|The Defects of the Pre-Aristotelian Systems←||TOC||→Conclusion|
A Criticism of Plato
For the present, then, we may dismiss the subject of the Pythagoreans; the foregoing brief mention of them will be found adequate. As for (990 b) those who assume the Ideas as causes, in the first place, in the attempt to discover the causes of the entities of the actual world125 they introduced the notion of a second class of entities equally numerous with them.126 This is just as if one who wished to count certain things should fancy that while they remain fewer he will not succeed, but should first multiply them and then count. For the Ideas are pretty nearly as numerous as, or not fewer than, the things by inquiring into whose causes they advanced from actual objects to Ideas. For there is something synonymous corresponding to every group not only of substances but of all other things in which there is a One over the Many,127 both in this world of actual things and in that of eternal things.
Again, none of the methods of argument by which we try to prove the existence of the Ideas really establishes the conclusion. From some of them no necessary conclusion follows; from others, it follows that there would also be Ideas in cases where we do not believe in them. According to the arguments drawn from the sciences,128 there will be Ideas of all things of which there are sciences. According to that based on the One over the Many,129 there must be Ideas also of negatives, and according to that based on our ability to conceive of what has perished,130 Ideas of perishable things; for there is a memory-image131 of them. Besides, his most exact132 arguments partly lead to Ideas of relatives, of which there is, according to us, no self-existing class, and partly bring the "third man”133 into the argument. And, speaking generally, the arguments for the Ideas lead to the denial of things134 whose reality we Platonists are even more concerned to maintain than that of the Ideas. For it follows from them that it is not the Dyad but number which is logically primary, that the relative is prior to the absolute, and all the other inconsistencies between the consequences which have been drawn from the theory of Ideas and its principles. Further, according to the conviction on which our Ideal theory is based, there will be Ideas not only of substances, but of much else (for there are common concepts not only in the case of substances but in other cases, and sciences not only of substances but of other entities, and there is a host of similar consequences). But according to rigid logic, and the accepted theory of the Ideas, if things are related to the Ideas by "participation" there can be Ideas only of substances. For things do not "partake" of them per accidens; they only partake of each Idea in so far as it is not predicated of something else as a substitute. What I mean is, e.g., that if anything partakes of the Idea of "double" it also partakes of something eternal, but only per accidens, for it is an accident of the Idea of "double" to be eternal.135 Hence the Ideas must be of substances.136 But the same terms which denote substance here137 denote it also there; or what else can be meant by saying that there is (991 a) besides the actual things here something which is the unity corresponding to their multiplicity? And if the Ideas and the things which partake of them are members of the same class, they will have something in common. For why should duality be one and the same thing in the case of the perishable pairs and that of the pairs which though many are eternal, and not equally so in the case of the Idea of duality and a particular pair of things?138 But if they are not members of the same class, they can have nothing but their name in common, and it is much as if one called both Callias and a wooden image men, without reference to any community of character in them.139
Above all, it would be difficult to explain what the Ideas contribute to sensible things, whether to those which are eternal140 or those which undergo generation and dissolution. For they are not the causes of any movement or change in them. But, once more, they are also of no assistance for the knowledge of other things (for the Ideas are not the substance of things; if they were, they would be in the things); nor do they contribute to their Being, since they are not present in the things which partake of them.141 If they were, they might perhaps be thought to be causes in the sense in which an admixture of white is the cause that something is white. But this line of thought, which was first enunciated by Anaxagoras, and repeated later by Eudoxus and others, is easily refutable, for it is an easy task to collect many impossible consequences in opposition to such a doctrine.142
Once more, other things are not derived from the Ideas in any of the established senses of the term "derivation"; to call them "archetypes" and to say that other things "partake" of them is to employ empty words and poetical metaphors. For what is the agency which actually constructs things with the Idea as its model? A thing may both be and become like something else without being imitated from it. Thus whether Socrates exists or not, there may equally be some one like Socrates, and it is clear that the case would not be altered even if Socrates were eternal. Also, there will be many archetypes, and consequently many Ideas, for the same thing; e.g., "animal" and "biped" will be archetypes in the case of man, as well as the "Idea of Man." Further, the Ideas will be archetypes not only of sensible things, but of Ideas themselves, e.g., the genus will be the archetype of the species contained in it. So one and the same thing will be both archetype and copy.143 Besides, it may surely be regarded as (991 b) an impossibility that the substance of a thing and the thing of which it is the substance should be separated. So, how can the Ideas, if they are the substances of things, be separate from them?
In the Phaedo144 we are told that the Ideas are causes both of Being and of Becoming. And yet, even if the Ideas exist, the things which partake of them do not come into being unless there is something to set the process in motion; and many other things come into being, e.g., a house, a ring, of which we Platonists say there are not Ideas. Hence, clearly, it is possible for other things as well both to exist and come into being through the agency of causes of the same kind as those of the objects just referred to.145
Further, if the Ideas are numbers, how can they be causes? Perhaps, because things are a second set of numbers; e. g., this number is Man, that Socrates, that again Callias. But why, then, are the first set of numbers considered the causes of the others? For it will make no difference that the one are eternal and the others not. But if the explanation is that things here are ratios between numbers -- e. g., a musical concord – plainly, there is some one thing of which they are ratios. Now, if there is such a thing, viz., matter, manifestly the numbers themselves must be ratios of one thing to a second. I mean that, e.g., if Callias is a numerical ratio of fire, earth, water, and air, the Idea, too, must be a number of some other things which are its substrate, and the "Ideal Man," whether a number or not, still will be a numerical ratio of certain things, and not simply a number, nor does it follow on these grounds that he will be a number.146
Again, one number can be composed of many other numbers, but how can one Idea be formed of many Ideas? If you say it is not composed of the numbers themselves, but of the units contained in them, e.g., those of the number 10,000, what is the relation between these units? If they are all homogeneous, many paradoxical consequences must follow; if they are not homogeneous, neither those of the same number with one another nor all with all, what can make the difference between them, seeing that they have no qualities?147 Such thinking is neither rational nor consistent.
Again, it becomes necessary to construct a second kind of number which is the object of Arithmetic and all the studies which have been called " intermediate." How or out of what principles can this be constructed? And on what grounds must it be regarded as "intermediate" between things here and the ideal numbers?148 Again, each of the units in the Dyad149 must be derived from a prior Dyad; but this is impossible. Again, why is a number of units when formed (992 a) into a collection one thing?150 Again, in addition to all this, if the units differ, the Platonists have followed the example of those who maintain four or two elements. Each of these thinkers gives the name of element not to their common substrate, e.g., body -- but to fire and earth, whether they have a common substrate, viz., body, or not. But the One is in fact spoken of as if it were as homogeneous as fire or water. But if it is homogeneous in this sense, the numbers cannot be substances; rather, it is manifest that if there is a self-existing One and this One is a first principle, "one" is an equivocal term.151 In any other case it is an impossibility.
When we152 wish to refer our substances to their principles we derive length from the Short and Long, a special case of the Small and Great, the plane from the Broad and Narrow, body from the High and Low. Yet, how can the line be contained in the plane, or the line and plane in the solid? The Broad and Narrow is a different genus from the High and Low. So, just as numbers are not contained in these classes, because the Many and Few is a different class from them, clearly no other of the higher genera will be contained in the lower.153 Nor, again, is the Broad the genus of which the High is a species, for if it were so, body would be a kind of plane.
Again, how will it be possible for points to "be in" figures? Plato, in fact, rejected this class of entities as a mere fiction of the geometers. He used to speak of them as the "beginning of the line," for which he often employed the expression "indivisible line." But even these lines must have a limit, so that the same argument which proves the existence of the line proves, also, that of the point.154
To speak generally, though it is the business of wisdom to discover the cause of visible things, we have neglected that task (for we have nothing to say about the cause by which change is initiated), but in the fancy that we are describing their substance we assert the existence of a second class of substances, though our explanation of the way in which they are substances of the former set is empty verbiage, for "participation," as I have said, is nothing at all. Nor do the Ideas stand in any connection with the kind of cause which we observe in the practical155 sciences, the cause for the sake of which all Mind and all Nature act, and which we have included among our first principles. Mathematics has been termed by our present-day thinkers into the whole of Philosophy, in spite of their declaration that it ought to be studied for the sake of something further.156 (992 b.)
Besides, we may fairly regard the entity which they assume as matter as being more properly of a mathematical kind, and as being rather a predicate and a specific difference of substance and matter than identical with matter itself. I mean the Great and the Small; just as the physicists, when speaking of rarity and density, say that these are the primary specific differences of the material substrate, for they are a kind of excess and defect. And as to motion, if these elements157 are to constitute motion, plainly the ideas will be in motion;158 if they are not to constitute it, whence has it come? Thus the whole study of physical Nature is abolished. And even the proof, which is fancied to be so easy, that all things are one, does not follow. Their method of "exposition,"159 even if one grants all their assumptions, does not prove that all things are one, but only that there is a self-existing One, and does not even prove this unless it is granted that the universal is a genus; but in some cases that is impossible. And as for the objects they consider logically posterior to the numbers, viz., lines and planes and solids, no rational grounds can be produced to show how they exist or can exist, nor what character they possess. They cannot be Ideas (for they are not numbers), nor the "intermediate" class of objects (for these are mathematical figures), nor yet can they be identical with perishable things. Manifestly, we have here, again, a fresh and a fourth class of objects.160
In general, it is impossible to discover the elements of existing things if one does not first distinguish the different "senses of existence," especially when the inquiry is directed towards the problem of what elements existing things are composed. For one certainly cannot discover what are the elements of which activity or passivity or straightness is composed. If the problem is soluble at all, it is only soluble in the case of substances.161 So it is an error to ask after, or to think one has found, the elements of everything. How, indeed, could one possibly learn the elements of everything? For it is clear that one could not possibly have been in previous possession of any information at all. Just as he who is learning geometry may very well have previous knowledge about other things, but has no previous acquaintance with the truths which belong to that science, and which he is about to learn, so it is in all other cases. So, if there is, as some assert, a universal science of everything, he who learns it must have no previous acquaintance with anything. And yet all learning is effected through previous acquaintance with some or all of the matters concerned. This is true both of learning from demonstration and of learning from definitions. The parts which compose the definition must be previously known and familiar. The same is true, also, of learning from induction. But if it (993 a) be suggested that this knowledge is really innate, it is surely a mystery how we can possess the most excellent of sciences and yet be unconscious of the fact. Besides, how are we to recognize what existence consists of? How can the result be established?162 There is a difficulty implied here, since the same doubt might be suggested as about certain syllables. Some say that the syllable ΖΑ consists of Σ, Δ and Α, others that it is a distinct sound, different from those already familiar. Besides, how could one become acquainted with the objects of sense-perception, without possessing the corresponding form of sense-perception? Yet, this ought to be possible if all things are composed of the same constituent elements,163 as composite articulate sounds are composed of their own special elements.164
(Taylor's footnotes have been converted to endnotes)
. τῶνδὶ τῶν ὄντων, "entities here" in the actual world perceptible by sense, as contrasted with things there, i. e., in the "intelligible world" of Plato's Ideas.
 The rest of the critique of Plato down to 991b 7, "of which we Platonists say there are not Ideas," appears again in Metaphysics, M, chs. 4, 5, in a form which is almost verbally identical with the present chapter, except that there Aristotle does not, as here, affect by the use of the pronoun of the first person plural to be speaking as a critic from within the Platonic circle itself. This repetition is one of many indications that the Metaphysics is in no sense a literary "work," prepared by its author for circulation.
. The "One over the Many" (ἕν ἕπὶ πολλῶν) is the single class-concept predicable of each severally of a plurality of individuals.
. This is the argument that, since there is exact and absolute truth, there must be a corresponding class of objects of knowledge, viz., the eternal, immutable Ideas. It occurs in Plato, e.g., at Republic, 478a; Timaeus, 51. Aristotle objects that there are sciences of objects for which the Platonists themselves did not postulate corresponding Ideas, viz., negatives, relatives, artificial products. These limitations do not, however, occur in the Platonic dialogues. We read in the Cratylus (389) of an Idea of shuttle, in the Republic (597) of an Idea of bed--artificial products; in the Phaedo and Parmenides of Ideas of equality and bigness--relations; in the Parmenides of an Idea of inequality--a negative.
. This appears to be what we might call the argument from the existence of a Limit, i. e., the inference of Phaedo, 74 ff, that there must, e.g., be such a thing as absolute equality, which is never actually exhibited but only suggested as an ideal limit by the examples of approximate equality presented by sensuous perception. Aristotle's rather shallow objection would be most strikingly expressed by putting it in the form that since 0 is one of the most familiar instances of a limit, the argument from the existence of a limit requires that 0 should exist.
. The "argument from our ability to conceive what has perished" is best illustrated by Aristotle's own previous observation in ch. 6, that Plato held that the objects referred to in definitions cannot be sensible objects, since the definition is always equally true, but all sensible things are mutable.
. These, according to Alexander, are the arguments which make the relation between the Idea and the corresponding class of sensible objects more definite by saying that the Idea is the Original (παράδειγμα) of which the sensible thing is a copy (ὁμοίωμα or μίμημα), i. e., the arguments in which the Idea appears as an Ideal Limit or Standard.
. The "third man" is the difficulty known in modern logic as the "indefinite regress." We learn from Alexander that it had been originally raised by the sophist, Polyxenus. Plato himself alludes to it in Republic, 597, and explicitly states it in Parmeuides, 132, though without formally indicating his answer to it. It runs thus: If the likeness between Socrates, Plato, and other persons proves that they are all "copies" of a common archetype, the "Idea of Man," then the likeness between this Idea and Socrates must also prove that both Socrates and the Idea are "copies" of another common archetype, which will be a second and more ultimate Idea of Man; and the likeness between the first and second Ideas of Man proves the existence of a third Idea, which is their common archetype, and so on in indefinitum. (The real solution of the puzzle is that the relation between Socrates and "man" is not the same as the relation between Socrates and Plato. Socrates and Plato are both members of the class men; "man" is not a member of the class "men." Hence the argument of Polyxenus and Aristotle is a sophism, and the difficulty about the "regress" does not arise except in the case of those classes which can be members of themselves. On these classes, see Russell, Principles of Mathematics, I., ch. X., and Appendix B.)
. The "things" in question, Alexander explains, are the constituent elements of the Ideas themselves, the One and the Dyad of the Great and Small. Aristotle contends that the theory of Ideas leads to consequences which are incompatible with the initial assumption as to these elements. e.g., if the Great and Small is one of the two constituents of every Idea, it must be a simpler notion presupposed in every Idea and thus logically prior to all the Ideas. Therefore it must, of course, be prior to the Idea of Number. But, since you can say, e.g., "The Great and Small are a pair of entities" or "are two entities," and two is a number, number should be the class, or universal, of which the Dyad is one instance, and it ought to follow that number is logically prior to what Plato regards as one of its simple constituents. (The reader will readily perceive that this, again, is a sophism, turning on the identification of the Indeterminate Dyad or "Variable" with the number 2. The repeated instances of this identification which occur both in this chapter and throughout book M afford a striking illustration of Aristotle's deficiency in exact mathematical thought.) He further goes on to object that Plato's theory makes the "relative" prior to the "absolute." This is because the fundamental concepts of that theory, "number" and "archetype," are relative terms. (Every number or archetype is a number or archetype of something.)
. The point is this: You can say, e.g., "a right-hand glove and a left-hand glove are two gloves"; thus in Platonic phrase, the gloves "partake of the Idea of " two. But though the Idea of two, like all Ideas, is eternal, you cannot say "these two gloves are eternal," for gloves, as we know, wear out. In the terminology of Aristotelian logic the relation of "participation," if it exists, must be between the sensible thing and the substance of the corresponding Idea, not between the thing and the accidents of the Idea.
 Reading with Bonitz in his Commentary, p. 114, and apparently with Alexander, οὐσίας in line 34 for MSS. ουσία which Christ keeps. The MSS. text gives the sense, "the Ideas must be substances," but this is throughout assumed by Aristotle as admitted.
. "Here" = among sensible things, "there" = among the Ideas, in the "intelligible" world, a mode of expression which became afterward technical with the Neoplatonists.
. The many pairs of things which are eternal are, of course, the instances of couples which occur in pure Mathematics (e. g., pairs of conjugate diameters, pairs of asymptotes). The argument is our old friend, the "third man." "To be a couple," he contends, is predicable alike of the Idea of "two" and of a sensible couple. You can say: "The Idea of ‘two' and this pair of gloves are two couples." Therefore, on Platonic principles, there must be a second more ultimate Idea of "two," in which both the first Idea of "two" and the gloves "participate." The sophistical character of the reasoning becomes obvious when we reflect that the Idea of "two" is not itself two things, but one thing. Do not confuse this Idea of "two" with the Indeterminate Dyad.
. At this point the parallel passage of book M (1079b 3) adds the following paragraph:
But if we assume that in general the universal concept coincides with the Idea (e. g., the qualification "plane figure" and the other constituents of the definition with the "Idea of the circle"), but that, in the case of the Idea, it must be further specified of what this Idea is the archetype, one has to consider whether this addition is not purely empty. To which constituent of the definition is it to be added? To "center," to "plane," or to all alike? For all the constituents of the essence are Ideas, e.g., "animal" and "biped." [I. e., in the definition of man as a two-footed animal. Tr.] Besides, clearly it [i. e., the proposed extra qualification by which the Idea is to be distinguished from a mere universal generic concept, viz., that it is "the archetype of a class of sensible things." Tr.] must itself be an entity, just as "plane" is an entity which must be present as a genus in all the species. [i. e., he argues that the same grounds which lead the Platonists to say that there is an Idea of "plane" of which circles, ellipses, and all the other plane figures "partake" would equally lead to the view that there is an Idea of "archetype" of which all the other Ideas "partake"--a fresh application of the "third man." Tr.]
. i. e., the heavenly bodies, which, according to Aristotle, are ungenerated and incorruptible.
 This is the essence of Aristotle's most telling objection to the Platonic doctrine, viz., that Plato regarded the Ideas as "separable" from the sensible things which, nevertheless, depend on them for their Being. In modern terminology the point is, that Plato holds that what we mean to assert in a typical proposition of the form "X is a Y" (e.g., "Socrates is a man'') is a relation between X (Socrates) and a second entity Y ("humanity," the "Idea of Man"). Aristotle regards this as an impossible analysis.
. Plato's friend, Eudoxus of Cnidus, the astronomer, had attempted to meet the objection just mentioned by saying that things are a "mixture" in which the Idea is one ingredient. Aristotle regards this as analogous to the doctrine of Anaxagoras, according to which every thing contains some degree of all the contrasted qualities of matter, but exhibits to our senses only those of which it has most. The "consequences" are, no doubt, of the same kind as those urged in ch. 8, against Anaxagoras. Alexander says that Aristotle had developed them more at length in his lost work, "On Ideas."
. He means that if from ‘Socrates is a man’ you can infer the existence of an "Idea of Man" of which Socrates "partakes," you ought equally from "Man is an animal" to infer an "Idea of Animal" of which "the Idea of Man" partakes.
. Phaedo, 100d: "When I am told that anything is beautiful because it has a goodly colour or shape, or anything else of the kind, I pay no attention to such talk, for it only confuses me. I cling simply, plainly, perhaps foolishly, to my own inner conviction that nothing makes a thing beautiful but the presence, or communication, whatever its nature may be, of that Ideal Beauty. Without any further assertion as to the nature of this relation, I assert merely that it is through Beauty that all beautiful things are beautiful."
. The argument has two branches. (1) The mere existence of the Idea is not enough to guarantee that of a corresponding group of sensible things. (e.g., the existence of an "Idea of Man" does not secure the existence of Socrates. Socrates must have had parents, and his existence depends on certain acts of those parents.) (2) And artificial products, on the other hand, certainly come into being. Yet the Platonists, according to Aristotle, say that there are no Ideas of such products. Why then, if houses and rings can come into being, though there are no Ideas of them, may the same not be true of everything else?
. The paragraph develops further the contention that numbers are relative terms. The argument is as follows: He suggests that Plato may have reconciled the assertions that the Ideas are Numbers and that they are the causes of things by the view that a sensible thing (e.g., the organism of Callias) is a combination of certain materials in accordance with a definite numerical law. This law would be, in Aristotelian phrase, the "form" or "formal" cause of the thing in question. Only, in that case, the thing in question (the body of Callias) is not merely a numerical law, but a law of the combination of certain specific material. Consequently, if the sensible thing (the body of Callias) is a copy of a certain archetype (the "Idea of Man"), this archetype also must contain something corresponding to the material factor in the thing, and thus even on Plato's own principles, the Idea will not be merely a "number" but a numerical law of the combination of certain material. There seems to be an allusion to the formation of the human organism out of materials which are definite compounds of the four "elements," as described in the Timaeus.
 Aristotle's point is, that any two numbers can be added together and their sum will be a third number of the same kind. But Ideas, or class-concepts, he thinks, cannot be added. If they are numbers, they must be numbers composed of units which, unlike those of Arithmetic, are not all of the same kind, and therefore cannot always be added so as to produce a resultant of the same kind as the factors. He thinks that you may then suppose either that each of the units which compose one and the same "Ideal number" may be of the same kind as all the other units of that number, but different in kind from any of the units of a different "Ideal number," or that even the units of one and the same "Ideal number" may be all different in kind from one another, the former being the more natural hypothesis. The two forms of the supposition, which are here curtly dismissed, are discussed at length in M, ch. 7, 8, 1081a 1-1083a 20. The reader will see that Aristotle's philosophy of number is doubly defective, since (1) he has no conception of the dependence of arithmetical addition on the more fundamental process of logical addition (for which see Russell, Principles of Mathematics, I., ch. XII.); (2) and he has, also, no conception of any class of numbers except the integers. (On this point, see Milhaud, Les Philosophes-Géomètres de la Grèce, pp. 359-365, who well asks by what addition of integers Aristotle could have obtained such numbers as √2, √3)
. For a detailed attack on the conception of mathematical objects as "intermediate" between Ideas and sensible things, see M, ch. 2, p.1076a 37 ff.
. I. e., the Indeterminate Dyad of the Great and Small. The argument is, that since this is a dyad or "pair," it must consist of two members; whence, then; are these derived? (You must not say that they are repetitions of the other element, the One, because in the Platonic system the "Great and Small" is regarded as being logically no less ultimate and elementary than the One.) Here, again, we have, as Bonitz observes, an unfair identification of the "Indeterminate Dyad" with the number 2. It is only the latter, not the former, which can be said to consist of two units. And even in the case of the latter such an expression is a loose and inaccurate way of saying that 2 is the number determined by the addition of 1 to 1, or the number of the terms of a class formed by uniting in one class the terms of the classes a and b, when a and b each have only one term and their terms are not identical.
. i. e., each Idea is one thing or unit, an entity corresponding to one determinate class or type. How then, can it also be a number, which is a collection of units? Cf. H, 1044a 2, where the same complaint is made that the Platonists cannot explain what it is that makes a number one thing, and M, 1082a 15, where he asks, "how can the number 2 be an entity distinct from its two units?" This and many other passages of M show how very literally and naïvely Aristotle conceives of integers as formed by addition. What he does not see is, that "addition is not primarily a method of forming numbers, but of forming classes or collections. If we add B to A we do not obtain the number 2, but we obtain A and B, which is a collection of two terms, or a couple." (Russell, op cit. p. 135.)
. i. e., the kind of number meant by the Platonists when they speak of their Ideas as numbers must be something quite different from what the arithmetician means by number.
. i. e., "we Platonists."
. The argument is aimed at the Platonic application of the principles of the One and the Great and Small to define geometrical extension in one, two, three dimensions. The point is, that whereas, according to Aristotle, a solid contains surfaces, a surface lines, and a line points, this could not be the case on the Platonic principles, according to which each of the three dimensions consists of magnitudes of a different kind. (Cf. M, 9, 1085a 7-31.) Hence, he holds, a Platonist ought not to be able to define a plane in terms of the definition of a straight line, nor a solid in terms of the definition of a plane, or vice versa. Now, Aristotle holds that you can do the latter. A plane is, e.g., the boundary of a solid; a straight line is the boundary of a plane (as we should say, the intersection of two planes). This is what he means by planes being "in" solids, and lines "in" planes. He does not, of course, mean that, as the Pythagoreans had thought, a solid is actually made up of superposed laminae, or a plane of juxtaposed strips. The argument is, however, fallacious; since, e.g., a plane may quite well be, as the Platonists held, a different kind of magnitude from a straight line and yet be definable in terms of the definition of a straight line. Aristotle has, in fact, been led astray by his inadequate theory of definition as being exclusively by genus and difference. "Higher" genera means, of course, those which require for their conception a higher degree of abstraction and analysis.
 Aristotle is referring to a view, known from the commentators to have been held by Xenocrates, and here attributed by him to Plato himself, that there are really no such entities as points, what we call a point being, in fact, not a magnitude but the "starting point" (ἀρχή) or "beginning" of a magnitude, viz., of the line. There is no trace of this doctrine in the dialogues of Plato, and the imperfect tense (ἐκάλει) shows that Aristotle is referring not to any Platonic passage, but to verbal statements made by Plato in his lectures. Since the view in question was adopted by Xenocrates, the actual president of the Academy during Aristotle's activity in Athens as a teacher, it is natural that he should have treated it to special criticism; among the extant works ascribed to him there is, in fact, a special tract, "On Indivisible Lines." Plato's difficulty, no doubt, was that the point has no dimensions; it is a zero magnitude. The error of refusing to admit the point, or zero dimension, is exactly analogous to the universal error of Greek arithmeticians in regarding 1, not 0, as the first of the integers. Though, since the definition of a point, often cited by Aristotle as a "unit having position," seems to come from Pythagorean and Platonic sources (Cf. M, 8, 1084b 26, 33), it seems possible that Aristotle (and Xenocrates?) may have misunderstood what Plato meant by calling the point an "indivisible line," as is maintained by Milhaud, op. cit. p. 341-2. The reader will note that, though Aristotle's conclusion that Geometry requires the point is sound, his argument is a petitio principii, since it assumes the existence of the limit.
. I follow Zeller in making the necessary addition of ποιητικάς before επιστήμας in line 29.
. The reference is specially to the place assigned to Mathematics as a propaedeutic to the study of the Ideas in Republic, VII., particularly to 531d: "All these are mere preludes to the hymn which has to be earned. For you surely do not consider those who are proficients in them as dialecticians."
. viz., the Great and Small.
. Because the Great and Small is a constituent of every Idea. That the Ideas should "be in motion" is impossible, on Platonic principles, because one chief characteristic of them is their immutability.
. The method here and elsewhere called by Aristotle "exposition" (ἔκθεσις) is the familiar Platonic procedure of inferring from the existence of many individual things possessing some common predicate the existence of a single supersensible entity, the Idea, which is their common archetype. He objects (1) that the argument, in any case, does not prove that all the individual things are one thing, but only that, beside them, there is one ideal archetype of which they are all copies; (2) it does not even prove this unless the common predicate is the name of a "real kind" or genus. This is a corollary from his previous conclusion that if there are Ideas they can only be Ideas of substances.
 The point is this: The Platonists hold that the many lines, planes, solids, of Geometry are copies of certain single archetypal entities--the line, the plane, the solid. These are the "objects posterior to the Numbers" here spoken of. But what are these objects? Not Ideas (since they are not numbers, and every Idea is a number); not geometrical figures (since geometrical figures are copies of them); not physical things, since they are immutable. Thus they must be a fourth class of objects, not provided for in the Platonic classification of objects into Ideas, mathematical objects, and sensible things. This argument is further developed in great detail in ch. 2, of book M, given in appendix D.
 He concludes his polemic by an attack on the general theory of the nature of science which is tacitly implied in the Platonic doctrine, viz., that the objects of all the sciences are composed of the same constituent elements. He has already explained that Plato thought that the elements of the Ideas are the elements of everything. It follows that there is ultimately only one science, viz., Dialectic, which, as we learn from Republic, VI., 511, cognizes the ultimate axioms from which all scientific truth can be deduced. Aristotle holds that there is no such supreme science of first principles; every science has its own special subject-matter, and consequently its own special axioms (Analytica Posteriora, I., 76a 16). In this passage he urges two objections to the Platonic view. (1) Analysis into constituent elements is only possible in the case of substances. In a substance you have always the two constituent logical elements of matter and form (which appear in its definition as genus and difference), but these elements cannot be found in a quality, an action, or a state. Cf. H. 1044b 8: "Things which exist in nature, but are not substances, have no matter, but their substrate is their substance. e.g., what is the cause of an eclipse? What is its matter? There is none, but the moon is the thing affected." He means, then, that Plato thinks that in the end all objects of knowledge are made of the same ingredients, and therefore there is only one science of them all; but Aristotle says there is no sense in asking what qualities or activities are made of. (2) The second objection depends on the principle that all learning of anything depends on and requires previous knowledge. (See Appendix A.) To learn the truth by demonstration, you must previously know the premises of the proof; to learn it from a definition, you must know the meaning of the terms employed; to learn it by induction, i. e., comparison of instances, you must previously be acquainted with the individual instances. Hence if all truths constituted a single science, before learning that science you would know no truths at all, and therefore the process of learning itself would be impossible. To meet the retort which a Platonist, who held with Plato that all knowledge is really recollection, would be sure to make, viz., that the knowledge of the ultimate axioms is "innate," and not acquired at all (Cf. Plato, Meno, 81c, etc.), he argues that if we had such innate cognitions we could not be unconscious of having them--the same argument afterward employed by Locke.
As an argument against the doctrine of an all-embracing science the reasoning seems a pure petitio principii, since it merely goes to prove the necessity of some self-evident truths.
. i. e., even when you have analysed everything back into its simple elements, how are you to recognize the fact that they are simple and that the analysis cannot be carried any further?--an objection which, one might think, is as much or as little applicable to Aristotle's own analysis of a thing into matter and form as to Plato's analysis of everything into the One and the Great and the Small. The illustration about the analysis of a syllable into its simple constituent sounds is from Plato, Theaetetus, 203a, where, however, the application of it is rather different, Aristotle's point is, that while some grammarians regard the sound of the Greek letter Z (which appears to have been equivalent to our ds) as simple, others hold that it can be analysed further into the two sounds of Σ and Δ.
. i. e., if, for instance, a visible object, such as a shade or color, is ultimately constituted by a combination of purely logical categories, like the One and the Great and Small (as must be the case if the "elements of the Ideas are the elements of all things"), a Platonic philosopher, even though blind from birth, ought to be able to have "pure anticipated cognitions" of all the colors of the spectrum.
. This clause is plausibly regarded by Christ as a misplaced gloss on the words of the sentence: "Some say that the syllable–Σ,Δ, and Α," above.
Created for Peithô's Web from Aristotle on his predecessors; being the first book of his Metaphysics; tr. from the text edition of W. Christ, with introd. and notes by A. E. Taylor. Chicago, Open Court, 1907.Taylor's footnotes have been converted to endnotes. Greek unicode text entered with Peithô's Younicoder.
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