Routledge History of Philosophy Volume IV

DESCARTES: METHODOLOGY 171
have been used with slightly different meanings by different writers, but it is
basically that part of the mathematical process in which one proves deductively,
perhaps from first principles, what one has discovered or shown the truth of
by analysis. In the case of theoretical analysis, one needs synthesis, because in
the analysis what we have done is to show that a true theorem follows from a
theorem whose truth we wish to establish, and what we must now do is to show
that the converse is also the case, that the theorem whose truth we wish to
establish follows from the theorem we know to be true. The latter demonstration,
whose most obvious form is demonstration from first principles, is synthesis. A
synthetic proof is, in fact, the ‘natural order’ for Greek and Alexandrian
mathematicians, the analysis being only a ‘solution backwards’. So what we are
invariably presented with are the ‘naturally ordered’ synthetic demonstrations:
there is no need to present the analysis as well. Descartes objects to such
procedures. He accuses the Alexandrian mathematicians Pappus and Diophantus
of presenting only the synthesis from ulterior motives:
I have come to think that these writers themselves, with a kind of
pernicious cunning, later suppressed this mathematics as, notoriously,
many inventors are known to have done where their own discoveries were
concerned. They may have feared that their method, just because it was so
easy and simple, would be depreciated if it were divulged; so to gain our
admiration, they may have shown us, as the fruits of their method, some
barren truths proved by clever arguments, instead of teaching us the
method itself, which might have dispelled our admiration. 31
In other words, analysis is a method of discovery, whereas synthesis is merely a
method of presentation of one’s results by deriving them from first principles.
Now it is true that in many cases the synthetic demonstration will be very
straightforward once one has the analytic demonstration, and indeed in many
cases the latter is simply a reversal of the former. Moreover, all equations have
valid converses by definition, so if one is dealing with equations, as Descartes is
for example, then there is no special problem about converses holding. But the
synthetic demonstration, unlike the analytic one, is a deductively valid proof, and
this, for the ancients and for the vast majority of mathematicians since then, is
the only real form of proof. Descartes does not accept this, because he does not
accept that deduction can have any value in its own right. We shall return to this
issue below.
The case of problematical analysis with a positive outcome is more
complicated, for here there was traditionally considered to be an extra reason
why synthesis was needed, namely the production of a ‘determinate’ solution. In
rejecting synthesis in this context, Descartes is on far stronger ground. Indeed,
one of the most crucial stages in the development of algebra consists precisely in
going beyond the call for determinate solutions. In the case of geometry, analysis
provides one with a general procedure, but it does not in itself produce a