Part 14

With the remarks about the nature of the challenge I have thrown down I heartily agree. May I however suggest that I have a right to expect that criticism should be, not only “competent and candid,” but careful? It is a difficult subject, and _I_ at least am not always able to express myself in such a way that my meaning cannot be misunderstood by any one. I think if your reviewer looks at what I have said again, with the aid of what I say further in my _Essay on Reasoning_, he will see that his criticisms have really originated in misunderstandings, and perhaps he will alter his judgment that I have “come short of the high result to which I aspired.”[18]

But my chief object in writing to you to-day is to bring specially to your notice my ideas on the nature of so-called “necessary truths.” I am not quite clear how far you will find my views harmonise with your own. To a great extent I am inclined to think they are simply a further analysis of the views you express in _The Monist_ and in your _Fundamental Problems_. I will briefly sketch my own ideas and you can then judge whether they are yours also or not.

In my _Essay on Reasoning_ I classify assertions as Truisms (assertions whose truth depends solely on the definitions of their terms) and Real Assertions, which convey some real subjective or objective information. I show that the validity of all purely formal knowledge depends on the fact that it is deduced from definitions alone, which are laid down _arbitrarily_ and that the supposed peculiar certainty of the theorems of pure mathematics is merely due to the fact that they are all truisms. Thus, I think it a misnomer to call such theorems “necessary” truths. It would be nearer the mark to call them “arbitrary” truths.

There is no _necessity_ whatever about the theorem “twice two is four.” “Two” is defined as 1 + 1; “twice,” as the operation of adding a thing to itself. It follows from this that “twice two” is 1 + 1 + 1 + 1; and this, by _definition_, is “four.” If “four” were defined as 1 + 1 + 1, (and there is no “necessary” reason why it should not be,) then “twice two” would _not_ be “four.” The assertion “twice two is four” conveys no real information whatever—at best it could only tell us what one of its terms meant if they had not all been previously defined.

I cannot insist too strongly on the importance of a proper understanding and use of logical definition. If you desire to know whether a given assertion is true or false, _a priori_ or _a posteriori_, the first step in the investigation MUST be to find out how its terms were defined. If it turns out that the truth (or falsehood) of the assertion can be formally deduced from these definitions, then the assertion is a truism (or contradiction in terms): in either case it can give no real information, and even if true cannot be a “necessary” truth. Only if the definitions of the terms are both independent and consistent is it open to discussion how we might come to a knowledge of the fact it expresses.

I may briefly indicate here how I think the problem ought to be attacked. “Objective facts” can only be established by induction. I do not mean by that term necessarily the process described by Mill, but some similar process, based ultimately on _inductio per enumerationem simplicem_. Now no such process can ever lead to a necessary truth. The most fundamental and certain induction which can be made, that which induces us to believe in the objectivity of our environment, does not lead to a “necessary truth”; and much less can any other induction based upon this one do so. “Objective facts” then may be established with greater or less probability, but can never be _necessarily_ true. But all inductions are based on our perceptions, that is ultimately on our subjective sensations. And a man can, nay, must be, absolutely certain of the reality of his own sensations though he cannot be certain of the interpretations he puts upon them. If I have a toothache I cannot be absolutely certain that I have a tooth, but, at least while the pain lasts, I am _absolutely_ certain that I have an ache. And so of any subjective sensation.

I can similarly be absolutely certain that I entertain a given concept, while that concept is before my mind; though of course it is possible that if I assert the possession of that concept I may do so in language which may be misunderstood by the person I am addressing. If then a man has certain concepts which he can call up at will, the reality of those concepts, _qua_ concepts, is to him a _necessary truth_. He may lay down such necessary truths as axioms, and by their aid he may give real subjective import to a symbolic argument, and so obtain new and complicated assertions which are also to him necessary truths. This is what I do in my subjective theory of geometry. That theory might be regarded as purely symbolic—the axioms might have been left out, and all its conclusions looked upon as mere truisms. The conclusions of geometry of four or more independent directions can perhaps _only_ be regarded as truisms. But by the aid of the axioms, geometry of two and three independent directions can be given real subjective import, and its conclusions therefore regarded as necessary truths, as long as they are only taken subjectively. They may further be applied objectively by the aid of objective facts established by induction, but in this case their validity is no greater than that of the primary facts, the counterparts of the subjective axioms, which are employed to give the theory objective import.

I confess I have not studied Kant sufficiently to say that his views differ, materially from mine, though I always thought they did until I read your interpretations of them. Perhaps I misunderstood the sense in which Kant used the term _a priori_. The term has been used in so many different senses that I prefer myself to drop it altogether. If it merely refers to priority in time there can be no practical doubt that, whether in the case of the human race or of an individual thinker, a large amount of sense-experience must have preceded even so simple an _a priori_ judgment as “twice two is four.” If the term merely refers to priority in logical validity it seems to me better to say that “such and such assertions are not dependent upon experience.” But Kant says of the assertion “7 + 5 = 12” that it is not only “_a priori_” but “synthetic”. By the latter term he means that its truth was _not_ deduced from definitions alone, and that the assertion therefore conveys real information. In this I believe he was wrong, and though he afterwards declares that “all knowledge _a priori_ is empty and cannot give information about things,” unless the true nature of _a priori_ knowledge is made more clear, people will inevitably continue to believe the contrary—and to believe moreover that Kant taught so.

Any language which seems to imply that there is some dread necessity about mathematical truths—that they could not be otherwise if they would—is very misleading. Of course it is necessarily true that _if_ you have seven objects and add five more to them you will have in all twelve objects. But the whole objective difficulty is begged by the supposition. “Much virtue in if!”

As I understand it the essence of the “laws” of pure mathematics is that they are verbal, that is they are only abbreviated expressions of the results of certain verbal processes. If the processes are repeated and the results similarly expressed, the results must always be the same. Our reason cannot “inform us about the form of existence” unless it is first given, as the _data_ or facts which correspond to the definitions of our symbolic arguments. It is only because our reasoning faculties are limited that symbolic arguments are necessary at all—that it is not evident to us at once that the conclusions of the most intricate mathematical calculations are given to us along with the _data_. Given the data, then in all possible worlds the conclusions must indeed follow, but only because they really are already _in_ the data which were given.

It may be that you will not only agree with all I have said, but have already said much of it yourself. But there are some passages in your _Fundamental Problems_ which seem to imply otherwise. I think the great objection I have to urge against Kant, and also perhaps against you, is that you do not distinguish as clearly as I could wish between symbolic argument and real, though subjective, knowledge. And the only way to distinguish between them is by inquiring into the definitions of the terms.

For example, on p. 165 of _Fundamental Problems_ you say that to four-dimensional beings Kepler’s third law “would most probably appear as ‘the cubes of their times of revolution being proportional to their mean distances to the fourth power.’”

Now what sort of assertion do you take Kepler’s law to be? Originally it was a purely empirical law obtained by pure induction. If the four-dimensional people obtained their law the same way why should the result appear different to them? Or do you conceive the law to be deduced from Newton’s theory of gravitation? But even so the law of the inverse square was obtained empirically. If you think that law can be explained (as the analogous law for the distribution of light can) by the supposition that the integral of the force over all points at a given distance from the origin is constant, still this supposition is purely gratuitous unless established by induction from experience. If you grant any one of these suppositions you can by symbolic argument obtain the law corresponding to Kepler’s for a four-dimensional space. But I may mention that in no case does the result you anticipate come out. On the first two suppositions the law would be unaltered. On the last supposition the law of gravity would be changed to the inverse cube; but after that the solution of the problem has nothing to do with four dimensions—it is a two-dimensional problem only. The result is that in general planets could not move in closed orbits at all. They might conceivably revolve in circles, but such a condition would be unstable, and if it obtained their periodic times would vary as the squares of their distances.

Again you say (p. 74) “the doctrine of the ‘conservation of matter and energy,’ although it has been discovered with the assistance of experience, can be proved in its full scope by pure reason alone.” I should very much like to see your proof (which I cannot find in _Fundamental Problems_). How do you define the terms of the doctrine? Do you deduce the proof from these definitions—that is do you make it a truism? Or do you base it upon subjective axioms as I do my geometry? Or if you base it on objective facts, how do you prove those facts by pure reason alone? And if it is purely a subjective proof, how, can you say the doctrine is proved “in its full scope”? Surely objective applications come within its scope?

It would not be fair in me to ask you to publish my reply to your reviewer’s criticisms, though if that reply is justified the criticism must have done the prospects of my book some injury, seeing from what a quarter it comes. But I hope you will see your way to publishing the latter part of this letter in _The Monist_, together with your reply to it, if you think it worthy of such a distinction.

I have just come across, in this month’s _Nineteenth Century_, another remarkable instance of reasoning which seems to be rendered entirely nugatory by the want of proper definitions. It is asserted that conceptual thought is impossible without language. At first sight this would certainly appear to be a real assertion. It follows from it that since dogs have no language they have no “conceptual thought.” But it may be plainly shown that dogs do entertain “general notions,” which in ordinary English would be included under the head of “conceptual thought.” The apparent contradiction is however explained when it appears that the author distinguishes general notions as “concepts” or “recepts,” according as they _are_ or _are not named_. This being his definition of “conceptual thought” as opposed to other thought, it appears that the assertion is only a truism after all, and conveys no real information whatever. To discuss it further is then mere waste of time. The author of the assertion doubtless _wished_ it to convey some information, but he did not attend to his definitions and so failed to attain his object.

EDWARD T. DIXON.

FOOTNOTES:

[18] The reviewer of Mr. Dixon’s book has read these remarks on his criticism (_The Monist_, Vol. II, No. 1, p. 126) and has given them what seems to him full consideration. He confesses that he misunderstood what Mr. Dixon meant by “a direction.” (See the article “Logic as Relation-Lore” to be published in a subsequent number.) In regard to the requirements for a logical definition he must still abide by his former opinion. The need of a definition arises either from the inaccuracy in the application of a term or from a supposed lack of knowledge as to its signification. Hence to use the term itself in its own definition is to import into the definition the same vagueness or ignorance which it is the very office of a definition to correct. When Mr. Dixon says that it is requisite for a logical definition that the defining assertions “must not be _independent_ of the meaning of the term defined,” what is that but to say that the same must be _dependent_ upon that meaning? which, unless the reviewer again misunderstands the author, is to say that we must understand the meaning of the term before we can understand the definition.

ρσλ.

MATHEMATICS A DESCRIPTION OF OPERATIONS WITH PURE FORMS.

IN REPLY TO MR. EDWARD DIXON.

It is true, as Mr. Dixon says, that “Any language which seems to imply that there is some dread necessity about mathematical truths is very misleading.” But to say, as Mr. Dixon does in another passage, that the truisms of mathematics are arbitrary truths, is more misleading still. The theorems of the formal sciences are not “assertions whose truth depends solely on the definition of their terms.” They are “real assertions which convey some real subjective or objective information.”

Mr. Dixon objects to Kant’s assertion that 7 + 5 = 12 is not only _a priori_ but also synthetic. He declares, in contradistinction to Kant, that it is deduced from definitions alone; that therefore it is empty, and cannot give any information about things. This latter proposition, which is a phrase of Kant’s, appears in this context as an inconsistency of Kant’s. And it would be an inconsistency, if it had to be understood in the sense in which Mr. Dixon quotes it. We construe Kant’s phrase that “the _a priori_ is empty, and cannot give information about things,” in a different way. We think that Kant intends to say that the _a priori_ imparts real information concerning relations and forms; but that it does not impart real information concerning substances or the materiality of things. It is apparent that the assertion 7 + 5 = 12 cannot be derived from the definitions of 7 or 5. Similarly, the ideas of higher mathematics are not deduced from the few definitions of elementary mathematics that tell what points, lines, parallels, etc., are. Of what, then, are these complex theorems of mathematics, products? They are not derived from sense-experience, nor from the definitions of their ultimate elements.

Is their origin mysterious? Here Kant leaves us in the lurch; he simply declares that formal truth is _a priori_ and transcendental; and those of his disciples who call themselves, with preference, transcendentalists, have ample occasion to introduce in this _lacuna_ of Kantian thought, all the mysticism they please.

The problem of the origin of the truths of formal sciences is not so difficult as it is sometimes represented. The theorems of higher mathematics are the products of certain _operations_ performed with the elementary forms described in the definition with which the mathematician starts. These operations are not arbitrary; they are not merely verbal processes; they are realities of highest importance. Not material realities, but realities, nevertheless. They are functions, and mathematics deals with the products of functions. It is true that we might call twice two by any other name than four; we might call it _vier_, or _quatre_, but the operation 1 + 1 + 1 + 1 would remain the same, by whatever name we call its result. Mathematical truths, accordingly, are not empty in the sense that they are meaningless; for they are significant in the highest degree. They give real information, not about things, but about certain relations that obtain among things. They describe certain operations in which formal relations are traced. And they describe them exhaustively, so that the result is, as the Germans call it, _eindeutig bestimmt_, and the result will, under all circumstances, be the same. Twice two _will always be_ the same as 1 + 1 + 1 + 1. This “it will always be,” is called necessary. There is nothing dreadful about it, nor is there any mystery connected with it. It is not an awful fate that decrees it, but it is the nature of sameness, that the same is and will be the same, so long as it remains the same.

It is often overlooked that every number in arithmetic is the result of an operation which is symbolised by a certain figure. Numbers are not concrete things; and as soon as we forget that they are products of a function, we are liable to lapse into mistakes. This happens most frequently with the numbers “zero” and “infinite.” The latter of these two symbols is often looked upon as a concrete thing; and because the infinite, with actual reality, is, in its completeness, inconceivable, it has made, of every one who stumbled over this stone of offence, a mystic, and many a radical, fearless thinker bows down to worship before the idea of infinitude-function as it would be if it were a real thing.

Says Mr. Dixon, “Our reason cannot inform us about the form of existence, unless it is first given.” This is very true. The form is given, and formal systems such as the numerical system and the lines and figures of mathematics are mental constructions built of the stones quarried out of the relational given in experience.

Form being given, we can reason about the form of existence in general. We can have ready in our minds systems of pure forms to apply to all the various cases of our experience. And this will help us in unravelling the problems of reality, and in extending our knowledge in those fields with which we are little acquainted. Far be it from us to consider the definitions, the operations, and the results of the formal sciences as purely verbal; if they were, mathematics would lose all the great importance which it undeniably possesses, and become mere verbiage.

I confess that I do not understand Mr. Dixon when he says: “It is only because our reasoning faculties are limited that symbolic arguments are necessary at all.” In my opinion all mental activity is symbolic. Every idea is a symbol that signifies something. It is not because our reasoning faculties are limited that symbolic arguments are necessary at all, but symbolism is the nature of our mind, and symbols are the elements with which our reasoning faculties have to deal. In this sense, every argument is symbolic. If it symbolises sense-experience, it represents our knowledge of what may be called the materiality of things. If it symbolises operations with pure forms, it represents the purely formal relations of mathematics logic, algebra, etc.

The doctrine of the conservation of matter and energy in reality means nothing more or less than that there is no increase or decrease in the world at large. Nothing originates out of, and nothing disappears into, nothing. It means that twice 1 + 1 is 1 + 1 + 1 + 1, neither more nor less; or, in other words, it means that all events are transformations. New things originate, but their newness consists in their forms. In this sense the law of the conservation of matter and energy would have to be called, from Mr. Dixon’s standpoint, other differences neglected, a truism. It is not a truism, in the sense of being arbitrary, but in the sense of being a purely formal truth, as are all mathematical theorems.

Mr. Dixon refers to a passage in _Fundamental Problems_, in which I say that “To four-dimensional beings, Kepler’s third law would most probably appear as the cubes of the times of revolutions of the planets being proportional to their mean distances to the fourth power.” His questions, “What sort of an assertion do you take Kepler’s law to be?” and “Why should the result appear different to them?” show that Mr. Dixon has overlooked the condition on which this proposition was made. The first sentence of this paragraph begins with the words, “If space inhered, as Kant maintains, in the thinking subject only, spatial relations and laws would appear different to four-dimensional beings.” Space relations are not subjective, in my opinion, but objective. Therefore, since space relations do not inhere in the thinking subject only, because they are a feature of the objective world, and inhere in the thinking subject in so far as it is at the same time an object in the objective world, Kepler’s law would appear to four-dimensional beings, if they could exist at all, just the same as it does to us three-dimensional beings.

P. C.

BOOK REVIEWS.

SOCIAL STATICS. Abridged and Revised; together with THE MAN VERSUS THE STATE. By _Herbert Spencer_. New York: D. Appleton & Co., 1892.

JUSTICE. Being Part IV of the Principles of Ethics. By _Herbert Spencer_. New York: D. Appleton & Co., 1891.

Among Mr. Spencer’s most important books are those entitled _Justice_ and _Social Statics_. The latter, which first appeared in 1850, has just been republished in about one-half of the original size, some parts having been transferred to the book on _Justice_, and others omitted altogether. “One difference,” as he says, “is that what there was in my first book of supernaturalistic interpretation has disappeared, and the interpretation has become exclusively naturalistic—that is, evolutionary.” Another change is that a demonstration of the injustice of socialism is substituted for his former arguments, plainly repudiated in _Justice_, against private ownership of land. Equally important is the omission of the chapter asserting “The Right to Ignore the State.”