “deduction”, which is supposed to demonstrate necessity, can only start from, and proceed in terms of, observation of actuality. Similarly when he says (p. 62) that “those properties [of the form of externality] which can be deduced from its mere function of rendering experience of interrelated diversity possible, are to be regarded as *a priori*”, his position is quite illogical. The properties of interrelated diversity can be discovered only by examining situations which exhibit interrelated diversity; so that not only are the premises and the conclusions of the supposed deductions *identical* (viz., all things which render experience of interrelated diversity possible are X, Y, Z), but nothing is said to show that Euclid's axioms are not equally “*a priori*”, since Euclid claims that they indicate properties which he finds in such situations, i.e., in the only interrelated diversity he knows. In fact, the question is solely of “empirical laws”. This is partly obscured, not only by the reference to “deduction”, but also by the reference to “experience” of interrelated diversity. But all the propositions in question are about what is diverse and interrelated, and nothing about experienc*ing* really enters into the argument.

Russell makes a great point of the “logical consistency” of non-Euclidean systems. Here again he is assuming that he knows “all about” such systems, or that he has the peculiar privilege of declaring what is to be regarded as assailable and what is unassailable. We have to note

― 11 ―

two distinct senses in which consistency and inconsistency are spoken of. There is inconsistency in fact; two propositions are said to be inconsistent when one, with a fact or a number of facts, implies the falsity of the other, i.e., when the two together, with or without certain facts, imply a false proposition. This cannot be determined by taking the propositions by themselves but only in relation to facts. But two propositions, inconsistent in this sense, may be perfectly consistent in the other sense, viz., that neither by itself implies the falsity of the other. Now consistency in the latter sense is of the very slightest importance as a description of a group of propositions. Limiting ourselves to that group we find no member of it disproved by any other or collection of others. But there is nothing scientific about limiting ourselves to such a group, allowing them to “define” a science. We ought, on the contrary, to bring them into relation with every available fact, so that any real inconsistency will appear. Russell cannot say that both Euclidean and non-Euclidean geometries are consistent in the broader sense; so that the consistency he claims for non-Euclidean geometry is a barren distinction.

We conclude, then, that geometry is, like all others, an empirical or experimental science dealing with things of a certain sort, that there is nothing “*a priori*” about it but that it is concerned throughout with fact. When Russell says (Principles of Mathematics, p. 5) that pure mathematics asserts “merely that Euclidean propositions follow from the Euclidean axioms, i.e., it asserts an implication; any space which has such and such properties has also such and such other properties”, he is again using “implication” in his characteristically loose way, and he omits to indicate that these facts can be discovered only if we can examine a space having “such and such properties”. Geometry, we may say, is concerned with empirical characters and relations of things in space and is a practical science, and Euclidean geometry consists not of “implications” but of propositions (connected to some extent, of course, by argument) which are either true or false. We can say that, if there were no externality, no geometrical propositions would be true, just as we can say that if there were no distinction between truth and falsity, no propositions whatever would be true. But these statements do not help us in the least to discover any proposition, geometrical or other, which is true. To call them, therefore, statements of the *implications* of the form of externality and of the principle of contradiction is the sheerest absurdity. We must rather say that, since these “principles” have no practical consequences, *there are no such principles*. Our sole concern in science is with facts, and we can attach no meaning to the suppositions “if there were no externality” and “if there were no distinction between truth and falsity”; they cannot even be *conceived* to be facts, that is, they cannot be supposed.