Philosophies which have laid most stress on the a priori elements in knowledge have in general tended to turn for supporting arguments to mathematics, and, conversely, systems which seem primarily to draw inspiration from mathematical models have usually been led into elaborating theories of the a priori. In neo-Platonism the ways in which mathematics were regarded, and the uses to which it was able validly to be applied by different individuals and at varying times, were extremely diverse; it frequently meant little more than a fecund source for metaphysical analogies, emblemising and providing "insights" into moral and theological mysteries. Leaving aside for the moment the question of the influence of Platonism on mathematical development as an independent scientific discipline, and also that relating to the special status given to the "mathematicals" by Plato, which as reproduced by Dee, will be discussed in detail in a study of the Preface, the present paragraphs are intended only to illustrate some of the philosophical conclusions regarding mathematics, or those which mathematics was drawn on to support, characteristic of a neo-Platonic tradition. Many of these are crucial, for in the sixteenth century opinions on the philosophical significance of mathematics, and its relation to the natural sciences, might be taken, and there was very wide contemporary conscious recognition of this being so, as providing the most fundamental line of demarcation between "Aristotelian" and "Platonic" philosophies. A typical summary of the position is Mazzini's of 1597: "Creditit Plato Mathematicas ad speculationes physicas apprime esse accomodatas. Qua propter passim eas adhibet in reserandis mysteriis physicis. At Aristoteles omnino secus sentire videtur, erroresque Platonis adscribet amori mathematicarum."(192) Many peripatetics, Galileo complained, even "dissuade their disciplines from studying the Mathematicks, as Sciences that vitiate the reason, and render it less apt for contemplation"; and far from denying this Simplicicus replies with a justification along the usual lines: "These Mathematical subtleties Salviatus are true in abstract but applied to sensible and Physical matter they hold not good."(193) Similarly Bruno calls Aristotle an enemy to mathematics - "mathematicorum inimicus, logicus magis Aristoteles (qui plus arguendo quam argumentando valet)" adding, as Dee and many others pointed out, often with unconcealed triumph, that, whenever Aristotle touched on any profound mysteries of nature or religion he was driven to use mathematical analogy to express his meaning (194).

For Plato mathematical "objects provided (since they were knowable, being functional parts of an intelligibly connected system, and since however they might be conceived of as "existing," it could clearly not be as sensible intuitions) the most apt illustrations, by reason of their familiarity, to assist the mind in the difficult task of grasping something of the nature of those higher regions where true reality was to be sought. "For the colourless formless and intangible truly existing essence with which all true knowledge is concerned holds this region and is visible only to the mind, the pilot of the soul."(195) "That fixed and pure and true, and what we call unalloyed, knowledge" says the Philebus (196) "has to do with the things which are eternally the same without change or mixture, or with that which is most akin to them, and all other things are to be regarded as secondary and inferior." Thus "the arithmetical and metrical arts far surpass the others," and, moreover, when these "are stirred up by the impulse of the true philosopher," and proceed logically, and are not confined to merely physical things, they "are immeasurably superior in accuracy and truth about measures and numbers."(197) The Arts are classified according to the amount of mathematics they can employ, from which they gain their certainty (198): for "if arithmetic and the sciences of measurement were taken away from all the arts what was left of any of them would be so to speak, pretty worthless...All that would be left for us would be to conjecture and to drill the perceptions by practice and experience with the additional use of the powers of guessing, which are commonly caled arts and acquire their efficacy by practice and toil."(199) Hardie concludes, "The science that was not mathematical could for him (Plato) hardly be more than opinion."(200) In the Statesman measuring is called "the Kingly Art" and when intellectual activity is divided into Judging and Commanding, calculation is used as an example, and possibly meant as the type pattern of the first (201). Measurement is not merely our sole chief instrument for rectifying illusory appearances (202) but is also applicable to ethical questions, for Plato also develops the mathematical implications of the concept of "the mean" which had for long figured in classic writings (as in Hesiod's Works and Days) often with rather inexact associative, emotive reference, as the canon of ethics, and aesthetics, and the real principle of good, the well-proportioned nature producing the body's health etc. From the participation of the Unlimited - the indefinite and unconditioned - in the class of the Limited, of separated magnitude and number, arises a third class of the things making up the natural world, insofar as it is knowable; "The class of equal and double and everything that puts an end to the difference between opposites and makes them commensurable and harmonious by the introduction of number." The addition of Limit introduces moderation and allows of perfection "and thence arise the seasons and all the beauties of our world."(203) The "science of measurement" governs the investigation of pleasure and pain judged in their relation to the Good; it is a "study of their excess and defect and equality in relation to each other" (204) while objects such as "the straight line, and the circle, and the plane and solid figures" themselves are sources of pure and "unmixed" pleasure, and while other beauties may be exhibited as imitative "the beauty of these is not relative like that of other things but they are always absolutely beautiful by nature."(205)

Such speculations however are by no means the chief feature of the Dialogue. Subsequent developments of Platonic thought gave them a more overtly fundamental position. Speusippus, according to Aristotle, "feeling it necessary to assume some direct knowledge as to the principle from which the mind proceeds to discursive thought, posited the numbers of the decad, whose propositions carry immediate convicion, and because within the decad he found the pattern of all relations and proportions of existence."(204) To affirm a thoroughgoing Pythagoreanism was a commonplace of much neo-Platonism, thus Plutarch writes in the Platonic Questions, that "the intelligence of these Ideas and forms by subtraction, deduction and division of bodies, is ranged answerable to the order of the Mathematicks, arising from Arithmetick...unto Geometry...to Astrology...and...Harmonicae....Moreover of intellectual things there is no other judge but the understanding in the mind; for cogitation or intelligence is no other thing but the understanding, so long as it is applied unto Mathematicals, wherein things intellectual appear as within mirrours."(207)

Those who treat number as separable, summarised Aristotle: "assume that it exists and is separable because the axioms will not apply to sensible objects, whereas the statements of mathematics are true and appeal to the soul."(208) This represents fairly a perennial line of argument - that mathematical procedures and the "objects" they deal with cannot be drawn from, and are clearly independent of, the sensible, but are nevertheless wholly certain, and moreover seem not to be subject to alteration by any fantasy of the mind, i.e., cannot be conceived as being other than they are discovered to be. The rigour of mathematical demonstrations, Proclus argues, is a sign that numbers and geometrical shapes are not "abstractions" from things; knowledge of them is a process of "discovery," rather than a crative act of the soul, since as the mind has not full control over them they must possess "a spontaneous substance," and if so, the soul must contain these forms, otherwise it could not know its discoveries, intellectually represented in conventional signs, as true: "Mais si l'ame fait naitre ceux-ci tout en possedant les modeles en substance, ses productions sont les emissiom des formes qui pre-existent en elle, et en disant cela, nous serions du parti de Platon et aurions trouve la veritable substance des mathematiques."(209) Augustine similarly asserts that that on which numbers are dependent, Unity, is a concept not drawn from things; that the origin of truths apprehended by the reason must not be sought at a level below the reason; and that since the conclusions of mathematics are necessary, immutable and eternal, they present the very type of the propositions formulable by the intellect which are to be distinguished as "Truths."(210)

A typical sixteenth century "Platonic" statement - such as may be paralleled in Dee's writings - on the "a priori" nature of number is Guy le Fevre de la Boderie's

"Pource les Nombres nus d'essence toute pleine,

Plus simple que tout corps, voire que l'ombre vaine,

Ne sont percus des yeux, ni de l'air penetrant

La taye du cerveau et par l'oreille entrant:

Ils sont donc en l'Esprit qui les contient ensemble,

Car le sense ne recoit qui ce qui lui ressemble." (211)

The Cambridge Platonists reproduce the familiar arguments. In mathematics, Smith affirms, the imagination is wholly slave to reason, in considering its truths the mind is active in the midst of realities native to itself, and the soul "converseth only with its own being."(212) He, as well as Cudworth, More and Culverwell, at times reverses the application of the argument, and argue that mathematical knowledge demonstrates the immateriality and immortality of the soul, on the principle that like is only known to like. "Immoveable essense" (such as a triangle exemplifies) were, according to Cudsworth, the only true objects of knowledge, and "the essence of geometrical figures and all the system of truth that flow from that esssence are involved in the very nature of rational thought and could not be arbitrarily brought into existence or changed by our thinking."(213) A similar argument, suggested by geometrical figures, was used by Descartes: "Quodque hic masime considerandum puto, invenio apud me innumeras ideas quarundam rerum, quae etiamsi extra me fortasse nullibi existant, non tamen dici possunt nihil esse; et quamvis a me quodammodo ad arbitium cogitentur, non tamen a me finguntur, sed suas habent veras et immutabiles naturas: ut cum, exempli causa, triangulum imaginor, etsi fortasse talis figura nullibi gentium extra cogitationem meam existat, nec unquam existet, est tamen profecto determinata quaedam ejus natura, sive essentia, sive forma, immutabilis et aeterna, quae a me non efficta est, nec a mente mea dependet, ut patet ex eo quod demonstrari possint variae proprietates de isto triangulo...."(214) The strength of the appeal of such arguments is witnessed by the fact that while they, and the position they support, were closely associated with confessed neo-Platonism during the Renaissance, both have survived essentially the same, through large changes in philosophical and religious context. There are not lacking today examples of prominent mathematicians who have thus defended the "objective reality" of their science (215).

Dee's position in this, and in many other respects, was consciously approximated to that of Roger Bacon; whose reputation he championed against suspicions of necromancy, and who, perhaps of all Dee's predecessors, received the most frequent and unqualified praise from him. Fastening on the incident of the slave in the Meno, which he had found reproduced in Cicero, Bacon argued from it "wherefore since this knowledge (of mathematics) is almost innate and as it were precedes discovery and learning or at least is less in need of them than other sciences, it will be first among sciences and will precede others disposing us towards them." Distinguishing between our inferential, and mediated knowledge of nature and its processes in general, and the truth of actual occurences in these which is in general only apparent to God, he asserts - foreshadowing a famous declaration of Galileo's - "but in mathematics only....are the same things known to us and to nature simply."(216) He extends the familiar position that, as Synesius expressed it (217) mathematical propositions are certain above all others, and theother sciences are consequently proud to borrow whenever possible demonstration from them, into the grandiose aim of "per vias mathematicae verficare omnia quae in Naturalibus scientiis sunt necessaria."(218) This must be achieved because "in other sciences the assistance of mathematics being excluded there are so many doubts, so many opinions, so many errors on the part of man that these sciences cannot be unfolded," which is so, since, Bacon observes, in natural sciences and in much of metaphysics reliance has otherwise to be placed on arguments which are doubtful, since they proceed from effects to causes, and cannot demonstrate - as is the only certain method - from an initial grasp of "necessary causes" - adding "and likewise neither in matter pertaining to logic nor in grammar, as is clear, can there be very convincing demonstration because of the weak nature of the materials which those sciences treat." Thus investigation by "dialectical and sophistical argument as commonly introduced" must be rejected; mathematics must "enter into the truths and activities of the other sciences, regulating them, without which they cannot be made clear nor taught nor learned," and this "simply amounts to establishing definite methods of dealing with all sciences."(219) A recurrent theme is concerned with logic. Bacon, by exhibiting the implicit syllogisms in the natural actions of children or animals, claims that logic, however formally it can be expressed, is inherent in nature, and known innately (220), but that it is less fundamental in this respect than mathematics, and should be formally derived from it. He is reported in notes of lectures delivered by him, recently discovered, as declaring, after describing the interconnection of all sciences, the key to which is mathematics, "Volo probare quod non potest homo docere Logicam nisi sit optime in mathematicis imbutus....quoniam mathematicas docens potest et descendere ad Logicalia et grammaticalia quia in eis instructus est"; and he proceeds to a consideration of the categories, claiming that the meaning of each is gained through what is only clearly known quantitatively, concluding "dico ergo quod liber Posteriorum sine Arismetics et Geometria non scietur."(221)

Similar themes appear in Cusa, whose works Dee also seems to have prized highly and whose statements on mathematics were often quoted in prefaces to sixteenth century editions of Euclid. The mind, declared Cusa (222) intending it to be no more metaphysical expression, was a measuring instrument, to know was to ascertain the true proportions of things. In Idiota (223) a supposed etymological connection of "mens" and words relating to measuring is urged in support of the view that the true function of thought is to endow all things with limit and measure. All judgment, the first chapter of de Docta Ignorantia claims, is a comparison of a thing to something presupposed, and hence the determination of a proportion between them; all conclusions are therefore comparative, while all things susceptible of proportion are included in the scope of number. The spread of such views in the Renaissance gained impetus from the increasing reimportation and emphasis on the importance of logical rigour into mathematics, a consequence of the revival of the full text of the Elements, the discovery of the writings of Archimedes and the development of Algebra. This was in sharp contrast withthe impression that the form in which Boethius transmitted Euclid to the middle ages, or the number theory as set out by Theon, might give; for there, propositions, while they might be admitted as certain and undeniable, appeared rather as isolated truths, and the secret of their essential connection was an open question for metaphysical speculation. "Perfect" and "Amicable" numbers might still be so regarded by mathematicians of the Renaissance, but their attention was now orientated chiefly towards the way in which conclusions could be proved to be true, by strictly mathematical methods, even though apparently intuitively recognisable as so being, or discoverable by "experience," rather than to presenting mathematics as a collection of individual "factual statements" about numbers or figures. Declarations that mathematics exemplify Reason, and provide the pattern for philosophical procedure, and even its matter, become more widespread and have a somewhat different implication in the late fifteenth and sixteenth centuries than in the earlier times, and even when used apparently purely rhetorically are not without significance.(224)