The Intelligent Troglodyte’s Guide to Plato’s Republic, 2016a

65 Numbers as Summoners
See 521c-526c. How can the rational part of the soul be awakened and made to
turn toward the forms? Socrates notes that there are certain objects of sense
perception – “summoners,” he calls them – that “strike the relevant sense at the
same time as do their opposites.” For instance, the same thing can appear both big
and small, hard and soft, thick and thin, light and heavy, and so on. Each of these
properties is, of course, relational. For something to appear both big and small, it
has to appear big in relation to one thing, and small in relation to something else.
Socrates’ point is that the question “Is it big?” leads the mind to a further question
– “What is it to be big?” – and this question cannot be answered simply on the
basis of sense perception. One has to stop looking and start thinking. Socrates
suggests that the numerical properties of things are similarly problematic and
thought-provoking. His discussion of this is unclear, but he seems to have in mind
questions such as these: Is a baseball team one or nine? Is a slice of pie one or
one eighth? When one lump of clay is rolled in with another is the result one or
two? These questions, of course, have answers. But again, the answers call for
more than mere sense perception. The rational part of the soul needs to wake up
and consider what it means to be one, nine, one eighth, and so on, and then
consider how these numbers “that are accessible only in thought” are relevant in
particular contexts. Numbers are some of the simplest and most accessible of
forms, and people who make a regular exercise of studying them “become
generally sharper than they were.” It therefore makes sense to begin the study of
forms with the study of numbers.
How are mathematical truths (such as that the three interior angles of a
triangle are equivalent to two right angles) similar in nature to definitional
truths (such as that green is a color)?
How are mathematical truths different in nature from observational truths
(such as that there is beer in the refrigerator)?
Are mathematical questions better at turning the mind toward forms than
other sorts of questions?