Abstract
I offer a fresh reading of the proof for the existence of number at Parmenides 143d–144a. I identify the main apparent problems with the proof as it has been read from Cornford and Allen forward—above all, that Parmenides’ four classes of numbers omit the primes, that two of these classes are redundant, and that there are ambiguities in class membership. All of these problems disappear, however, if we read Parmenides as deriving his classes from the treatment of square and oblong numbers in the so-called pebble arithmetic that Plato inherits from the Pythagoreans. I close by pointing out ways in which this reading appears to fit with Plato’s teaching, as reported by Aristotle in Metaphysics A6, that numbers can be “readily derived” from the Great and the Small.
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Notes
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My thanks to Rachel Kitzinger for illuminating conversation about nuances of the Greek.
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Not Plato’s great successor. Not only is the dramatic date of the dialogue about 450 BCE, but this dramatis persona is identified as “the man who later became one of the Thirty” (127d), the pro-Spartan aristocrats who, with Lysander’s support, seized control of Athens in 404–403.
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This objection was made as early as Aristotle, at least on the widely accepted reading of ἔξω τῶν πρώτων at Metaphysics A6: 987b34. I shall offer a possible alternative to this reading in Sect. 9.4 below.
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The exceptional status of one in ancient Greek arithmetic is widely acknowledged. If number is defined as “a multitude of units” (see Euclid, Elements VII, Def.2), then one is not a number; rather, as Aristotle, for whom the notion of a multitude of countable units figures in all of his definitions of number (see Heath 1921, 70), writes, one is “the measure” (τὸ μέτρον) and the “starting-point” (or “principle,” ἀρχή) of number (Metaphysics N1: 1088a5–7) and, so, stands apart from and is prior to the rest of the series of integers. See n8 below.
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Philolaus fr. 5, discussed in M.E. Hager (1962, 1–2).
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That Plato grants to his dramatis persona Parmenides the recognition that one ought not be classified as odd or even is evidenced by his having Parmenides jump over one in silence and declare two to be the first even number and three to be the first odd number (143d7–8).—It is also striking that in the Phaedo when Socrates needs a paradigmatic participant in the form Oddness, Plato has him cite three, not one. The only text that appears to differ from the Parmenides and the Phaedo on this question is Hippias Major 302a. As Knorr (1975, 167) writes, “the Hippias-passage utilizes a conception of the unit (as an odd number) not recognized in the arithmetic theories known to Plato, Archytas, and Philolaus.” I am persuaded by Allen’s observation (1974, 711n37) that Plato there has Socrates “treat one as an odd number” as a “dialectical” tactic, not “a matter of abstract number theory.”
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This appears to be the approach of Cornford 1939, 141n2 (for objections to his formulations, see Allen 1974, 713n44 and 1983, 310n158), of Brumbaugh (1961, 98 (“The primes, omitted in the cross-classification [of odds and evens], are presumably generated by addition.”), and of Sayre (1996, 170–171). Scolnicov (2003) dissents, arguing that Plato conceives number “primarily as proportions and structures, and only secondarily as (denumerable) collections” (106); he denies that “primes greater than three” can be established “by the addition of a unit to a previously derived collection, for this would make them συνθέσεις μονάδων” (105). Scolnicov’s interpretation of Plato’s conception of number is, in my view (Miller 1999, 76–83), a good insight into the notion of number as it emerges in the final four members of the five mathematical studies that Plato has Socrates prescribe for the would-be philosopher in Republic VII, for in the several forms of geometry and in harmonic theory number always appears in and as ratio, even when, as with the paradigm case of the relation of the side and the diagonal of the square, there is no common unit of measure; but as I shall argue, it misses the notion of number that governs the first study, “logistic and arithmetic” (525a, also 522c), and that is the conception in force in Parmenides 143d–144a (see Sect. 9.3 below). An outlier is Turnbull (1998), whose ingenious notions of the “two machine” and the “three machine” appear, if I follow his constructions, to derive from Parmenides’ pairs and triads at 143c–d operations of selective grouping that, in opportune combinations, can yield prime numbers of the things grouped; as he acknowledges, however, “[t]here is nothing in [143a–144e] or elsewhere in the Parmenides to back up [this] procedure for generating the primes” (79).
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This is strongly implied by Allen when he writes that “the existence of any number implies the existence of every number” (1974, 714)—but note Sayre’s observant reservation at 1996, 343n17—and, again, when he writes that “any of Parmenides’ methods—multiplication of even numbers, of odd numbers, or of odd and even numbers—will suffice to prove the existence of any number, since any of these methods, by repeated application, will prove the existence of a number larger than any number desired” (1983, 228).
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For an illuminating discussion of the “pebble arithmetic” and its mathematical power, see Knorr (1975, ch. V).
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This is also, arguably, a problem with the familiar translation of the suffix -άκις as “times” in ἀρτιάκις (“even times”) at 143e7 and ff. and in περιττάκις (“odd times”) at 144a1 and ff. To the English ear this suggests multiplication, and while this meaning is quite possible, it is too specific to stand without contextual support; -άκις need signify no more than “[taken] a number of times,” without respect to the multiplication of a number by another number and the product this yields. Hence my more neutral translations of ἀρτιάκις as “an even number of times” and of περιττάκις as “an odd number of times” in Sect. 9.1. See LSJ under ἀρτιάκις and περιττάκις.
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Knorr (1975, 145–146). Indirect evidence of the preservation of pebble arithmetic in the Academy is Speusippus’ tract “On Pythagorean Numbers.”
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In short, the student moves from logistic and arithmetic to the three geometrical studies by dropping the units in the figurative arrays of logistic and arithmetic and focusing his attention on the figures, as such, that the arrays of units express; analogously, he moves from the three geometrical studies to harmonics by dropping the figures and focusing his attention on the ratios, as such, that the figures express. On the pedagogical value of these refocusings for the student preparing for dialectic, see the work cited in n17.
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I owe a great debt to a fine paper, “Revisiting Plato’s Generation of Number (Parmenides 143c1–144a5),” prepared for an SAGP meeting in 2008 by Michael Barkasi, then an undergraduate at Kutztown who later went on to receive his doctorate in philosophy at Rice. My own earlier work on the Parmenides (Miller 1986 [1991] and 1995) and on the pebble arithmetical representation of numbers in the first of the five studies in Republic VII (Miller 1999) moved me to seek him out when I first came across an announcement of his paper, and while he was writing it, we shared a rich correspondence from which I benefited greatly. Unfortunately, he did not go on to publish his paper, so one of the services of the present chapter, above all in Sect. 9.3, is to make available, with his advance approval but in my own more informal manner of presentation, much of what we shared in the course of our correspondence.
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The original referent of the Greek γνώμων was a carpenter’s square, the tool by which the carpenter marked a right angle; Knorr (1975) gives its mathematical sense as “that number which, when added to a term in a given class of consecutive figured numbers, produces the next term in that class” (143). I have indicated each of the gnomons in my diagrams by highlighting the asterisks, or “ones,” that constitute them.
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My thanks to Glenn Johnson of Penn State for help with the design of this and the following diagrams.
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An alternative but equivalent mode of representation is to “wrap” each new gnomon “around” (cf. περιτιθεμένων, Aristotle Physics 203a13) the preceding figure, setting the pebbles that make it up on the two sides of the figure opposite to the sides on which the preceding gnomon was placed; thus the figures would be expanded in opposite directions, first downward and to the right, then upward and to the left, then downward and to the right, then upward and to the left, and so on. I have chosen the mode of expanding each figure in the same two directions simply for convenience’s sake.
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We must add the obvious caveat that in the series of oblongs the preservation of the shape is inexact: since one dimension (in our diagrams, the width) of the oblong exceeds the other (the height) by one, preserving the oblong as we expand it involves a regular diminishing of the ratio of that excess; we move from a “three twice” to a “four thrice” to a “five four-times,” and so on.
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This is not to deny, of course, that we mightn’t equally well represent the first oblong number as “three twice” (as we in fact did), hence as a member of “odds an even number of times,” or as “two thrice” (as we might have but did not), hence as a member of “even an odd number of times.” But whichever way we choose to begin, the oblong produced by the next expansion will be a case of the reverse pairing, and the series of expansions will proceed as an alternation of the two classes. If we take our first pairing to be “three twice,” that is, a member of “odds an even number of times,” its first expansion will yield its reversal, a member of “evens an odd number of times,” and the expansion of that member of “evens an odd number of times” will yield its reversal, a member of “odds an even number of times.” And the analogous alternation will result if we begin, instead, with “two thrice,” that is, with a member of “evens an odd number of times,” and expand it to yield a member of “odds an even number of times,” and so on.
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In the following pages I restrict myself to the question of the process of deriving the numbers from the Great and the Small as Aristotle raises it at Metaphysics A6: 987b32–988a1; I have treated the distinct question of the status of the One and the Great and the Small as the joint causes of the being of the series of integers, as this is suggested by Parmenides 144b–e, in Miller (1995, 612–614). For the all-important distinction between Aristotle’s report that the One and the dyad are responsible for the being of the series of integers and the (supposed—but, I have argued, widely misinterpreted) report that they are responsible for the being of the forms, see Miller (1995, 599–600 and 622–629).
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It is important to stress that by the reflections in this section I do not mean to imply that in Parmenides 143d–144a Plato has Parmenides offer a deduction or derivation of the being of number; Parmenides’ argument is, as Allen (1970, 1974, 712–714; and 1983, 227–228) and Schofield (1972, 103) rightly stress, an existence proof, not a deduction or derivation from prior principles. But this does not prevent us from recognizing, in Parmenides’ four classes, the conceptual terms by which, when their members are set in the right order and taken as stages of an endlessly iterable sequence, the series of integers might be derived.
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Needless to say, on this interpretation of Aristotle’s sentence, not only the idea of “something malleable” but also the idea of an “engendering,” a γεννᾶσθαι, of the numbers must be given the non-literal sense of a metaphor for “derivation.”
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Does reading τῶν πρώτων as “the primary [ones],” that is, “the primary [numbers],” imply that Plato as we are interpretively reconstructing Aristotle’s report of his teachings violates the principle that, as Aristotle formulates it, “one” is not a number but rather the “starting-point” (ἀρχή) and “measure” (μέτρον) of number? (Recall n6.) Our suggested reading of Parmenides 143d–144a puts us in position to answer in the negative. If it is right that Plato’s Parmenides is drawing on the pebble arithmetical account of the expansions of the square and of the oblong, then, precisely as the bases for the very formation of these figures, “one” and “two” are not mere members of the series generated by their gnomons; rather, to repeat Aristotle’s language, they are “starting-points” and “measures” of the series of numbers. To this we might add that it is striking that Plato does not have Parmenides attempt a metaphysical account of what is basic to “one” and to “two”; instead, as we saw at the outset of these reflections and have just noted again, Plato has Parmenides find the being of “two” to be exhibited by any pairing of the eidetic characters “one” and “being” and “different,” and find the being of “one” to be exhibited by the members of any such “two.” This leaves it open to us to ask, what might such a metaphysical account, were Plato to offer one, consist in? In the context of Aristotle’s report in Metaphysics A6 it is of course tempting to return to the starting point of this postscript and ponder the relations of “one” to “the One” (τὸ ἕν) and of “two” to the so-called dyad. But undertaking this difficult reflection would require a wider textual basis and a deeper array of ontological possibilities than we have had occasion to consider in this chapter. For a beginning, see Miller (1995).
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Miller, M. (2022). Parmenides 143d–144a and the Pebble-Arithmetical Representation of Number. In: Bloom, D., Bloom, L., Byrd, M. (eds) Knowing and Being in Ancient Philosophy. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-98904-0_9
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