thought, and the Idols of the Cave the perturbations resulting from
individual idiosyncrasies, there are other Idols "formed by the
intercourse and association of men with each other," which Bacon calls
"Idols of the Market Place, on account of the commerce and consort of
men there." By reason of its manifold and necessary imperfections--its
looseness, variability, ambiguity, and inadequacy--the language we are
forced to employ for the embodiment and interchange of ideas plays
ceaseless havoc with our thought, not only introducing confusion and
misconception into discussion, but often, "like the arrows from a
Tartar bow," reacting seriously upon our minds. A large part of the
vocabulary to which we must perforce have recourse, even when dealing
with the most abstruse and delicate subjects, is made up of words
taken over from vulgar usage and pressed into higher service; they
carry with them long trains of vague connotations and suggestions; the
superstitions of the past are often imbedded in them; no one can ever
be absolutely certain of their intellectual values. While, therefore,
they may do well enough for the rough needs of daily life, they prove
sadly defective when required for careful and exact reasoning. And
even with that small and comparatively insignificant portion of our
language which is not inherited from popular use, but fabricated by
philosophers themselves, the case is not much better. Every word, no
matter how cautiously employed, inevitably takes something of the tone
and color of the particular mind through which it passes, and when put
into circulation fluctuates in significance, meaning now a little
more and now a little less.[25] What wonder, then, that "the high and
formal discussions of learned men" have so often begun and ended in
pure logomachy, and that in discussions which are neither high nor
formal and in which the disputants talk hotly and carelessly the
random bandying of words is so apt to terminate in nothing beyond the
darkening of counsel and the confusion of thought?
[Footnote 25: The need of a language of rigid mathematical
precision for the purposes of philosophic thought and discussion
has long been the subject of remark. Hence Bishop Wilkins's Essay
toward a real character and a philosophic language (1668), and
the earlier Ars Signorum of George Dalgarno--boldly presented by
its inventor as a "remedy for the confusion of tongues, as far as
this evil is reparable by art." We may give these ingenious
authors full credit for the excellent intentions with which they
set out on impossible undertakings. A philosophic language may
perhaps be attained in the millennium, but then probably it will
be no longer needed. Meanwhile readers interested in the history
of the mad scheme called Volapük may find some curious matter in
these rare works.]
Bacon notes two ways particularly in which words impose on the
understanding--they are employed sometimes "for fantastic suppositions
... to which nothing in reality corresponds," and sometimes for actual
entities, which, however, they do not sharply, correctly, and
completely describe. The eighteenth century speculated at length on a
state of Nature and the social contract, unaware that it was deluding
itself with unrealities, and we have not yet done with such
abstractions as the Rights of Man, Nature (personified), Laws of
Nature (conceived as analogous to human laws), and the Vital
Principle. The more common and serious danger of language, however,
lies in the employment of words not clearly or firmly grasped by the
speaker or writer--words which, in all probability, he has often heard
and used, and which he therefore imagines to represent ideas to him,
but which, closely analyzed, will be found to cover paucity of
knowledge or ambiguity of thought. Cause, effect, matter, mind, force,
essence, creation, occur at once as examples. Few among those who so
glibly rattle them off the tongue have ever taken the trouble to
inquire what they actually mean to them, or whether, indeed, they can
translate them into thought at all.
Among the Idols of the Market Place we must also class the evils
arising from the tendency of words to acquire, through usage and
association, a reach and emotional value not inherent in their
original meanings. This is what Oliver Wendell Holmes happily
described as the process of polarization. "When a given symbol which
represents a thought," said the Professor at the Breakfast Table, "has
lain for a certain length of time in the mind it undergoes a change
like that which rest in a certain position gives to iron. It becomes
magnetic in its relations--it is traversed by strange forces which
did not belong to it. The word, and consequently the idea it
represents, is _polarized_." The larger part of our religious and no
small portion of our political vocabulary consist of such polarized
words--words which, on account of their acquired magnetism, unduly
attract and influence the mind. We can never hope to think calmly and
clearly while the very symbols of our thoughts thus possess a kind of
thaumaturgic power over us, which in turn readily transfers itself to
our ideas.
If, then, "words plainly force and overrule the understanding and
throw all into confusion and lead men away into numberless empty
controversies and idle fancies," it behooves us to watch closely the
interrelations of language and thought. To put it in the vernacular,
we must at all times make sure that we know what we are talking about
and say what we mean. To this end the study of language itself is
useful, but the habits of precise thought and expression will never be
acquired by linguistic exercise alone. To use no word without a
distinct idea of what it means to us as we speak or write it; to
check, when necessary, the process of thought by constant redefinition
of terms; to depolarize all language that has become, or threatens to
become, magnetic, thus translating familiar ideas into "new, clean,
unmagnetic" phraseology, these may be set down as first among the
rules to which we should tolerate no exception.
We now come to the last group of Idols--those "which have immigrated
into men's minds from the various dogmas of philosophies, and also
from wrong laws of demonstration." These Bacon calls Idols of the
Theater, "because in my judgment all the received systems are but so
many stage-plays, representing worlds of their own creation after an
unreal and scenic fashion." And perhaps this conceit carries further
than Bacon himself intended, for it not only suggests the
unsubstantial character of philosophic speculations, but also reminds
us how, in the world's history, these airy fabrics have succeeded each
other as on a stage, some to be hissed and some applauded, but all
sooner or later to drop out of popular favor and be forgotten.
Dealing with these Idols of the Theater, or of Systems (of which there
are many, "and perhaps will be yet many more"), Bacon takes the
opportunity of criticising, briefly but incisively, the methods and
results of ancient and mediæval philosophers. His classification of
false systems is threefold: The sophistical, in which words and the
finespun subtilties of logic are substituted for "the inner truth of
things"; the empirical, in which elaborate dogmas are built up out of
a few hasty observations and ill-conducted experiments; and the
superstitious, in which philosophy is corrupted by myth and
tradition. Under the first head, Bacon again instances Aristotle, whom
he accuses of "fashioning the world out of categories"; under the
second he glances especially at the alchemists; and under the third he
refers to Pythagoras and Plato. To follow Bacon into these historic
issues does not belong to our present purpose. Suffice it to notice
the continued vitality of these three classes of speculative error.
Bacon's judgment of Aristotle--that "he did not consult experience as
he should have done, in order to the framing of his decisions and
axioms; but, having first determined the question according to his
will, he then resorts to experience, and, bending her into conformity
with his placets, leads her about like a captive in a procession"--is
at least equally applicable to thinkers like Hegel and his followers.
Empiricism has by no means been eliminated from the scientific or
would-be scientific world. And as for the philosophy which is
corrupted by myth and tradition, the countless attempts that are still
made to "reconcile" the facts of science with the data and
prepossessions of theology are enough to prove that, _mutato nomine_,
the methods of Pythagoras and Plato and of those who in Bacon's day
sought "to found a system of natural philosophy on the first chapter
of Genesis, on the book of Job, and other parts of the sacred
writings," are as yet far from obsolete.
It is hardly necessary to call attention to the fact that there is a
close similarity between systematic empiricism and some of the dangers
brought out in connection with the Idols of the Tribe, for in each
case stress must be laid on the tendency to generalize hastily, depend
on scattered and inadequate data, and seek for light in the
"narrowness and darkness" of insufficient knowledge. This matter is
important only as showing how a common weakness may be caught up and
dignified in a philosophic system and rendered more dangerous by the
adventitious weight and influence which it gains thereby. Another
point, not distinctly dealt with by Bacon, calls, however, for special
remark. While the various Idols of the Theater, or of Systems,
exercise their own peculiar and characteristic influences for evil,
they all tend to the debasement of thought by reason of the authority
which they gradually acquire. Associated with great names, promulgated
by schools, officially expounded by disciples and commentators, they
finally settle into a creed which is regarded as having oracular and
dogmatic supremacy. The formula "Thus saith the Master" closes
discussion. Not the fact itself, but what this or that teacher has
said about the fact, comes at last to be the all-important question.
In the condition of mind thus engendered there is no chance for
intellectual freedom, self-reliance, growth. Lewes related an anecdote
of a mediæval student "who, having detected spots in the sun,
communicated his discovery to a worthy priest. 'My son,' replied the
priest, 'I have read Aristotle many times, and I assure you that there
is nothing of the kind mentioned by him. Go rest in peace, and be
certain that the spots which you have seen are in your eyes, and not
in the sun.'"[26] Such an incident forms an admirable commentary on
the saying of the witty Fontenelle that Aristotle had never made a
true philosopher, but he had spoiled a great many. The position
assumed is simple enough: Aristotle _must_ be right, therefore
whatever does not agree with the doctrines of the Stagirite must be
wrong. Are your facts against him, then revise your facts. Come what
may of it, you must quadrate knowledge with accepted system. Here is
the theological method in a nutshell. And the theological method has
only too often been the method also of the established philosophic
schools.
[Footnote 26: History of Philosophy, vol. ii, pp. 95, 96.]
In our own relations with these Idols of the Theater the first and
last thing to remember is that all systems are necessarily partial and
provisional. "They have their day and cease to be," and at the best
they only mark a gradual progress toward the truth. There can be no
finality, no closing word authoritatively uttered. Our attitude toward
the systems of the past and the present, toward long-accepted
traditions, and dogmatically enunciated conclusions, must be an
attitude of firm and steady--of respectful, it may be, but still firm
and steady--independence. We must resist the tendency to passive
acquiescence, and endeavor to combine with generous hospitality to all
ideas the habit of not accepting anything merely because it is stated
_ex cathedra_, or is backed by an influential name, or can "plead a
course of long observance for its use." Perhaps to wean ourselves from
this particular form of idolatry there is nothing so helpful as a wide
and constant study of the history of thought. The pathway of
intellectual development is strewn with outgrown dogmas and exploded
systems. How fatuous, then, to accept, whole and untested, the
doctrine of any master, new or old, believing that his word will give
us complete and undiluted truth!
* * * * *
So much, then, we may say with Bacon "concerning the several classes
of Idols and their equipage, all of which must be renounced and put
away with a fixed and solemn determination, and the understanding
thoroughly freed and cleansed; the entrance into the kingdom of man,
founded on the sciences, being not much other than the kingdom of
heaven, whereinto none may enter except as a little child." It may
perhaps be urged that the result of such a survey as we have taken of
the obstacles to clear thought is to leave the mind dazed and
discouraged, partly because the suggestions made for the conquest of
these obstacles, though easily formulated in theory are difficult and
sometimes impossible in practice, and partly because the general if
not expressed tendency of our analysis is (it may be said) in the
direction of that Pyrrhonic skepticism which "doomed men to perpetual
darkness." To the former objection I have only to reply that it is one
to which all discussions of the principles and problems of conduct are
necessarily open. "If to do were as easy as to know what were good to
do, chapels had been churches, and poor men's cottages princes'
palaces."[27] None the less, to state as lucidly as we can what were
good to do under certain circumstances is properly regarded as part of
the business of ethics. The other point is touched upon by Bacon
himself in words which it would be impertinent to seek to better: "It
will also be thought that by forbidding men to pronounce and set down
principles as established until they have duly arrived through the
intermediate steps at the highest generalities, I maintain a sort of
suspension of the judgment, and bring it to what the Greeks call
_acatalepsia_--a denial of the capacity of the mind to comprehend
truth. But in reality that which I meditate and propound is not
_acatalepsia_, but _eucatalepsia_; not denial of the capacity to
understand, but provision for understanding truly; for I do not take
away authority from the senses, but supply them with helps; I do not
slight the understanding, but govern it. And better surely it is that
we should know all that we need to know, and yet think our knowledge
imperfect, than that we should think our knowledge perfect, and yet
not know anything we need to know."
[Footnote 27: This quotation is _not_ from Bacon.]
MATHEMATICS FOR CHILDREN.
BY M. LAISANT.
Except with persons having specially favorable surroundings, I believe
that the vast majority of parents have a feeling of dread at the
thought of putting their children to the study of mathematics. They
know that the child must learn something about it in order to pass his
examinations; but with this knowledge goes an apprehension of loading
his mind with those ideas which are so complicated and hard to
acquire, and we put off the dreaded moment of setting him to work as
late as possible.
While I believe it is wise to spare the child all useless overwork, I
am persuaded also that the best way of sparing him is not to shrink
from initiating him into hard work, if that can be done in a rational
way.
I regard all the sciences as, at least to a certain extent,
experimental, and, notwithstanding the views of those who would regard
the mathematical sciences as a series of operations in pure logic,
resting upon strictly ideal conceptions, I believe that we may affirm
that there does not exist a mathematical idea that can enter our brain
without the previous contemplation of the outer world and the facts it
offers to our observation. This affirmation, the discussion of which
now would carry us too far, may help to a clear idea of the way we
should try to convey the first mathematical ideas to the mind of the
child.
The outer world is the first thing the child should be taught to
regard and concerning which he should be given as much information as
possible--information which he will have no trouble in storing, we may
well believe, and from this outer world the first mathematical notions
should be borrowed; to these should succeed later an abstraction,
which is less complicated than it seems.
Our primary teaching of arithmetic now follows in the tracks of that
of grammar, as we might as well say that the teaching of grammar
follows in the tracks of that of arithmetic. That is, in either case
we teach the child a number of abstract and confusing definitions
which he can not comprehend, imposing on him a series of rules to
follow under the pretext of giving him a good practical direction, and
we force him to learn and memorize these rules whether they are good
for anything or not.
When the child has grown older he is given two or three short lessons
a week in science, nine tenths of which, with his fleeting memory, he
forgets before the next week's lessons come on. He can not relish
anything that is taught him in that way, and it would be vastly better
to give him no scientific ideas at all than to scatter them around in
such a way, for all teachers agree that a fresh pupil is more easily
dealt with and can be taught more satisfactorily and thoroughly than
one who has been mistaught.
When the student has passed through it all and has established himself
in life he is apt to look back upon his experiences under such
teachings in no very amiable mood, and to regard such matters in the
light of barriers that were set up to prevent his getting his diploma
with too little work; and even if his profession is one that calls for
applications of mathematics he prepares himself with sets of formulas
that enable him to dispense with the imperfect instruction he has
received.
When we think of giving a child a mathematical education we are apt to
ask whether he has special aptitudes fitting him to receive it. Do we
ask any such questions when we talk of teaching him to read and write?
Oh, no! we all acknowledge that reading and writing are useful,
practical, and indispensable arts, which every human being not infirm
or defective should learn. Now, elementary mathematics, which
represents a tolerably extended equipment, is no less useful and
indispensable than the knowledge of reading and writing, and I assert
further, what may seem paradoxical to many, that it can be assimilated
with much less fatigue than the earliest knowledge of reading and
writing, provided always that instead of proceeding in the usual way
and giving lessons bristling with formulas and rules, appealing to the
memory, imposing fatigue, and producing nothing but disgust, we adopt
the philosophical method of conveying ideas to the child by means of
objects within reach of his senses. The teaching should be wholly
concrete and applied only to the contemplation of external objects and
their interpretation, and the instruction should be given continually,
especially during the primary period, under the form of play. Nothing
is easier than this, then, in arithmetic; for instance, to use dice,
beans, balls, sticks, etc., and by their aid give the child ideas of
numbers.
Do we do anything of this kind? When I was taught to read and write I
knew how to write the figure 2 before I had any idea of the number
two. Nothing is more radically contrary to the normal working of the
brain than this. The notion of numbers--up to 10, for example--should
be given to the child before accustoming him to trace a single
character. That is the only way of impressing the idea of number
independently of the symbol or the formula which is only too ready to
take the place in the mind of the object represented by it.
When a child has learned to count through the use of such objects as I
have mentioned he may be taught what is called the addition table.
This table can be learned by heart easily enough, but when we reach
the multiplication table we come upon one of the tortures of
childhood. Would it not be simpler and easier to make the children
construct these tables, instead of making them learn them?
[Illustration: FIG. 1.]
Let us first take the addition table, and suppose that we trace ten
columns on suitably ruled paper, at the top of which we write the
first ten numbers, for example, and then write them again at the
beginning of a certain number of horizontal lines (Fig. 1). Let us
suppose, too, that we have a box divided into compartments arranged
like the squares in our table, into which we put heaps of balls,
beans, or dice corresponding to the numbers indicated in the table.
The child will take, for example, two balls from one compartment and
three from another, will put them together and place his five balls in
the case corresponding with the point where the lines of two and three
will meet, and will thus gradually accustom himself to the idea that
two added to three are equal to five, four and two to six, etc.,
before he knows how to write the corresponding figures. As soon as he
has learned how to write them he can himself make the table with
figures (Fig. 2), showing that one and one make two, one and three
four, etc.
[Illustration: FIG. 2.]
This will be all the easier for him because he will only have to write
the figures in their order in the lines and the columns. This
furnishes an excellent writing exercise after the children have begun
to write figures, and affords besides a certain method of teaching
them the addition table up to nineteen at least. I insist that all
this can be done even before the child knows how to write the figures
by means of an arrangement like a printer's case, and that it will be
as a play, rather than a study, to the child. Hardly anything more
will be required than to bring the toy to the child's notice and leave
him to himself after he has been started with it, and he will get
along the faster the less he is bothered.
A similar process may be adopted with the multiplication table. With a
case like the other, it is only necessary to tell the child that if he
wants to know how much are three times four he has only to make heaps
of four things each, take three of them and put them in the box at the
intersection of the line three and the column four. If he can write
the figures he will write 12, instead of gathering up the twelve
objects that represent the product. When he has played at this for
some time he may become acquainted with all the products up to ten
times ten or beyond without having to make any abnormal effort of
memory.
The idea of numeration, which is usually put off till a later period,
should also be given at the beginning. Children soon understand the
decimal numeration and learn to write 10 for ten, and other numbers
composed of one of the nine ciphers and zero. But the fact which,
however, though quite essential to know, receives very little
attention is that there is nothing particular about this number ten,
and that systems of numeration can be devised resting on any basis
that may be taken; that the principle of every system of numeration
consists in taking a certain number of units and grouping them. Take,
for example, a system having five as its basis. All the numbers of
such a system can be represented with the figures 1, 2, 3, and 4, the
symbol 10 standing in this case for five. To construct a number we
have only to group the units by fives and observe the result.
To learn decimal numeration by this process we put tens of objects
into little boxes, tens of little boxes into larger ones, and so on.
The child can in this way acquire an exact idea of the units of
successive order in any system that may be desired.
This method of teaching was developed in a remarkable way about
thirty years ago by Jean Macé in a little book entitled
_L'Arithmétique du Grand-Papa_--Grandpa's Arithmetic--which made some
impression when it appeared, but has been substantially forgotten.
In this method I attach much importance to giving these exercises a
form of play. I believe that nothing in primary instruction should
savor of obligation and fatigue. It would, on the other hand, be
better to try to induce the child to desire himself to go on, and it
would always be well to try to give him the illusion, in all stages of
instruction, that he is the discoverer of the facts we wish to impress
upon his mind.