2.1 The Aristotelian Division of Mathematics
Aristotle (probably preceded by Pythagoras) defined mathematics as the study of quantity and distinguished between discrete
and continuous quantities, which resulted in a division of mathematics into arithmetic and geometry, the sciences of multitude
and magnitude.
From the discovery of incommensurables, magnitudes which were not ratios of numbers, the science of magnitude was considered
more fundamental (though not universally in classical Greece, Aristotle considering arithmetic more fundamental).
All numbers and ratios corresponded to magnitudes, but not all magnitudes corresponded to a number or ratio of numbers (numbers
for the Greeks were just the positive integers).


2.2 Traditional Ideals of Proof
Detlefsen calls the ideal of proof in classical Greece the "constructive ideal", which had a genetic and a constructional component.
On the genetic component Aristotle says:
We suppose ourselves to possess unqualified scientific knowledge of a thing, as opposed to knowing it in the accidental way
in which the sophist knows, when we think that we know the cause on which the fact depends, as the cause of that fact and
of no other, and, further, that the fact could not be other than it is.
[You would not guess from the context of this quotation in the book that it immediately leads into material espousing the
syllogism as the best method of proof.
Though this is Aristotle's main message here, the syllogism is nowhere mentioned by Detlefsen as a standard of proof.
This may be realistic, it is unlikely that many proofs even in mathematics, were formally syllogistic.
But if we doubt that Aristotle's views in relation to the syllogism represent standard practice, why should we believe the
cited passage gives a widely accepted and practiced standard of proof?]
In the case of mathematics it is the formal cause which is relevant, and the formal cause of a mathematical object is its definition.
[Syllogistic demonstration is one way of establishing this kind of causal relationship, the premises of a syllogism being
regarded as the causes of its conclusion.]
The constructional component is an ideal for the realisation of the genetic component which specifies that the best kind of definition for a
geometric object is one which indicates how it is constructed.
[Detlefsen's material here is not wholly convincing on the matter of this constrctional component being a widely accepted
standard of proof going beyond certain kinds of geometric constructions.]

