In the course of that inquiry my attention was forcibly arrested by the fact that the immense variety of subjects upon which people reason distribute themselves in a general way between two extremes, at one of which the process of reasoning is conducted exclusively by means of representative or typical imagery (not necessarily visual), symbols being either absent or appearing merely as habitual but useless adjuncts, while at the other extreme representative or typical images are either absent or, if present, are useless, and the process is wholly dependent upon the use of symbols.1 The distinction does not by any means exactly correspond to that usually understood in the contrast implied by the terms abstract and concrete as applied to subjects of thought. Moreover, it appeared to me (for reasons which I cannot even briefly indicate here) that the more dependent we are on symbols for the conduct of any process of reasoning, the greater is the risk we run of illusions of judgement as to the real import of the process, of lapsing into a mystical attitude of mind with regard to it. Such an attitude of mind is no doubt quite as much and quite as often the result of indefiniteness or instability of the conceptions or ideas which form the groundwork of the subject discussed, as of the circumstance that the subject is one which demands the use of symbols for the conduct of the ratiocinative process. I am very far from saying that all mystical attitude is traceable to ratiocination by symbols, but that the necessity of ratiocination by symbols does carry in its train a tendency to mysticism from which even the keenest intellects cannot always free themselves.2

1 Usually, but not necessarily, words. The term 'symbol', it need hardly be said, is employed here in a restricted and technical sense; not, e.g., in the sense of Goblet d'Alviella's Migration of Symbols. The symbols of that work are what would here be described as conventionalized representative images, a middle form between the purely representative and the purely symbolic.

2 The use to which I propose to put the term 'mysticism' is, I must admit, somewhat arbitrary; but I do not know of any other name which could be less inappropriately employed to denote the kind of illusion I have in mind. What this is will become plain enough from the examples of it given in the body of the work, but even at this stage I may give some indication of the general sense in which I use the term. Some thinkers have spoken of, or alluded to, a certain ' reaction of language upon thought which results in error or confusion, but, so far as I know, without giving any explanation of the nature of this process or any indication of the conditions under which it takes place. It is to this kind of illusion or, to

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Now if we except pure or synthetic geometry there is no subject in which the conduct of the ratiocinative process is more dependent upon the employment of symbols than pure mathematics, and in which therefore, according to my view, we might more confidently expect to find evidence of the mystical tendency. On the other hand, there is no subject in which the ideas reasoned about are more definite and stable; and, certainly, mysticism, illusion as to the real import of a process of thought, seems to be utterly incompatible with definiteness and stability of the conceptions which underlie it. Merely to harbour the suspicion that mathematicians may not have developed their science in entire freedom from this tendency seems thus to betray an almost irrational leaning to scepticism. Yet, dismiss all a priori theoretical considerations; there are not wanting others of a practical nature which make such a suspicion at least plausible. It has long been a common-place of observation, that many men of great intellectual ability, capable in general of handling abstract subjects of thought with uncommon ease, are nevertheless apparently quite unable to learn mathematics. There is something in the subject, or in the manner of teaching it, which revolts them. I am reminded of a friend who, having taken up and dropped the study of pure mathematics, explained to me that he had abandoned the subject because, as he expressed it (with an obviously intentional touch of humour), he found that it required a kind of low cunning which he was destitute of. Expressed seriously, without whimsical exaggeration, there is in the orthodox exposition of mathematical symbolism much which, to many people, seems to be mere sophistry, paradox, and wordplay. They know, indeed, that it cannot in fact be so, since contact with the practical soon makes an end of all conclusions founded upon sophistry, paradox, and word-play; and the conclusions of mathematics are in general of eminent practical application. They conclude therefore that they have essentially ' unmathematical' minds. This really means no more than that they are unable to learn mathematics, and leaves us quite

be more exact, to the mental attitude which makes it possible, that I refer when I speak of mysticism, the mystical tendency or bias. I need scarcely add that my choice of the term implies no sort of indirect judgement respecting the nature or value of the religious or spiritual experiences which have been called mystical.

in the dark as to why people with logical heads should suppose themselves incompetent to reason logically about the very few, definite, and stable concepts which are the subject-matter of the science.

With the object of reasoning commodiously and swiftly about these concepts, mathematicians have invented and gradually perfected a special symbolic system, a special language. But the construction of this language is itself a work of logic in a far more exact sense than is the case with ordinary language. The study of such a language, one would suppose, ought not to present insurmountable difficulties to minds otherwise trained to think logically. There must be something in the orthodox exposition which, if it cannot be called downright sophistry and paradox, is at least inimical to plain straightforward thinking; and the modest avowal of incompetence on the part of those who are unable to follow it is at once a tribute and a sacrifice to intellectual integrity.

In considerations such as these, no less than in those which suggested themselves to me in the course of protracted reflection on the nature of language and its functions in the process of thought, originated the supposition that even in pure mathematics, conspicuously the domain of definite and stable conceptions, careful investigation might be found to reveal evidence of an infiltration. of the mystical. That this is in fact the case is what-after the preliminary discussion about language and symbolic expression in general-I have endeavoured to show; and I believe that this fact, if so it be, in some measure accounts for the almost insuperable difficulty which many strenuous minds find in mastering even the first principles of the subject, as well as for the purely intellectual repugnance with which they regard it.

The reader who is pressed for time but yet thinks the main subject might interest him, would perhaps do well to omit Chapters II and III as not being absolutely essential to the understanding of what comes after them. But I hesitate to recommend him to follow this course, because in doing so it is not unlikely that certain objections or difficulties may suggest themselves to him and unfavourably affect his judgement of the views expressed: objections or difficulties which probably would not suggest themselves at all after a careful perusal of these preliminary chapters.

CHAPTER II

THOUGHT AND LANGUAGE

Presupposition involved in conscious expressive action.-Thought and Meaning.—Contrast of mental attitude in the origination and in the acquisition of Language.-Beliefs suggested in the process of Languageacquisition, and more or less modified in the course of Experience.Meaning and Definition.-Opposed views of Definition (J. S. Mill and Taine): Due to not clearly distinguishing between its nature and its function. Max Müller's doctrine of the Identity of Thought and Language.

UNDERLYING and prompting my action in writing the words now actually flowing from my pen, there is a belief or presupposition that, upon being read, they will evoke, in the reader's mind, thoughts similar to those which, in mine, seek this mode of expression. These thoughts, the reader's and mine, more or less similar as I suppose them to be, dissimilar as possibly they may be, are meanings of the written words.

This is to affirm, generally, that a meaning is a thought in an individual mind; is not a something having an existence, somewhere and somewhen, independent of any mind. But unless we agree, which I do not, to make synonyms of the words thought' and' meaning', this is not to affirm the converse, that a thought or process of thinking is necessarily a meaning. And if the question arises: What differentiates a meaning from a thought? we may answer generally that a thought is a meaning just in so far as it is associated in a mind with a word or words, with marks or signs of some kind. The fact of association is what constitutes the associated things respectively meaning and symbol.

The fact itself that I thus draw attention to this presupposition or belief is evidence to others that it is at the present moment definite and explicit in my mind. This, however, is not generally the case. We commonly address one another, whether in speech or writing, without definite consciousness of the presupposition ; certainly, at least, without its being explicitly present, silently embodied or formulated in words. Given the desire to communicate our thoughts, however, the act of speaking or writing, as

a means to that end, would be inexplicable without the presence of the presupposition in the form of an habitual and implicit expectation, born of the long-continued experience of the actions (including speech) of those whom we address, or who are addressed by others. The habitual commonly fails to command attention, and by contrast with that which does command it, sinks into the subconscious. This mental condition has its parallel in ordinary perception. I perceive a tree; I am said to perceive that it is solid, resistant; my visual perception of solid objects is, apart from the visual element, a subconscious expectation that what is thus seen would prove to be resistant. There is no question here of deliberate inference that this would be the case; there may have been, probably there was, more or less definitely, conscious expectation when I was learning to perceive. Although we may attend to, or reflect upon, a habit of this kind, the habit itself is, in its very nature, undeliberate.

Whatever view we may take of the origin of language; whether we incline to believe in the onomatopoetic, interjectional, 'synergastic' or other original character of words, we cannot fail to recognize that language is now, and has been throughout the historic period of man's development, essentially conventional. Or, lest we risk by this statement to seem to ignore what philology has discovered for us as to the growth, decay, and transmutation of words in accordance with observed laws, it may be safer to assert that it is the association between word and thought which is essentially conventional; the result, not of any natural affinity between certain thoughts and certain articulate sounds, but of what may by metaphor be called repeated mechanical juxtaposition. No one, I believe, would venture to assert that there is any more obvious or natural connexion between any one than any other of the words 'dog', 'chien', 'Hund', and our conception of that animal. Nor can we discover in the constitution and use of language any reason whatever for supposing that any other articulate sound would not have served equally well with any of those in actual use as a name for the 'friend of man'. And it would in no wise modify this view if to-morrow any one should succeed in showing that these three names are indisputably traceable to some articulate sound originally mimetic in character. For it is not argued that language was conventional in its origination, but that it has become so in the course of its development.