BEYOND THE EDGE OF CERTAINTY: Essays in Contemporary Science and Philosophy

148 Pages·1965·7.056 MB·English

by  Robert G. Colodny (ed.)

BEYOND
THE
EDGE
OF
CERTAINTY
Essays in Contemporary Science
and Philosophy
NORWOOD RUSSELL HANSON
Editor
BRIAN ELLIS ROBERT G. COLODNY
HILARY PUTNAM
DAVID HAWKINS
PHILIP MORRISON
PAUL K. FEYERABEND
NICHOLAS RESCHER
PRENTICE-HALL, INC., Englewood Cliffs, New Jersey
Prentice-Hall International, Inc,, London
Prentice-Hall of Australia, Pty„ Ltd., Sydney
Prentice-Hall of Canada, Ltd., Toronto
Prentice-Hall of India (Private) Ltd., New Delhi
Prentice-Hall of Japan, Inc., Tokyo
BEYOND THE EDGE OF CERTAINTY;
Preface
Essays in Contemporary Science and Philosophy
Editor, Robert G. Colodny
© 1965 by
PRENTICE-HALL, INC.,
Since 1960, the Center for Philosophy of Science at the University
Englewood Cliffs, N.J.
of Pittsburgh has annually presented a number of public lectures
on various current topics in the philosophy of the physical, biologi­
All rights reserved. cal, and social sciences as a permanent part of its program of instruc­
No part of this book may be reproduced tion and research. The lectures delivered during 1960-1961 by scholars
in any form, by mimeograph or any other means,
from diverse institutions were published in 1962 by the University
without permission in writing from the publisher.
of Pittsburgh Press under the editorship of Professor Robert G.
Colodny in Frontiers of Science and Philosophy. The present book
of lectures is the second in the series of volumes published under
the Center’s auspices.
Library of Congress Catalog Card No.: 65-10037
Second printing........September, 1965 Adolf Grunbaum
Andrew Mellon Professor of Philosophy
and
Printed in the United States of America Director of the Center for Philosophy of Science
University of Pittsburgh
C-07603
Contents
Introduction
Robert G. Colodny 1
Newton’s First Law: A Philosopher’s Door into Natural
Philosophy Norwood Russell Hanson ..................... 6
The Origin and Nature of Newton’s Laws of Motion
Brian Ellis ......................................................................... 29
A Response to Ellis’s Conception of Newton’s First Law
Norwood Russell Hanson .............................................. 69
A Philosopher Looks at Quantum Mechanics
Hilary Putnam ................................................................. 75
The Thermodynamics of Purpose
David Hawkins ................................................................... 102
The Physics of the Large
Philip Morrison 118
Problems of Empiricism
Paul K. Feyerabend 145
The Ethical Dimension of Scientific Research
Nicholas Rescher 261
Index of Names ......................................................................... 279
Index of Topics 283
BEYOND THE EDGE OF CERTAINTY:
Essays in Contemporary Science and Philosophy
Robert G. Colodny
University of Pittsburgh
Introduction
Our age is possessed by a strong urge towards the criticism of traditional
customs and opinions. A new spirit is arising which is unwilling to accept
anything on authority, which does not so much permit as demand
independent, rational thought on every subject, and which refrains
from hampering any attack based upon such thought, even though it
be directed against things which formerly were considered to be as
sacrosanct as you please. In my opinion this spirit is the common cause
underlying the crisis of every science today. Its results can only be
advantageous: no scientific structure falls entirely into ruin: what is
worth preserving preserves itself and requires no protection.
—Erwin Schrodinger, Science and the Human Temperament
In an age of astounding scientific creativity, it is extremely difficult
to obtain a synoptic view of the entire advancing wave of knowledge.
The difficulty arises not only from the sheer quantity of published
material and the intrinsic complexity of experimental and theoret­
ical findings, but also from the necessity of mapping the “edge of
certainty” against some body of knowledge assumed to be established,
orthodox, part of a consensus of the international community of
scholars. Despite these intractable problems, the modest claim may be
made that certain general properties of the process of scientific inno­
vation may be discerned. Without asserting any exhaustiveness or
hierarchical ranking, these include:
An increasing tendency of hitherto separate disciplines to flow to­
gether, producing conceptual aggregates which are more than the sum
of the parts.
As a consequence of the above, as witnessed by such specialties as
biophysics, astrophysics, space biology, geochemistry, etc., a compre­
hensive, unified vision of the cosmos begins to exhibit clearer features.
2 Beyond the Edge of Certainty
Introduction S
Unsuspected links of unity between separate theories are uncovered; the im­
past, but from a conviction that many of the most agonizing issues con­
plications of older, sometimes half-forgotten, hypotheses emerge, and both
fronting philosopher-scientists arise from the accelerating development of
pioneering scientists and philosophers of science become conscious of how
elements of “classical” scientific thought are embedded in contemporary distinct branches of sciences which experienced contingent junctures and
theoretical constructs. then proceeded in more or less incomplete fusion, carrying with them
It is at junctures such as these that a daring innovator such as Schro- unresolved and partly concealed epistemological and metaphysical prob­
dinger will write: “The old links between philosophy and physical science,
lems ingredient in the earlier epochs.
after having been frayed in many places, are being more closely renewed.
The relationship between classical mechanics and many divisions of
The further physical science progresses, the less it can dispense with phil­
osophical criticism. But at the same time, philosophers are increasingly contemporary science (often referred to in the pages that follow) will il­
obliged to become intimately acquainted with the sphere of research, to lustrate this point. As is well known, Newtonian mechanics made irrevers­
which they undertake to prescribe the governing laws of knowledge. ible the process of mathematization of the physical sciences begun by
Assuming the fulfillment of Schrodinger’s rigorous conditions, the world
Athenian and Alexandrian scholars of antiquity. The varied elements of
picture presented by the cosmologically oriented scientists and by phi­
the universe were analyzed in terms of- the basic concepts of space, time,
losophers of science will tend to differ mainly with respect to the “fine
grain.” Philosophy of science and “science” in the traditional sense become matter, force, and energy. The comprehensive nature of these elements of
complementary enterprises. dynamical discourse and the elegant mathematical formalisms created in
To the extent that an ensemble of problems is analyzed in depth by a the formulation of their mutual relations endowed the early dynamical
group of philosopher-scientists, the emergent findings will comprise parts of
laws with the characteristics of perfect examples of a law of nature, and
a unified vision, a characteristic of all great ages of scientific achievement.
the general method of analysis set forth in the Principia and elaborated
The reader of these pages will therefore expect to find that the Han- further by Laplace set a style and a tactic for scientific research that ad­
son-Ellis discussions of Newtonian mechanics, Putnam’s analysis of the vanced from success to success in an ever-widening range of phenomena,
status of quantum mechanics, Hawkins’s analysis of thermodynamics and until thermodynamics and electrodynamics bore the unmistakable imprint
biological teleology, and Morrison’s account of astrophysics not only share of a particular scientific and philosophic vision.
common historical backgrounds, but also reflect aspects of that unity of Here it may be of value to note that in the progression from the
physical nature that the various disciplines have been articulating for the Fuler-Lagrange formalism to the Hamilton-Jacobi, the domain of the
last three centuries. It is also of interest to note that the papers of Haw­ dynamical ideas had increased and embodied extremal principles based
kins and Morrison are excellent examples of that fruitful stimulus to con­ upon the work of Fermat and Maupertuis and fully exploiting various
ceptual novelty occasioned by the confluence of several distinct disci­ conservation theories. There thus emerged a more powerful synthesis of
plines. physical knowledge whose supreme instrument was the variational cal-
Feyerabend’s comprehensive critique of empiricism will be found to culus.2 This led to a renewed philosophical interest in the foundations of
illuminate the procedures of all of the foregoing essayists inasmuch as it physical science, an interest associated with the critiques of Hertz, Mach,
draws upon similar scientific content and hypotheses for elucidating the and Poincar^, and which has been revitalized in the epoch of quantum
nature of scientific progress from the point of view both of discovery and and relativistic mechanics.
of validation. These extend in time from Newtonian mechanics to wave That the transfer of elements from the Newtonian-Hamiltonian world
mechanics, thus clarifying many of the essential problems of the scientific to the Finsteinian-Minkowskian-Weylian has been accomplished only
and philosophical dialogue. This does not, however, entail philosophical through the closest cooperation of mathematicians, physicists, and philoso­
agreement with the other writers, but rather clarification of the problems phers is one of the enduring monuments of modern intellectual history.
posed and the methodology of discourse. Here again one should recall that a rather common feature in the history
Finally, Rescher’s probe of a long-neglected aspect of the scientific of science is the preparation by one age of the mathematical and techno­
enterprise—its ethical dimensions—anchors this volume in its most hu­ logical instruments to be exploited fully by later generations of scientists
mane context: the moral dilemmas inherent in that range of activities who will work in a philosophical climate that may be completely differ­
which culminate in knowledge and which arise from the necessary social- ent. Thus magnificent scientific constructs repeatedly reveal unsuspected
political setting in which scientists are, like other beings, inextricably en­ flaws when subjected to a different mode of philosophical scrutiny. What
meshed. It is also of interest to note that most of the essays in this work a purely formal logical analysis of theories sometimes fails to reveal is the
locate particular scientific and philosophical problems in an historical fact that a complex physical theory is often compounded of elements
context. This decision arises not out of any antiquarian interest in the from quite distinct historical epochs, each of which may have arisen un­
4 Beyond the Edge of Certainty Introduction 5
der radically different canons of rigor and different sets of submerged They know that even the most abstract and remote ideas may one day be­
come of great practical importance—like Einstein’s law of equivalence of
metaphysical presuppositions. Awareness that elements of the scientific
mass and energy. They have begun to organize themselves and to discuss the
world picture reveal in their lack of perfect fit the very disjointed histori­
problem of their responsibility to human society. It should be left to these
cal processes that created them in the first place may well be one of the organizations to find a way to harmonize the security of the nations with
reasons for a sense of perpetual crisis in those advancing disciplines that the freedom of research and publication without which science must
have crossed the “edge of certainty.” stagnate.6
Finally, considering the experimental and tentative nature of the Notes
formulations of both science and philosophy of science, one is driven to
1. Erwin Schrbdinger, Science and the Human Temperament, James
inquire into the climate of opinion that permits the unhampered devel­
Murphy, trans. (London: George Allen 8c Unwin, Ltd., 1935).
opment of scientific and philosophic ideas. Here all the contributors to
2. For an elementary, but beautifully lucid account of this, see A. d’Abro,
this volume express a consciousness of and a revulsion to any form of au­ The Evolution of Scientific Thought, Chap. 34 (New York: Dover Publications,
thoritarian orthodoxy, be it grounded in the scientific community or aris­ Inc., 1950).
ing external to it. That this is not an issue arising only in alien and 3. Science, Vo\. CXLI (Sept. 13, 1963), 1017. Cf. T. S. Kuhn, “The Function
enemy lands is well exemplified by Polanyi’s assertion that of Dogma in Scientific Research,” in A. C. Crombie, ed.. Scientific Change (New
York: Basic Books, Inc., 1963).
Scientific method is, and must be, disciplined by an orthodoxy which 4. Scientific American, Vol. CCVIII (May, 1963), 47.
can permit only a limited degree of dissent, and . . . such dissent is fraught
5. Ibid., 48. Emphasis added. Dirac adds;
with grave risk to the dissenter. . . . The authority of current scientific
It seems to me one of the fundamental features of nature that funda­
opinion is indispensable to the discipline of scientific institutions: ... its
mental physical laws are described in terms of mathematical theory of great
functions are invaluable even though its dangers are an unceasing menace
beauty and power, needing quite a high standard of mathematics for one to
to scientific progress.^
understand it. You may wonder: Why is nature constructed on these lines?
That this represents a minority opinion concerning the relationship of One can only answer that our present knowledge seems to show that nature
scientific creativity and orthodoxy is epitomized by the recent remarks of is so constructed. . . . We simply have to accept it. One could perhaps
describe the situation by saying that God is a mathematician of a very high
that magnificent Pythagorean, Paul Dirac. In his article “The Evolution
order and He used very advanced mathematics in constructing the Uni­
of the Physicist’s Picture of Nature,” Dirac, after discussing the circum­
verse. . . .
stances that were prologue to the discovery of Schrodinger’s wave equa­ A good many people are working on the mathematical basis of quantum
tion, states: “I think there is a moral to this story, namely that it is more theory, trying to understand the theory better and to make it more powerful
important to have beauty in one’s equations than to have them fit experi­ and more beautiful. If someone can hit on the right lines along which to
make this development, it may lead to a future advance in which people will
ment.”^ After an analysis of the later developments of quantum theory,
first discover the equations and then, after examining them, gradually learn
Dirac adds:
how to apply them. To some extent that corresponds with the line of de­
velopment that occurred with Schrodinger’s discovery of his wave equation.
I should like to suggest that one not worry too much about this con­
Schrbdinger discovered the equation simply by looking for an equation with
troversy. . . . The present stage of physical theory is merely a steppingstone
mathematical beauty.
towards the better stages that we will have in the future. One can be quite
sure that there will be better stages simply because of the difficulties that 6. Max Born, The Restless Universe, W. M. Deans, trans. (New York: Dover
occur in the physics of today.^ Publications, Inc., 1951), 308-309.
In a scientific community that has as tools such onetime nonortho­
dox inventions as Riemannian geometry, Hamilton’s noncommutative
quaternions, the interconvertability of mass and energy, non-Aristotelian
logics, etc., etc., the Diracian point of view as to the audacity of concep­
tual innovation and a fearless research strategy needs no amens.
The last words belong to Max Born:
The subordination of fundamental research to political and military
authorities is detrimental. The scientists themselves have learned by now
that the period of unrestricted individualism in research has come to an end.
Newton’s First Law: A Philosopher’s Door into Natural Philosophy
Section I
Newton’s first law of motion reads:
N orwood R ussell Hanson EVERY BODY FREE OF IMPRESSED FORCES EITHER PER­
Yale University SEVERES IN A STATE OF REST OR IN UNIFORM RECTI­
LINEAR MOTION AD INFINITUM.^
This is not axiomatic in the ancient sense of being “self-evident.”
Giants from Aristotle through Archimedes, Ptolemy, Copernicus, and on
to Galileo stood against all or part of this claim. Thus Aristotle:
Newton’s First Law:
All movement is either natural or enforced, and force accelerates natural
A Philosopher’s Door motion (e.g., that of stone downwards), and is the sole cause of unnatural.2
Again:
into Natural Philosophy
It is clear, then, that in all cases of local movement there will be noth­
ing between the mover and the moved, if it can be shown that the pushing
or pulling agent must be in direct contact with the load. But this follows
directly from our definitions, for pushing moves things away (either from
the agent or from something else) to some other place, and pulling moves
things from some other place either to the agent or to something else.^
Not the state of rest, but the states of uniform translation form an And again:
objectively distinguished class of motions, and this puts an end to the
For it is only when something external moves a thing, or brings it to
substantial ether. Finally, and fourthly, the general relativity theory
rest against its own internal tendency, that we say this happens by force;
re-endows this metric world structure with the capacity of reacting to the
otherwise we do not say that it happens by force.^
forces of matter. Thus, in a sense, the circle is closed.
The only counterthesis to Aristotle’s dicta is that of Philoponus (6th
—Hermann Weyl, Philosophy of Mathematics and Natural Science
century a.d.), who attacked the idea of antiperistasis by insisting that, so
far from being essential to the motion of a projectile, the medium acts
merely to resist such motion. The arrow does not continue its flight be­
Too often the intellectual excitement of science seems geared only to
cause of the parted air burbling around behind the tail of the shaft;
research in contemporary physics. Philosophers continually discuss
rather, the arrow would go much further were it not for the resistance
cosmology, relativity, and microphysics. In such areas one’s ideas
of this very burbling air. Thus:
are stretched and strained beyond what our ancestors might have
anticipated. Historians of science also have focused attention on past But a certain additional time is required because of the interference of
events only by remarking their analogies and similarities with per­ the medium. For the pressure of the medium and the necessity of cutting
through it make motion through it more difficult.''’
plexities in physics today. They do not always do this of course, but
it happens enough to warrant comment. And again:
However, there are statements, hypotheses, and theories of “clas­
Rather it is necessary to assume that some incorporeal motive force is
sical” science that are rewarding in themselves—without having to
imparted by the projector to the projectile, and that the air set in motion
be referred to the agonies that now confound quantum theory and contributes either nothing at all or else very little to this motion of the
cosmology. Specifically, Newton’s first law of motion—the “law of projectile.®
inertia”—which has everything a logician of science could desire.
However, before Buridan, the Aristotelian outlook prevailed—and
Understanding the complexities and perplexities of this fundamental
was fairly reasonable, given the available observations. The tugging and
mechanical statement is in itself to gain insight into what theoretical
pushing of carts seemed to indicate clearly that a steady force was con­
physics in general really is.
8 Beyond the Edge of Certainty Newton's First Law: A Philosopher’s Door into Natural Philosophy y
tinually necessary to keep a cart moving^ This law of Newton's, however,
Again:
pronounces that, once moving, the cart (ideally) will continue to move ad
indefinitum; the tugging horse functions only to overcome frictional re­ The spirit of motion, evoked by the child, exists invisibly in the top;
it stays in the top for a longer or shorter time according to the strength of
sistance, wind resistance, and the like. But nothing observed by the an­
the impression by which this virtue has been communicated; as soon as the
cients could have substantiated such a “law,” That is why Philoponus’s
spirit ceases to enliven the top, the top falls.i^
“anti-Aristotelian” views about motion did not carry the day.
But the progress toward Newton’s first law is not unbroken; even the
If one wishes to draw a line of separation between the realm of ancient great Kepler regards continued force as being necessary merely to main­
and modern science, it must be drawn at the instant when Jean Buridan
tain the planetary motions:
conceived his theory of momentum, when he gave up the idea that stars
are kept in motion by certain divine intelligences, and when he proclaimed The body of the sun is circular and magnetic and is turned in its space
that both celestial and earthly motions are subject to the same mechanical [path?] carrying the sphere of its influence which is not attractive but
laws.8 [rather] propulsive. (Solis corpus est circulariter magneticum et convertitur
in suo spado, transferens orbem virtutis suae, quae non est attractoria sed
Thus Buridan:
promotoria.y^
With respect to the heavens as a whole, one should envisage one con­
Galileo’s teacher, Buonamici, writes:
tinual influence penetrating all the way to the center: nevertheless that in­
fluence has another property and power near the heavens and far off. And Still, whenever a moving object does not itself generate the force in
heavy and light bodies arrange themselves in this lower world because of question, it is then said to be moved by force. That is, while acquiring its
this influence which is diversified in power [force?] above and below. And ultimate position, it does not itself possess a “moving propensity” since it
this should not be denied [simply] because we fail to perceive this influence does not achieve motion from itself. (Vi autem moveri ilia dicuntur quan-
—since we also do not perceive that which is diffused from a magnet to iron docunque id quod movetur non confert vim, hoc est non habet illo pro-
through a medium which, nevertheless, is of great force. (Debemus imaginari pensionem, quo movetur, quia. s. non perficiatur ex eo motu, locum ilium
a toto caelo unam infiuentiam continuam usque ad centrum; tamen ilia in- adipiscens in quo conservetur.y^
fiuentia prope caelum et remote habet aliam proprietatem et virtutem, et
propter illam infiuentiam sic virtualiter diversificatam superius et inferius Galileo himself is far from clear about precisely how to express his
ordinant se gravia et levia in hoc mundo inferiori. Et non debet hoc negari brilliant insight:
ex eo quod illam infiuentiam non percipimus sensibiliter, quia etiam non
percipimus illam quae de magnete multiplicatur per medium usque ad fer- . . . Therefore the impetus, ability, energy, or, as one might say, the
rum, quae tamen est magnae virtutis.)^ momentus of descent of the moving body is diminished by the plane upon
which it is supported and along which it rolls. (L’impeto, il talento, I’energia,
Buridan is here generating nothing less than the impetus theory, o vogliamo dire il momento del discendere. . .
which is itself genetically connected with Newton’s first law:
Benedetti had already formulated the law of inertia as early as 1585:
One can say that God, when creating the world, has moved, as He
Once moving they are never at rest unless impeded. . . . Once in
pleased, each of the celestial orbits; He has given to each of them an im­
motion only an outside force can restrain them. (Mota semel nunquam
petus which kept them moving since then. . . . Thus He could rest on the
quiescunt, nisi impediantur. . . . Quod semel movetur semper . . . move­
seventh day from the work He had done.^®
tur dum ab extrinseco impediatur.y*
Albert of Saxony reiterates this theory of circular impetus:
Indeed, several natural philosophers—e.g., Descartes, Gassendi, Ba-
When God created the celestial spheres. He put each of them in motion liani—were actively exploring along these “inertial” lines. But it was the
as He pleased; and they continue in their motion still today by virtue of the considerable thoroughness of Galileo’s inquiries which finally succeeded
impetus which He impressed on them; this impetus is not subjected to any
in reversing the philosopher’s verdict.
diminution, since the mobile has no inclination which could oppose the
impetus, as no corruption there exists.^
Section II
Nicholas of Cusa comments:
Galileo argued that a sphere, after rolling down a plane on one side
... It is not you nor your spirit who move immediately the globe
of a room, would cross the floor and ascend an inclined plane on the op­
which is now rotating in front of you. It is, however, you who initiate this
posite side. Moreover (ignoring friction), the sphere will ascend the sec­
motion, since the impulsion of your hand, following your will, produced an
impetus and as long as this impetus endures the globe continues to move.12 ond plane to precisely that height above the floor from which it had been
10 Beyond the Edge of Certainty Newton's First Law: A Philosopher’s Door into Natural Philosophy 11
released onto the first inclined plane. What the sphere acquires in its de­ then be drawn through all those possible inclinations to which we just
scent is thus exactly equal to driving it back up to its original height on referred.
the second slope. This is not self-evident either. But intuitively it seems
somehow more plausible than any bald statement of the law of inertia
which itself, apparently, is now easily generated.
Suppose we incline the second plane less and less steeply. The obser­
vation still holds; the sphere will still seek a height on that second plane
equal to that from which it started down the first. The second plane is
lowered and gets closer to the floor; the sphere travels further and further
along the plane “in order to” attain again its original height. (Think of a
soapbox derby racer crossing the finish line; it will either come to a stop
in a short distance by climbing a steep hill or else travel further by as­ Consider the intersection of A-B with each of these inclined planes.
cending a shallow hill.) Now, as the angle between the floor and the sec­ At each point of intersection ( ) note the deceleration of the ascending
ond plane gets closer to zero, the distance the sphere will travel along it sphere—i.e., the rate at which its velocity is falling off as it climbs this
and up it will increase; it will, indeed, proceed toward an infinite length secondary plane. We would expect such instantaneous decelerations to be
of travel as the angle inclination proceeds toward zero. (Thus the derby greater on a steeply inclined plane than at the corresponding point on a
racer would proceed to infinity were its path unobstructed with hills, steep plane of shallow inclination. And as this plane is “flattened” from its
or shallow, or with friction, wind resistance, or gravity.)^® original inclination down to where it coincides with the floor, so similarly
the value of the deceleration variable as it “moves” along the line A-B
will itself decrease. Ultimately, when the plane does join the floor, the
deceleration will be absolutely zero (for now the plane and A-B will “in­
tersect” only at infinity). A subsidiary argument will apply in the case of
acceleration: since there is nowhere in this Gedankenexperiment the pos­
sibility that the sphere might gain in its accelerations as it ascends the
second slope, the first argument is decisive. The point here is that the uni­
formity of a body’s (force free) motion can be argued for; it need not be
This much alone indicates that a perfect sphere in motion on an ideal
(frictionless) floor will move along a straight line to infinity. Only some­ merely assumed.®^
All of which suggests that “rectilinearity,” “motion ad infinitum/’
thing like our second inclined plane would prevent this by taking from
“uniform,” and “force free” are interdependent conceptions within classi­
the rolling sphere just that which distinguished it from a motionless
cal mechanics. It is possible to treat the idea of uniform, rectilinear mo­
sphere resting on the floor.
tion ad infinitum as itself built into the notion “force free”—as part of
Galileo’s reflections, and those of most physics texts, stop at this
the latter’s semantical content. Thinking of a body free of impressed
point.^® Doubtless, this much does constitute a convincing suggestion as
forces, then, is merely thinking of a body either at rest or else in uniform,
to the nonterminating, rectilinear character of force free motion. But that
rectilinear motion ad infinitum. But this same semantical game can also
such motion will be uniform is usually assumed to follow qualitatively
be played by packing “force free” into one of the other concepts: uni­
from such a “thought experiment”; either that or it seems to follow from
form, or rectilinear, or motion ad infinitum. And so on.
the very definition of “force free motion.”^®
Thus the first thing one learns about Newton’s first law is that its
But why assume what is itself demonstrable? Why discuss qualita­
terms are semantically linked. The meaning of some of its constituent
tively what can be proved quantitatively? Think of a hypothetical class
terms “unpacks” sometimes from one or two of the others; but then some­
of all those inclined planes that could be nested within the angle between
times the meaning of these others unpacks from that of the first. Which
our secondary plane and the floor. As our second plane is inclined less
are the “contained,” and which the semantical “containers,” can affect
and less steeply, every possible intervening angle with the floor will ulti­
the logical exposition of any mechanical theory built thereupon. In just
mately have been traversed by the plane, and (ideally) by the ascending
this way one can distinguish the elegant mechanical theories of Lagrange
sphere. Now imagine a line parallel with the floor, but lower than the
and Hertz. Archimedes had required an immovable platform away from
original height from which the sphere descended. The line (A-B) could