Music and the Origins of Ancient Science 15
counts as one and what as two. Thus, though a child can parrot counting numbers, under-
standing them requires discernment.
This deeper knowledge leads to the four sister studies, each of which illuminates some
essential aspect of math ē matik ē , the master art of grasping all learnable things. All four
studies are necessary for the quest Socrates calls philosophy: each of these studies “ appears
to compel soul to use thought by itself for purposes of truth by itself. ”^19 Arithmetic relies
on the most fundamental insight into being and otherness and how they can be combined
into more or less inclusive wholes when considered as a one, or distinguished into the
various numbers as many. The primal category of multitude is manifest in whole numbers,
each one called in Greek an arithmos , meaning a countable multitude of countable things,
an integer whose wholeness is essential to its integrity, its very essence. The word arithmos
comes from the Indo-European root common to our word rite , as in the counting essential
to the performance of sacred ritual, as well as the fundamental sense of rightness (in
Sanskrit ṛ ta ) or rhythm (in Greek rhythmos ) underlying the cosmos as an ordered whole.
A shape or pattern conveying motion, such as the fluid pose of a dancer or of a sculpted
figure, was also called rhythmos , which the Romans commonly translated as numerus ,
number itself. 20
Arithmetic also concerns how whole numbers can be connected by a ratio, a logos. In
Homer, the verb legein means “ to gather together, ” as when the grieving Achilles tells his
comrades “ let us gather up [ leg ō men ] the bones of Patroklos ” in preparation for his funeral
rites.^21 By extension, this word for gathering or collecting also came to signify speaking,
recounting, telling, and reasoning, implying that all these are deeply forms of bringing
together , hence of connected expressions. In that sense, only by means of logoi — the
primal relations between integers — do the counting numbers really become fully the object
of accounts and reason, of logos manifest in what the Greeks therefore called logic. Logos
also has the specific meaning of a musical interval, hence suggesting that, as logoi , musical
intervals may be deeper even than the integers whose relation they express. One might
daringly suggest that the intervals (such as 1:2, 2:3, 3:4) come before the integers them-
selves, which remain profoundly isolated until we express their relation. Can we under-
stand the concepts of two or three only if we grasp each in relation to the unit of which
they are implicitly composed (2:1, 3:1)? If so, arithmetic may implicitly rely on musical
ratios to ground our awareness of number.^22
In contrast, geometry deals with magnitude ( megathos ), which Plato (and Greek math-
ematicians in general) considered deeply different from multitude ( pl ē thos ). The Pythago-
reans were credited with the crucial insight that showed the full extent of the distinction
between arithmetic and geometry: in general, geometric lines cannot be expressed as any
number or as any ratio of finite numbers. Most famously, the diagonal of a square is not
commensurable with its side: if its side is a unit length, no ratio of whole numbers m : n
can express the length of the diagonal, called irrational ( alogon , not having a logos ) or
unspeakable ( arrh ē ton ) because not expressible in terms of finite numbers. A beautifully