Copyright © 2013
Avello Publishing Journal
ISSN: 2049 – 498X
Issue 1 Volume 3:
Principia Mathematica
Newton's Diagrams
Philip Catton
University of Canterbury, New Zealand.
Isaac Newton — theologian, alchemist, raging recluse, icon, mathematician,
practical man, magician, measurer extraordinaire, and most profoundly,
progenitor of physics as exact science — comes into coherent focus for us
as he was in himself only when we recognise why his diagrams in
Philosophiæ Naturalis Principia Mathematica are made by him to be not
merely pedagogical, but instead indispensable to the reasoning itself.
§1. Diagrams and natural philosophy.
When in his published writing Newton reasons out for us what he terms the
mathematical principles of natural philosophy, he does so with diagrams. How
essentially? Supposedly, not essentially at all. Allegedly1 Newton had actually
developed Principia by roundly analytical means, which then he hid. And indeed, the
following facts are scarcely in contention. Roughly two decades previous to the
publication of Principia, Newton had secured for his own use calculus. When Principia
was first published, the calculus was a secret art as yet largely private to him; in the
use of the calculus, Newton was prodigiously adept. If Newton in the Principia had
made direct use of the calculus as we know it now, then this would have simplified the
mathematics there, and reduced the function of diagrams to mere pedagogy; and,
mathematical contemporaries of Newton’s were accustomed to making out the diagram
as not merely pedagogical, but rather as indispensable part to the reasoning itself.
1
Niccolò Guicciardini (1999, pp. 110 ff.) argues that Newton himself promulgated this view, in
order to claim that Principia evidences his priority over Leibniz in the discovery of the calculus.
Guicciardini’s scholarship is roundly magisterial, yet I will resist subtly, below, his perspective on this
point.
Moreover, taken all together, these facts precipitate the rumour that Newton sought in
Principia as simply the best way to show his mathematical genius off to his
contemporaries. That alone supposedly is why diagrammatic reasoning is left essential
within that work.
In fact, however, the rumour that Newton faked his dependence upon diagrams
nourishes a litany of confusions, confusions that one will endeavour here to identify
and redress. It is true that in Principia Newton stays, and stays quite deliberately, the
inchoate symbolism of his pregnant but still private calculus. Instead of producing
there any calculative, algebraical, symbolically-styled reasoning about what is
infinitesimal, Newton instead literally draws out for us in diagrams what is momentary
about motion. Yet these executions of his are to his mind analysis at its surest.
Newton in no way hides analysis from us in Principia. Rather Newton practices
analysis there boldly, in the diagrammatic form he is deeply disposed to think most
apt.
The understanding that the surest way of reasoning mathematically is essentially
diagrammatic is deep-lying within the tradition of mathematical inquiry coming down
from the Ancient Greeks.2 Newton made a difference to what mathematics can
comprehend, not by departing from the Greek view of the diagram but by compounding
2
For a more thorough study of this point in both philosophical and historical perspective see
Catton & Montelle 2012.
commitment to that very view. For, Newton added to Greek practices in diagrammatic
reasoning his own new flourish, of drawing out what is momentary about motion.
Within the diagram-based tradition of mathematical thinking, there had long existed an
understanding of analysis. Newton cites in this connection the Greek mathematician of
late antiquity Pappus of Alexandria (c. 290 – c. 350 C.E.). When Newton represents
within diagrams what is momentary about motion, he departs not at all from Pappus’s
view of what analysis comes to diagrammatically. We connect with Newton therefore
when we note the following. Analysis, on Pappus’s understanding of it, is a creative
process, often even productive of new concepts. Pappus-styled analysis is not merely
symbolical or calculative, and does not reduce what is complex to its simpler elements;
instead it builds a novel insight. Pappus writes in his Mathematical Collection (c. 340
C.E.) as follows about analysis and synthesis:3
Analysis … takes that which is sought as if it were admitted and passes from it
through its successive consequences to something which is admitted as the
result of synthesis: for in analysis we admit that which is sought as if it were
already done and we inquire what it is from which this results, and again what
is the antecedent cause of the latter, and so on, until by so retracing our steps
we come upon something already known or belonging to the class of first
principles, and such a method we call analysis as being solution backwards.
But in synthesis, reversing the process, we take as already done that which
was last arrived at in the analysis and, by arranging in their natural order as
3
Quoted, as it (very commonly) is quoted, as in Heath’s Euclid [1956], volume 1, p. 138. For
more, see Pappus of Alexandria [1986], pp. 82 ff.
consequences what before were antecedents, and successively connecting them
one with another, we arrive finally at the construction of what was sought; and
this we call synthesis.
Moderns consider with some bewilderment what Pappus could possibly have
meant by these words, but in the context of diagrammatic reasoning his meaning
becomes plain. In this connection, the way in which, to our ears, ‘analysis’ links
with the logical, calculative, algebraic, or symbolical, becomes spurious. Bearing
in mind that in Pappus’s day algebra was not yet invented and so had in no way
been brought to geometry, it is essential for us to think that Pappus means by
‘analysis’ neither more nor less than he says he means by it. Analysis concerns
diagrams. To consider Pappus’s meaning, here is an illustration:
The illustration is the simplest that I can think of to make Pappus’s meaning
clear, and concerns the Euclidean demonstration (Elements I.32) that the internal
angles of a triangle sum to a straight line. Synthetically the proposition is
demonstrated for a given triangle by “producing” one of the sides and constructing the
parallel to another, as above, observing then the equality of the exterior angle with the
sum of the two opposite internal angles and thus that the sum of the three internal
angles is a straight line.
By analysis however one could discover more or less directly how such a
synthesis should be accomplished. The trick, as Pappus says, is “to take that which is
sought as if it were admitted” i.e. to assume that the exterior angle is the sum of the
opposite internal angles so that the three internal angles will sum to a straight line:
One then quickly finds oneself inserting the parallel to the opposite side, and then the
needed synthesis becomes clear.4 One might even say that from the phenomenon that
the internal angles of triangles do sum to a straight line, by analysis one deduces the
cause of that’s being necessarily so. (Pappus himself in the above quotation even
introduces causal talk to us this way.)
4
Admittedly, how quickly depends on whether it is as a first or second step that one inserts the
angles as depicted, which ordering alone is helpful towards the discovery of the starting point for
synthesis.
Every phase of analysis in Newton’s Principia can be thought of as deducingfrom-a-phenomenon-its-cause. The converse phase of discussion, synthesis, deduces
from-a-cause-the-resulting-phenomenon. Principia contains in spades both these
phases. To illustrate this, let us consider the very first propositions of Book 1 of
Principia, Propositions 1 and 2 (which are converses of one another, and so
respectively consider the converse phases of synthesis and analysis). These
propositions and their accompanying diagram are perhaps more often discussed than
any others of Newton’s. Yet our present purpose is to link them to Pappus and that
purpose is novel. We need the Pappus distinction between analysis and synthesis in
order to understand Newton fairly. Newton writes:
• Proposition 1 The areas which bodies made to move in orbits describe by radii
drawn to an unmoving centre of force lie in unmoving planes and are
proportional to the times.
• Proposition 2 Every body that moves in some curved line described in a plane
and, by a radius drawn to a point, either unmoving or moving uniformly
forward with a rectilinear motion, describes areas around that point
proportional to the times, is urged by a centripetal force tending toward
that same point.
Newton reasons to these conclusions in the following diagrammatic fashion:
Consider first the degenerate case where the sun-centred (S-directed) accelerative
force is zero. This, degenerately, remains one way that an accelerative force could
obtain in the direction of S — the force in question is simply zero in magnitude. In that
case a body (a planet, say) that begins at A and sweeps to B across an initial moment
will carry on inertially and sweep to c in a second equal moment. Notice that in this
reasoning we are led to draw out for ourselves what is momentary about motion. We
reason that AB and Bc are equal, the motion of a body from A to B and then from B to
c across successive moments being an inertial one. We thereby draw into connection
with the momentary significant geometrical structure.5 Proceeding as discussed within
5
We give effect, in other words, to a transcendentalist faith. By a drawing out for ourselves,
grossly, of something in fact infinitely finer than we can draw, we admit the grossness of our practical
powers and the finitude of our intellect. Yet we assume that nevertheless, by our reason we are made in
the image of God. Just as an unavoidable human grossness means that we can only ever deal
approximately with breadthless lines or dimensionless points, so also, when we draw out what is
momentary about motion, we do so grossly. The expectation however is that the ideal to which we are
the present diagram, we readily recognise the equality of the areas swept out about S
during the two successive moments of time. For triangles SAB and SBc can be made
out as having equal bases (viz., AB and Bc) from each of which the perpendicular
height to S is the same. Therefore, triangles SAB and SBc are equal in area. Yet on
next considering what will happen if from one moment to the next the planetary body is
deflected by the sun (at S) — say from B not to c, but rather to C — we then make out
geometrically the following. The planetary body, thus deflected, will sweep out in the
second moment an area equal to that swept out in the first moment, if and only if the
triangles SBC and SBc are equal in area. For, we already know that the area of
triangle SBc is identical to that of triangle SAB. And yet as the triangles SBC and SBc
share the base SB, they are equal in area if and only if they are constructed between
parallel lines, that is to say, if and only if Cc parallels SB. But this will hold if and only
if the deflection that the planetary body has suffered is in the direction of S, the sun.
Thereby, in one fell diagrammatic swoop, Newton convinces us, on the one hand,
that if — or to the extent that — Kepler’s Second Law (the law that planets sweep out
equal areas in equal times) is true, then the planets participate in an acceleration
always directed to the sun, and thus are subjected continually to a sun-centred force;
and, on the other hand, that by being subject to a sun-centred force, the planets are
required to sweep out equal areas in equal times about the sun, and thus satisfy
beholden in these practices can itself be innocent of this grossness of ours. The ideal itself can be risen
higher than any possible accomplishment by us.
Kepler’s Second Law. The first phase of this is analysis: it deduces from a
phenomenon (that of Kepler’s Second Law holding) its cause (the force upon the
planets being sun-centred). The second phase is synthesis: it deduces from the cause
(the force upon the planets being sun-centred) the phenomenon to be explained (that
Kepler’s Second Law holds). Synthetic reasoning was
traditionally received as mathematically superior, in as much as synthetic
understanding is the endpoint. Perhaps this is why Newton orders synthesis first (as
Proposition 1). A larger but related reason for Newton’s prioritising the synthesis
could concern God. Prowess in analysis is not far separated from being able to fathom
what is so very ultimate in the order of things that it requires a supernatural cause.
That would mean that prowess in analysis is not far separated from being adept in
magical arts, a point that we will pause to consider again, later. Be that as it may
however, Newton is notably completely up-front about delivering within Principia the
analytic phase (as Proposition 2), not only the synthetic one (as Proposition 1).
Newton does not hide the analysis, but simply performs it diagrammatically.
Newton makes diagrammatic performance essential to the reasoning itself.
Present-day mathematicians and philosophers struggle to comprehend that this is
possible. They consider diagrams to represent mere visualization. Diagrams seem to
them at best of pedagogical or heuristic value. This perspective reflects their
commitment to symbolical reasoning. Formalism or logicism as one’s philosophy of
mathematics are pinnacle expressions of this commitment. The age of computers just
is an age of formalising and logicising and to that extent of formalism and logicism.
This is not to say that such philosophies pan out. We contemporaries need in any
case to recall that our perspective is new and even rather peculiar. At one time the
practice of mathematics made diagrams or diagrammatic performances essential to the
reasoning itself.6 And to participants in of mathematical inquiry back then, diagrams
certainly did not seem a mere form of visualization.
The visual cognitively pales compared to the practical. A diagram relates to the
practical. In Euclid, insight is produced when practice becomes intuitive, when reason
commends as necessary every step of a sequence of steps through the overall
6
Catton and Montelle 2012 marshals the case for this contention.
performance of which one achieves an end. Quite typically, propositions in Euclid are
things that it is proposed to do, that is to say, practical ends that Euclid forms for
mathematicians to make their own. And then the demonstration is a rationally perfect
execution of diagrammatic steps that most excellently achieve the desired end. Just as
‘to diagram’ is a richer verb than ‘to picture’ and embodies a practical connection,
geometers garner insights not merely visually but by practically drawing out into
diagrams the conditions for a kind of perfection of agency. From what alone allows the
relevant manual practices to achieve their ends rationally harmoniously arise
constraints upon the form of visualisations. The conditioning is not the other way
around, from the form of the visual upon the possibilities for the practical. It is very
much from the practical upon the visual. (We must mature significantly as agents
before we truly see.) To make rationally perfected a practical grasp of space is to
fathom constraints upon possible motion. We are brought by Euclid to understand
spatial structure in terms of constraints upon possible motion. This is what prioritises
the practical to the visual, and argues that the visual can have its form only by virtue
of practical conditions upon the motions that are possible.
§2. Vicissitudes of the diagram and of analysis.
Many philosophers expect that reason more or less centrally is logic. Logic is
the study of the calculative function of reason. In the context of calculation, a diagram
is but a heuristic aid. Any diagram simply helps initiate the thinking, thinking that will
then be entirely borne by the symbolism. The diagram is then for pedagogy; it
becomes propaedeutic to the reasoning rather than any principal part of it. We live in
an age that much links analysis and calculation. Given this identification, which does
however step us far away from Pappus, or thus far away from the whole ancient
mathematical culture of diagrammatic reasoning, and thus far away from Newton, we
may regard logic as the study of the analytic function of reason. Few present-day
philosophers would complain against this understanding. But recall that logic leapt
forward late in the nineteenth century or even early in the twentieth. Logic only lately
has achieved the power that it now possesses. By the discovery of just such power,
the potentialities of calculation became fully charted. It is by the discovery of their
scope that we have been propelled into a computer age. Computers epitomize the new
power that logic has attained. Newton, we shall see, stands as a signal influence upon
this development. But he was himself of the earlier age.
For a while (early in his career), Newton himself helped to make out his methods
of mathematical analysis as calculative. Later, he returned himself determinedly to
diagrammatic forms of rumination and professed them to be superior. Newton
participated to the extent that he did in the development of calculative methods
because just such a shift in the culture of mathematical rumination was on the make.
Within it eventually issues of the rigor or calculus would be addressed. The rigorizers
of the calculus would clarify the mode for logical reasoning itself. Via their
accomplishments humans would be propelled into a computer age. To the extent that
Newton’s mathematical analysis is calculative, it concerns infinitesimals. In what he
called the ‘analytic method of fluxions’, Newton introduced a symbol for just such
quantities. Yet infinitesimals have seemed doubtful. Empiricist philosophy in
particular undermines any conviction that the term ‘infinitesimal’ is even meaningful.
Was there anything really to which Newton’s symbol might refer? In decades not long
after Newton, empiricist philosophers such as Berkeley and Hume were significant in
sounding alarm concerning infinitesimals, though neither of these figures was himself
even remotely a mathematician. Neither remotely possessed strong enough logical
theory to license what each himself proposed should be understood by ‘mathematical
analysis’. As empiricists these philosophers had an understanding of analysis that
differed from Pappus’s. Analysis to them was about breaking things down to elements.
Analysis was not the creative diagrammatic performance that Pappus describes. For
empiricists, analysis is about finding original elements that enter our situation merely
as givens. The contentfulness of such elements would need to be from how they are
given to the mind. Yet in that case, infinitesimals are entirely suspect. Newton’s
symbol for the infinitesimal seems liable not really to refer. For how could what is
infinitely small have any content before our minds?
Neither Berkeley nor Hume himself possessed even the remote beginnings of a
workable philosophy of mathematics. The worries of Berkeley and Hume about
infinitesimals generalise easily to breadthless lines and extensionless points, and thus
to the conditions for Euclid’s geometry to have become exact science. How could the
notion of a truly breadthless line, or a truly extensionless point, have any content
before our minds? To answer this question positively one must look away from
empiricist accounts of meaning, and recognise a connection between meaning and
ideals for practices. Truly, meaning consists not in a content or in a way of that
content’s being given to the mind. Meaning links to ideals — for example, to ideals to
which our practices with straight edge and compass are beholden, when we fashion
geometry an exact science. What words such as ‘point’ or ‘line’ mean depends not
on the images we create on paper or in our heads, but rather upon our practical
directedness, and what perfection in those practices would be. Whether mathematical
analysis possesses integrity as Newton deploys it in Principia reduces not at all to
whether Newton’s symbol for the infinitesimal has a referent, and depends not at all on
whether of this referent a clear and contentful image can be delivered to the mind.
By the nineteenth century however, confidence about spatial intuition was much
unravelled for a variety of reasons. Critical concerns about the meaningfulness of
mathematical language by empiricist philosophers were but one set of such reasons.
Mathematicians became concerned for many other reasons besides to reduce to zero
any reliance upon intuition in their reasoning.
Thus I have scarcely begun to detail what pushed the diagram from the fore of
mathematics to the rear, or rather, from being indispensable to the reasoning itself in
mathematics to being but a pedagogical aid. However, for my present purposes it
suffices to observe that, two centuries after Newton, his calculus itself spurred
reconsideration of the very nature of reason. (Two centuries is a long time, and yet,
even by the present day, three and a half centuries after Newton first developed the
calculus, the ruminations are by no means concluded.) Nineteenth-century efforts to
rigorise mathematical analysis culminated, around the beginning of the twentieth, in
the advancement in logic above discussed. After this great advancement in logic there
followed bold reinvention of the philosophical spirit itself. Not as a coincidence with
this but rather as a consequence of it, the possibility of the computer was also
unleashed. Linking in no small measure back to Newton are reasons why we live in a
computer age, and also why we have just had a century that was (most especially in
English speaking lands) heyday for analytic philosophy.
Newton himself I make out as closer in his conception of reason itself to
rationalist philosophers than to empiricists. Rather than linking reason with logic
(which is an empiricist trait), Newton significantly resembles Plato, in instead linking
reason with rational self-determination in practice. Euclid, for example, mobilizes
reason to render what is practical as in its every step harmoniously necessitated. Let
me briefly mention some consequences, philosophically, of philosophy’s by now having
stepped so far away from this view. Today’s philosophers are poles apart from Plato
when (as is utterly common) they call necessity — for example, mathematical necessity
—a modality. For they mean by that, that necessity is a mode of being-true. Indeed
with different emphasis we better expose their perspective: they mean to say that
necessity is a mode of being-true. These philosophers of the present day rather
overlook the practical, so that, in pondering necessity, they consider only the
theoretical. The so-called ‘analytic’ orientation in philosophy especially encourages
such forgetfulness of the practical and a resulting overconcentration on the
theoretical. In order to limit necessity to the theoretical, a philosopher needs what is
assertoric to take over the universe of reason (as it were). Analytic philosophers
typically willingly stage just such a take-over. Necessity is but a way-of-being-true
that some assertions have and others lack, they claim. What is imperatival rather than
assertoric, and thus what is practically rather than theoretically necessary, they would
simply like to abjure, or explain away. Yet one thing we can learn by considering
Newton, and his mathematics, and indeed the fit in his mind of that mathematics to
physics, is that the mere theoretical conception of necessity (as a modality) cannot
justly illuminate necessity by his lights.
In any case if we consider diagrams merely as visualizations, then we fail to bring
out a certain duality of the diagram that much fascinated the Greeks. Euclid purposes
the diagram to portray a practical deed. Yet the diagram also epitomizes timelessness
of form for perfectly executing that deed. If, fundamentally, a proposition for Euclid is
something practical that it is proposed to do, and moreover rational demonstration for
Euclid means moving straightedge and compass perfectly through specific ideal steps in
order to do what has been proposed, then for Euclid proposition and demonstration
both involve motion. But the diagram epitomizes timelessness. Therefore the diagram
sports a profound duality between temporality and timelessness.
In order for proposition-and-demonstration mathematics, such as Euclid’s in his
Elements, or Newton’s in Principia, to be science, it must be rationally perfectable,
and in order for it to be rationally perfectable, it must sport an ideality that transcends
what humans can possibly accomplish (and that rather, in its transcendental finality for
all cognisers, rises quite beyond what is merely immanent or human). The Greek route
to such ideality is through that which is drawn. Newton adopts the same path as his
own and steps even further along it. Inhering in the actual materiality of straightedge, compasses, and geometer, there are doubtless impediments to performative
perfection in the execution of diagrams. But, to make geometry a science, you ignore
the impediments. You ignore, for example, that if two drawn curves truly intersect
punctually, then it will be infinitely involving just to push a straight edge up, or set
down the point of a compass, exactly there. Over against the actual impossibility, you
just do it, ignoring the imperfections that must actually inhere in your practice. By
drawing out for us what is momentary about motion, Newton redoubles that kind of
ideality that the diagram must have, but at the same time also redoubles the
connection, via the diagram, between insight, and the grasp of a timeless constraint
upon motion.
Thus Newton states at the outset in Principia that whereas “mechanics is
distinguished from geometry by the attribution of exactness to geometry and of
anything less than exactness to mechanics … [y]et the errors do not come from the
art but from those who practice the art”. Consequently, in the limit of a transcendent
perfection of the artisan, geometry and mechanics merge as one single science after
all. “Anyone who works with less exactness is a more imperfect mechanic, and if
anyone could work with the greatest exactness, he would be the most perfect mechanic
of all” —namely, God (Principia 1999 [1687], Author’s Preface to the Reader; p. 381).
Newton’s own human efforts with diagrams in Principia concern of course how
the world moves. Yet for Newton that is a delving not just down towards the
exactitude of which God alone is capable (although it is that). Also it delves down
towards the actual on-going work of God. By diagrams themselves humans fathom
formal timelessness within the manner by which material things change. As God is
geometer, He brings timeless form to how the world goes.
§3. Analysis of continuous change: Newton’s artful rumination upon
forces.
Some background: in some further reasoning of Newton’s that I next briefly
discuss, Newton significantly uses Galileo’s law of free fall (that in freely falling motion
ignoring resistance the vertical distance that a body traverses from rest is proportional
to the square of the time taken, irrespective of the body’s mass or state of horizontal
motion). Newton significantly uses Kepler’s three laws of planetary motion. These
are: First, that secondaries orbit their primaries (e.g. the planets orbit the sun) in
elliptical orbits, with the primary at one focus; Second, the law of areas, above
described — that secondaries sweep out about their primaries equal areas in equal
times; and Third, that among the secondaries of a common primary, such as among the
planets relatively to the sun, R3/T2 is a constant, where R is a secondary’s average
distance from its primary, and T its orbital period. Newton also very significantly uses
his own three laws of motion: First, the law of inertia, Second, F = ma (which Newton
himself hardly expresses thus algebraically!), and Third, the law of the equality of
action and reaction. Newton, Lucasian Professor of Mathematics, in that capacity also
stands deeply immersed in the tradition of diagrammatic reasoning that propagated
down the centuries from Euclid. It is true that Viète and Descartes had produced
symbolically-styled forms of geometric reasoning and Newton helped compound the
power invested in these inchoate symbolisms. I call the symbolisms inchoate because
they long antedate any adequate theory of a numerical continuum, yet they are boldly
beholden to some such future success. To Newton’s mind,7 symbolically-styled
reasoning is untrustworthy, whereas diagrammatic demonstration is by contrast
entirely apt to the qualities of mathematical necessity and insight. This assessment by
Newton of the situation in his own day was in fact perfectly apt, and fair.
Consider then as far as this sketched background makes possible Newton’s
reasoning, early in Book 1 of Principia, surrounding the diagram for his Proposition 6
there:
7
See for example Newton on geometry and algebra in Universal Arithmetic, tr. J. Raphson,
London, 1769 (2nd edition), pp. 465–470.
Here, Newton draws out as PR what the momentary motion would be of a planet that is
not acted upon at all by the sun and instead moves inertially, and draws out as PQ
what, by virtue of the sun’s action upon the planet, that planet’s motion actually
comes to in the self-same moment. QR is taken to visualize the sun’s deflecting
impulse upon the planetary body at P, and, because sun-centred, is thus parallel to
SP. Under constraints imposed by (i) Galileo’s law of free-fall, which becomes
accurately or very nearly true in the momentary since then the distances to S are
approximately all the same, (ii) Newton’s own laws of motion, and (iii) the knowledge
from Proposition 1 that the area SQP is proportional to the time taken, Newton walks
us through various geometrical considerations that together imply that Kepler’s First
Law obtains if and only if the force that QR helps us to visualize is inversely as the
square of the planet’s distance SP from the sun. Thus in their analytic phase these
reasonings deduce from the phenomenon of ellipticality the cause of that very
ellipticality, namely that there is a sun-centred, inverse-square force. In their
synthetic phase these reasonings deduce from the cause (viz., there being a sun-
centred, inverse-square force) the phenomenon (viz. that planets’ orbits are elliptical,
as stated by Kepler as his First Law).
Orthodoxy holds that rigor was brought to Newton’s calculus only by the
imposition of a theory of limits. It is even said8 that Newton points us in Principia to
just such limit-theoretic understandings. I am cautious about this view. Limit
theory’s whole purpose becomes, in the nineteenth century, to suppress intuition in
mathematics and heighten the purchase of logic. Dealing limit-theoretically with
motion is in my view the involute of Newton’s approach. Newton draws out for us
what is momentary about motion, whereas limit theory instead carries what it analyses
back away from us, to just beyond the horizon of our explicit attention. In this way it
much nullifies intuition and heightens the purchase of algebra and logic. The fruit of
this effort is the nineteenth-century delivery of a coherent, cogent theory of the
numerical continuum. The numerical continuum must contain all its limit points, that
is to say, any limit of a sequence of ratios of numbers must also be a number. Then
the workings of the calculus can be made out as secure. This way to ‘rigourise’
mathematical analysis has significantly dominated the mathematical attention of the
twentieth. Again, the patterns of mathematical attention that dominate in the present
day form obstacles to our understanding Newton. Significantly it was Newton’s
calculus that the nineteenth-century mathematicians sought to rigourise; but this is
8
Among others, by Guicciardini (1999, 2009), whom I discuss below.
one instance of many where the enormity of Newton’s effect on us makes problems for
our today understanding Newton as Newton in his own century was.
Guicciardini’s in-most-ways-magisterial 1999 is plain in its pronouncement that
Newton himself treats of limits. Commonly the diagrams of Newton’s that I have
above introduced are in the present day thought unreadable except in such a way as
implies the ‘taking of a limit’. A first, apparently verbal move against this conception
which I will argue is actually robustly telling, is to note that Newton writes not of
‘limits’, but of ‘first and ultimate ratios’. There is something further to spot: the
ratios are not numerical, but rather are of geometrical quantities. I shall next quote
from Guicciardini, 1999, about this (pp. 43-44):
Newton … points out that the method of first and ultimate ratios rests on the
following Lemma 1:
Quantities, and also ratios of quantities, which in any finite time constantly tend
to equality, and which before the end of that time approach so close to one
another that their difference is less than any given quantity, become ultimately
equal.
Newton’s ad absurdum proof runs as follows:
If you deny this, let them become ultimately unequal, and let their ultimate
difference be D. Then they cannot approach so close to equality that their
difference is less than the given difference D, contrary to the hypothesis.
This principle might be regarded as an anticipation of Cauchy’s theory of limits, but
this would certainly be a mistake, since Newton’s theory of limits is referred to a
geometrical rather than a numerical model. The objects to which Newton applies his
‘synthetic method of fluxions’ or ‘method of first and ultimate ratios’ are
geometrical quantities generated by continuous flow.
Here I dispute not Guicciardini’s resistance to the idea that in Newton there is
anticipation of Cauchy’s theory, which resistance is perfectly cogent, but rather
Guicciardini’s crediting to Newton any ‘theory of limits’ at all. My objection is
subtle. The quantities of which ratios, first and ultimate, are to be considered, are as
Guicciardini points out geometric. (Even in some ways by Guicciardini himself,
translations of Newton’s Latin expressions for chord lengths, areas etc. ease us over
to thinking numerically, as for us is the familiar way to think, when in order to grasp
Newton’s meaning we need to be thinking geometrically; so, holding on to
Guicciardini’s insight here that Newton is not Cauchy can be difficult for us.) The
“ultimate ratio” of geometric quantities is still geometric, yet in a way that no human
can depict; and it remains a ratio. Cauchy helps us towards the conception that the
limit of a convergent sequence of numerical ratios just is, not a ratio, but a number.
That is a different point of view. What is ultimate for Newton is not thus numerical,
but is rather irreducibly a ratio of geometrical quantities, quantities infinitely smaller
than we can draw or depict, yet geometrical for all of that. We can draw out only
grossly what is thus ultimate, but that is no more to say that the ultimate is other than
what we draw out, than we would call it impossible to draw a geometrical line, or to
diagram a geometrical point. Properly to grasp Newton’s intent about the drawing
out, is I believe to see in limit theory but the involute of Newton’s approach.
Now, as is well known9, the Proposition 6 reasoning of Newton’s above,
concerning the way that (with other considerations) ellipticality of orbits implies an
inverse-square distance-dependency of the force, is the less compelling so far as
human knowledge is concerned, because astronomers in Newton’s day could not attest
empirically that Kepler’s First Law is in all respects “accurately or very nearly true”.
To determine empirically the exact figure of a planet’s orbit relatively to the sun is
prodigiously difficult. So while the astronomers of Newton’s day could empirically
attest to the high accuracy of Kepler’s Second and Third Laws, in respect of the First
Law they could attest to the high accuracy only of a special implication of the First
Law, viz., that planets will approach the sun nearest always on the same side of the
sun with respect to the fixed stars. So in the order of our own knowledge, Newton
quite deliberately privileges, later in Principia, some more special considerations which
(following W. Harper, but interpolating my own emphasis on analysis and synthesis) I
spell out as follows:
9
See Curtis Wilson, 1970.
Note incidentally the remarkable consilience within Newton’s reasonings: by two
very different measurement inferences from quite utterly disparate astronomical
phenomena (quiescence of perihelia, harmonic law), Newton is able to deduce one and
the same theoretical conclusion, viz. that the distance variation of the sun-centred
force acting on the planets is inversely as the square.
Given that these later considerations are, within the order of our own knowledge,
more telling, why does Newton set out so early in Principia the diagram
with which (by diagrammatic analysis) to link ellipticality and the inverse-square
character of the force? In Newton’s order of exposition, an analysis occurs early that
is scarcely as telling for our knowledge as are the exacting demonstrations that occur
in Principia later on. Newton therefore departs in his order of exposition from the
order of human knowledge. Why?
Much magic, not a little theology, and yet at the same time the ignition of
science itself, all devolve upon the answer that is needed, here. Rich clues to what is
going on with Newton, Newton himself delivers to us in the very Preface to Principia.
Here, Newton mentions Pappus six words in. The opening paragraph of the Preface is
as follows:
Since the ancients (according to Pappus) considered mechanics to be of the
greatest importance in the investigation of nature and science and since the
moderns  rejecting substantial forms and occult qualities  have undertaken
to reduce the phenomena of nature to mathematical laws, it has seemed best in
this treatise to concentrate on mathematics as it relates to natural philosophy.
The ancients divided mechanics into two parts: the rational, which proceeds
rigorously through demonstrations, and the practical. Practical mechanics is
the subject that comprises all the mechanical arts, from which the subject of
mechanics as a whole has adapted its name. But since those who practice an
art do not generally work with a high degree of exactness, the whole subject of
mechanics is distinguished from geometry by the attribution of exactness to
geometry and of anything less than exactness to mechanics. Yet the errors do
not come from the art but from those who practise the art. Anyone who works
with less exactness is a more imperfect mechanic, and if anyone could work
with the greatest exactness, he would be the most perfect mechanic of all. For
the description of straight lines and circles, which is the foundation of
geometry, appertains to mechanics. Geometry does not teach how to describe
these straight lines and circles, but postulates such a description. For
geometry postulates that a beginner has learned to describe lines and circles
exactly before he approaches the threshold of geometry, and then it teaches
how problems are solved by these operations. To describe straight lines and to
describe circles are problems, but not problems in geometry. Geometry
postulates the solution of these problems from mechanics and teaches the use
of the problems thus solved. And geometry can boast that with so few
principles obtained from other fields, it can do so much. Therefore geometry is
founded on mechanical practice and is nothing other than that part of universal
mechanics which reduces the art of measuring to exact propositions and
demonstrations. But since the manual arts are applied especially to making
bodies move, geometry is commonly used in reference to magnitude, and
mechanics in reference to motion. In this sense rational mechanics will be the
science, expressed in exact propositions and demonstrations, of motions that
result from any forces whatever and of the forces that are required for any
motions whatever. The ancients studied this part of mechanics in terms of the
five powers that relate to the manual arts [i.e., the five mechanical powers] and
paid hardly any attention to gravity (since it is not a manual power) except in
the moving of weights by these powers. But since we are concerned with
natural philosophy rather than manual arts, and are writing about natural
rather than manual powers, we concentrate on aspects of gravity, levity, elastic
forces, resistance of fluids, and forces of this sort, whether attractive or
impulsive. And therefore our present work sets forth mathematical principles
of natural philosophy. For the basic problem [lit. whole difficulty] of
philosophy seems to be to discover the forces of nature from the phenomena of
motions and then to demonstrate the other phenomena from these forces. It is
to these ends that the general propositions of books 1 and 2 are directed, while
in book 3 our explanation of the system of the world illustrates these
propositions. For in book 3, by means of propositions demonstrated
mathematically in books 1 and 2, we derive from celestial phenomena the
gravitational forces by which bodies tend toward the sun and toward the
individual planets. Then the motions of the planets, the comets, the moon,
and the sea are deduced from these forces by propositions that are also
mathematical. If only we could derive the other phenomena of nature from
mechanical principles by the same kind of reasoning! For many things lead me
to have a suspicion that all phenomena may depend on certain forces by which
the particles of bodies, by causes not yet known, either are impelled toward
one another and cohere in regular figures, or are repelled from one another and
recede. Since these forces are unknown, philosophers have hitherto made trial
of nature in vain. But I hope that the principles set down here will shed some
light on either this mode of philosophizing or some truer one.
Armed, in part, with the above thinking concerning Pappus and diagrammatic
reasoning, One gathers from this Preface conclusions that are largely novel to
Newtonian scholarship. It is true that in recent decades Newton scholarship has
significantly far-redressed an earlier resistance to the idea that Newton could have
produced deductions from phenomena at all. Philosophers of science (who evidently
had not read their way even the tiniest distance into Newton’s Principia) for some
decades treated with staged derision Newton’s concept of deductions from
phenomena, signaling that very concept as indication (and they effected to find others
as well) that Newton was philosophically feeble. Yet by now Principia has been better
investigated. William L. Harper, George E. Smith, and many others, have far
advanced our understanding of both the mathematical integrity, in Principia and
elsewhere, surrounding Newton’s deductions from phenomena, and the exemplariness
as science of just such measurement inferences. Yet within this new literature I know
of no-one who duly emphasizes Pappus, or the diagram with its practical connection,
or the theological suggestion that Newton advances in his Preface, that, by his
diagrams in Principia, Newton delves as deeply as a human might into the working out
by God of the movement of the world. Thus Proposition 6 achieves as synthesis deepgoing explanation: the cause of the orbit’s ellipticality is the inverse-square distancedependency of the force. As Pappus-styled analysis, that yields the starting point for
just such synthesis, the reasoning treats as given that the orbit is elliptical, and
deduces the cause, viz., the inverse-square distance-dependency of the force.
Many contemporary scholars of Newton seem convinced however that Newton
suppressed analysis from his Principia. To draw as they do a blank about analysis in
Principia and the relation that the Preface tells us is there between analysis,
deductions from phenomena, the movement of the world and God, all represents a
grave misstep in my view. How large a misstep, I believe can be measured by
Newton’s rages.
§4. Magic, natural philosophy and God.
Newton rages like no other, at precisely those ordinary men in his midst who
dare distrust his delvings into the ultimate nature of things. Who is Robert Hooke (for
example), to question Newton’s penetration mathematically of the phenomena of light,
or Newton’s mathematical priority in the discernment of inverse-square centripality of
the celestial force? Newton’s rage against Hooke is the stuff of wizardly combat. A
magician delves deep into hidden connections in the universal order. This confers
upon the magician powers that ordinary mortals do not possess. Question the
magician’s deep-going discernment and you question the very art. Newton is, and
aims to be. thoroughly a magician of the dark arts of alchemy and the occult. His
rages tell us all. But Newton at the same time redefines aspirations for knowledge, in
ways that have precipitated us into a scientific age. To this, mathematics is key. A
scientist delves deep into hidden connections in the universal order. This confers
upon the scientist powers that ordinary mortals do not possess. Truly we need not to
read the magic out of Newton’s Principia, but rather to read magic into the
subsequent science.10 Remove the dross and render it serious, and magic takes the
form that we today recognise as science. Mathematics is key to this process. And
similarly we need not to read religion out of Principia, but to understand instead simply
that, in its every diagram, Newton seeks after a closest human approximation to the
intellection of God. Moreover, something like the transcendent standard that Newton
presupposes is presupposed still by science in the West.
Why people understand Principia as serious science is much because of its
mathematical brilliance. To call Principia exemplary is both true and deeply ironic.
Scarcely anyone can even read the book, let alone directly emulate it. Yet it shifted
fundamentally what humans would consider epistemic knowledge even to be. Its direct
influence is less on the understanding than on the understanding of the understanding.
Principia profoundly reshaped humanity’s aspirations for thought. Indeed, the work,
despite its scarcely ever being read, is often picked, rightly as I believe, as the most
influential single work of its millennium.
10
To write these words, I electronically transferred files between home and office and depended
on LCD screens and laser printers. Such technologies are for all of that pretty profound realisations of
the ambitions associated formerly with the magical arts.
Shocking to the world it was therefore to discover Newton’s ‘magician’ side.
Scholars have nonetheless discovered in Newton’s papers vast evidence of alchemy
and passionate religious intellection. Even among the scholars who have most studied
these dimensions of Newton however, and certainly among those who base their
impression of Newton mostly on the published works, the effort that has been shown is
not quite to consider equally parts of serious science by Newton his prodigious
alchemical researches, or the Biblical exegesis and endeavour to reinterpret religion
which were far vaster still. However serious Newton himself may have been about
them, these elements of his genius are relegated by some to the status of having been
frivolous, or by others to the argument that the 17th century was not yet anywhere a
century of science.11 Even those scholars who study Newton most intently (and are
consequently in greatest sympathy with him) seem inclined to pass with some
embarrassment for him over these far vaster reaches of his intellection, and simply to
thank goodness that they did not prevent Newton from producing for us his Principia.
Clearly therefore the aspirations for knowledge by which Newton is taken to have
redefined his millennium are understood to connect neither with alchemy nor with
religion never mind that these pursuits together were well over half of that man. The
11
Thus, J. E. Maguire and P. M. Rattansi in their pivotal 1966 argue (p. 138) that “Isaac Newton
… was not a ‘scientist’ but a Philosopher of Nature. In the intellectual environment of his century, it
was a legitimate task to use a wide variety of material to reconstruct the unified wisdom of Creation.”
Very true: and yet by his diagrammatic analysis as a magician’s art with theological intent, Newton
nonetheless sets the defining example for the ignition of science.
epistemic aspirations that we are understood to have acquired from Newton relate to
mathematical brilliance in the investigation of nature and concern Principia chiefly and
little else that that man was about. Of course this attitude implies that Principia
comes apart from both alchemy and religion. When this attitude is countered by E.
Maguire and P. M. Rattansi (1966), B. J. T. Dobbs (1975, 1991), and even in some
ways by R. S. Westfall (1971, 1980), their point is not so much as I would wish it to
be, that Newton brings out his overall concerns brilliantly within the mathematics that
he prosecutes. The proper appreciation of diagrammatic reasoning is key to our taking
this needed step.
To study Principia in the spirit in which it was written is almost impossible today,
for the mathematical culture it is roundly a part of is significantly alien to our own.
Yet Newton’s many dimensions together may be the basis for surmounting the
problem. For example, his theology can illuminate the high literal-mindedness that he
not only appreciates in Euclid but practices similarly in his physics. The intellection of
the West is regulated by the thought that, while you can never actually manage to be
God (since God is perfect beyond possible human accomplishment), yet, ideally you
would be God. The West lays upon your shoulders the search for The View from
Nowhere.12 High literal-mindedness is the essence of this search. For example,
Newton lays out in the Scholium to the Definitions in his Principia reasons to be literal
12
Thomas Nagel’s 1986 by this title is in my estimation one of the great books of the West.
minded about temporal durations.13 Treating of duration as universal in its significance
and possessed of an objective measure involves, however, in our own efforts at
measurement of duration, a transcendental finality, a limiting goal that transcends what
we humans can ever accomplish practically. Any material process short of the
dynamical unfolding of the cosmos as a whole will be interfered with gravitationally if
not in other ways by other material processes, so that any material process to which
we might look practically as a clock, if adopted as our standard for duration, would
lead us to conclude inconveniently that order everywhere is subtly defiled. We would
discover in the wider universe processes sometimes speeding up a little and sometimes
slowing down a little, for no physical reason that can be discovered in those processes
themselves. Consider then the measure, the adoption of which as a standard for
duration is consistent with finding that every other process ever only speeds up or
slows down for an identifiable physical reason. Newton showed definitively why we
must admit that this measure is transcendent of our possible practical grasp. Nothing
short of the cosmos as a whole, or God’s all-encompassing insight, can pull into view
quite that perfect measure of duration. Consequently those ‘who confuse true
quantities with their relations and common measures’ both ‘do violence to the
Scriptures’ i.e. violence to the glory of God and ‘no less corrupt mathematics and
philosophy’ (Newton 1999 [1687], p. 414). We cannot practicably ground in
13
The emphasis on literal mindedness is my own, but see DiSalle 2002 in relation to the following.
measurement our own literal-mindedness about duration, but in order to possess exact
science, we must nevertheless help ourselves to the notion (as given by God).
It is hubris perhaps for the West so far originally to have wrapped the infinite
into its cultural fold, but we do at least have burgeoning exact science to show for it.
Consider, then, what cultural conditions there are for our possession of the very
notion theoretically of temporal duration. And behold the vaulting commitment to
reason-in-the-world that lies within that cultural form. That there is a measure
(however transcendental) of duration, the adoption of which as a standard for duration
is consistent with all processes everywhere ever speeding up or slowing down only for
a reason lying within those processes themselves, is an idea that palpably involves
vaulting commitment to reason-in-the-world. In order for there to be temporal
duration in such literal respects, every last process must be exquisitely rationally
harmonized with all the other processes; everything must unfold as it does for reasons,
reasons across which there is likewise harmony or coherence. We are a hair’s breadth
from theism in the making of such a vaulting assumption. Cross, moreover, a
philosopher or a scientist or a mathematician, on the question whether literalmindedness is itself an acceptable commitment, and in the glowering agony of their
response, you will see how palpably you do insult to an article of faith. I am myself
disposed to view our vaulting commitment to reason-in-the-world “critically” in
Immanuel Kant’s sense of that word. The God concept may, to deep-going degree, be
regulative of Western culture including Western philosophy and science, but I frame no
belief with it on that account of anything beyond the natural. Only on account of my
culture’s rationalism is the very concept of nature or the natural fully available to me,
so that in its very culturally emergent quality, the concept of nature will not be able to
cover quite everything. Some commitments stand as conditions for the possibility of
that cultural form, and to wrap into philosophy or science the task of evidencing or
otherwise warranting those commitments will on that account fail. However strongly
disposed I am, given my culture, to think that nature completes my situation, still I will
struggle to fit meaning itself or mindedness or ethics or mathematics, or anything of
which the original touch of infinity is defining, quite under its fold. Naturalizing
intentionality, naturalizing ethics, naturalizing mathematics, all to me seem fraught
philosophical projects, however fully the impulses resonate with me that draw other
philosophers into such pursuits. Yet the agony here is to me no inducement to have
truck with the supernatural. Rather I see it as an original instability of my culture (and
yet a creative one). Some aspects of the vaulting investment in reason itself are much
like faith, and God as metaphor is even helpful in some degree as explanation of their
directedness. Yet the condition in question, being thoroughly cultural, seems to me
not in the end to tell about the world as it is in itself. It tells at most to tell about us,
who are in the circumstance of western philosophy, and science.
Among the difficulties that stand in the way of comprehension of Newton by
contemporary Newton interpreters is the relation he believes he possesses to God.
Here, however, recognition of the ideality of mathematical practice can be of some
help: we can at least remind ourselves usefully, that in the West, and as a seeming
necessary condition for the full blossoming of exact science here, humans have
relinquished expecting that the ideals to which human practices are subject are even
possibly realizable by humans. Whereas the East has conceived the ideal as humanly
realizable (albeit ever so infrequently, e.g. by a Buddha or a Confucius), and whereas
the East sports little or no truck with the monotheist’s God, in the West the ideal for
human practice has been made out as transcendent of possible accomplishment by
humans. Newton may deny the literal divinity of Christ, but insofar as this rivalry (of
Newton with Jesus himself) concerns who stands the closer to God, still the
transcendence of God is admitted. That there is, beyond the humanly practicable,
such a transcendent standard, is as needs be, Newton maintains. That the ideal for
human accomplishment is transcendent of possible human accomplishment is
necessary, he maintains, in order for mathematics, in Newton’s understanding of that
word, to fall out as it does as exact science. While we may balk in other ways about
Newton’s theology, we must concede that he is correct on this point. A standard that
is transcendent of possible human accomplishment is also necessary in order for
natural philosophy to admit of its mathematical principles. Newton understands this
clearly, and if we want for ourselves his own exact science, again we must concede the
correctness of this view of his.
Newton’s early calculus far further advances the algebraically situated ‘analytic
art’ of Viète, rendered into analytic geometry by Descartes, and then generalized in
its application by many subsequent mathematicians, including some whose concerns
were with infinite series. Powerful though Newton’s advancements were in these
methods, by the mid 1670s he began to compare them invidiously with insights of the
ancients. Leading up to this change in his views, he had deeply studied the seventh
book of Pappus’s Collectiones. I propose to think that Newton as mathematician
shifted from symbol back to diagram, for the reason that here he better connected in
his own estimation with God. Scholars such as B. J. T. Dobbs have shown that from
the early 1670s Newton connected with alchemy his quest, which was theological, to
restore ancient knowledge. The ancients’ orientation to the diagram was something to
sift and weigh and consider, and doing so promptly paid Newton an inspiring dividend.
By diagrammatic analysis that was newly turned upon motion or physical change,
Newton found that he could discover what is so deep-lying in the causal order as to
seem but the work of God. In the diagram, the ancients had something right, which
the moderns had lost.
Diagrammatic reasoning sorts out how with straightedge and compass one should
as mathematician act, and so its burden is partly muscular. This makes it apt for
physics. Diagrams inform us about constraints on motion. Indeed, while calculus and
nineteenth-century conceptions of rigor chased the diagram for a long while from the
fore in mathematics to the rear, in physics allure for the diagram has been steadier.
Newton as mathematician might seem singular to have concocted the calculus yet
cleaved to the diagram. But we need to remember that Newton was also physicist.
Significantly much as physicist was Newton seeking special closeness to God.
§5. Plato, Newton, and the diagram.
Developments which are significantly recent, like the nineteenth-century
creation of the theory of real numbers, or the breakthrough discoveries about logic
itself which have made the computer age possible, at the same time conspire to
obscure the vision we might otherwise possess back onto the culture of mathematical
inquiry of which Newton is a part. Yet in all the ways by now remarked and more, we
need clearly to appreciate Newton’s creativity within that culture if we are accurately
to understand Newton’s place within, and overall impact upon, the philosophy of the
West. This impact is large, in part because Newton vindicates the ancient Greek
Plato, signally. He thus epitomizes (no less signally than problematically) a certain
kind of triumph of the West. Newton vindicates Plato’s conception that physics or
natural philosophy, if mature, must be mathematical. This accomplishment of
Newton’s also demonstrates how natural philosophy can come together as a science.
In short, Newton begins to transition the West profoundly from one epoch to another,
from an epoch of significantly unfulfilled aspiration for science to an epoch of
burgeoning accomplishment in it. (Before Newton, western philosophy relentlessly
asked the question whether a science of nature is possible. After Newton, western
philosophy asked how science is possible. Newton accomplished something so
significant that, in its light, doubt could scarcely be maintained whether science is
actual.) Yet we are blocked first from understanding Plato, and thence from properly
understanding Newton, precisely by the scale of that change.
Notably a synthetic philosopher, Plato takes vast inspiration from the culture of
mathematical inquiry in his day. Plato considers the opposition between presocratic
philosophers Parmenides and Heraclitus. Mathematics is key to how Plato surmounts
this opposition. Parmenides makes out the real as timeless and necessary and utterly
unitary. Heraclitus considers with equal rationality what conclusions to draw about
the very idea of order in the real, granting that, as experience teaches us, change, or
flux, obtains. Heraclitus concludes that change could not but be radically ubiquitous if
it obtains at all, and therewith the very idea of order in the real seems to Heraclitus
forfeit. These views are thesis and antithesis to Plato’s synthesis. Plato coaxes us to
think that deep-lying rationally apprehensible form in the real might after all be
discoverable by humans, provided (or to the extent) that people can but fathom formal
timelessness within the manner by which material things change. Necessity of
practical imperatives must precede necessity of truths. Plato considers diagrammatic
demonstration of geometrical propositions actually to involve, and thereby to
epitomize, such insight. It is no wonder then that Plato expects, otherwise wholly
presciently, that physics, if mature, must be mathematical.
The Greek mathematical diagram has as its whole purpose to epitomize
timelessness in the manner in which the instruments of geometrical construction can
most perfectly be moved through certain ideal steps. And just as Plato’s philosophy
of mathematics cannot be separated from the diagram, or therewith the perfection of
practice, so correspondingly, neither can Newton’s accomplishment. To fathom well
why diagrams are for Newton no mere aids to pedagogy but rather fully essential to his
reasoning in the Principia is to begin to comprehend Newton not in pieces but rather
as one whole.
Heterogeneous by present-day lights, the many sides of Newton pose a challenge
to us. Alchemist, theologian, magician, raging recluse, icon, mathematician, practical
man, progenitor of physics as exact science — why must we struggle so, simply to
understand the sides of this singular genius as forming one magnificent (if estranged,
solitary) self? The key to solving this dilemma is the diagram. For to appreciate the
diagram is to fathom many points all at once of separation of us from Newton. To
appreciate the diagram is thus to be reminded of many points all at once of how
Newton has affected our thought. The diagram therefore potentially steps us towards
connecting with Newton comprehendingly, of him not in pieces but as one whole.
B. J. T. Dobbs powerfully argues that Newton is to be considered as a coherent
whole. I very much agree, yet I believe I bring the physicist well forward within this
whole whereas Dobbs allows the physicist in Newton to be much eclipsed. The key to
seeing Newton as one whole and as prominently physicist is the diagram, recalling how
it mathematises motion, reveals deep-lying causes, is practically connected, supports
wizardry in deep-going discovery by those adept in its use, and connects, in the
practical ideal to which it is beholden, ultimately to God. Dobbs (here quoted as by
Margaret J. Osler) contends that Newton’s accomplishment was
not just of ‘the mathematical principles of natural philosophy.’ On the
contrary, it was to have been a grand unification of natural and divine
principles, and it included a vision of God’s activity not only in this
world as we know it but also at the world’s beginning and at its
apocalypitic end and renovation. It was a vision in which the Aryan
Christ, as God’s ‘Agent’ throughout time, always putting the will of the
Supreme God into effect, kept God intimately connected both with the
physical world and with humanity: that was Newton’s ultimate answer to
the twin spectres of deism and atheism that had always haunted him. It
is also a vision that forces one to the conviction that one must give a
religious interpretation not only to Newton’s alchemy but to all of
Newton’s work, including the Principia and Opticks, since Newton
himself was apparently motivated to study ‘the frame of nature’ in order
to learn of God’s activity.14
I say yea here to all this, but contend as well that diagram clinches this unity; about
whether the mathematical principles of natural philosophy are key Dobbs demurs, yet
that removes the linchpin of the unity in my view.
§6. Attitudes.
When physicists direct their minds to the pinnacle of the human condition, they
of course discover physicists there. Newton is one whom they see there. Physicists
are typically very proud that they possess Isaac Newton as one of their own. To some
extent, Newton’s own intellectual purposes play on among physicists, even now. Yet
14
Dobbs 1991, p. 254; my eye was drawn to this in Osler 2004, p. 10. For a broader picture
than concerns Newton’s times alone or the science / theology connection in him alone, see Osler 2010.
See also Snobelen, whose unpublished n.d. has helped me here.
when, by the enormous force of his mind, Newton fulfilled his intellectual purposes
pointedly, the whole human historical epoch changed. Physics as exact science
ignited. An age of science began. Of course this iconic example of intellectual
virtuosity by a physicist is validating for physicists. And their satisfaction in Newton
the present study in no way seeks to undermine. But clearly I am attempting
nonetheless to broaden, and in this way to challenge, how physicists think about
Newton. Newton was a creature of a very different century. Twenty-first-century
physicists do still know a profound part of Newton it is true, yet they do not know
even a tenth of what he was. If they consider Newton as at all strictly one of their
own, they fall, I believe, into a mistaken view of how their own scientific discipline first
became possible at all.
Artists, not only physicists, view Newton through the lens of textbook physics.
Artists on that account see a stunted figure. I do not endorse their understanding of
Newton, but I do endorse their conviction that on that understanding he makes a less
than well rounded figure. Hollow yet muscular and largely empty-headed, Salvador
Dali’s Newton possesses a vast personal energy focused into what is perfectly
symbolically describable about the real: the swing of a pendulum. The pendulum and
the pendulum alone commands Newton’s inside.
nothing about Newton,
Seemingly in this artist’s view,
certainly not his paltry genitals, befit a man-of-the-world. Dali’s depiction of Newton
seems to be an essay in contempt. And famously William Blake paints Newton no
more favourably. In his poetry accusing Newton of ‘single vision’, Blake depicts the
man crouched over his diagrams, his back turned to the wider universe. Muscularity is
again symptomatic of an ill-focused energy, and the world seems to have been
darkened, not illuminated, by Newton’s effort.
My plea to physicists — to consider the part that they know of Newton to be not
even a tenth of the man — is at the same time a plea to the artists, to consider Newton
a far more enchanting mind by an artist’s own lights than contemporary textbook
physics reveals. Only by dint of many further dimensions could Newton begin to have
the historical significance that he has. Let us get real, and get over taunts that
Newton is somehow fundamentally feeble. And surely let us understand that Newton is
extraordinary in his practical dimensions: his mathematics is art, its necessity
aesthetic, its rational force a balance of analysis and synthesis.
When philosophers direct their minds to the pinnacle of the human condition,
they of course discover philosophers there. Philosophers: in your case this foible
cannot be helped. On the contrary, this foible is deeply over-determined by
conditions upon the cultural form generally within which the philosophical activity first
emerges. Philosophy requires a high literal-mindedness such as alone also makes
wider theoretical pursuits truly possible. Once literal minded theorizing is launched,
philosophy is inescapable. Philosophy consists in a deepest-going literally minded
effort to consider what to believe, what to value — and why. To understand that
pursuit modestly, as playing perhaps second fiddle compared to other things that
people can do, abandons prioritizing reflection as highest human calling of all. But
that makes the person no longer a philosopher. Philosophers ask the hardest and most
deep-going questions. They are required to be pinnacle humans in their own eyes,
lest they be left doubting (quite against their calling) whether reflection intellectually is
highest. However far symptomatic high regard for reflection is of a wider orientation of
the culture, its pinnacle status is a unique burden for philosophers professionally to
bear. The resulting foible, of over-determined immodesty, occasionally bends the
philosophical intellect into an unfortunately obdurate state. People talk of a ‘gaggle’
of geese, a ‘flock’ of sheep, a ‘school’ of fish, a ‘herd’ of cattle, and an ‘arrogance’
of philosophers.
Newton is himself key to how philosophy swung wide into an analytic phase.
Certain obdurate qualities follow into the philosophical imagination by virtue of the
analytic philosophical orientation, that quite preclude philosophy’s currently
appreciating Newton’s intellect as it was. The force of Newton’s discoveries was
epoch-making. He precipitated profound intellectual change. We are left by the
effect of his genius the less able to comprehend clearly the qualities themselves that
that genius possessed. In order to grasp Newton we must understand both our
separation from him and in that, a connection. This is a challenge, and so in order to
accomplish it, obdurate about our contemporary philosophical dispositions is just what
we need not to be.
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