Since the works of Dee which will be described in subsequent chapters are concerned mainly with more particularised problems, the foregoing sections have been designed to illustrate some of the generalities which helped to frame the philosophic background of his thought, both from sources utilised by Dee himself or from those, such as the Cambridge Platonists, of following generations whose doctrines, though conveniently more explicit, are closely concordant with his own, and if more developed are yet related philosophical formulations within a single intellectual tradition. Something must also be said however of the ways in which the views and activities of such thinkers as Dee influenced or proved complementary to the outlook and methods of the "new" science of the day. One contrast between this and the older orthodox natural philosophy has been succinctly shown by Collingwood: "For the Aristotelian doctrine that change is an expression of tendency, the sixteenth century substituted the Platonic doctrine - strictly the Pythagorean doctrine for in essence it is pre-Socratic - that change is a function of structure."(225) A consideration in terms of structure, that is, the attribution of causal efficacy only to preceding and therefore probably determinable structural relationships, suggests at least the possibility of a complete analysis of apparently disparate entities or "events," on some common quantitative scale; for structure is amenable, as "innate tendency," dependent upon the whole synthetic form of the object in process of change, is not, to mathematical description. The differences in the changing object, though accompanied by widely contrasted qualitative accidents, may in this way perhaps be reduced to a matter of comparable degrees; it becomes a not impossible speculation that all observable characteristics may result from the imposition of a certain intelligible order upon some primary undifferentiated stuff, of which the only necessary property is to be "susceptible of arrangement," and that the secrets of all generation and decay, the reality of all natural phenomena may be exhaustively expressed, as Dee never wearied of asserting, in terms of "Number, Measure and Weight." The sphere in which teleological explanation was admitted - a philosophy which did not abandon a spiritually significant eschatology could clearly not dispense with it wholly - became more narrowly restricted, and action through time increasingly accepted, as being for the purposes of the natural sciences, irreversibly unidirectional. The "mechanism" that this suggests as an almost inevitable consequence may indeed be no more than a teleology a rebours, but at least possessed the advantage as a working procedural guide that it did not assume in advance as a working procedural guide that it did not assume in advance that any of the factors determining change were, ipso facto, inaccessible to observation.

The evolution of a concept of "structure" in nature, with such implications, in the Renaissance is best viewed in relation to two themes, in many respects only artificially separable. The revival of "the neo-Platonic interpretation of Plato dominated by the passion for a fully articulated vision of the world as a structural entity,"(226) and the attempt to extend as far as possible methods of quantitative analysis to natural questions which accompanied the rapid progress in the theoretical development, and the spread of knowledge of mathematics. The former body of doctrine stood firmly on Plato's declaration that "we are sure of this....that whatever is completely, is completely knowable, and what in no way is in everyway unknowable," and, by refusing to separate Being from a logical account, developed such a theory of knowledge as Cusa's: that the function of thought was not to discover and examine particular discrete and self existing entities and to build the world from the abstractions drawn from a grouping of their natures so considered, but to establish intelligible relations, and a hierarchy of more and more comprehensive laws; that "vere soire est per nexus soire." The neo-Platonic picture of the universe of St. Augustine had not in general statement been very different from that now revived - "une matiere impregnee d'intelligibilite par les idees divines: tout y est ordre mesure et nombre, les formes des corps se reduisent a de certaines proportions numeriques et les operations de la vie se deroulent, elles aussi, selon les lois intelligibles des nombres. En droit l'univers est intelligible pour une pensee capable de la connaitre comme tel."(227) But its revival in the world of the Renaissance had novel consequences and led to, and was used in defence of, "a habit of detailed and accurate observation, based on the postulate that everything in nature however minute and apparently accidental, is permeated with rationality and therefore significant and valuable."(228)

The Timaeus hinted at the metaphysically satisfactory account of the mutual relationships between meaning or purpose, and discoverable mechanical causality, which was most compatible with such views. The latter was the temporal unfolding in discursively intelligible form and therefore inevitably limited, and bound by necessity arising from the nature of the selected medium of its expression, of timeless values. These, though "causes," in the sense of providing explanation required by the soul, did not intrude actively into the temporal flow of events; though Reason was an agent there, conceived of as operating as man's mind appears to allow him to do, rearranging and controlling events, but only in accordance with Necessity and natural law. The ultimate values could be thought of as the limit defined by the automatic universal development. "We must accordingly," says the Timaeus, "distinguish two types of cause, the necessary and the divine. The divine we should search out in all things for the sake of a life of such happiness as our nature admits; the necessary for the sake of the divine, reflecting that apart from the necessary those other objects of our serious study cannot by themselves be perceived or communicated, nor can we in any other way have part or lot in them."(229) It is probably on this section of the Timaeus that Boethius bases his version of this doctrine, developed in terms of Fate and Providence: "For Providence is the very Divine reason itself seated in the highest Prince which disposeth all things. But Fate is a disposition inherent in changeable things by which Providence connecteth all things in their due order. For Providence embraceth all things together, though diverse, though infinite, but Fate putteth things by which Providence connecteth all things in their due order. For Providence embraceth all things together, though diverse, though infinite, but Fate putteth every particular thing into motion being distributed by places forms and times....which although they be diverse yet the one dependeth upon the other....Providence is an immoveable and simple determination of those things which are to be done and Fate a moveable connexion and temporal order of those things which the Divine simplicity hath disposed to be done."(230) Such a teaching Cusa probably refers to when discussing The Soul - or Form - of the Universe. He says that some have called the general type of causality (that superior nature, which is responsible for all change and process), Spirit, or intelligence, others the Soul of the World, others destiny concealed in substance, but the followers of Plato call it a necessity of structural constitution.(231) Leonardo, who held that the body is the first great work of the soul, being its realisation through matter of its ideal of the human form, found in such views the reconciliation between this thesis, which involves a hierarchical superiority of the spiritual over the material, and a theory of nature as exhibiting only mechanically describable, controllable and mechanically reproducible phenomena, from which last position he developed his method of investigation. Charbonnel, who on this point finds Leonardo's thought recalls both Plotinus and Bruno, thus reproduced it: "La necessite est le lien eternel, la regle de la nature mais en definitive elle se confond avec la raison. Ce qu'il y a de primitif, c'est l'intelligible, c'est la raison vivante souveraine, dont la nature est l'expression visible. Savoir c'est approfondir l'esprit et ses lois. Par l'etude des faits nous sommes ramenes du dehors au dedans, de la realite a l'idee...."(232)

The elevation of mathematics to the position of the primary instrument of the intellect for the analysis of nature also found theoretical inspiration in the Timaeus, even though its particular exemplifications of the results obtainable by such a method might not themselves be taken over as dogmas. Atomism always represents the pursuing of uniformity into the very elements of things, and the geometrical atomism of the Timaeus exhibited the apparently primary substances of earth, air, fire and water, still figuring as ultimates in an Aristotelian physics, as constructs of intelligible concepts; it had also attempted physical explanation in mathematical terms of such qualitative phenomena as colour diversity. It suggested a general framework for thought capable of embracing all things and a uniform language for the expression of all conclusions, thereby intelligibly and firmly connecting them together. Its great power and promise lay in abolishing what in Aristotelian thought sometimes appeared unbridgeable gulfs between "things," "facts" and "laws"; for, though on different levels all these must in essence always be abstractions, they were now abstractions made on identical principles, and their interrelations capable of clear examination. What the Timaeus precisely meant for the scientific mathematicism which began its modern career in the Renaissance, Brunschwieg has thus described: "En utilisant par un raisonnement assez insolite ( ) les maigres ressources de la science de son temps. Flaton fait oeuvre de prophete plus que de precurseur: il delimite du dehors le terrain ou s'elevera l'edifice de la pensee moderne."(233)

Among a large body of Renaissance thinkers an increasing dissatisfaction is discernible with the Aristotelian logic insofar as it laid claim to be of assistance to constructive thought. A reduction to syllogistic form might be a useful touchstone for consistency in thinking after any creative process was over, but its impotence in all other respects became increasingly remarked: "Logic, it appears to me," declared Galileo, comparing it unfavourably with geometrical method, "teaches us how to test the conclusiveness of any argument or demonstration already discovered and completed, but I do not believe that it teaches us to discover correct arguments and demonstrations."(234) The syllogism employed, as its extreme advocates recommended, as an engine for the investigation of nature involved from an empirical standpoint a palpably vicious circle, for then interpreted purely extensively (as forming part of a scheme of class inclusions), the transition from the universal to the particular could bring forth no new knowledge, since if the former represented no more than the invariable association of a set of characteristics abstracted represented no more than the invariable association of a set of characteristics abstracted from observation of particulars, these must previously have been identified as present in the particular, which figures in the conclusion, when it was subsumed under the universal in the premise. The form of Aristotelian induction, associated with such a logical method, was found equally unsatisfactory. Only induction by a simple and exhaustive enumeration could be formally valid, but to escape the difficulty here by asserting that although this could rarely be done with particulars it might be with species, or else to rely on the rational intuitions, possessed by gods (or rather separated spiritual substances such as angels, for Christian Aristotelians) and men, as being able properly to extract from a few cases a universal of this kind that could operate as a middle term in argument, proved especially in purely physical questions (relating to motion for instance) merely misleading. Again though all definitions amount to the enunciation of some particular properties selected as characteristic, yet those of mathematics by permitting an exhaustive "construction" of the entity described, involved implicitly all other truths about it, which though initially unknown or unobserved, might thereupon be validly deduced from the implications of the definition itself - at the same time, since often many alternative definitions of equal adequacy were here possible a practical refutation of Aristotle's claim that all things had a single "essence" that could be expressed in one "true definition" only was provided. On the other hand if deduction claiming to establish genuine new knowledge were attempted syllogistically, employing synthetic definitions of the kind Aristotle recommended then it would in fact appear successful only when an unjustified importation of "meaning" into the terms figuring in the premisses, in excess of that attributed to them by definition, had unobserved, by psychological sleight, taken place. (An attack on the claims for the syllogism made by Aristotelians, along these lines occurs as Morsenne's Verite des Sciences - it embodies, Morsenne argues, no implications of ideas or causes but only a formal arrangement of elements observed as usually empirically associated. If man possessed Intuition equal to that of the Angels, both the minor and the conclusion would be seen to be unnecessary, they would be known immediately as present in the major; as it is, this last is established only by an induction from particulars, a procedure which has not been universal, else there would be no new particular to appear in the minor to which the conclusion relates - and hence remains always to some degree uncertain (235).) The "purely verbal" nature of syllogistic reasoning which while by its form it seemed to compel the reason, yet in its matter all too frequently remained thoroughly unconvincing for the understanding, became glaringly apparent in the light of many of the acute and genuine problems of sixteenth century thought.

It may well be, as F.C.S. Schiller insisted, that the sole alternative to a logic which comes to terms with psychology (thus ultimately abandoning any claim to be absolute and eternal) is one enslaved to grammar (236). It would seem that much of the popularity of Ramus' logic, among Platonists, humanists, and the new scientists, was due to the impression it gave at the time (as distinct from any final historical judgment on it as a system) of being a method directly drawn from, and remaining much closer than the Aristotelian to, observable natural mental process, of truing to base itself on the way terms are in fact used in thinking, and of being a new and profound attempt to investigate actual meanings, and evolve a formal method which should take into account that these are always relative to purpose. Nevertheless a logic "enslaved to grammar" offers the great advantage of providing absolute, demonstrable certainty through formal manipulation made in accordance with merely syntactical considerations. Such a system, having such advantages, the Renaissance found in mathematics, and this moreover apparently resolved the antinomy which syllogistic reasoning was incapable of doing, that if a logic is to be of value it must be able to produce conclusions which are of intelligibility - an example is Malebranche's declaration that that could only be accepted as a true cause, between which and its effect the mind perceived a necessary connection as clearly as it did the equivalence that a series of substitutions could establish between two terms of an equation in algebra (239). It was an ideal suggested by mathematics and which applied to physical questions - and it is problems of method in natural philosophy that Malebranche was here treating of - it led inevitably, though not immediately, to mechanism ("Le mecanisme" it has been claimed "prend comme terrain solide de construction l'unite profonde de l'intelligible et de l'experience, du pensable et du representable, du rationnel et du perceptible"(240)). However much it might represent the limit of already acquired knowledge in some subjects, an inference of cloven hoofs from the observation "horned" by way of the form "ruminant" functioning as cause, was no longer in even remote conformity with the, ideally, demanded pattern of scientific procedure. The Cambridge Platonists were to receive more encouragement from the practical science of their day in believing that the division into "forms" and "matter" could be dispensed with yet in rejecting these as false and artificial, and replacing them by the terms "body" and "spirit" and removing all teleological considerations from "body," they were developing the position of many predecessors in the Renaissance. But conscious life which could be observed to act in relation to an end - though only in accordance with the prior representation the mind framed of this, which alone and not the "end" itself could therefore properly be regarded as "causal" - this class of conscious volitional actions, taken together with that of physical phenomena mechanically explicable, did not suffice to entirely account for natural change and generation. The solution usually adopted by those who forsook Aristotelianism was to ascribe what remained to the "seminal reasons," which the stoics had employed and which combined the advantages and avoided some of the difficulties of both the Democritean atoms and the Ideas of Plato (certainly in the original form in which these had been put forward). Augustine had made large use of this concept and it had subsequently had a lengthy and respectable history in philosophic thought. Sir Thomas Browne still found it a useful hypothesis, offering, for want of better, a plausible shadow of explanation (241). It at least avoided the dangers of a theory which would have directed speculation towards considering the final entelechy of a form rather than observation towards conditions of development.

Aristotle had thought it a cogent argument to urge against the Pythagoreans that number could not be admitted as the essence of things nor the cause of the form, for the reason that essence was always a ratio (as the essence of skin and bone was to be three parts fire and two of earth) and number was merely adjectival, signifying an amount of matter, an intensity of quality (242). This could not serve as valid criticism against the procedure of the Renaissance neo-Platonists, who were not solely, or perhaps chiefly concerned in subsuming a thing under some one cardinal number, binding them together in a mesh of allegory, and so interpreting nature through the supposed mystic properties of the numeral system. Rather they accepted this statement (implied for example in Dee's treatment of medicine graduation), but turned its force against Aristotle, by asserting that all that could be certainly known of the nature of the terms figuring in the ratios compounding essence was in turn merely other determinable ratios that they observably entered into. The key to nature was therefore still mathematical, but it lay in the application of what Dee always referred to as "The Divine Science of Proportion" - a title originated by Paccioli. The change of orientation that was initiated in the Renaissance, and has since, evolving persistently, maintained ascendancy by this new approach, has been delineated by Blanche: "En l'adoptant, la pensee transformait du meme coup sa propre structure et s'obligeait a une revision intellectuelle totale. La hierarchie allait desormais se renverser entre les sens et l'entendement, entre les choses et les rapports. L'idealisme de la relation allait se substituer au realisme de la substance et a l'idealisme de la representation, et une forme nouvelle de realite se faire une place entre le monde anthropomorphique de la perception et le monde transcendant de la speculation, entre le phenomene et la chose en soi....Le concret ce n'est plus le qualitatif, dans sa diversite pittoresque, et d'ou l'abstrait se forme par reduction, c'est un entrecroisement de determinations intellectuelles...Le reel n'est plus au point de depart, mais au terme de la connaissance. Il n'est plus donne a la sensibilite par les qualites d'une substance il est construit par l'intelligence comme un systeme de rapports."(243) The Renaissance mathematicisation of the world already contains in embryo the definition of scientific knowledge of Eddington - that any account of the external world "involves unknowable actors executing unknowable actions," which are nonetheless a vehicle for "The knowledge we can acquire [which] is a knowledge of a structure or patter, contained in the actions."(244)

The new methods in action and the visible manifestation of their power are more clearly apparent in a generation later than Dee's. But the same doctrines, for which Dee out of faith was a largely theoretical propagandist, were held by the much greater and more creative Galileo, who refashioned the world out of them in a way that could not be ignored. Mathematics, Galileo asserted in the Dialogues of the Two Principle Systems was the one type of knowledge in which man's knowledge though less in extent was necessarily the same in kind as God's (245); it was therefore a domain in which reality and God's thoughts could be immediately attained. "La filosofia," he declared, "e scritta in questo grandissimo libro, che continuamente ci sta aperto innanzi a gli occhi (io dico l'universo) ma non si puo intendere se prima non simpara a intender la lingua e conoscer i caratteri, ne'quali e scritto. Egli e scritto in lingua matematica e i caratteri son triangoli, cerchi ed altri figure geometriche, senza i quali mezzi e impossibile a intenderne umanamente parol, senza questi e un aggirarsi vanamente per un oscuro laberinto."(246) He seems to have consciously accepted without serious doubts, that the mathematical methods he employed were inseparable from a Platonic theory of knowledge and metaphysics (247); and recent research has corrected older views on the empirical elements in his work and its supposed contrast with earlier science, in showing that where ever he rejected the Aristotelian physics or method, Galileo's "did not revert to mere observation but on the contrary reverted to a more vigorous form of Platonic rationalism, in which observed nature is merely an instance of a universal geometric order of infinite complexity."(248) "On pourrait dire," Professor Koyre has written of Galileo, "...qu'il n'a aucune confiance en une observations non verifiee theoriquement. L'epistemologie galileenne n'est pas positiviste. Elle est archimedienne."(249) A statement strikingly illustrated by the true version of the Pisa episode, which exhibits Galileo as unshaken in his faith in the reasoning which demonstrates contradiction in the Aristotelian position - two equal weights tied together cannot double their rate of fall merely by being so conjoined if their single rates were equal when they were separate, since the velocity of neither can increase that of the other; while a lighter weight attached to a heavier, should on the Aristotelian premisses both result in a faster rate of fall, and by composition of their respective velocities, a slower - and presents him as merely referring to this argument when his opponents claim to vindicate Aristotle by experiment from the leaning tower, dropping wood and lead of equal volume, the lead reaching the ground invariable three cubits in advance of the wood (250). Galileo's attitude is exactly that expressed in the declaration made by Descartes after dealing with the laws of shock: "Et les demonstrations de tout ceci sont si certaines qu'encores que l'experience nous sembleroit faire voir le contraire, nous serions neanmoins obliges d'ajouter plus de foi a notre raison qu'a nos sens."(251)

An inspiration and a model for Galilean science was the successful and apparently a priori method followed by Archimedes in mechanics. Indeed those very features were fixed on in the Renaissance as being of the highest value in it that have been singled out for criticism by a later positivism; thus Mach, from the general thesis that mathematical reasoning is powerless to discover experimental facts, criticises Archimedes' treatment of the lever, for disguising his premises as definitions and axioms claiming to be self evident, when they can be no more than the results of general resumes of and abstraction from common experience (252). Nevertheless it was through the use of similar methods, that Galileo, Stevin and others were able to complete the foundations on which the edifice of all later mechanics has been raised. In contrast to the vast collections of "cases" recommended by Baconian induction, the small number of experiments that was here found necessary is significant, and many of these were, or in the unfamiliar formal exposition could be replaced by, such as were of an entirely "Ideal" character - that it was not necessary to realise, since the result was evident as soon as their conditions were stated. Such an experiment is that by which Stevin demonstrated the principle of the composition of forces, after which "le developpement complet de la statique n'etait plus desormais qu'un probleme purement geometrique et analytique."(253) The general appeal to experience made by Aristotelianism, only analysed in accordance with concepts based on synthesized perceptions, cannot, as these purely theoretical demonstrations can, be granted the title of "experiment," in the same manner in which the term is applied today to scientific technique in even its most empirical aspects.