To substitute "structure" for tendency, or rather to regard "tendency" as a feature arising from structure as neo-Platonists and mathematical philosophers were led to do in the Renaissance, is of course, ultimately, to replace "quality" by "quantity" as the most important feature of nature, and as the determinant in causal operations; and this was widely recognised at the time. But such a step was compatible with Platonic, as it was not with Aristotelian, thought. The Platonists, Aristotle had argued at length, as much as Atomists, reduce all things to quantitative relations (254). The Timaeus gave support to later scientific theory, by rigidly separating matter from form, which was its intelligible and ]i arrangement (thus surpassing qualitative atomism); "Durch diese Scheidung kommt er (Plato) zu dem, was den ersten Stofftheorien der Ionier als Ahnung zugrunde lag: er fasst die Elemente als Erscheinungsformen als Aggregatzustande eines einzigen unwandelbaren und qualitatslosen Grundstoffes. Das ist die grosse Leistung Platons, die der Geschichte der Physik angehort."(255) Matter, argues Plotinus, has not qualities in itself; they are imposed by the mind and recognised in the illumination of it by mind (as air can be made luminous while remaining still itself invisible). Magnitude, limit, and number are what are first imposed upon it before any other forms may be, and its participation in these determines of what other forms it may then be the recipient (256). Physics, Avicenna held, and for this Averroes attacked him, must study bodies not as "being," or "matter" and "form," but solely spatially and as they are in motion or at rest (257). Crescas in his criticism of Aristotle, states many positions held by Renaissance Platonists to whom his work was well known, distinguishes sharply between the "separable" and "inseparable," or the "external" and "inherent" accidents of bodies, and as necessary accidents (as against Aristotle's list of nine) he allows only the mathematical three - quantity, figure and position (258). Such a view, which amounts to a distinction into primary and secondary qualities, is native to a Platonic philosophy, long before seventeenth century thinkers build their new systems in terms of it. Again, the doctrines of absolute space - an existent, geometrically propertied framework of the universe - reaches Newton through Gassendi and Henry More, and had been previously advocated by Campanella, and by Patrizzi, who declared Space to be the first thing created by God extra se, that it was prior to all nature, and therefore the study of mathematics must precede the study of nature, which must be investigated through them (259). Dee also employs such assumptions, and indeed they appear in Crescas who attacks Aristotle for holding that "dimensions" are the property of bodies, claiming they are a property of space, which underlies even the void (which he accepts) and are thus "immoveable."(260)

Proclus had declared that all forms of mathematics could be reduced to the same principles of operation, and these extended to the interpretation of all things that are (261). Dee did not question this. He and other thinkers, who then also advocated the penetration of all natural science by mathematics, are indeed as explicit as Kant: "Ich behaupte aber, dass in jeder besonderen Naturlehre nur so viel eigentliche Wissenschaft angetroffen werden Konne, als darin Mathematik anzutruffen ist," and their supporting reasons, though expressed in a different idiom, may be without undue distortion correlated with his (262). Nevertheless it was, understandably, inevitable that to dethrone quality for quantity and to profess a belief as Dee would seem to have done, that the world ought to be fully describable by means of "Number, Measure and Weight," and that it was only human deficiencies, limitations, and illusions, some correctable, some ineradicable, that prevent man doing so or fully conceiving reality in these terms, was in the sixteenth century frequently branded as mere perversity. For it is all too obvious that all that seems to be ever directly experienced by man takes the form of qualitative percepts, and on experienced quality, on the world, as directly known by sense and synthetic image, solidly rested the Aristotelian physics; it was propped by the whole weight of every man's ordinary every day experience and "common sense," and in accordance with the standards, and from the data these provided, it began its inductions. To hold the opposing view involved a more than extraordinary faith in mathematics. Copernicus' particular form of saving the appearances, for example, could appeal only to that minority who were prepared to make formidable philosophical, and, as it emerged, perhaps religious change ("I have turned the whole frame of the world, and am thereby almost a new Creator" are the words Donne puts into his mouth (263)) subordinate to the doubtful advantage of an aesthetically simpler and more harmonious mathematical account of the tracks of some few distant points of light across the night sky, an account moreover which violated generally accepted physical assumptions in the interests of a purely a priori principle, seemingly largely to be of merely academic interest, the principle of "equal motion." Moreover "The three leading senses," as Lord Brooke pertinently commented, "have confuted Copernicus these many years, for the ye seeth the circulations of the heavens, we feele ourselves upon a stable and firm foundation and our eares heare not from the volutations of the Earth such a black Cant as her heaving rowlings would rumble forth: and yet now if we will believe our new Masters sense hath done as sense will do, misguided our reason."(264)

Dee, while he never in print committed himself to the Copernican system, championed the principles from which it took its source, and made a lifelong defence of the virtues of mathematics. The gradual triumph of such views was to change the whole appearance of the scientific universe, and already in the succeeding century "just those qualities are seized upon as fundamental in nature which it has been found possible by Newton's time to handle by the exact mathematical method."(265) For Galileo a physical object is merely a mass in space and time, all else is contributed by the observer. Man's sensations of things (light is an instance) Descartes argued, are, as compared with their reality, exactly what the mere sound of words are to their meanings, they are the conventional signs of nature (266). "We conceive this atomick Physiology," writes Cudworth, who declares it to be "Mosaical," "as to the essential thereof to be, unquestionably true, viz that the only principles of bodies are magnitude, figure, site, motion and rest; and that the qualities and forms of inanimate bodies are really nothing but several combinations of these, causing several phancies in us," (267) and thus held that "knowledge of the real qualities of the external world must be sought in the application of rational mathematical principles to the world of pure extension," for the principles of such knowledge would be purely native to the mind, clear, universal and necessary, arising from the same logic of God which created the world (268). It is clear from such seventeenth century statements that the claims of intuition have already been largely surrendered; future change and subtilization in the picture of the world which science thereafter takes as "real" become more and more purely a function of the development of an accepted methodology in its inevitable self-unfolding. "Things" Russel has declared "are those series of aspects which obey the laws of physics," (269) and Jeans: "The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete."(270) The fundamental thesis of such a position is apparently the equation of the "reality" of an object or event with the form of description which embodies all the operations necessary for an exact reproduction of it, e.g., that, since an algebraic equation, involving three Cartesian co-ordinates, may be constructed which will fully and exactly describe and permit the perfect duplication of, any surface whatsoever, such a formula could be given, and be regarded as the scientific essence of the Venus de Milo; or again, a piece of music issues from a gramophone as an effect of the curve described by the needle on the disc, therefore "les qualities si complexes que les auditeurs attribuent a la composition et a l'execution appartiennent a cette courbe, puisque cette courbe suffit a les fair revivre."(271) Nevertheless a grave problem arises, "a very little reflection," wrote G.H. Hardy, "is enough to show that the physicists' reality whatever it may be has few or none of the attributes which common sense ascribes instinctively to reality. A chair may be a collection of whirling electrons or an idea in the mind of God, each of these accounts of it may have its merits, but neither conforms at all closely to the suggestions of common sense."(272) The long history of creative achievements of science has perhaps largely palliated the disturbing effect of this dichotomy for the practical scientist, and engendered a confidence which can view the problem as a mere irrelevance, certainly not pressingly requiring resolution. It was far more acute for the would-be Pythagoreanisers of the Renaissance, and the novelty of their methods could not find adequate excuse from past successful use, but was driven for its apology to Philosophy, while to hold that in abstract thought only mathematical demonstration, in physical questions only what could be exactly measured and reduced to quantitative expression, were wholly certain was a thesis which although itself easily defensible, could, especially at this time, only teach either a lesson in scepticism or imply the necessity of some Pythagorean, perhaps mystical metaphysic: and on the whole, the temper of the Renaissance was not one inclined to rest in mere phenomenalist scepticism (273).

The new attention to mathematics was not totally lacking in immediate advantages. Certain hoary problems of the older physics could now be simply solved. A sign of its rapidly increasing connections with, and importance for, the needs of contemporary society is the large number of navigators, instrument makers, surveyors and others of the swelling technical artisan classes, who sympathised with, consulted, received instruction from, or acknowledged a debt to Dee and other theoreticians. But such factors can be easily over estimated (274). Mathematical "discoveries" do not run parallel in time with their applications. The initial stimulus to such research is seldom some known physical need; the "use" of the results, if and when it occurs, appears as a second, almost "independent" discovery (275). But while no practical end may be available as motive for these pursuits, the lack of one is sometimes felt. A modern mathematician writes on this subject: "I suppose that every mathematician is sometimes depressed, as certainly I often am myself, by this feeling of helplessness and futility....It is possible that the life of a mathematician is one which no perfectly reasonable man would elect to live."(276) Mathematicians in the sixteenth century were to some extent perhaps protected from such a "feeling of futility," it is true, by the spontaneous enthusiasm accompanying the discovery in ancient and Arab writings of vast new territories for mathematical exploration, but also very often and more fundamentally by some philosophy, usually of a kind that can be clearly associated with neo-Platonism, which taught that number relationships possessed in themselves transcendental value, and could, as it were, speak significantly to the soul. This had its own dangers.

Such philosophies offered a new criterion of Reality, in judging the external world. There may of course be an indefinite number of possible criteria of this kind, but the use of some or other of them is invariably indispensable, for the term "Reality" insofar as it is thought to denote anything positive or is attached as a supposedly meaningful description to any entity or occurrence, whenever it is employed in contradiction to that which is rejected as "illusion" or "mere appearance" is, while masking as an ontological statement, clearly dependent on value, and merely indicates that some evaluative, discriminatory operation has been made. "The real," Professor Dingle has summarised, "is that to which we attach most importance"(277) - one may compare Nietzsche's "To know is only to work with one's favourite metaphors." In daily life such discriminations in appearance are made to a large extent automatically and by acquired habit; they are necessary to the simplest form of practical thought; for all the aspects and qualities which make up the concrescence of what is distinguished as an "object," even of the most primitive kind, can never be presented to the sense together at the same time, while the presentations offered at differing times will involve contradiction if all are to be unconditionally and uncritically accepted. Distinctions of levels of reality among such various presentations are hence unavoidably always made (the penny is round though it appears elliptical from every point of view but one), and made in accordance with some Idea of the thing, constructed from selected privileged aspects, accepted as representing its permanent, or true nature. These as made by common sense in every day life, produce small scale, local, coherency, but without reference to, and often totally discordantly with, any large scale or overall coherency, for which no pressing need is felt; but to establish such a scheme, to supply a logical account of all appearance, is one of the primary problems all philosophies have found confronting them. But though the universal initial motive of all such systems may well be to "save the appearances," an orderly scheme of thought cannot be constructed which admits all of these on an egalitarian basis, or rather the more it would attempt to do so the more it is driven to explain them in terms of something else, if intelligibility is to be achieved at all (the Heraclitean honey which is both sweet and bitter, is something far different from any experienceable honey merely because it pretends to be only a compound of all actual experiences). "The heart of the new scientific metaphysics," as it was developed in the seventeenth century, Burtt observes, "is to be found in the ascription of ultimate reality and causal efficacy to the world of mathematics, which world is identified with the realm of material bodies moving in space and time."(278) Some illustrations of the relationship of such a view with neo-Platonism have already been given; mathematics exhibited an a priori certainty, so that here the philosopher could be sure as Macrobius, and other Platonists had generally stressed ,that he was dealing with "metaphysical existences that are, while things corporeal appear to be."(279) For the peripatetics mathematics was, whenever it could be legitimately applied, only an approximate, because abstract, account of the external world, whose reality was far differently founded; to Platonists the world was rather an approximation to a mathematical truth by which it was ordered.

One of the strongest arguments at the disposal of the new thought in the sixteenth century was the apparently exact correspondence between the results to be gained from empirical evidence, intuition and logic, offered by Euclidean geometry - a "fact" which could, validly, be considered of the highest importance to philosophy until the mid-nineteenth century (280). But just as the germ of all the mystical Pythagorean theology may have been the factual observation of the correspondence maintaining between the lengths of string producing the notes forming the consonant intervals of the scale and the first four cardinal numbers, so on the observed correspondences between mathematics and the external world Renaissance Platonists founded speculations of a very far-reaching kind, intended to embrace not only the natural, but the intellectual and spiritual worlds. Nevertheless the chief appeal of their theories insofar as they were proposed as guides and criteria for scientific investigation, promising an immense simplification and unification of the view of the universe was, and perforcedly remained so throughout the Renaissance, as opposed to the practical or utilitarian inducements for their adoption which were still almost entirely hypothetical, largely "aesthetic." "C'est un fait curieux, mais indeniable," writes de Broglie, "que ce sentiment sert souvent de guide dans l'elaboration des theories de la philosophie naturelle...Une doctrine qui parvient d'un seul coup a realiser une vaste synthese en montrant l'analogie profonde de phenomenes en apparence etrangers les uns aux autres produit incontestablement sur l'esprit du theoricien une impression de beaute et l'incline a croire qu'elle renferme une grade part de vertite. Il ne s'agit pas ici de la fameuse economie de pensee....Leur beaute [that of such theories] ne vient pas de ce qu'elle sont simples ou compendieuses mais de ce qu'elles nous revelent une harmonie cachee derriere la diversite des apparences, de ce qu'elles nous permettent de ramener la multiplicite des phenomenes a une sort d'unite organique."(281). In this respect the mathematician of the Renaissance, however theoretical, had the advantage over older "scientific" hypotheses. For in the case of colour, if qualities are the ultimate units revealed by analysis, diverse colours can the only be united by considering them as specific cases of some unimaginable, inaccessible and totally uninformative idea of Colour in general; to postulate on the other hand, that the qualitative differences in colours are but irrelevantly contrasting perceptual by-products (being wholly relative to the individual observer) of some purely numerical difference in some one common "stuff," characterised only by its being susceptible of sustaining such patterning, at once effects a more intelligible unification, offers at least a plausible illusion of explanation, and opens a path to experimental investigation (282). Such an interpretation of colours - as for example seen in the rainbow - the ancient atomists made, and Aristotle attempted to refute, attacking them for making the existence of the quality of colours dependent on the sight (and "sweetness" and "bitterness" on the taste etc.) (283). Thus Plato attributes to Protagoras in the Theaetetus the view that colours as appearance are produced by differing "motions" proceeding from variously modified objects impinging on the eye - and perhaps does not dispute the account as far as it goes; he later employs it himself in the Timaeus. For atomism and a mathematical neo-Platonism it was a naturally arising, firmly held view, that qualitative aspects of things were merely results of the character of the sense organ rather than intrinsic properties of things themselves; and as a dogma proved in various ways of great utility to science, long before Johannes Muller in the nineteenth century produced, probably the first, directly empirical, supporting evidence (284).

Dee believed that an abstract scheme could be correlated with the world of appearance, but he regarded this not as a mere fictional construction of the intellect, a scaffolding for thought and artificial device justified only as it led back to and assisted in effecting changes upon phenomena (285), but as a nearer approach to the ultimate realities of things, which lay concealed in the mathematical harmonies themselves. Today it may be necessary to admit opposing scientific theories, as, in practice, being only equally valid ways of ordering information, classifying and predicting experience (286) and to hold that formally "the plurality of equally cogent systems...dispels the indispensability in what is logically prior."(287) But such is a reflectively critical view rather than one which assists to creation, and probably arises only in the presence of several known working hypotheses, rather than at a time, as is the case with the sixteenth century, marked by the endeavour to achieve one at all. (The attitude that some thinkers adopted towards the Ptolemaic, Copernican and Tychonic cosmologies is a partial singular exception, and even so indicates rather the recognition of a supposedly temporary limitation of available evidence, than a positive philosophical position.) Moreover to those who embrace, or extensively work with, particular theories, they appear frequently to be something more than such a view would allow, and seem rather to provide ways of "seeing into" the universe, of showing by abstract concepts or symbolic analogy what things really are, and not merely how they behave. Such were their theories to the Renaissance neo-Platonic scientists. The Timaeus stated the fundamentals of their approach; sense perception involving contradictions and being entirely impermanent and in flux must, uninterpreted, spell only illusion; the true nature of the physical world could only be discovered by considering it as made up of mathematical, not experienced, intelligible not perceived, elements. To Kepler it seemed that the astonishing "harmonies" he discovered in the solar system could only mean that the mathematics of the universe were independent of it, bodies were subject to this mathematical scheme but it did not appear to be a material necessity of body itself (288). The special dangers of such a theory, practically speaking, and these account for many of Dee's mystical "aberrations," were that once it was accepted that all things could be adequately represented in numbers, a view increasingly justified in action, a natural implication seemed to be to reverse the application, and assert that number relationships when they accorded with no observed facts, still had ontological significance, were either themselves transcendental realities, or represented in essence sublime and celestial mysteries. Thus in the Republic it is urged that as all material things fall short of the truth - of which they are distant and imperfect imitations - and therefore also of the reality of number and geometrical figures, one should not employ, for example, mathematics merely to aid the science of astronomy (light brained astronomers who did this, studying the heavens only as phenomena, are reincarnated as birds in the Timaeus) but rather the heavens should be taken as a model to aid man in the study of mathematics for spiritual purposes (289).