Philosophical Works of the Late James Frederick Ferrier/Lectures on Greek Philosophy (1888)/Pythagoras



1. The notices of the Pythagorean philosophy which have been transmitted to us, whether in its earlier or in its later manifestations, are scanty and extremely obscure. With the later manifestations we need not trouble ourselves. They are founded on spurious data, or at least on data which are not sufficiently authenticated. They are mystical in the extreme, and their symbolism is utterly incomprehensible. The earlier form of the philosophy, in so far as it is extant, is preserved in the fragments of Philolaus, and in a few notices by Aristotle. Philolaus was a contemporary of Socrates, and flourished about 420 B.C. Aristotle was a good deal later: so that there was an interval of nearly a hundred years between Pythagoras, who was in his prime about the year 540, and the earliest expositor of his opinions with whom we are acquainted. These two, Philolaus and Aristotle, are the principal sources of our knowledge of the Pythagorean philosophy in its earlier form. For the later manifestations of this philosophy Sextus Empiricus, who lived in the first half of the third century A.D., must be studied.

2. Aristotle lays down the general principle of the Pythagoreans in the following terms: "Number," he says, "is, according to them, the essence of all things; and the organisation of the universe, in its various determinations, is a harmonious system of numbers and their relations.” "The boldness of such an assertion," says Hegel, "impresses us as very remarkable; it is an assertion which strikes down at one blow all that our ordinary representations declare to be essential and true. It displaces sensible existence, and makes thought and not sense to be the criterion of the essence of things. It thus erects into substance and true being something of a totally different order from that form of existence which the senses place before us." (Werke, xiii. 237-38.)

3. What Pythagoras and his followers meant precisely by number it is not easy to say. One point seems to be certain, that number, in the Pythagorean sense, denoted law, order, form, harmony. It is said that Pythagoras was the first who called the world κόσμος, or order, thereby indicating that order was the essence of the universe—that law or number, or proportion or symmetry, was the universal principle in all things.

4. If we compare this position with that occupied by the Ionic philosophers, we shall perceive that it is an advance, an ascent, to some extent at least, from sense to reason. In fact, the great distinction between sense and reason is now beginning to declare itself. To revert for a few moments to the Ionic philosophy. This philosophy is an advance on ordinary thinking; ordinary thinking is held captive by the senses. It accepts their data implicitly, or without question. In the estimation of ordinary thinking, things are precisely as they appear: and their diversity is more attended to than their unity. In a word, ordinary thinking has eyes only for the particular, and is blind, or nearly so, to the universal. The Ionic philosophy rose into a higher position. It aimed at unity: it sought for a universal amid the diversity of sensible things; and this was an advance, a step in the right direction. The Ionic philosophy stood on a platform somewhat higher than that of ordinary thinking. But still this platform is far from being the platform of reason. The unity which the Ionic philosophers sought for among sensible things was sought for by means, and under the direction, of sense itself. It was a mere sensible universal; water, or infinite matter, or air; in short, it was something in itself material, and therefore something which, instead of being itself the universal in all things, did itself require to be brought under a universal, or reduced to unity under a higher principle. It was, in fact, a particular universal, in other words, a contradiction. The Ionic school, we may say, never rose above the region of sense, although within that region they certainly rose into a stratum of atmosphere elevated above that of ordinary thinking.

5. Let us now pass to the Pythagorean philosophy. Whether the Pythagoreans emancipated themselves completely from the thraldom of the senses, or whether such an emancipation be either practicable or desirable, I shall not now attempt to determine; but this is certain, that their speculations shot up higher into the region of pure reason than did those of their Ionic predecessors. Number is more an object of reason, and less an object of sense, than either water or air; and therefore we say that, while the position of the Ionic school is more that of sense than that of reason, the position of the Pythagorean school is more that of reason than that of sense.

6. Number is a truer universal than either water or air, or any other sensible thing. It is possible that the conception of number may not be an adequate conception of the universal in all things; that it may not be identical or coincident with the conception of the ultimately and absolutely real; but it is certainly a nearer approximation to this than any conception which we find set forth in the systems of the Ionic philosophers. The test of which is this: Suppose you had to explain something about the universe to an intelligence different from man's, unless that intelligence had senses similar to man's, he could not understand what you meant by water, or air, or earth, or fire, or colour, or sound, or heat, or cold: but whatever his senses were, or whether he had any senses or not, I conceive he would understand what you meant by number, he would know what one meant, and what many meant. He would not understand intuitionally what a tree was, and he could not be made to understand it intuitionally: but he might understand it symbolically, by being informed that it and everything else was a unity which admitted of being resolved into multiplicity, and that each of the fractions was again a unity. Unless he could be made to understand this—in short, unless he could form some conception of number—it seems to me that he would not be an intelligence at all. And therefore it may be said that number is a true universal, that is to say, it is a necessary thought; it expresses something which is the truth for all, and not merely the truth for some, intelligence. At any rate, it is a wider and truer universal than either water or air, or any other sensible thing.

7. We are now able to understand the apparently very paradoxical assertion of the Pythagoreans, namely, that number is the substance of things, the essence of the universe; and we are able, moreover, to perceive in what sense this doctrine is true. The whole paradox is resolved, the whole difficulty is cleared, by attending to the distinction to which I have so often directed your thoughts, the distinction between truth for all, and truth for some, or otherwise expressed, the distinction between the universal faculty in man, and the particular faculty in man. If we hold that the substance of things is to be found in that which is the truth for some, in other words, that it is to be apprehended by the particular faculty in man, in that case we shall certainly not hold that number is the substance of things; on the contrary, we shall hold that earth, or water, or air, or matter generally, is the substance of things, because this is what falls under the apprehension of the particular faculty in man. But if we hold that the substance of things is to be found in that which is the truth for all, that the essence of things centres in that which is the truth for all intelligence, in other words, that the essence of things is to be apprehended by the universal faculty in man; in that case we shall certainly not hold that earth or water, or matter generally, is the substance of things, for this is not necessarily the truth for all intellect; on the other band, we shall experience no great difficulty in holding that number is the substance of things, because number is the truth for all, and is that which falls under the apprehension of the universal faculty in man. You can thus readily understand the Pythagorean doctrine, even though you may be not quite willing to assent to it, that number is the essence of the universe, the ultimately and absolutely real. Number is this, because number is the truth of the universe for all intelligence; matter and its qualities are not the essence of the universe, not the ultimately and absolutely real, because they are not the truth for all, but only the truth for some intelligence, that is, for intelligence constituted with senses like ours.

8. To clear up this philosophy still further, it is right that I should state to you the grounds on which I hold that number is an object of reason, that is, of the universal faculty in man; in other words, is an object of all reason, and is not an object of sense, or of the particular faculty in man; in other words, is not an object merely of some intelligence. My reason, then, for holding that number is an object of pure thought rather than of sense is this; that every sense has its own special object, and is not affected by the objects of the other senses. For instance, sight has colour for its object, and can take no cognisance of sound. In the same way hearing apprehends sound, and takes no cognisance of colour. In like manner we cannot touch colours or sounds, but only solids. Neither can any man taste with his eyes, or smell with his ears. If number, then, were an object of sense, it would be the special object of some one sense; but it is not this. It accompanies our apprehension of all the objects of the senses, and is not appropriated to any sensible objects in particular. It is not like all the other objects of sense, the special object of any one sense, and therefore I conclude that it is not an object of sense at all, but an object of thought or reason. When we look at one colour, what we see is colour, what we think is one, i.e., number; when we look at many colours, what we see is colour, what we think is many, i.e., number. This distinction, the distinction by which number is assigned to reason and not to sense, is, I think, an important aid towards understanding the Pythagorean philosophy.

9. Number is a necessary form of thought under which we place or subsume whatever is presented to the mind. Hence form, which is another name for number, and not matter, is the essence of all things, at least of all intelligible things. It is the truth and substance of the universe—its truth and substance, not only in so far as it exists for us, but in so far as it exists for intelligence generally. Without number they are absolutely incomprehensible to any intelligence. Take away number, that is to say, let the universe and its contents be neither one nor many, and chaos, or worse, is come again. We are involved in contradictory nonsense. Number, then, or form, and not matter, as the Ionic philosophers contended; number, and not the numberless, or ἄπειρον of Anaximander, is the true universal, the common ground, the ultimately real in all things. With Pythagoras form or number is the essential, matter the unessential: with the Ionics matter is the essential, and form or number the unessential. In their respective positions the two schools stand diametrically opposed. But the Pythagorean is certainly a stage in advance of the Ionic.

10. In the account which I have hitherto given you of the Pythagorean philosophy, I have taken the statement of its principle from Aristotle, and, founding on his text, I have endeavoured, by means of a few critical reflections of my own, to impart to it some intelligibility, and to show you that there is some meaning, and also some truth in the assertion, that number is the essence of all things. I go on to speak of the Pythagorean philosophy as represented by Philolaus. Philolaus was probably the first of the Pythagoreans who committed to writing any of their master's doctrines; for neither the founder of the school, nor his immediate disciples, appear to have put their opinions on record. Philolaus was, as I said, a contemporary of Socrates. He wrote a work on the Pythagorean system, with which Plato seems to have been acquainted. Some fragments of this work are extant, and were collected and published in 1819 by a German scholar, Augustus Boeckh.

11. In this work we find these words: "Everything," says Philolaus, "which is known has in it number, for it is impossible either to think or know anything without number." He thus makes number the source and condition of intelligence, and the ground of the intelligible universe. But the following is even more important: "It is necessary," says Philolaus, "that everything should be either limiting or unlimited, or that everything should be both limiting and unlimited. Since, then, it appears, that things are not made up of the limiting only, nor of the unlimited only, it follows that each thing consists both of the limiting and the unlimited, and that the world, and all that it contains, are in this way formed or adjusted." This is a remarkable extract, for it shows that the Pythagoreans had to some extent anticipated the great principle of Heraclitus, namely, that every thing and every thought is the unity or conciliation of contraries; a principle, the depth and fertility of which have never to this day been rightly apprehended or appreciated, far less fathomed and exhausted.

12. In his dialogue entitled Philebus, Plato touches on this Pythagorean doctrine. For the word περαίνοντα, which is Philolaus's expression for the limiting, he substitutes πέρας, the limit; and the union of the two (the limit and the unlimited) he calls μικτόν, the mixed. So that, according to Pythagoras (and Plato seems to approve of the doctrine), everything is constituted out of the πέρας, and the ἄπειρον, the limit and the unlimited, and the result is the μικτόν, that is, the union of the two. This principle, afterwards applied to morals, led to Aristotle's doctrine of the μεσότης, or of virtue as a mean between two extremes. The πέρας; in the physical world was a limit or law imposed on the infinite lawlessness of nature: the πέρας or μεσότης in the moral world was a limit imposed on the infinite lawlessness of passion.

13. To get a little further insight into this matter, let us consider the conception of the μικτόν. This, I conceive, is equivalent to the limited. Now, let us ask what it is, in any case, that is limited? Perhaps you will say that it is the limited that is limited. But that would be an inept answer. What would be the sense of limiting the limited, the already limited? That would be a very superfluous process. Therefore, if the limit is to answer any purpose, it must be applied, not to the limited, but to the unlimited; and this, accordingly, is the way in which the Pythagoreans apply it. The limit is an element in the constitution of the limited; the unlimited being the other element.

14. Here is another way of putting the case. Take any instance of the limited, any bounded or limited thing, a book, for example. No one can say that the book is without limits. The limit, then, is certainly one element in its constitution. But is the limit the only element? Does the book consist of nothing but limits? That certainly cannot be maintained. There is something in the book besides its mere limits. What is that something? Is it the limited? Clearly it is not; because the limited is the total subject of our analysis; and, therefore, to hold that the limited is the other element, would be equivalent to holding that the whole subject of the analysis was a mere part or element of the analysis. The limited (the book) is what we are analysing, and therefore it would be nonsense to say that the limited was one factor in the analysis, while the limit was the other factor. This would be analysing a total thing into that total thing and something else. But if the limited cannot be the other term of the analysis, that other term must be the unlimited. What else can it be? The limited, then—in this case the book—consists of the limit and of the unlimited, and these are the two elements which go to the constitution of everything. Suppose the limits—for example, the two ends of a line—taken away, and no ends left, that which would remain would be the unlimited. But that cannot be conceived, you will say. Certainly it cannot. But it can be conceived to this extent, that if that part of a line which we call its ends or limits be taken away, and no new limits posited, then the remaining part considered in and by itself, is necessarily the unlimited. This element, which truly cannot be conceived without the other element, is the ἄπειρον of the Pythagoreans; and it cannot be conceived for this reason, that conception is itself constituted by the union or fusion of these two elements, the limit and the unlimited. Such is the Pythagorean doctrine, and it seems to me to be not only perfectly intelligible, but also perfectly true.

15. Another form which the Pythagoreans employed to express their principle was the expression μονὰς, the one, and ἀόριστος δυὰς, the indeterminate or indefinite two. Of these terms, the latter, in particular, is very obscure, and has been very insufficiently explained. I will endeavour to throw what light upon them I can out of my own reflections. First of all, these terms seem to be merely another form of expression for the πέρας and the ἄπειρον; the μονὰς; or one is the πέρας or limit; the ἀόριστος δυὰς is the ἄπειρον, the unlimited and indeterminate. Everything in being limited is one. This is expressed by the term μονὰς, which stands for the sameness or identity in things; but the diversity of things is inexhaustible; and this capacity of infinite diversity is indicated by the term ἀόριστος δυὰς, indefinite difference; so that, according to the Pythagoreans, the general scheme of the universe, as regarded by pure reason, is identity, combined with a capacity of infinite diversity. Neither of the terms has any meaning out of relation to the other. But let us for a moment consider each term by itself; ἀόριστος δυὰς, taken by itself, stands for absolute diversity. Everything in the universe is absolutely different from every other; all things are particular, and they are held together by no universal. The ἀόριστος δυὰς, in short, signifies, when taken by itself, the unbounded and inexhaustible particular. The μονὰς, again, taken by itself, stands for their unity; it signifies their feature of agreement. In a word, it is their genus, just as the ἀόριστος δνὰς is a general expression for their difference. Μονὰς is the Pythagorean term for the universal; ἀόριστος δυὰς is the Pythagorean term for the particular; and neither of these is capable of being conceived without the other. The true conceivable limit, whether considered as a thought or a thing, is the result of their combination.

16. We shall perhaps get more light thrown on these terms if we consider them under a purely arithmetical point of view. It might be thought that these words, πονὰς and ἀόριστος δυὰς, simply signified one and two, or one and indeterminate two. But that is not at all the meaning which the Pythagoreans attached to them. According to the Pythagoreans, every number consisted of these two parts; the μονὰς and the δυὰς were not numbers, but were the mere elements of number. This seems a perplexing position, yet it is susceptible, I think, of explanation. For example, every number is different from every other number; 1 is different from 5, 5 is different from 10, 10 is different from 20, and from 100, and so on. But every number also agrees with every number; and in what respect is it that all numbers agree? They all agree in this respect, that every number is once, or one times that number, whatever it may be; 5 and 10 and 20, and so on, agree in being once 5, once 10, and once 20. Each of these is one times what it is, so that they all agree in containing the μονὰς, or one. If you were to say five, or five ones, and did not mean once five, or one times five ones, your words would have no meaning. Neither you yourself nor any one else would know what you meant. But when you say once five, and then once ten, you not only express an agreement, you also express a difference between five and ten. Now, the general term for this difference is ἀόριστος δυὰς, and this δυὰς or diversity is said to be ἀόριστος or indefinite, because it varies indefinitely—once 10, once 20, once 30, once 40, once 1000, once 1,000,000—the once term, the μονὰς, never varies, but the other term, the δυὰς, as expressed by 20, 30, 40, 1000, 1,000,000, varies indefinitely, and its variations are inexhaustible; hence it is called ἀόριστος. Perhaps the simplest translation of ἀόριστος would be the indefinite any; ἀόριστος δυὰς any particular number. I conceive that in this way the Pythagorean doctrine, that the μονὰς and the ἀόριστος δυὰς are the elements of number, may be explained. Neither is the number one any exception; it, too, is composed of the μονὰς and the ἀόριστος δυὰς. One, like all other numbers, is different from any other number. In what respect does it differ from all other numbers? It differs from them in being one. In what respect does it agree with them? It agrees with them in being once one, or one times one, or one one. When we say "one," we usually mean "one one;" but we do not always or necessarily mean this, but may just as well mean 100 or 1000. One, viewed strictly, stands for once any number; and therefore, when it stands for the numerical one, it should be, and it is, construed to the mind as one one. One one, then, is the first arithmetical number, and, if so, we must be able to show that its elements are the μονὰς and the δυὰς for these are, according to Pythagoreans, the elements of all number without exception; and this can be shown without much difficulty. One one: which word, in that expression, stands for the monad, the point of agreement in all numbers? The first one does so. We say one one, one five, one ten, one hundred. All these numbers agree in being onei.e., once what they are. Then, again, which word, in the expression one one, stands for the duad—the diversity, the point in which one one differs from all other numbers? The second one does so. One one, one five, one hundred. The second word in each of these expressions expresses the difference of each of these numbers. One one is different from one five in its second term, but not in its first. From these remarks it appears, I think, that even number one is no exception to the Pythagorean law, which declares that the elements of all number are the monad and the duad, and that these are not themselves numbers. Thus, by considering numbers, we obtain light as to the constitution of the universe. Everything in the universe has some point in which it resembles everything else, and it has some point or points in which it differs from everything else; just as every number has some point in which it resembles all other numbers, and some in which it differs from all other numbers.

17. The monad and the duad being the elements of number must be viewed as antecedent to number. There is thus a primary one which is the ground or root out of which all arithmetical numbers proceed. And there is also a primary duad from which numbers derive their diversity. These two enter into the composition of every number (even into the composition of the numerical one), the one of them giving to all numbers their unity, or agreement, or identity; the other of them giving to all numbers their diversity. The primitive numbers, the numbers antecedent, as we may say, to all arithmetical numbers, are the Pythagorean monad and the Pythagorean duad. Of these, the former expresses the invariable and universal in all number; the latter, the variable and particular. And inasmuch as the particular is inexhaustible and indefinite, the duad is called ἀόριστος or indeterminate. Better to hold them elements of number than numbers.

18. As an illustration of the spirit of this philosophy, let me show you how a solid, or rather the scheme of a solid, may be constructed on Pythagorean principles. Given a mathematical point and motion, the problem is to construct a geometrical solid, or a figure in space of three dimensions, that is, occupying length, breadth, and depth. Let the point move—move its minimum distance, whatever that may be; this movement generates the line. Now let the line move. When you are told to let the line move, your first thought probably is that the line should be carried on in the same direction—should be produced; but you see at once (the moment it is pointed out to you) that such a movement is not a movement of the line, but is still merely a movement of the point. You cannot move the line, then, by continuing it at one or at both ends. To move the line you must move it laterally. That alone is the movement of the line. The lengthening of the line is, as I said, merely the movement of the point. The movement of the line then generates a surface. Now, move the surface. Here, too, you must be on your guard against continuing your lateral motion, for that is merely a continuation of the motion of the line; and this is not what is required. You are required to move not the line, but the surface; you must therefore move the surface either up or down into the third dimension of space, namely, depth; and these three movements give you the scheme of solid. You have merely to suppose this scheme filled with visible and palpable matter, that is, with something which is an object for the particular faculty in man, to obtain a solid atom; and out of atoms you can construct the universe at your discretion.

19. It seems at first sight a marvellous piece of foolishness that a philosopher should ascribe to empty unsubstantial number a higher degree of reality than he allows to the bright and solid objects which constitute the universe of matter. The apparent paradox is resolved when we consider the kind of truth which the philosopher is in quest of. He is not searching for truth as it presents itself to intellects constituted in a particular way, furnished, for example, with such senses as ours. If that were what he was in quest of, he would very soon find what he wanted in the solid earth and the glowing skies. But that is not what he is in quest of. He is seeking for truth as it presents itself to intellect universally, that is, to intellect not provided with human senses. And this being his aim, he conceives that such truth is to be found in the category of number, while it is not to be found in stocks and stones, and chairs and tables, for these are true only to some minds, that is, to minds with human senses; but the other is true to all minds, whatever senses they may have, and whether they have any senses at all or not. Slightly changed, the line of Pope might be taken as their motto by the Pythagoreans,

"We think in numbers, for the numbers come."

They come whether we will or not. Whatever we think, we think of under some form either of unity or multiplicity. Number seems to be a category of reason and universality.

20. This explanation seems to relieve the Pythagorean principle from all tincture of absurdity, and to render it intelligible, if not convincing; admit that truth and reality are rather to be found in what is true for all minds, than in what is true for some minds; and admit further, that number is true for all minds, and that material things are not true for all minds (but only for minds with senses); and what more is required to prove that truth and reality are rather to be found in number than in material things? The whole confusion and misapprehension with which the Pythagorean and Platonic, and many other systems, have at all times been overlaid, have their origin in an oversight as to the kind of truth which philosophy aims at apprehending. Philosophers themselves have seldom or never explained the nature of the end which they had in view, even when they were most intently bent on its attainment. Hence they seem to run themselves into absurdities, and hence their readers are bewildered or repelled. But let it be borne in mind that the end which philosophy pursues is the truth as it exists for intellect universal, and not for intellect particular—for intellect unmodified, and not for intellect modified—for intellect whether with senses like ours, or with senses totally different; and the apparent paradoxes of the Pythagorean and other ancient philosophies will be changed generally into articles of intelligible belief, and will stand out for the most part as grand and unquestionable verities, at any rate, as nearer approximations to absolute truth than anything which the mere senses can place before us.