9
(7.522c)
The little matter of distinguishing one, two, and three—in a word, number and calculation: do not all arts and sciences necessarily partake of them?:
“Number” (
arthmos
) and “calculation” (
logismos
) lead “the soul toward being” (7.523a) because they help people make sense of confusing appearances and thus lead them to look beyond mere appearances (7.523b). All of the mathematical disciplines that Socrates goes on to describe, beginning with
arithmetikê
(the “science of number”), are essential to the philosopher’s training since it helps him (or her) “rise out of the sea of change and lay hold of true being” (7.525b).
10
(7.523a)
It appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection, but never to have been rightly used; for the true use of it is simply to draw the soul toward being:
Compare 7.527a-b and 7.529a-531c for critiques of the methods and emphases of those who currently study geometry, astronomy, and harmonics.
11
(7.524d)
And to which class do unity and number belong?:
That is, do “unity and number” belong to the class of impressions that are not innately confusing and require no “calculation” or to the class that requires abstract reasoning to be properly understood? The phrase “unity and number” reflects the fact that Plato did not consider “one” a number.
12
(7.524e)
“What is absolute unity?”:
Any single object in the phenomenal world (that is, a visible “one”) is actually both “one” and “many”; for example, one flower has many petals, one piece of fruit has many seeds. The realization that every visible “one” is in fact both “one” and “many” accordingly leads one to wonder about the “absolute unity” that is not also “many,” and to realize eventually that this “absolute unity” is not to be found in the phenomenal world.
13
(7.525d-e)
absolute unity:
That is, the unit that is adopted for the purpose of a given calculation and which is, for the purpose of that calculation, indivisible. Such a unit is, strictly speaking, hypothetical.
14
(7.527a-b)
They have in view practice only, and are always speaking ... of squaring and extending and applying and the like
—
they confuse the necessities of geometry with those of daily life; whereas knowledge is the real object of the whole science:
In the fifth century B.C.E., some geometers tried their hand at town planning; for example, Hippodamus of Miletus designed the grid-iron layouts of streets in Piraeus and the Athenian colony of Thurii (in Italy). Such undertakings seem to have struck many people as “ridiculous”; in Aristophanes’
Birds
(produced in 414 B.C.E.), the geometer Meton is comically represented as a pompous would-be town planner.
15
(7.527d)
I am amused, I said, at your fear of the world, which makes you guard against the appearance of insisting upon useless studies:
Socrates’ gentle admonition resonates with what has been established about the “un-philosophic” nature of the many and their current prejudice against philosophy; see, for example, 6.488e-489a and 6.494a.
16
(7.528a)
Then take a step backward, for we have gone wrong in the order of the sciences:
Solid geometry, which Socrates proposes as the logical follow-up for the study of plane geometry, is less complex and abstract than astronomy, which is the study of “solid objects in revolution [that is, motion].” The subjects of the entire preparatory curriculum (that is,
arithmetikê,
plane geometry, solid geometry or stereometry, astronomy, harmonics) are increasingly complex and abstract, and they are organized so as to make future philosophers ready for the supremely difficult and wholly abstract operations of dialectic.
17
(7.528b)
but so little seems to be known as yet about these subjects:
Problems of solid geometry had concerned several theoreticians (for example, Anaxagoras, Democritus, some in the Pythagorean school) in the fifth century. What Glaucon apparently means here is that solutions to complex stereometrical problems (that is, beyond simple problems such as the doubling of a cube) had not yet been discovered.
18
(7.529b)
whether a man gapes at the heavens or blinks on the ground, seeking to learn some particular of sense, I would deny that he can learn, for nothing of that sort is matter of science:
As in the case of geometry (7.527a), Socrates argues that astronomy ought not be pursued for the sake of understanding physical phenomena (that is, the movements of heavenly bodies), but rather as abstract geometry in four dimensions that is concerned with “true motions of absolute slowness and absolute swiftness.” So too “harmonics,” which is concerned with the motion of sounds (7.530d), is to be studied for the sake of understanding abstract “harmonies” of number as opposed to those that can be physically heard (7.531c).
Plato perhaps has Socrates specifically disavow interest in problems dealing with “some particular of sense” in order to distance him further from the figure of “Socrates” in Aristophanes’ Clouds, whose “Think-Factory” sponsors ridiculous research in “astronomy” and other fields. See note 22 on 2.378b and note 10 on 7.523a.
19
(7.530d)
There is a second, I said, which is the counterpart of the one already named:
Astronomy, which has been “already named,” is the study of the motion of solid bodies in space; its “sister science” is harmonics, the study of the motion of sound. For the comparison of the functions of the ear and eye, see 6.507c-d.
20
(7.530d)
as the Pythagoreans say:
The reference is to the followers of Pythagoras, the religious thinker and theorist who settled in Croton in southern Italy during the sixth century B.C.E. Pythagoreans were known in the classical period for their ascetic way of life, their belief in the reincarnation of the soul, and their mathematically based study of music and harmonics.
21
(7.531a)
The teachers of harmony compare the sounds and consonances which are heard only, and their labor, like that of the astronomers, is in vain:
This is most likely a reference to the Pythagoreans.
22
(7.531a)
condensed notes, as they call them: Pyknomata
in Greek is a technical term that apparently refers to combinations of two quarter-tone (or semi-tone) intervals.
23
(7.531b)
one set of them declaring that they distinguish an intermediate note and have found the least interval which should be the unit of measurement:
For example, the quarter-tone. Other theorists (perhaps including Plato himself) posited the semi-tone as “the least interval,” with a view not only to simplifying the analysis of music, but also encouraging simplicity in musical composition.
24
(7.531b)
plectrum:
The plectrum was the pick by which the strings of an instrument were plucked. The metaphor developed in the first sentence of this paragraph plainly alludes to the torturing and beating of slaves.
25
(7.531d)
the prelude, or what?: Prooimion
in Greek refers broadly to the introductory section of a song (or poem or speech). The musical metaphor is continued in the following paragraphs, where Socrates refers to the
nomos,
or “song,” that dialectic performs.
26
(7.531d-e)
For you surely would not regard the skilled mathematician as a dialectician?:
Compare 6.510b-511d. Unlike the mathematician who, for example, assumes the existence of “absolute unity” (7.525d-526a,) the dialectician’s study of “the idea of one” would not be complete until he had “ascended to first principles” (6.511a-b) and showed by argument that such a concept is essential to the rational understanding of both the intelligible and phenomenal worlds.
27
(7.532d)
This, however, is not a theme to be treated of in passing only.... And
so,
whether our conclusion be true or false, let us assume all this, and proceed at once from the prelude or preamble to the chief strain, and describe that in like manner:
Socrates’ reticence in describing dialectic mirrors the reserve he displays when discussing the idea of the good, which is the ultimate object of dialectic. Compare 7.517b. The word translated here as “chief strain” is
nomos,
which also means “law” and/or “custom.”
28
(7.533d)
Custom terms them sciences:
That is,
epistemai
(the plural of
epistemê).
Whereas the term
epistemai
is “customarily” used to refer to disciplines, or fields of study, Socrates has earlier used
epistemê
to designate the cognitive faculty by which objects in the intelligible world are apprehended.
29
(7.534d)
you would not allow the future rulers to be like posts, having no reason in them, and yet to be set in authority over the highest matters?:
Jowett’s interpretation, that
grammai
in this context refers to the lines at the start of racecourses (hence his translation “posts”), is disputable. The reference is more likely to
alogoi grammai—that
is, “irrational quantities” such as the square root of negative one. If so, there would be a witty play on the adjective
alogos
(“irrational,” “having no reason”) since those who are unable to approach the idea of the good through dialectic are unable to give an account (
logon dounai
) of it. Compare 7.531e, where Socrates and Glaucon agree that mathematicians are incapable of reasoning (literally, unable to
logon
dounai
)
.
30
(7.536c)
Certainly not, I said; and yet perhaps, in thus turning jest into earnest
I
am equally ridiculous.... I had forgotten ... that we were not serious, and spoke with too much excitement. For when I saw philosophy so undeseruedly trampled under foot of men. . . . :
Socrates painted a vivid (and somewhat playful) picture of philosophy bereft of legitimate “suitors” and forced to “marry ... a bald little tinker” at 6.495c-496a. His statement that he and his interlocutors “are not serious” may come as a surprise, given the stress repeatedly laid on the importance of the issues raised in
Republic;
see note 14 on 1.344e. Nonetheless, it is in keeping with the many advertisements concerning the provisional nature of the conversation, and it perhaps should stand as a reminder that Plato did not hope to accomplish anything truly “serious” in this or any other dialogue. For the fundamentally “playful” nature of writing, see
Phaedrus
274c-278e.
31
(7.537e)
Do you not remark, I said, how great is the evil which dialectic has introduced?:
Socrates’ acknowledgment that dialectic is potently destabilizing, since it inevitably causes one to devalue that which one prized in the past (and which others may still prize) and to develop “a questioning spirit” that “asks what is fair and honorable” (7.538d), paves the way for his distinction between dialectic, as practiced for socially constructive ends by the true philosopher, and “antilogic” and “eristic,” as pursued by those who carelessly engage in arguments simply for the sake of winning (7.539b-d; see note 6 on 5.454a). It also reinforces his point that philosophy (that is, dialectic) is not suitable for young, restless people, no matter how gifted (7.539a-b; compare 6.498b-c).
Book
8
1
(8.544c)
the four governments of which I spoke ... are first, those of Crete and Sparta, which are generally applauded:
In contrast to Athens’ democracy, the governments of Crete and Sparta strictly limited political franchise. Critics of Athenian democracy praised the Spartan constitution in particular, and Plato is careful to have Socrates associate it with “timarchy,” which is characterized by “love of honor” (see 545b), and not with the more degenerate “oligarchy,” in which wealth alone determines qualification for political franchise and leadership.
2
(8.545d)
Clearly, all political changes originate in divisions of the actual governing power; a government which is united, however small, cannot be moved:
Compare the concern for
stasis
(factionalism and civil strife) at 4.444b.
3
(8.545e)
Shall we imagine them in solemn mockery, to play and jest with us as if we were children, and to address us in a lofty tragic vein, making believe to be in earnest?:
As above at 7.536b-c, Plato has Socrates call attention to the fundamentally “playful” nature of the discussion, and he seems to invite his readers not to take what is claimed in 8.546a-c too literally.
4
(8.546a)
In plants that grow in the earth, as well as in animals that move on the earth’s surface, fertility and sterility of soul and body occur when the circumferences of the circles of each are completed, which in short-lived existences pass over a short space, and in long-lived ones over a long space:
Socrates envisions the “cycle” of fertility for each species of living thing as represented by a circle; for a short-lived species, the representative circle is small and so is its circumference, whereas the circle is larger for the longer-lived. The notion advanced here—that the cycles of fertility and sterility in individual species are mathematically comprehensible—anticipates what is suggested in 8.546b—c about the existence of a rational, mathematically comprehensible order governing the cosmos as a whole.
5
(8.546b)
but the period of human birth is comprehended in a number ... :
Adopting the voice of the Muses, Socrates presents a calculation for the number that comprehends, or governs, human births. The number ( 12,960,000) is in fact a “master number” that comprehends the area of two great figures, one of which is a square with sides of 3600 units, and the other a rectangle with sides of 4800 and 2700 units.
The “number of Plato” is a notoriously difficult passage that has garnered a great deal of scholarly attention. Critics have offered a number of interpretations of its significance. It is possible that the number and its calculation are indebted to Pythagorean mathematical theories, and it is also possible that the two figures comprehended in the master number 12,960,000 represent two periods in the “lifetime” of the cosmos with which human births, if they are to be “goodly and fortunate” (8.546c), must somehow be in accord. All such interpretations are speculative, however. As suggested in note 3 on 8.545e, the fact that Socrates ascribes the calculation of the number to the Muses (who, he imagines, “play and jest with us as if we were children”) should make us cautious about attempting to interpret the number and its calculation in an overly precise manner. Nonetheless, it seems reasonable to suppose that the passage in 8.546a-c is intended to convey, in general and suggestive terms, two facts: (1) that there is a rational order governing the cosmos, which can be expressed in mathematical terms, and (2) that this order is extremely difficult to comprehend. However its philosophical significance is interpreted, the passage puts Plato’s mathematical sophistication on full display.