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knowledge of any language can be acquired in a
short time or in a few lessons. Possibly some easy
authors might be read profitably by means of inter-
linear translations before commencins: a series of
such exercises as those of OllendrofF. A o^ood teacher
might impart in this way a knowledge of pronuncia-
tion, the meaning of many words, and some idea of
construction, all of which would be very advanta-
geous in learning the Grammar.

Pupils might begin the study of a language like
French or German by commencing with its Gram-
mar ; but the teacher will find it very difiicult to
interest pupils in the study of the abstract Grammar
of a foreign language, and, besides, it is scarcely
possible to acquire the ability to speak a language
in this manner.

After a course of elementary instruction in which
pupils have learned to speak, read, and write a
foreign language with some facility, and possess a
good knowledge of its Grammar, they may com-
mence with profit the reading of authors. Easy
authors must be first chosen, and afterwards those
more difiicult. Translations should be required and
questions be asked upon the subject-matter in much
the same way as has already been described in
speaking of methods of teaching the Dead Lan-
guages.



CHAPTER III.

INSTRUCTION IN THE FORMAL SCIENCES.

The Formal Sciences treat of the necessary /on7?«
in which truth presents itself or by which truth is
conceived. They may be divided into two great
classes, Mathematies and Logic.

Mathematics is the science of pure quantity. Its
principles have no dependence upon material things.
All its calculations and demonstrations may be made
without reference to them. But its formulae express
the conditions under which matter exists in space
and time.

Logic is the science of pure thought. Its ^Y\n-
ciples are not derived from the manner in which
thinking is done, but they show how it must he done.
Its formulae express the relations between the several
parts of the thinking process.

The sciences of Mathematics and Logic are
called Formal Sciences, because they relate to truth
only in its abstract or ideal condition. The prin-
ciples of both would be true if matter had no
existence.

The following quotation from Sir William Ham-
ilton will show that the object-matter of the F-ormal
Sciences is exhausted by Mathematics and Logic.
He says, " Formal Knowledge is of two kinds ; for
it regards either the conditions of the Elaborative

(294)



THE FOKMAL SCIENCES IN GENERAL. 295

Faculty — the Faculty of Thought Proper — or the
conditions of the Presentations or Representations
of oxteroal things ; that is, the intuitions of Space
and Time. The former of these sciences is Pure
Logic — the science which considers the laws to
which the Understanding is astricted in its elabora-
tive operations, without inquiring what is the ob-
ject — what is the matter, to which these operations
are applied. The latter of these sciences is Mathe-
matics, or the science of Quantity, which considers
the relations of Space and Time, without inquiring
whether there be any actual reality in space or time.
Formal truth will, therefore, be of two kinds — •
Logical and Mathematical."

The Formal Sciences are evolved from certain
ideas and are founded upon certain axioms of which
it is not their province to treat. These belong to
the domain of Philosophy — a Rational Science.

If now we have correctly apprehended the nature
of Mathematics and Logic, methods of instruction
adapted to impart a knowledge of them must have
much in common ; and, therefore, it may be well
before discussing the particular principles of instruc-
tion which a2:)ply to each separately, to speak of the
general principles which apply to both alike.

I. The Formal Sciences in GeneraL

The object-matter of a Formal Science admits
division into three classes, as follows : 1. Definitions
and AxiomB ; 2. Deductions' and Demonstrations ; 3.
Applications. Its applications are not properly a
part of the science ; but they are very important in



296 INSTRUCTION IN THE FOHMAL SCIENCES.

the work of teaching: to illustrate and enforce scien-
tific principles.



'a



1. Definitions and Axioms. — Definitions, in the
sense here intended, express the necessary limita-
tions of particular conceptions. This is their mean-
ing whether they relate to the explication of a term
or to the nature of a thing.

Axioms, in the sense here intended, express
the necessary relations of particular conceptions.
Axioms in Mathematics express relations in space
and time, and Axioms in Logic express the rela-
tions of one part of the thinking process to another.

It is exceedingly important that teachers should
be careful in teaching Definitions in the Formal
Sciences, where no real object can be presented to
illustrate their meaning. We must understand the
meaning of terms before we can use them properly.
An object of thought must stand out before the
mind distinct in itself, and separate from everything
else, before one sure step can be taken in the inves-
tigation of its relations. Imperfect Definitions
vitiate processes of reasoning, and it is to be feared
that much of our teaching is defective in not requir-
ing pupils to define fully, distinctly, and adequately.

The following are the most important laws to
which Definitions must conform. Their meaning is
sufiftciently plain without any explanation.

1st. A definition must be a truthful representation
of the conception defined. It must contain nothing
that does not belong to it.

2d. A definition must be an adequate representa-
tion of the whole conception. It must contain all
that belongs to it.



THE FOKMAL SCIEXCES IX GEXEEAL. 297

3d. All that is contained in a Definition should
be self-evident. A Definition should not need
defining.

4th. A Definition should be an affirmative prop-
osition. Showing what a thing is not does not
always reveal w^hat it is.

5th. A conception cannot be defined by using the
same terms in which the conception is expressed.
In such a case, the unknown terms which darkened
the conception would also darken the definition
of it.

6th. Definitions should be stated in the briefest,
-Strongest, and most expressive form of w^ords.

Let pupils study, closely the Definitions of the
text-book, let them test them, and make others for
themselves. They may commit them to memory,
but it is much more important that they should
understand them. If properly conducted, exercises
in learning Mathematical and Logical definitions
will prove an exceedingly valuable discipline for
the mind.

All reasoning would be impossible without certain
fixed principles from which to start. 'No man could
ever convince another with regard to a truth or an
error, if there were not some common point of
agreement between them. Hence the necessity of
Axioms in the economy of thought. And, although
a formal statement of them is not always made, they
constitute the bases of all sciences, and are espe-
cially prominent in the sciences of Mathematics
and Logic.



298 INSTRUCTION IN THE FOEMAL SCIENCES.

As previously stated, the Formal Sciences borrow
their Axioms from the Rational Sciences. From
Axioms in general it may be their province to select
such as belong to them; but they have nothing to
do in determining the nature of Axioms, the tests
by which they are to be distinguished, their number
or their classification.

Mathematical Axioms are so well known that it
seems unnecessary to enumerate them. They un-
derlie as well the sciences which treat of number as
those which treat of form.

Among Logical Axioms the following may be
named —

1st. All thinking is governed by law.

2d. Every universal is composed of particulars.

3d. Every particular is comprehended in a uni-
versal.

4th. Whatever may be predicated of a universal
may be predicated of all the particulars of which it
is composed.

5th. Whatever may be predicated of all the par-
ticulars composing a universal may be predicated
of the universal.

6th. If two terms agree with the same third term
they agree with each other.

7th. If of two terms, the one agrees and the other
disagrees with the same third term, they disagree
with each other.

This enumeration is not intended to exhaust the
Axioms belonging to the science of Logic, but
simply to show that there are such Axioms.

With respect to pupils old enough to comprehend



THE FORMAL SCIENCES IN GENERAL. 299

Axioms, the method of teaching them presents no
difficulty. Their simple statement will secure as-
sent, and nothing more is needed. The discus-
sion of their use in building up a science belongs
further on.

2. Deductions and Demonstrations. — Deduction
may be defined as the process of drawing out a
particular from a universal truth by simple inspec-
tion or by a single step of reasoning. Demonstra-
tion may be defined as the method of finding new
truths by the process of comparing definitions,
axioms, and established propositions with one
another. The first has the form of a direct infe-
rence or a single syllogism, while the second con-
sists of a train of reasoning or a series of syllo-
gisms. As the method of both is substantially the
same, both may be considered together under the
name Demonstration. This may be the case also if
Deduction be used to designate a general method of
reasoning, and Demonstration, an application of it.

In Pure Mathematics, all that cannot be learned
directly, by intuition, must be learned by Demonstra-
tion. Inductive reasoning has no place in Mathe-
matics.

In Pure Logic the same is true, for although an
Inductive Syllogism may be used, yet, in a pure
form, the conclusion must be just as much a posi-
tive truth as it is in a Deductive Syllogism. In Ap-
plied Logic as in Applied Mathematics, the conclu-
sions are not always either certain or exact.

The Demonstrations of Logic consist essentially
in showing the relations between the conclusions



^00 INSTRUCTION IN THE FORMAL SCIENCES.

of syllogisms and their premises. In general, but a
single step is necessary to be taken, away from the
first principles upon which the science rests.

Demonstrations in Mathematics, although like
those of Logic in the circumstance that they con-
cern pure conceptions and not the conceptions of
material objects, differ from them in several parti-
culars. In Mathematics it is not the doctrine of the
sjdlogism as an exposition of the laws of thought
that is to be demonstrated, but the relations of num-
bers and forms by means of syllogisms. Mathema-
tics is a formal application of Logic to the concep-
tions of time and space. The student of Mathe-
matics therefore cannot select any premises but he
must select the right premises, lie cannot often
find the truth he seeks at the end of a single syllo-
gism, but must frequQjitly trace it through a long
series of syllogisms.

So far as methods of teaching them are concerned,
however, the Demonstrations of Mathematics and of
Logic may be considered together ; and the point now
is to find the governing principles of those methods.

One who would become skilful in demonstrating
must attend to the following rules : —

1st. Understand the proix)sition to be demon-
strated and its relations to the definitions and
axioms upon which it depends and to the propo-
sitions which may have preceded or are to follow it.

2d. Observe a rigid logical order in the successive
steps of the demonstration.

3d. Argue closely and clearly.

4th. Attain positive conclusions.

5th. Use appropriate language.



THE FORMAL SCIENCES IN GENERAL. 801

These rules are sufficiently obvious without ex-
planation. If any one of them is disregarded no
perfect demonstration can be secured. They apply,
however, to the demonstration of independent prop-
ositions. The object-matter of a Formal Science
is composed of several kinds of propositions w^hicli
must be divided according to certain laws, among
which the following are the most important —

1st. The divisions should exclude one another.

2d. The order of the divisions should be deter-
mined by their logical relations.

3d. In the arrangement of particular propositions
the simple and the independent should precede the
complex and the dependent.

A child first learns to reason in connection with
objects. The steps he takes are very short and very
easy. Properly instructed, his skill rapidly improves
until he can appreciate the abstract relations of
things or thoughts. For first efibrts at formal
demonstration, easy propositions should be given
him, and then those more difiicult. Eventually he
may be able to follow the most abstruse reasoning
incident to Mathematics or Logic.

If teachers reason skilfully, their pupils will be
likely to be benefited by their example.

Practice in detecting the diiFerent kinds of fal-
lacies in arguments will be a good exercise.

A wise teacher will lead his j)upils to discover
their own errors in reasoning rather than correct
them himself The method Socrates so successfully
practiced against the Sophists of his day may be
just as usefully applied now.

26



802 INSTEUCTION IN THE FORMAL SCIENCES

3. Applications. — Mind and matter are correla-
tive. For every ideal truth there must be a real
thing — for every form of thought there must be
matter to fill the form, or the creation would not
harmonize. The world within must envisage the
world without, or God could not have created it.
Hence all abstract formulae must be adapted to some
concrete phenomena ; or every Formal Science must
have its Applications.

Mathematical principles may be applied to all
things that appear under the conditions of space and
time.

Logical principles are of universal application, for
all things may be thought about.

In making an application of Formal truths three
things are necessary : 1st, To have attained a clear
conception of the truths themselves ; 2d, To have
carefully observed and colligated facts ; 3d, To be
able to apply the right ideas to the right facts.

Formal truths so far as they are not axiomatic are
attained by the process of demonstration as already
shown.

The collection and colligation of facts belong to
the department of Empirical science and are to be
treated of in the proper place.

The Applications of the Formal Sciences consist
in fitting the right ideas to the right facts. This
may be more a work of art than of science, but
nature presents no more important work for human
efibrt to perform. He who deals only with Formal
thought is apt to become impractical and visionary.
He may build up systems which seem beautiful, but
at a touch they vanish into the thin air of which



MATHEMATICS. 303

tliej were composed. He who absorbs all liis time
in collecting facts, who with eyes cast clown to earth
never looks heavenward, but occupies himself in
examining animals, and plants, and .stones, and
fossils, until the eye of faith grows dim and matter
seems omnipotent, does even less for himself and
mankind than the speculative dreamer. But he
who accustoms himself to apply the right ideas to
the right facts, to prove his reasonings, to verify his
theories, will be in no danger of becoming an im-
practical idealist on the one hand or a coarse mate-
rialist on the other. He finds that every fact rests
in an idea; that each jewel has its casket in the
crown of nature ; that forms of thought existed in
the God-mind and He made matter to fill them.

As hints to teachers giving instruction in the
Applications of the Formal Sciences, it may be
stated that sometimes facts may be given and pupils
required to find principles, and sometimes principles
may be given and pupils required to find facts ; that
easy applications should always precede those more
difficult; that numerous examples and abundant
illustrations should be furnished, arranged both with
reference to specific principles and miscellaneously ;
and that close explanations should be exacted in
all cases.

IL Mathematics.

After what has now been said respecting the
nature of the Formal Sciences in General and the
methods of teaching them, it is not deemed necessary
to treat specially of methods of teaching Mathe-
matics. Besides, what should be said specifically



304 INSTRUCTION IN THE FORMAL SCIENCES.

with respect these methods will appear in speaking
of methods of teaching Arithmetic, Algebray and
Geometry.

Something will be expected, however, in regard
to the advantages to be derived from the study of
Mathematics.

Mathematics has occupied a prominent place in
courses of instruction for the j^oung from the earliest
times. Some have thought that its disciplinary ad-
vantages were greater than could be derived from
any other branch of instruction, while others have
maintained that its study was rather hurtful than
otherwise. In the hope of contributing something
toward the settlement of the question, it is proposed
here briefly to consider the value of Mathematical
studies: 1. In the?nselves ; 2. In their objective rela-
tions ; 1. In their effects upon the mind.

1. The Value of Mathematical Studies in themselves.
— All truth is worthy of study for its own sake. To
decide otherwise would be to question the wisdom
of God who created it. All kinds of truth, however,
may not be of equal value, and the inquiry might
be made as to the relative value of Mathematical
truth. Truth may be divided into three kinds:
ideal truth, formal truth, and real truth. Ideal truth
is the truth which we know by simple intuition,
which furnishes the basis upon which all other truth
rests, and the criteria by which it is judged. Formal
truth expresses the necessary forms in which all
truth presents itself or by wdiich it is conceived.
Real truth is the harmonious relation between
things or between thought and things. In compar-



MATHEMATICS. 305

ing tlie value of these several kinds of triitli, no
reasons appear why formal truth is not of as much
worth as. either of the other kinds. It seems as
noble in itself, is of as much use, and manifests as
fully the glory of the Creator. But formal truth is
of two kinds. Mathematical and Logical, and we
seek to know only the value of Mathematical truth.

As has been already shown, Logic contains a
larger body of truth than Mathematics and is of
wider application, but I can find no standard by
which it can be determined that a truth in the one
science is more valuable than a truth in the other.

Mathematics is a noble science. Many of its
principles are exceedingly beautiful, and some of
them almost sublime. It has won the admiration
of great men in all ages, and his education must be
considered incomplete among whose acquisitions a
knowledge of Mathematics is not found.

2. The Value of 3IatJiematical Studies in their Ob-
jective Relations. — 'Eo other science is so generally
connected with the aifairs of business as Mathe-
matics. Arithmetic is used in keeping accounts and
in all the transactions of buying and selling. In
connection with Geometry, it is used in all me-
chanical employments. Geometry, Algebra, Trigo-
nometry, Conic Sections, &c., cannot be dispensed
with in the construction of machinery, nor in any
of the departments of Engineering. All this, how-
ever, is so generally understood that it seems hardly
necessary to mention it.

Mathematics is the hand-maid of the sciences.
Working by means of this potent instrument mod-

26*



306 INSTRUCTIOX IX THE FORMAL SCIENCES.

erii pliilosopliers Lave been able to make rapid
advances in many departments of physical science.
To it, we are indebted for what is most valuable in
Mechanics, Optics, Pneumatics, Thermotics, As-
tronomy, and other sciences like these. It has its
uses in Geography, Chemistry, Geology, and even
Political Economy. Matter everywhere presents
itself to us under Mathematical conditions. Laws
that find their expression in Mathematics rule all
that moves in the heavens, all that ilies in the air, all
that swims in the waters, all that springs up from
the earth or that falls upon its surface, and the firm
earth itself. Yonder yellow leaf that is lifted from
its stem by the autumn wind, and after innumerable
gyrations in the air, falls upon the surface of the
stream and is borne onward by the current, makes
no movement but in obedience to such laws.
^Mathematics has principles great enough to sweep
the Universe, and hold suns and planets in their
grasp, and delicate enough to poise the smallest
atom on a point much too fine for human con-
ception.

3. Tlte Value of Mathematieal Studies in their
Effects upon the Mind. — One of the most important
objects of study is to secure mental discipline.
"What is the value of Mathematics in this respect ?
In discussing this point. Sir William Hamilton says :
"If we consult reason, experience, and the common
testimony of ancient and modern times, none of our
intellectual studies tend to cultivate a smaller number
of the faculties, in a more partial or feeble manner than
Mathematics.'' In proof of this opinion, he quotes



MATHEMATICS. 807

a large lumiber of authorities, a few of wliom I sliall
take the liberty of citing here :

" Bernhardt a celebrated Prussian educator, says :
'It is asked — Do 3fathematics awaken the judgment,
the reasoning faculty, and the understanding to an all-
sided activity? We are compelled to answer — No.'

"'This also shows me,' says Goethe, 'more and
more distinctly, what I have long in secret been
aware of, that the cultivation afforded by the Mathe-
matics is, in the highest (^egree one-sided and con-
tracted.'

'•''Descartes stated in a letter in 1630, 'That he had
renounced the study of Mathematics for many years,
and that he was anxious not to lose any more of his
time in the barren operations of Geometry and
Arithmetic, studies which never lead to anything
important.'

"'Thus it is rare,' says Pascal, 'that Mathemati-
cians are observant, or that observant minds are
Mathematical.'

" Dugald Stewart says, ' When the Mathematician
reasons upon subjects unconnected with his favorite
studies, he is apt to assume, too confidently certain
intermediate principles as the foundation of his
arguments.' And again, ' I have never met with
a mere Mathematician who was not credulous to a
fault.'

*•' Bayle says, 'It cannot be disputed, that it is
rare to find 'much devotion in persons who have
once acquired a taste for^ the study of Mathe-
matics.'

"i>e Stael, to the same effect, 'The Mathematics
lead us to lay out of account all that is not proved.' "



308 INSTEUCTION IN THE FORMAL SCIENCES.

Sir "William's argument against the use of Mathe-
matics as a discipline for the mind is summed up in
the following sentence. "We are thus disqualified
for observation, either internal or external^ for abstrac-
tion, and generalization, for common reasoning, nay,
disposed to the alternative of blind credulity or of
irrational skepticism. '' This argument he supports
at much length with great ability and greater learn-
ing. When closely examined, however, the whole
argument will be found to bear not so much against
the use of Mathematics as a disciplinary study in
its proper place, as against the injudicious claims
advanced in its behalf in that regard.

'Eo one should claim for the study of Mathema-
tics that it disciplines the ordinary powers of obser-
vation. It is not concerned with either material or
mental phenomena. Its province is not to collect
facts. Pure Mathematics is quite indifferent to the
existence of matter. There is a kind of observing
power, how^ever, wdiich the study of Mathematics
does cultivate — that power which sees truth in de-
finitions and axioms and without which all demon-
strations w^ould be blind and unproductive of fruit.

Abstraction and generalization as used in the Em-
pirical sciences have no place in Mathematics, and
therefore that study cannot develop and strengthen
the mental powers by which those processes are
performed. But in another sense all of Pure Ma-
thematics is abstract, and • surely Mathematical
truths admit classification and generalization. In
every branch of Mathematics there are forms of
demonstration which are true in particular cases,
and there are others which must be true in all



MATHEMATICS. 309

cases. In teaching, pupils nriay be made to advance
from particular examples to general principles.

If hy common reasoning is meant that kind of
reasoning in which the conclusions arrived at are
probable but not positive, it must be admitted that



Online LibraryJames Pyle WickershamMethods of instruction .. → online text (page 19 of 31)