Quantity

Plato requires the study of arithmetic of anyone who would become a philosopher, but he distinguishes between the arithmetic of the merchant and the arithmetic of the mathematician. The one counts for the sake of gain, the other for the sake of truth. In the seventh book of the Republic, Socrates prescribes five mathematical disciplines for the education of the guardians, beginning with the science of number, on the ground that each of them compels the soul to turn from the things that come to be toward that which is. The argument rests on a contrast between what the senses report and what number determines. Two fingers held up may be judged large or small according to what they are set beside, and the eye returns no settled answer; but the number two is not more or less two under any comparison. Where sensation delivers contrary reports, number admits of none.

The Philebus approaches quantity from another side. Socrates there divides whatever exists into the unlimited, the limit, the mixture of the two, and the cause of the mixture. Pleasure and the states of the body fall with the unlimited, since they admit of the more and the less and slide along a continuum without any fixed term. Measure, ratio, and number fall with limit, and it is by the imposition of limit upon the unlimited that anything ordered comes to be. On this view the good life is a measured life, and the well-ordered soul is one whose desires have received number and proportion. The account of quantity therefore bears upon ethics as well as upon mathematics, a connection treated more fully under the ideas of Temperance and Good and Evil.

Whether the objects of mathematics exist separately, after the manner of substances, is a question Plato's account leaves to his successors. Number, for him, is not an attribute abstracted from sensible things but something the mind grasps in its own right, and things are intelligible so far as they participate in definite number and ratio. The Pythagorean saying that all things are number stands behind this, though Plato does not repeat it in that form. He treats the mathematical sciences rather as preparation for dialectic, a training without which the mind cannot make the ascent to the Good. The dispute over whether numbers and figures are beings in separation or abstractions from bodies is taken up by Aristotle, and it belongs also to the discussions collected under the ideas of Form and Mathematics.

Plato's treatment establishes for the tradition the principle that mathematics discloses the structure of what is, rather than serving as an instrument of practical convenience. Aristotle inherits both the problem and much of the vocabulary, though he places quantity among the attributes of substance rather than among separate forms.

Aristotle places quantity among the categories, the highest genera under which whatever is may be said to be. It stands second in the list, after substance, and the order is not accidental, for quantities are attributes of bodies and not beings that subsist by themselves. In his theory the categories are strictly indefinable, since to define a term is to state its genus and its difference, and there is no genus above quantity. What can be given instead are the marks by which quantity is recognized. A quantum, he says in the Metaphysics, is that which is divisible into constituent parts each of which is by nature a one and a this. The question how much, or how many, is answered by way of quantity, and the two forms of the question correspond to the two principal kinds.

Quantity is either discrete or continuous. Number and speech are instances of the discrete; lines, surfaces, solids, and also time and place are instances of the continuous. The principle of the division appears to be the presence or absence of a common boundary at which the parts join. The syllables of an utterance are separate, each distinct from the rest, and no boundary unites them; the parts of a line meet at a point, the parts of a plane at a line, the parts of a solid at a line or a plane. Aristotle adds two further marks. Quantity has no contrary, for nothing is opposed to a length of three cubits as hot is opposed to cold; and quantity admits of no variation in degree, since one thing is not two cubits long in a greater degree than another. He holds, further, that the most distinctive mark of quantity is equality and inequality, for only things compared quantitatively are said to be equal or unequal.

These marks do more than sort terms. If quantity has no contrary, then increase and diminution differ in kind from alteration, which proceeds between contraries; a thing that grows does not pass from one opposite to another but acquires more of the same. Upon this the account of change in the Physics depends, and the point is pursued under the idea of Change. The continuity of magnitude carries a further consequence, that every magnitude is divisible into magnitudes and that nothing continuous can be composed of indivisible parts. Whether the divisibility so asserted is actual or only potential is a question treated under the idea of Infinity. As for the objects of mathematics, Aristotle takes them to be quantities considered apart from sensible matter, abstracted from body rather than existing in separation from it.

Aristotle's analysis supplies the terms in which quantity is discussed for many centuries afterward, and the divisions of magnitude and multitude, of the continuous and the discrete, remain in use among writers who reject much else in his philosophy. The question his account leaves open is whether quantity is a real attribute of substances or a construction of the mind from the materials of experience. Descartes presses the first alternative to the point of identifying body with its quantity, while Kant takes the second, making quantity a condition under which appearances are given rather than something found in things.

Aquinas receives the Aristotelian doctrine of the categories and puts it to work in natural philosophy and in theology alike. Quantity, on his account, is the first accident of corporeal substance, first in the sense that the other bodily accidents presuppose it. A color requires a surface, a surface requires extension, and extension is quantity. Matter considered in itself is pure potency, without shape or distinction of part from part; quantity is what first determines it as extended, giving it parts outside of parts and rendering it divisible. He distinguishes further between the quantities that inhere in bodies and those the mathematician considers. Number, dimension, and figure can be thought apart from sensible qualities, and this is to abstract them from sensible matter; but they cannot be thought apart from the corporeal substance which is their subject. A mathematical solid has three dimensions and yet occupies no place.

Beside the division of quantity into the continuous and the discrete, Aquinas proposes another, between dimensive and virtual quantity. Quantity of bulk is found only in corporeal things and has therefore no place in God. Quantity of virtue is measured according to the perfection of some nature or form, and it is in this sense that Augustine's remark is to be understood, that in things which are great but not in bulk, to be greater is to be better. The two are not commensurable with one another, since the very meaning of measure changes when one turns from the dimensions of a body to the perfections of a being. A parallel division holds for number. Division of the continuous is material and yields number, a species of quantity found only in material things; division by opposed forms is formal and yields a multitude which belongs to no genus but is transcendental, in the sense in which being is divided by one and many. The number of persons in the Trinity is a multitude of this second sort.

The doctrine of the Eucharist gives the analysis its most particular application. If the substance of the bread is changed while its accidents remain, something must sustain those accidents, which would otherwise inhere in nothing. Since dimensive quantity is the first accident and the others depend upon it, quantity is their subject in this case, and the whiteness, the taste, and the shape remain in the dimensions of the bread. Quantity here performs an office ordinarily belonging to substance. The philosophical point should not be lost in the theological one. If quantity is what renders matter divisible, then an account of matter waits upon an account of quantity, and the relation between the two is pursued under the idea of Matter.

Through Aquinas the Aristotelian treatment of quantity passes to the Latin West in a form fitted to natural philosophy and to theology at once. His doctrine that quantity is the first accident of body, together with his distinction between dimensive and virtual quantity, sets out a framework which Descartes, Leibniz, and Locke each undertake in different ways to revise or to replace.

Hobbes proposes that reasoning is nothing but reckoning, the adding and subtracting of the consequences of general names. The mind, on this view, neither contemplates separate forms nor ascends by dialectic; it computes. Whoever reasons conceives a sum from the addition of parcels, or a remainder from the subtraction of one sum from another, and does so with names and propositions as the arithmetician does with numbers. Where Aristotle had placed quantity among ten categories, Hobbes makes the operation performed upon quantities the whole of thought. It follows that error in reasoning is miscalculation, and that the remedy for a dispute is the remedy for a disputed account, namely to reckon again from settled definitions.

The doctrine of reason rests upon a doctrine of body. Hobbes holds that whatever exists is body, and that body has quantity necessarily. He asks whether there could be "matter that had not some determined quantity, when quantity is nothing else but determination of matter; that is to say, of body, by which we say that one body is greater or less than another by thus or thus much." The sensible qualities are not in things but in us, produced by the motions of external bodies upon the organs of sense. What the understanding has to work with, then, is magnitude, motion, and figure. Terms answering to nothing measurable he calls insignificant speech, and much of the scholastic vocabulary falls under that description in his judgment. The relation of quantity to the sensible qualities is treated more fully under the idea of Quality.

The same manner of reckoning is carried into moral and civil philosophy. The commonwealth is constituted by a computation of powers: the authority of the sovereign is reckoned from the transfer of right by each subject, justice consists in the keeping of covenants, and injury in their violation. The passions themselves admit of quantitative description, desire and aversion being motions toward and away from objects and differing in degree rather than in kind. Whether a science of morals and politics can be constructed on this model, in the manner of geometry, is a question the tradition takes up at length, and it bears upon the discussions gathered under the ideas of Justice and Law.

The identification of reasoning with computation stands at the beginning of a long line of attempts to mechanize thought, from Leibniz's calculating machine and his project of a universal characteristic to the formal logic of later centuries. Hobbes's restriction of knowledge to what can be measured also prepares the empiricist inquiry which Locke undertakes into the manner in which the mind comes by its ideas of number and extension.

Descartes carries the reduction of body to quantity further than his predecessors by identifying corporeal substance with extension itself. Nothing at all belongs to the nature or essence of body, he observes, except that it is a thing with length, breadth, and depth, admitting of various shapes and various motions. Shape and motion are only modes, which no power could make to exist apart from extension, and gravity, hardness, the power of heating, and the other qualities we experience in bodies consist solely in motion or its absence and in the configuration and situation of parts. Where Aquinas held quantity to be the first accident of corporeal substance, Descartes leaves nothing for the accident to be an accident of. The clear and distinct idea of matter is the idea of a geometrical solid, and physics accordingly becomes a kind of geometry.

He is careful, nonetheless, about the sense in which extension may be spoken of apart from body. Considering the statement that body possesses extension, he remarks that we frame no two images in the imagination, one of body and one of extension, but a single image of extended body, so that the statement says no more than that the extended is extended. When we say instead that extension is not body, the term takes on another meaning and answers to no image in the imagination; it becomes an abstraction, which may properly occupy the geometer but which should not be treated as though it had independent reality. The distinction between physical and mathematical quantity is thus preserved even where the distinction between matter and extension is not.

Several consequences follow directly from the identification. There can be no vacuum, since space without extension is nothing and extension without body is impossible; the world is a plenum, filled throughout with extended substance in motion. Neither can there be atoms, for whatever is extended remains divisible. Motion must therefore be the displacement of one body by another in a full medium, and the planets are carried in vortices of subtle matter rather than moving freely through a void. Descartes also gives an account of dimension broad enough to cover more than the three of space. By dimension, he writes, he understands nothing but the mode and aspect according to which a subject is considered to be measurable, so that weight is a dimension in terms of which heaviness is estimated and speed a dimension of motion. Any division of a whole into parts of identical nature, whether it exist among things or be the work of the understanding, furnishes such a dimension. These physical quantities and their measurement are treated more fully under the idea of Mechanics.

The identification of matter with extension carries the reduction of the physical world to quantity as far as it has been taken. What it leaves unexplained is the union of mind and body, since thought and extension share no attribute through which the one might act upon the other. The philosophers of the following generation divide over the point. Leibniz denies that extension is the essence of substance and puts force in its place; Locke grants that extension belongs to bodies but denies that the real essence of matter is known to us at all.

Locke takes up quantity not as an attribute of things but as a question about the origin of our ideas. How does the mind come by the ideas of number, extension, and measure? The answer, on his principles, is that they enter by sensation and reflection and by no other door. We perceive particular bodies of particular bulk; we observe particular collections of particular things; and from these, by comparing, abstracting, and combining, the general ideas of quantity are formed. Nothing here is stamped upon the mind at birth. A child learns to count by handling things, and the idea of an endless number is not given but framed, by observing that to any number a unit may still be added. The manner in which the mind enlarges its ideas of quantity beyond anything found in nature belongs to the discussion collected under the idea of Infinity.

Among these ideas number holds the first place. Locke calls unity the simplest of all our ideas and the one suggested to the mind by the greatest variety of ways, since every thing that exists is one, and every collection of things is some number. Extension belongs only to bodies, but number applies to whatever can be thought, to sounds and to moments as much as to solids. The simple modes of number are for him the most distinct of all, since the least variation, being a unit, marks off one combination from its nearest neighbor as clearly as from the most remote. He contrasts this distinctness with what obtains among ideas whose differences are counted by degrees rather than by quantity, where we have no such accurate discrimination and no way of measuring their equality. The demonstrative certainty commonly attributed to mathematics alone he traces to this distinctness, rather than to any privilege belonging to the ideas of number, extension, and figure as such.

Locke keeps the distinction between primary and secondary qualities, counting solidity, extension, figure, motion or rest, and number among the attributes bodies retain however far they are divided, while colors, sounds, tastes, and odors exist only in the perceiver. He uses the word quality for the first group, though extension, figure, and number are the traditional objects of the mathematical sciences and were traditionally reckoned quantities rather than qualities. What he will not grant is the Cartesian inference from extension to essence. Since the real essences of substances lie beyond our knowledge, we cannot say that extension exhausts the nature of matter; for all we can determine, God might have superadded a power of thinking to matter. The clear and distinct idea, upon which Descartes rested, does not by itself reach the constitution of things.

Locke's account moves the discussion of quantity from the constitution of bodies to the constitution of our knowledge of them. The question it leaves is how mathematics, if its ideas are drawn from experience, can be certain and necessary rather than merely general. Hume presses the difficulty, and Kant frames his own inquiry as an answer to it.

Leibniz agrees with Descartes that natural philosophy must be mathematical, and disagrees with him about what extension is. Extension, he argues, is a resultant rather than a primitive attribute; it presupposes something repeated and something that does the repeating, and cannot on that account be the nature of a substance. What is substantial is force, the capacity to act, and bodies are aggregates of simple substances, or monads, whose ordered perceptions give rise to the appearance of magnitude and figure. Extension stands to the monads somewhat as the rainbow stands to the drops of water. It is not nothing, and it is not a substance.

Quantity in all its forms belongs accordingly to what Leibniz calls the well-founded phenomenon. Space, time, number, and continuous magnitude are ideal, not fictitious; they are appearances grounded in a reality that is itself neither spatial nor temporal. The monads do not act upon one another across an interval, having no windows through which anything might pass, and their agreement is pre-established. Spatial and temporal relations are the way that order presents itself to finite perception, so that space is an order of coexistences rather than a thing or a container, a position he argues at length against the Newtonians and one taken up further under the ideas of Space and Time.

The mathematical consequence is a division between the continuous and the discrete which Leibniz treats as marking the difference between the ideal and the actual. In the ideal order the whole is prior to its parts, and a continuous magnitude is divisible without end; in the actual order the parts are prior, and matter stands actually divided to infinity rather than divisible in potency alone. The difficulties arising from a confusion of the two orders he called the labyrinth of the continuum. The infinitesimal calculus, of which he is one of the inventors, works within the ideal order and does not require that infinitely small quantities be found among things. Whether the infinitesimal is a quantity at all, or a manner of speaking about limits, is a question treated under the idea of Infinity.

In demoting quantity from an attribute of substance to a well-grounded appearance, Leibniz opens the way to the position Kant afterward takes. If space and time are not determinations of things as they are in themselves but conditions under which things are presented, then mathematics holds necessarily of appearances while saying nothing of what lies beyond them. Kant retains the conclusion and rests it on another ground, locating the ideality of space and time in the constitution of the knowing subject rather than in the confused perception of a monadic order.

Kant frames the question of quantity as a question about the possibility of mathematics. Mathematical propositions are not analytic, since no analysis of the concepts of seven and five yields the concept of twelve; the sum must be constructed in intuition. Neither are they generalizations from experience, since they hold necessarily and admit of no exception. Kant concludes that they are synthetic and a priori, and that this is possible only if space and time are not properties of things in themselves but the forms under which anything whatever is given to sense. Geometry then holds of every object of possible experience because space is the form of outer intuition, and arithmetic because time is the form of inner intuition.

The principle stated in the Axioms of Intuition is that all intuitions are extensive magnitudes. A magnitude is extensive when the representation of the parts makes the representation of the whole possible: to represent a line one must draw it in thought, running through its parts and gathering them into a whole. Every appearance, being given in space or in time, requires this successive synthesis, and is quantitative before any question of its particular character arises. Quantity, on this account, is not discovered among things but is a condition under which anything can be an object for us at all. Kant distinguishes it from the intensive magnitude treated in the Anticipations of Perception, where what is graduated is the degree of a sensation rather than an aggregate of parts, and this distinction connects his treatment of quantity with his treatment of quality.

Quantity also names one of the four principal groupings in the table of the categories, under which Kant places unity, plurality, and totality. These answer to the division of judgments into universal, particular, and singular in the table of judgments, the categories being drawn from the logical functions of judging. Like the other categories, quantity is for him indefinable, a transcendental concept of the understanding presupposed in all determination of objects rather than derived from any. The upshot he describes as empirical realism joined with transcendental idealism. Whatever can be experienced is measurable, and mathematics applies to it without exception; whether anything is quantitative apart from the conditions of its appearing is a question that cannot be settled, since we have no access to objects except through those conditions.

Kant's solution composes the dispute between those who derived quantity from the nature of things and those who derived it from experience, by locating it in the constitution of the knowing subject. The cost, as Hegel afterward objects, is that knowledge is confined to appearances and the thing in itself placed permanently beyond reach. The bearing of this on what can be known at all is treated under the idea of Knowledge.

Hegel treats quantity not as one category standing beside others but as a moment in the movement through which the categories are generated. The Science of Logic begins with pure being, which passes into nothing, and whose unity is becoming; becoming settles into determinate being, and determinate being is quality. Quality is the determinateness that is one with being, so that to alter it is to make the thing another thing. Quantity arises when this determinateness is sublated, at once cancelled and preserved, the determination remaining real but becoming indifferent to what the thing is. A field is a field whether it measures one acre or ten. This indifference of the determination to its subject is what Hegel takes to be the character of quantity.

A quantum is quantity with a determinate limit, a magnitude having some definite value, and the value is external to the concept in the sense that quantity as such does not prescribe which value shall obtain. Continuity and discreteness, which Aristotle had taken as dividing quantity into two kinds, Hegel treats instead as two moments of one concept, each requiring the other. Continuity is the self-sameness of quantity, its indifference to any internal boundary; discreteness is the presence of the unit, the one that can be counted. A continuum without units would be nothing determinate, and units without continuity would be so many separate things rather than a magnitude. Number contains both, being at once an amount of units and the unity of that amount.

Quantity passes over into measure, which is the unity of quality and quantity. In measure a quantitative alteration, continued far enough, issues in a qualitative change: water warmed by degrees does not grow gradually more vaporous but at a certain point boils. Hegel calls the points at which such transitions occur nodal lines of measure, and he draws examples from chemistry, from the ratios of musical intervals, and from the constitution of states, where a magnitude indifferent within limits becomes decisive once the limits are passed. The indifference of quantity to quality, taken as absolute, therefore does not hold. The logic of quantity, pursued to its end, returns to quality in a richer form.

Hegel's treatment brings to a close the long engagement with the category that begins with Aristotle's placing of quantity among the highest genera. Where Aristotle distinguished quantity from quality by its want of contrariety and of degree, and Kant made it a condition under which appearances are given, Hegel undertakes to show that the separation cannot be maintained and that each passes into the other. The doctrine of nodal transitions was afterward taken up by Marx and Engels as a principle of historical change, a use discussed under the idea of Dialectic.