THE ORGANISATION OF THOUGHT
(_Presidential Address to Section A, British Association, Newcastle,
1916_)
THE subject of this address is the organisation of thought, a topic
evidently capable of many diverse modes of treatment. I intend more
particularly to give some account of that department of logical science
with which some of my own studies have been connected. But I am
anxious, if I can succeed in so doing, to handle this account so as to
exhibit the relation with certain considerations which underlie general
scientific activities.
It is no accident that an age of science has developed into an age
of organisation. Organised thought is the basis of organised action.
Organisation is the adjustment of diverse elements so that their
mutual relations may exhibit some predetermined quality. An epic poem
is a triumph of organisation, that is to say, it is a triumph in the
unlikely event of its being a good epic poem. It is the successful
organisation of multitudinous sounds of words, associations of words,
pictorial memories of diverse events and feelings ordinarily occurring
in life, combined with a special narrative of great events: the whole
so disposed as to excite emotions which, as defined by Milton, are
simple, sensuous, and passionate. The number of successful epic poems
is commensurate, or rather, is inversely commensurate, with the obvious
difficulty of the task of organisation.
Science is the organisation of thought. But the example of the epic
poem warns us that science is not any organisation of thought. It is
an organisation of a certain definite type which we will endeavour to
determine.
Science is a river with two sources, the practical source and the
theoretical source. The practical source is the desire to direct our
actions to achieve predetermined ends. For example, the British nation,
fighting for justice, turns to science, which teaches it the importance
of compounds of nitrogen. The theoretical source is the desire to
understand. Now I am going to emphasise the importance of theory in
science. But to avoid misconception I most emphatically state that I
do not consider one source as in any sense nobler than the other, or
intrinsically more interesting. I cannot see why it is nobler to strive
to understand than to busy oneself with the right ordering of one's
actions. Both have their bad sides; there are evil ends directing
actions, and there are ignoble curiosities of the understanding.
The importance, even in practice, of the theoretical side of science
arises from the fact that action must be immediate, and takes place
under circumstances which are excessively complicated. If we wait for
the necessities of action before we commence to arrange our ideas, in
peace we shall have lost our trade, and in war we shall have lost the
battle. Success in practice depends on theorists who, led by other
motives of exploration, have been there before, and by some good chance
have hit upon the relevant ideas. By a theorist I do not mean a man who
is up in the clouds, but a man whose motive for thought is the desire
to formulate correctly the rules according to which events occur. A
successful theorist should be excessively interested in immediate
events, otherwise he is not at all likely to formulate correctly
anything about them. Of course, both sources of science exist in all
men.
Now, what is this thought organisation which we call science? The
first aspect of modern science which struck thoughtful observers was
its inductive character. The nature of induction, its importance, and
the rules of inductive logic have been considered by a long series of
thinkers, especially English thinkers: Bacon, Herschel, J. S. Mill,
Venn, Jevons, and others. I am not going to plunge into an analysis
of the process of induction. Induction is the machinery and not the
product, and it is the product which I want to consider. When we
understand the product we shall be in a stronger position to improve
the machinery.
First, there is one point which it is necessary to emphasise. There
is a tendency in analysing scientific processes to assume a given
assemblage of concepts applying to nature, and to imagine that the
discovery of laws of nature consists in selecting by means of inductive
logic some one out of a definite set of possible alternative relations
which may hold between the things in nature answering to these obvious
concepts. In a sense this assumption is fairly correct, especially
in regard to the earlier stages of science. Mankind found itself in
possession of certain concepts respecting nature--for example, the
concept of fairly permanent material bodies--and proceeded to determine
laws which related the corresponding percepts in nature. But the
formulation of laws changed the concepts, sometimes gently by an added
precision, sometimes violently. At first this process was not much
noticed, or at least was felt to be a process curbed within narrow
bounds, not touching fundamental ideas. At the stage where we now
are, the formulation of the concepts can be seen to be as important
as the formulation of the empirical laws connecting the events in the
universe as thus conceived by us. For example, the concepts of life,
of heredity, of a material body, of a molecule, of an atom, of an
electron, of energy, of space, of time, of quantity, and of number. I
am not dogmatising about the best way of getting such ideas straight.
Certainly it will only be done by those who have devoted themselves to
a special study of the facts in question. Success is never absolute,
and progress in the right direction is the result of a slow, gradual
process of continual comparison of ideas with facts. The criterion of
success is that we should be able to formulate empirical laws, that is,
statements of relations, connecting the various parts of the universe
as thus conceived, laws with the property that we can interpret the
actual events of our lives as being our fragmentary knowledge of this
conceived interrelated whole.
But, for the purpose of science, what is the actual world? Has science
to wait for the termination of the metaphysical debate till it can
determine its own subject-matter? I suggest that science has a much
more homely starting-ground. Its task is the discovery of the relations
which exist within that flux of perceptions, sensations, and emotions
which forms our experience of life. The panorama yielded by sight,
sound, taste, smell, touch, and by more inchoate sensible feelings,
is the sole field of activity. It is in this way that science is the
thought organisation of experience. The most obvious aspect of this
field of actual experience is its disorderly character. It is for
each person a _continuum_, fragmentary, and with elements not
clearly differentiated. The comparison of the sensible experiences of
diverse people brings its own difficulties. I insist on the radically
untidy, ill-adjusted character of the fields of actual experience from
which science starts. To grasp this fundamental truth is the first
step in wisdom, when constructing a philosophy of science. This fact
is concealed by the influence of language, moulded by science, which
foists on us exact concepts as though they represented the immediate
deliverances of experience. The result is, that we imagine that we have
immediate experience of a world of perfectly defined objects implicated
in perfectly defined events which, as known to us by the direct
deliverance of our senses, happen at exact instants of time, in a space
formed by exact points, without parts and without magnitude: the neat,
trim, tidy, exact world which is the goal of scientific thought.
My contention is, that this world is a world of ideas, and that its
internal relations are relations between abstract concepts, and that
the elucidation of the precise connection between this world and the
feelings of actual experience is the fundamental question of scientific
philosophy. The question which I am inviting you to consider is this:
How does exact thought apply to the fragmentary, vague _continua_
of experience? I am not saying that it does not apply: quite the
contrary. But I want to know how it applies. The solution I am
asking for is not a phrase, however brilliant, but a solid branch of
science, constructed with slow patience, showing in detail how the
correspondence is effected.
The first great steps in the organisation of thought were due
exclusively to the practical source of scientific activity, without
any admixture of theoretical impulse. Their slow accomplishment was
the cause and also the effect of the gradual evolution of moderately
rational beings. I mean the formation of the concepts of definite
material objects, of the determinate lapse of time, of simultaneity,
of recurrence, of definite relative position, and of analogous
fundamental ideas, according to which the flux of our experience is
mentally arranged for handy reference: in fact, the whole apparatus of
commonsense thought. Consider in your mind some definite chair. The
concept of that chair is simply the concept of all the interrelated
experiences connected with that chair--namely, of the experience of the
folk who made it, of the folk who sold it, of the folk who have seen
it or used it, of the man who is now experiencing a comfortable sense
of support, combined with our expectations of an analogous future,
terminated finally by a different set of experiences when the chair
collapses and becomes firewood. The formation of that type of concept
was a tremendous job, and zoologists and geologists tell us that it
took many tens of millions of years. I can well believe it.
I now emphasise two points. In the first place, science is rooted in
what I have just called the whole apparatus of commonsense thought.
That is the _datum_ from which it starts, and to which it
must recur. We may speculate, if it amuses us, of other beings in
other planets who have arranged analogous experiences according to
an entirely different conceptual code--namely, who have directed
their chief attention to different relations between their various
experiences. But the task is too complex, too gigantic, to be revised
in its main outlines. You may polish up commonsense, you may contradict
it in detail, you may surprise it. But ultimately your whole task is to
satisfy it.
In the second place, neither commonsense nor science can proceed with
their task of thought organisation without departing in some respect
from the strict consideration of what is actual in experience. Think
again of the chair. Among the experiences upon which its concept is
based, I included our expectations of its future history. I should
have gone further and included our imagination of all the possible
experiences which in ordinary language we should call perceptions of
the chair which might have occurred. This is a difficult question, and
I do not see my way through it. But, at present, in the construction of
a theory of space and of time there seem insuperable difficulties if we
refuse to admit ideal experiences.
This imaginative perception of experiences, which, if they occurred,
would be coherent with our actual experiences, seems fundamental in our
lives. It is neither wholly arbitrary, nor yet fully determined. It
is a vague background which is only made in part definite by isolated
activities of thought. Consider, for example, our thoughts of the
unseen flora of Brazil.
Ideal experiences are closely connected with our imaginative
reproduction of the actual experiences of other people, and also
with our almost inevitable conception of ourselves as receiving our
impressions from an external complex reality beyond ourselves. It
may be that an adequate analysis of every source and every type of
experience yields demonstrative proof of such a reality and of its
nature. Indeed, it is hardly to be doubted that this is the case. The
precise elucidation of this question is the problem of metaphysics. One
of the points which I am urging in this address is, that the basis of
science does not depend on the assumption of any of the conclusions of
metaphysics; but that both science and metaphysics start from the same
given groundwork of immediate experience, and in the main proceed in
opposite directions on their diverse tasks.
For example, metaphysics inquires how our perceptions of the chair
relate us to some true reality. Science gathers up these perceptions
into a determinate class, adds to them ideal perceptions of analogous
sort, which under assignable circumstances would be obtained, and this
single concept of that set of perceptions is all that science needs;
unless indeed you prefer that thought find its origin in some legend of
those great twin brethren, the Cock and Bull.
My immediate problem is to inquire into the nature of the texture of
science. Science is essentially logical. The nexus between its concepts
is a logical nexus, and the grounds for its detailed assertions
are logical grounds. King James said, "No bishops, no king." With
greater confidence we can say, "No logic, no science." The reason for
the instinctive dislike which most men of science feel towards the
recognition of this truth is, I think, the barren failure of logical
theory during the past three or four centuries. We may trace this
failure back to the worship of authority, which in some respects
increased in the learned world at the time of the Renaissance. Mankind
then changed its authority, and this fact temporally acted as an
emancipation. But the main fact, and we can find complaints[2] of it
at the very commencement of the modern movement, was the establishment
of a reverential attitude towards any statement made by a classical
author. Scholars became commentators on truths too fragile to bear
translation. A science which hesitates to forget its founders is lost.
To this hesitation I ascribe the barrenness of logic. Another reason
for distrust of logical theory and of mathematics is the belief that
deductive reasoning can give you nothing new. Your conclusions are
contained in your premises, which by hypothesis are known to you.
In the first place this last condemnation of logic neglects the
fragmentary, disconnected character of human knowledge. To know one
premise on Monday, and another premise on Tuesday, is useless to you
on Wednesday. Science is a permanent record of premises, deductions,
and conclusions, verified all along the line by its correspondence
with facts. Secondly, it is untrue that when we know the premises we
also know the conclusions. In arithmetic, for example, mankind are not
calculating boys. Any theory which proves that they are conversant with
the consequences of their assumptions must be wrong. We can imagine
beings who possess such insight. But we are not such creatures.
Both these answers are, I think, true and relevant. But they are
not satisfactory. They are too much in the nature of bludgeons,
too external. We want something more explanatory of the very real
difficulty which the question suggests. In fact, the true answer is
embedded in the discussion of our main problem of the relation of logic
to natural science.
It will be necessary to sketch in broad outline some relevant features
of modern logic. In doing so I shall try to avoid the profound general
discussions and the minute technical classifications which occupy the
main part of traditional logic. It is characteristic of a science in
its earlier stages--and logic has become fossilised in such a stage--to
be both ambitiously profound in its aims and trivial in its handling of
details.
We can discern four departments of logical theory. By an analogy
which is not so very remote I will call these departments or sections
the arithmetic section, the algebraic section, the section of
general-function theory, the analytical section. I do not mean that
arithmetic arises in the first section, algebra in the second section,
and so on; but the names are suggestive of certain qualities of
thought in each section which are reminiscent of analogous qualities
in arithmetic, in algebra, in the general theory of a mathematical
function, and in the mathematical analysis of the properties of
particular functions.
The first section--namely, the arithmetic stage--deals with the
relations of definite propositions to each other, just as arithmetic
deals with definite numbers. Consider any definite proposition; call
it "_p_." We conceive that there is always another proposition which
is the direct contradictory to "_p_"; call it "not-_p_." When we have
got two propositions, _p_ and _q_, we can form derivative propositions
from them, and from their contradictories. We can say, "At last one of
_p_ or _q_ is true, and perhaps both." Let us call this proposition
"_p_ or _q_." I may mention as an aside that one of the greatest living
philosophers has stated that this use of the word "or"--namely, "_p_ or
_q_" in the sense that either or both may be true--makes him despair of
exact expression. We must brave his wrath, which is unintelligible to
me.
We have thus got hold of four new propositions, namely, "_p_ or _q_,"
and "not-_p_ or _q_," and "_p_ or not-_q_," and "not-_p_ or not-_q_."
Call these the set of disjunctive derivatives. There are, so far,
in all eight propositions, _p_, not-_p_, _q_, not-_q_, and the four
disjunctive derivatives. Any pair of these eight propositions can be
taken, and substituted for _p_ and _q_ in the foregoing treatment.
Thus each pair yields eight propositions, some of which may have been
obtained before. By proceeding in this way we arrive at an unending set
of propositions of growing complexity, ultimately derived from the two
original propositions _p_ or _q_. Of course, only a few are important.
Similarly we can start from three propositions, _p_, _q_, _r_, or
from four propositions, _p_, _q_, _r_, _s_, and so on. Any one of the
propositions of these aggregates may be true or false. It has no other
alternative. Whichever it is, true or false, call it the "truth-value"
of the proposition.
The first section of logical inquiry is to settle what we know of the
truth-values of these propositions, when we know the truth-values of
some of them. The inquiry, so far as it is worth while carrying it,
is not very abstruse, and the best way of expressing its results is a
detail which I will not now consider. This inquiry forms the arithmetic
stage.
The next section of logic is the algebraic stage. Now, the difference
between arithmetic and algebra is, that in arithmetic definite
numbers are considered, and in algebra symbols--namely, letters--are
introduced which stand for any numbers. The idea of a number is also
enlarged. These letters, standing for any numbers, are called sometimes
variables and sometimes parameters. Their essential characteristic is
that they are undetermined, unless, indeed, the algebraic conditions
which they satisfy implicitly determine them. Then they are sometimes
called unknowns. An algebraic formula with letters is a blank form. It
becomes a determinate arithmetic statement when definite numbers are
substituted for the letters. The importance of algebra is a tribute to
the study of form. Consider now the following proposition--
The specific heat of mercury is 0·033.
This is a definite proposition which, with certain limitations, is
true. But the truth-value of the proposition does not immediately
concern us. Instead of mercury put a mere letter which is the name of
some undetermined thing: we get--
The specific heat of _x_ is 0·033.
This is not a proposition; it has been called by Russell a
propositional function. It is the logical analogy of an algebraic
expression. Let us write ƒ(_x_) for any propositional function.
We could also generalise still further, and say,
The specific heat of _x_ is _y_.
We thus get another propositional function, F(_x_, _y_),
of two arguments _x_ and _y_, and so on for any number of
arguments.
Now, consider ƒ(_x_). There is the range of values of _x_,
for which ƒ(_x_) is a proposition, true or false. For values of
_x_ outside this range, ƒ(_x_) is not a proposition at all,
and is neither true nor false. It may have vague suggestions for us,
but it has no unit meaning of definite assertion. For example,
The specific heat of water is 0·033
is a proposition which is false; and--
The specific heat of virtue is 0·033
is, I should imagine, not a proposition at all; so that it is neither
true nor false, though its component parts raise various associations
in our minds. This range of values, for which ƒ(_x_) has sense, is
called the "type" of the argument _x_.
But there is also a range of values of _x_ for which ƒ(_x_)
is a true proposition. This is the class of those values of the
argument which _satisfy_ ƒ(_x_). This class may have no
members, or, in the other extreme, the class may be the whole type of
the arguments.
We thus conceive two general propositions respecting the indefinite
number of propositions which share in the same logical form, that is,
which are values of the same propositional function. One of these
propositions is,
ƒ(_x_) yields a true proposition for each value of _x_ of
the proper type;
the other proposition is,
There is a value of _x_ for which ƒ(_x_) is true.
Given two, or more, propositional functions ƒ(_x_) and ϕ(_x_)
with the same argument _x_, we form derivative propositional
functions, namely,
ƒ(_x_) or ϕ(_x_), ƒ(_x_) or not-ϕ(_x_),
and so on with the contradictories, obtaining, as in the arithmetical
stage, an unending aggregate of propositional functions. Also each
propositional function yields two general propositions. The theory
of the interconnection between the truth-values of the general
propositions arising from any such aggregate of propositional functions
forms a simple and elegant chapter of mathematical logic.
In this algebraic section of logic the theory of types crops up, as we
have already noted. It cannot be neglected without the introduction of
error. Its theory has to be settled at least by some safe hypothesis,
even if it does not go to the philosophic basis of the question. This
part of the subject is obscure and difficult, and has not been finally
elucidated, though Russell's brilliant work has opened out the subject.
The final impulse to modern logic comes from the independent discovery
of the importance of the logic variable by Frege and Peano. Frege went
further than Peano, but by an unfortunate symbolism rendered his work
so obscure that no one fully recognised his meaning who had not found
it out for himself. But the movement has a large history reaching back
to Leibniz and even to Aristotle. Among English contributors are De
Morgan, Boole, and Sir Alfred Kempe; their work is of the first rank.
The third logical section is the stage of general-function theory.
In logical language, we perform in this stage the transition from
intension to extension, and investigate the theory of denotation. Take
the propositional function, ƒ(_x_). There is the class, or range
of values for _x_, whose members satisfy ƒ(_x_). But the same
range may be the class whose members satisfy another propositional
function ϕ(_x_). It is necessary to investigate how to indicate
the class by a way which is indifferent as between the various
propositional functions which are satisfied by any member of it, and of
it only. What has to be done is to analyse the nature of propositions
about a class--namely, those propositions whose truth-values depend on
the class itself and not on the particular meaning by which the class
is indicated.
Furthermore, there are propositions about alleged individuals
indicated by descriptive phrases: for example, propositions about "the
present King of England," who does exist, and "the present Emperor
of Brazil," who does not exist. More complicated, but analogous,
questions involving propositional functions of two variables involve
the notion of "correlation," just as functions of one argument involve
classes. Similarly functions of three arguments yield three-cornered
correlations, and so on. This logical section is one which Russell has
made peculiarly his own by work which must always remain fundamental.
I have called this the section of functional theory, because its
ideas are essential to the construction of logical denoting functions
which include as a special case ordinary mathematical functions, such
as sine, logarithm, etc. In each of these three stages it will be
necessary gradually to introduce an appropriate symbolism, if we are to
pass on to the fourth stage.
The fourth logical section, the analytic stage, is concerned with the
investigation of the properties of special logical constructions,
that is, of classes and correlations of special sorts. The whole of
mathematics is included here. So the section is a large one. In fact,
it is mathematics, neither more nor less, but it includes an analysis
of mathematical ideas not hitherto included in the scope of that
science, nor, indeed, contemplated at all. The essence of this stage is
construction. It is by means of suitable constructions that the great
framework of applied mathematics, comprising the theories of number,
quantity, time, and space, is elaborated.
It is impossible, even in brief outline, to explain how mathematics
is developed from the concepts of class and correlation, including
many-cornered correlations, which are established in the third section.
I can only allude to the headings of the process, which is fully
developed in the work, _Principia Mathematica_, by Mr. Russell
and myself. There are in this process of development seven special
sorts of correlations which are of peculiar interest. The first sort
comprises one-to-many, many-to-one, and one-to-one correlations.
The second sort comprises serial relations, that is, correlations
by which the members of some field are arranged in serial order, so
that, in the sense defined by the relation, any member of the field
is either before or after any other member. The third class comprises
inductive relations, that is, correlations on which the theory of
mathematical induction depends. The fourth class comprises selective
relations, which are required for the general theory of arithmetic
operations, and elsewhere. It is in connection with such relations that
the famous multiplicative axiom arises for consideration. The fifth
class comprises vector relations, from which the theory of quantity
arises. The sixth class comprises ratio relations, which interconnect
number and quantity. The seventh class comprises three-cornered and
four-cornered relations which occur in geometry.
A bare enumeration of technical names, such as the above, is not very
illuminating, though it may help to a comprehension of the demarcations
of the subject. Please remember that the names are technical names,
meant, no doubt, to be suggestive, but used in strictly defined
senses. We have suffered much from critics who consider it sufficient
to criticise our procedure on the slender basis of a knowledge of
the dictionary meanings of such terms. For example, a one-to-one
correlation depends on the notion of a class with only one member, and
this notion is defined without appeal to the concept of the number one.
The notion of diversity is all that is wanted. Thus the class α has
only one member, if (1) the class of values of _x_ which satisfies
the propositional function,
_x_ is not a member of α,
is not the whole type of relevant values of _x_, and if (2) the
propositional function,
_x_ and _y_ are members of α, and _x_ is diverse from
_y_
is false, whatever be the values of _x_ and _y_ in the
relevant type.
Analogous procedures are obviously possible for higher finite cardinal
members. Thus, step by step, the whole cycle of current mathematical
ideas is capable of logical definition. The process is detailed and
laborious, and, like all science, knows nothing of a royal road of
airy phrases. The essence of the process is, first, to construct
the notion in terms of the forms of propositions, that is, in terms
of the relevant propositional functions, and secondly, to prove the
fundamental truths which hold about the notion by reference to the
results obtained in the algebraic section of logic.
It will be seen that in this process the whole apparatus of special
indefinable mathematical concepts, and special _a priori_
mathematical premises, respecting number, quantity, and space, has
vanished. Mathematics is merely an apparatus for analysing the
deductions which can be drawn from any particular premises, supplied
by commonsense, or by more refined scientific observation, so far as
these deductions depend on the forms of the propositions. Propositions
of certain forms are continually occurring in thought. Our existing
mathematics is the analysis of deductions which concern those forms and
in some way are important, either from practical utility or theoretical
interest. Here I am speaking of the science as it in fact exists. A
theoretical definition of mathematics must include in its scope any
deductions depending on the mere forms of propositions. But, of course
no one would wish to develop that part of mathematics which in no sense
is of importance.
This hasty summary of logical ideas suggests some reflections. The
question arises, How many forms of propositions are there? The
answer is, An unending number. The reason for the supposed sterility
of logical science can thus be discerned. Aristotle founded the
science by conceiving the idea of the form of a proposition, and by
conceiving deduction as taking place in virtue of the forms. But he
confined propositions to four forms, now named A, I, E, O. So long as
logicians were obsessed by this unfortunate restriction, real progress
was impossible. Again, in their theory of form, both Aristotle and
subsequent logicians came very near to the theory of the logical
variable. But to come very near to a true theory, and to grasp its
precise application, are two very different things, as the history of
science teaches us. Everything of importance has been said before by
somebody who did not discover it.
Again, one reason why logical deductions are not obvious is, that
logical form is not a subject which ordinarily enters into thought.
Commonsense deduction probably moves by blind instinct from concrete
proposition to concrete proposition, guided by some habitual
association of ideas. Thus commonsense fails in the presence of a
wealth of material.
A more important question is the relation of induction, based on
observation, to deductive logic. There is a tradition of opposition
between adherents of induction and of deduction. In my view, it
would be just as sensible for the two ends of a worm to quarrel.
Both observation and deduction are necessary for any knowledge worth
having. We cannot get at an inductive law without having recourse to a
propositional function. For example, take the statement of observed
fact,
This body is mercury, and its specific heat is 0·033.
The propositional function is formed,
Either _x_ is not mercury, or its specific heat is 0·033.
The inductive law is the assumption of the truth of the general
proposition, that the above propositional function is true for every
value of _x_ in the relevant type.
But it is objected that this process and its consequences are so simple
that an elaborate science is out of place. In the same way, a British
sailor knows the salt sea when he sails over it. What, then, is the use
of an elaborate chemical analysis of sea-water? There is the general
answer, that you cannot know too much of methods which you always
employ; and there is the special answer, that logical forms and logical
implications are not so very simple, and that the whole of mathematics
is evidence to this effect.
One great use of the study of logical method is not in the region of
elaborate deduction, but to guide us in the study of the formation of
the main concepts of science. Consider geometry, for example. What are
the points which compose space? Euclid tells us that they are without
parts and without magnitude. But how is the notion of a point derived
from the sense-perceptions from which science starts? Certainly points
are not direct deliverances of the senses. Here and there we may see or
unpleasantly feel something suggestive of a point. But this is a rare
phenomenon, and certainly does not warrant the conception of space as
composed of points. Our knowledge of space properties is not based on
any observations of relations between points. It arises from experience
of relations between bodies. Now a fundamental space-relation between
bodies is that one body may be part of another. We are tempted to
define the "whole and part" relation by saying that the points occupied
by the part are some of the points occupied by the whole. But "whole
and part" being more fundamental than the notion of "point," this
definition is really circular and vicious.
We accordingly ask whether any other definition of "spatial whole and
part" can be given. I think that it can be done in this way, though,
if I be mistaken, it is unessential to my general argument. We have
come to the conclusion that an extended body is nothing else than the
class of perception of it by all its percipients, actual or ideal. Of
course, it is not any class of perceptions, but a certain definite sort
of class which I have not defined here, except by the vicious method
of saying that they are perceptions of body. Now, the perceptions of a
part of a body are among the perceptions which compose the whole body.
Thus two bodies _a_ and _b_ are both classes of perceptions; and _b_ is
part of _a_ when the class which is _b_ is contained in the class which
is _a_. It immediately follows from the logical form of this definition
that if _b_ is part of _a_, and _c_ is part of _b_, then _c_ is part of
_a_. Thus the relation "whole to part" is transitive. Again, it will
be convenient to allow that a body is part of itself. This is a mere
question of how you draw the definition. With this understanding, the
relation is reflexive. Finally, if _a_ is part of _b_, and _b_ is part
of _a_, then _a_ and _b_ must be identical. These properties of "whole
and part" are not fresh assumptions, they follow from the logical form
of our definition.
One assumption has to be made if we assume the ideal infinite
divisibility of space. Namely, we assume that every class of
perceptions which is an extended body contains other classes of
perceptions which are extended bodies diverse from itself. This
assumption makes rather a large draft on the theory of ideal
perceptions. Geometry vanishes unless in some form you make it. The
assumption is not peculiar to my exposition.
It is then possible to define what we mean by a point. A point is the
class of extended objects which, in ordinary language, contain that
point. The definition, without presupposing the idea of a point, is
rather elaborate, and I have not now time for its statement.
The advantage of introducing points into geometry is the simplicity
of the logical expression of their mutual relations. For science,
simplicity of definition is of slight importance, but simplicity of
mutual relations is essential. Another example of this law is the way
physicists and chemists have dissolved the simple idea of an extended
body, say of a chair, which a child understands, into a bewildering
notion of a complex dance of molecules and atoms and electrons and
waves of light. They have thereby gained notions with simpler logical
relations.
Space as thus conceived is the exact formulation of the properties of
the apparent space of the commonsense world of experience. It is not
necessarily the best mode of conceiving the space of the physicist.
The one essential requisite is that the correspondence between the
commonsense world in its space and the physicists' world in its space
should be definite and reciprocal.
I will now break off the exposition of the function of logic in
connection with the science of natural phenomena. I have endeavoured
to exhibit it as the organising principle, analysing the derivation
of the concepts from the immediate phenomena, examining the structure
of the general propositions which are the assumed laws of nature,
establishing their relations to each other in respect to reciprocal
implications, deducing the phenomena we may expect under given
circumstances.
Logic, properly used, does not shackle thought. It gives freedom, and
above all, boldness. Illogical thought hesitates to draw conclusions,
because it never knows either what it means, or what it assumes, or
how far it trusts its own assumptions, or what will be the effect of
any modification of assumptions. Also the mind untrained in that part
of constructive logic which is relevant to the subject in hand will be
ignorant of the sort of conclusions which follow from various sorts of
assumptions, and will be correspondingly dull in divining the inductive
laws. The fundamental training in this relevant logic is, undoubtedly,
to ponder with an active mind over the known facts of the case,
directly observed. But where elaborate deductions are possible, this
mental activity requires for its full exercise the direct study of the
abstract logical relations. This is applied mathematics.
Neither logic without observation, nor observation without logic, can
move one step in the formation of science. We may conceive humanity
as engaged in an internecine conflict between youth and age. Youth is
not defined by years but by the creative impulse to make something. The
aged are those who, before all things, desire not to make a mistake.
Logic is the olive branch from the old to the young, the wand which in
the hands of youth has the magic property of creating science.
FOOTNOTES:
[Footnote 2: _E.g._ in 1551 by Italian schoolmen; cf. Sarpi's
_History of the Council of Trent_, under that date.]