Wilbur Knorr, Incommensurable Magnitudes and Early Greek Geometry [end of year Aristotle series]
Aristotle and mathematics, with more hints about how this might have impacted his methods or terminology for the study of Ethics and Politics..
Wilbur Richard Knorr wrote:
Introduction:
One may … profitably explore the thesis that the dialecticians assumed the role of critics; that through the arguments of such philosophers as Parmenides and Zeno, Anaxagoras, Democritus, Plato and others, the mathematicians came to recognize the nature of the foundational difficulties, were forced to take the challenge seriously, and were thus induced to engage in a new form of mathematical creativity, the study and reorganization of the logical structure of geometry.
Under such a view, the impulse toward foundational studies would be external and gratuitous, as far as the actual mathematicians were concerned. But on the basis of one assumption I will find it possible to perceive the motives for foundational work within the Greek mathematical discipline itself. The assumption is that the study of incommensurable magnitudes was a field of interest in its own right. Even the briefest exposure to Book X of the Elements suffices to convince one that no other assumption could account for the creation and completion of a theory as large and intricate as this one is. Now, there is a special aspect of the study of incommensurability of interest to us: unlike other branches of mathematics, this field can make no appeal to practice for supporting its claims.
Many mathematical traditions confronted the computational anomaly posed by such quantities as √2; there was little problem in producing approximations which would satisfy any given context. But the Greeks advanced to the assertion that such quantities were irrational. This is a statement of an entirely different theoretical order. Its justification can be based only upon logical deduction, and a theory of such quantities can have but one criterion of validity: logical consistency. Hence, if within such a theory one has call for geometric constructions, for theorems on proportionality or for other mathematical materials, these must be brought into conformity with the logical standards of the whole theory before their introduction is possible. For the technical invalidity of any step must invalidate the whole argument in which it occurs. …
[MGH: With limited knowledge √2 or J2 is the best I can do to represent this symbol.]
… The Greeks never introduced standard arithmetical notation or dictions for them; our '√2' was called the 'side of the square 2' or was studied via the ratio of the side and diameter of the square; its arithmetic approximation was never far from the actual geometric constructions.
Hence, if we are even to begin to grasp the sense of the Greek theory of incommensurability, we must banish the modern connotations of the number-concept. The frequently-heard question: "Why didn't the Greeks construct the irrational numbers?" may have some interest. But it is much like asking why the Greeks never learnt to speak English. It will lead us away from an accurate understanding of what they were attempting to do. And it will tempt us to blame them for what were really among their most brilliant theoretical achievements in mathematics: the theory of number, the theory of proportion and the theory of incommensurable magnitudes.
Chapter 2
The Side and the Diameter
… By contrast with Iamblichus, who associates the irrational with the dode cahedron and the lines in extreme and mean ratio, the fourth-century writers Plato and Aristotle always discuss incommensurability in the context of the side and diameter of the square. Aristotle's uses of the example of the incommensurability of these latter lines show it to be a result familiar to his audience. It had, presumably, already entered the textbook tradition of geometry by his time. But he never credits its discovery to the Pythagoreans, despite his frequent discussions of Pythagorean doctrines. On the contrary, the central dogma which Aristotle ascribes to the Pythagoreans, "all things are (or partake of) numbers", is incompatible with the acceptance of the irrational. From Plato one might draw the impression that the dissemination of knowledge about incommensurable magnitudes was at his time fairly recent. But since he seems to credit Theodorus with advances in the theory of incommensurability, and since Plato most likely learned of this work during his voyage to Cyrene sometime after 390 B.C., we may safely claim that incommensurability had been discovered earlier than this. But it is difficult to determine how much earlier. …
Aristotle, in addition to showing us that the paradigm instance of incommensurability was that of the side and the diameter of the square, gives us a clue to the nature of the proof by which this incommensurability was established.
In his exposition of the technique of reasoning per impossibile (Prior Analytics 1.23, 41a29) he [Aristotle] refers to the proof in this way: if the side and diameter are assumed commensurable with each other, one may deduce that odd numbers equal even numbers; this contradiction then affirms the incommensurability of the given magnitudes. Now, just such a proof is given as an appendix to the Tenth Book of the Elements …
Chapter 3
The Role of Diagrams
From Aristotle we learn that the function of the diagram is to make the truth of a fact or a theorem obvious. For instance, “this will be clear from the diagram to those who examine it”. …The geometer makes no inferences on the lines he draws, but rather reasons on the basis of the facts made clear by means of them … Moreover, he observes that the process of constructing a diagram can be useful for the purposes of instruction (de Caelo 280 a3). Thus, constructions were not mere accessories to mathematical arguments; their purpose was to make evident the truth of the theorem under investigation.
So close was this association of theorem and diagram that the two terms might be used as synonyms.We may cite two explicit passages to this effect. When Aristotle refers to 'diagrams' in Categories 14 a39, Ammonius comments that "diagrams and theorems are the same” … On a similar usage in Metaphysics 998 a26, Asclepius remarks that 'diagrams' are synonymous with 'theorems in geometry’ …Similar instances where 'diagrams' are associated with 'geometrical propositions' are found at Sophistical Refutations 175 a27 and Metaphysics 1051 a22.
Chapter 5
The Pythagorean Arithmetic
Pythagorean philosophy maintained that being was to be understood in terms of dualities, and a primary manifestation of duality was that of the odd and the even. Philolaus, for instance, spoke of the odd and the even as the “two proper forms of number”. A further indication of the importance of this dichotomy comes in the list of contraries preserved by Aristotle; here the dichotomy of 'odd-even' is second only to that of 'limit-unlimited', and stands just above that of 'one-many'. The Pythagoreans, according to Aristotle, took the "elements of number to be the elements of all things ... and the elements of number to be the odd and the even.”
This comment, showing the importance the Pythagoreans assigned to the concept of the odd and the even, is reminiscent of passages in Plato's works, in which the disciplines of arithmetic and logistic are defined in terms of the odd and the even. Logistic deals as it were with the odd and even, that is, how they are related in multitude with respect to themselves and with respect to each other." "What is the subject matter of the art of arithmetic? ... It is the odd and the even, regardless of the quantity of either." "The arithmetic art is a hunt for knowledge of every odd and even." Jacob Klein reasonably interprets these passages to indicate Plato's intent to define with full scientific rigor the fields of the theory of number (i.e., logistic and arithmetic) without reference to the concepts of number or numbered things. In other words, the concepts of the odd and even are viewed by Plato as prior to the concept of number itself. Aristotle's remarks in the Metaphysics enable us to assign this conception of the status of the odd and even to the Pythagoreans. We can conclude that the Pythagoreans viewed the odd and the even as the fundamental starting-point in the study of numbers.
Chapter 8
Geometry of Incommensurability: Theaetetus and Eudoxus
… the proponents of the existence of a pre-Euclidean theory of proportion based on anthyphairesis have missed the single most important buttress of their thesis: the theorem cited by Aristotle in the Topics to illustrate the anthyphairetic definition of proportion is one of the few fundamental lemmas on proportionality requisite for the theory of incommensurable magnitudes, the theory which owed its initial organization to Eudoxus' predecessor Theaetetus. The same theorem, it is true, is necessary at one step or another for the completion of some of the theorems within the anthyphairetic proportion theory. But that is not its primary significance and would not have been sufficient to induce Aristotle to single it out as an example. The fact that Aristotle uses a different term, antanairesis, may merely be evidence of the informal state of the theory at his time, in which anthyphairesis and antanairesis (and presumably other suitable synonyms) were freely interchanged to designate the division algorithm. …
… As far as Eudoxus' mathematical researches are concerned, Aristotle remained aware of the different stages of the developing theory of proportions. At Topics 158 b29 he introduces the anthyphairetic theory, which must have characterized the work in the Theaetetean theory of proportion and of irrationals still being conducted in the Academy when he entered. At Physics 206 b6 he discusses the two forms of the condition of comparability (V, Def. 4 and X,l), which we have argued represent the meeting-point of the anthyphairetic and Eudoxean conceptions of proportion. At Posterior Analytics 74 al 7 he [Aristotle] contrasts the two theories of proportion in respect of the generality of their proofs. We may thus gather that within Aristotle's long career the theory of proportions passed through its transitional stages, dominated by Eudoxus' innovations, and achieved its completed form no later than c. 330. The elementary theory of incommensurability was of course thoroughly familiar both to Plato and Aristotle; but Eudemus' comment on Theaetetus' definitions of the irrationals and the remark in De Lineis Insecabilibus 968 b19 suggest that the theory of irrationals, in the complete form as in Elements X, did not exist long before c.330. …
Chapter 9
Conclusions and Synthesis
… Aristotle, in the Posterior Analytics, once seems to accept integers as a type of magnitude (75b 4). But in this frequently misconstrued comment he in fact does no such thing. Rather, he is clarifying his observation that each discipline, e.g. arithmetic and geometry, has its own definitions and subject matter and that it is invalid to employ the techniques of one in the proofs of assertions within the province of the other. "For instance," he says, "you cannot prove a geometric theorem by the principles of arithmetic, unless the magnitudes are integers." (Note, not: "unless the integers are magnitudes.") Far from suggesting that integers are a type of magnitude, he argues that in special cases magnitudes may be treated as in tegers. We know of several instances of such studies: metrical geometry is one; the theory of incommensurable magnitudes is another; and a third is that form of number theory assisted by geometrical constructions. …
… In fact, Aristotle always recognizes the disciplines of arithmetic and geometry as distinct. There are, however, mathematical fields in which the two disciplines are co-applicable. The study of incommensurability is such. Is this study to be classified as arithmetic or as geometry? Aristotle assigns [it] to geometry (76b 9); certainly this is justified in view of the necessity of geometric constructions of the irrational lines. But there can be no study of irrationals without the introduction of arithmetic distinctions. The very notion of 'commensurability' immediately entails an arithmetic equivalent: 'having the ratio of integers' (X,5). It is certainly for this reason that Plato had set these studies within the realm of arithmetic; he even attempted to use the study of incommensurability as a way of unifying all the mathematical disciplines under the heading of number theory (cf. Chapter III, Section VII).
Neither Aristotle's answer nor Plato's can be strictly correct. For this theory needs the techniques of both disciplines, and thus must be viewed as a composite of arithmetic and geometry. …
The Source:
Wilbur Richard Knorr*, The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry, D. Reidel Publishing 1975
*b.1945–d.1997
Evolutions of social order from the earliest humans to the present day and future machine age.