No More Learning

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.

In physics as commonly set forth, sense-data appear as functions of
physical objects: when such-and-such waves impinge upon the eye, we
see such-and-such colours, and so on. But the waves are in fact
inferred from the colours, not vice versa. Physics cannot be regarded
as validly based upon empirical data until the waves have been
expressed as functions of the colours and other sense-data.

Thus if physics is to be verifiable we are faced with the following
problem: Physics exhibits sense-data as functions of physical objects,
but verification is only possible if physical objects can be exhibited
as functions of sense-data. We have therefore to solve the equations
giving sense-data in terms of physical objects, so as to make them
instead give physical objects in terms of sense-data.


II. CHARACTERISTICS OF SENSE-DATA

When I speak of a "sense-datum," I do not mean the whole of what is
given in sense at one time. I mean rather such a part of the whole as
might be singled out by attention: particular patches of colour,
particular noises, and so on. There is some difficulty in deciding
what is to be considered _one_ sense-datum: often attention causes
divisions to appear where, so far as can be discovered, there were no
divisions before. An observed complex fact, such as that this patch of
red is to the left of that patch of blue, is also to be regarded as a
datum from our present point of view: epistemologically, it does not
differ greatly from a simple sense-datum as regards its function in
giving knowledge. Its _logical_ structure is very different, however,
from that of sense: _sense_ gives acquaintance with particulars, and
is thus a two-term relation in which the object can be _named_ but not
_asserted_, and is inherently incapable of truth or falsehood, whereas
the observation of a complex fact, which may be suitably called
perception, is not a two-term relation, but involves the propositional
form on the object-side, and gives knowledge of a truth, not mere
acquaintance with a particular. This logical difference, important as
it is, is not very relevant to our present problem; and it will be
convenient to regard data of perception as included among sense-data
for the purposes of this paper. It is to be observed that the
particulars which are constituents of a datum of perception are always
sense-data in the strict sense.

Concerning sense-data, we know that they are there while they are
data, and this is the epistemological basis of all our knowledge of
external particulars. (The meaning of the word "external" of course
raises problems which will concern us later. ) We do not know, except
by means of more or less precarious inferences, whether the objects
which are at one time sense-data continue to exist at times when they
are not data. Sense-data at the times when they are data are all that
we directly and primitively know of the external world; hence in
epistemology the fact that they are _data_ is all-important. But the
fact that they are all that we directly know gives, of course, no
presumption that they are all that there is. If we could construct an
impersonal metaphysic, independent of the accidents of our knowledge
and ignorance, the privileged position of the actual data would
probably disappear, and they would probably appear as a rather
haphazard selection from a mass of objects more or less like them. In
saying this, I assume only that it is probable that there are
particulars with which we are not acquainted. Thus the special
importance of sense-data is in relation to epistemology, not to
metaphysics. In this respect, physics is to be reckoned as
metaphysics: it is impersonal, and nominally pays no special attention
to sense-data. It is only when we ask how physics can be _known_ that
the importance of sense-data re-emerges.


III. SENSIBILIA

I shall give the name _sensibilia_ to those objects which have the
same metaphysical and physical status as sense-data, without
necessarily being data to any mind. Thus the relation of a _sensibile_
to a sense-datum is like that of a man to a husband: a man becomes a
husband by entering into the relation of marriage, and similarly a
_sensibile_ becomes a sense-datum by entering into the relation of
acquaintance. It is important to have both terms; for we wish to
discuss whether an object which is at one time a sense-datum can still
exist at a time when it is not a sense-datum. We cannot ask "Can
sense-data exist without being given? " for that is like asking "Can
husbands exist without being married? " We must ask "Can _sensibilia_
exist without being given? " and also "Can a particular _sensibile_ be
at one time a sense-datum, and at another not? " Unless we have the
word _sensibile_ as well as the word "sense-datum," such questions are
apt to entangle us in trivial logical puzzles.

It will be seen that all sense-data are _sensibilia_. It is a
metaphysical question whether all _sensibilia_ are sense-data, and an
epistemological question whether there exist means of inferring
_sensibilia_ which are not data from those that are.

A few preliminary remarks, to be amplified as we proceed, will serve
to elucidate the use which I propose to make of _sensibilia_.

I regard sense-data as not mental, and as being, in fact, part of the
actual subject-matter of physics. There are arguments, shortly to be
examined, for their subjectivity, but these arguments seem to me only
to prove _physiological_ subjectivity, i. e. causal dependence on the
sense-organs, nerves, and brain. The appearance which a thing presents
to us is causally dependent upon these, in exactly the same way as it
is dependent upon intervening fog or smoke or coloured glass. Both
dependences are contained in the statement that the appearance which a
piece of matter presents when viewed from a given place is a function
not only of the piece of matter, but also of the intervening medium.
(The terms used in this statement--"matter," "view from a given
place," "appearance," "intervening medium"--will all be defined in the
course of the present paper. ) We have not the means of ascertaining
how things appear from places not surrounded by brain and nerves and
sense-organs, because we cannot leave the body; but continuity makes
it not unreasonable to suppose that they present _some_ appearance at
such places. Any such appearance would be included among _sensibilia_.
If--_per impossibile_--there were a complete human body with no mind
inside it, all those _sensibilia_ would exist, in relation to that
body, which would be sense-data if there were a mind in the body. What
the mind adds to _sensibilia_, in fact, is _merely_ awareness:
everything else is physical or physiological.


IV. SENSE-DATA ARE PHYSICAL

Before discussing this question it will be well to define the sense in
which the terms "mental" and "physical" are to be used. The word
"physical," in all preliminary discussions, is to be understood as
meaning "what is dealt with by physics. " Physics, it is plain, tells
us something about some of the constituents of the actual world; what
these constituents are may be doubtful, but it is they that are to be
called physical, whatever their nature may prove to be.

The definition of the term "mental" is more difficult, and can only be
satisfactorily given after many difficult controversies have been
discussed and decided. For present purposes therefore I must content
myself with assuming a dogmatic answer to these controversies. I shall
call a particular "mental" when it is aware of something, and I shall
call a fact "mental" when it contains a mental particular as a
constituent.

It will be seen that the mental and the physical are not necessarily
mutually exclusive, although I know of no reason to suppose that they
overlap.

The doubt as to the correctness of our definition of the "mental" is
of little importance in our present discussion. For what I am
concerned to maintain is that sense-data are physical, and this being
granted it is a matter of indifference in our present inquiry whether
or not they are also mental. Although I do not hold, with Mach and
James and the "new realists," that the difference between the mental
and the physical is _merely_ one of arrangement, yet what I have to
say in the present paper is compatible with their doctrine and might
have been reached from their standpoint.

In discussions on sense-data, two questions are commonly confused,
namely:

(1) Do sensible objects persist when we are not sensible of them? in
other words, do _sensibilia_ which are data at a certain time
sometimes continue to exist at times when they are not data? And (2)
are sense-data mental or physical?

I propose to assert that sense-data are physical, while yet
maintaining that they probably never persist unchanged after ceasing
to be data. The view that they do not persist is often thought, quite
erroneously in my opinion, to imply that they are mental; and this
has, I believe, been a potent source of confusion in regard to our
present problem. If there were, as some have held, a _logical
impossibility_ in sense-data persisting after ceasing to be data, that
certainly would tend to show that they were mental; but if, as I
contend, their non-persistence is merely a probable inference from
empirically ascertained causal laws, then it carries no such
implication with it, and we are quite free to treat them as part of
the subject-matter of physics.

Logically a sense-datum is an object, a particular of which the
subject is aware. It does not contain the subject as a part, as for
example beliefs and volitions do. The existence of the sense-datum is
therefore not logically dependent upon that of the subject; for the
only way, so far as I know, in which the existence of _A_ can be
_logically_ dependent upon the existence of _B_ is when _B_ is part of
_A_. There is therefore no _a priori_ reason why a particular which is
a sense-datum should not persist after it has ceased to be a datum,
nor why other similar particulars should not exist without ever being
data. The view that sense-data are mental is derived, no doubt, in
part from their physiological subjectivity, but in part also from a
failure to distinguish between sense-data and "sensations. " By a
sensation I mean the fact consisting in the subject's awareness of the
sense-datum. Thus a sensation is a complex of which the subject is a
constituent and which therefore is mental. The sense-datum, on the
other hand, stands over against the subject as that external object of
which in sensation the subject is aware. It is true that the
sense-datum is in many cases in the subject's body, but the subject's
body is as distinct from the subject as tables and chairs are, and is
in fact merely a part of the material world. So soon, therefore, as
sense-data are clearly distinguished from sensations, and as their
subjectivity is recognised to be physiological not psychical, the
chief obstacles in the way of regarding them as physical are removed.


V. "SENSIBILIA" AND "THINGS"

But if "sensibilia" are to be recognised as the ultimate constituents
of the physical world, a long and difficult journey is to be performed
before we can arrive either at the "thing" of common sense or at the
"matter" of physics. The supposed impossibility of combining the
different sense-data which are regarded as appearances of the same
"thing" to different people has made it seem as though these
"sensibilia" must be regarded as mere subjective phantasms. A given
table will present to one man a rectangular appearance, while to
another it appears to have two acute angles and two obtuse angles; to
one man it appears brown, while to another, towards whom it reflects
the light, it appears white and shiny. It is said, not wholly without
plausibility, that these different shapes and different colours cannot
co-exist simultaneously in the same place, and cannot therefore both
be constituents of the physical world. This argument I must confess
appeared to me until recently to be irrefutable. The contrary opinion
has, however, been ably maintained by Dr. T. P. Nunn in an article
entitled: "Are Secondary Qualities Independent of Perception? "[29] The
supposed impossibility derives its apparent force from the phrase:
"_in the same place_," and it is precisely in this phrase that its
weakness lies. The conception of space is too often treated in
philosophy--even by those who on reflection would not defend such
treatment--as though it were as given, simple, and unambiguous as
Kant, in his psychological innocence, supposed. It is the unperceived
ambiguity of the word "place" which, as we shall shortly see, has
caused the difficulties to realists and given an undeserved advantage
to their opponents.
Two "places" of different kinds are involved in
every sense-datum, namely the place _at_ which it appears and the
place _from_ which it appears. These belong to different spaces,
although, as we shall see, it is possible, with certain limitations,
to establish a correlation between them. What we call the different
appearances of the same thing to different observers are each in a
space private to the observer concerned. No place in the private world
of one observer is identical with a place in the private world of
another observer. There is therefore no question of combining the
different appearances in the one place; and the fact that they cannot
all exist in one place affords accordingly no ground whatever for
questioning their physical reality. The "thing" of common sense may in
fact be identified with the whole class of its appearances--where,
however, we must include among appearances not only those which are
actual sense-data, but also those "sensibilia," if any, which, on
grounds of continuity and resemblance, are to be regarded as belonging
to the same system of appearances, although there happen to be no
observers to whom they are data.

An example may make this clearer. Suppose there are a number of people
in a room, all seeing, as they say, the same tables and chairs, walls
and pictures. No two of these people have exactly the same sense-data,
yet there is sufficient similarity among their data to enable them to
group together certain of these data as appearances of one "thing" to
the several spectators, and others as appearances of another "thing. "
Besides the appearances which a given thing in the room presents to
the actual spectators, there are, we may suppose, other appearances
which it would present to other possible spectators. If a man were to
sit down between two others, the appearance which the room would
present to him would be intermediate between the appearances which it
presents to the two others: and although this appearance would not
exist as it is without the sense organs, nerves and brain, of the
newly arrived spectator, still it is not unnatural to suppose that,
from the position which he now occupies, _some_ appearance of the
room existed before his arrival. This supposition, however, need
merely be noticed and not insisted upon.

Since the "thing" cannot, without indefensible partiality, be
identified with any single one of its appearances, it came to be
thought of as something distinct from all of them and underlying them.
But by the principle of Occam's razor, if the class of appearances
will fulfil the purposes for the sake of which the thing was invented
by the prehistoric metaphysicians to whom common sense is due, economy
demands that we should identify the thing with the class of its
appearances. It is not necessary to _deny_ a substance or substratum
underlying these appearances; it is merely expedient to abstain from
asserting this unnecessary entity. Our procedure here is precisely
analogous to that which has swept away from the philosophy of
mathematics the useless menagerie of metaphysical monsters with which
it used to be infested.


VI. CONSTRUCTIONS VERSUS INFERENCES

Before proceeding to analyse and explain the ambiguities of the word
"place," a few general remarks on method are desirable. The supreme
maxim in scientific philosophising is this:

_Wherever possible, logical constructions are to be substituted
for inferred entities. _

Some examples of the substitution of construction for inference in the
realm of mathematical philosophy may serve to elucidate the uses of
this maxim. Take first the case of irrationals. In old days,
irrationals were inferred as the supposed limits of series of
rationals which had no rational limit; but the objection to this
procedure was that it left the existence of irrationals merely
optative, and for this reason the stricter methods of the present day
no longer tolerate such a definition. We now define an irrational
number as a certain class of ratios, thus constructing it logically by
means of ratios, instead of arriving at it by a doubtful inference
from them. Take again the case of cardinal numbers. Two equally
numerous collections appear to have something in common: this
something is supposed to be their cardinal number. But so long as the
cardinal number is inferred from the collections, not constructed in
terms of them, its existence must remain in doubt, unless in virtue of
a metaphysical postulate _ad hoc_. By defining the cardinal number of
a given collection as the class of all equally numerous collections,
we avoid the necessity of this metaphysical postulate, and thereby
remove a needless element of doubt from the philosophy of arithmetic.
A similar method, as I have shown elsewhere, can be applied to classes
themselves, which need not be supposed to have any metaphysical
reality, but can be regarded as symbolically constructed fictions.

The method by which the construction proceeds is closely analogous in
these and all similar cases. Given a set of propositions nominally
dealing with the supposed inferred entities, we observe the properties
which are required of the supposed entities in order to make these
propositions true. By dint of a little logical ingenuity, we then
construct some logical function of less hypothetical entities which
has the requisite properties. This constructed function we substitute
for the supposed inferred entities, and thereby obtain a new and less
doubtful interpretation of the body of propositions in question. This
method, so fruitful in the philosophy of mathematics, will be found
equally applicable in the philosophy of physics, where, I do not
doubt, it would have been applied long ago but for the fact that all
who have studied this subject hitherto have been completely ignorant
of mathematical logic. I myself cannot claim originality in the
application of this method to physics, since I owe the suggestion and
the stimulus for its application entirely to my friend and
collaborator Dr. Whitehead, who is engaged in applying it to the more
mathematical portions of the region intermediate between sense-data
and the points, instants and particles of physics.

A complete application of the method which substitutes constructions
for inferences would exhibit matter wholly in terms of sense-data, and
even, we may add, of the sense-data of a single person, since the
sense-data of others cannot be known without some element of
inference. This, however, must remain for the present an ideal, to be
approached as nearly as possible, but to be reached, if at all, only
after a long preliminary labour of which as yet we can only see the
very beginning. The inferences which are unavoidable can, however, be
subjected to certain guiding principles. In the first place they
should always be made perfectly explicit, and should be formulated in
the most general manner possible. In the second place the inferred
entities should, whenever this can be done, be similar to those whose
existence is given, rather than, like the Kantian _Ding an sich_,
something wholly remote from the data which nominally support the
inference. The inferred entities which I shall allow myself are of two
kinds: (_a_) the sense-data of other people, in favour of which there
is the evidence of testimony, resting ultimately upon the analogical
argument in favour of minds other than my own; (_b_) the "sensibilia"
which would appear from places where there happen to be no minds, and
which I suppose to be real although they are no one's data. Of these
two classes of inferred entities, the first will probably be allowed
to pass unchallenged. It would give me the greatest satisfaction to be
able to dispense with it, and thus establish physics upon a
solipsistic basis; but those--and I fear they are the majority--in
whom the human affections are stronger than the desire for logical
economy, will, no doubt, not share my desire to render solipsism
scientifically satisfactory. The second class of inferred entities
raises much more serious questions. It may be thought monstrous to
maintain that a thing can present any appearance at all in a place
where no sense organs and nervous structure exist through which it
could appear. I do not myself feel the monstrosity; nevertheless I
should regard these supposed appearances only in the light of a
hypothetical scaffolding, to be used while the edifice of physics is
being raised, though possibly capable of being removed as soon as the
edifice is completed. These "sensibilia" which are not data to anyone
are therefore to be taken rather as an illustrative hypothesis and as
an aid in preliminary statement than as a dogmatic part of the
philosophy of physics in its final form.


VII. PRIVATE SPACE AND THE SPACE OF PERSPECTIVES

We have now to explain the ambiguity in the word "place," and how it
comes that two places of different sorts are associated with every
sense-datum, namely the place _at_ which it is and the place _from_
which it is perceived. The theory to be advocated is closely analogous
to Leibniz's monadology, from which it differs chiefly in being less
smooth and tidy.

The first fact to notice is that, so far as can be discovered, no
sensibile is ever a datum to two people at once. The things seen by
two different people are often closely similar, so similar that the
same _words_ can be used to denote them, without which communication
with others concerning sensible objects would be impossible. But, in
spite of this similarity, it would seem that some difference always
arises from difference in the point of view. Thus each person, so far
as his sense-data are concerned, lives in a private world. This
private world contains its own space, or rather spaces, for it would
seem that only experience teaches us to correlate the space of sight
with the space of touch and with the various other spaces of other
senses. This multiplicity of private spaces, however, though
interesting to the psychologist, is of no great importance in regard
to our present problem, since a merely solipsistic experience enables
us to correlate them into the one private space which embraces all our
own sense-data. The place _at_ which a sense-datum is, is a place in
private space. This place therefore is different from any place in the
private space of another percipient. For if we assume, as logical
economy demands, that all position is relative, a place is only
definable by the things in or around it, and therefore the same place
cannot occur in two private worlds which have no common constituent.
The question, therefore, of combining what we call different
appearances of the same thing in the same place does not arise, and
the fact that a given object appears to different spectators to have
different shapes and colours affords no argument against the physical
reality of all these shapes and colours.

In addition to the private spaces belonging to the private worlds of
different percipients, there is, however, another space, in which one
whole private world counts as a point, or at least as a spatial unit.
This might be described as the space of points of view, since each
private world may be regarded as the appearance which the universe
presents from a certain point of view. I prefer, however, to speak of
it as the space of _perspectives_, in order to obviate the suggestion
that a private world is only real when someone views it. And for the
same reason, when I wish to speak of a private world without assuming
a percipient, I shall call it a "perspective. "

We have now to explain how the different perspectives are ordered in
one space. This is effected by means of the correlated "sensibilia"
which are regarded as the appearances, in different perspectives, of
one and the same thing. By moving, and by testimony, we discover that
two different perspectives, though they cannot both contain the same
"sensibilia," may nevertheless contain very similar ones; and the
spatial order of a certain group of "sensibilia" in a private space of
one perspective is found to be identical with, or very similar to, the
spatial order of the correlated "sensibilia" in the private space of
another perspective. In this way one "sensibile" in one perspective is
correlated with one "sensibile" in another. Such correlated
"sensibilia" will be called "appearances of one thing. " In Leibniz's
monadology, since each monad mirrored the whole universe, there was in
each perspective a "sensibile" which was an appearance of each thing.
In our system of perspectives, we make no such assumption of
completeness. A given thing will have appearances in some
perspectives, but presumably not in certain others. The "thing" being
defined as the class of its appearances, if ? is the class of
perspectives in which a certain thing ? appears, then ? is a member of
the multiplicative class of ? , ? being a class of mutually exclusive
classes of "sensibilia. " And similarly a perspective is a member of
the multiplicative class of the things which appear in it.

The arrangement of perspectives in a space is effected by means of the
differences between the appearances of a given thing in the various
perspectives. Suppose, say, that a certain penny appears in a number
of different perspectives; in some it looks larger and in some
smaller, in some it looks circular, in others it presents the
appearance of an ellipse of varying eccentricity. We may collect
together all those perspectives in which the appearance of the penny
is circular. These we will place on one straight line, ordering them
in a series by the variations in the apparent size of the penny. Those
perspectives in which the penny appears as a straight line of a
certain thickness will similarly be placed upon a plane (though in
this case there will be many different perspectives in which the penny
is of the same size; when one arrangement is completed these will form
a circle concentric with the penny), and ordered as before by the
apparent size of the penny. By such means, all those perspectives in
which the penny presents a visual appearance can be arranged in a
three-dimensional spatial order. Experience shows that the same
spatial order of perspectives would have resulted if, instead of the
penny, we had chosen any other thing which appeared in all the
perspectives in question, or any other method of utilising the
differences between the appearances of the same things in different
perspectives. It is this empirical fact which has made it possible to
construct the one all-embracing space of physics.

The space whose construction has just been explained, and whose
elements are whole perspectives, will be called "perspective-space. "


VIII. THE PLACING OF "THINGS" AND "SENSIBILIA" IN PERSPECTIVE SPACE

The world which we have so far constructed is a world of six
dimensions, since it is a three-dimensional series of perspectives,
each of which is itself three-dimensional. We have now to explain the
correlation between the perspective space and the various private
spaces contained within the various perspectives severally. It is by
means of this correlation that the one three-dimensional space of
physics is constructed; and it is because of the unconscious
performance of this correlation that the distinction between
perspective space and the percipient's private space has been blurred,
with disastrous results for the philosophy of physics. Let us revert
to our penny: the perspectives in which the penny appears larger are
regarded as being nearer to the penny than those in which it appears
smaller, but as far as experience goes the apparent size of the penny
will not grow beyond a certain limit, namely, that where (as we say)
the penny is so near the eye that if it were any nearer it could not
be seen. By touch we may prolong the series until the penny touches
the eye, but no further. If we have been travelling along a line of
perspectives in the previously defined sense, we may, however, by
imagining the penny removed, prolong the line of perspectives by
means, say, of another penny; and the same may be done with any other
line of perspectives defined by means of the penny. All these lines
meet in a certain place, that is, in a certain perspective. This
perspective will be defined as "the place where the penny is. "

It is now evident in what sense two places in constructed physical
space are associated with a given "sensibile. " There is first the
place which is the perspective of which the "sensibile" is a member.
This is the place _from_ which the "sensibile" appears. Secondly there
is the place where the thing is of which the "sensibile" is a member,
in other words an appearance; this is the place _at_ which the
"sensibile" appears. The "sensibile" which is a member of one
perspective is correlated with another perspective, namely, that which
is the place where the thing is of which the "sensibile" is an
appearance. To the psychologist the "place from which" is the more
interesting, and the "sensibile" accordingly appears to him subjective
and where the percipient is. To the physicist the "place at which" is
the more interesting, and the "sensibile" accordingly appears to him
physical and external. The causes, limits and partial justification of
each of these two apparently incompatible views are evident from the
above duplicity of places associated with a given "sensibile. "

We have seen that we can assign to a physical thing a place in the
perspective space. In this way different parts of our body acquire
positions in perspective space, and therefore there is a meaning
(whether true or false need not much concern us) in saying that the
perspective to which our sense-data belong is inside our head. Since
our mind is correlated with the perspective to which our sense-data
belong, we may regard this perspective as being the position of our
mind in perspective space. If, therefore, this perspective is, in the
above defined sense, inside our head, there is a good meaning for the
statement that the mind is in the head. We can now say of the various
appearances of a given thing that some of them are nearer to the thing
than others; those are nearer which belong to perspectives that are
nearer to "the place where the thing is. " We can thus find a meaning,
true or false, for the statement that more is to be learnt about a
thing by examining it close to than by viewing it from a distance. We
can also find a meaning for the phrase "the things which intervene
between the subject and a thing of which an appearance is a datum to
him. " One reason often alleged for the subjectivity of sense-data is
that the appearance of a thing may change when we find it hard to
suppose that the thing itself has changed--for example, when the
change is due to our shutting our eyes, or to our screwing them up so
as to make the thing look double. If the thing is defined as the class
of its appearances (which is the definition adopted above), there is
of course necessarily _some_ change in the thing whenever any one of
its appearances changes. Nevertheless there is a very important
distinction between two different ways in which the appearances may
change. If after looking at a thing I shut my eyes, the appearance of
my eyes changes in every perspective in which there is such an
appearance, whereas most of the appearances of the thing will remain
unchanged. We may say, as a matter of definition, that a thing changes
when, however near to the thing an appearance of it may be, there are
changes in appearances as near as, or still nearer to, the thing. On
the other hand we shall say that the change is in some other thing if
all appearances of the thing which are at not more than a certain
distance from the thing remain unchanged, while only comparatively
distant appearances of the thing are altered. From this consideration
we are naturally led to the consideration of _matter_, which must be
our next topic.


IX. THE DEFINITION OF MATTER

We defined the "physical thing" as the class of its appearances, but
this can hardly be taken as a definition of matter. We want to be able
to express the fact that the appearance of a thing in a given
perspective is causally affected by the matter between the thing and
the perspective. We have found a meaning for "between a thing and a
perspective. " But we want matter to be something other than the whole
class of appearances of a thing, in order to state the influence of
matter on appearances.

We commonly assume that the information we get about a thing is more
accurate when the thing is nearer. Far off, we see it is a man; then
we see it is Jones; then we see he is smiling. Complete accuracy would
only be attainable as a limit: if the appearances of Jones as we
approach him tend towards a limit, that limit may be taken to be what
Jones really is. It is obvious that from the point of view of physics
the appearances of a thing close to "count" more than the appearances
far off. We may therefore set up the following tentative definition:

The _matter_ of a given thing is the limit of its appearances as their
distance from the thing diminishes.

It seems probable that there is something in this definition, but it
is not quite satisfactory, because empirically there is no such limit
to be obtained from sense-data. The definition will have to be eked
out by constructions and definitions. But probably it suggests the
right direction in which to look.

We are now in a position to understand in outline the reverse journey
from matter to sense-data which is performed by physics. The
appearance of a thing in a given perspective is a function of the
matter composing the thing and of the intervening matter. The
appearance of a thing is altered by intervening smoke or mist, by blue
spectacles or by alterations in the sense-organs or nerves of the
percipient (which also must be reckoned as part of the intervening
medium). The nearer we approach to the thing, the less its appearance
is affected by the intervening matter. As we travel further and
further from the thing, its appearances diverge more and more from
their initial character; and the causal laws of their divergence are
to be stated in terms of the matter which lies between them and the
thing. Since the appearances at very small distances are less affected
by causes other than the thing itself, we come to think that the limit
towards which these appearances tend as the distance diminishes is
what the thing "really is," as opposed to what it merely seems to be.
This, together with its necessity for the statement of causal laws,
seems to be the source of the entirely erroneous feeling that matter
is more "real" than sense-data.

Consider for example the infinite divisibility of matter. In looking
at a given thing and approaching it, one sense-datum will become
several, and each of these will again divide. Thus _one_ appearance
may represent _many_ things, and to this process there seems no end.
Hence in the limit, when we approach indefinitely near to the thing
there will be an indefinite number of units of matter corresponding to
what, at a finite distance, is only one appearance. This is how
infinite divisibility arises.

The whole causal efficacy of a thing resides in its matter. This is in
some sense an empirical fact, but it would be hard to state it
precisely, because "causal efficacy" is difficult to define.

What can be known empirically about the matter of a thing is only
approximate, because we cannot get to know the appearances of the
thing from very small distances, and cannot accurately infer the limit
of these appearances. But it _is_ inferred _approximately_ by means of
the appearances we can observe. It then turns out that these
appearances can be exhibited by physics as a function of the matter
in our immediate neighbourhood; e. g. the visual appearance of a
distant object is a function of the light-waves that reach the eyes.
This leads to confusions of thought, but offers no real difficulty.

One appearance, of a visible object for example, is not sufficient to
determine its other simultaneous appearances, although it goes a
certain distance towards determining them. The determination of the
hidden structure of a thing, so far as it is possible at all, can only
be effected by means of elaborate dynamical inferences.


X. TIME[30]

It seems that the one all-embracing time is a construction, like the
one all-embracing space. Physics itself has become conscious of this
fact through the discussions connected with relativity.

Between two perspectives which both belong to one person's experience,
there will be a direct time-relation of before and after. This
suggests a way of dividing history in the same sort of way as it is
divided by different experiences, but without introducing experience
or anything mental: we may define a "biography" as everything that is
(directly) earlier or later than, or simultaneous with, a given
"sensibile. " This will give a series of perspectives, which _might_
all form parts of one person's experience, though it is not necessary
that all or any of them should actually do so. By this means, the
history of the world is divided into a number of mutually exclusive
biographies.

We have now to correlate the times in the different biographies. The
natural thing would be to say that the appearances of a given
(momentary) thing in two different perspectives belonging to different
biographies are to be taken as simultaneous; but this is not
convenient. Suppose _A_ shouts to _B_, and _B_ replies as soon as he
hears _A's_ shout. Then between _A's_ hearing of his own shout and his
hearing of _B's_ there is an interval; thus if we made _A's_ and _B's_
hearing of the same shout exactly simultaneous with each other, we
should have events exactly simultaneous with a given event but not
with each other. To obviate this, we assume a "velocity of sound. "
That is, we assume that the time when _B_ hears _A's_ shout is
half-way between the time when _A_ hears his own shout and the time
when he hears _B's_. In this way the correlation is effected.

What has been said about sound applies of course equally to light. The
general principle is that the appearances, in different perspectives,
which are to be grouped together as constituting what a certain thing
is at a certain moment, are not to be all regarded as being at that
moment. On the contrary they spread outward from the thing with
various velocities according to the nature of the appearances. Since
no _direct_ means exist of correlating the time in one biography with
the time in another, this temporal grouping of the appearances
belonging to a given thing at a given moment is in part conventional.
Its motive is partly to secure the verification of such maxims as that
events which are exactly simultaneous with the same event are exactly
simultaneous with one another, partly to secure convenience in the
formulation of causal laws.


XI. THE PERSISTENCE OF THINGS AND MATTER

Apart from any of the fluctuating hypotheses of physics, three main
problems arise in connecting the world of physics with the world of
sense, namely:

1. the construction of a single space;
2. the construction of a single time;
3. the construction of permanent things or matter.

We have already considered the first and second of these problems; it
remains to consider the third.

We have seen how correlated appearances in different perspectives are
combined to form one "thing" at one moment in the all-embracing time
of physics. We have now to consider how appearances at different times
are combined as belonging to one "thing," and how we arrive at the
persistent "matter" of physics. The assumption of permanent substance,
which technically underlies the procedure of physics, cannot of course
be regarded as metaphysically legitimate: just as the one thing
simultaneously seen by many people is a construction, so the one thing
seen at different times by the same or different people must be a
construction, being in fact nothing but a certain grouping of certain
"sensibilia. "

We have seen that the momentary state of a "thing" is an assemblage of
"sensibilia," in different perspectives, not all simultaneous in the
one constructed time, but spreading out from "the place where the
thing is" with velocities depending upon the nature of the
"sensibilia. " The time _at_ which the "thing" is in this state is the
lower limit of the times at which these appearances occur. We have now
to consider what leads us to speak of another set of appearances as
belonging to the same "thing" at a different time.

For this purpose, we may, at least to begin with, confine ourselves
within a single biography. If we can always say when two "sensibilia"
in a given biography are appearances of one thing, then, since we have
seen how to connect "sensibilia" in different biographies as
appearances of the same momentary state of a thing, we shall have all
that is necessary for the complete construction of the history of a
thing.

It is to be observed, to begin with, that the identity of a thing for
common sense is not always correlated with the identity of matter for
physics. A human body is one persisting thing for common sense, but
for physics its matter is constantly changing. We may say, broadly,
that the common-sense conception is based upon continuity in
appearances at the ordinary distances of sense-data, while the
physical conception is based upon the continuity of appearances at
very small distances from the thing. It is probable that the
common-sense conception is not capable of complete precision. Let us
therefore concentrate our attention upon the conception of the
persistence of matter in physics.

The first characteristic of two appearances of the same piece of
matter at different times is _continuity_. The two appearances must be
connected by a series of intermediaries, which, if time and space form
compact series, must themselves form a compact series. The colour of
the leaves is different in autumn from what it is in summer; but we
believe that the change occurs gradually, and that, if the colours are
different at two given times, there are intermediate times at which
the colours are intermediate between those at the given times.

But there are two considerations that are important as regards
continuity.

First, it is largely hypothetical. We do not observe any one thing
continuously, and it is merely a hypothesis to assume that, while we
are not observing it, it passes through conditions intermediate
between those in which it is perceived. During uninterrupted
observation, it is true, continuity is nearly verified; but even here,
when motions are very rapid, as in the case of explosions, the
continuity is not actually capable of direct verification. Thus we can
only say that the sense-data are found to _permit_ a hypothetical
complement of "sensibilia" such as will preserve continuity, and that
therefore there _may_ be such a complement. Since, however, we have
already made such use of hypothetical "sensibilia," we will let this
point pass, and admit such "sensibilia" as are required to preserve
continuity.