some uses of proportion in newtons principia, book i
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EMILY GROSHOLZ
SOME USES
OF
PROPORTION IN NEWTON'S
PRINeIPIA, BOOK I: A CASE STUDY IN APPLIED
MATHEMATICS
DURING the 17th century, a gradual transition occurs in which proportions
between ratios
:
B
::
C :
D
are treated more and more as equations between
quotients B
=
C
D
In this essay, I will examine the significance of
this transition for an understanding of the relation of mathematics and physics
in the period, illustrating my arguments with a detailed analysis of
Propositions I, VI and XI from Book I of Newton's Principia, In particular,
I want to urge
that
Newton's use of proportion-notation was appropriate and
indeed advantageous to his mathematical physics; and that it reveals a complex
interplay of mathematical
and
physical elements which illuminates the vexed
philosophical problem of how to understand scientific idealization.
I
In her essay entitled Compounding Ratios ,
2
Edith Sylla argues
persuasively that the seventeenth century inherited two distinct traditions
concerning ratios and proportions. The first, arguably Euclidean tradition,
views a ratio as
a
kind of relation in respect of size between two magnitudes
of the same
kind, and
a proportion as an assertion of similitude between
two such relations. This view is expounded in Book V
of
the Elements; it is
applicable to magnitudes of every kind. In Book VI the general theory of ratio
is applied to ratios between geometrical magnitudes (incommensurable as well
as commensurable), and in Book VII and succeeding books to ratios between
numbers (integers).
*Department of Philosophy, 246 Sparks Building, The Pennsylvania State University,
University
Park, PA
16802, U.S.A.
Received
19
March 1986.
'Citations are taken from Andrew Motte's translation of the Principia revised by Florian Cajori
(Berkeley: University of California Press, 1934).
Transformatian and Tradition in the Sciences, Everett Mendelsohn, (ed.) (Cambridge:
Cambridge University Press, 1984), pp. 11- 43.
l The Thirteen OOks Euclid s Elements, Sir Thomas Heath (ed.) (New York: Dover, 1956),
Book V, Defn. 3.
Ibid.,Book
V,
Defns, 6 8
Stud. Hist. Phil.
sa
Vo . 18, No. 2, pp.
209-220,
1987.
Printed in Great Britain.
9
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210 Studies in History and Philosophy of Science
The leading principle of this theory of ratio is the Eudoxian or Archimedian
axiom, which states: magnitudes are said to have a ratio to one another
which are capable, when multiplied, of exceeding one
another.
In other
words, one can form a ratio
A
B
(where
A
B
if f
there is a positive integer
n
such that
nA
>
B
This principle illustrates a salient feature of classical
mathematics, noted by Jacob Klein
6
and Ernst Cassirer, in that the terms
(magnitudes) are first in importance and are treated in analogy with
substances. Terms exist as heterogeneous kinds, each with its own peculiar
integrity (numbers; line lengths, shaped areas, shaped volumes;
commensurable and incommensurable magnitudes; finite and infinitesimal
magnitudes). This integrity must be accounted for (as in, e.g. the Definitions,
Postulates and Common Notions of Book I of the Elementst; and respected.
Specifically, when terms are associated in the formation
of
ratios and
proportions, their association must be carefully justified, and may sometimes
be proscribed. Thus, the Eudoxian axiom allows the conjunction of two
numbers, or two line segments, or two areas in a ratio, but proscribes that of a
number and a geometrical magnitude, of two geometrical magnitudes of
different dimensions, and of a finite and infinitesimal magnitude. In a sense,
this enforcing of a certain homogeneity between terms in a ratio entails a
segregation of mathematical magnitudes and prevents various strategies of
generalization which appear only later in the work of Vieta, Ferrnat and
Descartes,
But there is another side to conceiving of a ratio as a relation between terms.
Since the terms maintain their distinctness and dominate the relation, they do
not disappear into the relation, and the relation may be thought of as a virtual,
not
actually completed, operation. Thus, the Eudoxian axiom is also famous
for allowing a mathematics of ratios to include incommensurable or irrational
magnitudes. The ratio of the diagonal of a square to its side can figure in the
framing and solution of problems even though the operation : cannot be
carried out as it can in the case of the ratio 4 : 2. This tradition can therefore
accommodate a certain heterogeneity among terms within ratios, and so
facilitate certain strategies
of
generalization. Indeed, in the work of Newton,
as we shall see, it accommodates the introduction of non-Eudoxian (or non
Archimedean) magnitudes.
Since terms are preeminent in this tradition, terms are strictly distinguished
from relations or ratios between terms, and the heterogeneity of terms may
Ibid.
Book V,
Def n,
4.
Oreet: Mathematical Thought
n
the Origin 01Algebra
(Boston:
M.LT.
Press, 1968),
Chap.
9. _
7Substance and Function W. C. and M. C. Swabey (trans.) (New York: Dover, 1953), Chap. 2.
Heath. op. cit.
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Uses of Proportion in Newton s Principia
211
infect, as it were, the ratios which they constitute. Thus the unification of
ratios in proportions, on the one hand, and through the operation of
compounding, on the other, also requires careful justification and
proscriptions. Proportions, for Euclid, are not equations between ratios but
assertions of similitude between one kind of ratio and another, e.g. between a
ratio of two line segments and a ratio of two volumes.
Euclid does not say that the proposition
A
B
e :
D
can be constituted
when A x D B x e because A and D and Band C, potentially
heterogeneous, may be such that it does not make sense to take the product of
the two terms. In classical mathematics, multiplication is not a closed
operation; the multiplication
of
two line segments, e.g., produces an area.
Thus there is no natural interpretation for the product of two volumes, or two
areas. Instead, in Book V, Definition 5 of the
Elements.
Euclid generalized
the Eudoxian axiom, stating that proportions between non-continuous ratios,
A : B ::
e:
D
can be formed
if f
for all positive integers
m
n, when nA
mB
then correspondingly, ne
mD.
Note that this definition excludes
infinitesimal or infinite magnitudes, and tolerates incommensurable
magnitudes, as terms; and allows for the possibility that (mutually
homogeneous)
A
and
B
may be of a kind different from (mutually
homogeneous) e and D
The compounding of ratios in this tradition is carefully restricted; this is
Sylla s main point in her exposition of it in reference to
Elements
V,
definitions 9, 10, 17 and 18, Boethius
De institutione arithmetica
Bradwardine s
De proportionibus velocitatum in motibus
and Oresme s
De
porportionibus proportionum. The operation of compounding was only
performed on continuous ratios, in pairs or series (e.g. A B B:
e: D .
Since continuous ratios share terms, this condition insures that all terms and
hence all relations between terms (ratios) will be homogeneous. Such series of
ratios are compounded by forming the ratio of the antecedent of the first term
and the consequent of the last
A
D
in the example given above). This
operation is considered a kind
of
addition, as if the ratio
A
D
were composed
of A B, B
e and e :
D,
since small components form a whole additively.
Sylla notes that in the first edition of the Principia and even afterwards,
Newton refers to compounding as addition, and adheres to the first tradition
of compounding in many of his calculations.
To conclude, the first tradition takes terms as preeminent; allows for their
possible heterogeneity and accordingly puts restrictions on how they are
unified in ratios, proportions and compoundings; subordinates relations to
Ibid.
Sylla, op. cit. pp 18 22
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212
Studies in History and Philosophy
of
Science
their relata, and proportions to the relations they relate; and may treat : as
a virtual operation.
The second tradition appears to originate with Theon, a commentator on
Ptolemy s
lmagest
and is transmitted in the Middle Ages by Jordanus
Nemorarius, Campanus and Roger
Bacon.
This tradition countenances the
compounding of non-continuous ratios, and associates with each ratio a size
or denomination such that in compounding, denominations are multiplied
together. Thus, it tends to treat ratios as numbers (the size
of
the ratio) and
proportions as equations between numbers. In other words, terms are
subordinated to (swallowed up in) relations, and relations to equations. This
development is strongly associated with the rise of analytic geometry and
infinitesimal analysis, and forms an important strand in what Cassirer calls the
functionalization
of
mathematics and science.
In general, this tradition homogenizes the magnitudes occurring in ratios
and proportions, treating them uniformly as numbers. The assumption of such
homogeneity played a central role in the systematization and unification of
algebra and geometry in analytic geometry and later in infinitesimal analysis.
At the same time, at least in the initial stages exemplified by the work
of
Oresme and Descartes, it tended to limit the conception of number to the
rationals by requiring that division be an operation which could actually be
carried out in familiar ways.
In a deeper sense, too, it buried certain important questions and possibilities
of conceptualization particularly germane to the mathematization
of
science.
The distinctions between finite and infinitesimal magnitudes, between
numbers and geometrical elements, between geometrical and physical
elements, and among lines, curves, areas and volumes were, as I shall argue,
still quite significant in the context of the application of geometry to physics.
Newton worked with and exploited these distinctions, in part because he
retained an allegiance to the first tradition. More generally, philosophers of
science cannot afford to forget certain fundamental questions which the
second tradition suppresses. What
justifies
the claim that physical magnitudes
(like distance, or time) stand in relations to each other similar to relations
between numbers? Or that geometrical magnitudes do likewise? Or that
instances of one kind of physical magnitude stand in relations to each other
similar to relations between instances of a wholly different kind of physical
magnitude? Or that we can form ratios between two wholly different kinds
of
physical magnitudes (like velocity and time) to form new kinds
of
physical
magnitudes? These questions cannot be answered in a trivial way, nor without
appeal to the history of mathematics and science.
Ibid pp .
22 - 26.
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Uses ofProportion in Newton s Principia
213
e
-
·· ·t
v i
E.
./
.
..
: :
s ~ .
Fig.
I.
Proposition XI of Book I of the Principia is central to Newton s project.
It states If
a body revolves in an ellipse; it is required to find the law
of
the
centripetal force tending to the focus of the ellipse and concludes that the
required law is an inverse square law. The statement of the problem and its
solution depend in essential ways on two prior results, Propositions ID and
VI 4 which I will discuss first in order to better exhibit certain features of
Proposition XI.
Proposition I is Newton s version
of
Kepler s Law
of
Areas: The areas
which revolving bodies described by radii drawn to an immoveable centre
of
force do lie in the same immovable planes, and are proportional to the times in
which they are described. Newton s
proof
of this claim is illustrated by Fig. 1.
S is the center of force. A body proceeds on an inertial path from to in an
Newton op. cit., pp.
56-57.
Jbid.,
pp.
40-42.
One might debate whether Newton countenanced actual infinitesimals.
Without wanting to reopen the debate here,
I
would observe that his ways of using evanescent
magnitudes are Leibnizian enough to support that contention, despite the Archimedean air of
Principia I, Lemmas and his reservations about indivisibles in the following Scholium.
Ibid. pp.
48-49.
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Studies in History and Philosophy
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interval of time; if not deflected, it would continue on in a second, equal
interval of time along the virtual path
Be
However, Newton continues,
when
the body is arrived at
suppose th t a centripetal force acts at once with a
great impulse so that the body arrives
not
at c, but at C. Then cC
V
represents the deflection
of
the body due to the force; indeed, as we shall see,
cC RV becomes the geometrical representative of the force. The perimeter
ABCDEF represents the trajectory of the body as it is deflected at the
beginning
of
each equal interval of time by discrete and instantaneous
impulsions from S. Newton then uses the Euclidean theorem that triangles
with equal bases and equal elevations have equal areas, to show that area
~ S
area
~ S c
area b SBC; this equality extends to triangles SDC
SED SFE . . by the same reasoning, so that equal areas are described in equal
times. We have only, Newton concludes,
to
let the number of those triangles
be augmented,
nd
their breadth diminished in infinitum for this result to
apply to a continuously acting force and a curved trajectory.
There
re
two related points which I want to make about this diagram (Fig.
I). First of all, it is thoroughly ambiguous, and must be read in two apparently
incompatible ways, as a collection of finite lines and areas (where the
perimeter is composed of rectilinear line segments) and as a collection of
infinitesimal as well as finite lines and areas (where the perimeter is a curve).
The first reading allows the application of Euclidean theorems to the problem;
the second reading makes the problem applicable to the kind
of
motion and
force Newton is interested in. For this reason, the diagram, whose meaning
and intent cannot be deciphered unless it is read in both ways, could not have
arisen in Euclidean geometry.
Secondly, the diagram would not have arisen in Euclidean geometry alone
even if it is read as a finite configuration. The theorem about triangles of equal
area provides no motivation for studying triangles of equal area multiplied and
linked in this particular fashion. Neither does this resemble a problem of
quadrature, for in that case (a) the perimeter would be a known curve, (b)
there would be no reason to single out the point S and (c) there would be no
reason to study the line segments Be and cC BV In other words, though this
diagram can be treated by Euclidean means, no Euclidean geometer would
have formulated it, or been interested in it if it were presented to him. This
diagram is a problem about force, motion and time, distances and areas. It
arose within physics, and geometry enters as an auxiliary means to help solve
it.
The formulation which I have just given of the status of this problem in
physical-geometry, however, does not yet do justice to the complex interplay
of geometry and physics in this context. I will refine it as I discuss Propositions
VI
nd
XI,
nd
their.attendant diagrams
nd
proofs. The general train of my
argument will be the following. In these propositions, Newton is dealing with
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Uses of Proportion in Newton s Principia
Fig. 2.
215
problems which arise within physics (the problem of motion in general and
planetary motion in particular; the problem of force) but cannot be solved
within physics alone. The sterility
of
Aristotelian physics in the seventeenth
century testifies to this. Geometry is brought in as an auxiliary field to help
solve the problems. However, physics has already been penetrated and
structured by geometry (in the work of Kepler, Galileo, Descartes and
Huygens) so that what counts as a physical problem for Newton is already to
an extent formulated in geometrical terms. Moreover, geometry itself is
already in a process of revision and expansion, in response to the problems
concerning motion and force which physics has presented to it. That is,
there has already been a long process of assimilation between physics and
geometry. Although each retains items and problems alien to the other, the
common ground between them has steadily increased, with the shape of each
field altering accordingly. And Newton profits from his exploitation of this
common ground, the hybrid techniques, items and problems of
physical-geometry.
Proposition XI is a special case of the general result which Newton works
out in Proposition VI, l7 where he shows that for any kind of revolution Q
of a body P around a center of force S the centripetal force will be inversely
proportional to the quantity
sp
2
QT
2
QR
In Fig. 2, R is the virtual inertial trajectory the body would have followed if it
had not been attracted by S, and by Law I it is directly proportional to the time
t. By Kepler s Law
of
Areas (Proposition I) the curvilinear area SPQ is also
j don t
mean to exclude here the mathematization
of
physics already accomplished in the work
of
Euclid, Ptolemy, Archimedes, and the author
of Mechanical Questions.
I ·For a full-dress defense of this claim, see Francois DeGandt s Mathematlques et realite
physique au XVII<siecle , in
Penser tes mathematiques
(Editions du Seuil, 1982), pp. 167
-194.
My exposition here is heavily indebted to an unpublished essay by Francois DeGandt, Le
traitment geometrique des forces centrales dans les
rincipi
de Newton , forthcoming in the
Graduate Faculty Philosophy Journal
New School for Social Research.
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216 Studies in History and Philosophy of Science
proportional to t; since
2SPQ
SP x QT it is proportional to
SP
x
as
well. (The fact that
SP
x QT is an area, and an infinitesimal area, is thus
physically significant. Kepler s insight, so important for the development of
astronomy, was that the area swept
out
by
SP
serves as a reliable uniform
measure of time. That the area is infinitesimal signals that here Newton is
deploying a mathematics developed to treat the problem of motion, non
uniform as well as uniform, through an analysis involving instantaneous
velocity, time and force.) The segment
QR
represents the virtual trajectory of
the body as a result
of
the centripetal force exerted by S;
it
is thus directly
proportional to the force. t
is also directly proportional to 2 , by Lemma X S
in which Newton generalizes Galileo s result that in free fall the space
traversed is proportional to the square of the time, to hold for all cases
of
a
body attracted by a constant or continuously varying force provided
that
one
considers only the first instant of motion. Then since
F
2 F
QR
QR
er X
SP
x
QT
Sp2 X QT2
or, as Newton prefers to write t so
that
the ratio has three dimensions,
Sp
2
x
QT
2
Fa
QR
Newton uses the diagram in Fig. 2 to find a way of representing the
centripetal force at S, and concludes
that
t is inversely as
Sp2
x
QT
2
QR
Note that very little in either the diagram or the reasoning about it is of
Euclidean provenance: only
that
SP is a line segment, PY is the tangent to the
curve at and the area of a triangle is half the product of its base and
altitude. (But the application of the latter theorem is curiously unfiuclidean.)
What the diagram is for and how it should be read is determined for the most
part
by the way it represents a physical situation. Why QR and SP x
QT
should be chosen as particularly significant, and how and why they are related
can only be explained by theorems
of
physics developed by Kepler, Galileo,
Descartes and Newton. Of course, these theorems also geometrize physics,
providing techniques for representing physical magnitudes by lines and areas,
and
therefore Newton can set forth his results in such a diagram. But the
diagram of Fig. 2 could only have arisen, and is only intelligible, within the
context of physical-geometry. Finally, to reiterate my earlier point, Newton s
expression of his result as the centripetal force will be inversely as the solid
Sp2 x QT
2
QR
10
Ibid. pp. 4 35.
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Uses of Proportion in Newton s Principia
Fig. 3.
217
allows him to exploit the proportion-idiom of the first tradition to relate and
yet discriminate heterogeneous physical magnitudes lines and areas and finite
and infinitesimal magnitudes.
Since Proposition VI is a general result and since the expression for force is
given in terms of infinitesimal magnitudes Newton must apply it to cases in
which the trajectory PQis specified in such a way
that
he can replace
SP
2
QT
2
QR
with another expression derived from the specific trajectory of and
involving finite magnitudes. In Proposition XI he chooses the case where the
trajectory is an ellipse clearly a crucial step in the application of his physical
principles to the solar system in Book
ll
Figure 3 combines the physical-geometrical schema
of
Fig. 2 with the p ure
geometry
of
the ellipse but
is instructive to examine the combination in
detail. The latus rectum L
=
and the diameters of the ellipse BC S
DK and PG have no physical import. But the presence
of
the elements of Fig. 2
impose physical import on other parts of the ellipse: the perimeter PBDGKis
also the orbit of a revolving body S is also the center of force SP also the
distance
of
the revolving body to the center of force. By contrast the auxiliary
lines PH
IH
and PF like the ellipses s diameters enter the reasoning only
insof ar as they are geometrical. Thus. though Fig. 3 contains more purely
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Studies in History and Philosophy of Science
geometrical elements than Fig. 2, the problem
and
solution which it represents
are thoroughly hybrid, both geometrical and physical.
The proof of Proposition XI proceeds by establishing proportions between
line segments
and
products
of
line segments by means
of
theorems
about
similar triangles, isosceles triangles
and
ellipses, and culminates in an elaborate
compounding of these ratios. Newton first proves that
EP :;
AC using
auxiliary lines HI and HP; H is the other focus. Since MHS is similar to
MCS
SE :;
El
and EP :;
Yz PS Pl or Yz PS PH since tJlIH is
isosceles.
PS PH :; ZAC
by the nature of ellipse-construction, so
EP :;
AC.
Then Newton begins to establish certain extended proportions. L
X
QR
:
L x Pv :: QR : Pv, This seems to be a straightforward bit
of
reasoning about
ratios,
but
note
that
all the magnitudes involved except L may be construed as
infinitesimal, because they are being used for the analysis of motion. Newton
is profiting
from
: as a virtual operation to manipulate non-Archimedean
magnitudes. QR Px
:
Pv
:: PE :
PC because
tJlxv
is similar to tJlEC.
This is a Euclidean result employed in a highly non-Euclidean way to relate a
finite to an infinitesimal triangle. Finally, PE : PC :: AC : PC by the first
result, so
L
x
QR : L
x Pv
:: AC: PC.
Next, Newton asserts L x
v
Gv x Pv
::
L
:
Gv Pv here is infinitesimal
and
that
Gv
x Pv :
v
::
p : CIY
a fact
about
ellipses, except
that
Pv
and
Qv
are infinitesimal
magnitudes,
Actually, it is only at this point in the
proof
that
Newton says explicitly let Q-P. One might then take the foregoing
reasoning as about, not infinitesimal, but small finite quantities, so that the
application
of
Euclidean results is straightforward. However, as we shall see,
the final compounding of ratios evenhandedly combines ratios established
before and after the step where
Q-P.
Newton depends on the ambiguity of his
diagram: read as finite, it allows the application
of
Euclidean results; read as
infinitesimalistic, it provides a mathematical schema for motion.
When points P and Q coincide, Newton claims that Qy2 ;
Qx
z
and
so Qx
z
Qvl
:
QT
2
::
Ep
z
:
pp2
since the infinitesimal triangle
QxT
is similar to
tJlEF.
Then
Ep
z
: pp2::
CA
z:
ppz by the first result,
and CA
z:
PP:: CIY:
cj by a previously established result
about
ellipses. Thus Qx
z
Qvl
:
QT
z
::
CIY: ce. Newton is now ready to carry
out
the final compounding, which
Sylla summarizes in the following perspicuous array. 20
L
x QR :
L
x Pv :: AC : PC
L X Pv : Gv
x
Pv
::
L : Gv
Gv x Pv : Qy2 .. p :
CIY
Qv
2
:Qx2
.. 1 I
x
:
Q P
cir
:
ce
19 Ibid. Lemma XII,
p.
53.
2IlSylla,
op. cit.,
p. 16.
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Uses of Proportion in Newton s Principia
219
Sylla notes that Newton has set up the left-hand ratios as a continuous series,
and compounds them according to the first tradition, taking the extreme terms
and forming the new ratio L x QR
:
QT
2
• The right-hand ratios he
compounds according to the second tradition, by multiplication:
AC
x
L
x
PC
1 x x .
Gv
et» x I x Cg or (substituting
Bc:aIACfor
and cancelling) PC
:
Gv.
Our prior investigation into the geometrical
physical nature of Newton s diagrams and procedures sheds further light on
this process
of
compounding. For the magnitudes on the left-hand side have
physical import and are largely infinitesimal. That is, they are just the kind of
magnitudes which ought to be treated in the first tradition, which respects the
heterogeneity of terms and, given the virtual nature of : allows for the
manipulation of infinitesimals. The magnitudes on the right-hand side, if we
recall
that
Gv
GP
as
Q- -P,
are all constant finite geometrical line lengths
(some of which are squared) with no physical import. Having no reason to
think of them otherwise than as numbers, Newton can handle them according
to the second tradition, multiplying them like quotients.
The compounding thus yields the proportion,
L
x
QR
: QT
2
PC
:
Gv.
As
Q- -P, PC Gv,
so
L
x
QR: QT
2
: :
I : I, so
L
x
QR
is proportional to
Sp
2
. Sp
2
QT
2
.
QT
2
• Multiplying both these terms by
[QR]
we find that QR IS
proportional to L x
Sp
2,
or, since L is a constant, to SP2. Thus, the central
force in this problem is inversely proportional to the square
of
the distance
SP
IV
What does our examination of these Newtonian propositions reveal about
the much-debated topic of idealization? can help us reject two misleading
models of idealization, and in so doing formulate some positive observations
on the topic.
In Proposition XI, Newton does not begin with a physical problem,
translate it into a mathematical analogue, solve the mathematical problem and
then translate the solution back into physical terms. This account would leave
the unresolved puzzle, how is it possible to find analogical structures in physics
and mathematics? And, how can the analogy itself be justified?
I have said that Newton brings geometry in as an auxiliary field to help solve
problems which arise in physics but could
not
be solved by physics alone, and
21
This view of applied mathematics is presented in Stephan Korner, The Philosophy of
Mathematics (London: Hutchinson, 1960), pp. 176
182.
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220
Studies in istory and Philosophy Science
cited the exhaust ion of Aristotelian physics in the seventeenth century as
evidence
for
this incapacity of physics. But notice
that
the interplay
of
geometry
and
physics as a result of this unification of fields is much more
complex
than
the model sketched above suggests. Newton s statement
of
the
problem in Proposition XI in reference back to Proposition VI), while
evidently the statement
of
a physical pr oble m
about
the relation between
motions, times, distances
and
forces, is already couched in mathematical
terms. he mathematical reasoning in the
proof
reveals at every step that
Newton is using a mathematics adapted to and extended by its employment in
the s ol ut io n
of
physical problems,
and the
solution is given in terms
of
physical-geometry. In the overlap between the two fields where Newton is
carrying out his investigation, geometry has reformulated physics, and physics
has changed the shape
of
geometry. The geometrical-physical tools
of
investigation have been developed precisely to solve the geometrical-physical
problems,
and
it makes no sense to try to disentangle the geometrical from
the
physical elements and
then
ask why they match.
or does it m ak e sense to say that Newton developed this physical-geometry
and then
applied it to physical reality tout court so
that
the puzzle
of
idealization is
then
to figure
out
how he could do this. So stated, the puzzle has
no sol ut ion. since physical reality has been g rant ed no cognitive s tructure,
which could
then match
up somehow with the structure of physical-geometry.
Newton obviously intended Pr oposition XI to apply to planetary
motion
around the sun, and just as obviously realized that it was only one stage of that
appl icati on. But the o bj ect of application here is not planetary motion tout
court but p lanetary m ot ion al ready cognitively s tructu red as an o bj ect of
science.
My general po int here is
that
the philosophical problem
of
idealization, in
particular, of how mathematics can be applied to physical reality, sh oul d be
addressed in terms of
the
history
of
science. A series
of
case studies like the
one
contained in this essay would, I believe, provide grounds
for
local
and
piecemeal generalization
and
also reveal striking differences in the nature
and
justification
of
that application f rom one era to the next.
hus
I also believe
that
no interesting global solution to the problem exists.
What
history reveals
in any given era e.g, the seventeenth century) is a physics already partiall y
adapted to mathematics and a mathematics similarly elaborated in the service
of
physics. The application of mathemati cs to physics in a certain case e.g.
Newton s rincipia will then consist in taking that assimilation of distinct but
partially correlated domains one step further. he business
of
the philosopher
is to elucidate the peculiar grounds
and
presuppositions of that assimilation,
and
to cautiously generalize
on
the
basis
of
her
or
his analysis wit hou t
succumbing to philosophy s instinctive desire
for
the last w or d.
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