2. Greeks and ‘origins’
Socrates: Then as between the calculating and measurement employed in building or commerce and
the geometry and calculation employed in philosophy—well, should we say there is one sort of each, or
should we recognize two sorts?
Protarchus: On the strength of what has been said, I should give my vote for there being two.
(Plato,Philebus, tr. in Fauvel and Gray 2.E.4, p. 75)1. Plato and theMeno
One feature of mathematics which has remained fairly constant from the earliest times to the
present day is a general view that its aim is to use ‘numbers’ to solve problems which arise in the
world. However, another idea has been widespread among mathematicians at least since the time of
the ancient Greeks, and its statement dates back to Plato, whose different view is summarized above:
there is a down-to-earth mathematics which you use for accounts and measuring, and there is a
superior mathematics, which I use for some other purpose. What we know of this view, what its
implications were, and its early history are the subject of this chapter. Plato—a philosopher, whose
dates are usually given as roughly 427–348bceand who was mostly writing in the early fourth
century—is one of the central figures in the history of Greek mathematics. There are a number of
reasons for this. A simple one is that Plato dealt in some detail with mathematical questions in his
works; and, while mathematics had supposedly been practised for 200 years before his time, his
Dialoguesare the earliest first-hand documents which we have. Almost equally important is that,
as the quotation indicates, Plato defined a particular view of what mathematics was, or should be.
A rough characterization is that real mathematics is more abstract—numbers are no longer num-
bers of ‘things’ or measurements of length, area, or time, but have an independent existence as
objects which you reason with. As Socrates says earlier in the same dialogue:
The ordinary arithmetician, surely, operates with unequal units; his ‘two’ may be two armies or two cows or two
anythings from the smallest thing in the world to the biggest; while the philosopher will have nothing to do with him,
unless he consents to make every single instance of his unit equal to every other of its infinite number of instances.
(Plato,Philebus, tr. in Fauvel and Gray 2.E.4, p. 75)
And while not all of his successors agreed with this approach, those who did were those who
had most influence. This is especially true of some very late writers (after 300ce) who are the main
authorities for what we know of the history.
One of Plato’s longest and clearest mathematical discussions, often referred to, is in the dialogue
called theMeno. (For the mathematical part, see Fauvel and Gray 2.E.1 (pp. 61–67); Fowler (1999)
has text with variations of his own construction; and the whole dialogue is online for example,