7.5: Numbers as Summoners

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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?