The last two papers consist, to a great extent, of controversial matter about Mr. Mill. It would hardly become me to maintain that Mr. Mill was infallible, or that controversy with him was necessarily uninstructive, or that the plan of singling out a definite antagonist for definite attack is not recommended by many considerations. On the contrary, I think that nothing can give greater spirit, vigour, and precision to discussions which are only too apt to become vague. All this, however, is consistent with a sense of the inconveniences of such a method of procedure, the greatest of which is that it does not follow that because you confute A. B., you establish what A. B. denied. Whether Dr. Ward succeeds in finding real flaws in Mr. Mill's views I shall not inquire. But however that may be, he fails to convince me of the truth of his own opinion, for reasons which I now proceed to assign.
One of the great points which Dr. Ward labours in the papers in question is thus stated in syllogistic form on p. 18 of the last of them:—
Whatever the existent cognitive faculties of mankind testify is instinctively known by mankind as certainly true.
But the existent cognitive faculties of mankind testify that any given mathematical axiom is self-evidently necessary.
Ergo, it is instinctively known by mankind as certainly true that any given mathematical axiom is self-evidently necessary.
To say that I deny the major and the minor and the conclusion of this syllogism is an imperfect way of expressing my dissent from it. I feel that its author speaks a language different from mine, and lives, so to speak, in a different intellectual world. The words know," "true," "necessary," and many others, must, I suppose, mean to him something which they do not mean to me. Apart, however, from this, the syllogism appears to me to exemplify in a striking manner the defect which Mr. Mill attributed, as I think justly, to all syllogistic reasoning. The major and minor premisses could never be affirmed unless the truth of the conclusion was independently known. Indeed, they are simply the conclusion stated in terms of increasing generality. Dr. Ward gets a conclusion to start with by supposing that there is something special in mathematical knowledge. He gets a minor by supposing that the special characteristic of mathematical knowledge is that it is obtained by a direct act of some special faculty of the mind, and the major is obtained by generalising the minor.
This appears more clearly upon examining the terms of the syllogism. The major proposition appears to me simply to repeat six times over the words "We know." Each of the six expressions "existent cognitive faculties," "testify," "instinctively," "known," "certainly," " true," asserts or implies the same thing, and the whole syllogism amounts to this:— "We know something. We know Euclid. Therefore we know Euclid." This appears to me a cumbrous way of saying “We know Euclid."
Again, I dissent from the psychological theory implied in the use of this language. A man, according to this syllogism, has existent cognitive faculties, and he has also other faculties by which ho instinctively knows. Besides these two faculties, he is acquainted, I suppose, otherwise, with the meaning of the words "certainly true." The first set of faculties "testify." Thereupon the second set of faculties inform the common owner of the two sets that what the first set of faculties say is "certainly true." It occurs to me that the faculties which "instinctively know" require a voucher, as well as the "cognitive faculties" which "testify." After all, what are a man's faculties except the man himself when engaged in a certain act? and what meaning is there in the assertion that one set of his faculties corroborated another through a third? When all is said, what does it mean, except that people have certain ways of gaining knowledge which, from the nature of the case, they are obliged to trust. And did any one ever deny it? The whole apparatus of cognitive faculties, instinctive knowledge, and certain truth is only, as it seems to me, an expansion of the words "we know," and carries us no further.
Upon the whole, the substance and purport of the syllogism appears to me to be this :—There are two kinds of knowledge, or perhaps I should say we know two kinds of truths, contingent truths and necessary truths. The first class, namely, the class of contingent truths, includes all common facts, such as that so many persons, dressed in such a way, are sitting round a table at a given time and place. The other class, namely, the class of necessary truths, consists of general propositions, of which those which relate to time, space, and number are specimens. We can distinguish between contingent and necessary truths by an unfailing test. A contingent truth might be imagined to be, and might be, other than it is, but a necessary truth cannot; or to put the same thing in a different way, Omnipotence could alter the one, and cannot alter the other. Dr. Ward, I have no doubt, would accept the following illustration, though I do not give it as an exact quotation. Omnipotence could make white gold or cold fire, but could not make a quadrangular figure contained by three straight sides. I deny the existence of this distinction, and if I am right, Dr. Ward's syllogism is either wrong or unmeaning. If no truths are necessary it is wrong. If all truths are necessary it is unmeaning.
The expression "necessary truth" may have one of two different meanings. It may mean a fact which could not have been otherwise than it is, or it may mean a truth affirmed by the very use of certain words. Now, I say that if you mean by necessary truths facts which could not have been otherwise than they are, I am by no means sure that all truths whatever are not necessary, and I am much disposed to think they are. It is a truth that these lines were written on blue paper with a quill pen, by a man standing in front of a desk in the hollow of a window looking into the Inner Temple Garden, on the 16th of February, 1874. I can easily imagine any one of these circumstances having been different, but the assertion of their existence is as true as that two and two make four, and I was when they occurred equally unable to doubt of any one of them. Being past, they are unalterable (I suppose) even by Omnipotence, and in order that they might have happened otherwise, it might, for aught I can tell, have been necessary for the whole constitution of the universe to have been slightly altered from all eternity. What, then, is the meaning of the assertion that any fact whatever is contingent? Every fact whatever is. It would not be a fact, if it did not exist; and if it exists, and comes under my "existent cognitive faculties," it is to me necessary. How can any power of imagining its absence, and the substitution for it of a similar but slightly different state of things, afford me any sort of evidence as to the possibility of its not having happened? When a man says, "This ink might just as well have been blue as black," all that he really means is that he can easily imagine the absence of the black ink and the presence of blue ink in its place, but for aught I know to the contrary, the presence of the black ink was determined by causes reaching far beyond Adam.
If, on the other hand, you mean by necessary truths, truths which are implied by the very use of certain words, then I say that facts come first, and that words ought to be made to fit them; and that when you describe the properties of space, time, and number as necessary truths, all that you ought to mean, all that you can prove, is that certain propositions about them (e.g., that two straight lines cannot enclose a space,) describe in perfectly clear and adequate language facts which we learn by experience, just as we learn by experience that a given member of the Metaphysical Society reads a paper about necessary truths under given circumstances.
An illustration will show how very much the difference between contingent and necessary truth (using the word "necessary" in the second sense) is a difference as to the use of words. It is, we are told, a contingent truth that gold is yellow, and the reason is because God could make white gold, but could not make a quadrangular trilateral. It seems to me that the truth of the assertion that God could make white gold entirely depends on the meaning which men choose to attach to the word "gold." If by the word "gold" I mean a metal of a certain specific gravity, malleable, not liable to rust, and of a yellow colour, then God can no more make white gold than he can make a square triangle. If by the word "gold" I mean a metal of a certain specific gravity, malleable, and not liable to rust, whatever may be its colour, then God can (I suppose) make gold of any colour, but I know not why I should not annex the meaning " yellow" to the word "gold," as well as the. meaning "metal." Dr. Ward somewhere observes that it would be easy for Omnipotence to make cold fire. All that I can say to that is that if Omnipotence made something which sparkled, and crackled, and smoked, but did not burn, I should not call it fire.
The difference, and the only difference which I can perceive, between the class of truths which relate to the properties of time, space, and number; and propositions as to common objects and occurrences, is this: The words which relate to time, space, and number are perfectly simple and adequate to that which they describe, whereas the words which relate to common objects are in nearly every case complex, often to the highest degree; the words "straight," " line," "plane," "surface," " angle," "circle," "triangle," have no ambiguity at all. A line means a line, add the idea of breadth, or thickness, or specific colour, or weight, and the word becomes inappropriate; but the words "paper," "stick," "book," "man," "fire," "gold," and so on, mean a collection of many qualities which may be varied by imagination, without destroying the general resemblance between the image raised by the word used and the thing signified. If I hear of red gold, for instance, I understand a metal having all the other properties of gold as commonly known, except the quality of yellowness, for which redness is substituted. When I am told of a black swan, I mean a bird like a white swan, but of a different colour. But when I hear of straight lines two of which would enclose a space, or of a figure contained by three straight lines making four angles with each other, I know that if the words employed are employed in their usual senses the propositions into which they are introduced are not true.
I will consider immediately the manner in which we get our knowledge of the qualities of space, but before doing so I will make an observation on the character of the words in which we embody that knowledge, and of the thing to which they apply. Space has, as far as we know, no qualities or properties at all, except qualities and properties which the words used by us express with perfect clearness and adequacy; and this I take to be the reason why the propositions which we make about space do not admit of being varied, and cannot even be imagined to be false. When we speak of a straight line, we mean an imaginary line resembling the lines popularly called straight, but distinguished from them by having no breadth, no thickness, and no deviation whatever from the apparent general direction. We can imagine a substance like gold in all respects except its colour, or its specific gravity, or its malleability, or its exchangeable value; and we can think of gold with exclusive reference to any one or more of these qualities. We thus find no difficulty in applying the word " gold" to numerous imaginary substances, differing from each other in many respects, but resembling each other in the particular matters of which we think when we use the word. Moreover, all the words which describe the qualities of gold admit of degrees. There are numerous shades of colour, for instance, to which the word "yellow" applies. Space, on the other hand, has no qualities at all, except the qualities of figure, and these qualities are described in words which have one meaning, and no more. Hence we cannot vary either our mental image of space itself, or the meaning of the words in which we describe it. If we tried to do so, we should speak without a meaning, and reduce the subject of our speech to the condition to which that eminent logician, Crambe, reduced his abstract Lord Mayor. When Martinus Scriblerus said that he could not conceive of a Lord Mayor without his gold chain or his turtle, Crambe replied that he could conceive of a Lord Mayor without gold chain, turtle, fur gown, swordbearer, chaplain, coach, office, body, soul, or spirit, which, he submitted, was the abstract idea of a Lord Mayor. Martinus, I am sorry to say, called Crambe an impudent liar, which, though rude, was not, I think, wholly unnatural; but seriously speaking, I think that to try to conceive of space as being other than it is is like trying to conceive of red as being blue. You can substitute one colour for another, but the sole property of any given colour is to be itself. Alter it, and you destroy it. It is the same of space. We cannot modify it in imagination, because there is nothing in it to modify, and because we have no experience of anything else, not quite the same, but very like it, which we can substitute for it.
I now come to the question by what means is our knowledge of the characteristics of Space acquired, and to this I reply it is acquired in precisely the same way as our knowledge of any common fact,—the fact, for instance, that a particular sheet of paper is blue, and not white,—namely, by the use of our senses. Now if this is the case, either all truths are necessary, or mathematical truths depend upon experience, like others, and may thus be called contingent.
The question,— What is the nature of time, space, and number? is quite independent of the question,— How do we become aware of their properties? I am not myself able to attach any meaning
to the words "Space " and " Number " apart from distinct objects existing in space, and of faculties capable of perceiving them as so existing, nor can I attach any meaning to the word "Time" apart from the faculty of memory; but whether space, time, and number are objective or subjective, whether they are the colour, so to speak, of the things we look at, or of glasses through which we are obliged to look, it is undeniable that our knowledge of them is entirely dependent upon our senses and our memory. A person who passed his life in dreamless sleep, so that he had no external perceptions at all, and whose mind was conscious of no succession of thoughts or impressions, would know nothing of space, number, or time. On the other hand, the instant a person begins to use his senses or his memory he becomes aware of space, time, and number, and he continues to be made aware of them at every instant at which he uses his faculties through the whole of his life. His early ideas on the subject are exceedingly confused, but by experience, especially if it is guided by instruction, they become perfectly clear and systematic; and when they have once reached a clear and systematic condition subsequent experience adds nothing to them. He knows them as well as they admit of being known, just as a lad of 14 knows his multiplication-table and his alphabet as well as he will ever know them, if he lives to be a hundred. The experience by which we learn to understand the words "before" and "after" is so early and simple that no one remembers its acquisition, but I should suppose most people remember learning the multiplication-table and the first elements of geometry. If I were to generalise from my own experience, I should say that we begin with exceedingly confused notions upon the matter, and that after a time, longer or shorter, as it may be, we see that the matter really is as we are told that it is,—that is to say, we see that our impressions of external objects really are summed up by the multiplication-table and geometrical axioms.
I distinctly remember the first day when I really understood the first proposition of the first book of Euclid, and how I demonstrated it to myself over and over again many times with extreme delight and satisfaction. It was exactly the same sort of feeling as one which I have often experienced in later life,—the feeling of discovering one's way about a place. If a man takes up his abode in a new neighbourhood, and proceeds to explore it, he will find (at least I have often found) that at first he is very much astray, even if he has maps to help him. By degrees he begins to find his way, he mentally connects one road with another, and sees what are the relative positions of such and such woods, hills, houses, and other objects. The whole at last takes its place in his mind, sometimes by a kind of crisis which enables him, with striking distinctness and rapidity, to say, " Now I know where I am." When this happens he knows the country, and if he lives in it fifty years his knowledge will not alter, though, of course, it may become more detailed and minute.
Our acquaintance with the relations of space in general is, if I am not mistaken, of precisely the same nature as our acquaintance with particular portions of space. We learn the general meaning of the words "line," "surface," "solid," "point," "round," "square," and the like, as we learn the meaning of other common words. A nurse or a mother tells a child that the marks which she makes on a piece of paper are lines, just as she tells it that the creature which lies on the rug is a dog. I suppose no one ever yet studied Euclid who did not know perfectly well before he read a word of it what a straight and a crooked line, a round thing and a square thing, look like, nor can any one have seen a board, or a table, or a sheet of paper, without having received the impression of parallel lines. I should further suppose that no one ever learnt to walk without learning what is meant by a short cut from place to place. Experience teaches every human being who is not an idiot, and indeed every animal, that it saves time to cut a corner, and the difference between this homely proposition and the proposition that two sides of a triangle are greater than the third is only a difference of expression.
If it is denied that matters of this sort are learnt by experience, it appears to me that it ought, in consistency, to be denied that anything whatever is learnt by experience. It appears to me just as clear that experience teaches us to compare together the length of lines in general, and, in particular, to compare the length of a straight and crooked line terminating at the same points, as that it teaches us to compare together the lengths of any two specific lines. I see no difference whatever between the process by which we learn that the word "straight" means a line of a peculiar kind, like that which is apparently formed by a string tightly stretched, and that such lines are the shortest way from point to point; and the process by which we learn that the Oxford Road is proximately straight, and that that road forms a shorter connection between Victoria Gate and the Marble Arch than the road which goes all round Hyde Park. The difference between the two propositions is simply that one of them refers to one particular corner of the contents of space, and the other to space, or which is the same thing, to the contents of space in general.
It is sometimes asked how you are enabled by any number of observations on particular parts of space to make general observations on it? I think Dr. Ward asks in one place what right Mr. Mill had to suppose that the conditions of space in Sirius were the same as they are here. The answer appears to me to be, that by the word "space" we mean that enormous apparent blue vault which appears to our senses to contain the earth, the solar system, and innumerable other systems, nebulae, and fixed stars. All these things we see with our eyes, and we picture space to ourselves as an enormous expanse or cavity in which they are all contained. No one, I suppose, will deny that experience enables us to draw an imaginary line between two trees or two book-cases, between which, if we pleased, we could draw a real line; or that it informs us that if we represent these lines on paper, we can reason about the relations of the objects to each other as well as we could if we confined our attention to the things themselves, and indeed, in many instances, much better. If the possibility of making and using maps is not a fact taught by experience, then experience teaches nothing at all. If it is, then when we draw imaginary lines from star to star, and argue about their distances, upon data which we have gathered from our local experience of space, we are proceeding upon experience, the experience upon which we proceed being that of our own eyesight, which assures us that fixed stars do exist in space, and that that which we call space is a vast homogeneous vault for them to exist in. I do not see how this can be denied by any one who does not confine the word "experience" to experience by touch. At this moment, I see a sparrow sitting on a tree, perhaps ten yards off. Behind the sparrow and through the fog, I see the sun, and I have identically the same reason for believing that the sun and the sparrow both exist in space.
I maintain, on the whole, that we learn the characteristics of Space by looking at things in it and by moving about in it, just as we learn the shape of a room and the position of the articles of furniture in it by the very same process; and I say that both or neither of the matters thus learnt are learnt by experience.
I have already attempted to explain the reason why it is practically impossible for us to imagine or conceive (I think the only difference between the two operations is in the greater distinctness of imagination, and its application to matters of which we are informed by the eye or ear,) any alteration in space, time, or number, their properties and relations, the reason being that our ideas of them are simple-ideas, and therefore cannot be altered without being destroyed. But I will pursue the matter a little further, with the view of showing two things, —first, that no inference whatever can be drawn from the extent of our power of imagining or conceiving; and secondly, that though we cannot imagine or conceive of an alteration of the qualities of space, time, or number, we can readily imagine facts which, if they existed, would prevent us from forming our present ideas of space, time, and number, and would show that those ideas, if formed, were incorrect.
The first point may, I think, be established very shortly. The processes of imagining and conceiving consist, as far as we know, in combining together, in our own minds, things which we have seen apart in nature. When we imagine a centaur, we imagine part of the body of a horse combined with part of the body of a man, and so of everything else. Now time, space, and number enter into nearly every imagination of our minds. There may be some thoughts which have no relation to them, but these I need not at present consider. Now there is but one space, one series of numbers, and one course or stream of time, and our idea of each of the three is a perfectly simple idea, independent of everything else, and continually present to our minds. How, then, can we modify it in imagination? It is as impossible to do so as to imagine a new colour, or to think out the common expression, "If I were you." Thus our incapacity to imagine or conceive certain things proves simply that we have no experience which enables us to do so. It neither proves, nor to my mind does it tend to prove, that what we cannot imagine or conceive cannot be conceived or imagined by any other intelligent being, even if he is omnipotent. To me the expression "space of four dimensions " conveys no meaning whatever, but I am far from denying that it might convey a meaning to a being with faculties differently constituted, and I believe mathematicians would be able to give grounds for supposing that it would.
As to the second point, I say, that though we cannot picture to ourselves a state of things, in which the conditions of time, space, and number differ from those with which we are acquainted, in the sense of forming a complete and coherent mental picture of it, we can easily imagine facts which would prevent us from forming our present ideas about time, space, and number, or would show that if formed, those ideas were false. If, then, such facts existed, our present ideas as to time, space, and number would not exist, or if they did, would be regarded as false. Hence their truth depends upon the continued non-existence of facts readily imaginable, and hence we must conclude either that they might be otherwise, or that no one fact which we observe could be otherwise, and in either case there is an end of their special character as necessary truths.
Not to trouble you longer, I will conclude with a single illustration of this. Dr. Ward says:—" Let there be 16 rows of pebbles, each containing 18. It is a necessary truth that the whole number is 288. Omnipotence could divide one pebble into two or create new pebbles, but it is beyond the sphere of Omnipotence to effect that, so long as there remain 16 rows of 18 pebbles each, the whole number of pebbles should be either more or less than two hundreds, eight tens, and eight units." There is, I believe, a superstition in Wiltshire that no one can count the stones at Stonehenge, but that if you pass your life in counting you will always bring out a different result. Now suppose this were the fact, and suppose it were a fact commonly observed, that if you counted Dr. Ward's pebbles over and over again, arranging them each time in a different order, you always brought out a different result, would it not follow that the multiplication-table was not true? That table assumes, and so implicitly asserts, that there are things which retain their identity for a certain time, and that they do not lose it by the alteration of their position. I do not see why this truth should not be otherwise, why there should not be a world in which the act of putting two pairs of things together should reduce the number to three, just as the juxtaposition of two drops of water produces one drop. It is true that the one drop contains as much water as the two contained, but this is very far from being immediately obvious, or from being incapable of being disproved by experience. Every proposition in the multiplication-table is indeed either merely arbitrary, or else it is a statement of the fact that by varying the arrangement
of groups of objects you do not vary their number, which is a property of matter learnt by experience. When you say three times three is nine, you either give a name to three groups of threes, which name might just as well be eleven or seven as nine, or else you affirm that the juxtaposition or rearrangement of three groups of three things does not affect their number, which is perfectly true, but is necessary only in the general sense already referred to.
Upon the whole, it appears to me that the one type of truth and knowledge is the proposition,—"This sheet of paper which I hold in my hand is blue, and that other is white," and that all other assertions are reducible to this type. Truth thus means the correspondence between the thoughts or images raised by words, and the thoughts or images raised by the joint action of the senses and the mind directed to the things to which the words refer. Whether such truth is called "necessary" or not is to me matter of indifference. The essential point is that when we say that statements are true, we mean only that they correspond either with present perception, or with a present recollection of past perception. When we say that they are certain, we mean only that we do not, in fact, doubt them at the time when we make them. Truth and certainty never can be freed from any errors which may be inherent in our faculties or our memory, and every assertion which we make is, or ought to be, made subject to a tacit reservation in respect of such errors. - You cannot have anything truer than truth or more certain than certainty, in the senses of truth and certainty just stated.
Read at a meeting of the Metaphysical Society, March 10. 1874.
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