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AMERICAN PHILOSOPHY |
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by Ferdinand Canning Scott Schiller
It is amazing what a spell the ideal of exactness has cast upon the philosophic mind. For hundreds, nay thousands, of years philosophers seem to have been yearning for exactness, and hoping that, if only could attain it, all their troubles would be over. All the pitfalls in the way of philosophic progress would be circumvented, and every philosophic science, from psychology and logic to the remotest heights of metaphysics, would become accessible to the meanest understanding. Yet what a gap there is between these professions and the practice of philosophers! Despite of their zeal for exactness, what body of learned men is more careless in their terminology and more contemptuous of all the devices which seem conducive to exactness? Experience shows that it is quite impossible to pin any philosophic term down to any single meaning, even for a little while, or even to keep its meaning stable enough to avoid gross misunderstanding. Even the most express and solemn definitions are set at naught by the very writers who propounded them. The most famed philosophers are the very ones who have been the worst offenders. For example, Kant's fame rests in no small measure on the tricks he played with words like "a priori," "category," "object," and his systematic confusion of "transcendental" and "transcendent." There is hardly a philosophy which does not juggle thus with ambiguous terms. If the theories of philosophers may be interpreted in the light of their practice, they should be the last persons in the world to laud "exactness." On the other hand, they might fairly be expected to inform us what "exactness" means, or at least what they wish it to mean. I do not find, however, that they are at all eager to do this. Apparently they are content to refer to mathematics as an "exact" science, and to admonish philosophy to respect and aspire to the mathematical ideal. To understand exactness, therefore, we must go to mathematics and inquire whether and in what senses mathematics are "exact." Now it is clear that mathematics are not exact in the sense that mathematical objects exactly reproduce physical realities; nor do physical realities exactly exemplify mathematical ideals. There are no straight lines nor circles to be found in nature, while all the physical constants, like the year, month, and day, are inexact. Plato knew this, but yet thought God as a mathematician; he should have added that if God geometrizes, He does so very inexactly. Hence, if the relation between realities and mathematical ideals is conceived as a copying or reproduction, it cannot possibly be "exact." Which is the archetype, and which the copy, does not matter; alike whether the real copies the mathematical ideal, or the latter is moulded upon the former, no exactness can be found. There is, however, a sense in which exactness depends on definition; and mathematicians take great pride in the exactness of their definitions. A definition can be exact, because it is a command addressed to nature, and it sounds quite uncompromising. If the real will not come up to the definition, so much the worse for the real! In so far therefore as exactness depends on definitions, mathematics can be exact. It can be as exact as anything defined exactly. But there appear to be limits to the exactness thus attainable. The exactness of a definition is limited by two difficulties. (a) In the first place things must be found to which the definition, when made, does actually apply. And secondly, (b) the definition has to be maintained against the growth of knowledge. Both these difficulties may easily prove fatal to exactness. As to (a), it is clear that we cannot arbitrarily "define" the creatures of our fancy, without limits. Definitions which apply to nothing have no real meaning. The only sure way, therefore, of securing that a definition will be operative and will have application to the real, is to allow the real, idealized if necessary, to suggest the definition to the mathematician. The mathematician was sensible enough to adopt this procedure. He allowed a ray of sunlight to suggest the definition of a straight line, and this assured to Euclidean geometry a profitable field of application. But it did not render the definition immutable, and immune to the growth of knowledge. The mathematical definition remains dependent on the behavior of the real. If, therefore, rays of light are found to curve in a gravitational field, a far-reaching doubt is cast on the use of Euclidean geometry for cosmic calculations. As to (b), the definer retains the right to revise his definitions. So the very framing of his definition may suggest to the mathematician the idea of developing it in some promising and interesting direction. But this procedure may entail a further definition, or redefinition, which destroys the exactness of the first formula. Thus when he has accomplished the "exact" definition of a circle and an ellipse, it may occur to a mathematician that after all a circle may be taken as a special case of an ellipse, and that it would be interesting to see what happens if he followed out this line of thought. He does so, and arrives at "the points at infinity," with their paradoxical properties. Again the development of non-Euclidean geometries has rendered ambiguous and inexact the Euclidean concepts, e.g., of "triangle." Even so elementary and apparently stable a conception as that of the unit of common arithmetic undergoes subtle transformations of meaning as others beyond the original operation of addition are admitted. In mathematics then, as in the other sciences, it is inevitable that the conceptions used should grow. It is impossible to prohibit their growth, and to restrict them to the definitions as they were conceived at first. Indeed the process of stretching old definitions so as to permit of new operations is even particularly evident in mathematics. The method by which it is justified is that of analogy. If an analogy can be found which promises to bridge a gap between one notion and another, their identity is experimentally assumed. And if the experiment works for the purposes of those who made it, the differences between them are slurred over and ignored. If it were not possible to take the infinitesimal, now as something, now as nothing, what would be left of the logic of the calculus? But the logician at least should remind himself that analogy is not an exact and valid form of argument. Can exactness be said to inhere in the symbols used by mathematicians? Hardly. + and -, and even =, have many uses, and therefore senses, even in the exactest mathematics. The truth is that mathematical definitions cannot be more exact than our knowledge of the realities to which, sooner or later, directly or indirectly, they refer. Nor can mathematical symbols be more exact than words. It is sheer delusion to think otherwise. And what about words? Whence do they get their meanings, and how are they stabilized and modified? Words get their meaning by being used successfully by those who have meanings to convey. Verbal meaning, therefore, is derivative from personal meaning. Once a verbal meaning is established and can be presumed to be familiar, personal meaning can employ a word for the purpose of transmitting a new meaning judged appropriate to a situation in which a transfer of meaning to others is judged necessary or desirable. Thus a transfer of meaning is always experimental, and generally problematic and inexact. Moreover the situation which calls for it is always more or less new. Hence a successful transfer, that is the understanding of a meaning, always involves an extension of an old meaning; and in the course of time this may result in a complete reversal of the initial definition. For example, when the "atom" was first imported into physics, it was defined as the ultimate and indivisible particle of matter. Now, notoriously, it has been subdivided so often that there seems to be room in it for an unending multitude of parts; and its exploration is the most progressive part of physics. The word remains, but its definition has been radically changed. For the scientist always has an option when he finds that his old words are no longer adequate: he can either change his terms, or else his definitions. But there is, and can be, no fixity and no exactness about either. There is a further difficulty about definitions. All words cannot be defined. Wherever the definer begins, or ends, he makes use of terms not yet defined, or has recourse to definitions revolving in a circle. So, if he hankers after exactness, he declares that some terms are indefinable and need no definition. This subterfuge is utterly unworthy of an exact logician. For if he holds that these indefinables are yet intuitively understood or apprehended, he enslaves his "logic" to psychology. If he admits that he cannot guarantee that any two reasoners will understand the indefinables alike, he explodes the basis of all exactness. Thus even the exactest definitions are left to float in a sea of inexactitude. The situation grows still more desperate if the logician realizes that, to achieve exactness, he must eradicate and overcome the potential ambiguity of words. He must devise words which exactly fit the particular situation in which the words are used. For otherwise the same word will be permitted to mean one thing in one context, another in another. It will be what logicians have been wont to call "ambiguous." In this, however, they may have been mistaking for a flaw the most convenient property of words, namely their plasticity and capacity for repeated use as vehicles of many meanings. For the alternative of demanding a one-one correspondence between words and meanings, seems incomparably worse. I remember this was tried once by Earl Bertrand Russell, in a sportive mood. It was not long after the war, and he had just emerged from the dungeon to which he had been consigned for an ill-timed jest, that he came to Oxford to read a paper to a society of undergraduate philosophers, on what he called "vagueness." I was requested to "open the discussion" on this paper, and so obtained what in Hollywood is called a "preview" of it. What was my amazement when I found that Russell's cure for "vagueness," that is, the applicability of the same word to different situations, was that there should be distinctive words for every situation! Certainly that would be a radical cure; but in what a state would it leave language! A language freed from "vagueness" would be composed entirely of nonce words, "hapax legomena," and almost wholly unintelligible. When I pointed out this consequence, Russell cheerfully accepted it, and I retired from the fray. Russell had rightly diagnosed what was the condition of exactness. But he had ignored the fact that his cure was impracticable, and far worse than the alleged disease. Nor had he considered the alternative, the inference that therefore the capacity of words to convey a multitude of meanings must not be regarded as a flaw, but that a distinction must be made between plurality of meanings and actual ambiguity. It is vital to logic that the part words play in transmitting meaning from one person to another should be rightly understood; but does not such understanding reduce the demand for "exactness" to a false ideal? What finally is the bearing of these results on the pretensions of logistics? It seems to reduce itself to a game with fictions and verbal meanings. (1) It is clear that it is a fiction that meanings can be fixed, and embodied in unvarying symbols. (2) It is clear that the verbal meanings to be fixed are never the personal meanings to be conveyed in actual knowing. The assumption that they can be identified is just a fiction too. (3) There appears to be no point of contact between the conventions of this game and the real problems of scientific knowing. This is the essential difference between logistics and mathematics. Pure mathematics is a game too, but it has application to reality. But logistics seems to be a game more remote from science than chess is from strategy. For in a science the meanings concerned are those of the investigators, that is, are personal. They are also experimental. They respond to every advance in knowledge, and are modified accordingly. Their fixity would mean stagnation, and the death of science. Words need only enough stability of meaning, when they are used, for the old senses (which determine their selection) to yield a sufficient clue to the new senses to be conveyed, to render the latter intelligible. In their context, not in the abstract. In the abstract they may remain infinitely "ambiguous," that is, potentially useful. This does no harm, so long as it does not mislead in actual use. And when an experimenter ventures on too audacious innovations upon the conventional meanings of his words, the right rebuke to him is not "You contradict the meaning of the words you use," but "I do not understand; what do you mean?" I am driven then to the conclusion that logistics is an intellectual game. It is a game of make-believe, which mathematically trained pedants love to play, but which does not on this account become incumbent on every one. It may have the advantage that it keeps logisticians out of other mischief. But I fail to see that it has either any serious significance for understanding scientific knowing or any educational importance for sharpening wits!
Excerpted from Acts of the Eighth International Congress of Philosophy, 1934 |
Formal Logic, a Scientific and Social Problem, by Ferdinand Canning Scott Schiller The ethics of pessimism Author, by Ferdinand Canning Scott Schiller Humanism, by Ferdinand Canning Scott Schiller
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