remarkable instance of reasoning which seems to be rendered entirely
nugatory by the want of proper definitions. It is asserted that
conceptual thought is impossible without language. At first sight this
would certainly appear to be a real assertion. It follows from it that
since dogs have no language they have no “conceptual thought.” But it
may be plainly shown that dogs do entertain “general notions,” which
in ordinary English would be included under the head of “conceptual
thought.” The apparent contradiction is however explained when it appears
that the author distinguishes general notions as “concepts” or “recepts,”
according as they _are_ or _are not named_. This being his definition of
“conceptual thought” as opposed to other thought, it appears that the
assertion is only a truism after all, and conveys no real information
whatever. To discuss it further is then mere waste of time. The author of
the assertion doubtless _wished_ it to convey some information, but he
did not attend to his definitions and so failed to attain his object.
EDWARD T. DIXON.
FOOTNOTES:
[18] The reviewer of Mr. Dixon’s book has read these remarks on his
criticism (_The Monist_, Vol. II, No. 1, p. 126) and has given them what
seems to him full consideration. He confesses that he misunderstood
what Mr. Dixon meant by “a direction.” (See the article “Logic as
Relation-Lore” to be published in a subsequent number.) In regard to the
requirements for a logical definition he must still abide by his former
opinion. The need of a definition arises either from the inaccuracy in
the application of a term or from a supposed lack of knowledge as to its
signification. Hence to use the term itself in its own definition is to
import into the definition the same vagueness or ignorance which it is
the very office of a definition to correct. When Mr. Dixon says that it
is requisite for a logical definition that the defining assertions “must
not be _independent_ of the meaning of the term defined,” what is that
but to say that the same must be _dependent_ upon that meaning? which,
unless the reviewer again misunderstands the author, is to say that we
must understand the meaning of the term before we can understand the
definition.
ρσλ.
MATHEMATICS A DESCRIPTION OF OPERATIONS WITH PURE FORMS.
IN REPLY TO MR. EDWARD DIXON.
It is true, as Mr. Dixon says, that “Any language which seems to imply
that there is some dread necessity about mathematical truths is very
misleading.” But to say, as Mr. Dixon does in another passage, that
the truisms of mathematics are arbitrary truths, is more misleading
still. The theorems of the formal sciences are not “assertions whose
truth depends solely on the definition of their terms.” They are “real
assertions which convey some real subjective or objective information.”
Mr. Dixon objects to Kant’s assertion that 7 + 5 = 12 is not only _a
priori_ but also synthetic. He declares, in contradistinction to Kant,
that it is deduced from definitions alone; that therefore it is empty,
and cannot give any information about things. This latter proposition,
which is a phrase of Kant’s, appears in this context as an inconsistency
of Kant’s. And it would be an inconsistency, if it had to be understood
in the sense in which Mr. Dixon quotes it. We construe Kant’s phrase that
“the _a priori_ is empty, and cannot give information about things,” in
a different way. We think that Kant intends to say that the _a priori_
imparts real information concerning relations and forms; but that it does
not impart real information concerning substances or the materiality
of things. It is apparent that the assertion 7 + 5 = 12 cannot be
derived from the definitions of 7 or 5. Similarly, the ideas of higher
mathematics are not deduced from the few definitions of elementary
mathematics that tell what points, lines, parallels, etc., are. Of what,
then, are these complex theorems of mathematics, products? They are not
derived from sense-experience, nor from the definitions of their ultimate
elements.
Is their origin mysterious? Here Kant leaves us in the lurch; he simply
declares that formal truth is _a priori_ and transcendental; and those of
his disciples who call themselves, with preference, transcendentalists,
have ample occasion to introduce in this _lacuna_ of Kantian thought, all
the mysticism they please.
The problem of the origin of the truths of formal sciences is not
so difficult as it is sometimes represented. The theorems of higher
mathematics are the products of certain _operations_ performed with
the elementary forms described in the definition with which the
mathematician starts. These operations are not arbitrary; they are not
merely verbal processes; they are realities of highest importance. Not
material realities, but realities, nevertheless. They are functions,
and mathematics deals with the products of functions. It is true that
we might call twice two by any other name than four; we might call
it _vier_, or _quatre_, but the operation 1 + 1 + 1 + 1 would remain
the same, by whatever name we call its result. Mathematical truths,
accordingly, are not empty in the sense that they are meaningless; for
they are significant in the highest degree. They give real information,
not about things, but about certain relations that obtain among things.
They describe certain operations in which formal relations are traced.
And they describe them exhaustively, so that the result is, as the
Germans call it, _eindeutig bestimmt_, and the result will, under all
circumstances, be the same. Twice two _will always be_ the same as 1 + 1
+ 1 + 1. This “it will always be,” is called necessary. There is nothing
dreadful about it, nor is there any mystery connected with it. It is not
an awful fate that decrees it, but it is the nature of sameness, that the
same is and will be the same, so long as it remains the same.
It is often overlooked that every number in arithmetic is the result
of an operation which is symbolised by a certain figure. Numbers are
not concrete things; and as soon as we forget that they are products
of a function, we are liable to lapse into mistakes. This happens most
frequently with the numbers “zero” and “infinite.” The latter of these
two symbols is often looked upon as a concrete thing; and because the
infinite, with actual reality, is, in its completeness, inconceivable, it
has made, of every one who stumbled over this stone of offence, a mystic,
and many a radical, fearless thinker bows down to worship before the idea
of infinitude-function as it would be if it were a real thing.
Says Mr. Dixon, “Our reason cannot inform us about the form of existence,
unless it is first given.” This is very true. The form is given, and
formal systems such as the numerical system and the lines and figures of
mathematics are mental constructions built of the stones quarried out of
the relational given in experience.
Form being given, we can reason about the form of existence in general.
We can have ready in our minds systems of pure forms to apply to all the
various cases of our experience. And this will help us in unravelling
the problems of reality, and in extending our knowledge in those fields
with which we are little acquainted. Far be it from us to consider the
definitions, the operations, and the results of the formal sciences
as purely verbal; if they were, mathematics would lose all the great
importance which it undeniably possesses, and become mere verbiage.
I confess that I do not understand Mr. Dixon when he says: “It is only
because our reasoning faculties are limited that symbolic arguments
are necessary at all.” In my opinion all mental activity is symbolic.
Every idea is a symbol that signifies something. It is not because our
reasoning faculties are limited that symbolic arguments are necessary
at all, but symbolism is the nature of our mind, and symbols are the
elements with which our reasoning faculties have to deal. In this sense,
every argument is symbolic. If it symbolises sense-experience, it
represents our knowledge of what may be called the materiality of things.
If it symbolises operations with pure forms, it represents the purely
formal relations of mathematics logic, algebra, etc.
The doctrine of the conservation of matter and energy in reality means
nothing more or less than that there is no increase or decrease in the
world at large. Nothing originates out of, and nothing disappears into,
nothing. It means that twice 1 + 1 is 1 + 1 + 1 + 1, neither more nor
less; or, in other words, it means that all events are transformations.
New things originate, but their newness consists in their forms. In this
sense the law of the conservation of matter and energy would have to
be called, from Mr. Dixon’s standpoint, other differences neglected, a
truism. It is not a truism, in the sense of being arbitrary, but in the
sense of being a purely formal truth, as are all mathematical theorems.
Mr. Dixon refers to a passage in _Fundamental Problems_, in which I say
that “To four-dimensional beings, Kepler’s third law would most probably
appear as the cubes of the times of revolutions of the planets being
proportional to their mean distances to the fourth power.” His questions,
“What sort of an assertion do you take Kepler’s law to be?” and “Why
should the result appear different to them?” show that Mr. Dixon has
overlooked the condition on which this proposition was made. The first
sentence of this paragraph begins with the words, “If space inhered, as
Kant maintains, in the thinking subject only, spatial relations and laws
would appear different to four-dimensional beings.” Space relations are
not subjective, in my opinion, but objective. Therefore, since space
relations do not inhere in the thinking subject only, because they are a
feature of the objective world, and inhere in the thinking subject in so
far as it is at the same time an object in the objective world, Kepler’s
law would appear to four-dimensional beings, if they could exist at all,
just the same as it does to us three-dimensional beings.
P. C.
BOOK REVIEWS.
SOCIAL STATICS. Abridged and Revised; together with THE MAN VERSUS THE
STATE. By _Herbert Spencer_. New York: D. Appleton & Co., 1892.
JUSTICE. Being Part IV of the Principles of Ethics. By _Herbert Spencer_.
New York: D. Appleton & Co., 1891.
Among Mr. Spencer’s most important books are those entitled _Justice_ and
_Social Statics_. The latter, which first appeared in 1850, has just been
republished in about one-half of the original size, some parts having
been transferred to the book on _Justice_, and others omitted altogether.
“One difference,” as he says, “is that what there was in my first book of
supernaturalistic interpretation has disappeared, and the interpretation
has become exclusively naturalistic—that is, evolutionary.” Another
change is that a demonstration of the injustice of socialism is
substituted for his former arguments, plainly repudiated in _Justice_,
against private ownership of land. Equally important is the omission of
the chapter asserting “The Right to Ignore the State.”
The demand for Woman Suffrage has also been withdrawn from the new
edition of _Social Statics_, though it retains the original protest
against “the reign of man over woman,” and asserts an “equality of
rights in the married state.” Here again, Spencer’s final position must
be sought in his _Justice_ where it is urged that women cannot justly
have equal powers with men unless they have equal responsibilities. They
cannot serve their country as men do; and if they take an equal share
in the government, “their position is not one of equality but one of
supremacy.” Even in time of peace, they are, he thinks, too impulsive
to vote judiciously, too sympathetic to oppose “fostering the worse at
the expense of the better,” and too fond of “a worship of power under
all its forms” to protect individual liberty against the encroachment of
authority. This objection seems particularly strong, because there is
still great danger of the growth of state despotism at the expense of
personal freedom, even in republics. Many recent instances are given by
Spencer in “The Man versus the State,” now reprinted in the same volume
with _Social Statics_; and it is urged in _Justice_, that even in the
United States “universal suffrage does not prevent an enormous majority
of consumers from being heavily taxed by a protective tariff for the
benefit of a small minority of manufacturers and artisans.”
Our voters are much too ready to follow hasty impulses and unscrupulous
leaders; and both faults are most common among the most ignorant. How
strongly education encourages independence was acknowledged by those
slave-holders who said, “Our negroes shall not learn to read, for that
makes them run away.” Public schools have found their worst enemies among
Popes and Czars, and their best friends in the statesmen most honored by
republics. There is no other institution for whose advantages Americans
are practically unanimous. The necessity of popular education at the
public cost is acknowledged by Huxley, Mill, and other advanced thinkers
so generally, that Spencer’s exceptionally hostile opinion ought not to
be taken as a self-evident truth.
Mr. Spencer’s examination of this subject does not appear to have been
so thorough as the occasion demands. In denying that education prevents
crime, he relies mainly on Joseph Fletcher, who, as stated in both
editions of _Social Statics_, “has entered more elaborately into this
question than perhaps any other writer of the day,” and who admits that
there is a “superficial evidence against instruction.” Spencer takes
no notice of Fletcher’s having succeeded completely in breaking down
this superficial evidence. In elaborate papers, published in the tenth,
eleventh, and twelfth volumes of _The Journal of the London Statistical
Society_, and illustrated by many tables and maps, Fletcher shows that
the proportion to the population, in various parts of England, of people
unable to sign their names, corresponded everywhere to the proportion
of illegitimate births as well as of commitments for crime. Separating
these latter into classes according to degree of guilt, he proves that
the worst crimes are most common where there is the most ignorance.
Thus he is enabled to say, “The conclusion is therefore irresistible
that education is essential to the security of modern society.” That
this testimony of Spencer’s principal witness is really the truth can
be further proved by the statistics in the _Encyclopædia Britannica_,
showing that between 1841 and 1876, while the percentage of illiterates
to population in England and Wales was reduced one-half, that of
criminals was reduced to one-third of what it was originally. (Vol. VIII,
pp. 221 and 249-251.)
Spencer also refers to the fact that schools have sometimes been carried
on in the interest of despotism; but most kinds of food are easily
adulterated; and education is valuable, notwithstanding, as food for
liberty. This last consideration disposes completely of his comparison of
state-churches with state-schools; and the fact, mentioned in the revised
but not in the original edition of _Social Statics_, that opinions differ
about the best methods of education, is really an additional instance of
the encouragement given by our system of public schools to independence
of thought.
Spencer’s chief objection to this system is that it does not fit his
theory that “the liberty of each, limited by the like liberties of
all, is the rule in conformity with which society must be organised,”
(p. 45). Such a “law of right social relationships,” (p. 55) would, he
admits, require us to repeal our laws against indecency, abolish our
Boards of Health, and close our poorhouses, postoffices, banks, and
lighthouses, except in so far as these institutions, like our streets and
roads, might be cared for by benevolent individuals. He does not tell us
how a government, thus limited to managing the police, army, and navy,
could keep up a fire-department, nor how new streets, roads, railways,
or canals could be opened, in case the owners of land put their prices
too high for the projectors; but the most unfortunate application of his
theory would be to close our public schools.
There is no danger of this, however; and the principal evil likely to
result from his pushing his theory so far, is that he prevents people
from seeing its real value, as indicating the direction in which our
race has advanced and must make all further progress. We shall keep on
diminishing the power of the state over the man, as well as that of
the man over the child, but neither authority will ever be abolished
entirely. We shall dispense, sooner or later, with some of the public
institutions which Spencer condemns; but our common schools will, I
think, last as long as government itself. The abolitionists helped the
slave to freedom by pointing out the North Star; but they did not advise
him to quit solid earth. This mistake, although we grant that Spencer
shows us our North Star, is sometimes made in _Social Statics_.
Timely help, too, is given by him, in a thoroughly practical way, to
those reformers who are passing out from under the cloud with a silver
lining into a Cleveland summer and a fair prospect of a Harrison fall.
Among the words best worth putting into actions at once, are these: “The
right of exchange is as sacred as any other right, and exists as much
between members of different nations as between members of the same
nation. Morality knows nothing of geographical boundaries.” ... “Hence,
in putting a veto upon the commercial intercourse of two nations, or in
putting obstacles in the way of that intercourse, a government trenches
upon men’s liberties of action, and by so doing directly reverses its
function. To secure for each man the fullest freedom to exercise his
faculties, compatible with the like freedom of all others, we find to be
the state’s duty. Now trade prohibitions and trade restrictions not only
do not secure this freedom, but they take it away. So that in enforcing
them the state is transformed from a maintainer of rights into a violator
of rights.” ... “Whether it kills, or robs, or enslaves, or shackles
by trade regulations, its guilt is alike in kind, and differs only in
degree.” (_Social Statics_, ed. of 1850, pp. 326, 327; ed. of 1892, p.
137).
F. M. H.
AN ESSAY ON REASONING. By _Edward T. Dixon_. Cambridge (Eng): Deighton,
Bell, & Co. 1891. Pp. 88.
Some years ago the author of this essay made public certain views of his,
on “Geometry of Four Dimensions.” He was surprised to find that though
his arguments were received with incredulity they were not refuted.
This result appeared to him to be due to the fact that he was not
understood, that his views on geometry of even two and three dimensions
being different from those commonly entertained, he had failed of being
understood, because he had not begun his explanation at the beginning.
He therefore set to work to analyse those views and ultimately published
a book on the subject. This book, _The Foundations of Geometry_, was
reviewed by us in _The Monist_ of October, 1891. But now again the author
regards himself as not understood. He rested the positions and arguments
of his book upon certain views of logic and especially of definition,
which depart from the orthodox views, and he misjudged the fullness of
explanation that would therefore become needful. Hence this little essay.
The proper approach to the views of the author is through his doctrine of
definition. Usually definition is regarded as finding its main motive and
utility in the convenience of social converse. The meaning of any term is
regarded as resting not in the choice of him who utters it, but in the
suppositions of those who are addressed. It is true that a license is
accorded to any one upon a sufficient occasion to give a special intent
to some word, but only upon condition that that intent shall be made
sufficiently express, in other words well understood by those addressed.
Hence definition is usually taken to mean the recital or the precision of
the meaning of a term by means of language naturally apt for that end.
There is no good sense in pretending to effect either one of these ends
by language that lacks natural ability on that behalf.
Now Mr. Dixon holds, if we understand him, that conventional usage is
of very subordinate consequence in this matter, that it pertains to
the prerogative of an author to throw upon those whom he addresses the
task of gathering his meanings as best they can; that even when he
professes to explain his meanings he need not seek and employ any plain,
direct speech, but may supply his instruction indirectly: may ask his
audience to solve a problem, or to rightly guess what certain hints
mean; may require them to extract the meaning in question out of a set
of assertions that involve the same in a collateral way only. This he
calls “implicit definition.” It is analogous, he tells us, to an unsolved
equation or set of equations in algebra. So far as we are aware no one
can claim priority of the author in respect to this expedient. He seems
to regard it as of great importance, and proposes by its aid to overcome
the difficulties that environ the fundamentals of geometry.
We think that the author is led to put undue confidence in his implicit
definition, by his peculiar views upon propositions. He holds that all
propositions can, without loss or gain in the meanings as originally
stated, be reduced to statements of strict identity. This done,
propositions can, as he thinks, be operated upon after the fashion usual
with equations. But we submit that between a logical proposition and an
algebraic equation there is a difference that is in general irreducible.
For example take this proposition, Every parent loves children. To alter
this to, Every parent is identical with some [or every] person that loves
children, as is, we think, the prescription of Mr. Dixon, will not
serve; for by reading our identity in the reverse order we have: Some [or
every] person that loves children is identical with every parent.
Mr. Dixon’s views in respect to terms and to the doctrine of denotation
and connotation depart as widely from the suppositions usually held,
as do his views regarding propositions and definition. To follow out
the consequences of his proposed innovations in any adequate fulness is
forbidden to us by lack of space. We feel sure that further reflection
will lead him to much modification of his doctrines.
ρσλ.
OUTLINES OF A PHILOSOPHY OF RELIGION. By _Hermann Lotze_. Edited by _E.
C. Conybeare_, M.A. London: Swan, Sonnenschein, & Co. New York: Macmillan
& Co. 1892. Pp. 176.
This book is an excellent translation of one of the most important works
of a prominent philosopher, who made an unusually strong impression
upon the minds of his contemporaries. Almost every line of this clean,
accurate, and charming translation betrays the translator’s devotion to
the subject, for he has taken the utmost care to bring out the ideas
of the author in the same brilliant style for which Professor Lotze is
justly famous.
The translator says in the preface: “I have completed and venture to
publish the following translation of Hermann Lotze’s _Lectures upon the
Philosophy of Religion_ in the same hope in which it was undertaken by
my late wife, that it may be of use to some who cannot read the German
original, and yet desire a concise statement of the form in which one
of the clearest-minded of our later thinkers put to himself those great
questions—as to the origin and destiny of the spirit of man, as to life
in general, and the meaning of the material universe—which occupy us
all at some time or another, many of us as soon as we have won food and
shelter for our bodies.”
We do not share Mr. Conybeare’s and his deceased wife’s enthusiasm for
the author. Although we are not blind to the great deserts of Professor
Lotze, his amiable personality, the depth of his religious and emotional
nature, the breadth of his scholarly erudition, and the brilliancy of his
ingenious, not to say poetical, presentation of philosophical subjects,
we cannot conceive that his work is come to abide. On the contrary, we
consider his philosophy as antiquated in many respects. He considers
problems that originate from a mere confusion of ideas, as being
insolvable in their nature, and attempts the solution of other problems
with inadequate methods. His thoughts still remind us of the ontological
spirit of past philosophies, and his principles are not in agreement with
positivism and the methods of scientific research.
As an instance, we quote the following passage: “We must ever set aside
any attempt to describe in positive terms, or to construct in thought,
the process by which this absolute being came to be not only one, and
that unconditionally, but at the same time a many of things which
condition one another reciprocally.”