Chapter XXVI: Part 26

Mr. Spencer in admitting that “the general drift of the passages quoted makes it tolerably clear that Kant must have believed in the operation of natural causes ... in producing organic forms,” adds:

“He does not, however, extend the theory of natural genesis to
the exclusion of the theory of supernatural genesis.”

How does Mr. Spencer prove his statement? Does he quote a passage from Kant which expresses his belief in supernaturalism? No, Mr. Spencer does not quote Kant, and it would be difficult to find a passage to suit that purpose. Mr. Spencer adduces a few unmeaning phrases gleaned at random and torn out of their context, and from these phrases he concludes that Kant believed in the supernatural. Kant spoke somewhere of “the wisdom of nature” who has things so arranged that the species might be preserved. If the wisdom of nature in preserving the species is to be taken literally, the phrase might prove that Kant believed nature to be a wise old woman. Kant spoke further of “the germ of reason placed in man whereby he is destined to social intercourse.” Does the usage of the word “destined” really “imply divine intervention,” as Mr. Spencer says? Mr. Spencer adds:

“And this [viz. Kant’s usage of these phrases] shows that I was
justified in ascribing to him the belief that Space and Time,
as forms of thought [sic!], are supernatural endowments.”

What might we not prove by this kind of loose argumentation!

Kant did not introduce any supernatural explanations; on the contrary, he proposed to exclude “supernatural genesis.” He says e. g. in a passage of the “Critique of Judgment” quoted on page 41 of the appendix:

“If we assume occasionalism for the production of organised
beings, nature is thereby wholly discarded ... therefore it
cannot be supposed that this system is accepted by anyone who
has had to do with philosophy.”

And furthermore Kant rejects the partial admission of the supernatural, saying:

“As though it were not the same to make the required forms
arise in a supernatural manner at the beginning of the world as
during its progress.”

Mr. Spencer charges Kant with inconsistency. We do not intend to say that Kant was in all the phases of his development consistent with himself. But we do say that the charge of Mr. Spencer against Kant consists in this: the real Kant has said things which are incompatible with Mr. Spencer’s view of Kant.

This settles the sixth point.

VII.

Mr. Spencer’s reply to my criticism is a very strange piece of controversy and I have actually been at a loss, how to account for it.

The situation can be explained only by assuming that Mr. Spencer, being an impatient reader, when finding out that he disagreed with my propositions, could go no further and wrote his reply to me without having read my articles. This is very hard on a critic who, carefully avoiding everything that might look like fault-finding, is painstakingly careful in giving to the author criticised every means of investigating the truth himself and helps him in a friendly way to correct his errors.

There is only one consolation for me, which is, that I am in good company. The great thinker of Koenigsberg is very severely censured in almost all of Mr. Spencer’s writings for ideas which he never held. And now Mr. Spencer confesses openly and with ingenuous sincerity, that his knowledge of Kant’s writings is extremely limited. But why he condemns a man of whom he knows so little Mr. Spencer does not tell us.

Mr. Spencer says:

“My knowledge of Kant’s writings is extremely limited. In 1844
a translation of his “Critique of Pure Reason” (then I think
lately published) fell into my hands, and I read the first few
pages enunciating his doctrine of Time and Space: my peremptory
rejection of which caused me to lay the book down.

“Twice since then the same thing has happened; for, being
an impatient reader, when I disagree with the cardinal
propositions of a work I can go no further.

“One other thing I knew. By indirect references I was made
aware that Kant had propounded the idea that celestial bodies
have been formed by the aggregation of diffused matter. Beyond
this my knowledge of his conceptions did not extend; and my
supposition that his evolutionary conception had stopped short
with the genesis of sun, stars, and planets was due to the
fact that his doctrine of Time and Space, as forms of thought
[sic] anteceding experience, implied a supernatural origin
inconsistent with the hypothesis of natural genesis.”

Kant has been a leader in thought for the last century. It is very important to criticise his ideas wherever they are wrong, but his errors cannot be conquered by _ex cathedra_ denunciations.

Darwin’s habits in investigating and weighing the pro and con of a question were very different from Mr. Spencer’s, and Darwin’s success is in no small degree due to the sternness with which he adhered to certain rules of reading and studying. We find in his “Autobiography” certain reminiscences labeled “important” from which the following is most instructive:

“I had also, during many years, followed a golden rule, namely,
that whenever a published fact, a new observation or a thought,
came across me, which was opposed to my general results, to
make a memorandum of it without fail, for I had found by
experience that such facts and thoughts were far more apt to
escape from the memory than favorable ones.”

Experience teaches that we can learn most from those authors with whom we do not agree. The ethics of reading and studying demand other habits than laying a book down when we disagree with its cardinal propositions. Such habits prevent progress and create prejudices.

* * * * *

Mr. Spencer has not answered my criticism at all. Mr. Spencer did not even take into consideration the passages quoted from Kant. He republished all the false statements of Kant’s views, so inconsiderately made, together with all the perverse opinions based upon them. The assurance with which Mr. Spencer makes statements which have no foundation whatever is really perplexing even to a man who is well informed on the subject, and it will go far to convince the unwary reader. What, however, shall become of the general tenor of philosophical criticism and controversy if a man of Mr. Spencer’s reputation is so indifferent about being informed concerning the exact views of his adversary, if he is so careless in presenting them, if he makes positively erroneous statements on confessedly mere “supposition,” and finally, if in consequence thereof he is flagrantly unjust in censuring errors which arise only from his own too prolific imagination?

We feel confident that Mr. Spencer will explain his side of the question satisfactorily. His mistakes being undeniable, we do not believe that he will seek to deny them. Yet we trust that Mr. Spencer as soon as he finds himself at fault, will not even make an attempt at palliation, that he will not blink the frank acknowledgment of his misstatements and also of having treated Kant with injustice. A man who has devoted his life to the search for truth will not suffer any blot to remain on his escutcheon.

EDITOR.

FOOTNOTES:

[69] This article “Herbert Spencer on the Ethics of Kant” was electrotyped at the time it appeared in _The Open Court_. It is appended to this number of _The Monist_ as documentary evidence of the fact, that there is not even so much as an occasion in the article for confounding “consciousness” and “conscientiousness.”

[70] See _Fundamental Problems_, pp. 197-206; and _The Ethical Problem_, p. 32, seq., especially p. 33, lines 18-20.

[71] Second edition reads “_ihnen_” in place of “_ihr_,” viz. _der Zeit_. The word “_ihnen_” refers to _Theilvorstellungen der Zeit_.

WHAT DOES ANSCHAUUNG MEAN?

Mr. Spencer’s erroneous statement that Kant conceives space and time as forms of thought instead of forms of intuition induces me to make a few explanatory remarks concerning the term _Anschauung_.

Kant means that space and time are immediately given in experience and not inferences drawn from the data of experience; they are not thoughts, but objects of direct perception.

Sense-impressions are data, they are prior to ideas, the latter being constructions made out of sense-impressions. Sense-impressions are facts, but ideas are of an inferential nature; they are (to use Lloyd Morgan’s excellent term) constructs. Now Kant claims that space and time are in the same predicament: they also are immediately given, they also are _Anschauungen_. Kant did not trouble himself much to prove that they are forms; he seems to have taken that for granted. But he was very careful to show that they are not ideas, not thoughts, not abstractions, not generalisations, but that they are as direct data as are sense-impressions and he calls the knowledge which man has by directly facing the object of knowledge “_Anschauung_.”

The conclusion which Kant draws from this may be characterised as follows:

Sensations are not things but appearances; they are subjective, not objective, they are not the objects themselves but what our sensibility makes of objects. Space and time being _Anschauungen_, Kant argues that they are of the same kind as the sense-data of knowledge, that they are inherent in our nature. Thus Kant maintains: “Sensations are the products of our sensibility, and space and time are the forms of our sensibility.”

The word _Anschauung_ has been a _crux interpretum_ since translations have been made from Kant, and it is quite true that no adequate word to express it, exists in English. I enjoyed of late a discussion on the subject with Mr. Francis C. Russell who called my attention to several notes in _The Journal of Speculative Philosophy_. The following is from the pen of Dr. W. T. Harris (Vol. II, p. 191):

“Through a singular chance, the present number of the journal
contains two notes from two contributors on the proper
translation of the German word _Anschauung_. Mr. Kroeger holds
that the word _Anschauung_, as used by Fichte and also by Kant,
denotes an act of the Ego which the English word _Intuition_
does not at all express, but for which the English word
‘contemplation’ is an exact equivalent. Mr. Peirce suggests
that no person whose native tongue is English will translate
_Anschauung_ by another word than _Intuition_. Whether there
is a failure to understand English on the one hand or German
on the other, the Editor does not care to inquire. It is
certain that while intuition has been adopted generally as an
equivalent for the word under consideration both by English
and French translators, yet it was a wide departure from the
ordinary English use of the term. Besides this, we have no
English verb _intuite_ (at least in the Dictionaries), and
the reader will find that the verb used by Meiklejohn (in the
translation of Kant’s _Kritik_) for it, is _contemplate_,
and the same rendering is given by Smith in his excellent
translation of Fichte’s Popular Works (London, 1849).”

Mr. Charles S. Peirce says:

“No person whose native tongue is English will need to be
informed that contemplation is essentially (1) protracted (2)
voluntary, and (3) an action, and that it is never used for
that which is set forth to the mind in this act. A foreigner
can convince himself of this by the proper study of English
writers. Thus, Locke (Essay concerning Human Understanding,
Book II., chap. 19, § 1) says, ‘If it [an idea] be held
there [in view] long under attentive consideration, ’tis
_contemplation_”; and again, (_Ibid._, Book II., chap. 10, § 1)
‘Keeping the _Idea_, which is brought into it [the mind] for
some time actually in view, which is called _Contemplation_.’
This term is therefore unfitted to translate _Anschauung_;
for this latter does not imply an act which is necessarily
protracted or voluntary, and denotes most usually a mental
presentation, sometimes a faculty, less often the reception of
an impression in the mind, and seldom, if ever, an action.

“To the translation of _Anschauung_ by intuition, there is,
at least, no such insufferable objection. Etymologically the
two words precisely correspond. The original philosophical
meaning of intuition was a cognition of the present manifold
in that character; and it is now commonly used, as a modern
writer says, ‘to include all the products of the perceptive
(external or internal) and imaginative faculties; every act of
consciousness, in short, of which the immediate object is an
_individual_, thing, act, or state of mind, presented under the
condition of distinct existence in space and time.’ Finally, we
have the authority of Kant’s own example for translating his
_Anschauung_ by _Intuitus_; and, indeed, this is the common
usage of Germans writing Latin. Moreover, _intuitiv_ frequently
replaces _anschauend_ or _anschaulich_. If this constitutes a
misunderstanding of Kant, it is one which is shared by himself
and nearly all his countrymen” (_ibid._ p. 152 et seqq.).

Mr. Peirce adds the following explanation concerning the term intuition in another note (_ibid._ p. 103):

“The word _intuitus_ first occurs as a technical term in St.
Anselm’s Monologium. He wished to distinguish between our
knowledge of God and our knowledge of finite things (and, in
the next world, of God, also); and thinking of the saying of
St. Paul, _Videmus nunc per speculum in œnigmate: tunc autem
facie ad faciem_, he called the former _speculation_ and the
latter _intuition_. This use of ‘speculation’ did not take
root, because that word already had another exact and widely
different meaning.

“In the middle ages, the term ‘intuitive cognition’ had two
principal senses, 1st, as opposed to abstractive cognition,
it meant the knowledge of the present as present, and this is
its meaning in Anselm; but 2d, as no intuitive cognition was
allowed to be determined by a previous cognition, it came to
be used as the opposite of discursive cognition (see Scotus,
In sentent. lib. 2, dist. 3, qu. 9), and this is nearly the
sense in which I employ it. This is also nearly the sense in
which Kant uses it, the former distinction being expressed
by his _sensuous_ and _non-sensuous_. (See Werke, herausg.
Rosenkrantz, Thl. 2, S. 713, 31, 41, 100, u. s. w.)

“An enumeration of six meanings of intuition may be found in
Hamilton’s Reid p. 759.”

If we have to choose between the two translations “intuition” and “contemplation,” we should with Mr. Peirce decidedly prefer the word “intuition.” The word contemplation corresponds to the German _Betrachtung_ and all that Mr. Peirce says against it holds good. But we must confess that the term intuition (as Mr. Peirce himself seems to grant) is not a very good translation either. The term intuition has other meanings which interfere with the correct meaning of _Anschauung_ and was actually productive of much confusion.

The English term intuition is strongly tinged with the same meaning that is attached to the German word _Intuition_. It means an inexplicable kind of direct information from some supernatural sources, which mystics claim to possess as the means of their revelations. In this sense Goethe characterises it satirically in Faust (Scene XIV). Mephistopheles describes the process as follows:

A blessing drawn from supernatural fountain!
In night and dew to lie upon the mountains;
All Heaven and Earth in rapture penetrating;
Thyself to Godhood haughtily inflating;
To grub with yearning force through Earth’s dark marrow,
Compress the six days’ work within thy bosom narrow,—
To taste, I know not what, in haughty power,
Thine own ecstatic life on all things shower,
Thine earthly self behind thee cast,
And then the lofty intuition [with a gesture] at last.

The satire is good on _Intuition_ but it would not apply to _Anschauung_, for the latter word excludes rigidly any mysticism or supernaturalism which the former essentially involves. To employ the term “intuition” for both ideas must necessarily weaken the meaning of _Anschauung_.

Besides we should bear in mind that the German _Anschauung_ is vernacular and should find a correspondent Saxon word. Such Latin words as intuition convey in English as much as in German the impression of being terms denoting something very abstract. Vernacular terms much more strongly indicate the immediateness and directness which is implied in _Anschauung_. In my conversation with Mr. Russell, we tried to coin a new word that should cover the meaning of _Anschauung_ as an act of “atlooking” and the word “atsight” readily suggested itself.

The word “atsight” is an exact English equivalent of the German _Anschauung_. It describes the looking at an object in its immediate presence. At the same time the word is readily understood, while philologically considered, its formation is fully justified by the existence of the words “insight and foresight.”

* * * * *

One of the most important of Kant’s doctrines is the proposition that all thought must ultimately have reference to _Anschauung_, i. e. to atsight. Through atsight only the objects of experience can be given us. All speculations not founded upon this bottom rock of knowledge are mere dreams. This is the maxim of positivism and it is the basis of all sound philosophy. Says Kant in the “Anhang” to his Prolegomena (in reply to a critic who had misunderstood his idealism) as a summary statement of his views:

“_Der Satz aller echten Idealisten, von der eleatischen Schule
an bis zum Bischof Berkley, ist in dieser Formel enthalten:
‘alle Erkenntnis durch Sinne und Erfahrung ist nichts als
lauter Schein, und nur in den Ideen des reinen Verstandes und
Vernunft ist Wahrheit.’_

“_Der Grundsatz, der meinen Idealismus durchgängig regiert
und bestimmt, ist dagegen: ‘Alles Erkenntnis von Dingen, aus
blossem reinen Verstande oder reiner Vernunft, ist nichts als
lauter Schein, und nur in der Erfahrung ist Wahrheit.’_”

“The doctrine of all genuine idealists from the Eleatic School
down to Bishop Berkeley is contained in this formula: All
cognition through the senses and experience is nothing but
illusion; and in the ideas of the pure understanding and reason
alone is truth.

“The principle, however, that rules and determines my idealism
throughout is this: All cognition out of pure understanding
or pure reason is nothing but mere illusion and in experience
alone is truth.”

Kant then proposes in order to avoid equivocation to call his views “formal or critical idealism,” adding that his idealism made any other idealism impossible. Criticism truly is the beginning of philosophy as an objective science. It gives the _coup de grace_ to those worthless declamations which still pass among many as philosophy. Says Kant:

“_So viel ist gewiss: wer einmal Kritik gekostet hat, den ekelt
auf immer alles dogmatische Gewäsche._”

“That much is certain: He who has once tasted critique will be
forever disgusted with all dogmatic twaddle.”

It is strange that in spite of Kant’s explicit declaration, which leaves no doubt about the positive spirit that pervades the principles of his philosophy, he is still misunderstood by his opponents no less than by those who profess to be his disciples.

* * * * *

There is no occasion now to treat the subject exhaustively, but it may be permitted to add a few remarks on Kant’s proposition that space and time are atsights.

We must distinguish three things:

1) Objective space.

2) Space as atsight, and

3) Space-conception.

Space as atsight is the datum. It is the immediate presence of relations among the sensory impressions. This, however, is not as yet that something which we generally call space. That which generally goes by the name of space is a construction built out of the relational data that obtain in experience and we propose to call it space-conception. Our space-conception, accordingly, (and here I include the mathematician’s space-conception) is based upon space as atsight, but it is more than atsight. It is an inference made therefrom, it is the product of experience. Space-conception, however, is as are all legitimate noumena, no mere subjective illusion, it possesses objective validity; it describes some real existence and this real existence represented in space-conception is what may be called objective space.

Objective space is the form of reality. Space as atsight is the form of sensibility. Space as space-conception is a construct of an abstract nature and serves as a description or plan of the form of reality.

The same is true of Time. Time as atsight is the relation of succession obtaining in the changes of experience. Time as time-conception is the noumenon constructed out of these data to represent that feature of reality which may for lack of a better term be called objective time.

Briefly: Space and Time are not things, not essences, not entities, but certain features of existence. They are the forms of reality. When existence finds a representation in the feelings of a sentient being, time and space appear as their forms, and these forms furnish the material out of which are built the conceptions of Space and Time.

EDITOR.

THE LAW OF MIND.

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 emphasised 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 specialised 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 signalised 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. III.); 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 recognised 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 bonds 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 recognised, or known as known before, except only the last moment, which of course will be absolutely unrecognisable to itself. Indeed, even this last moment will be recognised 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, DeMorgan, 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, DeMorgan, 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 _xⁿy⁰_ 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_xᵖy𐞥_. Then, this term with the two following will be

=Transcriber’s Note:= Unicode has no subscript q character, so the Greek subscript phi character ᵩ is used in these formulæ to represent it. Italics have been removed for readability.

Axᵖy𐞥 + A(ᵖ⁄ᵩ₊₁)xᵖ⁻¹y𐞥⁺¹ + A(ᵖ⁄ᵩ₊₁)(ᵖ⁻¹⁄ᵩ₊₂)xᵖ⁻²y𐞥⁺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 _xᵖy𐞥⁺¹_ and the third will be the only one giving a term in _xᵖ⁻¹y𐞥⁺²_; and when we multiply by _y_ the first will be the only term giving a term in _xᵖy𐞥⁺¹_, and the second will be the only term giving a term in _xᵖ⁻¹y𐞥⁺²_. Hence, adding like terms, we find that the coefficient of _xᵖy𐞥⁺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 _xᵖ⁻¹y𐞥⁺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⁄ᵩ₋₁)]xᵖy𐞥⁺¹ + A(ᵖ⁄ᵩ₊₁)[1+(ᵖ⁻¹⁄ᵩ₋₂)]xᵖ⁻¹y𐞥⁺²

= A(ᵖ⁺𐞥⁺¹⁄ᵩ₊₁)xᵖy𐞥⁺¹ + A(ᵖ⁺𐞥⁺¹⁄ᵩ₊₁). (ᵖ⁄ᵩ₊₂)xᵖ⁻¹y𐞥⁺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.[72]

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 recognised 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 coördinates 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 he 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.

Kant’s definition expresses one simple property of a continuum; but it allows of gaps in the series. To mend the definition, it is only necessary to notice how these gaps can occur. Let us suppose, then, a linear series of points extending from a point, _A_, to a point, _B_, having a gap from _B_ to a third point, _C_, and thence extending to a final limit, _D_; and let us suppose this series conforms to Kant’s definition. Then, of the two points, _B_ and _C_, one or both must be excluded from the series; for otherwise, by the definition, there would be points between them. That is, if the series contains _C_, though it contains all the points up to _B_, it cannot contain _B_. What is required, therefore, is to state in non-metrical terms that if a series of points up to a limit is included in a continuum the limit is included. It may be remarked that this is the property of a continuum to which Aristotle’s attention seems to have been directed when he defines a continuum as something whose parts have a common limit. The property may be exactly stated as follows: If a linear series of points is continuous between two points, _A_ and _D_, and if an endless series of points be taken, the first of them between _A_ and _D_ and each of the others between the last preceding one and _D_, then there is a point of the continuous series between all that endless series of points and _D_, and such that every other point of which this is true lies between this point and _D_. For example, take any number between 0 and 1, as 0.1; then, any number between 0.1 and 1, as 0.11; then any number between 0.11 and 1, as 0.111; and so on, without end. Then, because the series of real numbers between 0 and 1 is continuous, there must be a _least_ real number, greater than every number of that endless series. This property, which may be called the Aristotelicity of the series, together with Kant’s property, or its Kanticity, completes the definition of a continuous series.

The property of Aristotelicity may be roughly stated thus: a continuum contains the end point belonging to every endless series of points which it contains. An obvious corollary is that every continuum contains its limits. But in using this principle it is necessary to observe that a series may be continuous except in this, that it omits one or both of the limits.

Our ideas will find expression more conveniently if, instead of points upon a line, we speak of real numbers. Every real number is, in one sense, the limit of a series, for it can be indefinitely approximated to. Whether every real number is a limit of a _regular_ series may perhaps be open to doubt. But the series referred to in the definition of Aristotelicity must be understood as including all series whether regular or not. Consequently, it is implied that between any two points an innumerable series of points can be taken.

Every number whose expression in decimals requires but a finite number of places of decimals is commensurable. Therefore, incommensurable numbers suppose an infinitieth place of decimals. The word infinitesimal is simply the Latin form of infinitieth; that is, it is an ordinal formed from _infinitum_, as centesimal from _centum_. Thus, continuity supposes infinitesimal quantities. There is nothing contradictory about the idea of such quantities. In adding and multiplying them the continuity must not be broken up, and consequently they are precisely like any other quantities, except that neither the syllogism of transposed quantity, nor the Fermatian inference applies to them.

If A is a finite quantity and _i_ an infinitesimal, then in a certain sense we may write A + _i_ = A. That is to say, this is so for all purposes of measurement. But this principle must not be applied except to get rid of _all_ the terms in the highest order of infinitesimals present. As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares. Indeed, the latter, as stated in some books, involves propositions that are false; but this is not the case with the forms of the method used by Cauchy, Duhamel, and others. As they understand the doctrine of limits, it involves the notion of continuity, and therefore contains in another shape the very same ideas as the doctrine of infinitesimals.

Let us now consider an aspect of the Aristotelical principle which is particularly important in philosophy. Suppose a surface to be part red and part blue; so that every point on it is either red or blue, and, of course, no part can be both red and blue. What, then, is the color of the boundary line between the red and the blue? The answer is that red or blue, to exist at all, must be spread over a surface; and the color of the surface is the color of the surface in the immediate neighborhood of the point. I purposely use a vague form of expression. Now, as the parts of the surface in the immediate neighborhood of any ordinary point upon a curved boundary are half of them red and half blue, it follows that the boundary is half red and half blue. In like manner, we find it necessary to hold that consciousness essentially occupies time; and what is present to the mind at any ordinary instant, is what is present during a moment in which that instant occurs. Thus, the present is half past and half to come. Again, the color of the parts of a surface at any finite distance from a point, has nothing to do with its color just at that point; and, in the parallel, the feeling at any finite interval from the present has nothing to do with the present feeling, except vicariously. Take another case: the velocity of a particle at any instant of time is its mean velocity during an infinitesimal instant in which that time is contained. Just so my immediate feeling is my feeling through an infinitesimal duration containing the present instant.

ANALYSIS OF TIME.

One of the most marked features about the law of mind is that it makes time to have a definite direction of flow from past to future. The relation of past to future is, in reference to the law of mind, different from the relation of future to past. This makes one of the great contrasts between the law of mind and the law of physical force, where there is no more distinction between the two opposite directions in time than between moving northward and moving southward.

In order, therefore, to analyse the law of mind, we must begin by asking what the flow of time consists in. Now, we find that in reference to any individual state of feeling, all others are of two classes, those which affect this one (or have a tendency to affect it, and what this means we shall inquire shortly), and those which do not. The present is affectible by the past but not by the future.

Moreover, if state _A_ is affected by state _B_, and state _B_ by state _C_, then _A_ is affected by state _C_, though not so much so. It follows, that if _A_ is affectible by _B_, _B_ is not affectible by _A_.

If, of two states, each is absolutely unaffectible by the other, they are to be regarded as parts of the same state. They are contemporaneous.

To say that a state is _between_ two states means that it affects one and is affected by the other. Between any two states in this sense lies an innumerable series of states affecting one another; and if a state lies between a given state and any other state which can be reached by inserting states between this state and any third state, these inserted states not immediately affecting or being affected by either, then the second state mentioned immediately affects or is affected by the first, in the sense that in the one the other is _ipso facto_ present in a reduced degree.

These propositions involve a definition of time and of its flow. Over and above this definition they involve a doctrine, namely, that every state of feeling is affectible by every earlier state.

THAT FEELINGS HAVE INTENSIVE CONTINUITY.