I have appended to the article some remarks which Dr. Ward's reply has suggested. The reply convinces me that we look at the subject from such different points of view, that I have failed to make him understand the point of my argument. My additional observations may accordingly appear to be to some extent to a restatement, in a more explicit form, of the contents of the original paper. Those who interest themselves in these subjects will perhaps forgive this defect in the form of the article if it contributes anything to the clear apprehension of an obscure matter. The paper then was substantially as follows.
In a paper lately published in the “Dublin Review,” Dr. Ward states one of his principal doctrines as follows:—
“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 (without explaining them) 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 a theory about faculties 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 he instinctively knows. Besides these two sets of faculties, he is acquainted, I suppose, otherwise, with the meaning of the word “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 corroborates 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.
The substance and purport of the syllogism, however, appears 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. Those which relate to time, space, and number are specimens of them. 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, the minor is disproved. If all truths are necessary, the conclusion 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. The expression “could not have been otherwise,” is not in itself clear, as I shall show further on. If, as many people suppose, it is merely a way of describing facts, which might have been predicted by any one who had sufficient knowledge to make such a prediction, 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, in such a room, on such an hour, of such a day. 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 Not surely that I can prevent it from existing. If this were so, no fact would be 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 not contingent to me. How can any power of imagining its absence, and the substitution for it of a similar but slightly different state of things, afford 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 the 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. I could understand the meaning of calling these facts necessary, though I see no good in doing so; but the so-called truths are merely descriptions of the facts, and carry us no further than our perceptions of them.
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. 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” we 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” we mean a metal of a certain specific gravity, malleable, and not liable to rust, whatever may be its colour, then God can (we may suppose) make gold of any colour, but why should we 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. [“Let us take as an instance of a geometrical axiom the proposition that two parallel straight lines will never meet, and let us take as an instance of an obvious physical fact the warmth-giving property of fire. No one who reflects will doubt that an English child's experience of the latter truth is (to say the least) every whit as constant and uniform as his experience of the former. Yet when he comes to the age of reason, he pronounces that the former is a necessary truth ; whereas he would be simply amazed at the allegation that an Omnipotent Creator could not on any given occasion deprive fire of its warmth-giving property.” Dublin Review, Jan., 1874.] 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 complexity 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 we hear of red gold, for instance, we understand a metal having all the other properties of gold as commonly known, except the quality of yellowness, for which redness is substituted. When we are told of a black swan, we understand a bird like a white swan, but of a different colour. But when we 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, we know that if the words employed are employed in their usual senses, the propositions into which they are introduced cannot be true ; because the propositions deny the only quality which the words employed denote, and are thus contradictions in terms or mere nonsense. To talk of two straight lines which enclose a space is as much nonsense as to talk of two straight lines of which either one or both are crooked.
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 seems 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 thin parallelograms popularly called straight lines, but distinguished from them by having no breadth, no thickness, no specific colour, 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 2 is quite independent of the question,--How do we become aware of their properties 2 I am not myself able to attach any meaning to the words “Space” and “Number” apart from distinct objects existing in space, and 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 great 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. [On this passage Dr. Ward observes that the only inferences which we can draw from maps more readily than from the things themselves, are “those which have for their premisses (in addition to the data of the map) mathematical truths.” Surely this is not so. We can tell from a map much more readily than from actual observation, that Italy, resembles a boot, or that the Alps and Apennines run in certain directions, or that Great Britain and Ireland are contiguous islands; but how are these mathematical truths? Is the resemblance of a portrait to a face a mathematical truth and what is a map but a picture of a particular sort 2 I do not quite understand what the “data of the map’ are. The fundamental datum of every map is the fact that the apparent figure of an object can be represented to the eye by lines drawn on paper or similar materials. This is not a mathematical truth, but a fact shown to exist by experience. This fact is the basis of geometry.] 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,-- 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 representing to our minds, things which we have perceived by the combined operation of our senses and our intelligence. 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 they have no such special character as is denoted by the expression “necessary truths.”
Not to trouble you longer, I will conclude with a single illustration of this. Dr. Ward says: “Let there be sixteen rows of pebbles, each containing eighteen. It is a necessary truth that the whole number is two hundred and eighty-eight. 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 sixteen rows of eighteen 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. [Dr. Ward observes: “On such a supposition, if the inhabitants possessed reason, they would know with absolute certainty that two and two make neither more nor less than four, and they would know that some power was constantly at work destroying material objects which had existed, or uniting material objects which had been distinct. As Mr. Stephen has not assigned any reasons for his opinions on this head, I need not assign any reasons for mine.” I think I have assigned a reason. My argument is that the multiplication table assumes, that by changing the position of things, their number is unaffected. This is a fact which might be imagined to be otherwise. Suppose it were otherwise, where, I ask, would you get your multiplication table? How would it ever occur to the mind 2 Dr. Ward simply asserts that it would do so. I should be curious to see his reasons for this opinion. Till he gives them, the matter stands thus. We agree that the multiplication table represents and assumes a fact proved by continual experience. I affirm, and he denies, that its truth is dependent on the continuance of the experience. In reference to illustrations similar to this published by me years ago, Professor Clifford observed in a paper published in the Contemporary Review for October, that in such a world the word “number '' would have quite a different meaning from the one which it bears with us. No doubt it would. The very point which I wish to prove is that propositions about space and number, and the very meaning of those words, are as much dependent upon experience as any others, and in this Professor Clifford would (I suppose) agree.] 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 are 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 checked and corrected as far as possible. 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 are words of degree. They 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.
This was the paper to which Dr. Ward replied by a paper published as an article in the Dublin Review for July, 1874. The article contains much matter which I am content to leave without further remark to the judgment of those who read the two papers, but it is summed up by Dr. Ward himself in two theses, each of which he supports by two arguments. Before I proceed to the explanations which these arguments show to be necessary, I will state, in Dr. Ward's own words, each of the theses and each of the arguments by which it is supported, together with my answer to each argument. Dr. Ward states his theses and the arguments in favour of them very shortly, and I will imitate his brevity in my answers. The italics in every case, indeed all the italics in this article without exception, are Dr. Ward's.
FIRST ARGUMENT—Not one man in a million has observed the fact that trilaterals are triangular.
ANSWER.—If so, not one man in a million knows that trilaterals are triangular. Every man who does know it has either observed it as a fact for himself, or has had the fact pointed out to him by others, and knows it for that reason.
SECOND ARGUMENT-In the enormous majority of instances when the axiom [i.e., that all trilaterals are triangular] is first known by us, it is accepted as an entirely new proposition, and yet as being, notwithstanding its novelty, self-evidently true.
ANSWER.—The same may be said of every truth which is proved by experiment. The proposition that the words now under the reader's eye are printed on the page before him, is accepted by every reader who sees them for the first time as an entirely new proposition, and yet as being, notwithstanding its novelty, self-evidently true. Yet that proposition is proved by experience and observation only.
FIRST ARGUMENT.—I do not see how any one can deny—certainly Mr. Mill expressly admits—that the triangularity of all trilaterals can be known by purely mental experimentation, by the mere process of imagining a trilateral. The axiom, then, is self-evident, or, in other words, is known to be true by the mere process of being duly pondered [not pondered, but imagined, which is a different thing.]
ANSWER.—This also is true (subject to the qualification in brackets); but it is not inconsistent with the theory that belief in the doctrine in question is based upon experience. Having seen various lines and triangles we can imagine others, and argue about them as well as if they were represented by actual figures drawn on paper. Dr. Ward's argument requires some one who could imagine triangles without ever having learnt, by sight or touch, what a straight line is, without knowing by experience what is meant by space. Imagination is based on sensation, and sensation is one of the constituent elements of experience.” Mr. Mill's point is, that in this particular case imagination is a kind of experience.
SECOND ARGUMENT—“This second reason for my second thesis is based on that conviction of necessity which inevitably arises in our mind when we contemplate this or any other geometrical axiom. We pronounce at once, on the question being placed before us, that the triangularity of trilaterals is not simply a phenomenon which prevails within the region of our experience, but a truth which could not be otherwise, of which Omnipotence could not effect the contradictory. I allege this as a fact of which every one must be cognisant who carefully and fairly examines his own mind.”
Dr. Ward proceeds to say, that this “conviction of necessity cannot possibly be due to the mere frequent experience and observation of" (any mathematical) “axiom.” In proof of this, he returns to the illustration about fire. Every one experiences the heating power of fire as often at least as he perceives that a trilateral is triangular, and probably his attention is much more frequently directed to it, yet “we see no repugnance whatever in the notion that in some other planet a substance may be found which in every other respect resembles fire, but yet which does not possess this particular property of imparting warmth.”
ANSWERS.—(1.) It is not shown that a necessary truth (whatever that may be) cannot be known by observation and experience only; therefore, admitting, for the sake of argument, that what we are alleged to “pronounce at once” is pronounced at once, and is true, it does not follow that the truths so pronounced are not learnt by experience.
(2) No such “conviction of necessity” as is alleged to rise in “our minds” arises in my mind on contemplating such axioms. The only convictions which do arise in my mind with respect to them are that they appear at present to sum up the facts of external nature which are continually under my observation; that I have no ground to expect such an alteration in those facts as would falsify the axioms in question; and that I cannot form any consistent inherent picture of such a state of facts, though I can readily imagine isolated results, which if they existed would throw doubt upon such axioms. Hence, whenever I have occasion to think of time, space, or number, I imagine them as being what to my present experience they seem to be, and, if my memory is correct, always, have been.
(3.) I have never tried to account for the “conviction of necessity” which is said to attend our knowledge of mathematical axioms by the frequency with which we experience their truth. Our certainty of their truth (I avoid the phrase “conviction of necessity”) arises from their simplicity and the directness with which we observe the facts which mathematical axioms describe. ... The experience by which we learn the meaning of the words “straight line" and “crooked line,” is the experience by which we perceive the truth of the proposition two straight lines cannot inclose a space. The experience by which we learn the meaning of the words, “a blue, sheet of paper,” is the experience by which we perceive the truth of the proposition, “this sheet of paper is blue;” but the mere frequency with which we look at the sheet of paper, has nothing to do with our certainty of the truth of the proposition. One steady look is as good for the purpose of producing that certainty as ten thousand looks; but one is absolutely indispensable. So of the lines.
This shortly sums up what I have to say by way of rejoinder to Dr. Ward's reply. I now proceed to the further explanations which from his article appear to be required. I think that his theory is pervaded by two errors more or less connected together, which vitiate all his speculations. These errors are an obscure and imperfect conception of what is meant by experience, and a confusion of thought about necessity and possibility, which, as I shall try to show, leads him into strange inconsistencies. First, I will consider the subject of experience; and next, the subject of necessity and possibility.
Dr. Ward's reasoning seems to assume throughout that the acquisition of knowledge by experience must in all cases be a gradual process. He seems, for instance, to be under the impression that a man who speaks of learning from experience that two sides of a triangle are greater than the third, means that the assertion is generalized from the observation of a vast number of individual triangles. If this is not Dr. Ward's impression, I do not understand such a passage as the following: “Imagine grave philosophers, telescope in hand, endeavouring to discern some trilateral in distant space in order that they may carefully count the number of its angles.” This, of course, is meant to suggest that those who think as I do,” ought in consistency to perpetrate the absurdity in question. [In Dr. Ward's language “phenomenists,” Dr. Ward being a “necessist.” I may in passing disclaim these nicknames. I dislike Dr. Ward's habit of coining words. Surely the common English of every day life is quite capable of expressing any proposition which has a distinct meaning.] The best way of answering this will be to show, by an example, what I mean by learning from experience the proposition to which Dr. Ward continually recurs about triangles having three sides.
First. What is the proposition? Dr. Ward says:—“The axiom which throughout my articles I have chosen for the purpose of illustrating this question has been the axiom that “all trilateral figures are triangular.” I certainly never heard that this proposition was “an axiom " at all, but this is of little importance. It is more important to remark that as stated the proposition is not even true. For instance, a capital Z or N is a trilateral figure, but it has two and not three angles. If the three sides were zigzags the figures might still be called trilateral, but they would have many more than three angles. To make the so-called axiom true, it must be worded in the following or in equivalent terms:— “If a portion of a plane superficies is enclosed by three straight lines they will form three angles with each other, and no more.” Now I assert that this proposition is learnt from experience and nothing else; and I further assert that experience, and experience alone, enables us to assert that this proposition is true of every part of space—that it is as true in Sirius as it is in London.
Any one who wished to teach a person the proposition just stated could do so by drawing a triangle and pointing out that it had three angles, and no more. He might then proceed to show in various obvious ways that if the three sides were arranged in any other way, they would not enclose space. One obvious mode of doing this would be to tell the student to imagine any one of the three sides turned round on either of its extremities as a centre. So long as it continued to enclose any portion of space it would cut the other two lines at two points, and as soon as it ceased to do so the three lines would cease to enclose space. This surely is experience in the strictest sense of the word, and the result is to show the student that the only way in which those particular three lines could be made to enclose space is by cutting each other at three points. If further proof were wanting, he might be challenged to do it in any other way. I cannot conceive in what other manner the proposition could be established, and I think Dr. Ward himself would own that this was not merely experience, but experience in the form of a crucial experiment.
I suppose Dr. Ward would say that such a proof would apply to only one triangle, or set of triangles, and that the difficulty is to show how experience could establish it with reference to all possible triangles in every part of space.
The experiment in question might readily be so managed as to apply to all possible triangles. By making each of the three lines revolve on its extremity, each of them is made to point in every direction to which any straight line in the plane of the paper can point. By making the paper revolve on its axis, each triangle is made to occupy successively all the planes into which space can be divided. Thus, with a single triangle and a single sheet of paper, we can perform experiments which show that the proposition in question is not affected either by the direction of the lines or by the plane in which they are placed. Equally simple experiments would show that it is unaffected by the length of the lines.
If a man was so unimaginative as to require such illustrations, it would be easy to show him that the result was the same whether the sides of the triangle were an inch long, or were drawn by the imagination between three fixed stars situated in remote parts of the sky. It can hardly be said that this is not an appeal to experience, and it appears to me equally idle to deny that the proper inference from the experience in question is that the proposition applies to every part of space where there are or may be straight lines. We believe triangles in Sirius to be like triangles in London, because our eyes, tell us that Sirius is included in the vast vault which we call space, and because the acquaintance with the three dimensions of space which we gain by looking at it and moving about in it assures us that a straight line is a straight line, whatever way it points, and whatever its length may be.
Indeed the very terms of the proposition, when correctly stated, are such as to show its truth when they are compared with the things which they denote. This may be easily shown. The proposition is as follows when correctly stated:—“If a portion of a plane superficies is enclosed by three straight lines they will form three angles with one another, and no more.” What is a plane superficies? Anything flat-this sheet of paper, for instance. What do you mean by the words “enclose a portion of a plane superficies 2" Drawing lines in different directions, but in the same plane, so arranged as to return to the point from which you begin. What is an angle : The figure made by the meeting of two straight lines going in different directions, or a bent line each of the parts of which is straight. Draw three straight lines in different directions in such a manner that the third line ends where the first line began.
In either of these cases the three lines enclose space, and meet each other in three points. Thus the proposition described as “a necessary truth" comes, when properly stated and explained, to this plain statement of two matters of fact — first: figures enclosing space can be drawn with three sides and three angles, and they are commonly called triangles. Secondly, no one ever yet has been able to imagine or to suggest a way in which three straight lines can be made to cut each other in more than three places. This is really all that the proposition that all trilaterals are triangular means. If any one thinks that it means more, I would recommend to his notice the article by Professor Clifford already referred to. At the conclusion of that article, the author states his conviction that we do not know that mathematical axioms are universally true. Whether he is right or wrong in this I do not pretend to say. It is enough for my argument that a man of the highest scientific attainments deliberately makes such an assertion. How Dr. Ward can reconcile the fact that Professor Clifford has expressed such an opinion with his own theory of necessary truth I cannot imagine. The article in question directly contradicts, by its very existence, Dr. Ward's assertion that a “conviction of necessity inevitably arises in our minds when we contemplate any geometrical axiom.” Unless Professor Clifford deceives himself on a matter of which no one else can judge, no such conviction arises in the mind of at least one very eminent mathematician. [The following is an extract from the article referred to:—“I am driven to conclude in regard to every apparently universal statement either that it is not really universal, but a particular statement about my nervous system, about my apparatus of thought, or that I do not know that it is true, and to this conclusion . . . I shall endeavour to lead you.”]
I am almost ashamed to labour a point which to my mind is so clear that to enforce it is like burning daylight, but experience, the universal teacher, shows that it is not equally clear to every one. Perhaps this question may throw some light on the subject. Is there one single proposition about time, space, or number, of which we can affirm that its truth would be known to a being who had no sensations whatever? If not, sensation—and so experience— is essential to knowledge, and Dr. Ward's fundamental thesis, that certain truths “are cognizable by us quite independently of experience,” is disproved.
So much for the question of experience. I pass to the confusion of thought about necessity, to which I have already referred. It appears to me that Dr. Ward's views on this matter may be shown to expose him to the following dilemma. Either he must give up the whole doctrine of necessary truth, which, as he would himself admit, forms an essential point of the philosophical foundation upon which he wishes to place Roman Catholic theology, or he must accept it in a form which would reduce all mysteries to nonsense, and render all miracles impossible. He is aware of the danger, and makes an effort to avoid it, which I will examine in its place, but I must first show what his opinion is. In his last paper he gives the following explanation of the expression “necessary truth”—“a truth which could not be otherwise, of which Omnipotence could not effect the contradictory.”
The second part of this definition is the really important member of it. If it were left out the first part would fall of itself. What is the meaning of “could not” or “cannot?” Power, so far as we know, can be exerted only by voluntary agents. The statement that a man cannot enclose a space with two straight lines is both intelligible and true. The statement that two straight lines “cannot.” enclose a space taken strictly is as unmeaning as the statement that they cannot paint a picture. Three straight lines “cannot ” in this sense enclose a space any more than two, though any man can do it with three straight lines. I think therefore that Dr. Ward was perfectly right in adding the second to the first branch of his definition, “A necessary truth is a truth which could not be otherwise,” is a definition which tells us nothing unless the words “could not be" are connected with some voluntary agent. This is the reason why my paper assumes that a “truth which could not be otherwise” means a truth which we cannot imagine to be otherwise; and I have already given reasons which I need not repeat for thinking that the mere fact that men are unable to imagine the falsehood of geometrical axioms, proves nothing more than the fact that they are unable to alter any one fact which they perceive. Hence the really important part of Dr. Ward's definition of necessary truth is that they are truths of which Omnipotence cannot effect the contradictory. The result of it is, that in order to know whether or not a truth is necessary, we must know what God can and what he cannot do.
As I have already shown, there is a sense in which the power of God is limited by the language of man. Define gold as a metal of a yellow colour, and God cannot make red gold. Define a straight line as a line which is not bent, and God cannot make a straight line which is bent. This, however, is mere quibbling. The substantial question is whether we can learn anything from asking ourselves whether God can or cannot bring about particular results capable of being more or less intelligibly described by human language. To me such an inquiry appears wholly absurd and monstrous. If a bookworm had somehow or other arrived at the conclusion that a Bible was probably produced by a being who possessed whatever the bookworm meant by intelligence, and if having arrived at that conclusion it were to go on to inquire what this being could and could not do, in order to get a measure of the comparative value of different propositions which its fellow bookworms had laid down about eating the leaves of books, it would act very like a man who affects to know what can and what cannot be done by a Being capable of doing everything which displays marks of design, of arranging the stars, making men and women, animals, and insects discoverable only by the microscope.
Dr. Ward can hardly take this view. It is essential to his whole system that he should measure the power of God, and when examined it will distinctly appear that he does, in fact, measure it by the powers of his own mind. He does, as a matter of fact, argue upon the supposition that God cannot do certain things because the human imagination stands in the way of it, and that God can do everything which the human imagination can conceive or depict.
Of course Dr. Ward does not, and could not, hold this theory consciously. Of course he repudiates it when it is ascribed to him. In his last paper upon the subject, he says:–“Imagine a Catholic of all men committing himself to such an argument! Imagine a Catholic implying that what is inconceivable is necessarily false! Did any one, e.g., ever dream of imagining that human beings on earth can conceive in its integrity the dogma of the Blessed Trinity? Of course I heartily agree with my critic that things utterly inconceivable by the human intellect may to beings of a higher nature be the simplest of truths.”
That a Catholic or any other man should be led by the necessities of his argument to contradict himself, and that he should be prevented from seeing this by his own verbal subtlety, is nothing at all surprising, and any one who has read Dr. Ward's articles must, I think, perceive that no man is more likely to be led into such a position; for no writer of our day is so fond of coining new words and devising verbal distinctions. I think that in the present case it can be shown that he has found himself compelled by the necessities of his argument to take the human faculties as being the measure of God's omnipotence in some cases, whilst in other cases which depend upon the same principle he arbitrarily refuses to do so.
The first part of this proposition is proved by passages already quoted for another purpose.
His first argument in support of the thesis that mathematical axioms are necessarily true is this: “The triangularity of all trilaterals can be known by purely mental experimentation, by the mere process of imagining a trilateral. By this act of imagination we know infallibly that” [any] “trilateral is triangular, or, in other words, that it is outside the sphere of Omnipotence to make a trilateral which shall not be triangular.” Thus by a mere act of imagination we learn what God cannot do. The second argument in support of the same thesis is very much to the same effect. It “is based on that conviction of necessity which inevitably arises in our mind when we contemplate this” (the triangularity of trilaterals) “or any other geometrical axiom. We pronounce at once that the triangularity of trilaterals is not simply a phenomenon which prevails within the region of our own experience, but a truth which could not be otherwise, of which Omnipotence could not effect the contradictory.” In fewer words God cannot alter mathematical axioms, because we have a conviction that God cannot alter them. A feeling of ours, the “conviction of necessity arising in our minds,” is the negative limit of God's power. He cannot do what we feel that he cannot do.
In other places, Dr. Ward uses the human imagination to show positively what God can do. For instance, he says that “an Omnipotent Creator could, on any given occasion, deprive fire of its warmth-giving property,” “support stones in the water,” “alter the taste of beetroot,” “divide one pebble into two, or create new pebbles,” and so forth. In a passage referred to above, he says in effect that there may be a substance like fire in all other respects in some other planet, because “we see no repugnance whatever” in the notion. That is, the existence of such a body is possible because we can imagine it as existing. Thus, the power of causing innumerable events is ascribed to God, simply, as far as I can see, because man can imagine their occurrence. We thus find that Dr. Ward believes God to be able to bring about any result which man can distinctly imagine, and that he also believes on the strength of acts of his imagination and feelings in his own mind that there are other things which God cannot do. It would be natural to conclude from this that Dr. Ward makes the powers of his own mind, his power of conceiving or imagining, the measure of God's Omnipotence; and I believe that this inference is just, though, as the paragraph above quoted shows, he repudiates it, and regards it with something approaching to horror.
I now proceed to consider the means by which he tries to avoid it. His opinions on the subject are to be found in an article published in the Dublin Review for July, 1871, called “The Rule and Motive of Certitude.” The point of that article, as far as it affects the present question, may be stated very briefly in the following propositions, which are almost in Dr. Ward's own words.”
[The passage is so important that I give Dr. Ward's very words, though they are not very conveniently arranged, and are encumbered with matter immaterial to the present argument. They are these: P. 57, “But we think there are propositions which may most fitly be called inconceivable and unthinkable, yet which all theists regard as indubitably true. We refer to religious mysteries.” P. 59, “We implied a few pages back that a proposition is necessarily false which contradicts what is known by the very conception of its ‘subject.” We should here explain that this does not at all conflict with what we have just been saying about mysteries. The reason is this. When the archetype is apprehended by me as indefinitely transcending my conception there of various propositions are not ‘known by its very conception,” which otherwise would be.”]
1. A proposition is necessarily false which contradicts what is known by the very conception of its subject.
2. If the subject is apprehended as infinitely transcending the conception thereof, various propositions are not known by its very conception which otherwise would be [so known].
3. Therefore proposition (1) is consistent with the assertion that many propositions are indubitably true, though “they may most fitly be called inconceivable and unthinkable.”
I do not understand what is meant by, “knowing by the very conception of a subject.” A man knows that the leaf under his eyes is green, not by his conception of it, but by looking at it; nor do I understand how the fact, that a leaf or anything else has many other qualities besides those denoted by the word leaf, prevents us from understanding those which are so denoted, or entitles us to talk nonsense about them. A man may know that a leaf is green, that it has a particular shape, and occupies a particular portion of space. He may also know that it has an internal structure, a set of organs which “infinitely transcend "his knowledge of them ; but he would not therefore believe the most learned botanist in the world if he were to assert things “inconceivable and unthinkable” about the leaf affecting its shape and colour: if, for instance, he were to say that it was both green and also bright scarlet, and that it was often in two places at once; the reply would be, “I can judge of that as well as you.” Leaving these dark sayings as they stand, let us see how they apply to particular cases.
Dr. Ward repeats again and again in a variety of forms of words that God cannot make two straight lines enclose a space.
The catechism put forward by all the Roman Catholic bishops in England as a simple statement of their creed contains these questions and answers —Q. What is the Holy Eucharist? —A. It is the true body and blood of Christ under the appearances of bread and wine. Q. How are the bread and wine changed into the body and blood of Christ?—A. By the power of God, to whom nothing is impossible or difficult. Q. When is the change made? —A. When the words of consecration ordained by Jesus Christ are pronounced by the priest in the Mass.
When Mass is performed a quantity of wafers are consecrated at once, each of which is declared to be the true body of Christ, and Masses are being performed all over the world at the same moment.
Hence if the statement in the catechism is true, the true body of Christ is in many places at one and the same moment of time. Hence God can cause a body to be in two or more places at once. Yet says Dr. Ward he cannot cause two straight lines to enclose a space. The one operation is a mystery, “utterly inconceivable by the human intellect,” no doubt, but indubitably true. The other contradicts that which is “cognized” as a “necessary truth,” and God himself cannot do that. How can distinctions about “knowing by the very conception of a subject,” and any other kind of knowing, meet a case like this? What intelligible distinction is it possible to draw between the state of our minds as to the proposition, “two straight lines cannot enclose a space,” and the proposition, “a body cannot be in two places at once?" Dr. Ward says that by the mere act of imagining a trilateral, we know infallibly that every trilateral must be triangular, and that God himself could not make a four-cornered one. What answer can he give to a person who says that by the act of imagining a “true body” he is satisfied that God himself cannot put it in two places at once, because he “knows infallibly by this act of imagination” that every body fills at every time one determinate part of space, or, in other words, that “it is outside the sphere of Omnipotence to cause it to be in more portions of space than one’ at any time?
The more this result is considered the more amazing it will appear to be. That all trilaterals are triangular is a necessary truth, which God himself cannot alter. It is known “by the very conception of the subject.” That a body cannot be in two places at once, is not known by the very conception of the subject, probably because body is apprehended by us as infinitely transcending our conception of it, therefore God can cause a body to be in two places at once, or even in three or more. If so, surely he can make a four-cornered figure of three sides—for the true body may as well be straight as of any other figure—and if it can be in two places at once it can make a trilateral with four corners. Two ordinary straight lines would form one of the angles, and the transcendent body being in two places at once would form three others with them and with itself. We, therefore, thus get a three-sided figure with four angles, which contradicts the necessary truth cognized by Dr. Ward. Thus the necessary and contingent truth may be brought into collision; and what is to happen then?
Upon the whole, it would seem that we are not much aided in our search after necessary truth, by being told to ask ourselves what God cannot do, and the difficulty is, if possible, increased by the information given in a very cautious and elaborate way, that he can accomplish some apparent impossibilities but not others; and that the test by which the two classes of apparent impossibilities may be distinguished, is that those which cannot be accomplished are and that the others are not known to be impossible “by the very conception of the subject.”
Do we then learn more as to the nature of necessary truths by approaching the test proposed from the other side, that is to say, by considering the question, What God can do Dr. Ward does not explicitly answer this question, but the illustrations already referred to show that he considers that God can bring about any result which man can distinctly imagine. It is almost as difficult to reconcile the doctrine of necessary truths with this assumption as to reconcile it with the belief in mysteries. I do not understand what is meant by knowing by the very conception of a subject, but be this as it may, we all know somehow or other that iron is hard and solid, and that it occupies space. Does Dr. Ward affirm, and if so on what grounds, that God can separate two links of an iron chain without breaking either of them? A being like a man, only much stronger and more dexterous, might probably be able to open one of the links, take out the other, and solder up the opening so quickly, that the human eye could not follow the operation; but this is another matter. My question is, whether God could make the one piece of iron pass through the other without dividing it? Whether, in other words, he could cause two pieces of metal to occupy the same space, at the same time, let the time be as short, and the space as small as you please? If the answer is yes, then God can “effect the contradictory” of a truth which to me at least appears (to use Dr. Ward's phrase) as necessary as any geometrical axiom whatever. To speak of two iron rods as occupying let us say the same cubic inch of space at the same moment of time, is to talk nonsense; just as much as to talk of two straight lines enclosing a space. I can attach no more meaning to the one statement than to the other. If the answer is no, then God's powers are only an exaggeration of human powers. God, like man, must command nature by obeying it. His operations, like ours, must be limited by the properties of matter. Such a conception is of course inconsistent with the whole of Dr. Ward's theology. In particular it would make creation impossible. To make something out of nothing is a feat which no imaginable extension of human skill and power would even tend to effect. Once admit the doctrine of necessary truth, and it will inevitably follow that unless it can be shown how a given result might be brought about by a man sufficiently strong and skilful, it can never be positively affirmed that God can bring it about, for a necessary truth may stand in the way.
Dr. Ward's utterances about fire are a good illustration of this. In the paper printed above I said that Dr. Ward would probably admit that God “could make cold fire.” In his reply Dr. Ward says, “We are constantly experiencing and observing the warmth-givingness of fire . . . . Yet there is no kind of conviction existing in our mind as to the necessity of this fact; we see no repugnance whatever in the notion that in some other planet a substance may be found which in every other respect resembles fire . . . . but yet which does not possess this particular property of imparting warmth.” In the Dublin. Review for January, 1874, he says in substance (in the passage quoted above) that it is clear that “an Omnipotent Creator could on any occasion deprive fire of its warmth-giving property.” These passages throw a light on Dr. Ward's theories, of which it is impossible to overrate the importance. He will not go so far as to say in terms that God can make cold fire. He probably feels that to make such an assertion is very like saying that God can make a crooked straight line. He thinks, however, that God can go very near making cold fire. He can make something exactly like fire in every other respect except that of giving warmth. Moreover, he can deprive fire of its “warmth-givingness” on any particular occasion.
Each of these assertions is very strange, and more particularly the last. If God can deprive fire of its “warmth-givingness” on any particular occasion, why might he not go on doing so continually, and thus make cold fire 2 Apart from this, however, what right has Dr. Ward to say that the “warmth-givingness of fire” is not a necessary truth? which he must say before he can assert that God can “effect its contradictory.” The only ground on which he can say so is that he can imagine the other qualities of fire combined in one substance, this one being left out. No doubt he can, but what does that prove? How can any man undertake to assert that everything which he can imagine may exist? Nothing is more easy than to imagine a man enjoying perpetual youth, and living for millions of ages, floating about in the air, crossing the sea on a cloak, or walking about with his head in his hand; but before we can undertake to say that these things are possible, we must show how they can be effected consistently with what we know of the properties of matter. It is one thing to admit, as I should, that we cannot deny that they might be done by a being of immeasurable power and skill, but it is quite another to affirm, as Dr. Ward impliedly does, that they certainly can be done.
If we could analyse all the facts which are referred to by the proposition “fire heats the human body” as distinctly as we can analyse the facts referred to by the proposition “the square of the hypothenuse of a right-angled triangle is equal to the sum of the squares of the sides enclosing the right angle,” we might probably discover that to speak of depriving fire on any given occasion of its “warmth-givingness,” or to speak of constructing a substance resembling fire in every other particular than its capacity of giving warmth, is exactly like speaking of causing the squares of the hypothenuses of a few right-angled triangles to be not quite so large as the sum of the squares of the sides for a few days, or of constructing a triangle similar to a right-angled triangle in every other property except this. The progress of physical science seems to me to make it probable in the highest degree that this actually is the case. Every fresh discovery seems to point to the general conclusion that the qualities of space which we can grasp with perfect distinctness, are only particular cases of a principle which extends to matter in all its forms, that nothing but our ignorance prevents us from exhibiting all the phenomena of animal life for instance, in the form of conclusions connected by demonstrations as strict as Euclid, with definitions, axioms, and postulates having as good a claim as his to the title of necessary truths. No one can positively affirm this, however probable it may appear; but until any one can affirm the contrary, until it can be shown that no assertion about the qualities of matter is a necessary truth, Dr. Ward cannot on his own principles justify his implied assertion that God can do whatever man can imagine. The utmost that he can properly assert is that God can probably do whatever man could do if he were much stronger and more skilful than he is.
For these reasons I say that the doctrine of necessary truth cannot be stated in any coherent or intelligible form except a form which turns all mysteries into nonsense, and reduces all miracles to the type of such a discovery as the electric telegraph.
Upon the whole, it appears to me that the word necessary as applied to truth is unmeaning. “Necessary truth,” in short, is nothing but truth disfigured by an unnecessary adjective which obscures the real question at issue between Dr. Ward and his opponents. This question is, “What is truth?” When we affirm that a proposition is true, can we affirm more than that the words of which it consists raise in our minds, thoughts, images, conceptions, ideas, or whatever else you please to call them, corresponding more or less distinctly and completely to those which are now, or which, as we now think, formerly were or hereafter will be raised in our minds by the direction of our bodily senses to external objects, or by our own internal feelings? Can we, in short, leap off our own shadows? Can we make any affirmation at all which is not at bottom an affirmation about ourselves? Is not the whole of our knowledge subject to all the limitations, and liable to all the imperfections which beset language, sense, memory, anticipation, the process of drawing inferences, and, in a word, every human operation whatever ? In one word, is not truth relative to man?
That this is so is an assertion hardly to be distinguished from the doctrine, that all our knowledge is founded upon experience, and that all our opinions on matters external to ourselves, which we can neither see, hear, touch, nor otherwise perceive by the exercise of our senses, are of the nature of more or less probable inferences founded upon what we can see, hear, touch, or otherwise perceive.
It is not, and indeed it cannot be denied by any one, that this is so with respect to almost all departments of knowledge. But it is vehemently contended by a school of which Dr. Ward is perhaps the most prominent Roman Catholic representative in England, that the principal doctrines of religion and morals rest upon a different basis.
And it is almost conceded by them that this opinion cannot be maintained unless mathematical and especially geometrical truths can be shown to be based on the foundation on which, as they say, moral and religious truths are founded. They feel that it would be almost absurd to ask a man to “intue” (to use their strange dialect) the truth of the proposition “there is a God,” unless they can make out that he is accustomed to “intue” the proposition, two straight lines cannot enclose a space, and others of the same sort. This is the reason why the doctrine of necessary truth is asserted under all sorts of different names with such persistency and such an expenditure of needless ingenuity.
For my own part, I can only regret, as a waste of power, the passionate efforts which are continually being made to get at some superior kind of truth, by poring over the operations of the mind. As Coleridge in the last generation made the distinction between reason and understanding the foundation of a great part of his philosophy, so Dr. Ward attempts to leap off his own shadow by all manner of strange phrases about necessary truth and contingent truth, “cognizing,” “intuing,” “ontologism,” “phenomenism,” “objectivism,” the imaginable and the unimaginable, the conceivable and the inconceivable, the thinkable and the unthinkable, knowing pure and simple, and “knowing by the very conception of the subject.” To me all such speculations are simply an attempt to coin ignorance into a superior sort of knowledge by shaking up hard words in a bag. I am very far indeed from asserting that the human body, as we see and know it, is the whole of the human being; that we are nothing more than the aggregates of the various organs and powers which can be seen, touched, weighed and measured. I think it highly probable that a being with appropriate faculties for that purpose would perceive in us much that we cannot perceive in ourselves. A man who could not see his own eyes or those of other people, would learn very little about them from pondering on the sensation of winking; but he would be guilty of equal and opposite errors, if he either concluded that he had no organ by which he was able to see, or devised elaborate theories about its nature and properties from the very trifling indications of its character afforded to him by its use. This is an exact parallel to metaphysics, and seems to me to explain their barrenness. As regards the operations of our own minds, we resemble persons who, being precluded by circumstances from the study of human and comparative anatomy, speculate on the internal organs of their own bodies. If persons so situated were to attempt to construct a system of anatomical knowledge out of the obscure feelings of their own hearts, brains, lungs, and stomachs, and if this system turned out, on examination, to be composed of slight metaphors derived from their observation of their hands, feet, eyes, ears, and mouths, they would, as it seems to me, occupy precisely the same position as Dr. Ward and many others, who follow methods dependent on the same principle. It is by no means improbable that they would, more or less unconsciously, struggle to conceal even from themselves the true character of their undertaking by inventing new and unfamiliar words at every turn, and by so disguising real ignorance under an appearance of unusual profundity.
If we consider the words by which the different operations of the mind are described we shall find that they supply proof that we know nothing about our minds; that our conjectures about them and their operations are,to the last degree, vague and unsatisfactory; and that our language on the subject is no more than a set of metaphors obviously incomplete, and in many respects incorrect.
Look, first of all, at the names which we give to the mind and its principal faculties, if they are to be regarded as something different from the mind. “I,” “mind,” “spirit,” “soul,” “reason,” understanding,” &c. We cannot give the derivation of all these words. Some of them, like ancient coins, have passed from hand to hand so often, that the original stamp is worn out. This, however, is not the case with all. “Spirit” is breath. Whatever “soul” may mean, “âme " its equivalent, is “anima,” and that again is breath. “Reason" and “understanding” are really no more than two metaphors which express the same thing in different ways. “That which counts or reckons,” “that which stands under,” as a table stands under the things laid upon it, and so if it were conscious would “understand” their relative position. “Verstand” is a similar metaphor, but rather less distinct. The French “entendement" is equally instructive, though the metaphor is different. The sense of hearing is given as a name to the faculty which understands when the ear hears. “Intelligo is ‘intus, or ‘inter’ ‘lego.’”
Look next at the names of the different mental operations. “Imagine,” “conceive,” “think,” “attend,” “intend,” “apprehend.” “comprehend.” The last four are pbvious metaphors—“stretch to,” “stretch towards,” “lay hold of,” “grasp.” If this were doubtful, it might be proved by reference to a passage in Cicero's “Academics,” in which the author illustrates the stoical doctrine as to the different degrees of knowledge—“visum,” which may perhaps be called perception; “assensus,” which comes very near to apprehension, “comprehensio,” and “scientia.”
“Hoc quidem Zeno gestu conficiebat. Nam quum extensis digitis adversam manum ostenderat, ‘Visum inquieb.at hujusmodi est.’ Deinde quum paulum digitos constrinxerat, ‘Assensus hujusmodi.' Tum quum plane compresserat pugnumque fecerat comprehensionem illam esse dicebat. Qua ex similitudine etiam nomen ei rei quod ante non fuerat καταληψω imposuit. Quum autem laevam manum admoverat et illum pugnum arcte vehementerque compresserat, scientiam talem esse dicebat, cujus compotem nisi sapientem esse neminem.”—Academ. I. ii. 47.The other three words, “think,” “imagine,” “conceive,” are equally metaphorical. Dr. Ward is very particular in drawing distinctions between imagination and conception, but surely they mean the same thing, though the meaning is denoted by different metaphors. If “think” (as Horne Tooke supposed) is connected with “thing,” it is difficult to distinguish it from “imagine.” The difference between imagination and conception is the difference between drawing a picture and taking-or, as we should rather say—putting together, the parts of a whole. The one metaphor is clearer and more lively; the other more general and better suited to words which do not denote objects of sight. The words, “A white horse, sixteen hands high, with brown spots, a long tail, and no shoe on his off fore foot," raise a set of images. The words, “although I had known him for some years, I was not aware that he was married,” might more properly be said to be conceived or understood.
The difference between the two sentences is simply that one relates wholly to objects which can be seen, the other to periods of time (“some years ”), legal relations (“married"), mental operations (“know,” “known"), persons indicated but not described (“he.” “I”), and forms of speech like “although,” which by a mental shorthand refers to a great number of other things. (This man was my acquaintance. I did not know he was married. I do know of most of my acquaintances whether they are married or not. It is singular that I did not know it in respect of him.) It is more natural, certainly, to speak of putting together such thoughts in one's mind than to speak of drawing a picture of them; but each separate thought might be the subject of mental images, and the act of putting or taking several things together is itself an image, and a very expressive one. An illustration of the fact that language about the mind and its operations is metaphorical, is given by Dr. Ward himself, and is all the more instructive and interesting on account of the unconsciousness with which it is given. He quotes with approbation the following passage from Mr. James Martineau —
“You may deny the idea of the infinite as not clear; and clear it is not, if nothing but the mental picture of an outline deserve that word. But if a thought is clear when it sits apart without danger of being confounded with another, when it can exactly keep its own in speech and reasoning without forfeit and without encroachment— if, in short, logical clearness consists not in the idea of a limit but in the limit of the idea—then no sharpest image of any finite quantity is clearer than the thought of the infinite.”Here a protest against taking imagination as a test of clearness is made by means of a series of images much more lively than appropriate. A thought is something which can sit, and it can sit either in company or alone; it can be in danger, it can forfeit its own territory, and encroach on the territory of other thoughts. It is “clear”—i.e., bright; or “obscure,” that is dark: and its brightness stands together in the line drawn round the image (consists in the limit of the idea). It would not be easy to crowd a greater number of metaphors into a given number of words. The passage also shows how easily able men are run away with by their metaphors. The “idea of a limit” is contrasted with “the limit of an idea,” as if there was a difference in the sense because there is a difference in the sound of the two expressions. Retranslate the metaphors into pictures, and it is obvious that the two phrases mean the same thing. The “idea of a limit” is the picture of a boundary. “The limit of an idea.” is the boundary of a picture. Now, as the boundary of the picture must be part of the picture, and must be itself depicted, it is obvious that these expressions mean one and the same thing.
Upon the whole it seems to me that the difference between imagination, conception, thinking, reasoning, understanding, and all other words by which mental operations are described is simply that some of the metaphors which these words embody are more appropriate to thoughts upon some subjects, and some to thoughts upon other subjects, but that it is idle to attempt to distinguish, with any approach to accuracy, either between the processes indicated by these words or between the subjects to which those processes are applied. To return to the illustration given above, imagine the hopeless nonsense into which a man would fall who attempted, simply by studying his own sensations, to investigate the subject of digestion, and to say precisely what his heart and stomach were like, and how their operations affected each other. A man may describe his own bodily feelings accurately enough without possessing any anatomical or physiological knowledge at all, but he ought not to think of founding anything further on such a description. Surely upon the same principle a wise man ought to be content to describe his own thoughts as they arise within him without attempting to get beyond them by means of them. To think about thinking is to try to escape from metaphor by changing your metaphor; to try to avoid the imperfections of language by translating English into French, French into German, German into Spanish, and Spanish back into English. Some friends once discussed a question into which was introduced the phrase of “The Personality of the Absolute. One of the party excused himself from joining in the discussion on the ground that he saw no use in inquiring whether or not the Untied wears a Mask. Heap up upon the word “truth" such phrases as “absolute,” “necessary,” “eternal,” “instinctively known,” and the like for ever if you please. They do not affect in the slightest degree the following reflections:—
1. All our knowledge comes to us through faculties each and all of which are constantly liable to error which we cannot in all cases detect.
2. All our knowledge is expressed in language which, when closely examined, may be resolved into metaphors more or less inappropriate to the matter in hand, and capable of being misunderstood and perverted by any one who looks at it from a point of view a little different from our own.
3. All our knowledge includes an element of memory or anticipation, each of which is in the highest degree fallible.
4. All our anticipations involve an assumption utterly incapable of proof, that the future will resemble our present conception of the past.
5. Many of our anticipations involve an assumption which is probably false, that no new forces with which we are at present unacquainted will come into play and affect the results which we anticipate.
I cannot understand how any one of these assertions can be denied, or upon what grounds anyone who admits them can refuse to draw from them the conclusion that every assertion which we make should be coupled either expressly or tacitly with some such qualification as this:—“As at present advised, subject to further and better instructions, and upon the assumptions hereinbefore stated, I am of opinion------.” The opinion should further be dated, both in time and place, so as to show that a variation on these matters might affect its truth.
If we suppose (and surely it is at least probable enough to influence the conduct of reasonable men), that this life is only a stage in existence, and that death is as much the gate into a new life as birth was—should this be true, it is surely possible that death may resemble waking from sleep, and that many things which now appear to all of us truths, and to some of us necessary truths, may turn out after all to have been necessary fictions, which fuller experience will enable us to lay aside. Dreams are often founded upon realities, but when we wake the reality is seen to be altogether unlike what in our dreams we were compelled to believe it to be.
Contemporary Review, December 1874.