It was largely in questions relating
to mathematics, its general status, the validity of its methods,
its sphere of application, that much of the new science of the
Renaissance found itself in overt opposition to Aristotelianism.
Aristotle's own chief interests were not mathematical, and his
principal commentator, who had tried most faithfully to reconstruct
the original teachings of Aristotle, Averroes, was no more a mathematician
than his master, and, as distinguished from almost every other
leading Arab thinker, left no separate work on number. Mathematics
Aristotle seems to have regarded as merely a particular rational
scheme of fictions or abstractions, that might or might not apply
to certain parts of reality. "Mathematical accuracy,"
he had declared, "is not to be demanded in everything but
only in things which do not contain matter. Hence this method
is not that of natural science, because presumably all nature
is concerned with matter,"(99) and held it a defect and limitation
of harmony and optics that they did not study their objects "qua
sight or qua sound but qua lines and numbers"
with the aid of mathematics (100); and Italian Averroist scientists
of the sixteenth century, it has been shown, still maintained
a sharp distinction between the "analytic" method proper
to mathematics and the "resolutive" one of natural science,
which alone led to discovery and started always from the evidence
of the senses (101). In striking contrast is the declaration
of Dee's associate, Pedro Nunex, that he intends to abandon the
orthodox procedure in the natural sciences, and employ a method
of demonstration modelled on Euclid (102).
In many particular points the new
science found itself in direct opposition to Aristotelian doctrines.
Astronomers, whether or not they believed in the physical truth
of Copernicanism, widely admitted that Aristotle's transformation
of the geometrical hypotheses of Eudoxus into an intuitively conceivable
mechanism, had been unjustified and was not, anyway, in its original
form, to be accepted. Aristotle's determination of the speeds
of the planets, as being proportionate to their distance from
the zodiac, those farthest from perfection requiring the most
- the swiftest - motions (103), was, it was frequently pointed
out - as by Dee's acquaintance Jean de Peno, in his preface to
his edition of Euclid's Optics in 1556 - an exact reversal
of the facts, was the mistake of one ignorant or neglectful of
mathematical sciences. Cardan as a "naturalist" Aristotelian
still examines change and motion in terms of tendency (Appetitus)
(104) and prefaces a description of machines, by a discussion
of the distinction between natural and violent motions (105),
but the disadvantages, and even contradictions, inherent in such
an attitude become progressively more apparent, in numerous particular
sciences, as for instance in ballistical questions (106). A good
example of the conflict between Aristotelian doctrines, and direct
scientific findings of a kind which were increasingly regarded
as important, and the fundamental difference in their respective
attitudes to quantitative considerations is Hakewill's criticism
of the statement of Aristotle that the elements "as they
rise one above another in situation so they exceed one
another proportione decupla, by a tennefold proportion,"
which Hakewill denounces as nonsense since water is nowhere more
than two to three miles in depth, but the diameter of the earth
is seven thousand miles, while Nonius and Vitellio have shown
the air to be only fifty-two miles deep. In a later part of his
book Hakewill prints a reply he has received from an Aristotelian
Bishop (G.G.) on this point. It states: "I have often desired
you in theories not to bee exact in proportions, as if man's imagination
could apply a compasse and rule to measure out speculations; these
Mathematical punctilioes are not to be admitted in Philosophy,
yet it is necessary that in things which are most uncertaine,
wee should guesse at some certainty and be guided by one Rule,
and herein Aristotle hath done what the wit or endeavours of man
could effect." None the less G.G. goes on to cite numerous
"experiments" which will support the accuracy of the
tenfold proportion of the elements, all of which Hakewill patiently
explores and reveals their fallacies or unreality (107).
But besides the multitude of particular questions in dispute a more general one regarding knowledge and logical method was involved. Thomist Aristotelianism had effected a compromise with Anselm's position by admitting Anselm's definition of truth as referring to the adequation of the thing to the divine understanding though not to the human (108): its implications in regard to human understanding were clarified by Peter d'Ailly who then distinguished absolute knowledge, which is reducible to first principles but remains purely formal, and all intellectual knowledge of things, which must be derived from the sensations, and can, in consequence, never be more than probable. On the other hand, the geometry of Euclid or the statics of Archimedes seemed to present the sixteenth century with systems at once a priori and synthetic, and of complete certainty. Compared to these, formal syllogistic logic seemed a cumbersome unprofitable instrument, a possible if almost sterile method of organising what was already known, but which did not reflect the natural processes of thought or assist psychologically to discovery or creation. It was of course a pattern, though a very artificial one, to which all correct thought might eventually be reduced, even geometry (where the logical procedure, constantly suppressing the major premiss, is in effect enthymematical) though the resultant expression would be so lengthy and involved that it would obscure rather than illumine. There was indeed an attempt in the sixteenth century to set out Euclid as a series of syllogisms (109), but a multitude of others - some will be mentioned later - which attempted the reverse, and expressed various bodies of knowledge - even medicine - in Euclidean fashion. It was his lack of employment of such methods, and his attitude towards them (as well as his defence of such dogmas as the eternity of the world), that caused Campanella's Solarians to refuse, typically, the name of philosopher to Aristotle and to regard him as a mere logician.