CONTINUATION OF THE SAME SUBJECT.
269. It is even ridiculous to say that the cognitions of the sublimest
philosophers may be reduced to this equation: A is A. This, absolutely
speaking, is not only false, but contrary to common sense; but it is
neither contrary to common sense nor false to say that all cognitions
of mathematicians are perceptions of identity, which, presented under
different conceptions, undergoes infinite variations of form, and so
fecundates the intellect and constitutes science. For the sake of
greater clearness we will take an example, and follow one idea through
all its transformations.
270. The equation circle = circle (1) is very true, but not very
lucid, since it serves no purpose, because there is identity not only
of ideas but likewise of conceptions and expression. To have a true
progress in science we must not only change the expression, but also
vary in some way the conception under which the identical thing is
presented. Thus, if we abbreviate the above equation in this form: C
= circle (2), we make no progress, unless with respect to the purely
material expression. The only possible advantage of this is to assist
the memory, as instead of expressing the circle by a word, we express
it by its initial letter, C. Why is this? Because the variety is in
the expression, not in the conception. If, instead of considering the
identity in all its simplicity in both members of the equation, we
give the value of the circle with reference to the circumference, we
shall have C = circumference × 1/2 R (3), that is, the value of the
circle is equal to the circumference multiplied by one-half the radius.
In the equation (3) there is identity as in (1) and (2), because it
is affirmed in it that the value expressed by C is the same as that
expressed by circumference × 1/2 R; just as in the other two it is
expressed that the value of the circle is the value of the circle. But
is this equation different from the other two? It is very different.
What is the difference? The first two simply express the identity
conceived under the same point of view; the circle expressed in the
second member excites no idea not already excited by the first; but in
the last, the second member expresses the same circle indeed, but in
its relations with the circumference and radius; and, consequently,
besides containing a sort of analysis of the circle, it records the
analysis previously made of the idea of the circumference in relation
to the idea of radius. The difference is not, then, solely in the
material expression, but in the variety of conceptions under which the
same thing is presented.
Calling the value of the relation of the circumference with the
diameter N, and the circle C, the equation becomes: C = NR^2 (4). Here,
also, there is identity of value; but we discover a notable progress in
the expression of the second member, in which the value of the circle
is given, freed from its relations with the value of the circumference,
and dependent solely on a numerical value, N, and a right line, which
is the radius. Without losing the identity, and only by a succession of
perceptions of identity, we have advanced in science, and starting from
so sterile a proposition as circle = circle, we have obtained another,
by means of which we may at once determine the value of any circle from
its radius.
Leaving elemental geometry, and considering the circle as a curve
referred to two axes, with respect to which its points are determined,
we shall have Z = 2Bx-x^2(5); Z expressing the value of the ordinate;
B the constant part of the axis of abscissas; and x the abscissa
corresponding to Z. We have here a still more notable progress of
ideas: in both members we now express the value, not of the circle,
but of lines, by which we may determine all points of the curve;
and we easily conceive that this curve, which was contained in the
figure whose properties we determined in elemental geometry, may be
conceived under such a form as belongs to a genus of curves, whereof it
constitutes a species by the particular relations of the quantities 2x
and B; thus modifying the expression by adding a new quantity, combined
in this or that manner, we may obtain a curve of another species. If,
therefore, we would determine the value of the surface contained
in this circle, we may consider it, not solely with respect to the
radius, but to the areas comprised between the various perpendiculars
the extremities of which determine points of the curve and are called
ordinates. It results from this, that the same value of the circle may
be determined under various conceptions, although this value is at all
times identical; the transition from one conception to another is the
succession of the perceptions of identity presented under different
forms.
Let us now consider the value of the circle dependent on the radius:
this will give us C = function x (6). This equation enables us to
conceive the circle under the general idea of a function of its radius,
or of x, and consequently authorizes us to subject it to all the
laws to which a function is subject, and leads us to the properties
of their differentials, limits, and relations. By this equation we
enter into infinitesimal calculus, the expressions of which present
identity under a form which records a series of conceptions of long
and profound analysis. Thus, expressing the differential of the circle
by dc, and its integral by S. dc, we shall have C = S. dc, (7), an
equation in which are expressed the same values as in circle = circle,
but with this difference, that the equation (7) records immense
analytical labors: it results from a long succession of conceptions of
integral calculus, of differentials, and limits of the differentials
of the functions, of the application of algebra to geometry, and of
a multitude of elementary geometrical notions, algebraical rules and
combinations, and of whatever else was needed to arrive at this result.
Therefore, when we find the integral of the differential, and obtain
by integration the value of the circle, it would clearly be most
extravagant to affirm that the integral equation is nothing more than
the equation circle = circle; but it is not so to say that at bottom
there is identity, and that the diversity of expression to which we
have come, is the result of a succession of perceptions of the same
identity presented under different aspects. Supposing the conceptions,
through which it has been necessary to pass, to be A, B, C, D, E, M,
the law of their scientific connection may be thus expressed: A = B, B
= C, C = D, D = E, E = M; therefore A = M.
271. What we have just explained cannot be well understood unless
we recall some characteristics of our intellect, in which is found
the reason of so great anomalies. Our intellect is so weak as to
perceive things only successively: only after much study does it see
what is contained in the clearest ideas. Hence a necessity, to which
corresponds with admirable harmony a faculty that satisfies it: the
necessity is of conceiving under various, and different, as well as
distinct, forms, even the simplest things: the faculty is that of
decomposing the conception into many parts, and multiplying in the
order of ideas what in that of reality is only one. This faculty of
decomposition would be useless were not the intellect, in passing
through the succession of conceptions, to find means of connecting and
retaining them: otherwise it would continually lose the fruit of its
labors; it would slip from its hands as fast as it grasped it. Happily
it has this means in signs either written, spoken, or thought; those
mysterious expressions which at times not only designate an idea, but
also are the compendium of the labors of a whole life, and perhaps of
a long series of ages. When the sign is presented to us, we do not see
certainly and with full clearness all that it expresses, nor why the
expression is legitimate; but we know confusedly the meaning therein
contained; we know that in case of necessity, it is enough for us to
follow the thread of the perceptions through which we have passed,
thus going back even to the simplest elements of science. In making
calculations, the most eminent mathematician does not clearly see
the meaning of the expressions he uses, except as they relate to the
object before him; but he is certain that they do not deceive him, that
the rules by which he is guided are sure; because he knows that at
another time he established them by incontestible demonstrations. The
progress of a science may be compared to a series of posts on which the
distances of a road are marked: he who marked the numbers on the posts
uses them without necessity of recalling the operations which led him
to mark the quantity before him; he is satisfied with knowing that the
operations were well made, and that he wrote the result correctly.
272. The proof of this necessity of decomposition, besides being fully
established by the above example, is found in the elements of all
instruction, where, under a form of demonstration, it is necessary
to explain propositions which express simply the definitions or
axioms that have been before established. For example: we find in the
elementary works on geometry this theorem: all the diameters of a
circle are equal; and we must, if we would have beginners understand
it, give a demonstrative form to that which neither is nor can be any
thing more than an explanation, and is almost a repetition of the idea
of the circle. When we describe a circle, we fix a point around which
we revolve a line called the radius; since then the diameter is nothing
more than the sum of two radii continued in the same right line, the
mere enunciation of the theorem would seem sufficient to show that it
is evidently contained in the idea of the circle, and is as a sort of
repetition of the postulate, on which the construction of the curve is
founded: still it is not so, and it must be explained as if it were
a proof; we must show the diameter to be equal to two radii, these
radii to be equal, and at times repeat that this is supposed in its
construction: in a word, it is necessary to employ many conceptions to
show a truth, which ought to have been known by the simple intuition of
one alone, as is the case when the geometrical powers of the intellect
have acquired a certain strength and robustness.
273. We may now appreciate at its just value, the opinion of Dugald
Stewart, who, in his _Elements of the Philosophy of the Human Mind_,
says: "It may be fairly questioned, too, whether it can, with strict
correctness, be said of the simple arithmetical equation, 2 plus 2
= 4, that it may be represented by the formula A = A. The one is a
proposition asserting the equivalence of two different expressions;
to ascertain which equivalence may, in numberless cases, be an
object of the highest importance. The other is altogether unmeaning
and nugatory, and cannot, by any possible supposition, admit of the
slightest application of a practical nature. What opinion then shall we
form of the proposition A = A, when considered as the representative
of such a formula as the binomial theorem of Sir Isaac Newton?
When applied to the equation 2 plus 2 = 4, (which in its extreme
simplicity and familiarity is apt to be regarded in the light of an
axiom:) the paradox does not appear to be so manifestly extravagant;
but, in the other case, it seems quite impossible to annex to it any
meaning whatever."[22] This philosopher does not observe that the
pretended extravagance arises from his wrong interpretation of his
adversaries' opinion. No one ever thought of denying the importance
of the discoveries which prove different expressions equivalent: no
one doubts that Newton's formula of the binomial is a great advance
upon the formula A = A: but the question consists not in this, but in
seeing whether Newton's formula of the binomial is any thing more
than the expression of identical things; and whether even the merit of
the expression is or is not the fruit of a series of perceptions of
identity. Were the question presented under Dugald Stewart's point of
view, it would be unworthy of discussion: for philosophy should not
dispute upon things that are ridiculous as well as absurd.
[22] Part II., Chap. II., Sect. 3, § 2, pages 436-7.