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.