Dynamics versus Kinetics
At the present time the human mind is in danger of confusing the realm
of dynamic events, into which modern atomic research has penetrated,
with the world of the spirit; that is, the world whence nature is
endowed with intelligent design, and of which human thinking is an
expression in terms of consciousness. If a view of nature as a
manifestation of spirit, such as Goethe and kindred minds conceived it,
is to be of any significance in our time, it must include a conception
of matter which shows as one of its attributes its capacity to serve
Form (in the sense in which Ruskin spoke of it in opposition to mere
Force) as a means of manifestation.
The present part of this book, comprising Chapters VIII-XI, will be
devoted to working out such a conception of matter. An example will
thereby be given of how Goethe's method of acquiring understanding of
natural phenomena through reading the phenomena themselves may be
carried beyond his own field of observation. There are, however,
certain theoretical obstacles, erected by the onlooker-consciousness,
which require to be removed before we can actually set foot on the new
path. The present chapter will in particular serve this purpose.
*
Science, since Galileo, has been rooted in the conviction that the
logic of mathematics is a means of expressing the behaviour of natural
events. The material for the mathematical treatment of sense data is
obtained through measurement. The actual thing, therefore, in which the
scientific observer is interested in each case, is the position of some
kind of pointer. In fact, physical science is essentially, as Professor
Eddington put it, a 'pointer-reading science'. Looking at this fact in
our way we can say that all pointer instruments which man has
constructed ever since the beginning of science, have as their model
man himself, restricted to colourless, non-stereoscopic observation.
For all that is left to him in this condition is to focus points in
space and register changes of their positions. Indeed, the perfect
scientific observer is himself the arch-pointer-instrument.
The birth of the method of pointer-reading is marked by Galileo's
construction of the first thermometer (actually, a thermoscope). The
conviction of the applicability of mathematical concepts to the
description of natural events is grounded in his discovery of the
so-called Parallelogram of Forces. It is with these two innovations
that we shall concern ourselves in this chapter.
Let it be said at once that our investigations will lead to the
unveiling of certain illusions which the spectator-consciousness has
woven round these two gifts of Galileo. This does not mean that their
significance as fundamentals of science will be questioned. Nor will
the practical uses to which they have been put with so much success be
criticized in any way. But there are certain deceptive ideas which
became connected with them, and the result is that to-day, when man is
in need of finding new epistemological ground under his feet, he is
entangled in a network of conceptual illusions which prevent him from
using his reason with the required freedom.
A special word is necessary at this point regarding the term illusion,
as it is used here and elsewhere. In respect of this, it will be well
to remember what was pointed out earlier in connexion with the term
'tragedy' (Chapter II). In speaking of 'illusion', we neither intend to
cast any blame on some person or another who took part in weaving the
illusion, nor to suggest that the emergence of it should be thought of
as an avoidable calamity. Rather should illusion be thought of as
something which man has been allowed to weave because only by his own
active overcoming of it can he fulfil his destiny as the bearer of
truth in freedom. Illusion, in the sense used here, belongs to those
things in man's existence which are truly to be called tragic. It loses
this quality, and assumes a quite different one, only when man, once
the time has come for overcoming an illusion, insists on clinging to
it.
As our further studies will show, the criticism to be applied here does
not only leave the validity of measurement and the mathematical
treatment of the data thus obtained fully intact, but by giving them
their appropriate place in a wider conception of nature it opens the
way to an ever more firmly grounded and, at the same time, enhanced
application of both.
*
Our primary knowledge of the existence of something we call 'warmth' or
'heat' is due to a particular sense of warmth which modern research has
recognized as a clearly definable sense. Naturally, seen from the
spectator-standpoint, the experiences of this sense appear to be of
purely subjective value and therefore useless for obtaining an
objective insight into the nature of warmth and its effects in the
physical world. In order to learn about these, resort is had to certain
instruments which, through the change of the spatial position of a
point, allow the onlooker-observer to register changes in the thermal
condition of a physical object. An instrument of this kind is the
thermometer. In the following way an indubitable proof seems to be
given of the correctness of the view concerning the subjectivity of the
impressions obtained through the sense of warmth, and of the
objectivity of thermometrical measurement. A description of it is
frequently given in physical textbooks as an introduction to the
chapter on Heat.
To begin with, the well-known fact is cited that if one plunges one's
hands first into two different bowls, one filled with hot water and the
other with cold, and then plunges them together into a bowl of tepid
water, this will feel cold to the hand coming from the hot water and
warm to the hand coming from the cold. Next, it is pointed out that two
thermometers which are put through the same procedure will register an
equal degree of temperature for the tepid water. In this way the
student is given a lasting impression of the superiority of the
'objective' recording of the instrument over the 'subjective' character
of the experiences mediated by his sense of warmth.
Let us now test this procedure by carrying out the same experiment with
the help of thermometrical instruments in their original form, that is,
the form in which Galileo first applied them. By doing so we proceed in
a truly Goethean manner, because we divest the experiment of all
accessories which prevent the phenomenon from appearing in its primary
form.
To turn a modern thermometer into a thermoscope we need only remove the
figures from its scale. If we make the experiment with two such
thermoscopes we at once become aware of something which usually escapes
us, our attention being fixed on the figures recorded by the two
instruments. For we now notice that the two instruments, when
transferred from the hot and cold water into the tepid water, behave
quite differently. In one the column will fall, in the other it will
rise.
It is important to note that by this treatment of the two instruments
we have not changed the way in which they usually indicate temperature.
For thermometrical measurement is in actual fact never anything else
than a recording of the movement of the indicator from one level to
another. We choose merely to take a certain temperature level - that of
melting ice or something else - as a fixed point of reference and mark
it once for all on the instrument. Because we find this mark clearly
distinguished on our thermometers, and the scales numbered accordingly,
we fail to notice what lies ideally behind this use of the same zero
for every new operation we undertake.
What the zero signifies becomes clear directly we start to work with
thermometers not marked with scales. For in order to be used in this
form as real thermometers, they must be exposed on each occasion first
of all to some zero level of temperature, say, that of melting ice. If
we then take them into the region of temperature we want to measure, we
shall discern the difference of levels through the corresponding
movement of the column. The final position of the column tells us
nothing in itself. It is always the change from one level to another
that the thermometer registers - precisely as does the sense of warmth
in our hands in the experiment just described.
Hence we see that in the ordinary operation with the thermometers, and
when we use our hands in the prescribed manner, we are dealing with the
zero level in two quite different ways. While in the/two instruments
the zero level is the same, in accordance with the whole idea of
thermometric measurement, we make a special arrangement so as to expose
our hands to two different levels. So we need not be surprised if these
two ways yield different results. If, after placing two thermometers
without scales in hot and cold water, we were to assign to each its own
zero in accordance with the respective height of its column, and then
graduate them from this reference point, they would necessarily record
different levels when exposed to the tepid water, in just the same way
as the hands do. Our two hands, moreover, will receive the same
sense-impression from the tepid water, if we keep them in it long
enough.
Seen in this light, the original experiment, designed to show the
subjective character of the impressions gained through the sense of
warmth, reveals itself as a piece of self-deception by the
onlooker-consciousness. The truth of the matter is that, in so far as
there is any subjective element in the experience and measurement of
heat, it does not lie on the side of our sense of warmth, but in our
judgment of the significance of thermometrical readings. In fact, our
test of the alleged proof of the absolute superiority of
pointer-readings over the impressions gained by our senses gives us
proof of the correctness of Goethe's statement, quoted earlier, that
the senses do not deceive, but the judgment deceives.
Let it be repeated here that what we have found in this way does not
lead to any depreciation of the method of pointer-reading. For the
direct findings of the senses cannot be compared quantitatively. The
point is that the idea of the absolute superiority of physical
measurement as a means of scientific knowledge, in all circumstances,
must be abandoned as false.
*
We now turn to Galileo's discovery known as the theorem of the
Parallelogram of Forces. The illusion which has been woven round this
theorem expresses itself in the way it is described as being connected
ideally with another theorem, outwardly similar in character, known as
the theorem of the Parallelogram of Movements (or Velocities), by
stating that the former follows logically from the latter. This
statement is to be found in every textbook on physics at the outset of
the chapter on dynamics (kinetics), where it serves to establish the
right to treat the dynamic occurrences in nature in a purely kinematic
fashion, true to the requirements of the onlooker-consciousness.1
The following description will show that, directly we free ourselves
from the onlooker-limitations of our consciousness in the way shown by
Goethe - and, in respect of the present problem, in particular also by
Reid - the ideal relationship between the two theorems is seen to be
precisely the opposite to the one expressed in the above statement. The
reason why we take pains to show this at the present point of our
discussion is that only through replacing the fallacious conception by
the correct one, do we open the way for forming a concrete concept of
Force and thereby for establishing a truly dynamic conception of
nature.
*
Let us begin by describing briefly the content of the two theorems in
question. In Fig. 1, a diagrammatical representation is given of the
parallelogram of movements. It sets out to show that when a point moves
with a certain velocity in the direction indicated by the arrow a, so
that in a certain time it passes from P to A, and when it
simultaneously moves with a second velocity in the direction indicated
by
b, through which alone it would pass to B in the same time, its actual
movement is indicated by c, the diagonal in the parallelogram formed by
a and b. An example of the way in which this
theorem is practically applied is the well-known case of a rower who
sets out from P in order to cross at right angles a river indicated by
the parallel lines. He has to overcome the velocity a of the water of
the river flowing to the right by steering obliquely left towards B in
order to arrive finally at C.
It is essential to observe that the content of this theorem does not
need the confirmation of any outer experience for its discovery, or to
establish its truth. Even though the recognition of the fact which it
expresses may have first come to men through practical observation, yet
the content of this theorem can be discovered and proved by purely
logical means. In this respect it resembles any purely geometrical
statement such as, that the sum of the angles of a triangle is two
right angles (180°). Even though this too may have first been learnt
through outer observation, yet it remains true that for the discovery
of the fact expressed by it - valid for all plane triangles - no outer
experience is needed. In both cases we find ourselves in the domain of
pure geometric conceptions (length and direction of straight lines,
movement of a point along these), whose reciprocal relationships are
ordered by the laws of pure geometric logic. So in the theorem of the
Parallelogram of Velocities we have a strictly geometrical theorem,
whose content is in the narrowest sense kinematic. In fact, it is the
basic theorem of kinematics.
We now turn to the second theorem which speaks of an outwardly similar
relationship between forces. As is well nown, this states that two
forces of different magnitude and direction, when they apply at the
same point, act together in the manner of a single force whose
magnitude and direction may be represented by the diagonal of a
parallelogram whose sides express in extent and direction the first two
forces. Thus in Fig. 2, R exercises upon P the same effect as F1 and F2
together.
Expressed in another way, a force of this magnitude working in the
reverse direction (R') will establish an equilibrium with the other two
forces. In technical practice, as is well known, this theorem is used
for countless calculations, in both statics and dynamics, and indeed
more frequently not in the form given here but in the converse manner,
when a single known force is resolved into two component forces.
(Distribution of a pressure along frameworks, of air pressure along
moving surfaces, etc.)
It will now be our task to examine the logical link which is believed
to connect one theorem with the other. This link is found in the
well-known definition of physical force as a product of 'mass' and
'acceleration' - in algebraic symbols F=ma. We will discuss the
implications of this definition in more detail later on. Let us first
see how it is used as a foundation for the above assertion.
The conception of 'force' as the product of 'mass' and 'acceleration'
is based on the fact - easily experienced by anyone who cycles along a
level road - that it is not velocity itself which requires the exertion
of force, but the change of velocity - that is, acceleration or
retardation ('negative acceleration' in the sense of mathematical
physics); also that in the case of equal accelerations, the force
depends upon the mass of the accelerated object. The more massive the
object, the greater will be the force necessary for accelerating it.
This mass, in turn, reveals itself in the resistance a particular
object offers to any change of its state of motion. Where different
accelerations and the same mass are considered, the factor m in the
above formula remains constant, and force and acceleration are directly
proportional to each other. Thus in the acceleration is discovered a
measure for the magnitude of the force which thereby acts.
Now it is logically evident that the theorem of the parallelogram of
velocities is equally valid for movements with constant or variable
velocities. Even though it is somewhat more difficult to perceive
mentally the movement of a point in two different directions with two
differently accelerated motions, and to form an inner conception of the
resulting movement, we are nevertheless still within a domain which may
be fully embraced by thought. Thus accelerated movements and movements
under constant velocity can be resolved and combined according to the
law of the parallelogram of movements, a law which is fully attainable
by means of logical thought.
With the help of the definition of force as the product of mass and
acceleration it seems possible, indeed, to derive the parallelogram of
forces from that of accelerations in a purely logical manner. For it is
necessary only to extend all sides of an a parallelogram by means of
the same factor m in order to turn it into an F parallelogram. A single
geometrical figure on paper can represent both cases, since only the
scale needs to be altered in order that the same geometrical length
should represent at one time the magnitude a and on another occasion
ma. It is in this way that present-day scientific thought keeps itself
convinced that the parallelogram of forces follows with logical
evidence from the parallelogram of accelerations, and that the
discovery of the former is therefore due to a purely mental process.
Since the parallelogram of forces is the prototype of each further
mathematical representation of physical force-relationships in nature,
the conceptual link thus forged between it and the basic theorem of
kinematics has led to the conviction that the fact that natural events
can be expressed in terms of mathematics could be, and actually has
been, discovered through pure logical reasoning, and thus by the
brain-bound, day-waking consciousness 'of the world-spectator.
Justification thereby seemed to be given for the building of a valid
scientific world-picture, purely kinematic in character.
*
The line of consideration we shall now have to enter upon for carrying
out our own examination of what is believed to be the link between the
two theorems may seem to the scientifically trained reader to be of an
all too elementary kind compared with the complexities of thought in
which he is used to engage in order to settle a scientific problem. It
is therefore necessary to state here that anyone who wishes to help to
overcome the tangle of modern theoretical science must not be shy in
applying thoughts and observations of seemingly so simple a nature as
those used both here and on other occasions. Some readiness, in fact,
is required to play where necessary the part of the child in Hans
Andersen's fairy-story of The Emperor's New Clothes, where all the
people are loud in praise of the magnificent robes of the Emperor, who
is actually passing through the streets with no clothes on at all, and
a single child's voice exclaims the truth that 'the Emperor has nothing
on'. There will repeatedly be occasion to adopt the role of this child
in the course of our own studies.
*
In the scientific definition of force given above force appears as the
result of a multiplication of two other magnitudes. Now as is well
known, it is essential for the operation of multiplication that of the
two factors forming the product at least one should exhibit the
properties of a pure number. For two pure numbers may be multiplied
together - e.g. 2 and 4 - and a number of concrete things can be
multiplied by a pure number - e. g. 3 apples and the number 4 - but no
sense can be attached to the multiplication of 3 apples by 4 apples,
let alone by 4 pears! The result of multiplication is therefore always
either itself a pure number, when both factors have this property; or
when one of the two factors is of the nature of a concrete object, the
result is of the same quality as the latter. An apple will always
remain an apple after multiplication, and what distinguishes the final
product (apples) from the original factor (apples) is only a pure
number.
If we take seriously what this simple consideration tells us of the
nature of multiplication, and if we do not allow ourselves to deviate
from it for whatever purpose we make use of this algebraic operation,
then the various concepts we connect with the basic measurements in
physics undergo a considerable change of meaning.
Let us test, in this respect, the well-known formula which, in the
conceptual language of physics, connects 'distance' (s), 'time' (t),
and 'velocity' (c). It is written
c = s / t, or s = ct.
In this formula, s has most definitely the meaning of a 'thing', for it
represents measured spatial distance. Of the two factors on the other
side of the second equation, one must needs have the same quality as s:
this is c. Thus for the other factor, t, there remains the property of
a pure number. We are, therefore, under an illusion if we assume the
factor c to represent anything of what velocity implies in outer cosmic
reality. The truth is that c represents a spatial distance just as s
does, with the difference only that it is a certain unit-distance. Just
as little does real time enter into this formula - nor does it into any
other formula of mathematical physics. 'Time', in physics, is always a
pure number without any cosmic quality. Indeed, how could it be
otherwise for a purely kinematic world-observation?
We now submit the formula F=ma to the same scrutiny. If we attach to
the factor a on the right side of the equation a definite quality,
namely an observable acceleration, the other factor in the product is
permitted to have only the properties of a pure number; F, therefore,
can be only of the same nature as a and must itself be an acceleration.
Were it otherwise, then the equation F=ma could certainly not serve as
a logical link between the Velocity and Force parallelograms.
Our present investigation has done no more than grant us an insight
into the process of thought whereby the consciousness limited to a
purely kinematic experience has deprived the concept of force of any
real content. Let us look at the equation F=ma as a means of splitting
of the magnitude F into two components m and a. The equation then tells
us that F is reduced to the nature of pure acceleration, for that which
resides in the force as a factor not observable by kinematic vision has
been split away from it as the factor m. For this factor, however, as
we have seen, nothing remains over but the property of a pure number.
Let us note here that the first thinker to concern himself with a
comprehensive world-picture in which the non-existence of a real
concept of force is taken in earnest-namely, Albert Einstein - was also
the first to consider mass as a form of energy and even to predict
correctly, as was proved later, the amount of energy represented by the
unit of mass, thereby encouraging decisively the new branch of
experimental research which has led to the freeing of the so-called
atomic energy. Is it then possible that pure numbers can effect what
took place above and within Nagasaki, Hiroshima, etc.? Here we are
standing once again before one of the paradoxes of modern science which
we have found to play so considerable a part in its development.
To find an interpretation of the formula F=ma, which is free from
illusion, we must turn our attention first of all to the concepts
'force' and 'mass' themselves. The fact that men have these two words
in their languages shows that the concepts expressed by them must be
based on some experience that has been man's long before he was capable
of any scientific reflexion. Let us ask what kind of experience this is
and by what part of his being he gathers it.
The answer is, as simple self-observation will show, that we know of
the existence of force through the fact that we ourselves must exert it
in order to move our own body. Thus it is the resistance of our body
against any alteration of its state of motion, as a result of its being
composed of inert matter, which gives us the experience of force both
as a possession of our own and as a property of the outer world. All
other references to force, in places where it cannot be immediately
experienced, arise by way of analogy based on the similarity of the
content of our observation to that which springs from the exertion of
force in our own bodies.
As we see, in this experience of force that of mass is at once implied.
Still, we can strengthen the latter by experimenting with some outer
physical object. Take a fairly heavy object in your hand, stretch out
your arm lightly and move it slowly up and down, watching intently the
sensation this operation rouses in you.2 Evidently the experience of
mass outside ourselves, as with that of our own body, comes to us
through the experience of the force which we ourselves must exert in
order to overcome some resisting force occasioned by the mass. Already
this simple observation - as such made by means of the sense of
movement and therefore outside the frontiers of the
onlooker-consciousness - tells us that mass is nothing but a particular
manifestation of force.
Seen in the light of this experience, the equation F=ma requires to be
interpreted in a manner quite different from that to which scientific
logic has submitted it. For if we have to ascribe to F and m the same
quality, then the rule of multiplication allows us to ascribe to a
nothing but the character of a pure number. This implies that there is
no such thing as acceleration as a self-contained entity, merely
attached to mass in an external way.
What we designate as acceleration, and measure as such, is nothing else
than a numerical factor comparing two different conditions of force
within the physical-material world.
Only when we give the three factors in our equation this meaning, does
it express some concrete outer reality. At the same time it forbids the
use of this equation for a logical derivation of the parallelogram of
forces from that of pure velocities.
*
The same method which has enabled us to restore its true meaning to the
formula connecting mass and force will serve to find the true source of
man's knowledge of the parallelogram of forces. Accordingly, our
procedure will be as follows.
We shall engage two other persons, together with whom we shall try to
discover by means of our respective experiences of force the law under
which three forces applying at a common point may hold themselves in
equilibrium. Our first step will consist in grasping each other by the
hand and in applying various efforts of our wills to draw one another
in different directions, seeing to it that we do this in such a way
that the three joined hands remain undisturbed at the same place. By
this means we can get as far as to establish that, when two persons
maintain a steady direction and strength of pull, the third must alter
his applied force with every change in his own direction in order to
hold the two others in equilibrium. He will find that in some instances
he must increase his pull and in other instances decrease it.
This, however, is all that can be learnt in this way. No possibility
arises at this stage of our investigation of establishing any exact
quantitative comparison. For the forces which we have brought forth
(and this is valid for forces in general, no matter of what kind they
are) represent pure intensities, outwardly neither visible nor directly
measurable. We can certainly tell whether we are intensifying or
diminishing the application of our will, but a numerical comparison
between different exertions of will is not possible.
In order to make such a comparison, a further step is necessary. We
must convey our effort to some pointer-instrument - for instance, a
spiral spring which will respond to an exerted pressure or pull by a
change in its spatial extension. (Principle of the spring balance.) In
this way, by making use of a certain property of matter - elasticity -
the purely intensive magnitudes of the forces which we exert become
extensively visible and can be presented geometrically. We shall
therefore continue our investigation with the aid of three spring
balances, which we hook together at one end while exposing them to the
three pulls at the other.
To mark the results of our repeated pulls of varying intensities and
directions, we draw on the floor on which we stand three chalk lines
outward from the point underneath the common point of the three
instruments, each in the direction taken up by one of the three
persons. Along these lines we mark the extensions corresponding to
those of the springs of the instruments.
By way of this procedure we shall arrive at a sequence of figures such
as is shown in Fig. 3. This is all we can discover empirically
regarding the mutual relationships of three forces engaging at a point.
Let us now heed the fact that nothing in this group of figures reveals
that in each one of these trios of lines there resides a definite and
identical geometrical order; nor do they convey anything that would
turn our thoughts to the parallelogram of velocities with the effect of
leading us to expect, by way of analogy, a similar order in these
figures. And this result, we note, is quite independent of our
particular way of procedure, whether we use, right from the start, a
measuring instrument, or whether we proceed as described above.
*
Having in this way removed the fallacious idea that the parallelogram
of forces can, and therefore ever has been, conceived by way of logical
derivation from the parallelogram of velocities, we must then ask
ourselves what it was, if not any act of logical reason, that led
Galileo to discover it.
History relates that on making the discovery he exclaimed: 'La natura è
scritta in lingua matematica!' ('Nature is recorded in the language of
mathematics.') These words reveal his surprise when he realized the
implication of his discovery. Still, intuitively he must have known
that using geometrical lengths to symbolize the measured magnitudes of
forces would yield some valid result. Whence came this intuition, as
well as the other which led him to recognize from the figures thus
obtained that in a parallelogram made up of any two of the three lines,
the remaining line came in as its diagonal? And, quite apart from the
particular event of the discovery, how can we account for the very fact
that nature - at least on a certain level of her existence - exhibits
rules of action expressible in terms of logical principles immanent in
the human mind?
*
To find the answer to these questions we must revert to certain facts
connected with man's psycho-physical make-up of which the
considerations of Chapter II have already made us aware.
Let us, therefore, transpose ourselves once more into the condition of
the child who is still entirely volition, and thus experiences himself
as one with the world. Let us consider, from the point of view of this
condition, the process of lifting the body into the vertical position
and the acquisition of the faculty of maintaining it in this position;
and let us ask what the soul, though with no consciousness of itself,
experiences in all this. It is the child's will which wrestles in this
act with the dynamic structure of external space, and what his will
experiences is accompanied by corresponding perceptions through the
sense of movement and other related bodily senses. In this way the
parallelogram of forces becomes an inner experience of our organism at
the beginning of our earthly life. What we thus carry in the body's
will-region in the form of experienced geometry - this, together with
the freeing and crystallizing of part of our will-substance into our
conceptual capacity, is transformed into our faculty of forming
geometrical concepts, and among them the concept of the parallelogram
of movements.
Looked at in this way, the true relationship between the two
parallelogram-theorems is seen to be the very opposite of the one held
with conviction by scientific thinking up to now. Instead of the
parallelogram of forces following from the parallelogram of movements,
and the entire science of dynamics from that of kinematics, our very
faculty of thinking in kinematic concepts is the evolutionary product
of our previously acquired intuitive experience of the dynamic order of
the world.
If this is the truth concerning the origin of our knowledge of force
and its behaviour on the one hand, and our capacity to conceive
mathematical concepts in a purely ideal way on the other, what is it
then that causes man to dwell in such illusion as regards the
relationship between the two? From our account it follows that no
illusion of this kind could arise if we were able to remember
throughout life our experiences in early childhood. Now we know from
our considerations in Chapter VI that in former times man had such a
memory. In those times, therefore, he was under no illusion as to the
reality of force in the world. In the working of outer forces he saw a
manifestation of spiritual beings, just as in himself he experienced
force as a manifestation of his own spiritual being. We have seen also
that this form of memory had to fade away to enable man to find himself
as a self-conscious personality between birth and death. As such a
personality, Galileo was able to think the parallelogram of forces, but
he was unable to comprehend the origin of his faculty of mathematical
thinking, or of his intuitive knowledge of the mathematical behaviour
of nature in that realm of hers where she sets physical forces into
action.
Deep below in Galileo's soul there lived, as it does in every human
being, the intuitive knowledge, acquired in early childhood, that part
of nature's order is recordable in the conceptual language of
mathematics. In order that this intuition should rise sufficiently far
into his conscious mind to guide him, as it did, in his observations,
the veil of oblivion which otherwise separates our waking consciousness
from the experiences of earliest childhood must have been momentarily
lightened. Unaware of all this, Galileo was duly surprised when in the
onlooker-part of his being the truth of his intuition was confirmed in
a way accessible to it, namely through outer experiment. Yet with the
veil immediately darkening again the onlooker soon became subject to
the illusion that for his recognition of mathematics as a means of
describing nature he was in need of nothing but what was accessible to
him on the near side of the veil.
Thus it became man's fate in the first phase of science, which fills
the period from Galileo and his contemporaries up to the present time,
that the very faculty which man needed for creating this science
prevented him from recognizing its true foundations. Restricted as he
was to the building of a purely kinematic world-picture, he had to
persuade himself that the order of interdependence of the two
parallelogram-theorems was the opposite of the one which it really is.
*
The result of the considerations of this chapter is of twofold
significance for our further studies. On the one hand, we have seen
that there is a way out of the impasse into which modern scientific
theory has got itself as a result of the lack of a justifiable concept
of force, and that this way is the one shown by Reid and travelled by
Goethe. 'We must become as little children again, if we will be
philosophers', is as true for science as it is for philosophy. On the
other hand, our investigation of the event which led Galileo to the
discovery that nature is recorded in the language of mathematics, has
shown us that this discovery would not have been possible unless
Galileo had in a sense become, albeit unconsciously, a little child
again. Thus the event that gave science its first foundations is an
occurrence in man himself of precisely the same character as the one
which we have learnt to regard as necessary for building science's new
foundations. The only difference is that we are trying to turn into a
deliberate and consciously handled method something which once in the
past happened to a man without his noticing it.
Need we wonder that we are challenged to do so in our day, when mankind
is several centuries older than it was in the time of Galileo?
1 As to the terms 'kinetic' and 'kinematic', see Chapter II, page 30,
footnote.
2 For the sake of our later studies it is essential that the reader
does not content himself with merely following the above description
mentally, but that he carries out the experiment himself.