No More Learning

It would be difficult to give a precise answer to this question, but
certain characteristics may be noted as distinguishing the province of
philosophy from that of the special sciences.

In the first place a philosophical proposition must be general. It
must not deal specially with things on the surface of the earth, or
with the solar system, or with any other portion of space and time. It
is this need of generality which has led to the belief that philosophy
deals with the universe as a whole. I do not believe that this belief
is justified, but I do believe that a philosophical proposition must
be applicable to everything that exists or may exist. It might be
supposed that this admission would be scarcely distinguishable from
the view which I wish to reject. This, however, would be an error, and
an important one. The traditional view would make the universe itself
the subject of various predicates which could not be applied to any
particular thing in the universe, and the ascription of such peculiar
predicates to the universe would be the special business of
philosophy. I maintain, on the contrary, that there are no
propositions of which the "universe" is the subject; in other words,
that there is no such thing as the "universe. " What I do maintain is
that there are general propositions which may be asserted of each
individual thing, such as the propositions of logic. This does not
involve that all the things there are form a whole which could be
regarded as another thing and be made the subject of predicates. It
involves only the assertion that there are properties which belong to
each separate thing, not that there are properties belonging to the
whole of things collectively. The philosophy which I wish to advocate
may be called logical atomism or absolute pluralism, because, while
maintaining that there are many things, it denies that there is a
whole composed of those things. We shall see, therefore, that
philosophical propositions, instead of being concerned with the whole
of things collectively, are concerned with all things distributively;
and not only must they be concerned with all things, but they must be
concerned with such properties of all things as do not depend upon the
accidental nature of the things that there happen to be, but are true
of any possible world, independently of such facts as can only be
discovered by our senses.

This brings us to a second characteristic of philosophical
propositions, namely, that they must be _a priori_. A philosophical
proposition must be such as can be neither proved nor disproved by
empirical evidence. Too often we find in philosophical books arguments
based upon the course of history, or the convolutions of the brain, or
the eyes of shell-fish. Special and accidental facts of this kind are
irrelevant to philosophy, which must make only such assertions as
would be equally true however the actual world were constituted.

We may sum up these two characteristics of philosophical propositions
by saying that _philosophy is the science of the possible_. But this
statement unexplained is liable to be misleading, since it may be
thought that the possible is something other than the general, whereas
in fact the two are indistinguishable.

Philosophy, if what has been said is correct, becomes indistinguishable
from logic as that word has now come to be used. The study of logic
consists, broadly speaking, of two not very sharply distinguished
portions. On the one hand it is concerned with those general statements
which can be made concerning everything without mentioning any one
thing or predicate or relation, such for example as "if _x_ is a member
of the class ? and every member of ? is a member of ? , then _x_ is a
member of the class ? , whatever _x_, ? , and ? may be. " On the other
hand, it is concerned with the analysis and enumeration of logical
_forms_, i. e. with the kinds of propositions that may occur, with the
various types of facts, and with the classification of the constituents
of facts. In this way logic provides an inventory of possibilities, a
repertory of abstractly tenable hypotheses.

It might be thought that such a study would be too vague and too
general to be of any very great importance, and that, if its problems
became at any point sufficiently definite, they would be merged in the
problems of some special science. It appears, however, that this is
not the case. In some problems, for example, the analysis of space and
time, the nature of perception, or the theory of judgment, the
discovery of the logical form of the facts involved is the hardest
part of the work and the part whose performance has been most lacking
hitherto. It is chiefly for want of the right logical hypothesis that
such problems have hitherto been treated in such an unsatisfactory
manner, and have given rise to those contradictions or antinomies in
which the enemies of reason among philosophers have at all times
delighted.

By concentrating attention upon the investigation of logical forms, it
becomes possible at last for philosophy to deal with its problems
piecemeal, and to obtain, as the sciences do, such partial and
probably not wholly correct results as subsequent investigation can
utilise even while it supplements and improves them. Most
philosophies hitherto have been constructed all in one block, in such
a way that, if they were not wholly correct, they were wholly
incorrect, and could not be used as a basis for further
investigations. It is chiefly owing to this fact that philosophy,
unlike science, has hitherto been unprogressive, because each original
philosopher has had to begin the work again from the beginning,
without being able to accept anything definite from the work of his
predecessors. A scientific philosophy such as I wish to recommend will
be piecemeal and tentative like other sciences; above all, it will be
able to invent hypotheses which, even if they are not wholly true,
will yet remain fruitful after the necessary corrections have been
made. This possibility of successive approximations to the truth is,
more than anything else, the source of the triumphs of science, and to
transfer this possibility to philosophy is to ensure a progress in
method whose importance it would be almost impossible to exaggerate.

The essence of philosophy as thus conceived is analysis, not
synthesis. To build up systems of the world, like Heine's German
professor who knit together fragments of life and made an intelligible
system out of them, is not, I believe, any more feasible than the
discovery of the philosopher's stone. What is feasible is the
understanding of general forms, and the division of traditional
problems into a number of separate and less baffling questions.
"Divide and conquer" is the maxim of success here as elsewhere.

Let us illustrate these somewhat general maxims by examining their
application to the philosophy of space, for it is only in application
that the meaning or importance of a method can be understood. Suppose
we are confronted with the problem of space as presented in Kant's
Transcendental AEsthetic, and suppose we wish to discover what are the
elements of the problem and what hope there is of obtaining a solution
of them. It will soon appear that three entirely distinct problems,
belonging to different studies, and requiring different methods for
their solution, have been confusedly combined in the supposed single
problem with which Kant is concerned. There is a problem of logic, a
problem of physics, and a problem of theory of knowledge. Of these
three, the problem of logic can be solved exactly and perfectly; the
problem of physics can probably be solved with as great a degree of
certainty and as great an approach to exactness as can be hoped in an
empirical region; the problem of theory of knowledge, however, remains
very obscure and very difficult to deal with. Let us see how these
three problems arise.

(1) The logical problem has arisen through the suggestions of
non-Euclidean geometry. Given a body of geometrical propositions, it
is not difficult to find a minimum statement of the axioms from which
this body of propositions can be deduced. It is also not difficult, by
dropping or altering some of these axioms, to obtain a more general or
a different geometry, having, from the point of view of pure
mathematics, the same logical coherence and the same title to respect
as the more familiar Euclidean geometry. The Euclidean geometry itself
is true perhaps of actual space (though this is doubtful), but
certainly of an infinite number of purely arithmetical systems, each
of which, from the point of view of abstract logic, has an equal and
indefeasible right to be called a Euclidean space. Thus space as an
object of logical or mathematical study loses its uniqueness; not only
are there many kinds of spaces, but there are an infinity of examples
of each kind, though it is difficult to find any kind of which the
space of physics may be an example, and it is impossible to find any
kind of which the space of physics is certainly an example. As an
illustration of one possible logical system of geometry we may
consider all relations of three terms which are analogous in certain
formal respects to the relation "between" as it appears to be in
actual space. A space is then defined by means of one such three-term
relation. The points of the space are all the terms which have this
relation to something or other, and their order in the space in
question is determined by this relation. The points of one space are
necessarily also points of other spaces, since there are necessarily
other three-term relations having those same points for their field.
The space in fact is not determined by the class of its points, but by
the ordering three-term relation. When enough abstract logical
properties of such relations have been enumerated to determine the
resulting kind of geometry, say, for example, Euclidean geometry, it
becomes unnecessary for the pure geometer in his abstract capacity to
distinguish between the various relations which have all these
properties. He considers the whole class of such relations, not any
single one among them. Thus in studying a given kind of geometry the
pure mathematician is studying a certain class of relations defined by
means of certain abstract logical properties which take the place of
what used to be called axioms. The nature of geometrical _reasoning_
therefore is purely deductive and purely logical; if any special
epistemological peculiarities are to be found in geometry, it must not
be in the reasoning, but in our knowledge concerning the axioms in
some given space.

(2) The physical problem of space is both more interesting and more
difficult than the logical problem. The physical problem may be
stated as follows: to find in the physical world, or to construct from
physical materials, a space of one of the kinds enumerated by the
logical treatment of geometry. This problem derives its difficulty
from the attempt to accommodate to the roughness and vagueness of the
real world some system possessing the logical clearness and exactitude
of pure mathematics. That this can be done with a certain degree of
approximation is fairly evident If I see three people _A_, _B_, and
_C_ sitting in a row, I become aware of the fact which may be
expressed by saying that _B_ is between _A_ and _C_ rather than that
_A_ is between _B_ and _C_, or _C_ is between _A_ and _B_. This
relation of "between" which is thus perceived to hold has some of the
abstract logical properties of those three-term relations which, we
saw, give rise to a geometry, but its properties fail to be exact, and
are not, as empirically given, amenable to the kind of treatment at
which geometry aims. In abstract geometry we deal with points,
straight lines, and planes; but the three people _A_, _B_, and _C_
whom I see sitting in a row are not exactly points, nor is the row
exactly a straight line. Nevertheless physics, which formally assumes
a space containing points, straight lines, and planes, is found
empirically to give results applicable to the sensible world. It must
therefore be possible to find an interpretation of the points,
straight lines, and planes of physics in terms of physical data, or at
any rate in terms of data together with such hypothetical additions as
seem least open to question. Since all data suffer from a lack of
mathematical precision through being of a certain size and somewhat
vague in outline, it is plain that if such a notion as that of a point
is to find any application to empirical material, the point must be
neither a datum nor a hypothetical addition to data, but a
_construction_ by means of data with their hypothetical additions. It
is obvious that any hypothetical filling out of data is less dubious
and unsatisfactory when the additions are closely analogous to data
than when they are of a radically different sort. To assume, for
example, that objects which we see continue, after we have turned away
our eyes, to be more or less analogous to what they were while we were
looking, is a less violent assumption than to assume that such objects
are composed of an infinite number of mathematical points. Hence in
the physical study of the geometry of physical space, points must not
be assumed _ab initio_ as they are in the logical treatment of
geometry, but must be constructed as systems composed of data and
hypothetical analogues of data. We are thus led naturally to define a
physical point as a certain class of those objects which are the
ultimate constituents of the physical world. It will be the class of
all those objects which, as one would naturally say, _contain_ the
point. To secure a definition giving this result, without previously
assuming that physical objects are composed of points, is an agreeable
problem in mathematical logic. The solution of this problem and the
perception of its importance are due to my friend Dr. Whitehead. The
oddity of regarding a point as a class of physical entities wears off
with familiarity, and ought in any case not to be felt by those who
maintain, as practically every one does, that points are mathematical
fictions. The word "fiction" is used glibly in such connexions by many
men who seem not to feel the necessity of explaining how it can come
about that a fiction can be so useful in the study of the actual world
as the points of mathematical physics have been found to be. By our
definition, which regards a point as a class of physical objects, it
is explained both how the use of points can lead to important
physical results, and how we can nevertheless avoid the assumption
that points are themselves entities in the physical world.

Many of the mathematically convenient properties of abstract logical
spaces cannot be either known to belong or known not to belong to the
space of physics. Such are all the properties connected with continuity.
For to know that actual space has these properties would require an
infinite exactness of sense-perception. If actual space is continuous,
there are nevertheless many possible non-continuous spaces which will be
empirically indistinguishable from it; and, conversely, actual space may
be non-continuous and yet empirically indistinguishable from a possible
continuous space. Continuity, therefore, though obtainable in the _a
priori_ region of arithmetic, is not with certainty obtainable in the
space or time of the physical world: whether these are continuous or not
would seem to be a question not only unanswered but for ever
unanswerable. From the point of view of philosophy, however, the
discovery that a question is unanswerable is as complete an answer as
any that could possibly be obtained. And from the point of view of
physics, where no empirical means of distinction can be found, there can
be no empirical objection to the mathematically simplest assumption,
which is that of continuity.

The subject of the physical theory of space is a very large one,
hitherto little explored. It is associated with a similar theory of
time, and both have been forced upon the attention of philosophically
minded physicists by the discussions which have raged concerning the
theory of relativity.

(3) The problem with which Kant is concerned in the Transcendental
AEsthetic is primarily the epistemological problem: "How do we come to
have knowledge of geometry _a priori_? " By the distinction between the
logical and physical problems of geometry, the bearing and scope of
this question are greatly altered. Our knowledge of pure geometry is
_a priori_ but is wholly logical. Our knowledge of physical geometry
is synthetic, but is not _a priori_. Our knowledge of pure geometry is
hypothetical, and does not enable us to assert, for example, that the
axiom of parallels is true in the physical world. Our knowledge of
physical geometry, while it does enable us to assert that this axiom
is approximately verified, does not, owing to the inevitable
inexactitude of observation, enable us to assert that it is verified
_exactly_. Thus, with the separation which we have made between pure
geometry and the geometry of physics, the Kantian problem collapses.
To the question, "How is synthetic _a priori_ knowledge possible? " we
can now reply, at any rate so far as geometry is concerned, "It is not
possible," if "synthetic" means "not deducible from logic alone. " Our
knowledge of geometry, like the rest of our knowledge, is derived
partly from logic, partly from sense, and the peculiar position which
in Kant's day geometry appeared to occupy is seen now to be a
delusion. There are still some philosophers, it is true, who maintain
that our knowledge that the axiom of parallels, for example, is true
of actual space, is not to be accounted for empirically, but is as
Kant maintained derived from an _a priori_ intuition. This position is
not logically refutable, but I think it loses all plausibility as soon
as we realise how complicated and derivative is the notion of physical
space. As we have seen, the application of geometry to the physical
world in no way demands that there should really be points and
straight lines among physical entities. The principle of economy,
therefore, demands that we should abstain from assuming the existence
of points and straight lines. As soon, however, as we accept the view
that points and straight lines are complicated constructions by means
of classes of physical entities, the hypothesis that we have an _a
priori_ intuition enabling us to know what happens to straight lines
when they are produced indefinitely becomes extremely strained and
harsh; nor do I think that such an hypothesis would ever have arisen
in the mind of a philosopher who had grasped the nature of physical
space. Kant, under the influence of Newton, adopted, though with some
vacillation, the hypothesis of absolute space, and this hypothesis,
though logically unobjectionable, is removed by Occam's razor, since
absolute space is an unnecessary entity in the explanation of the
physical world. Although, therefore, we cannot refute the Kantian
theory of an _a priori_ intuition, we can remove its grounds one by
one through an analysis of the problem. Thus, here as in many other
philosophical questions, the analytic method, while not capable of
arriving at a demonstrative result, is nevertheless capable of showing
that all the positive grounds in favour of a certain theory are
fallacious and that a less unnatural theory is capable of accounting
for the facts.

Another question by which the capacity of the analytic method can be
shown is the question of realism. Both those who advocate and those
who combat realism seem to me to be far from clear as to the nature of
the problem which they are discussing. If we ask: "Are our objects of
perception _real_ and are they _independent_ of the percipient? " it
must be supposed that we attach some meaning to the words "real" and
"independent," and yet, if either side in the controversy of realism
is asked to define these two words, their answer is pretty sure to
embody confusions such as logical analysis will reveal.

Let us begin with the word "real. " There certainly are objects of
perception, and therefore, if the question whether these objects are
real is to be a substantial question, there must be in the world two
sorts of objects, namely, the real and the unreal, and yet the unreal
is supposed to be essentially what there is not. The question what
properties must belong to an object in order to make it real is one to
which an adequate answer is seldom if ever forthcoming. There is of
course the Hegelian answer, that the real is the self-consistent and
that nothing is self-consistent except the Whole; but this answer,
true or false, is not relevant in our present discussion, which moves
on a lower plane and is concerned with the status of objects of
perception among other objects of equal fragmentariness. Objects of
perception are contrasted, in the discussions concerning realism,
rather with psychical states on the one hand and matter on the other
hand than with the all-inclusive whole of things. The question we have
therefore to consider is the question as to what can be meant by
assigning "reality" to some but not all of the entities that make up
the world. Two elements, I think, make up what is felt rather than
thought when the word "reality" is used in this sense. A thing is real
if it persists at times when it is not perceived; or again, a thing is
real when it is correlated with other things in a way which experience
has led us to expect. It will be seen that reality in either of these
senses is by no means necessary to a thing, and that in fact there
might be a whole world in which nothing was real in either of these
senses. It might turn out that the objects of perception failed of
reality in one or both of these respects, without its being in any way
deducible that they are not parts of the external world with which
physics deals. Similar remarks will apply to the word "independent. "
Most of the associations of this word are bound up with ideas as to
causation which it is not now possible to maintain. _A_ is independent
of _B_ when _B_ is not an indispensable part of the _cause_ of _A_.
But when it is recognised that causation is nothing more than
correlation, and that there are correlations of simultaneity as well
as of succession, it becomes evident that there is no uniqueness in a
series of casual antecedents of a given event, but that, at any point
where there is a correlation of simultaneity, we can pass from one
line of antecedents to another in order to obtain a new series of
causal antecedents. It will be necessary to specify the causal law
according to which the antecedents are to be considered. I received a
letter the other day from a correspondent who had been puzzled by
various philosophical questions. After enumerating them he says:
"These questions led me from Bonn to Strassburg, where I found
Professor Simmel. " Now, it would be absurd to deny that these
questions caused his body to move from Bonn to Strassburg, and yet it
must be supposed that a set of purely mechanical antecedents could
also be found which would account for this transfer of matter from one
place to another. Owing to this plurality of causal series antecedent
to a given event, the notion of _the_ cause becomes indefinite, and
the question of independence becomes correspondingly ambiguous. Thus,
instead of asking simply whether _A_ is independent of _B_, we ought
to ask whether there is a series determined by such and such causal
laws leading from _B_ to _A_. This point is important in connexion
with the particular question of objects of perception. It may be that
no objects quite like those which we perceive ever exist unperceived;
in this case there will be a causal law according to which objects of
perception are not independent of being perceived. But even if this be
the case, it may nevertheless also happen that there are purely
physical causal laws determining the occurrence of objects which are
perceived by means of other objects which perhaps are not perceived.
In that case, in regard to such causal laws objects of perception will
be independent of being perceived. Thus the question whether objects
of perception are independent of being perceived is, as it stands,
indeterminate, and the answer will be yes or no according to the
method adopted of making it determinate. I believe that this confusion
has borne a very large part in prolonging the controversies on this
subject, which might well have seemed capable of remaining for ever
undecided. The view which I should wish to advocate is that objects of
perception do not persist unchanged at times when they are not
perceived, although probably objects more or less resembling them do
exist at such times; that objects of perception are part, and the only
empirically knowable part, of the actual subject-matter of physics,
and are themselves properly to be called physical; that purely
physical laws exist determining the character and duration of objects
of perception without any reference to the fact that they are
perceived; and that in the establishment of such laws the propositions
of physics do not presuppose any propositions of psychology or even
the existence of mind. I do not know whether realists would recognise
such a view as realism. All that I should claim for it is, that it
avoids difficulties which seem to me to beset both realism and
idealism as hitherto advocated, and that it avoids the appeal which
they have made to ideas which logical analysis shows to be ambiguous.
A further defence and elaboration of the positions which I advocate,
but for which time is lacking now, will be found indicated in my book
on _Our Knowledge of the External World_. [22]

The adoption of scientific method in philosophy, if I am not mistaken,
compels us to abandon the hope of solving many of the more ambitious
and humanly interesting problems of traditional philosophy. Some of
these it relegates, though with little expectation of a successful
solution, to special sciences, others it shows to be such as our
capacities are essentially incapable of solving. But there remain a
large number of the recognised problems of philosophy in regard to
which the method advocated gives all those advantages of division into
distinct questions, of tentative, partial, and progressive advance,
and of appeal to principles with which, independently of temperament,
all competent students must agree. The failure of philosophy hitherto
has been due in the main to haste and ambition: patience and modesty,
here as in other sciences, will open the road to solid and durable
progress.

FOOTNOTES:

[19] Bosanquet, _Logic_, ii, p. 211.

[20] _Some Problems of Philosophy_, p 124.

[21] _First Principles_ (1862), Part II, beginning of chap. viii.

[22] Open Court Company, 1914.




VII

THE ULTIMATE CONSTITUENTS OF MATTER[23]


I wish to discuss in this article no less a question than the ancient
metaphysical query, "What is matter? " The question, "What is matter? "
in so far as it concerns philosophy, is, I think, already capable of
an answer which in principle will be as complete as an answer can hope
to be; that is to say, we can separate the problem into an essentially
soluble and an essentially insoluble portion, and we can now see how
to solve the essentially soluble portion, at least as regards its main
outlines. It is these outlines which I wish to suggest in the present
article. My main position, which is realistic, is, I hope and believe,
not remote from that of Professor Alexander, by whose writings on this
subject I have profited greatly. [24] It is also in close accord with
that of Dr. Nunn. [25]

Common sense is accustomed to the division of the world into mind and
matter. It is supposed by all who have never studied philosophy that
the distinction between mind and matter is perfectly clear and easy,
that the two do not at any point overlap, and that only a fool or a
philosopher could be in doubt as to whether any given entity is mental
or material. This simple faith survives in Descartes and in a
somewhat modified form in Spinoza, but with Leibniz it begins to
disappear, and from his day to our own almost every philosopher of
note has criticised and rejected the dualism of common sense. It is my
intention in this article to defend this dualism; but before defending
it we must spend a few moments on the reasons which have prompted its
rejection.

Our knowledge of the material world is obtained by means of the
senses, of sight and touch and so on. At first it is supposed that
things are just as they seem, but two opposite sophistications soon
destroy this naive belief. On the one hand the physicists cut up
matter into molecules, atoms, corpuscles, and as many more such
subdivisions as their future needs may make them postulate, and the
units at which they arrive are uncommonly different from the visible,
tangible objects of daily life. A unit of matter tends more and more
to be something like an electromagnetic field filling all space,
though having its greatest intensity in a small region. Matter
consisting of such elements is as remote from daily life as any
metaphysical theory. It differs from the theories of metaphysicians
only in the fact that its practical efficacy proves that it contains
some measure of truth and induces business men to invest money on the
strength of it; but, in spite of its connection with the money market,
it remains a metaphysical theory none the less.

The second kind of sophistication to which the world of common sense
has been subjected is derived from the psychologists and
physiologists. The physiologists point out that what we see depends
upon the eye, that what we hear depends upon the ear, and that all our
senses are liable to be affected by anything which affects the brain,
like alcohol or hasheesh. Psychologists point out how much of what we
think we see is supplied by association or unconscious inference, how
much is mental interpretation, and how doubtful is the residuum which
can be regarded as crude datum. From these facts it is argued by the
psychologists that the notion of a datum passively received by the
mind is a delusion, and it is argued by the physiologists that even if
a pure datum of sense could be obtained by the analysis of experience,
still this datum could not belong, as common sense supposes, to the
outer world, since its whole nature is conditioned by our nerves and
sense organs, changing as they change in ways which it is thought
impossible to connect with any change in the matter supposed to be
perceived. This physiologist's argument is exposed to the rejoinder,
more specious than solid, that our knowledge of the existence of the
sense organs and nerves is obtained by that very process which the
physiologist has been engaged in discrediting, since the existence of
the nerves and sense organs is only known through the evidence of the
senses themselves. This argument may prove that some reinterpretation
of the results of physiology is necessary before they can acquire
metaphysical validity. But it does not upset the physiological
argument in so far as this constitutes merely a _reductio ad absurdum_
of naive realism.

These various lines of argument prove, I think, that some part of the
beliefs of common sense must be abandoned. They prove that, if we take
these beliefs as a whole, we are forced into conclusions which are in
part self-contradictory; but such arguments cannot of themselves
decide what portion of our common-sense beliefs is in need of
correction. Common sense believes that what we see is physical,
outside the mind, and continuing to exist if we shut our eyes or turn
them in another direction. I believe that common sense is right in
regarding what we see as physical and (in one of several possible
senses) outside the mind, but is probably wrong in supposing that it
continues to exist when we are no longer looking at it. It seems to me
that the whole discussion of matter has been obscured by two errors
which support each other. The first of these is the error that what we
see, or perceive through any of our other senses, is subjective: the
second is the belief that what is physical must be persistent.
Whatever physics may regard as the ultimate constituents of matter, it
always supposes these constituents to be indestructible. Since the
immediate data of sense are not indestructible but in a state of
perpetual flux, it is argued that these data themselves cannot be
among the ultimate constituents of matter. I believe this to be a
sheer mistake. The persistent particles of mathematical physics I
regard as logical constructions, symbolic fictions enabling us to
express compendiously very complicated assemblages of facts; and, on
the other hand, I believe that the actual data in sensation, the
immediate objects of sight or touch or hearing, are extra-mental,
purely physical, and among the ultimate constituents of matter.

My meaning in regard to the impermanence of physical entities may
perhaps be made clearer by the use of Bergson's favourite illustration
of the cinematograph. When I first read Bergson's statement that the
mathematician conceives the world after the analogy of a
cinematograph, I had never seen a cinematograph, and my first visit to
one was determined by the desire to verify Bergson's statement, which
I found to be completely true, at least so far as I am concerned.
When, in a picture palace, we see a man rolling down hill, or running
away from the police, or falling into a river, or doing any of those
other things to which men in such places are addicted, we know that
there is not really only one man moving, but a succession of films,
each with a different momentary man. The illusion of persistence
arises only through the approach to continuity in the series of
momentary men. Now what I wish to suggest is that in this respect the
cinema is a better metaphysician than common sense, physics, or
philosophy. The real man too, I believe, however the police may swear
to his identity, is really a series of momentary men, each different
one from the other, and bound together, not by a numerical identity,
but by continuity and certain intrinsic causal laws. And what applies
to men applies equally to tables and chairs, the sun, moon and stars.
Each of these is to be regarded, not as one single persistent entity,
but as a series of entities succeeding each other in time, each
lasting for a very brief period, though probably not for a mere
mathematical instant. In saying this I am only urging the same kind of
division in time as we are accustomed to acknowledge in the case of
space.
A body which fills a cubic foot will be admitted to consist of
many smaller bodies, each occupying only a very tiny volume; similarly
a thing which persists for an hour is to be regarded as composed of
many things of less duration. A true theory of matter requires a
division of things into time-corpuscles as well as into
space-corpuscles.

The world may be conceived as consisting of a multitude of entities
arranged in a certain pattern. The entities which are arranged I shall
call "particulars. " The arrangement or pattern results from relations
among particulars. Classes or series of particulars, collected
together on account of some property which makes it convenient to be
able to speak of them as wholes, are what I call logical constructions
or symbolic fictions. The particulars are to be conceived, not on the
analogy of bricks in a building, but rather on the analogy of notes
in a symphony. The ultimate constituents of a symphony (apart from
relations) are the notes, each of which lasts only for a very short
time. We may collect together all the notes played by one instrument:
these may be regarded as the analogues of the successive particulars
which common sense would regard as successive states of one "thing. "
But the "thing" ought to be regarded as no more "real" or
"substantial" than, for example, the role of the trombone. As soon as
"things" are conceived in this manner it will be found that the
difficulties in the way of regarding immediate objects of sense as
physical have largely disappeared.

When people ask, "Is the object of sense mental or physical? " they
seldom have any clear idea either what is meant by "mental" or
"physical," or what criteria are to be applied for deciding whether a
given entity belongs to one class or the other. I do not know how to
give a sharp definition of the word "mental," but something may be
done by enumerating occurrences which are indubitably mental:
believing, doubting, wishing, willing, being pleased or pained, are
certainly mental occurrences; so are what we may call experiences,
seeing, hearing, smelling, perceiving generally. But it does not
follow from this that what is seen, what is heard, what is smelt, what
is perceived, must be mental. When I see a flash of lightning, my
seeing of it is mental, but what I see, although it is not quite the
same as what anybody else sees at the same moment, and although it
seems very unlike what the physicist would describe as a flash of
lightning, is not mental. I maintain, in fact, that if the physicist
could describe truly and fully all that occurs in the physical world
when there is a flash of lightning, it would contain as a constituent
what I see, and also what is seen by anybody else who would commonly
be said to see the same flash. What I mean may perhaps be made plainer
by saying that if my body could remain in exactly the same state in
which it is, although my mind had ceased to exist, precisely that
object which I now see when I see the flash would exist, although of
course I should not see it, since my seeing is mental. The principal
reasons which have led people to reject this view have, I think, been
two: first, that they did not adequately distinguish between my seeing
and what I see; secondly, that the causal dependence of what I see
upon my body has made people suppose that what I see cannot be
"outside" me. The first of these reasons need not detain us, since the
confusion only needs to be pointed out in order to be obviated; but
the second requires some discussion, since it can only be answered by
removing current misconceptions, on the one hand as to the nature of
space, and on the other, as to the meaning of causal dependence.

When people ask whether colours, for example, or other secondary
qualities are inside or outside the mind, they seem to suppose that
their meaning must be clear, and that it ought to be possible to say
yes or no without any further discussion of the terms involved. In
fact, however, such terms as "inside" or "outside" are very ambiguous.
What is meant by asking whether this or that is "in" the mind? The
mind is not like a bag or a pie; it does not occupy a certain region
in space, or, if (in a sense) it does, what is in that region is
presumably part of the brain, which would not be said to be in the
mind. When people say that sensible qualities are in the mind, they do
not mean "spatially contained in" in the sense in which the blackbirds
were in the pie. We might regard the mind as an assemblage of
particulars, namely, what would be called "states of mind," which
would belong together in virtue of some specific common quality. The
common quality of all states of mind would be the quality designated
by the word "mental"; and besides this we should have to suppose that
each separate person's states of mind have some common characteristic
distinguishing them from the states of mind of other people. Ignoring
this latter point, let us ask ourselves whether the quality designated
by the word "mental" does, as a matter of observation, actually belong
to objects of sense, such as colours or noises. I think any candid
person must reply that, however difficult it may be to know what we
mean by "mental," it is not difficult to see that colours and noises
are not mental in the sense of having that intrinsic peculiarity which
belongs to beliefs and wishes and volitions, but not to the physical
world. Berkeley advances on this subject a plausible argument[26]
which seems to me to rest upon an ambiguity in the word "pain. " He
argues that the realist supposes the heat which he feels in
approaching a fire to be something outside his mind, but that as he
approaches nearer and nearer to the fire the sensation of heat passes
imperceptibly into pain, and that no one could regard pain as
something outside the mind. In reply to this argument, it should be
observed in the first place that the heat of which we are immediately
aware is not in the fire but in our own body. It is only by inference
that the fire is judged to be the cause of the heat which we feel in
our body. In the second place (and this is the more important point),
when we speak of pain we may mean one of two things: we may mean the
object of the sensation or other experience which has the quality of
being painful, or we may mean the quality of painfulness itself. When
a man says he has a pain in his great toe, what he means is that he
has a sensation associated with his great toe and having the quality
of painfulness. The sensation itself, like every sensation, consists
in experiencing a sensible object, and the experiencing has that
quality of painfulness which only mental occurrences can have, but
which may belong to thoughts or desires, as well as to sensations. But
in common language we speak of the sensible object experienced in a
painful sensation as a pain, and it is this way of speaking which
causes the confusion upon which the plausibility of Berkeley's
argument depends. It would be absurd to attribute the quality of
painfulness to anything non-mental, and hence it comes to be thought
that what we call a pain in the toe must be mental. In fact, however,
it is not the sensible object in such a case which is painful, but the
sensation, that is to say, the experience of the sensible object. As
the heat which we experience from the fire grows greater, the
experience passes gradually from being pleasant to being painful, but
neither the pleasure nor the pain is a quality of the object
experienced as opposed to the experience, and it is therefore a
fallacy to argue that this object must be mental on the ground that
painfulness can only be attributed to what is mental.

If, then, when we say that something is in the mind we mean that it
has a certain recognisable intrinsic characteristic such as belongs to
thoughts and desires, it must be maintained on grounds of immediate
inspection that objects of sense are not in any mind.

A different meaning of "in the mind" is, however, to be inferred from
the arguments advanced by those who regard sensible objects as being
in the mind. The arguments used are, in the main, such as would prove
the causal dependence of objects of sense upon the percipient. Now
the notion of causal dependence is very obscure and difficult, much
more so in fact than is generally realised by philosophers. I shall
return to this point in a moment. For the present, however, accepting
the notion of causal dependence without criticism, I wish to urge that
the dependence in question is rather upon our bodies than upon our
minds. The visual appearance of an object is altered if we shut one
eye, or squint, or look previously at something dazzling; but all
these are bodily acts, and the alterations which they effect are to be
explained by physiology and optics, not by psychology. [27] They are in
fact of exactly the same kind as the alterations effected by
spectacles or a microscope. They belong therefore to the theory of the
physical world, and can have no bearing upon the question whether what
we see is causally dependent upon the mind. What they do tend to
prove, and what I for my part have no wish to deny, is that what we
see is causally dependent upon our body and is not, as crude common
sense would suppose, something which would exist equally if our eyes
and nerves and brain were absent, any more than the visual appearance
presented by an object seen through a microscope would remain if the
microscope were removed. So long as it is supposed that the physical
world is composed of stable and more or less permanent constituents,
the fact that what we see is changed by changes in our body appears to
afford reason for regarding what we see as not an ultimate constituent
of matter. But if it is recognised that the ultimate constituents of
matter are as circumscribed in duration as in spatial extent, the
whole of this difficulty vanishes.

There remains, however, another difficulty, connected with space. When
we look at the sun we wish to know something about the sun itself,
which is ninety-three million miles away; but what we see is dependent
upon our eyes, and it is difficult to suppose that our eyes can affect
what happens at a distance of ninety-three million miles. Physics
tells us that certain electromagnetic waves start from the sun, and
reach our eyes after about eight minutes. They there produce
disturbances in the rods and cones, thence in the optic nerve, thence
in the brain. At the end of this purely physical series, by some odd
miracle, comes the experience which we call "seeing the sun," and it
is such experiences which form the whole and sole reason for our
belief in the optic nerve, the rods and cones, the ninety-three
million miles, the electromagnetic waves, and the sun itself. It is
this curious oppositeness of direction between the order of causation
as affirmed by physics, and the order of evidence as revealed by
theory of knowledge, that causes the most serious perplexities in
regard to the nature of physical reality. Anything that invalidates
our seeing, as a source of knowledge concerning physical reality,
invalidates also the whole of physics and physiology. And yet,
starting from a common-sense acceptance of our seeing, physics has
been led step by step to the construction of the causal chain in which
our seeing is the last link, and the immediate object which we see
cannot be regarded as that initial cause which we believe to be
ninety-three million miles away, and which we are inclined to regard
as the "real" sun.

I have stated this difficulty as forcibly as I can, because I believe
that it can only be answered by a radical analysis and reconstruction
of all the conceptions upon whose employment it depends.

Space, time, matter and cause, are the chief of these conceptions. Let
us begin with the conception of cause.

Causal dependence, as I observed a moment ago, is a conception which
it is very dangerous to accept at its face value. There exists a
notion that in regard to any event there is something which may be
called _the_ cause of that event--some one definite occurrence,
without which the event would have been impossible and with which it
becomes necessary. An event is supposed to be dependent upon its cause
in some way which in it is not dependent upon other things. Thus men
will urge that the mind is dependent upon the brain, or, with equal
plausibility, that the brain is dependent upon the mind. It seems not
improbable that if we had sufficient knowledge we could infer the
state of a man's mind from the state of his brain, or the state of his
brain from the state of his mind. So long as the usual conception of
causal dependence is retained, this state of affairs can be used by
the materialist to urge that the state of our brain causes our
thoughts, and by the idealist to urge that our thoughts cause the
state of our brain. Either contention is equally valid or equally
invalid. The fact seems to be that there are many correlations of the
sort which may be called causal, and that, for example, either a
physical or a mental event can be predicted, theoretically, either
from a sufficient number of physical antecedents or from a sufficient
number of mental antecedents. To speak of _the_ cause of an event is
therefore misleading. Any set of antecedents from which the event can
theoretically be inferred by means of correlations might be called a
cause of the event. But to speak of _the_ cause is to imply a
uniqueness which does not exist.

The relevance of this to the experience which we call "seeing the sun"
is obvious. The fact that there exists a chain of antecedents which
makes our seeing dependent upon the eyes and nerves and brain does not
even tend to show that there is not another chain of antecedents in
which the eyes and nerves and brain as physical things are ignored. If
we are to escape from the dilemma which seemed to arise out of the
physiological causation of what we see when we say we see the sun, we
must find, at least in theory, a way of stating causal laws for the
physical world, in which the units are not material things, such as
the eyes and nerves and brain, but momentary particulars of the same
sort as our momentary visual object when we look at the sun. The sun
itself and the eyes and nerves and brain must be regarded as
assemblages of momentary particulars. Instead of supposing, as we
naturally do when we start from an uncritical acceptance of the
apparent dicta of physics, that _matter_ is what is "really real" in
the physical world, and that the immediate objects of sense are mere
phantasms, we must regard matter as a logical construction, of which
the constituents will be just such evanescent particulars as may, when
an observer happens to be present, become data of sense to that
observer. What physics regards as the sun of eight minutes ago will be
a whole assemblage of particulars, existing at different times,
spreading out from a centre with the velocity of light, and containing
among their number all those visual data which are seen by people who
are now looking at the sun. Thus the sun of eight minutes ago is a
class of particulars, and what I see when I now look at the sun is one
member of this class. The various particulars constituting this class
will be correlated with each other by a certain continuity and certain
intrinsic laws of variation as we pass outwards from the centre,
together with certain modifications correlated extrinsically with
other particulars which are not members of this class. It is these
extrinsic modifications which represent the sort of facts that, in our
former account, appeared as the influence of the eyes and nerves in
modifying the appearance of the sun. [28]

The _prima facie_ difficulties in the way of this view are chiefly
derived from an unduly conventional theory of space. It might seem at
first sight as if we had packed the world much fuller than it could
possibly hold. At every place between us and the sun, we said, there
is to be a particular which is to be a member of the sun as it was a
few minutes ago. There will also, of course, have to be a particular
which is a member of any planet or fixed star that may happen to be
visible from that place. At the place where I am, there will be
particulars which will be members severally of all the "things" I am
now said to be perceiving. Thus throughout the world, everywhere,
there will be an enormous number of particulars coexisting in the same
place. But these troubles result from contenting ourselves too readily
with the merely three-dimensional space to which schoolmasters have
accustomed us. The space of the real world is a space of six
dimensions, and as soon as we realise this we see that there is plenty
of room for all the particulars for which we want to find positions.
In order to realise this we have only to return for a moment from the
polished space of physics to the rough and untidy space of our
immediate sensible experience. The space of one man's sensible objects
is a three-dimensional space. It does not appear probable that two men
ever both perceive at the same time any one sensible object; when they
are said to see the same thing or hear the same noise, there will
always be some difference, however slight, between the actual shapes
seen or the actual sounds heard. If this is so, and if, as is
generally assumed, position in space is purely relative, it follows
that the space of one man's objects and the space of another man's
objects have no place in common, that they are in fact different
spaces, and not merely different parts of one space. I mean by this
that such immediate spatial relations as are perceived to hold
between the different parts of the sensible space perceived by one
man, do not hold between parts of sensible spaces perceived by
different men. There are therefore a multitude of three-dimensional
spaces in the world: there are all those perceived by observers, and
presumably also those which are not perceived, merely because no
observer is suitably situated for perceiving them.

But although these spaces do not have to one another the same kind of
spatial relations as obtain between the parts of one of them, it is
nevertheless possible to arrange these spaces themselves in a
three-dimensional order. This is done by means of the correlated
particulars which we regard as members (or aspects) of one physical
thing. When a number of people are said to see the same object, those
who would be said to be near to the object see a particular occupying
a larger part of their field of vision than is occupied by the
corresponding particular seen by people who would be said to be
farther from the thing. By means of such considerations it is
possible, in ways which need not now be further specified, to arrange
all the different spaces in a three-dimensional series. Since each of
the spaces is itself three-dimensional, the whole world of particulars
is thus arranged in a six-dimensional space, that is to say, six
co-ordinates will be required to assign completely the position of any
given particular, namely, three to assign its position in its own
space and three more to assign the position of its space among the
other spaces.

There are two ways of classifying particulars: we may take together
all those that belong to a given "perspective," or all those that are,
as common sense would say, different "aspects" of the same "thing. "
For example, if I am (as is said) seeing the sun, what I see belongs
to two assemblages: (1) the assemblage of all my present objects of
sense, which is what I call a "perspective"; (2) the assemblage of
all the different particulars which would be called aspects of the sun
of eight minutes ago--this assemblage is what I define as _being_ the
sun of eight minutes ago. Thus "perspectives" and "things" are merely
two different ways of classifying particulars. It is to be observed
that there is no _a priori_ necessity for particulars to be
susceptible of this double classification. There may be what might be
called "wild" particulars, not having the usual relations by which the
classification is effected; perhaps dreams and hallucinations are
composed of particulars which are "wild" in this sense.

The exact definition of what is meant by a perspective is not quite
easy. So long as we confine ourselves to visible objects or to objects
of touch we might define the perspective of a given particular as "all
particulars which have a simple (direct) spatial relation to the given
particular. " Between two patches of colour which I see now, there is a
direct spatial relation which I equally see. But between patches of
colour seen by different men there is only an indirect constructed
spatial relation by means of the placing of "things" in physical space
(which is the same as the space composed of perspectives). Those
particulars which have direct spatial relations to a given particular
will belong to the same perspective. But if, for example, the sounds
which I hear are to belong to the same perspective with the patches of
colour which I see, there must be particulars which have no direct
spatial relation and yet belong to the same perspective. We cannot
define a perspective as all the data of one percipient at one time,
because we wish to allow the possibility of perspectives which are not
perceived by any one. There will be need, therefore, in defining a
perspective, of some principle derived neither from psychology nor
from space.

Such a principle may be obtained from the consideration of _time_.
The one all-embracing time, like the one all-embracing space, is a
construction; there is no _direct_ time-relation between particulars
belonging to my perspective and particulars belonging to another
man's. On the other hand, any two particulars of which I am aware are
either simultaneous or successive, and their simultaneity or
successiveness is sometimes itself a datum to me. We may therefore
define the perspective to which a given particular belongs as "all
particulars simultaneous with the given particular," where
"simultaneous" is to be understood as a direct simple relation, not
the derivative constructed relation of physics. It may be observed
that the introduction of "local time" suggested by the principle of
relativity has effected, for purely scientific reasons, much the same
multiplication of times as we have just been advocating.

The sum-total of all the particulars that are (directly) either
simultaneous with or before or after a given particular may be defined
as the "biography" to which that particular belongs. It will be
observed that, just as a perspective need not be actually perceived by
any one, so a biography need not be actually lived by any one. Those
biographies that are lived by no one are called "official. "

The definition of a "thing" is effected by means of continuity and of
correlations which have a certain differential independence of other
"things. " That is to say, given a particular in one perspective, there
will usually in a neighbouring perspective be a very similar
particular, differing from the given particular, to the first order of
small quantities, according to a law involving only the difference of
position of the two perspectives in perspective space, and not any of
the other "things" in the universe. It is this continuity and
differential independence in the law of change as we pass from one
perspective to another that defines the class of particulars which is
to be called "one thing. "

Broadly speaking, we may say that the physicist finds it convenient to
classify particulars into "things," while the psychologist finds it
convenient to classify them into "perspectives" and "biographies,"
since one perspective _may_ constitute the momentary data of one
percipient, and one biography _may_ constitute the whole of the data
of one percipient throughout his life.

We may now sum up our discussion. Our object has been to discover as
far as possible the nature of the ultimate constituents of the
physical world. When I speak of the "physical world," I mean, to begin
with, the world dealt with by physics. It is obvious that physics is
an empirical science, giving us a certain amount of knowledge and
based upon evidence obtained through the senses. But partly through
the development of physics itself, partly through arguments derived
from physiology, psychology or metaphysics, it has come to be thought
that the immediate data of sense could not themselves form part of the
ultimate constituents of the physical world, but were in some sense
"mental," "in the mind," or "subjective. " The grounds for this view,
in so far as they depend upon physics, can only be adequately dealt
with by rather elaborate constructions depending upon symbolic logic,
showing that out of such materials as are provided by the senses it is
possible to construct classes and series having the properties which
physics assigns to matter. Since this argument is difficult and
technical, I have not embarked upon it in this article. But in so far
as the view that sense-data are "mental" rests upon physiology,
psychology, or metaphysics, I have tried to show that it rests upon
confusions and prejudices--prejudices in favour of permanence in the
ultimate constituents of matter, and confusions derived from unduly
simple notions as to space, from the causal correlation of sense-data
with sense-organs, and from failure to distinguish between sense-data
and sensations. If what we have said on these subjects is valid, the
existence of sense-data is logically independent of the existence of
mind, and is causally dependent upon the _body_ of the percipient
rather than upon his mind. The causal dependence upon the body of the
percipient, we found, is a more complicated matter than it appears to
be, and, like all causal dependence, is apt to give rise to erroneous
beliefs through misconceptions as to the nature of causal correlation.
If we have been right in our contentions, sense-data are merely those
among the ultimate constituents of the physical world, of which we
happen to be immediately aware; they themselves are purely physical,
and all that is mental in connection with them is our awareness of
them, which is irrelevant to their nature and to their place in
physics.

Unduly simple notions as to space have been a great stumbling-block to
realists. When two men look at the same table, it is supposed that
what the one sees and what the other sees are in the same place. Since
the shape and colour are not quite the same for the two men, this
raises a difficulty, hastily solved, or rather covered up, by
declaring what each sees to be purely "subjective"--though it would
puzzle those who use this glib word to say what they mean by it. The
truth seems to be that space--and time also--is much more complicated
than it would appear to be from the finished structure of physics, and
that the one all-embracing three-dimensional space is a logical
construction, obtained by means of correlations from a crude space of
six dimensions. The particulars occupying this six-dimensional space,
classified in one way, form "things," from which with certain further
manipulations we can obtain what physics can regard as matter;
classified in another way, they form "perspectives" and "biographies,"
which may, if a suitable percipient happens to exist, form
respectively the sense-data of a momentary or of a total experience.
It is only when physical "things" have been dissected into series of
classes of particulars, as we have done, that the conflict between the
point of view of physics and the point of view of psychology can be
overcome. This conflict, if what has been said is not mistaken, flows
from different methods of classification, and vanishes as soon as its
source is discovered.

In favour of the theory which I have briefly outlined, I do not claim
that it is _certainly_ true. Apart from the likelihood of mistakes,
much of it is avowedly hypothetical. What I do claim for the theory is
that it _may_ be true, and that this is more than can be said for any
other theory except the closely analogous theory of Leibniz. The
difficulties besetting realism, the confusions obstructing any
philosophical account of physics, the dilemma resulting from
discrediting sense-data, which yet remain the sole source of our
knowledge of the outer world--all these are avoided by the theory
which I advocate. This does not prove the theory to be true, since
probably many other theories might be invented which would have the
same merits. But it does prove that the theory has a better chance of
being true than any of its present competitors, and it suggests that
what can be known with certainty is likely to be discoverable by
taking our theory as a starting-point, and gradually freeing it from
all such assumptions as seem irrelevant, unnecessary, or unfounded. On
these grounds, I recommend it to attention as a hypothesis and a basis
for further work, though not as itself a finished or adequate solution
of the problem with which it deals.

FOOTNOTES:

[23] An address delivered to the Philosophical Society of Manchester
in February, 1915. Reprinted from _The Monist_, July, 1915.

[24] Cf. especially Samuel Alexander, "The Basis of Realism," _British
Academy_, Vol. VI.

[25] "Are Secondary Qualities Independent of Perception? " _Proc.
Arist. Soc. _, 1909-10, pp. 191-218.

[26] First dialogue between Hylas and Philonous, _Works_ (Fraser's
edition 1901). I. p. 384.

[27] This point has been well urged by the American realists.

[28] Cf. T. P. Nunn, "Are Secondary Qualities Independent of
Perception? " _Proc. Arist. Soc. _, 1909-1910.




VIII

THE RELATION OF SENSE-DATA TO PHYSICS


I. THE PROBLEM STATED

Physics is said to be an empirical science, based upon observation and
experiment.

It is supposed to be verifiable, i. e. capable of calculating
beforehand results subsequently confirmed by observation and
experiment.

What can we learn by observation and experiment?

Nothing, so far as physics is concerned, except immediate data of
sense: certain patches of colour, sounds, tastes, smells, etc. , with
certain spatio-temporal relations.

The supposed contents of the physical world are _prima facie_ very
different from these: molecules have no colour, atoms make no noise,
electrons have no taste, and corpuscles do not even smell.

If such objects are to be verified, it must be solely through their
relation to sense-data: they must have some kind of correlation with
sense-data, and must be verifiable through their correlation _alone_.

But how is the correlation itself ascertained? A correlation can only
be ascertained empirically by the correlated objects being constantly
_found_ together. But in our case, only one term of the correlation,
namely, the sensible term, is ever _found_: the other term seems
essentially incapable of being found. Therefore, it would seem, the
correlation with objects of sense, by which physics was to be
verified, is itself utterly and for ever unverifiable.

There are two ways of avoiding this result.

(1) We may say that we know some principle _a priori_, without the
need of empirical verification, e. g. that our sense-data have _causes_
other than themselves, and that something can be known about these
causes by inference from their effects. This way has been often
adopted by philosophers. It may be necessary to adopt this way to some
extent, but in so far as it is adopted physics ceases to be empirical
or based upon experiment and observation alone. Therefore this way is
to be avoided as much as possible.

(2) We may succeed in actually defining the objects of physics as
functions of sense-data. Just in so far as physics leads to
expectations, this _must_ be possible, since we can only _expect_ what
can be experienced. And in so far as the physical state of affairs is
inferred from sense-data, it must be capable of expression as a
function of sense-data. The problem of accomplishing this expression
leads to much interesting logico-mathematical work.