CONTINUATION OF THE SAME SUBJECT.
304. If the Neapolitan philosopher's criterion be anywhere admissible,
it can only be in ideal truths; for as these are absolutely cut off
from existence, we may well suppose them to be known even by an
understanding which has not in reality produced them. So far as known
by the understanding they involve no reality, and consequently no
condition that exacts any productive force not referable to a purely
ideal order. In this order the human reason seems really to produce. If
we, for example, take geometry, we shall readily perceive that, even
in its profoundest parts and in its greatest complications, it is only
a kind of intellectual construction, wherein that only is to be found
which reason has placed there.
Reason it is which by force of perseverance has succeeded in uniting
elements and so disposing them as to attain that wonderful result, of
which it may say with truth: _this is my work_.
If we carefully observe the development of the science of geometry, we
shall perceive that the extended series of axioms, theorems, problems,
demonstrations and solutions, begins with a few postulates, and that
it goes on with the aid of the same, or others discovered by reason
according to the demands of necessity or utility.
What is a line? A series of points. The line, then, is an intellectual
construction, and involves only the successive fluxions of a point.
What is a triangle? An intellectual construction wherein the
extremities of three lines are united. What is a circle? Also an
intellectual construction; the space enclosed by a circumference
formed by the extremity of a line revolved around a point. What are
all other curves? Lines described by the movement of a point governed
by a certain law of inflexion. What is a surface? Is not its idea
generated by the motion of a line, just as that of a solid is generated
by the motion of a surface? And what are all the objects of geometry
but lines, surfaces, and solids of various kinds, combined in various
ways? Universal arithmetic, whether arithmetic properly so called, or
algebra, is a creation of the understanding. Number is a collection
of units, and it is the understanding that collects them. Two is only
one and one, and three only two and one; and thus with all numerical
values. The ideas expressing these values consequently contain a
creation of our mind, are its work, and include nothing not placed
there by it.
We have already observed that algebra is a kind of language. Its
rules are partly conventional, and its most complicated formulas may
be reduced to a conventional principle. Take one of the simplest: a^0
= 1: but why is it? Because a^0 = a^{n-n}; why? Because there is a
conventional usage to mark division by the remainder of the exponents;
and consequently a^n/a^n, which is evidently equal to one, may be
expressed a^n/a^n = a^{n-n} = 1.
305. These observations seem to prove Vico's system to be really true,
so far as pure mathematics, that is, science of the purely ideal order,
is concerned. Possibly also the same may be said of it in relation to
other science, as for example, metaphysics; but we shall not follow it
farther, since it is not easy to find a ground free from conflicting
opinions. Moreover, having shown how far Vico's system is admissible in
mathematics, we have thereby given a solution to difficulties to which
it is subject in its other branches.
306. That in a purely ideal order the understanding constructs is
undeniable, and the schools agree in this. There is no doubt that
reason supposes, combines, compares, deduces; operations which are
inconceivable without some kind of intellectual construction. The
understanding in this case knows what it makes, because its work is
present to it: when it combines it knows that it combines; when it
compares or deduces, it knows that it compares or deduces; when it
builds upon certain suppositions, which it has itself established, it
knows in what they consist, since it rests upon them.
307. The understanding knows what it makes; but this is not all that it
knows; for it has truths which neither are nor can be its works, since
they are the basis of all its works, as, for example, the principle of
contradiction. Can the impossibility of a thing being and not being at
the same time be said to be the work of our reason? Assuredly not.
Reason itself is impossible if this principle be not supposed; the
understanding finds it in itself as an absolutely necessary law, as a
condition _sine qua non_ of all its acts. Here, then, Vico's criterion
fails: "the understanding knows only the truth it makes:" and yet the
understanding knows but does not make the truth of the principle of
contradiction.
308. Facts of consciousness are known by reason, although they are not
its production. These facts are not only present to consciousness, but
are also objects of the combinations of reason: here, then, Vico's
criterion again fails.
309. Although in those things that are a purely intellectual work,
the understanding knows what it makes, it does not make whatever it
chooses; for then we should have to say that science is perfectly
arbitrary: instead of the geometrical results we now have, we might
have others as numerous as the individuals who deal in lines, surfaces,
and solids. This shows reason to be subject to certain laws, its
constructions to be connected with conditions which it cannot abstract.
One of these conditions is the principle of contradiction, which
would, were it to fail, annihilate all knowledge. True, by a series of
intellectual constructions one may ascertain the size of a sphere; but
can two understandings obtain two different values of it? They cannot,
for that would be an absurdity: they may choose different ways, or
express their demonstrations and conclusions in different terms; but
the value is the same: if there be any discrepancy, it is because one
or the other has fallen into an error.
310. If we thoroughly examine this matter, we shall perceive that the
intellectual construction, of which Vico speaks, is a fact generally
admitted. There are in this philosopher's system two new things, the
one good, the other bad; the good, is to have indicated one reason of
the certainty of mathematics; the bad, is to have exaggerated the
value of his criterion.
We have said that his system expressed a fact generally recognized,
but exaggerated by him. The understanding undoubtedly creates, in some
sense, ideal sciences; but in what sense? Solely by taking postulates,
and combining its data in various ways. Here ends its creative power,
for in these postulates and combinations it discovers truths not placed
there by itself.
What is the triangle in the purely ideal order? A creation of the
understanding, which disposes the lines in a triangular form, and,
preserving this form, modifies it in a thousand ways. Thus far there is
only one postulate and different combinations of it: but the properties
of the triangle flow by absolute necessity from the conditions of the
postulate: the understanding, however, does not make these properties,
it discovers them. The example of the triangle is applicable to all
geometry. The understanding takes a postulate; this is its free work,
but it must not come in conflict with the principle of contradiction.
From this postulate flow absolutely necessary consequences, independent
of intellectual action, and involving an absolute truth known by the
understanding itself. Consequently it is false to say of them that it
makes them. Suppose a man so to place a body, that, left to itself,
it will fall to the ground: is it the man who gives it the force to
fall? Certainly not, but nature. The man only supplies the condition
necessary for the force of gravity to produce its effect: when once
the condition is performed, the fall is inevitable. Here, then, is
a simile which shows clearly and exactly what happens in the purely
ideal order. The understanding performs the conditions; from them flow
other truths, _not made_, but known, by the understanding. This truth
is absolute, is as the force of gravity in the order of ideas. Hence
we see what is admissible, and what inadmissible in Vico's system.
The power of combination, a generally recognized fact, is admissible;
the exaggeration of this fact extended to all truths, when it only
comprises postulates in their various combinations, is inadmissible.
The rules of algebra are conventional inasmuch as they relate to
the _expression_, for this might evidently have been different.
Supposing, however, the expression, the development of the rules, is
not conventional, but necessary. In the expression a^n/a^n the number
of times the quantity has entered as factor might clearly have been
expressed in infinite ways; but supposing the present to have been
adopted, the rule is not conventional, but absolutely necessary; since
whatever the expression, it is always certain that the division of
a quantity by itself, with distinct exponents, gives for result the
diminution of the number of times it has entered as factor: this is
denoted by the remainder of the exponents; and consequently if the
number of times be equal in the dividend and the divisor, the result
will be = 0. Thus we see that even in algebra, what the understanding
has to do, is to perform the conditions, and express them as seems to
it best: but here its free work ends, for necessary truths result from
these conditions; and these it does not make, but only knows.
311. Vico's merit in this point consists in having expressed a very
clear idea of the cause of the greater certainty of the purely ideal
sciences. In these the understanding itself performs the conditions
upon which it has to build its edifice; it chooses the ground, forms
the plan, and raises the construction conformably to it. In the real
order this ground is already designated, just as are the plan of the
edifice and the materials for its construction. In both cases it is
subject to the general laws of reason, but with this difference, that
in the purely ideal order, it has to regard these laws and nothing
else; but in the real order, it cannot abstract the objects considered
in themselves, and is condemned to submit to all the inconveniences
they are of a nature to cause. We will explain these ideas by an
example. If we would determine the relation of the sides of a triangle
under certain conditions, we have only to suppose the conditions and
attend to them. The ideal triangle is in our understanding a perfectly
exact, and also a fixed, thing. If we suppose it to be an isosceles
triangle with the relation of the sides to the base as seven to five,
this ratio is absolute, immutable, so long as the supposition remains
unchanged. In all our operations upon these data, we are liable to
mistakes of calculation, but no error can arise from inexactness of
data. The understanding knows, indeed, for what it knows is its own
work. If the triangle be not purely ideal, but realized upon paper, or
on the ground, the understanding vacillates because those conditions,
which, in the purely ideal order, it fixes with all exactness, cannot
be transferred in like manner to the real order; and even were they
transferred, the understanding would have no means of appreciating
them. Therefore, Vico says, with great truth, that our cognitions lose
in certainty in the same proportion as they are removed from the ideal
order and swallowed up in the reality of things.
312. Dugald Stewart probably had in view this doctrine of Vico when he
explained the cause of the greater certainty of mathematical sciences.
It does not, he says, depend upon axioms, but upon definitions; that
is, he adopts, with a slight modification, the system of the Neapolitan
philosopher, that the mathematical are the most certain, because they
are an intellectual construction founded upon certain conditions placed
by the understanding and expressed by the definition.
This difference between the purely ideal and the real order did not
escape the scholastics. They were accustomed to say that there was
no science of contingent and particular, but only of necessary and
universal things. In the place of _contingent_ substitute _reality_,
since all finite reality is contingent; and instead of _universal_ put
_ideal_, since the purely ideal is all universal; and you will have
the same doctrine enunciated in distinct words. It is not easy to show
exactly how far modern philosophers have availed themselves of the
scholastic doctrine, in so far as the distinction between pure and
empirical cognitions is concerned; but it is certain that some very
clear passages upon these questions are to be found in the works of
the scholastics. It would not be strange if some moderns, particularly
Germans, whose laboriousness is proverbial, especially in matters of
erudition, had read them.(27)