from the human to the Divine mind, may look within and see if it be possible to conceive some further motive underlying even the moral law. Falteringly we may answer that but one conception seems capable of satisfying our minds, utterly inadequate as any idea of ours must necessarily be to respond to the inconceivable reality. That one conception, the conception which seems to take us even deeper into God's essence than the conception of right," is the correlated conception expressed by the sublimest and noblest of all words-'Love.'

The meaning of life, which it has thus been sought to extract from the combined interrogations of consciousness and consideration of natural phenomena, has given occasion to the enunciation of various maxims and principles, some of which have probably struck the reader as altogether abstract and devoid of practical utility. But the writer has purposely deferred the consideration of circumstances and limitations which must necessarily be entered upon in considering human life in the concrete. Here causes and principles have alone occupied us, but this is because the subject is to be completed in another paper devoted to the consideration, not of the aim and meaning of life, but of its ordering and government.

ST. GEORGE MIVART.

PHILOSOPHY OF THE PURE SCIENCES.1

III. THE UNIVERSAL STATEMENTS OF ARITHMETIC.

We have now to consider a series of alleged universal statements, the truth of which nobody has ever doubted. They are statements belonging to arithmetic, to the science of quantity, to pure logic, and to a branch of the science of space which is of quite recent origin, which applies to other objects besides space, and is called the analysis of position. I shall endeavour to show that the case of these statements is entirely different from that of the statements about space which I examined in my last lecture. There were four of those statements that the space of three dimensions which we perceive is a continuous aggregate of points, that it is flat in its smallest parts, that figures may be moved in it without alteration of size or shape, and that similar figures of different sizes may be constructed in it. And the conclusion which I endeavoured to establish about these statements was, that, for all we know, any or all of them may be false. In regard to the statements we have now to examine, I shall not maintain a similar doctrine; I shall only maintain that, for all we know, there may be times and places where they are unmeaning and inapplicable. If I am asked what two and two make, I shall not reply that it depends upon circumstances, and that they make sometimes three and sometimes five; but I shall endeavour to show, that unless our experience had certain definite characters, there would be no such conception as two, or three, or four, and still less. such a conception as the adding together of two numbers; and that we have no warrant for the absolute universality of these definite characters of experience.

In the first place it is clear that the moment we use language at all, we may make statements which are apparently universal, but which really only assign the meaning of words. Whenever we have called a thing by two names, so that every individual of a certain class bears the name A and also the name B, then we may affirm

This paper completes the publication of the substance of a series of three lectures delivered at the Royal Institution in 1873. The first two were published in the Contemporary Review while it was under the guidance of the present Editor of the Nineteenth Century. The third lecture contained-besides the substance of what is now here published—a statement of the doctrines afterwards set forth in a paper on the nature of things in themselves published in Mind, 1878.

the apparently universal proposition that every A is B. But it is really only the particular proposition that the name A has been conventionally settled to have the same meaning as the name B. I' may, for example, enunciate the proposition that all depth is profundity, and all profundity is depth. This statement appears to be of universal generality; and nobody doubts that it is true. But for all that it is not a statement of some fact which is true of nature as a whole; it is only a statement about the use of certain words in the English language. In this case the meaning of the two words is coextensive; one means exactly as much as, and no more than, the other. But if we suppose the word crow to mean a black bird having certain peculiarities of structure, the statement, All crows are black,' is in a similar case. For the word black has part of the meaning of the word crow; and the proposition only states this connection between the two words. Are the propositions of arithmetic, then, mere statements about the meanings of words? No; but these examples will help us to understand them. Language is part of the apparatus of thought; it is that by which I am able to talk to myself. But it is not all of the apparatus of thought; and just as these apparently general propositions, All crows are black,' 'All depth is profundity,' are really statements about language, so I shall endeavour to show that the statements of arithmetic are really statements about certain other apparatus of thought.

We know that six and three are nine. Wherever we find six things, if we put three things to them, there are nine things altogether. The terms are so simple and so familiar, that it seems as if there were no more to be said, as if we could not examine into the nature of these statements any further.

No more there is, if we are obliged to take words as they stand, with the complex meanings which at present belong to them. But the real fact is that the meanings of six and three are already complex meanings, and are capable of being resolved into their elements. This resolution is due-I believe equally and independently-to two great living mathematicians, by whose other achievements this country has retained the scientific position which Newton won for her at a time of fierce competition when no ordinary genius could possibly have retained it. The conception of number, as represented by that word and also by the particular signs, three, six, and so on, has been shown to embody in itself a certain proposition, upon the repetition of which the whole science of arithmetic is based. By means of this remark of CAYLEY and SYLVESTER, we are able to assign the true nature of arithmetical propositions, and to pass from thence by an obvious analogy to those other cases that we have to

consider.

What do I do to find out that a certain set of things are six in number? I count them; and all counting, like the names of

numbers, belongs first to the fingers. Now this is the operation of counting; I take my fingers in a certain definite order-say I begin with the thumb of each hand, and with the right hand. Then I lay my fingers in this order upon the things to be counted; or if they are too far away, I imagine that I lay them. And I observe what finger it is that is laid upon the last thing, and call the things by the name of this finger. In the present case it is the thumb of my left hand; and if we were savages, that thumb would be called six. At any rate, if the order of my fingers is settled beforehand, and known to everybody, I can quite easily make the statement, 'Here are six things,' by holding up the thumb of my right hand.

But, if I have only gone through this process once, there is already a great assumption made. For, although the order in which I use my fingers is fixed, there is nothing at all said about the order in which the things are touched by them. It is assumed that if the things are taken in any other order and applied to my fingers, the last one so touched will be the thumb of my left hand. If this were not true, or were not assumed, the word 'number' could not have its meaning. There is implied and bound up in that word the assumption that a group of things comes ultimately to the same finger in whatever order they are counted. This is the proposition of which I spoke as the foundation of the whole science of number. It is involved not only in the general term 'number,' but also in all the particular names of numbers; and not only in these words, but in the sign of holding up a finger to indicate how many things there are.

Let us now look in this light at the statement that six and three are nine. I have counted a group of things, and come to the conclusion that there are six of them. I have already said, therefore, that they may be counted in any order whatever and will come to the same number, six. I have counted another distinct group, and come to the conclusion that there are three of them. Then I put them all together and count them. Now, without seeing or knowing any more of the things than is implied in the previous statements, I can already count them in a certain order with my fingers. For I will first suppose the six to be counted; the last of them, by hypothesis, is attached in thought to the thumb of my left hand. Now I will count the other three; they are then attached, by hypothesis, to the first three fingers of my right hand. I can now go on counting the aggregate group by attaching to these three fingers the successive fingers of my left hand; for thus I shall attach the remaining three things to those fingers. I find in this way that the last of them comes to the fourth finger of my left hand, counting the thumb as first; and I know therefore that if the aggregate group has any number at all, that number must be nine.

But this is an operation performed on my fingers; and the statement that we have founded on it must therefore be, at least in part,

a statement about my counting apparatus. We may easily understand what is meant by saying that six and three are nine on my fingers, independently of any other things than these; this is a particular statement only. The statement we want to examine is that this is equally true of any two distinct groups whatever of six things and three things, which appears to be a universal statement. Now I say that this latter statement can be resolved into two as follows:1. The particular statement aforesaid: six and three are nine on my fingers.

2. If there is a group of things which can be attached to certain of my fingers, one to each, and another group of things which can be attached to certain other of my fingers, one to each, then the compound group can be attached to the whole set of my fingers that have been used, one to each.

Now this latter, it seems to me, is a tautology or identical proposition, depending merely upon the properties of language. The arithmetical proposition, then, is resolved or analysed in this way into two parts—a particular statement about my counting apparatus, and a particular statement about language; and it is not really general at all. But this, it is important to notice, is not the complete solution of the problem; there is a certain part of it reserved. For I only arrive at the number nine by certain definite ways of counting; I must count the six things first and then the three things after them. And I only arrive at the result that if the aggregate group of things has any number at all, that number is nine. It is not yet proved that they may be counted in any order whatever, and will always come to that number. Here, then, we are driven back to consider the nature of that fundamental assumption that the number of any finite group of distinct things is independent of the order of counting. Here is a proposition apparently still more general than any statement about the sum of two numbers. Do I or do I not know that this is true of very large numbers? Consider, for example, the molecules of water in this glass. According to Sir William Thomson, if a drop of water were magnified to the size of the earth, it would appear coarser-grained than a heap of small shot, and finer-grained than a heap of cricket-balls. We may therefore soon find that the number of molecules in this glass very far transcends our powers of conception. Do I know that if these molecules were counted in a certain order, and then counted over again in a certain other order, the results of these two countings would be the same? For the operations are absolutely impossible in anybody's lifetime. Can I know anything about the equivalence of two impossible operations, neither of which can be conceived except in a symbolic way? And if I do, how is it possible for this knowledge to come from experience?

I reply that I do know it; that such knowledge of things as there