Calculation and geometry

Use in the book

Calculation enters when Socrates asks what work temperance produces, as medicine produces health and house-building houses (165c–e). Critias, Charmides' cousin and guardian, protests that temperance is not by nature like the other kinds of knowledge, and asks in turn: "what work does the art of calculation have, or geometry, in the way that a house is the work of house-building, or a cloak of weaving ...?" "That's true," Socrates says, granting that they have none (165e–166a; see Art and craft).

But he can say what each is knowledge of: "Calculation, for instance, is surely of the even and the odd, how they stand in quantity to themselves and to each other, isn't it?" "Certainly," says Critias, and he grants that the odd and the even are "something other than calculation itself" (166a). "To themselves" can be read as within each class, even with even and odd with odd, beside the two classes with each other; calculation does not make a number more than itself, which Socrates will later find impossible (168c–e2). For geometry Socrates names no object. His second example is weighing, "the weighing of the heavier and the lighter weight", whose objects, the heavy and the light, are likewise "something other than weighing itself" (166b). Asked the same of temperance, Critias answers that it alone is knowledge "both of the other kinds of knowledge and of itself" (166c; see Knowledge of knowledge).

Neither calculation nor weighing is named in the argument about self-relation, but the objects of both return there. What is double of itself would also be half of itself; and Socrates asks whether "what is more than itself will also be less", and "what is heavier lighter", in the words that named weighing's objects (168c; see Double and half; Greater and smaller). Then he asks, "For with magnitudes and numbers and such things it is altogether impossible, isn't it?" and Critias answers, "Certainly" (168e). "Numbers" renders πλήθη, the word rendered "in quantity" in the definition of calculation (166a), and "magnitudes", μεγέθη (megethē), may stand for the objects of geometry, for which Socrates named none. The arts that showed a knowledge to be of something other than itself can thus be read as supplying, a little later, the plainest cases of a power that cannot be turned on itself.

Calculation returns once more near the end of the search, when Socrates asks which knowledge makes the man who knows everything happy. "Is it perhaps the one by which he knows draughts?" "Draughts?" says Critias. "What draughts?" "The one, then, by which he knows calculation?" "Not at all." Of the one by which he knows what is healthy Critias says, "More so"; and the one that does it most of all, he says, is the one by which he knows "good ... and bad" (174b; see Knowledgeable living).

In the Charmides calculation is the standing example of a knowledge that has no product but has an object of its own, other than itself. Critias' temperance is like it, on his own argument, in having no work, and unlike it in having no object but knowledge: it is knowledge "of no subject of learning" (168a), and so, without medicine, it cannot tell whether a man knows the healthy (170c–d).

Places to read

1. 165e–166aCritias asks what work calculation and geometry have, as a house is the work of house-building or a cloak of weaving; Socrates grants that they have none.
2. 166a–c2Calculation is of the even and the odd, weighing of the heavier and the lighter, each something other than the knowledge itself; Critias answers that temperance alone is knowledge of the other kinds of knowledge and of itself.
3. 168b–e2What is greater, double, more, heavier or older than itself would also be smaller, half, less, lighter or younger; with magnitudes and numbers it is altogether impossible.
4. 174a–b2Which knowledge makes the all-knowing man happy? Not draughts, not calculation; the knowledge of the healthy more so, the knowledge of good and bad most of all.

Translation in this edition

Λογιστική is "calculation" at 166a, both times, and so is τὸ λογιστικόν at 174b; Critias' τῆς λογιστικῆς τέχνης is "the art of calculation" (165e), the one place where the edition renders τέχνη as "art". Γεωμετρική is "geometry", and στατική "weighing" (166b). Πλῆθος is "in quantity" in the definition (166a) and "numbers" beside "magnitudes" (168e). At 165e B, the Bodleian manuscript written in 895, reads λογικῆς, an adjective from λόγος, "word" or "reason", that Plato's dialogues nowhere use; Burnet prints λογιστικῆς, "of calculation", with T and W, the Venice and Vienna manuscripts, and Socrates' answer at 166a, where the word stands without variant, bears the printed reading out by speaking of λογιστική.

ἄρτιος

Attested: ἀρτίου (artiou; 166a twice). Even, of numbers. With the odd it is the object of calculation, which considers how the two "stand in quantity to themselves and to each other" (166a). The adverb ἄρτι, "just now", is another word and is not listed. Lemma: ἄρτιος.

γεωμετρική

Attested: γεωμετρικῆς (geōmetrikēs; 165e). Geometry. Critias pairs it with calculation as a knowledge with no work like a house or a cloak (165e); Socrates, in reply, names an object for calculation and for weighing, but none for geometry (166a–b2). Lemma: γεωμετρική.

λογιστική

Attested: λογιστικῆς (logistikēs; 165e, 166a), λογιστική (logistikē; 166a). Calculation. Critias asks what work "the art of calculation" has (165e), where the Bodleian manuscript reads λογικῆς; Socrates answers with its object, the even and the odd, which are "something other than calculation itself" (166a). Lemma: λογιστική.

λογιστικός

Attested: λογιστικόν (logistikon; 174b). Belonging to calculation; with the article, calculation as a field of knowledge. It is Socrates' second suggestion for the knowledge that makes the all-knowing man happy, after draughts, and Critias answers, "Not at all" (174b). In the Republic the same neuter names the part of the soul by which it reasons (439d). Lemma: λογιστικός.

περιττός

Attested: περιττοῦ (perittou; 166a twice). Odd, of numbers. With the even it is the object of calculation, and is "something other than calculation itself" (166a). Lemma: περιττός.

πλῆθος

Attested: πλήθους (plēthous; 166a), πλήθη (plēthē; 168e). Number, quantity, multitude. Calculation considers how the even and the odd "stand in quantity to themselves and to each other" (166a); and with "magnitudes and numbers and such things", Socrates asks and Critias agrees, it is "altogether impossible" for a thing to have its power in relation to itself (168e). Lemma: πλῆθος.

Elsewhere in Plato

Critias' objection rests on a division Plato draws elsewhere. In the Gorgias Socrates divides the crafts into those that are mostly work and need little speech, like painting and sculpture, and those that achieve everything through speech and need little or no work, like arithmetic, calculation, geometry and draughts (450c–e); draughts and calculation come together again at Charmides 174b. Arithmetic is about the even and the odd, how many each is; calculation is about the same things, but examines how the odd and the even stand in quantity to themselves and to each other (451b–c), almost the words of Charmides 166a. What sets calculation apart is the question it asks, not a difference between practice and theory.

The unnamed visitor from Elea, in southern Italy, who leads the Statesman, divides all knowledge on the same line: arithmetic and its kindred arts are bare of action and supply only knowing, while carpentry and all handicraft have their knowledge bound up with their actions and bring into being bodies that did not exist before (258d–e). Calculation, once it knows the difference among numbers, has no further work (ἔργον, ergon, the word Critias uses at 165e) than to judge what it knows (259e; see Builder and house-building).

The Republic can be read as giving calculation and geometry a work after all, though in the soul. Those who are to share in the city's greatest affairs, Socrates tells Glaucon, Plato's brother, must pursue calculation until they see the nature of numbers by thought itself, not for buying and selling, as merchants do, but for war and to ease the soul's turning from becoming to truth and being (525b–c). Geometers speak like men at work, of "squaring" and "adding"; yet the study is pursued for the sake of knowing what always is, and so would draw the soul toward truth and be "productive of a philosopher's thought" (527a–b). The word for "productive", ἀπεργαστικόν (apergastikon), comes from the verb by which the Charmides asks what medicine, house-building and temperance "produce" (165c–e).