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)