AL I,4: "Every number is infinite; there is no difference."


Mathematical ideas involve what is called a continuum, which is, superficially at least, of a different character to the physical continuum. For instance, in the physical continuum, the eye can distinguish between the lengths of one-inch stick and a two-inch stick, but not between these which measure respectively one thousand miles and one thousand miles and on inch, though the difference in each case is equally an inch. The inch difference is either perceptible or not perceptible, according to the conditions. Similarly, the eye can distinguish either the one-inch or the two-inch stick from one of an inch and a half. But we cannot continue this process indefinitely -- we can always reach a point where the extremes are distinguishable from each other but their mean from neither of the extremes. Thus, in the physical continuum, if we have three terms, A, B, and C, A appears equal to B, and B to C, yet C appears greater than A. Our reason tells us that this conclusion is an absurdity, that we have been deceived by the grossness of our perceptions. It is useless for us to invent instruments which increase the accuracy of our observations, for though they enable us to distinguish between the three terms of our series, and to restore the theoretical Hierarchy, we can always continue the process of division until we arrive at another series: A', B', C', where A' and C' are distinguishable from each other, but where neither is distinguishable from B'.

On the above grounds, modern thinkers have endeavoured to create a distinction between the mathematical and the physical continuum, yet it should surely be obvious that the defect in our organs of sense, which is responsible for the difficulty, shows that our method of observation debars us from appreciating the true nature of things by this method of observation.
However, in the case of the mathematical continuum, its character is such that we can continue indefinitely the process of division between any two mathematical expressions so-ever, without interfering in any way with the regularity of the process, or creating a condition in which two terms become indistinguishable from each other. The mathematical continuum, moreover, is not merely a question of series of integral numbers, but of other types of numbers, which, like integers, express relations between existing ideas, yet are not measurable in terms of that series. Such numbers are themselves parts of a continuum of their own, which interpenetrates the series of integers without touching it, at least necessarily.

For example: the tangents of angles made by the separation of two lines from coincidence to perpendicularity, increases constantly from zero to infinity. But almost the only integral value is found at the angle of 45 degrees where it is unity.

It may be said that there is an infinite number of such series, each possessing the same property of infinite divisibility. The ninety tangents of angles differing by one degree between zero and ninety may be multiplied sixty fold by taking the minute instead of the degree as the co-efficient of the progression, and these again sixty fold by introducing the second to divide the minute. So on ad infinitum.

continues.....

A short digression is necessary. There may be some who are still unaware of the fact, but the mathematical and physical sciences are in no sense concerned with absolute truth, but only with the relations between observed phenomena and the observer. The statement that the acceleration of falling bodies is thirty-two feet per second, is only the roughest of approximation at the best. In the first place, it applies to earth. As most people know, in the Moon the rate is only one-sixth as great. But, even on earth, it differs in a marked manner between the poles and the equator, and not only so, but it is affected by so small a matter as the neighborhood of a mountain.
It is similarly inaccurate to speak of "repeating" an experiment. The exact conditions never recur. One cannot boil water twice over. The water is not the same, and the observer is not the same. When a man says that he is sitting still, he forgets that he is whirling through space with vertiginous rapidity.

It is possibly such considerations that led earlier thinkers to admit that there was no expectation of finding truth in anything but mathematics, and they rashly supposed that the apparent ineluctability of her laws constitutes a guarantee of their coherence with truth. But mathematics is entirely a matter of convention, no less so than the rules of Chess or Baccarat. When we say that "two straight lines cannot enclose a space", we mean no more than we are unable to think of them as doing so. The truth of the statement depends, consequently, on that of the hypothesis that our minds bear witness to truth. Yet the insane man may be unable to think that he is not the victim of mysterious persecution. We find that no reason for believing him. It is useless to reply that mathematical truths receive universal consent, because they do not. It is a matter of elaborate and tedious training to persuade even the few people when we teach of the truth of the simplest theorems in Geometry. There are very few people living who are convinced -- or even aware -- of the more recondite results of analysis. It is no reply to this criticism to say that all men can be convinced if they are sufficiently trained, for who is to guarantee that such training does not warp the mind?

But when we have brushed away these preliminary objections, we find that the nature of the statement itself is not, and cannot be, more than a statement of correspondences between our ideas. In the example chosen, we have five ideas; those of duality, of straightness, of a line, of enclosing, and of space. None of these are more than ideas. Each one is meaningless until it is defined as corresponding in a certain manner to certain other ideas. We cannot define any word soever, except by identifying it with two or more equally undefined words. To define it by a single word would evidently constitute a tautology.
We are thus forced to the conclusion that all investigation may be stigmatized as obscurum per obscurium. Logically, our position is even worse. We define A as BC, where B is DE, and C is FG. Not only does the process increase the number of our unknown quantities in Geometrical progression at every step, but we must ultimately arrive at a point where the definition of Z involves the term A. Not only is all argument confined within a vicious circle, but so is the definition of the terms on which any argument must be based.

It might be supposed that the above chain of reasoning made all conclusions impossible. But this is only true when we investigate the ultimate validity of our propositions. We can rely on water boiling at 100 degrees Centigrade,<<In revising this comment, I note with amusement that it had escaped me that 100 degrees C. is by definition the temperature at which water boils! I have seen it boil at about 84 degrees C. on the Baltoro Glacier, and determined my height above sea-level by observing the boiling point so often that I had quite forgotten the original conditions of Celsius.>> although, for mathematical accuracy, water never boils twice running at precisely the same temperature, and although, logically, the term water is an incomprehensible mystery.
To return to our so-called axiom; Two straight lines cannot enclose a space. It has been one of the most important discoveries of modern mathematics, that this statement, even if we assume the definition of the various terms employed, is strictly relative, not absolute; and that common sense is impotent to confirm it as in the case of the boiling water. For Bolyai, Lobatschewsky, and Riemann have shown conclusively that a consistent system of geometry can be erected on any arbitrary axiom soever. If one chooses to assume that the sum of the interior angles of a triangle is either greater than or less than two right angles, instead of equal to them, we can construct two new systems of Geometry, each perfectly consistent with itself, and we possess no means soever of deciding which of the three represents truth.

I may illustrate this point by a simple analogy. We are accustomed to assert that we go from France to China, a form of expression which assumes that those countries are stationary, while we are mobile. But the fact might be equally well expressed by saying that France left us and China came to us. In either case there is no implication of absolute motion, for the course of the earth through space is not taken into account. We implicitly refer to a standard of repose which, in point of fact, we know not to exist. When I say that the chair in which I am sitting has remained stationary for the last hour, I mean only "stationary in respect to myself and my house". In reality, the earth's rotation has carried it over one thousand miles, and the earth's course some seventy thousand miles, from its previous position. All that we can expect of any statement is that it should be coherent with regard to a series of assumption which we know perfectly well to be false and arbitrary.

It is commonly imagined, by those who have not examined the nature of the evidence, that our experience furnishes a criterion by which we may determine which of the possible symbolic representations of Nature is the true one. They suppose that Euclidian Geometry is in conformity with Nature because the actual measurements of the interior angles of a triangle tell us that their sum is in fact equal to two right angles, just as Euclid tells us that theoretical considerations declare to be the case. They forget that the instruments which we use for our measurements are themselves conceived of as in conformity with the principles of Euclidian Geometry. In other words, them measure ten yards with a piece of wood about which they really known nothing but that its length is one-tenth of the ten yards in question.

The fallacy should be obvious. The most ordinary reflection should make it clear that our results depend upon all sorts of condition. If we inquire, "What is the length of the thread of quicksilver in a thermometer?", we can only reply that it depends on the temperature of the instrument. In fact, we judge temperature by the difference of the coefficients of expansion due to heat of the two substances, glass and mercury.

Again, the divisions of the scale of the thermometer depend upon the temperature of boiling water, which is not a fixed thing. It depends on the pressure of the earth's atmosphere, which varies (according to time and place) to the extent of over twenty per cent. Most people who talk of "scientific accuracy" are quite ignorant of elementary facts of this kind.

It will be said, however, that having defined a yard as the length of a certain bar deposited in the Mint in London, under given conditions of temperature and pressure, we are at least in a position to measure the length of other objects by comparison, directly or indirectly, with that standard. In a rough and ready way, that is more or less the case. But if it should occur that the length of things in general were halved or doubled, we could not possibly be aware of the other so-called laws of Nature. We have no means so-ever of determining even so simple a matter as to whether one of two events happens before or after the other.

Let us take an instance. It is well known that the light of the sun requires some eight minutes to reach the earth. Simultaneous <<Simultaneity, closely considered, possesses no meaning soever. See A.A.Eddington, "Space, Time and Gravitation", 61.>> {WEH NOTE: SIC. This is page 51 in Eddington, op. cit. 1920 edition, 1959 reprint: "The denial of absolute simultaneity is a natural complement to the denial of absolute motion ..."} phenomena in the two bodies would therefore appear to be separated in time to that extent; and, from a mathematical standpoint, the same discrepancy theoretically exists, even if we suppose the two bodies in question to be only a few yards one more remote than the other. Recent consideration of these facts has show the impossibility of determining the fact of priority, so that it may be just as reasonable to assert that a dagger-thrust is caused by a wound as vice versa. Lewis Carroll has an amusing parable to this effect in "Through the Looking-Glass", which work, by the way, with its predecessor, is packed with examples of philosophical paradox. <<If I strike a billiard ball, and it moves, both my will and its motion have causes long antecedent to the act. I may consider both my Work and its reaction as twin effects of the eternal Universe. The moved arm and ball are part of a state of the Cosmos which resulted necessarily from its momentarily previous state, and so, back for ever. Thus, my Magical Work is only on of the cause-effects necessarily concomitant with the cause-effects which set the ball in motion. I may therefore regard the act of striking as a cause-effect of my original Will to move the ball, though necessarily previous to its motion. But the case of Magical work is not quite analogous. For I am such that I am compelled to perform Magick in order to make my Will to prevail; so that the cause of my doing the Work is also he cause of the ball's motion, and there is no reason why one should precede the other, See Book 4, Part III, for a full discussion. (Since writing the above, I have been introduced to "Space, Time and Gravitation", where similar arguments are adduced.)>>

We may now return to our text "Every number is infinite". The fact that every number is a term in a mathematical continuum is no more an adequate definition than if we were to describe a picture as Number So-and-So in the catalogue. Every number is a thing in itself,<<I regret to find myself in disagreement with the Hon. Bertrand Russell with regard to the conception of the nature of Number.>> possessing an infinite number of properties peculiar to itself.

Let us consider, for a moment, the numbers 8 and 9. 8 is the number of cubes measuring one inch each way in a cube which measures two inches each way; while 9 is the number of squares measuring one inch each way in a square measuring three inches each way. There is a sort of reciprocal correspondence between them in this respect.

By adding one to eight, we obtain nine, so that we might define unity as that which has the property of transforming a three-dimensional expansion of two into a two-dimensional expansion of three. But if we add unity to nine, unity appears as that which has the power of transforming the two-dimensional expansion of three aforesaid into a mere oblong measuring 5 by 2. Unity thus appears as in possession of two totally different properties. Are we then to conclude that it is not the same unity? How are we to describe unity, how know it? Only by experiment can we discover the nature of its action on any given number. In certain minor respects, this action exhibits regularity. We know, for example, that it uniformly transforms an odd number into an even one, and vice versa, but that is practically the limit of what we can predict as to its action.

We can go further, and state that any number soever possesses this infinite variety of powers to transform any other number, even by the primitive process of addition. We observe also how the manipulation of any two numbers can be arranged so that the result is incommensurable with either, or even so that ideas are created of a character totally incompatible with our original conception of numbers as a series of positive integers. We obtain unreal and irrational expressions, ideas of a wholly different order, by a very simple juxtaposition of such apparently comprehensible and commonplace entities as integers.

There is only one conclusion to be drawn from these various considerations. It is that the nature of every number is a thing peculiar to itself, a thing inscrutable and infinite, a thing inexpressible, even if we could understand it.

In other words, a number is a soul, in the proper sense of the term, an unique and necessary element in the totality of existence.

We may not turn to the second phrase of the text: "there is no difference". It must strike the student immediately that this is, on the face of it, a point blank contradiction of all that has been said above. What have we done but insist upon the essential difference between any tow numbers, and show that even their sequential relation is little more than arbitrary, being indeed rather a convenient way of regarding them for the purpose of coordinating them with out understanding than anything else? On a similar principle, we number public vehicles or telephones without implication even of necessary sequence. The appellation denotes nothing beyond membership of a certain class of objects, and is indeed expressly chosen to avoid being entangled in considerations of any characteristics of the individual so designated except that cursory designation.
when it is said that there is no difference between numbers (for in this sense I think we must understand the phrase), we must examine the meaning of the word 'difference'. Difference is the denial of identity in the first place, but the word is not properly applied to discriminate between objects which have no similarity. One does not ask, "What is the difference between a yard and a minute?" in practical life. We do ask the difference between two things of the same kind. The Book of the Law is trying to emphasize the doctrine that each number is unique and absolute. Its relations with other numbers are therefore in the nature of illusion. They are the forms of presentation under which we perceive their semblances; and it is to the last degree important to realize that these semblances only indicate the nature of the realities behind them in the same way in which the degrees on a thermoetric scale indicate heat. It is quite unphilosophical to say that 50 degrees Centigrade is hotter than 40 degrees. Degrees of temperature are simply conventions invented by ourselves to describe physical states of a totally different order; and, while the heat of a body may be regarded as an inherent property of its own, our measure of that heat in no way concerns it.

We use instruments of science to inform us of the nature of the various objects which we wish to study; but our observations never reveal the thing as it is in itself. They only enable us to compare unfamiliar with familiar experiences. The use of an instrument necessarily implies the imposition of alien conventions. To take the simplest example: when we say that we see a thing, we only mean that our consciousness is modified by its existence according to a particular arrangement of lenses and other optical instruments, which exist in our eyes and not in the object perceived. So also, the fact that the sum of 2 and 1 is three, affords us but a single statement of relations symptomatic of the presentation to us of those numbers.

We have, therefore, no means soever of determining the difference between any two numbers, except in respect of a particular and very limited relation. Furthermore, in view of the infinity of every number, it seems not unlikely that the apparent differences observed by us would tend to disappear with the disappearance of the arbitrary conditions which we attach to them to facilitate, as we think, our examination. We may also observe that each number, being absolute, is the centre of its universe, so that all other numbers, so far as they are related to it, are its appanages. Each number is, therefore, the totality of the universe, and there cannot be any difference between one infinite universe and another. The triangle ABC may look very different from the standpoints of A, B, and C respectively; each view is true, absolutely; yet it is the same triangle.

The above interpretation of the text is of a revolutionary character, from the point of view of science and mathematics. Investigation of the lines here laid down will lead to the solution of these grave problems which have so long baffled the greatest minds of the world, on account of the initial error of attaching them on lines which involve self-contradiction. The attempt to discover the nature of things by a study of the relations between them is precisely parallel with the ambition to obtain a finite value of Pi. Nobody wishes to deny the practical value of the limited investigations which have so long preoccupied the human mind. But it is only quite recently that even the best thinkers have begun to recognize that their work was only significant within a certain order. It will soon be admitted on all hands that the study of the nature of things in themselves is a work for which the human reason is incompetent; for the nature of reason is such that it must always formulate itself in proportions which merely assert a positive or negative relation between a subject and a predicate. Men will thus be led to the development of a faculty, superior to reason, whose apprehension is independent of the hieroglyphic representations of which reason so vainly makes use.