From the Beginning to Plato

FROM THE BEGINNING TO PLATO 147
development of mathematics as an abstract study. 51 This theory is propounded, it
seems, on the basis of an analysis of ordinary human knowledge and its
presuppositions.
Philolaus’ starting-point is gnōsis, the everyday activity of cognitive
‘grasping’ (individuation, identification, reidentification, reference) of ordinary
individual things. This ‘cognizing’ implies that its objects ‘have number’, i.e. are
in some sense measurable or countable. Quite generally, any cognizable object
must be marked off from everything else by a sharp, definite boundary. Whether
this boundary be spatial or temporal, the object within it will have some
measurable quantity (volume, time-duration). Also, a cognizable collection of
objects must have a number; indeed even a single object must be recognizable as
a single object and not a plurality, which implies a definite and practically
applicable method of counting. These points are recognizably related to some
arguments of Parmenides and Zeno. Zeno (see above, pp. 153–4) argues that a
‘many’ implies a definite number; but also that it implies definite, distinct units
and hence boundaries round these units. That what is must be a unit and have a
boundary is also argued in Parmenides (see above, p. 41).
The concept of a ‘boundary’ is central here. Philolaus’ analysis of the
presuppositions of cognition leads him to a logical separation of the contents of
the universe into ‘things which bound’ and ‘things unbounded’. Everything in
the cosmos, and that cosmos itself, is claimed manifestly to exhibit a structure
‘fitted together’ from the two kinds of thing. This dualism is obviously closely
related to views which Aristotle attributes to ‘the people called Pythagoreans’.
He reports that some of them set up two ‘columns of correlated opposites’, which
featured such items as limit/unlimited, odd/even, one/plurality, right/ left, male/
female, etc. (Metaphysics I. 5, 986a22–6).
Philolaus’ careful attempt to build up a general ontology on the basis of an
analysis of ordinary cognition, guided by mathematics, leads him naturally in the
direction of Aristotelian ‘form’ and ‘matter’. Whatever stuff an individual is
thought of as being ‘made of, is in itself not ‘bounded’; for it might be present in
any quantity. But for there to be an individual, there must be also a ‘bound’.
Further explication of just what is involved in this ‘fitting together’ is not
found, and it seems that Philolaus thought this question beyond the reach of
human knowledge. That conclusion is in conformity with his method. The
‘everlasting being’ of things, or ‘nature itself, is the subject of ‘divine cognition’
only. The ‘fitting together’ is achieved ‘in some way or other’. Mathematics,
clearly, cannot help; for it too exemplifies, rather than explains, the dualistic
structure. All that we can say is that even humble human cognition presupposes
such a structure of things in particular and in general; the first example, it has
been suggested, of a Kantian transcendental argument. 52