Giordano Bruno

by J. Lewis McIntyre

Uniqueness of all things.In the third place the atomic theory offers an explanation of the uniqueness of each natural existence, which Bruno’s philosophical theory already assumed. The ever moving atoms present a mechanism by which the infinite diversity and infinite succession of change in things may be brought about. The appearance of similarity, exactness, etc., is, as we have found, an illusion. Mathematically exact figures or bodies—a true circle, for example—are unattainable by sense, even if they exist in nature; but they do not exist in nature. Sense and knowledge.Sense is the primary faculty, through which the material of all others must pass, so that what has not entered through that window of the soul cannot be known at all. But a single point out of place on the circumference of a circle makes it cease to be a true circle, and our sense-apprehension is necessarily so confused and indistinct that we cannot distinguish between the true and the false, where truth depends upon so inappreciable a difference. Relativity.Moreover sense-knowledge is relative to the knowing subject, or to the subject’s position with regard to the object. What to the eye of one is too large is to another too small; a sound which is pleasant to one ear is not so to another; the food which to the hungry man tastes sweet, to the full man is nauseous; the ape to the ape is beautiful, but to the man is of laughter-inspiring ugliness. Hence the circumspect will not say “this has a good odour, taste, sound, this has a beautiful appearance,” but will add “to me,” “now,” “sometimes.” Nothing is good or evil, pleasant or painful, beautiful or ugly, simply and absolutely; but the same objects in relation to individual subjects receive from the senses contrary denominations, as they in fact produce contrary effects. In deciding what is to be called good or bad, honourable or base, nature and custom have been the chief agents, and alterations have issued from the slow rise and victory of different opinions. Among the Druids and Magi certain things were performed publicly at sacrifices which now, even when committed in privacy, are regarded as execrable, and are so by way of law, and in the present condition of affairs. Philosophy, as it teaches to abstract from particulars, to bring the nature and condition of things as far as possible under an absolute judgment, must define differently the useful and good in an absolute sense, from the useful and good as contracted to the human species. Objectively there is no definitely good or definitely evil, definitely true or definitely false, so that from one point of view we may say that all things are good; from another that all things are evil; from a third that nothing is good or evil, as neither of the contraries is true; from a fourth that all things are both good and evil, as each of the contraries is true. No sense deceives or is deceived: each judges of its proper object according to its own measure. There is no higher tribunal to which to refer its object, nor can reason judge of colour any more than can the ear; sensible truth does not follow any general or universal rule, but one which is particular, mutable, and variable. In the working of an external sense there may be different degrees of perfection or defect, but not of truth or falsity, which consist in the reference of the subject and predicate to one another. The faculty by which we judge this or that to be true colour or light, and distinguish from apparent colour or light, is not in the eye. To affirm that man is an animal, we must know both man and animal, know that animal nature is in man, and other things which, as means or circumstances, concur directly or indirectly in this knowledge. External sense can apprehend only one species or image of the object; from the colour and figure to pass to its name, its truth, its difference from other objects, belongs to a more inward faculty. Judgment based upon sensation.Yet the latter is always based upon sense;—a deaf man can neither imagine nor dream of sounds which he has never heard, nor a blind man of colours and figures which he has never seen.[402] This digression on the relativity of knowledge, and on the different functions of sense and reason, in which Bruno follows partly the teaching of Lucretius, partly the Peripatetic doctrine of knowledge, shows that even if a true or perfectly exact geometrical figure existed in nature, none of the faculties with which we are endowed could apprehend it, since it is not given by external sense.[403]

No exactness or similarity in composites.But in the second place[404] reason tells us that no true circle, or other figure, is possible in nature: for there is in nature no similarity except in the atoms; a true circle would imply the equality of all lines from the centre, but no two lines in nature are entirely and in all respects equal to one another. The circle or part of a circle which appears most perfect to us—the rainbow—is an illusion of the senses, due to the reflection of the light of the sun from the clouds; so the circles made by a stone falling into water cannot be perfect, for this would mean that the stone itself is perfectly spherical, that the water is everywhere of the same density, that no wind is playing upon its surface. Sound is not equally diffused owing to differences in the density and rarity of the air, nor is the horizon ever a perfect circle, owing to differences of clearness in different directions. Object and faculty alike are in continuous change; all natural things are continually altering their form or changing their position; therefore although they seem to sense to remain fixed for a time, we know that this is impossible, from the nature of things.[405] Whatsoever falls in the scope of sense-perception, even the distant sphere and stars, we judge to consist of the same elements, therefore to be subject equally to perpetual variability and vicissitude. Thus—the atoms alone being simple, and remaining ever the same—no composite thing can be the same for one moment even, as each is being altered continually in all parts and on all sides by the efflux and influx of innumerable atoms.[406] “Hence nothing is perfectly straight, nothing perfectly circular among composites, nothing absolutely solid but the atoms, nothing absolutely void but the spaces between them.” The facet of a diamond appears to be a perfect plane, perfectly compact, yet in reality it is rough and porous.[407] In matter no two lines or figures are entirely equal, nor can the same figure be repeated twice.[408] No man is twice of the same weight, the very instruments by which we measure and weigh things are themselves in constant change, and the flux of atoms is never equal, but now denser, now rarer. In general no two things are of the same weight, length, sound, or number, nor are two motions or parts of motion ever the same. To say that ten trees are equal to ten others is to speak merely from a logical point of view, for in fact each is one in a peculiar and special sense.[409] “Equality is only in those things which are permanent and the same; changing bodies are unequal to themselves at any two instants.”[410] “Nothing variable or composite consists at two moments of time wholly of the same parts and the same order of parts, since the efflux and influx of atoms is continuous, and therefore not even from the primary integrating parts will you be able to name a thing as the same twice.”[411]

Number itself is not an absolute, but a relative determination: it does not touch the nature of the thing itself. Nature has no difference of number, as we have, of odd and even, tens and hundreds; nor do the gods, spirits, or other rational beings define the numbers and measures of objects by the same series of terms. Both numbers and the methods of numbering are as diverse as are the fingers, heads, and mental equipment of the numberers. That which fits in with the numbers of nature will therefore never fit in with our numbers. Thus ten horses and ten men, although determined arithmetically by one and the same number, are in nature, or physically, wholly unequal to one another.[412]

The atoms.In order that men’s minds may be better disposed for the reception of truth, it is necessary first to demolish the foundations of error;[413] Bruno accordingly sets himself to disprove the infinite divisibility of the continuum.[414] It was the common belief that there were no limits set to the dividing power of either nature or art, so that, however small a part might be arrived at, it was possible to divide it into yet smaller parts, on the analogy of the division of a fraction into tens of thousands of parts. Bruno denied this analogy to be justifiable, as in the latter case we are concerned not with division but with multiplication or addition, not with a continuum, but with discrete quantities, and it was part of his general theory that the addition of discretes might be carried on ad infinitum; the inverse process he denied. He thus held opinions directly contrary to those of Aristotle, with whom the mass of the universe was finite, limited by its enclosing sphere, the parts of the universe unlimited. Aristotle had an upper but not a lower limit; Bruno a lower but not an upper. Time and space.So time and space, which Aristotle had treated as finite in duration or extent, but as infinitely divisible, like the universe itself, are regarded by Bruno as unlimited in their dimensions, but as consisting of discrete minimal parts. “In every point of duration is beginning without end, and end without beginning”; it is the centre of two infinities. Therefore the whole of duration is one infinite instant, both beginning and end, as immeasurable space is an infinite minimum or centre. “The beginning and source of all errors, both in physics and in mathematics, is the resolution of the continuous in infinitum. To us it is clear that the resolution both of nature and of true art, which does not advance beyond nature, descends from a finite magnitude and number to the atom, but that there is no limit to the extension of things either in nature or in thought, except in regard to the form of particular species. Everywhere and always we find the minimum, the maximum nowhere and never. The maximum and minimum, however, may in one sense coincide, so that we know the maximum to be everywhere, since from what has been said it is evident that the maximum consists in the minimum and the minimum in the maximum, as in the many is the one, in the one the many. Yet reason and nature may more readily separate the minimum from the maximum than the maximum from the minimum. Therefore the immeasurable universe is nothing but centre everywhere; eternity nothing but a moment always; immeasurable body an atom; immeasurable plane a point; immeasurable space the receptacle of a point or atom.”[415]

The chief source of error on the part of the Peripatetics was their failure to distinguish between the minimum as a part, and the minimum a terminus or limit. Hence their idea that no combination of physical minima would give a magnitude, since two or more would touch one another with their whole surface, i.e. would coincide:—otherwise the minimum would have parts, a part of each touching the other, and a part not touching. On their theory it would follow that magnitudes do not consist of parts, or at least not of elementary parts. This is inconsistent with nature, for existing magnitudes must have been built up out of nature’s elements, and with art, for art can measure only on the assumption of first parts. It is true that what is posited as first part in one operation may be the last result in another, for the minimum, as we have seen, is a relative conception, but some first part is always assumed in any operation. And as the operation of art is not infinite, so neither is there infinite subordination of parts.[416] When two minima touch one another, they do not do so with their whole body, or any part of it, but one with its terminus or limit may touch several others; no body touches another with the whole of itself or a part, but with either the whole or the part of its limiting surface. The terminus of a thing is therefore no part of it, and by implication not a minimal part. Hence there are two kinds of minima concerned—that of the touching body, or part, and the minimum of that by which the contact is effected, the terminus.[417] The atom, which is the minimal sphere, touches in the absolutely minimal point, the smallest terminus. Other spheres do not touch in a point simply, but in more than one, or in a plane circle.[418] By adding limit to limit we never obtain a magnitude; the terminus is no part, and therefore if in contact it would touch with its whole self, so that magnitude is not made up of termini, whether points, atoms, lines, or surfaces which are termini; and this was the false ground on which the Aristotelians denied the possibility of the atom. It remained to ask if the termini were infinite, since the atoms were not; but it was clear that their number was determined by that of the atoms. For two limits do not touch one another:—“They do not cohere or make a quantum, but through them others in contact with one another make a contiguum or continuum.”[419] It may be added that if the parts of a divisible body were infinite in number, the parts of the whole would be equalled by the parts of the half, for in the infinite there can be no greater and less. In the infinite, as we have seen above, there is no difference between palms, digits, miles, between units and thousands, nor in the infinite time that has elapsed are there more months than years, more years than centuries. If any one set of these were less than the others it would be finite, and if one finite number may be applied to the whole, then the whole is finite.[420] The force of the Achilles dilemma was derived from the false idea that the minimum of one kind had some relation to that of another kind, e.g. that of time to that of motion, that of impulsive force to that of the motion produced. A thing of one kind does not define or measure a thing of another, and the duration of one does not compare in the same sense with the duration of another. Parts of different things are only equivocally called parts, and minima are minima only according to their proper (and diverse) definitions; therefore one is not measured by another, except in a rough way, for practical purposes.[421]

The vacuum. Atoms spherical.As the atoms come into contact with one another, not in all points of their surface, but in a definite number, it follows that there is a space between them, in the interstices; it was this thought which led Democritus to posit a vacuum.[422] The figure of the corporeal minimum must be spherical, for any mass which has projections can always be thought of as smaller, when these projections have been removed; and nature itself suggests this, by the gradual rounding off of substances through time, and the apparent roundness and smoothness of rough and jagged bodies when the observer is at a distance.[423] Diversity of forms of composite bodies results easily from spherical atoms, through differences in situation and order, differing amounts of vacuum and solid; but a simple vacuum with solid bodies is not sufficient,—there must be a certain matter through which the latter cohere together.[424] Although all other determinations may be abstracted from, figure at least must be predicated of the atoms; quantity cannot be asserted of that which is thought to be unfigured. These determinations of the minimum, though not given to sense, may nevertheless be made object of thought, by analogy or inference from the combinations of sensible minima in larger composites, the same forms of aggregation being repeated in the higher which occur in the lower forms.[425]

From the consideration of mathematical figures as consisting of minima, Bruno attempted both to remodel and to simplify the existing mathematical theory, and, unfortunately fell foul of the new analytical mathematics, the theory of rationals and of approximations, which at that time was receiving marked extensions, and which has since justified itself so completely by results. It is true he did not entirely reject it, but he regarded it as merely an artifice for rough practical measurements. The true measure is always the minimum, inferred by analogy from the combinations of greater parts, which are perceived by sense. Thus the minimal circle, after the atom itself, consists of seven minima, the minimal triangle of three, and the minimal square of four, and as each figure increases not by the addition of one atom merely, but by a number determined by the original number of atoms in the figure, it follows that no one figure is ever equal to another. Thus the second triangle is of six minima, the second square of nine, the second circle of nineteen. The “squaring of the circle” is therefore impossible,[426] although it may be approximately reached through the ultimate coincidence of arc and chord, by which the circle becomes equal to a polygon with an infinite number of sides.[427] This, however, is only an approximation of sense, which fails to observe the infinitesimal differences that are caused by the existence of a few atoms, more or less, in a figure. They are visible to the eye of reason, which comprehends that no two figures in nature are ever exactly equal. In exact geometry the number of one species of figure has nothing in common with that of another. It is clear, however, that even on his own ground Bruno was in error in this regard; for example, the seventh triangle and the fifth square are each composed of thirty-six minima.[428] But it is hardly necessary to take seriously his teaching in this respect. He was wholly governed by the belief in the infinite diversity of nature, and the absolute incommensurability of any member of one species of beings with one of a different species. “Since a definite minimum exists, it is not possible either in reality or in thought for a square to be equalled by a circle, nor even a square by a pentagon, a triangle by a square, nor in fine any species of figure by a figure of another species; for difference in the number of sides implies also difference in the order and number of parts. As figures in this respect are as numbers, and one species of number cannot be equalled by another either ‘formally’ or fundamentally (i.e. either in idea or in fact), we can never make an equilateral figure of any kind equal to one of another by first parts.”[429] Where this transformation is apparently carried out, as where a cube of wax is moulded to another figure, the result is due to the varying degrees of density in the different parts of the material; no solid parts are added or subtracted, but the disposition and extent of the pores or vacua are altered. But no argument can be drawn from this rough method, for the principles of practice are different from those of science.[430]

The latter principles are then applied boldly to geometrical science: thus it is shown that an angle, although it may be multiplied indefinitely, can be divided only into two parts; all its lines, it is understood, consisting of fila or rows of atoms;[431] that the circle has not an infinite number of radii, for from the circumference to the centre only six such lines can be drawn;[432] that not every line can be divided into two equal parts, for the physical line or filum may, naturally, consist of an odd number of atoms;[433] in any case geometrical bisection can at best be a near approximation,—though the two halves be apparently equal, they may really differ by many atoms. On this basis, in the fourth and fifth books of the De Minimo, Bruno offers a simplification of the geometry of Euclid. As nature itself is the highest unification of the manifold, and the monad is the unity and essence of all number, so we are taught to pass “from the infinite forms and images of art to the definite forms of nature, which the mind in harmony with nature grasps in a few forms, while the first mind has at once the potentiality and the reality of all particular things in the (simple) monad.”[434] In accordance with the method of simplification suggested by this doctrine, Bruno sets himself to show that the greater part of Euclid may be intuitively presented in three complicated figures, named respectively the Atrium Appollinis, Atrium Palladis, and Atrium Veneris. He hoped that by this means, “if not always, for the most part at any rate, without further explanation, the demonstration and the very evidence of the thing might be presented to the senses of all, without numbers,—not after the partial method of others, who in considering a statue take now the foot, now the eyes, now the forehead, now other parts separately,—but explaining all in each and each in all.”[435] It is no part of the purpose of this book to go at length into the mathematics of Bruno, which unfortunately have not yet met with a competent exposition. Apart from the difficulty of the matter itself, the poetical form and setting of his theorems is an additional stumbling-block in the way of understanding. Bruno was put to many shifts in order to give a poetical colouring to the most prosaic of subjects.

We have gone thus fully into the detail of Bruno’s atomic theory, more so perhaps than its intrinsic value seems to demand, because this aspect of his doctrine is the most important philosophically, and has exercised the greatest influence upon the course of speculation. It also provides most clearly an exemplification of the return which was made, or thought to be made, by the Renaissance to the older pre-Aristotelian philosophy and science. The rejection by Aristotle and his scholastic followers of the atomic theory of Leucippus and Democritus had been based upon the identification of space and body. The possibility of a vacuum in the corporeal world was denied, on the ground that discreteness was inconsistent with the continuity which was felt to be a necessary condition of space. Accordingly, the reintroduction of the atom was possible only in one of two ways—either by the distinction between body and space, or by the application of the atomic constitution of body to space itself. The former and truer solution was not open to Bruno. His time was still too much under the domination of Peripatetic thought for him to be able to take the important step of critically separating these two notions. The latter way, therefore, was that which he followed. Hence the curious attempt to remodel mathematical theory on the basis of the atom, which we have described above, and the reduction of mathematical certainty to an illusion of sense. Figure is to be found only in the combinations of atoms; and owing to the spherical form of the atom, the infinite number of them existing in any body which is presented to sense, and the space which lies between their surfaces, mathematical equality and exactness are impossible. Neither straight line, therefore, nor perfect circle are to be found in reality. Mathematics, which should be based upon, or which presupposes, continuity, is confounded with physics, which presupposes the analysis of body into discrete, impenetrable atoms. Physical atomism finds its justification in the experienced fact of resistance, which is the primary quality of body as perceived by our senses. In mathematical space, on the other hand, we abstract from all qualities except that of dimension only. Resistance would be inexplicable were it possible to proceed ad infinitum in dividing matter; it implies an ultimate irreducible and indestructible unit, whether we regard this unit as a centre of force or as an inert substance merely.