PART II

LOVE AND CHANCE

I. THE ARCHITECTURE OF THEORIES[54]

Of the fifty or hundred systems of philosophy that have been advanced at

different times of the world’s history, perhaps the larger number have

been, not so much results of historical evolution, as happy thoughts

which have accidently occurred to their authors. An idea which has been

found interesting and fruitful has been adopted, developed, and forced

to yield explanations of all sorts of phenomena. The English have been

particularly given to this way of philosophizing; witness, Hobbes,

Hartley, Berkeley, James Mill. Nor has it been by any means useless

labor; it shows us what the true nature and value of the ideas developed

are, and in that way affords serviceable materials for philosophy. Just

as if a man, being seized with the conviction that paper was a good

material to make things of, were to go to work to build a _papier mâché_

house, with roof of roofing-paper, foundations of pasteboard, windows of

paraffined paper, chimneys, bath tubs, locks, etc., all of different

forms of paper, his experiment would probably afford valuable lessons to

builders, while it would certainly make a detestable house, so those

one-idea’d philosophies are exceedingly interesting and instructive, and

yet are quite unsound.

The remaining systems of philosophy have been of the nature of reforms,

sometimes amounting to radical revolutions, suggested by certain

difficulties which have been found to beset systems previously in vogue;

and such ought certainly to be in large part the motive of any new

theory. This is like partially rebuilding a house. The faults that have

been committed are, first, that the repairs of the dilapidations have

generally not been sufficiently thorough-going, and second, that not

sufficient pains had been taken to bring the additions into deep harmony

with the really sound parts of the old structure.

When a man is about to build a house, what a power of thinking he has to

do, before he can safely break ground! With what pains he has to

excogitate the precise wants that are to be supplied! What a study to

ascertain the most available and suitable materials, to determine the

mode of construction to which those materials are best adapted, and to

answer a hundred such questions! Now without riding the metaphor too

far, I think we may safely say that the studies preliminary to the

construction of a great theory should be at least as deliberate and

thorough as those that are preliminary to the building of a

dwelling-house.

That systems ought to be constructed architectonically has been preached

since Kant, but I do not think the full import of the maxim has by any

means been apprehended. What I would recommend is that every person who

wishes to form an opinion concerning fundamental problems, should first

of all make a complete survey of human knowledge, should take note of

all the valuable ideas in each branch of science, should observe in just

what respect each has been successful and where it has failed, in order

that in the light of the thorough acquaintance so attained of the

available materials for a philosophical theory and of the nature and

strength of each, he may proceed to the study of what the problem of

philosophy consists in, and of the proper way of solving it. I must not

be understood as endeavoring to state fully all that these preparatory

studies should embrace; on the contrary, I purposely slur over many

points, in order to give emphasis to one special recommendation, namely,

to make a systematic study of the conceptions out of which a

philosophical theory may be built, in order to ascertain what place each

conception may fitly occupy in such a theory, and to what uses it is

adapted.

The adequate treatment of this single point would fill a volume, but I

shall endeavor to illustrate my meaning by glancing at several sciences

and indicating conceptions in them serviceable for philosophy. As to the

results to which long studies thus commenced have led me, I shall just

give a hint at their nature.

We may begin with dynamics,—field in our day of perhaps the grandest

conquest human science has ever made,—I mean the law of the conservation

of energy. But let us revert to the first step taken by modern

scientific thought,—and a great stride it was,—the inauguration of

dynamics by Galileo. A modern physicist on examining Galileo’s works is

surprised to find how little experiment had to do with the establishment

of the foundations of mechanics. His principal appeal is to common sense

and _il lume naturale_. He always assumes that the true theory will be

found to be a simple and natural one. And we can see why it should

indeed be so in dynamics. For instance, a body left to its own inertia,

moves in a straight line, and a straight line appears to us the simplest

of curves. In _itself_, no curve is simpler than another. A system of

straight lines has intersections precisely corresponding to those of a

system of like parabolas similarly placed, or to those of any one of an

infinity of systems of curves. But the straight line appears to us

simple, because, as Euclid says, it lies evenly between its extremities;

that is, because viewed endwise it appears as a point. That is, again,

because light moves in straight lines. Now, light moves in straight

lines because of the part which the straight line plays in the laws of

dynamics. Thus it is that our minds having been formed under the

influence of phenomena governed by the laws of mechanics, certain

conceptions entering into those laws become implanted in our minds, so

that we readily guess at what the laws are. Without such a natural

prompting, having to search blindfold for a law which would suit the

phenomena, our chance of finding it would be as one to infinity. The

further physical studies depart from phenomena which have directly

influenced the growth of the mind, the less we can expect to find the

laws which govern them “simple,” that is, composed of a few conceptions

natural to our minds.

The researches of Galileo, followed up by Huygens and others, led to

those modern conceptions of _Force_ and _Law_, which have revolutionized

the intellectual world. The great attention given to mechanics in the

seventeenth century soon so emphasized these conceptions as to give rise

to the Mechanical Philosophy, or doctrine that all the phenomena of the

physical universe are to be explained upon mechanical principles.

Newton’s great discovery imparted a new impetus to this tendency. The

old notion that heat consists in an agitation of corpuscles was now

applied to the explanation of the chief properties of gases. The first

suggestion in this direction was that the pressure of gases is explained

by the battering of the particles against the walls of the containing

vessel, which explained Boyle’s law of the compressibility of air.

Later, the expansion of gases, Avogadro’s chemical law, the diffusion

and viscosity of gases, and the action of Crookes’s radiometer were

shown to be consequences of the same kinetical theory; but other

phenomena, such as the ratio of the specific heat at constant volume to

that at constant pressure, require additional hypotheses, which we have

little reason to suppose are simple, so that we find ourselves quite

afloat. In like manner with regard to light. That it consists of

vibrations was almost proved by the phenomena of diffraction, while

those of polarization showed the excursions of the particles to be

perpendicular to the line of propagation; but the phenomena of

dispersion, etc., require additional hypotheses which may be very

complicated. Thus, the further progress of molecular speculation appears

quite uncertain. If hypotheses are to be tried haphazard, or simply

because they will suit certain phenomena, it will occupy the

mathematical physicists of the world say half a century on the average

to bring each theory to the test, and since the number of possible

theories may go up into the trillions, only one of which can be true, we

have little prospect of making further solid additions to the subject in

our time. When we come to atoms, the presumption in favor of a simple

law seems very slender. There is room for serious doubt whether the

fundamental laws of mechanics hold good for single atoms, and it seems

quite likely that they are capable of motion in more than three

dimensions.

To find out much more about molecules and atoms, we must search out a

natural history of laws of nature, which may fulfil that function which

the presumption in favor of simple laws fulfilled in the early days of

dynamics, by showing us what kind of laws we have to expect and by

answering such questions as this: Can we with reasonable prospect of not

wasting time, try the supposition that atoms attract one another

inversely as the seventh power of their distances, or can we not? To

suppose universal laws of nature capable of being apprehended by the

mind and yet having no reason for their special forms, but standing

inexplicable and irrational, is hardly a justifiable position.

Uniformities are precisely the sort of facts that need to be accounted

for. That a pitched coin should sometimes turn up heads and sometimes

tails calls for no particular explanation; but if it shows heads every

time, we wish to know how this result has been brought about. Law is

_par excellence_ the thing that wants a reason.

Now the only possible way of accounting for the laws of nature and for

uniformity in general is to suppose them results of evolution. This

supposes them not to be absolute, not to be obeyed precisely. It makes

an element of indeterminacy, spontaneity, or absolute chance in nature.

Just as, when we attempt to verify any physical law, we find our

observations cannot be precisely satisfied by it, and rightly attribute

the discrepancy to errors of observation, so we must suppose far more

minute discrepancies to exist owing to the imperfect cogency of the law

itself, to a certain swerving of the facts from any definite formula.

Mr. Herbert Spencer wishes to explain evolution upon mechanical

principles. This is illogical, for four reasons. First, because the

principle of evolution requires no extraneous cause; since the tendency

to growth can be supposed itself to have grown from an infinitesimal

germ accidentally started. Second, because law ought more than anything

else to be supposed a result of evolution. Third, because exact law

obviously never can produce heterogeneity out of homogeneity; and

arbitrary heterogeneity is the feature of the universe the most manifest

and characteristic. Fourth, because the law of the conservation of

energy is equivalent to the proposition that all operations governed by

mechanical laws are reversible; so that an immediate corollary from it

is that growth is not explicable by those laws, even if they be not

violated in the process of growth. In short, Spencer is not a

philosophical evolutionist, but only a half-evolutionist,—or, if you

will, only a semi-Spencerian. Now philosophy requires thoroughgoing

evolutionism or none.

The theory of Darwin was that evolution had been brought about by the

action of two factors: first, heredity, as a principle making offspring

nearly resemble their parents, while yet giving room for “sporting,” or

accidental variations,—for very slight variations often, for wider ones

rarely; and, second, the destruction of breeds or races that are unable

to keep the birth rate up to the death rate. This Darwinian principle is

plainly capable of great generalization. Wherever there are large

numbers of objects, having a tendency to retain certain characters

unaltered, this tendency, however, not being absolute but giving room

for chance variations, then, if the amount of variation is absolutely

limited in certain directions by the destruction of everything which

reaches those limits, there will be a gradual tendency to change in

directions of departure from them. Thus, if a million players sit down

to bet at an even game, since one after another will get ruined, the

average wealth of those who remain will perpetually increase. Here is

indubitably a genuine formula of possible evolution, whether its

operation accounts for much or little in the development of animal and

vegetable species.

The Lamarckian theory also supposes that the development of species has

taken place by a long series of insensible changes, but it supposes that

those changes have taken place during the lives of the individuals, in

consequence of effort and exercise, and that reproduction plays no part

in the process except in preserving these modifications. Thus, the

Lamarckian theory only explains the development of characters for which

individuals strive, while the Darwinian theory only explains the

production of characters really beneficial to the race, though these may

be fatal to individuals.[55] But more broadly and philosophically

conceived, Darwinian evolution is evolution by the operation of chance,

and the destruction of bad results, while Lamarckian evolution is

evolution by the effect of habit and effort.

A third theory of evolution is that of Mr. Clarence King. The testimony

of monuments and of rocks is that species are unmodified or scarcely

modified, under ordinary circumstances, but are rapidly altered after

cataclysms or rapid geological changes. Under novel circumstances, we

often see animals and plants sporting excessively in reproduction, and

sometimes even undergoing transformations during individual life,

phenomena no doubt due partly to the enfeeblement of vitality from the

breaking up of habitual modes of life, partly to changed food, partly to

direct specific influence of the element in which the organism is

immersed. If evolution has been brought about in this way, not only have

its single steps not been insensible, as both Darwinians and Lamarckians

suppose, but they are furthermore neither haphazard on the one hand, nor

yet determined by an inward striving on the other, but on the contrary

are effects of the changed environment, and have a positive general

tendency to adapt the organism to that environment, since variation will

particularly affect organs at once enfeebled and stimulated. This mode

of evolution, by external forces and the breaking up of habits, seems to

be called for by some of the broadest and most important facts of

biology and paleontology; while it certainly has been the chief factor

in the historical evolution of institutions as in that of ideas; and

cannot possibly be refused a very prominent place in the process of

evolution of the universe in general.

Passing to psychology, we find the elementary phenomena of mind fall

into three categories. First, we have Feelings, comprising all that is

immediately present, such as pain, blue, cheerfulness, the feeling that

arises when we contemplate a consistent theory, etc. A feeling is a

state of mind having its own living quality, independent of any other

state of mind. Or, a feeling is an element of consciousness which might

conceivably override every other state until it monopolized the mind,

although such a rudimentary state cannot actually be realized, and would

not properly be consciousness. Still, it is conceivable, or supposable,

that the quality of blue should usurp the whole mind, to the exclusion

of the ideas of shape, extension, contrast, commencement and cessation,

and all other ideas, whatsoever. A feeling is necessarily perfectly

simple, _in itself_, for if it had parts these would also be in the

mind, whenever the whole was present, and thus the whole could not

monopolize the mind.[56]

Besides Feelings, we have Sensations of reaction; as when a person

blindfold suddenly runs against a post, when we make a muscular effort,

or when any feeling gives way to a new feeling. Suppose I had nothing in

my mind but a feeling of blue, which were suddenly to give place to a

feeling of red; then, at the instant of transition there would be a

shock, a sense of reaction, my blue life being transmuted into red life.

If I were further endowed with a memory, that sense would continue for

some time, and there would also be a peculiar feeling or sentiment

connected with it. This last feeling might endure (conceivably I mean)

after the memory of the occurrence and the feelings of blue and red had

passed away. But the _sensation_ of reaction cannot exist except in the

actual presence of the two feelings blue and red to which it relates.

Wherever we have two feelings and pay attention to a relation between

them of whatever kind, there is the sensation of which I am speaking.

But the sense of action and reaction has two types: it may either be a

perception of relation between two ideas, or it may be a sense of action

and reaction between feeling and something out of feeling. And this

sense of external reaction again has two forms; for it is either a sense

of something happening to us, by no act of ours, we being passive in the

matter, or it is a sense of resistance, that is, of our expending

feeling upon something without. The sense of reaction is thus a sense of

connection or comparison between feelings, either, _A_, between one

feeling and another, or _B_, between feeling and its absence or lower

degree; and under _B_ we have, First, the sense of the access of

feeling, and Second, the sense of remission of feeling.

Very different both from feelings and from reaction-sensations or

disturbances of feeling are general conceptions. When we think, we are

conscious that a connection between feelings is determined by a general

rule, we are aware of being governed by a habit. Intellectual power is

nothing but facility in taking habits and in following them in cases

essentially analogous to, but in non-essentials widely remote from, the

normal cases of connections of feelings under which those habits were

formed.

The one primary and fundamental law of mental action consists in a

tendency to generalization. Feeling tends to spread; connections between

feelings awaken feelings; neighboring feelings become assimilated; ideas

are apt to reproduce themselves. These are so many formulations of the

one law of the growth of mind. When a disturbance of feeling takes

place, we have a consciousness of gain, the gain of experience; and a

new disturbance will be apt to assimilate itself to the one that

preceded it. Feelings, by being excited, become more easily excited,

especially in the ways in which they have previously been excited. The

consciousness of such a habit constitutes a general conception.

The cloudiness of psychological notions may be corrected by connecting

them with physiological conceptions. Feeling may be supposed to exist,

wherever a nerve-cell is in an excited condition. The disturbance of

feeling, or sense of reaction, accompanies the transmission of

disturbance between nerve-cells or from a nerve-cell to a muscle-cell or

the external stimulation of a nerve-cell. General conceptions arise upon

the formation of habits in the nerve-matter, which are molecular changes

consequent upon its activity and probably connected with its nutrition.

The law of habit exhibits a striking contrast to all physical laws in

the character of its commands. A physical law is absolute. What it

requires is an exact relation. Thus, a physical force introduces into a

motion a component motion to be combined with the rest by the

parallelogram of forces; but the component motion must actually take

place exactly as required by the law of force. On the other hand, no

exact conformity is required by the mental law. Nay, exact conformity

would be in downright conflict with the law; since it would instantly

crystallize thought and prevent all further formation of habit. The law

of mind only makes a given feeling _more likely_ to arise. It thus

resembles the “non-conservative” forces of physics, such as viscosity

and the like, which are due to statistical uniformities in the chance

encounters of trillions of molecules.

The old dualistic notion of mind and matter, so prominent in

Cartesianism, as two radically different kinds of substance, will hardly

find defenders to-day. Rejecting this, we are driven to some form of

hylopathy, otherwise called monism. Then the question arises whether

physical laws on the one hand, and the psychical law on the other are to

be taken—

(_A_) as independent, a doctrine often called _monism_, but which I

would name _neutralism_; or,

(_B_) the psychical law as derived and special, the physical law alone

as primordial, which is _materialism_; or,

(_C_) the physical law as derived and special, the psychical law alone

as primordial, which is _idealism_.

The materialistic doctrine seems to me quite as repugnant to scientific

logic as to common sense; since it requires us to suppose that a certain

kind of mechanism will feel, which would be a hypothesis absolutely

irreducible to reason,—an ultimate, inexplicable regularity; while the

only possible justification of any theory is that it should make things

clear and reasonable.

Neutralism is sufficiently condemned by the logical maxim known as

Ockham’s razor, i.e., that not more independent elements are to be

supposed than necessary. By placing the inward and outward aspects of

substance on a par, it seems to render both primordial.

The one intelligible theory of the universe is that of objective

idealism, that matter is effete mind, inveterate habits becoming

physical laws. But before this can be accepted it must show itself

capable of explaining the tridimensionality of space, the laws of

motion, and the general characteristics of the universe, with

mathematical clearness and precision; for no less should be demanded of

every Philosophy.

[Illustration: Figure 6.]

Modern mathematics is replete with ideas which may be applied to

philosophy. I can only notice one or two. The manner in which

mathematicians generalize is very instructive. Thus, painters are

accustomed to think of a picture as consisting geometrically of the

intersections of its plane by rays of light from the natural objects to

the eye. But geometers use a generalized perspective.[57] For instance

in the figure let _O_ be the eye, let _A_ _B_ _C_ _D_ _E_ be the

edgewise view of any plane, and let _a_ _f_ _e_ _D_ _c_ be the edgewise

view of another plane. The geometers draw rays through _O_ cutting both

these planes, and treat the points of intersection of each ray with one

plane as representing the point of intersection of the same ray with the

other plane. Thus, _e_ represents _E_, in the painter’s way. _D_

represents itself. _C_ is represented by _c_, which is further from the

eye; and _A_ is represented by _a_ which is on the other side of the

eye. Such generalization is not bound down to sensuous images. Further,

according to this mode of representation every point on one plane

represents a point on the other, and every point on the latter is

represented by a point on the former. But how about the point _f_ which

is in a direction from _O_ parallel to the represented plane, and how

about the point _B_ which is in a direction parallel to the representing

plane? Some will say that these are exceptions; but modern mathematics

does not allow exceptions which can be annulled by generalization.[58]

As a point moves from _C_ to _D_ and thence to _E_ and off toward

infinity, the corresponding point on the other plane moves from _c_ to

_D_ and thence to _e_ and toward _f_. But this second point can pass

through _f_ to _a_; and when it is there the first point has arrived at

_A_. We therefore say that the first point has passed _through

infinity_, and that every line joins in to itself somewhat like an oval.

Geometers talk of the parts of lines at an infinite distance as points.

This is a kind of generalization very efficient in mathematics.

Modern views of measurement have a philosophical aspect. There is an

indefinite number of systems of measuring along a line; thus, a

perspective representation of a scale on one line may be taken to

measure another, although of course such measurements will not agree

with what we call the distances of points on the latter line. To

establish a system of measurement on a line we must assign a distinct

number to each point of it, and for this purpose we shall plainly have

to suppose the numbers carried out into an infinite number of places of

decimals. These numbers must be ranged along the line in unbroken

sequence. Further, in order that such a scale of numbers should be of

any use, it must be capable of being shifted into new positions, each

number continuing to be attached to a single distinct point. Now it is

found that if this is true for “imaginary” as well as for real points

(an expression which I cannot stop to elucidate), any such shifting will

necessarily leave two numbers attached to the same points as before. So

that when the scale is moved over the line by any continuous series of

shiftings of one kind, there are two points which no numbers on the

scale can ever reach, except the numbers fixed there. This pair of

points, thus unattainable in measurement, is called the Absolute. These

two points may be distinct and real, or they may coincide, or they may

be both imaginary. As an example of a linear quantity with a double

absolute we may take probability, which ranges from an unattainable

absolute certainty _against_ a proposition to an equally unattainable

absolute certainty _for_ it. A line, according to ordinary notions, we

have seen is a linear quantity where the two points at infinity

coincide. A velocity is another example. A train going with infinite

velocity from Chicago to New York would be at all the points on the line

at the very same instant, and if the time of transit were reduced to

less than nothing it would be moving in the other direction. An angle is

a familiar example of a mode of magnitude with no real immeasurable

values. One of the questions philosophy has to consider is whether the

development of the universe is like the increase of an angle, so that it

proceeds forever without tending toward anything unattained, which I

take to be the Epicurean view, or whether the universe sprang from a

chaos in the infinitely distant past to tend toward something different

in the infinitely distant future, or whether the universe sprang from

nothing in the past to go on indefinitely toward a point in the

infinitely distant future, which, were it attained, would be the mere

nothing from which it set out.

The doctrine of the absolute applied to space comes to this, that

either—

First, space is, as Euclid teaches, both _unlimited_ and _immeasurable_,

so that the infinitely distant parts of any plane seen in perspective

appear as a straight line, in which case the sum of the three angles of

a triangle amounts to 180°; or,

Second, space is _immeasurable_ but _limited_, so that the infinitely

distant parts of any plane seen in perspective appear as a circle,

beyond which all is blackness, and in this case the sum of the three

angles of a triangle is less than 180° by an amount proportional to the

area of the triangle; or,

Third, space is _unlimited_ but _finite_, (like the surface of a

sphere), so that it has no infinitely distant parts; but a finite

journey along any straight line would bring one back to his original

position, and looking off with an unobstructed view one would see the

back of his own head enormously magnified, in which case the sum of the

three angles of a triangle exceeds 180° by an amount proportional to the

area.

Which of these three hypotheses is true we know not. The largest

triangles we can measure are such as have the earth’s orbit for base,

and the distance of a fixed star for altitude. The angular magnitude

resulting from subtracting the sum of the two angles at the base of such

a triangle from 180° is called the star’s _parallax_. The parallaxes of

only about forty stars have been measured as yet. Two of them come out

negative, that of Arided (α Cycni), a star of magnitude 1-1/2, which is

—0.“082, according to C. A. F. Peters, and that of a star of magnitude

7-3/4, known as Piazzi III 422, which is —0.”045, according to R. S.

Ball. But these negative parallaxes are undoubtedly to be attributed to

errors of observation; for the probable error of such a determination is

about ± 0.“075, and it would be strange indeed if we were to be able to

see, as it were, more than half way round space, without being able to

see stars with larger negative parallaxes. Indeed, the very fact that of

all the parallaxes measured only two come out negative would be a strong

argument that the smallest parallaxes really amount to +0.”1, were it

not for the reflection that the publication of other negative parallaxes

may have been suppressed. I think we may feel confident that the

parallax of the furthest star lies somewhere between -0.”05 and +0.”15,

and within another century our grandchildren will surely know whether

the three angles of a triangle are greater or less than 180°,—that they

are _exactly_ that amount is what nobody ever can be justified in

concluding. It is true that according to the axioms of geometry the sum

of the three sides of a triangle are precisely 180°; but these axioms

are now exploded, and geometers confess that they, as geometers, know

not the slightest reason for supposing them to be precisely true. They

are expressions of our inborn conception of space, and as such are

entitled to credit, so far as their truth could have influenced the

formation of the mind. But that affords not the slightest reason for

supposing them exact.

Now, metaphysics has always been the ape of mathematics. Geometry

suggested the idea of a demonstrative system of absolutely certain

philosophical principles; and the ideas of the metaphysicians have at

all times been in large part drawn from mathematics. The metaphysical

axioms are imitations of the geometrical axioms; and now that the latter

have been thrown overboard, without doubt the former will be sent after

them. It is evident, for instance, that we can have no reason to think

that every phenomenon in all its minutest details is precisely

determined by law. That there is an arbitrary element in the universe we

see,—namely, its variety. This variety must be attributed to spontaneity

in some form.

Had I more space, I now ought to show how important for philosophy is

the mathematical conception of continuity. Most of what is true in Hegel

is a darkling glimmer of a conception which the mathematicians had long

before made pretty clear, and which recent researches have still further

illustrated.

Among the many principles of Logic which find their application in

Philosophy, I can here only mention one. Three conceptions are

perpetually turning up at every point in every theory of logic, and in

the most rounded systems they occur in connection with one another. They

are conceptions so very broad and consequently indefinite that they are

hard to seize and may be easily overlooked. I call them the conceptions

of First, Second, Third. First is the conception of being or existing

independent of anything else. Second is the conception of being relative

to, the conception of reaction with, something else. Third is the

conception of mediation, whereby a first and second are brought into

relation. To illustrate these ideas, I will show how they enter into

those we have been considering. The origin of things, considered not as

leading to anything, but in itself, contains the idea of First, the end

of things that of Second, the process mediating between them that of

Third. A philosophy which emphasizes the idea of the One, is generally a

dualistic philosophy in which the conception of Second receives

exaggerated attention; for this One (though of course involving the idea

of First) is always the other of a manifold which is not one. The idea

of the Many, because variety is arbitrariness and arbitrariness is

repudiation of any Secondness, has for its principal component the

conception of First. In psychology Feeling is First, Sense of reaction

Second, General conception Third, or mediation. In biology, the idea of

arbitrary sporting is First, heredity is Second, the process whereby the

accidental characters become fixed is Third. Chance is First, Law is

Second, the tendency to take habits is Third. Mind is First, Matter is

Second, Evolution is Third.

Such are the materials out of which chiefly a philosophical theory ought

to be built, in order to represent the state of knowledge to which the

nineteenth century has brought us. Without going into other important

questions of philosophical architectonic, we can readily foresee what

sort of a metaphysics would appropriately be constructed from those

conceptions. Like some of the most ancient and some of the most recent

speculations it would be a Cosmogonic Philosophy. It would suppose that

in the beginning,—infinitely remote,—there was a chaos of unpersonalized

feeling, which being without connection or regularity would properly be

without existence. This feeling, sporting here and there in pure

arbitrariness, would have started the germ of a generalizing tendency.

Its other sportings would be evanescent, but this would have a growing

virtue. Thus, the tendency to habit would be started; and from this with

the other principles of evolution all the regularities of the universe

would be evolved. At any time, however, an element of pure chance

survives and will remain until the world becomes an absolutely perfect,

rational, and symmetrical system, in which mind is at last crystallized

in the infinitely distant future.

That idea has been worked out by me with elaboration. It accounts for

the main features of the universe as we know it,—the characters of time,

space, matter, force, gravitation, electricity, etc. It predicts many

more things which new observations can alone bring to the test. May some

future student go over this ground again, and have the leisure to give

his results to the world.

Footnote 54:

_The Monist_, January, 1891.

Footnote 55:

The neo-Darwinian, Weismann, has shown that mortality would almost

necessarily result from the action of the Darwinian principle.

Footnote 56:

A feeling may certainly be compound, but only in virtue of a

perception which is not that feeling nor any feeling at all.

Footnote 57:

[The reader will find further light on the following illustration in

any text-book of projective geometry, e.g., Reye, _Geometry of

Position_, I, pp. 17-24, or _Encyc. Britannica_, XI, p. 689.]

Footnote 58:

[A more familiar example of this is the introduction of irrational or

absurd numbers like √2. After it was proved that no ratio of two

integers could possibly equal √2 the idea of number was generalized to

include the latter. Fractions and the so-called imaginary numbers

illustrate the same process of generalization for the sake of making

certain operations (i.e. division and finding the root) continuously

applicable.]

II. THE DOCTRINE OF NECESSITY EXAMINED[59]

In _The Monist_ for January, 1891, I endeavored to show what elementary

ideas ought to enter into our view of the universe. I may mention that

on those considerations I had already grounded a cosmical theory, and

from it had deduced a considerable number of consequences capable of

being compared with experience. This comparison is now in progress, but

under existing circumstances must occupy many years.

I propose here to examine the common belief that every single fact in

the universe is precisely determined by law. It must not be supposed

that this is a doctrine accepted everywhere and at all times by all

rational men. Its first advocate appears to have been Democritus, the

atomist, who was led to it, as we are informed, by reflecting upon the

“impenetrability, translation, and impact of matter (ἀντιτυπία καὶ φορὰ

καὶ πληγὴ τῆς ὕλης).” That is to say, having restricted his attention to

a field where no influence other than mechanical constraint could

possibly come before his notice, he straightway jumped to the conclusion

that throughout the universe that was the sole principle of action,—a

style of reasoning so usual in our day with men not unreflecting as to

be more than excusable in the infancy of thought. But Epicurus, in

revising the atomic doctrine and repairing its defences, found himself

obliged to suppose that atoms swerve from their courses by spontaneous

chance; and thereby he conferred upon the theory life and entelechy. For

we now see clearly that the peculiar function of the molecular

hypothesis in physics is to open an entry for the calculus of

probabilities. Already, the prince of philosophers had repeatedly and

emphatically condemned the dictum of Democritus (especially in the

“Physics,” Book II, chapters iv, v, vi), holding that events come to

pass in three ways, namely, (1) by external compulsion, or the action of

efficient causes, (2) by virtue of an inward nature, or the influence of

final causes, and (3) irregularly without definite cause, but just by

absolute chance; and this doctrine is of the inmost essence of

Aristotelianism. It affords, at any rate, a valuable enumeration of the

possible ways in which anything can be supposed to have come about. The

freedom of the will, too, was admitted both by Aristotle and by

Epicurus. But the Stoa, which in every department seized upon the most

tangible, hard, and lifeless element, and blindly denied the existence

of every other, which, for example, impugned the validity of the

inductive method and wished to fill its place with the _reductio ad

absurdum_, very naturally became the one school of ancient philosophy to

stand by a strict necessitarianism, thus returning to a single principle

of Democritus that Epicurus had been unable to swallow. Necessitarianism

and materialism with the Stoics went hand in hand, as by affinity they

should. At the revival of learning, Stoicism met with considerable

favor, partly because it departed just enough from Aristotle to give it

the spice of novelty, and partly because its superficialities well

adapted it for acceptance by students of literature and art who wanted

their philosophy drawn mild. Afterwards, the great discoveries in

mechanics inspired the hope that mechanical principles might suffice to

explain the universe; and though without logical justification, this

hope has since been continually stimulated by subsequent advances in

physics. Nevertheless, the doctrine was in too evident conflict with the

freedom of the will and with miracles to be generally acceptable, at

first. But meantime there arose that most widely spread of philosophical

blunders, the notion that associationalism belongs intrinsically to the

materialistic family of doctrines; and thus was evolved the theory of

motives; and libertarianism became weakened. At present, historical

criticism has almost exploded the miracles, great and small; so that the

doctrine of necessity has never been in so great vogue as now.

The proposition in question is that the state of things existing at any

time, together with certain immutable laws, completely determine the

state of things at every other time (for a limitation to _future_ time

is indefensible). Thus, given the state of the universe in the original

nebula, and given the laws of mechanics, a sufficiently powerful mind

could deduce from these data the precise form of every curlicue of every

letter I am now writing.

Whoever holds that every act of the will as well as every idea of the

mind is under the rigid governance of a necessity co-ordinated with that

of the physical world, will logically be carried to the proposition that

minds are part of the physical world in such a sense that the laws of

mechanics determine everything that happens according to immutable

attractions and repulsions. In that case, that instantaneous state of

things from which every other state of things is calculable consists in

the positions and velocities of all the particles at any instant. This,

the usual and most logical form of necessitarianism, is called the

mechanical philosophy.

When I have asked thinking men what reason they had to believe that

every fact in the universe is precisely determined by law, the first

answer has usually been that the proposition is a “presupposition” or

postulate of scientific reasoning. Well, if that is the best that can be

said for it, the belief is doomed. Suppose it be “postulated”: that does

not make it true, nor so much as afford the slightest rational motive

for yielding it any credence. It is as if a man should come to borrow

money, and when asked for his security, should reply he “postulated” the

loan. To “postulate” a proposition is no more than to hope it is true.

There are, indeed, practical emergencies in which we act upon

assumptions of certain propositions as true, because if they are not so,

it can make no difference how we act. But all such propositions I take

to be hypotheses of individual facts. For it is manifest that no

universal principle can in its universality be comprised in a special

case or can be requisite for the validity of any ordinary inference. To

say, for instance, that the demonstration by Archimedes of the property

of the lever would fall to the ground if men were endowed with

free-will, is extravagant; yet this is implied by those who make a

proposition incompatible with the freedom of the will the postulate of

all inference. Considering, too, that the conclusions of science make no

pretence to being more than probable, and considering that a probable

inference can at most only suppose something to be most frequently, or

otherwise approximately, true, but never that anything is precisely true

without exception throughout the universe, we see how far this

proposition in truth is from being so postulated.

But the whole notion of a postulate being involved in reasoning

appertains to a by-gone and false conception of logic. Non-deductive, or

ampliative inference, is of three kinds: induction, hypothesis, and

analogy. If there be any other modes, they must be extremely unusual and

highly complicated, and may be assumed with little doubt to be of the

same nature as those enumerated. For induction, hypothesis, and analogy,

as far as their ampliative character goes, that is, so far as they

conclude something not implied in the premises, depend upon one

principle and involve the same procedure. All are essentially inferences

from sampling. Suppose a ship arrives at Liverpool laden with wheat in

bulk. Suppose that by some machinery the whole cargo be stirred up with

great thoroughness. Suppose that twenty-seven thimblefuls be taken

equally from the forward, midships, and aft parts, from the starboard,

center, and larboard parts, and from the top, half depth, and lower

parts of her hold, and that these being mixed and the grains counted,

four-fifths of the latter are found to be of quality _A_. Then we infer,

experientially and provisionally, that approximately four-fifths of all

the grain in the cargo is of the same quality. I say we infer this

_experientially_ and _provisionally_. By saying that we infer it

_experientially_, I mean that our conclusion makes no pretension to

knowledge of wheat-in-itself, our ἀλήθεια, as the derivation of that

word implies, has nothing to do with _latent_ wheat. We are dealing only

with the matter of possible experience,—experience in the full

acceptation of the term as something not merely affecting the senses but

also as the subject of thought. If there be any wheat hidden on the

ship, so that it can neither turn up in the sample nor be heard of

subsequently from purchasers,—or if it be half-hidden, so that it may,

indeed, turn up, but is less likely to do so than the rest,—or if it can

affect our senses and our pockets, but from some strange cause or

causelessness cannot be reasoned about,—all such wheat is to be excluded

(or have only its proportional weight) in calculating that true

proportion of quality _A_, to which our inference seeks to approximate.

By saying that we draw the inference _provisionally_, I mean that we do

not hold that we have reached any assigned degree of approximation as

yet, but only hold that if our experience be indefinitely extended, and

if every fact of whatever nature, as fast as it presents itself, be duly

applied, according to the inductive method, in correcting the inferred

ratio, then our approximation will become indefinitely close in the long

run; that is to say, close to the experience _to come_ (not merely close

by the exhaustion of a finite collection) so that if experience in

general is to fluctuate irregularly to and fro, in a manner to deprive

the ratio sought of all definite value, we shall be able to find out

approximately within what limits it fluctuates, and if, after having one

definite value, it changes and assumes another, we shall be able to find

that out, and in short, whatever may be the variations of this ratio in

experience, experience indefinitely extended will enable us to detect

them, so as to predict rightly, at last, what its ultimate value may be,

if it have any ultimate value, or what the ultimate law of succession of

values may be, if there be any such ultimate law, or that it ultimately

fluctuates irregularly within certain limits, if it do so ultimately

fluctuate. Now our inference, claiming to be no more than thus

experiential and provisional, manifestly involves no postulate whatever.

For what is a postulate? It is the formulation of a material fact which

we are not entitled to assume as a premise, but the truth of which is

requisite to the validity of an inference. Any fact, then, which might

be supposed postulated, must either be such that it would ultimately

present itself in experience, or not. If it will present itself, we need

not postulate it now in our provisional inference, since we shall

ultimately be entitled to use it as a premise. But if it never would

present itself in experience, our conclusion is valid but for the

possibility of this fact being otherwise than assumed, that is, it is

valid as far as possible experience goes, and that is all that we claim.

Thus, every postulate is cut off, either by the provisionality or by the

experientiality of our inference. For instance, it has been said that

induction postulates that, if an indefinite succession of samples be

drawn, examined, and thrown back each before the next is drawn, then in

the long run every grain will be drawn as often as any other, that is to

say, postulates that the ratio of the numbers of times in which any two

are drawn will indefinitely approximate to unity. But no such postulate

is made; for if, on the one hand, we are to have no other experience of

the wheat than from such drawings, it is the ratio that presents itself

in those drawings and not the ratio which belongs to the wheat in its

latent existence that we are endeavoring to determine; while if, on the

other hand, there is some other mode by which the wheat is to come under

our knowledge, equivalent to another kind of sampling, so that after all

our care in stirring up the wheat, some experiential grains will present

themselves in the first sampling operation more often than others in the

long run, this very singular fact will be sure to get discovered by the

inductive method, which must avail itself of every sort of experience;

and our inference, which was only provisional, corrects itself at last.

Again, it has been said, that induction postulates that under like

circumstances like events will happen, and that this postulate is at

bottom the same as the principle of universal causation. But this is a

blunder, or _bevue_, due to thinking exclusively of inductions where the

concluded ratio is either 1 or 0. If any such proposition were

postulated, it would be that under like circumstances (the circumstances

of drawing the different samples) different events occur in the same

proportions in all the different sets,—a proposition which is false and

even absurd. But in truth no such thing is postulated, the experiential

character of the inference reducing the condition of validity to this,

that if a certain result does not occur, the opposite result will be

manifested, a condition assured by the provisionality of the inference.

But it may be asked whether it is not conceivable that every instance of

a certain class destined to be ever employed as a datum of induction

should have one character, while every instance destined not to be so

employed should have the opposite character. The answer is that in that

case, the instances excluded from being subjects of reasoning would not

be experienced in the full sense of the word, but would be among these

_latent_ individuals of which our conclusion does not pretend to speak.

To this account of the rationale of induction I know of but one

objection worth mention: it is that I thus fail to deduce the full

degree of force which this mode of inference in fact possesses; that

according to my view, no matter how thorough and elaborate the stirring

and mixing process had been, the examination of a single handful of

grain would not give me any assurance, sufficient to risk money upon

that the next handful would not greatly modify the concluded value of

the ratio under inquiry, while, in fact, the assurance would be very

high that this ratio was not greatly in error. If the true ratio of

grains of quality _A_ were 0.80 and the handful contained a thousand

grains, nine such handfuls out of every ten would contain from 780 to

820 grains of quality _A_. The answer to this is that the calculation

given is correct when we know that the units of this handful and the

quality inquired into have the normal independence of one another, if

for instance the stirring has been complete and the character sampled

for has been settled upon in advance of the examination of the sample.

But in so far as these conditions are not known to be complied with, the

above figures cease to be applicable. Random sampling and predesignation

of the character sampled for should always be striven after in inductive

reasoning, but when they cannot be attained, so long as it is conducted

honestly, the inference retains some value. When we cannot ascertain how

the sampling has been done or the sample-character selected, induction

still has the essential validity which my present account of it shows it

to have.

I do not think a man who combines a willingness to be convinced with a

power of appreciating an argument upon a difficult subject can resist

the reasons which have been given to show that the principle of

universal necessity cannot be defended as being a postulate of

reasoning. But then the question immediately arises whether it is not

proved to be true, or at least rendered highly probable, by observation

of nature.

Still, this question ought not long to arrest a person accustomed to

reflect upon the force of scientific reasoning. For the essence of the

necessitarian position is that certain continuous quantities have

certain exact values. Now, how can observation determine the value of

such a quantity with a probable error absolutely _nil_? To one who is

behind the scenes, and knows that the most refined comparisons of

masses, lengths, and angles, far surpassing in precision all other

measurements, yet fall behind the accuracy of bank-accounts, and that

the ordinary determinations of physical constants, such as appear from

month to month in the journals, are about on a par with an upholsterer’s

measurements of carpets and curtains, the idea of mathematical

exactitude being demonstrated in the laboratory will appear simply

ridiculous. There is a recognized method of estimating the probable

magnitudes of errors in physics,—the method of least squares. It is

universally admitted that this method makes the errors smaller than they

really are; yet even according to that theory an error indefinitely

small is indefinitely improbable; so that any statement to the effect

that a certain continuous quantity has a certain exact value, if

well-founded at all, must be founded on something other than

observation.

Still, I am obliged to admit that this rule is subject to a certain

qualification. Namely, it only applies to continuous[60] quantity. Now,

certain kinds of continuous quantity are discontinuous at one or at two

limits, and for such limits the rule must be modified. Thus, the length

of a line cannot be less than zero. Suppose, then, the question arises

how long a line a certain person had drawn from a marked point on a

piece of paper. If no line at all can be seen, the observed length is

zero; and the only conclusion this observation warrants is that the

length of the line is less than the smallest length visible with the

optical power employed. But indirect observations,—for example, that the

person supposed to have drawn the line was never within fifty feet of

the paper,—may make it probable that no line at all was made, so that

the concluded length will be strictly zero. In like manner, experience

no doubt would warrant the conclusion that there is absolutely _no_

indigo in a given ear of wheat, and absolutely _no_ attar in a given

lichen. But such inferences can only be rendered valid by positive

experiential evidence, direct or remote, and cannot rest upon a mere

inability to detect the quantity in question. We have reason to think

there is no indigo in the wheat, because we have remarked that wherever

indigo is produced it is produced in considerable quantities, to mention

only one argument. We have reason to think there is no attar in the

lichen, because essential oils seem to be in general peculiar to single

species. If the question had been whether there was iron in the wheat or

the lichen, though chemical analysis should fail to detect its presence,

we should think some of it probably was there, since iron is almost

everywhere. Without any such information, one way or the other, we could

only abstain from any opinion as to the presence of the substance in

question. It cannot, I conceive, be maintained that we are in any

_better_ position than this in regard to the presence of the element of

chance or spontaneous departures from law in nature.

Those observations which are generally adduced in favor of mechanical

causation simply prove that there is an element of regularity in nature,

and have no bearing whatever upon the question of whether such

regularity is exact and universal, or not. Nay, in regard to this

_exactitude_, all observation is directly _opposed_ to it; and the most

that can be said is that a good deal of this observation can be

explained away. Try to verify any law of nature, and you will find that

the more precise your observations, the more certain they will be to

show irregular departures from the law. We are accustomed to ascribe

these, and I do not say wrongly, to errors of observation; yet we cannot

usually account for such errors in any antecedently probable way. Trace

their causes back far enough, and you will be forced to admit they are

always due to arbitrary determination, or chance.

But it may be asked whether if there were an element of real chance in

the universe it must not occasionally be productive of signal effects

such as could not pass unobserved. In answer to this question, without

stopping to point out that there is an abundance of great events which

one might be tempted to suppose were of that nature, it will be simplest

to remark that physicists hold that the particles of gases are moving

about irregularly, substantially as if by real chance, and that by the

principles of probabilities there must occasionally happen to be

concentrations of heat in the gases contrary to the second law of

thermodynamics, and these concentrations, occurring in explosive

mixtures, must sometimes have tremendous effects. Here, then, is in

substance the very situation supposed; yet no phenomena ever have

resulted which we are forced to attribute to such chance concentration

of heat, or which anybody, wise or foolish, has ever dreamed of

accounting for in that manner.

In view of all these considerations, I do not believe that anybody, not

in a state of case-hardened ignorance respecting the logic of science,

can maintain that the precise and universal conformity of facts to law

is clearly proved, or even rendered particularly probable, by any

observations hitherto made. In this way, the determined advocate of

exact regularity will soon find himself driven to _a priori_ reasons to

support his thesis. These received such a socdolager from Stuart Mill in

his Examination of Hamilton, that holding to them now seems to me to

denote a high degree of imperviousness to reason; so that I shall pass

them by with little notice.

To say that we cannot help believing a given proposition is no argument,

but it is a conclusive fact if it be true; and with the substitution of

“I” for “we,” it is true in the mouths of several classes of minds, the

blindly passionate, the unreflecting and ignorant, and the person who

has overwhelming evidence before his eyes. But that which has been

inconceivable to-day has often turned out indisputable on the morrow.

Inability to conceive is only a stage through which every man must pass

in regard to a number of beliefs,—unless endowed with extraordinary

obstinacy and obtuseness. His understanding is enslaved to some blind

compulsion which a vigorous mind is pretty sure soon to cast off.

Some seek to back up the _a priori_ position with empirical arguments.

They say that the exact regularity of the world is a natural belief, and

that natural beliefs have generally been confirmed by experience. There

is some reason in this. Natural beliefs, however, if they generally have

a foundation of truth, also require correction and purification from

natural illusions. The principles of mechanics are undoubtedly natural

beliefs; but, for all that, the early formulations of them were

exceedingly erroneous. The general approximation to truth in natural

beliefs is, in fact, a case of the general adaptation of genetic

products to recognizable utilities or ends. Now, the adaptations of

nature, beautiful and often marvelous as they verily are, are never

found to be quite perfect; so that the argument is quite _against_ the

absolute exactitude of any natural belief, including that of the

principle of causation.

Another argument, or convenient commonplace, is that absolute chance is

_inconceivable_. (This word has eight current significations. The

_Century Dictionary_ enumerates six.) Those who talk like this will

hardly be persuaded to say in what sense they mean that chance is

inconceivable. Should they do so, it would easily be shown either that

they have no sufficient reason for the statement or that the

inconceivability is of a kind which does not prove that chance is

non-existent.

Another _a priori_ argument is that chance is unintelligible; that is to

say, while it may perhaps be conceivable, it does not disclose to the

eye of reason the how or why of things; and since a hypothesis can only

be justified so far as it renders some phenomenon intelligible, we never

can have any right to suppose absolute chance to enter into the

production of anything in nature. This argument may be considered in

connection with two others. Namely, instead of going so far as to say

that the supposition of chance can _never_ properly be used to explain

any observed fact, it may be alleged merely that no facts are known

which such a supposition could in any way help in explaining. Or again,

the allegation being still further weakened, it may be said that since

departures from law are not unmistakably observed, chance is not a _vera

causa_, and ought not unnecessarily to be introduced into a hypothesis.

These are no mean arguments, and require us to examine the matter a

little more closely. Come, my superior opponent, let me learn from your

wisdom. It seems to me that every throw of sixes with a pair of dice is

a manifest instance of chance.

“While you would hold a throw of deuce-ace to be brought about by

necessity?” (The opponent’s supposed remarks are placed in quotation

marks.)

Clearly one throw is as much chance as another.

“Do you think throws of dice are of a different nature from other

events?”

I see that I must say that _all_ the diversity and specificalness of

events is attributable to chance.

“Would you, then, deny that there is any regularity in the world?”

That is clearly undeniable. I must acknowledge there is an approximate

regularity, and that every event is influenced by it. But the

diversification, specificalness, and irregularity of things I suppose is

chance. A throw of sixes appears to me a case in which this element is

particularly obtrusive.

“If you reflect more deeply, you will come to see that _chance_ is only

a name for a cause that is unknown to us.”

Do you mean that we have no idea whatever what kind of causes could

bring about a throw of sixes?

“On the contrary, each die moves under the influence of precise

mechanical laws.”

But it appears to me that it is not these _laws_ which made the die turn

up sixes; for these laws act just the same when other throws come up.

The chance lies in the diversity of throws; and this diversity cannot be

due to laws which are immutable.

“The diversity is due to the diverse circumstances under which the laws

act. The dice lie differently in the box, and the motion given to the

box is different. These are the unknown causes which produce the throws,

and to which we give the name of chance; not the mechanical law which

regulates the operation of these causes. You see you are already

beginning to think more clearly about this subject.”

Does the operation of mechanical law not increase the diversity?

“Properly not. You must know that the instantaneous state of a system of

particles is defined by six times as many numbers as there are

particles, three for the co-ordinates of each particle’s position, and

three more for the components of its velocity. This number of numbers,

which expresses the amount of diversity in the system, remains the same

at all times. There may be, to be sure, some kind of relation between

the co-ordinates and component velocities of the different particles, by

means of which the state of the system might be expressed by a smaller

number of numbers. But, if this is the case, a precisely corresponding

relationship must exist between the co-ordinates and component

velocities at any other time, though it may doubtless be a relation less

obvious to us. Thus, the intrinsic complexity of the system is the same

at all times.”

Very well, my obliging opponent, we have now reached an issue. You think

all the arbitrary specifications of the universe were introduced in one

dose, in the beginning, if there was a beginning, and that the variety

and complication of nature has always been just as much as it is now.

But I, for my part, think that the diversification, the specification,

has been continually taking place. Should you condescend to ask me why I

so think, I should give my reasons as follows:

(1) Question any science which deals with the course of time. Consider

the life of an individual animal or plant, or of a mind. Glance at the

history of states, of institutions, of language, of ideas. Examine the

successions of forms shown by paleontology, the history of the globe as

set forth in geology, of what the astronomer is able to make out

concerning the changes of stellar systems. Everywhere the main fact is

growth and increasing complexity. Death and corruption are mere

accidents or secondary phenomena. Among some of the lower organisms, it

is a moot point with biologists whether there be anything which ought to

be called death. Races, at any rate, do not die out except under

unfavorable circumstances. From these broad and ubiquitous facts we may

fairly infer, by the most unexceptionable logic, that there is probably

in nature some agency by which the complexity and diversity of things

can be increased; and that consequently the rule of mechanical necessity

meets in some way with interference.

(2) By thus admitting pure spontaneity or life as a character of the

universe, acting always and everywhere though restrained within narrow

bounds by law, producing infinitesimal departures from law continually,

and great ones with infinite infrequency, I account for all the variety

and diversity of the universe, in the only sense in which the really

_sui generis_ and new can be said to be accounted for. The ordinary view

has to admit the inexhaustible multitudinous variety of the world, has

to admit that its mechanical law cannot account for this in the least,

that variety can spring only from spontaneity, and yet denies without

any evidence or reason the existence of this spontaneity, or else shoves

it back to the beginning of time and supposes it dead ever since. The

superior logic of my view appears to me not easily controverted.

(3) When I ask the necessitarian how he would explain the diversity and

irregularity of the universe, he replies to me out of the treasury of

his wisdom that irregularity is something which from the nature of

things we must not seek to explain. Abashed at this, I seek to cover my

confusion by asking how he would explain the uniformity and regularity

of the universe, whereupon he tells me that the laws of nature are

immutable and ultimate facts, and no account is to be given of them. But

my hypothesis of spontaneity does explain irregularity, in a certain

sense; that is, it explains the general fact of irregularity, though

not, of course, what each lawless event is to be. At the same time, by

thus loosening the bond of necessity, it gives room for the influence of

another kind of causation, such as seems to be operative in the mind in

the formation of associations, and enables us to understand how the

uniformity of nature could have been brought about. That single events

should be hard and unintelligible, logic will permit without difficulty:

we do not expect to make the shock of a personally experienced

earthquake appear natural and reasonable by any amount of cogitation.

But logic does expect things _general_ to be understandable. To say that

there is a universal law, and that it is a hard, ultimate,

unintelligible fact, the why and wherefore of which can never be

inquired into, at this a sound logic will revolt; and will pass over at

once to a method of philosophizing which does not thus barricade the

road of discovery.

(4) Necessitarianism cannot logically stop short of making the whole

action of the mind a part of the physical universe. Our notion that we

decide what we are going to do, if as the necessitarian says, it has

been calculable since the earliest times, is reduced to illusion.

Indeed, consciousness in general thus becomes a mere illusory aspect of

a material system. What we call red, green, and violet are in reality

only different rates of vibration. The sole reality is the distribution

of qualities of matter in space and time. Brain-matter is protoplasm in

a certain degree and kind of complication,—a certain arrangement of

mechanical particles. Its feeling is but an inward aspect, a phantom.

For, from the positions and velocities of the particles at any one

instant, and the knowledge of the immutable forces, the positions at all

other times are calculable; so that the universe of space, time, and

matter is a rounded system uninterfered with from elsewhere. But from

the state of feeling at any instant, there is no reason to suppose the

states of feeling at all other instants are thus exactly calculable; so

that feeling is, as I said, a mere fragmentary and illusive aspect of

the universe. This is the way, then, that necessitarianism has to make

up its accounts. It enters consciousness under the head of sundries, as

a forgotten trifle; its scheme of the universe would be more

satisfactory if this little fact could be dropped out of sight. On the

other hand, by supposing the rigid exactitude of causation to yield, I

care not how little,—be it but by a strictly infinitesimal amount,—we

gain room to insert mind into our scheme, and to put it into the place

where it is needed, into the position which, as the sole

self-intelligible thing, it is entitled to occupy, that of the fountain

of existence; and in so doing we resolve the problem of the connection

of soul and body.

(5) But I must leave undeveloped the chief of my reasons, and can only

adumbrate it. The hypothesis of chance-spontaneity is one whose

inevitable consequences are capable of being traced out with

mathematical precision into considerable detail. Much of this I have

done and find the consequences to agree with observed facts to an extent

which seems to me remarkable. But the matter and methods of reasoning

are novel, and I have no right to promise that other mathematicians

shall find my deductions as satisfactory as I myself do, so that the

strongest reason for my belief must for the present remain a private

reason of my own, and cannot influence others. I mention it to explain

my own position; and partly to indicate to future mathematical

speculators a veritable goldmine, should time and circumstances and the

abridger of all joys prevent my opening it to the world.

If now I, in my turn, inquire of the necessitarian why he prefers to

suppose that all specification goes back to the beginning of things, he

will answer me with one of those last three arguments which I left

unanswered.

First, he may say that chance is a thing absolutely unintelligible, and,

therefore, that we never can be entitled to make such a supposition. But

does not this objection smack of naïve impudence? It is not mine, it is

his own conception of the universe which leads abruptly up to hard,

ultimate, inexplicable, immutable law, on the one hand, and to

inexplicable specification and diversification of circumstances on the

other. My view, on the contrary, hypothetises nothing at all, unless it

be hypothesis to say that all specification came about in some sense,

and is not to be accepted as unaccountable. To undertake to account for

anything by saying boldly that it is due to chance would, indeed, be

futile. But this I do not do. I make use of chance chiefly to make room

for a principle of generalization, or tendency to form habits, which I

hold has produced all regularities. The mechanical philosopher leaves

the whole specification of the world utterly unaccounted for, which is

pretty nearly as bad as to boldly attribute it to chance. I attribute it

altogether to chance, it is true, but to chance in the form of a

spontaneity which is to some degree regular. It seems to me clear at any

rate that one of these two positions must be taken, or else

specification must be supposed due to a spontaneity which develops

itself in a certain and not in a chance way, by an objective logic like

that of Hegel. This last way I leave as an open possibility, for the

present; for it is as much opposed to the necessitarian scheme of

existence as my own theory is.

Secondly, the necessitarian may say there are, at any rate, no observed

phenomena which the hypothesis of chance could aid in explaining. In

reply, I point first to the phenomenon of growth and developing

complexity, which appears to be universal, and which though it may

possibly be an affair of mechanism perhaps, certainly presents all the

appearance of increasing diversification. Then, there is variety itself,

beyond comparison the most obtrusive character of the universe: no

mechanism can account for this. Then, there is the very fact the

necessitarian most insists upon, the regularity of the universe which

for him serves only to block the road of inquiry. Then, there are the

regular relationships between the laws of nature,—similarities and

comparative characters, which appeal to our intelligence as its cousins,

and call upon us for a reason. Finally, there is consciousness, feeling,

a patent fact enough, but a very inconvenient one to the mechanical

philosopher.

Thirdly, the necessitarian may say that chance is not a _vera causa_,

that we cannot know positively there is any such element in the

universe. But the doctrine of the _vera causa_ has nothing to do with

elementary conceptions. Pushed to that extreme, it at once cuts off

belief in the existence of a material universe; and without that

necessitarianism could hardly maintain its ground. Besides, variety is a

fact which must be admitted; and the theory of chance merely consists in

supposing this diversification does not antedate all time. Moreover, the

avoidance of hypotheses involving causes nowhere positively known to

act—is only a recommendation of logic, not a positive command. It cannot

be formulated in any precise terms without at once betraying its

untenable character,—I mean as rigid rule, for as a recommendation it is

wholesome enough.

I believe I have thus subjected to fair examination all the important

reasons for adhering to the theory of universal necessity, and have

shown their nullity. I earnestly beg that whoever may detect any flaw in

my reasoning will point it out to me, either privately or publicly; for

if I am wrong, it much concerns me to be set right speedily. If my

argument remains unrefuted, it will be time, I think, to doubt the

absolute truth of the principle of universal law; and when once such a

doubt has obtained a living root in any man’s mind, my cause with him, I

am persuaded, is gained.

Footnote 59:

_The Monist_, April, 1892.

Footnote 60:

_Continuous_ is not exactly the right word, but I let it go to avoid a

long and irrelevant discussion.

III. THE LAW OF MIND[61]

In an article published in _The Monist_ for January, 1891, I endeavored

to show what ideas ought to form the warp of a system of philosophy, and

particularly emphasized that of absolute chance. In the number of April,

1892, I argued further in favor of that way of thinking, which it will

be convenient to christen _tychism_ (from τύχη, chance). A serious

student of philosophy will be in no haste to accept or reject this

doctrine; but he will see in it one of the chief attitudes which

speculative thought may take, feeling that it is not for an individual,

nor for an age, to pronounce upon a fundamental question of philosophy.

That is a task for a whole era to work out. I have begun by showing that

_tychism_ must give birth to an evolutionary cosmology, in which all the

regularities of nature and of mind are regarded as products of growth,

and to a Schelling-fashioned idealism which holds matter to be mere

specialized and partially deadened mind. I may mention, for the benefit

of those who are curious in studying mental biographies, that I was born

and reared in the neighborhood of Concord,—I mean in Cambridge,—at the

time when Emerson, Hedge, and their friends were disseminating the ideas

that they had caught from Schelling, and Schelling from Plotinus, from

Boehm, or from God knows what minds stricken with the monstrous

mysticism of the East. But the atmosphere of Cambridge held many an

antiseptic against Concord transcendentalism; and I am not conscious of

having contracted any of that virus. Nevertheless, it is probable that

some cultured bacilli, some benignant form of the disease was implanted

in my soul, unawares, and that now, after long incubation, it comes to

the surface, modified by mathematical conceptions and by training in

physical investigations.

The next step in the study of cosmology must be to examine the general

law of mental action. In doing this, I shall for the time drop my

tychism out of view, in order to allow a free and independent expansion

to another conception signalized in my first _Monist_ paper as one of

the most indispensable to philosophy, though it was not there dwelt

upon; I mean the idea of continuity. The tendency to regard continuity,

in the sense in which I shall define it, as an idea of prime importance

in philosophy may conveniently be termed _synechism_. The present paper

is intended chiefly to show what synechism is, and what it leads to. I

attempted, a good many years ago, to develop this doctrine in the

_Journal of Speculative Philosophy_ (Vol. II.); but I am able now to

improve upon that exposition, in which I was a little blinded by

nominalistic prepossessions. I refer to it, because students may

possibly find that some points not sufficiently explained in the present

paper are cleared up in those earlier ones.

WHAT THE LAW IS

Logical analysis applied to mental phenomena shows that there is but one

law of mind, namely, that ideas tend to spread continuously and to

affect certain others which stand to them in a peculiar relation of

affectibility. In this spreading they lose intensity, and especially the

power of affecting others, but gain generality and become welded with

other ideas.

I set down this formula at the beginning, for convenience; and now

proceed to comment upon it.

INDIVIDUALITY OF IDEAS

We are accustomed to speak of ideas as reproduced, as passed from mind

to mind, as similar or dissimilar to one another, and, in short, as if

they were substantial things; nor can any reasonable objection be raised

to such expressions. But taking the word “idea” in the sense of an event

in an individual consciousness, it is clear that an idea once past is

gone forever, and any supposed recurrence of it is another idea. These

two ideas are not present in the same state of consciousness, and

therefore cannot possibly be compared. To say, therefore, that they are

similar can only mean that an occult power from the depths of the soul

forces us to connect them in our thoughts after they are both no more.

We may note, here, in passing, that of the two generally recognized

principles of association, contiguity and similarity, the former is a

connection due to a power without, the latter a connection due to a

power within.

But what can it mean to say that ideas wholly past are thought of at

all, any longer? They are utterly unknowable. What distinct meaning can

attach to saying that an idea in the past in any way affects an idea in

the future, from which it is completely detached? A phrase between the

assertion and the denial of which there can in no case be any sensible

difference is mere gibberish.

I will not dwell further upon this point, because it is a commonplace of

philosophy.

CONTINUITY OF IDEAS

We have here before us a question of difficulty, analogous to the

question of nominalism and realism. But when once it has been clearly

formulated, logic leaves room for one answer only. How can a past idea

be present? Can it be present vicariously? To a certain extent, perhaps;

but not merely so; for then the question would arise how the past idea

can be related to its vicarious representation. The relation, being

between ideas, can only exist in some consciousness: now that past idea

was in no consciousness but that past consciousness that alone contained

it; and that did not embrace the vicarious idea.

Some minds will here jump to the conclusion that a past idea cannot in

any sense be present. But that is hasty and illogical. How extravagant,

too, to pronounce our whole knowledge of the past to be mere delusion!

Yet it would seem that the past is as completely beyond the bounds of

possible experience as a Kantian thing-in-itself.

How can a past idea be present? Not vicariously. Then, only by direct

perception. In other words, to be present, it must be _ipso facto_

present. That is, it cannot be wholly past; it can only be going,

infinitesimally past, less past than any assignable past date. We are

thus brought to the conclusion that the present is connected with the

past by a series of real infinitesimal steps.

It has already been suggested by psychologists that consciousness

necessarily embraces an interval of time. But if a finite time be meant,

the opinion is not tenable. If the sensation that precedes the present

by half a second were still immediately before me, then, on the same

principle the sensation preceding that would be immediately present, and

so on _ad infinitum_. Now, since there is a time, say a year, at the end

of which an idea is no longer _ipso facto_ present, it follows that this

is true of any finite interval, however short.

But yet consciousness must essentially cover an interval of time; for if

it did not, we could gain no knowledge of time, and not merely no

veracious cognition of it, but no conception whatever. We are,

therefore, forced to say that we are immediately conscious through an

infinitesimal interval of time.

This is all that is requisite. For, in this infinitesimal interval, not

only is consciousness continuous in a subjective sense, that is,

considered as a subject or substance having the attribute of duration;

but also, because it is immediate consciousness, its object is _ipso

facto_ continuous. In fact, this infinitesimally spread-out

consciousness is a direct feeling of its contents as spread out. This

will be further elucidated below. In an infinitesimal interval we

directly perceive the temporal sequence of its beginning, middle, and

end,—not, of course, in the way of recognition, for recognition is only

of the past, but in the way of immediate feeling. Now upon this interval

follows another, whose beginning is the middle of the former, and whose

middle is the end of the former. Here, we have an immediate perception

of the temporal sequence of its beginning, middle, and end, or say of

the second, third, and fourth instants. From these two immediate

perceptions, we gain a mediate, or inferential, perception of the

relation of all four instants. This mediate perception is objectively,

or as to the object represented, spread over the four instants; but

subjectively, or as itself the subject of duration, it is completely

embraced in the second moment. (The reader will observe that I use the

word _instant_ to mean a point of time, and _moment_ to mean an

infinitesimal duration.) If it is objected that, upon the theory

proposed, we must have more than a mediate perception of the succession

of the four instants, I grant it; for the sum of the two infinitesimal

intervals is itself infinitesimal, so that it is immediately perceived.

It is immediately perceived in the whole interval, but only mediately

perceived in the last two-thirds of the interval. Now, let there be an

indefinite succession of these inferential acts of comparative

perception; and it is plain that the last moment will contain

objectively the whole series. Let there be, not merely an indefinite

succession, but a continuous flow of inference through a finite time;

and the result will be a mediate objective consciousness of the whole

time in the last moment. In this last moment, the whole series will be

recognized, or known as known before, except only the last moment, which

of course will be absolutely unrecognizable to itself. Indeed, even this

last moment will be recognized like the rest, or, at least, be just

beginning to be so. There is a little _elenchus_, or appearance of

contradiction, here, which the ordinary logic of reflection quite

suffices to resolve.

INFINITY AND CONTINUITY, IN GENERAL

Most of the mathematicians who during the last two generations have

treated the differential calculus have been of the opinion that an

infinitesimal quantity is an absurdity; although, with their habitual

caution, they have often added “or, at any rate, the conception of an

infinitesimal is so difficult, that we practically cannot reason about

it with confidence and security.” Accordingly, the doctrine of limits

has been invented to evade the difficulty, or, as some say, to explain

the signification of the word “infinitesimal.” This doctrine, in one

form or another, is taught in all the text-books, though in some of them

only as an alternative view of the matter; it answers well enough the

purposes of calculation, though even in that application it has its

difficulties.

The illumination of the subject by a strict notation for the logic of

relatives had shown me clearly and evidently that the idea of an

infinitesimal involves no contradiction, before I became acquainted with

the writings of Dr. Georg Cantor (though many of these had already

appeared in the _Mathematische Annalen_ and in _Borchardt’s Journal_, if

not yet in the _Acta Mathematica_, all mathematical journals of the

first distinction), in which the same view is defended with

extraordinary genius and penetrating logic.

The prevalent opinion is that finite numbers are the only ones that we

can reason about, at least, in any ordinary mode of reasoning, or, as

some authors express it, they are the only numbers that can be reasoned

about mathematically. But this is an irrational prejudice. I long ago

showed that finite collections are distinguished from infinite ones only

by one circumstance and its consequences, namely, that to them is

applicable a peculiar and unusual mode of reasoning called by its

discoverer, De Morgan, the “syllogism of transposed quantity.”

Balzac, in the introduction of his _Physiologie du mariage_, remarks

that every young Frenchman boasts of having seduced some Frenchwoman.

Now, as a woman can only be seduced once, and there are no more

Frenchwomen than Frenchmen, it follows, if these boasts are true, that

no French women escape seduction. If their number be finite, the

reasoning holds. But since the population is continually increasing, and

the seduced are on the average younger than the seducers, the conclusion

need not be true. In like manner, De Morgan, as an actuary, might have

argued that if an insurance company pays to its insured on an average

more than they have ever paid it, including interest, it must lose

money. But every modern actuary would see a fallacy in that, since the

business is continually on the increase. But should war, or other

cataclysm, cause the class of insured to be a finite one, the conclusion

would turn out painfully correct, after all. The above two reasonings

are examples of the syllogism of transposed quantity.

The proposition that finite and infinite collections are distinguished

by the applicability to the former of the syllogism of transposed

quantity ought to be regarded as the basal one of scientific arithmetic.

If a person does not know how to reason logically, and I must say that a

great many fairly good mathematicians,—yea, distinguished ones,—fall

under this category, but simply uses a rule of thumb in blindly drawing

inferences like other inferences that have turned out well, he will, of

course, be continually falling into error about infinite numbers. The

truth is such people do not reason, at all. But for the few who do

reason, reasoning about infinite numbers is easier than about finite

numbers, because the complicated syllogism of transposed quantity is not

called for. For example, that the whole is greater than its part is not

an axiom, as that eminently bad reasoner, Euclid, made it to be. It is a

theorem readily proved by means of a syllogism of transposed quantity,

but not otherwise. Of finite collections it is true, of infinite

collections false. Thus, a part of the whole numbers are even numbers.

Yet the even numbers are no fewer than all the numbers; an evident

proposition since if every number in the whole series of whole numbers

be doubled, the result will be the series of even numbers.

1, 2, 3, 4, 5, 6, etc.

2, 4, 6, 8, 10, 12, etc.

So for every number there is a distinct even number. In fact, there are

as many distinct doubles of numbers as there are of distinct numbers.

But the doubles of numbers are all even numbers.

In truth, of infinite collections there are but two grades of magnitude,

the _endless_ and the _innumerable_. Just as a finite collection is

distinguished from an infinite one by the applicability to it of a

special mode of reasoning, the syllogism of transposed quantity, so, as

I showed in the paper last referred to, a numerable collection is

distinguished from an innumerable one by the applicability to it of a

certain mode of reasoning, the Fermatian inference, or, as it is

sometimes improperly termed, “mathematical induction.”

As an example of this reasoning, Euler’s demonstration of the binomial

theorem for integral powers may be given. The theorem is that (_x_ +

_y_)^n, where _n_ is a whole number, may be expanded into the sum of a

series of terms of which the first is _x_^n_y_^o and each of the others

is derived from the next preceding by diminishing the exponent of _x_ by

1 and multiplying by that exponent and at the same time increasing the

exponent of _y_ by 1 and dividing by that increased exponent. Now,

suppose this proposition to be true for a certain exponent, _n_ = _M_,

then it must also be true for _n_ = _M_ + 1. For let one of the terms in

the expansion of (_x_ + _y_)^M be written A_x^p__y^q_. Then, this term

with the two following will be

A_x^p__y^q_ + A _p_/(_q_ + 1) _x_^{_p_ - 1} _y_^{_q_ + 1} + A

_p_/(_q_ + 1) (_p_ - 1)/(_q_ + 2) _x_^{_p_ - 2} _y_^{_q_ + 2}

Now, when (_x_ + _y_)^M is multiplied by _x_ + _y_ to give (_x_ +

_y_)^{M + 1}, we multiply first by _x_ and then by _y_ instead of by _x_

and add the two results. When we multiply by _x_, the second of the

above three terms will be the only one giving a term involving

_x^p__y_^{_q_ + 1} and the third will be the only one giving a term in

_x_^{_p_ - 1}_y_^{_q_ + 2}; and when we multiply by y the first will be

the only term giving a term in _x^p__y_^{_q_ + 1}, and the second will

be the only term giving a term in _x_^{_p_ - 1}_y_^{_q_ + 2}. Hence,

adding like terms, we find that the coefficient of _x^p__y_^{_q_ + 1} in

the expansion of (_x_ + _y_)^{M + 1} will be the sum of the coefficients

of the first two of the above three terms, and that the coefficient of

_x_^{_p_ - 1}_y_^{_q_ + 2} will be the sum of the coefficients of the

last two terms. Hence, two successive terms in the expansion of (_x_ +

_y_)^{M + 1} will be

A[1 + (_p_/(_q_ + 1))]_x^p__y_^{_q_+1} + A(_p_/(_q_ + 1))[1+ ((_p_ -

1)/(_q_ + 2))]_x_^{_p_-1}_y_^{_q_+2}

= A((_p_ + _q_ + 1)/(_q_ + 1))_x^p__y_^{_q_+1} + A((_p_ + _q_ +

1)/(_q_ + 1))(_p_/(_q_ + 2))_x_^{_p_-1}_y_{_q_+2}

It is, thus, seen that the succession of terms follows the rule. Thus if

any integral power follows the rule, so also does the next higher power.

But the first power obviously follows the rule. Hence, all powers do so.

Such reasoning holds good of any collection of objects capable of being

ranged in a series which though it may be endless, can be numbered so

that each member of it receives a definite integral number. For

instance, all the whole numbers constitute such a numerable collection.

Again, all numbers resulting from operating according to any definite

rule with any finite number of whole numbers form such a collection. For

they may be arranged in a series thus. Let F be the symbol of operation.

First operate on 1, giving F(1). Then, operate on a second 1, giving

F(1,1). Next, introduce 2, giving 3rd, F(2); 4th F(2,1); 5th, F(1,2);

6th, F(2,2). Next use a third variable giving 7th, F(1,1,1); 8th,

F(2,1,1); 9th, F(1,2,1); 10th, F(2,2,1); 11th, F(1,1,2); 12th, F(2,1,2);

13th, F(1,2,2); 14th, F(2,2,2). Next introduce 3, and so on, alternately

introducing new variables and new figures; and in this way it is plain

that every arrangement of integral values of the variables will receive

a numbered place in the series.[62]

The class of endless but numerable collections (so called because they

can be so ranged that to each one corresponds a distinct whole number)

is very large. But there are collections which are certainly

innumerable. Such is the collection of all numbers to which endless

series of decimals are capable of approximating. It has been recognized

since the time of Euclid that certain numbers are surd or

incommensurable, and are not exactly expressible by any finite series of

decimals, nor by a circulating decimal. Such is the ratio of the

circumference of a circle to its diameter, which we know is nearly

3.1415926. The calculation of this number has been carried to over 700

figures without the slightest appearance of regularity in their

sequence. The demonstrations that this and many other numbers are

incommensurable are perfect. That the entire collection of

incommensurable numbers is innumerable has been clearly proved by

Cantor. I omit the demonstration; but it is easy to see that to

discriminate one from some other would, in general, require the use of

an endless series of numbers. Now if they cannot be exactly expressed

and discriminated, clearly they cannot be ranged in a linear series.

It is evident that there are as many points on a line or in an interval

of time as there are of real numbers in all. These are, therefore,

innumerable collections. Many mathematicians have incautiously assumed

that the points on a surface or in a solid are more than those on a

line. But this has been refuted by Cantor. Indeed, it is obvious that

for every set of values of coördinates there is a single distinct

number. Suppose, for instance, the values of the coordinates all lie

between 0 and + 1. Then if we compose a number by putting in the first

decimal place the first figure of the first coördinate, in the second

the first figure of the second coördinate, and so on, and when the first

figures are all dealt out go on to the second figures in like manner, it

is plain that the values of the coördinates can be read off from the

single resulting number, so that a triad or tetrad of numbers, each

having innumerable values, has no more values than a single

incommensurable number.

Were the number of dimensions infinite, this would fail; and the

collection of infinite sets of numbers having each innumerable

variations, might, therefore, be greater than the simple innumerable

collection, and might be called _endlessly infinite_. The single

individuals of such a collection could not, however, be designated, even

approximately, so that this is indeed a magnitude concerning which it

would be possible to reason only in the most general way, if at all.

Although there are but two grades of magnitudes of infinite collections,

yet when certain conditions are imposed upon the order in which

individuals are taken, distinctions of magnitude arise from that cause.

Thus, if a simply endless series be doubled by separating each unit into

two parts, the successive first parts and also the second parts being

taken in the same order as the units from which they are derived, this

double endless series will, so long as it is taken in that order, appear

as twice as large as the original series. In like manner the product of

two innumerable collections, that is, the collection of possible pairs

composed of one individual of each, if the order of continuity is to be

maintained, is, by virtue of that order, infinitely greater than either

of the component collections.

We now come to the difficult question. What is continuity? Kant

confounds it with infinite divisibility, saying that the essential

character of a continuous series is that between any two members of it a

third can always be found. This is an analysis beautifully clear and

definite; but unfortunately, it breaks down under the first test. For

according to this, the entire series of rational fractions arranged in

the order of their magnitude, would be an infinite series, although the

rational fractions are numerable, while the points of a line are

innumerable. Nay, worse yet, if from that series of fractions any two

with all that lie between them be excised, and any number of such finite

gaps be made, Kant’s definition is still true of the series, though it

has lost all appearance of continuity.

Cantor defines a continuous series as one which is _concatenated_ and

_perfect_. By a concatenated series, he means such a one that if any two

points are given in it, and any finite distance, however small, it is

possible to proceed from the first point to the second through a

succession of points of the series each at a distance from the preceding

one less than the given distance. This is true of the series of rational

fractions ranged in the order of their magnitude. By a perfect series,

he means one which contains every point such that there is no distance

so small that this point has not an infinity of points of the series

within that distance of it. This is true of the series of numbers

between 0 and 1 capable of being expressed by decimals in which only the

digits 0 and 1 occur.

It must be granted that Cantor’s definition includes every series that

is continuous; nor can it be objected that it includes any important or

indubitable case of a series not continuous. Nevertheless, it has some

serious defects. In the first place, it turns upon metrical

considerations; while the distinction between a continuous and a

discontinuous series is manifestly non-metrical. In the next place, a

perfect series is defined as one containing “every point” of a certain

description. But no positive idea is conveyed of what all the points

are: that is definition by negation, and cannot be admitted. If that

sort of thing were allowed, it would be very easy to say, at once, that

the continuous linear series of points is one which contains every point

of the line between its extremities. Finally, Cantor’s definition does

not convey a distinct notion of what the components of the conception of

continuity are. It ingeniously wraps up its properties in two separate

parcels, but does not display them to our intelligence.