establish these it requires certain fundamental principles. These are
called in turn definitions, axioms, and postulates. Thus arithmetic
requires the number series, which beginning from unity, is obtained by
always adding one unit to the preceding number. Geometry requires the
conception of three dimensional spaces, with the postulates connected
with it. Mechanics requires certain fundamental laws, such as the
law of inertia, by which a body in motion, which is not submitted
to the action of other forces, covers in equal times equal spaces.
There has been much dispute as to whether these principles are _a
priori_ or _a posteriori,_ pure or experimental; but the dispute must
henceforth be considered settled in favour of the former alternative.
Even empiricists distinguish mathematical principles from natural or
empirical principles, as at least (to use their expression) _elementary
experiences,_ as experiences which man completes in his own spirit,
in isolation from external nature. This means, whether they like it
or no, that they too distinguish them profoundly from _a posteriori_
or experimental knowledge. The _a priori_ character of mathematical
principles is made manifest by every attack upon it.
[Sidenote: _Contradictory nature of these a priori principles. Their
unthinkability,_]
But when they are recognized as being not _a posteriori_ and empirical,
but _a priori,_ difficulties are not thereby at an end. The apriority
of those principles possesses other most singular characteristics,
which render them unlike the _a priori_ knowledge of philosophy,
the consciousness of universals and of values, for instance, of
logical or of moral value. For if it is impossible to think that
the concepts of the true and of the good are not true, on the other
hand it is _impossible to think that the principles of mathematics
are trice._ Indeed, when closely considered, they prove to be all of
them altogether false. The number series is obtained by starting from
unity and adding always one unit; but in reality, there is no fact
which can act as the beginning of a series, nor is any fact detachable
from another fact, in such a way as to generate a discrete series. If
mathematics abandons the discrete for the continuous, it comes out of
itself, because it abandons quantity for quality, the irrational,
which is its kingdom, for the rational. If it remains in the discrete,
it posits something unreal and unthinkable. Space is characterized
as constituted of three or more dimensions; but reality gives, not
this space, thus constituted, made up of dimensions, but spatiality,
that is to say, thinkability, intuitibility in general, living and
organic extension, not mechanical and aggregated. Its character is
not to have three dimensions, one, two, three, but to be spatiality,
in which all the other dimensions are in the one, and so there are
not distinguishable and enumerable dimensions. And if the three or
more dimensions as attributes of space prove to be unthinkable, and
also the point without extension, the line without superficies, and
the superficies without solidity--so too in consequence are all the
concepts derived from them, such as those of geometrical figures, none
of which has, or can have, reality. No triangle has, or can have,
the sum of its angles equal to two right angles, because no triangle
has existence. Hence those geometrical concepts are not completely
expressed in any real fact, since they are in none, thereby differing
from the philosophic concepts, which are all in every instant and are
not completely expressed in any instant. Similar results follow in the
case of the principles of Mechanics. No body can be withdrawn from the
action of external forces, because every body is connected with all the
others in the universe; hence the law of inertia is unthinkable.
[Sidenote: _and not intuitible._]
As they are unthinkable, so are the principles of mathematics
unimaginable; they have therefore been ill defined as imaginary
entities, for they would in that case lose such _a priori_ validity
as they have. They are _a priori,_ but without the character of
truth--they are organized contradictions. Had mathematics (said
Herbart) to die because of the contradictions of which it is composed,
it would have died long ago.[1] But it does not die of them, because it
does not set itself to think them, as a venomous animal does not die
of its own poison, because it does not inoculate itself. Were it to
pretend to think them and to give them as true, those contradictions
would all become falsities.
[Sidenote: _Identification of mathematics with abstract
pseudoconcepts._]
Now, a function which organizes theoretic contradictions without
thinking them, and so without falling into contradictions, is not a
theoretic, but a practical function, and is perfectly well known to
us as that particular productive form of the practical spirit which
creates pseudoconcepts. But since those contradictions are _a priori_
and not _a posteriori,_ pure and not representative, mathematics cannot
consist of those pseudoconcepts which are representative or empirical
concepts. It remains, therefore, that it consists of the other form of
pseudoconcepts, which are _abstract_ concepts, which we have already
defined as altogether void of truth and also void of representation,
as analytic _a priori_ and not synthetic _a priori._ And we have
demonstrated how, in the falsification or practical reduction of the
pure concept, concreteness without universality, that is to say, mere
generality, belongs to empirical concepts, and universality without
concreteness, that is to say, abstraction, to abstract concepts.
Such indeed are the fictions of mathematics;--they have universality
without concreteness, and therefore feigned universality. Inversely
to the natural sciences, which give the value of the concept to
representations of the singular, although they succeed in doing so
only by convention, mathematics gives the value of the single to
concepts, also succeeding in this only by convention. Thus it divides
spatiality into dimensions, individuality into numbers, movement into
motion and rest, and so on. It also creates fictitious beings, which
are neither representations nor concepts, but rather concepts treated
as representations. It is a devastation, a mutilation, a scourge,
penetrating into the theoretical world, in which it has no part, being
altogether innocuous, because it affirms nothing of reality and acts
as a simple practical artifice. The general purpose of that artifice
is known; it is to aid memory. And the particular mnemonic purpose of
this is at once evident; it is to aid the recall to memory of series of
representations, previously collected in empirical concepts and thus
rendered homogeneous. That is to say, they serve to supply the abstract
concepts, which make possible the judgment of enumeration; to construct
instruments for counting and calculating and for composing that sort of
false _a priori_ synthesis, which is the enumeration of single objects.
[Sidenote: _The ultimate end of mathematics: to enumerate and
consequently to aid the determination of the single. Its place._]
Applying thus to mathematics what has been said of the judgment of
enumeration, it is now clear that it facilitates the manipulation of
knowledge as to individual reality. Calculation indeed presupposes:
(i) perceptions (individual judgments); (2) classifications (judgments
of classification); and only by means of these latter does it attain
to the first. But it must attain to the first, because were there
no single things to recall to the mind, calculation would be vain.
Quantification would be sterile fencing, if it did not eventually
arrive at qualification.
Mathematics is sometimes conceived as the special instrument of the
natural sciences, _appendix magna_ to the natural sciences, as Bacon
called it; but from what has been said, we must not forget that both
taken together, because co-operating, constitute an _appendix magna_
or an _index locupletissimus_ to history, which is full knowledge of
the real. It is further altogether erroneous to present mathematics
as a prologue to all knowledge of the real, to philosophy and to the
sciences, for this confuses head with tail, _appendix_ and _index,_
with text and preface.
[Sidenote: _Particular questions concerning mathematics._]
It does not form part of the task that we have undertaken further to
investigate the constitution of mathematics and to determine whether
there be one or several mathematical sciences; if one be fundamental
and the others derived from it; if the Calculus include in itself
Geometry and Mechanics, or if all three can be co-ordinated and unified
in general mathematics; if Geometry and Mechanics be pure mathematics,
or if they do not introduce representative and contingent elements
(as seems to be without doubt the case in mathematical Physics); and
so on. Suffice it that we have established the nature of mathematical
science and furnished the criterion according to which it can be
discerned if a given formation be mathematics or natural science, if
it be pure or applied mathematics (concept or judgment of enumeration,
scheme of calculation, or calculation in the act). And for this reason
we shall not enter into the solution of particular questions, like
those concerning the number of possible fundamental operations of
arithmetic, or concerning the nature of the calculus of infinitesimals,
and whether, in this, there be any place for non-mathematical concepts,
that is, the philosophic, not the quantitative infinite, or, again,
concerning the number of the dimensions of space. As to the use of
mathematics, it concerns the mathematician who knows his business to
see what arbitrary distinctions it suits him to introduce, and what
arbitrary unifications to produce, in order to attain certain ends.
For the philosopher, these unifications and those distinctions, if
transported into philosophy, are all alike false, and all can be
legitimate, if employed in mathematics. If three dimensions of space
are arbitrary but convenient, four, five and _n_ dimensions will be
arbitrary, and the only question that can be discussed will be whether
they are convenient. Of this the philosopher knows nothing, as indeed
he is sure _a priori_ is the case.
[Sidenote: _Rigour of mathematics and rigour of philosophy. Loves and
hates of the two forms._]
Practical convenience suggests the postulates to mathematics; but the
purity of the elements that it manipulates gives to them the rigour
of demonstrations, the force of truth. It is a curious force, that
has a weakness for point of support,--the non-truth of the postulate,
and reduces itself to a perpetual tautology, by which it is recorded
that what has been granted has been granted. But the rigour of the
demonstrations and the arbitrariness of the foundations explain how
philosophers have been in turn attracted and repelled by mathematics.
Mathematics operating with pure concepts is a true _simia philosophiae_
(as it was said of the devil that he was _simia Dei_), and philosophers
have sometimes seen in it the absoluteness of thought and have saluted
it as sister or as the first-born of philosophy. Other philosophers
have recognized the devil in that divine form, and have addressed to it
the far from pleasant words that saints and ascetics used to employ on
similar occasions. Hence mathematics has been accused of not being able
to justify its own principles, notwithstanding its rigorous procedure;
and of constructing empty formulæ and of leaving the mind vacant. It
has been accused of promoting superstition, since the whole of concrete
reality lies outside its conventions, an unattainable mystery; and of
being too difficult for lofty spirits, just because it is too easy.[2]
Gianbattista Vico confessed that having applied himself to the study of
Geometry, he did not go beyond the fifth proposition of Euclid, since
"that study, proper to minute intellects, is not suitable to minds
already made universal by metaphysic."[3] But these accusations are not
accusations, and simply confirm the peculiar nature of those spiritual
formations, eternal as the nature of the spirit is eternal.
[Sidenote: _Impossibility of reducing the empirical sciences to
mathematics, and empirical limits of the mathematical science of
nature._]
The nature of mathematics being explained, we can now resume the thread
of the narrative, left hanging loose, and discover how inadmissible is
the claim for a mathematical science of nature, which should be the
true end and the inner soul of the empirical and natural sciences. It
is said that this mathematical science presides, as an ideal, over
all the particular natural sciences, but it should be added, as an
unrealized and unrealizable ideal, and therefore rather an illusion
and a mirage than an ideal. It is urged that this ideal has been
partially realized, and that therefore nothing prevents its being
altogether realized. But, indeed, whoever looks closely will see that
it has not been even partially realized, because mathematical formulæ
of natural facts are always affected by the empirical and approximate
character of the naturalistic concepts which they use, and by the
intuitive element upon which these are based. When it is sought to
establish in all its rigour the ideal of the mathematical science of
nature, it becomes necessary to assume as a point of departure elements
that are distinct, but perfectly identical and therefore unthinkable;
quantity without quality, which are nothing but those mathematical
fictions of which we have spoken. The idea of a mathematical science is
thus resolved into the idea simply of mathematics, and the much-vaunted
universality of that science is the universal _applicability_ of
mathematics, wherever there are things and facts to number, to
calculate and to measure. The natural sciences will never lose their
inevitable intuitive and historical foundation, whatever progress may
be made in the calculus and in the application of the calculus. They
will remain, as has been said, _descriptive_ sciences (and this time
it has been well said, as it prevents the failure to recognize the
intuitive elements, of which they are composed).
[Sidenote: _Decreasing utility of mathematics in the most lofty spheres
of the real._]
We have already illustrated the slight perceptibility of differences
(or the slight interest that we take in individual differences),
as we gradually descend into what is called nature or inferior
reality. On this is founded the illusion that nature is invariable
and without history. And it also explains why mathematics has seemed
more applicable to the _globus naturalis_ than to the _globus
intellectualis,_ and in the _globus naturalis,_ to mineralogy more
than to zoology, to physics more than to biology. Still, mathematics
is equally applicable to the _globus intellectualis,_ as, for
instance, in Economics and Statistics. And, on the other hand, it
is inapplicable to both spheres, when they are considered in their
effective truth and unity as the _history of nature_ or the _history
of reality,_ in which nothing is repeated and therefore nothing is
equal and identical. Beneath that difference of applicability there
is nothing but a consideration of utility. If the grains of sand on
which we tread can be considered (although they are not) equal to one
another, it happens less frequently that we regard those with whom we
associate and act in the same light. Hence the _decreasing utility_ of
naturalistic constructions (and of mathematical calculation), as we
gradually approach human life and the historical situation in which
we find ourselves. Decreasing but never non-existent, for otherwise,
neither empirical sciences (grammars, books on moral conduct,
psychological types, etc.) nor calculations (statistics, economic
calculations, etc.,) would continue in use. A constructor of machines
needs little intuition, but much physics and mechanics. A leader of
men needs very little mathematics, little empirical science, but much
intuitive and perceptive faculty for the vices and value of the human
individuals with whom he has to do. But both little and much are
empirical determinations; the Spirit, which is the whole spirit in
every particular man and at every particular instant of life, is never
composed of measurable elements.
[Footnote 1: _Introduction to Philosophy,_ Italian tr., Vidossich, p.
272.]
[Footnote 2: There is a curious collection of judgments adverse to
mathematics in Hamilton, _Fragments philosophiques,_ tr. Plisse, Paris,
1840, pp. 283-370.]
[Footnote 3: Autobiography in _Works,_ Ferrari, 2nd edition, iv. p.
336.]
VII
THE CLASSIFICATION OF THE SCIENCES
[Sidenote: _The theory of the forms of knowledge and the doctrine of
the categories._]
The explanations given as to the various forms of knowledge are
also explanations concerning the categories of the theoretic and
theoretic-practical spirit: the intuition, the concept, historicity,
type, number; and also quality and quantity and qualitative quantity,
space, time, movement, and so on. They form part of that doctrine of
the categories, in which the account of philosophy in the strict sense
is completed. To ask what mathematics or history is, means to search
for the corresponding categories; to ask what is the relation between
history and mathematics, and in general how the various forms of
knowledge are related to one another, means to develop genetically all
these forms, which is precisely what we have attempted.
[Sidenote: _The problem of the classification of the sciences and its
practical nature._]
But the difficult enquiry as to the forms of knowledge as categories
has not been much in favour in recent times. Another problem has, on
the other hand, acquired vogue. It has seemed more easy, but that is
not so, because though artfully disguised, it is at bottom identical
with the preceding problem. Instead of putting the question in the
manner indicated above, which implies seeking out the constitution
of the theoretic spirit, a modest request has been made for a
classification of the various forms of knowledge, a _classification of
the sciences._
Scant confidence in philosophic thought, and excessive confidence in
naturalistic methods, have so operated that, unable to renounce the
necessity of dominating the chaos of the various competing sciences and
not wishing to have recourse to philosophic systematization, an attempt
has been made to classify the sciences like minerals, vegetables, and
animals. Even now there exist writers occupying professorships who
claim to be specialists in classifying sciences. Volumes on this theme
appear with an unprofitable frequency and abundance.
[Sidenote: _False philosophic character that it assumes._]
Certainly, if such writers and professors were to proceed in an
altogether empirical manner, corresponding with their declarations,
nothing could be said against their labours, beyond advising them not
to discuss them philosophically in order that they may not waste time
in misunderstandings, and to recognize their slight utility. But, as
a fact, none of them contains himself within empirical limits, but
each gives some philosophic and rational basis to the classification
which he proposes. Thus there appear bipartitions of the sciences into
_concrete_ and _abstract,_ into _historical_ and _theoromatic_(or
nomotechnical), into sciences of the _successive_ and sciences of the
_coexistent,_ or into _real_ and _formal;_ or _tripartitions,_ into
sciences of _fact,_ of _law_ and of _value_; into _phenomenalist,
genetic_ and _systematic_ sciences; and into similar partitions and
groups, of which some are old acquaintances and correspond to functions
of the spirit that we have already distinguished, while others, on the
contrary, must be held to be false, because they confuse under the
same name functions that are different and divide functions that are
unique. But all of them, true or false, leave the empirical and direct
themselves to the problem of Logic and of theoretic Philosophy. This is
not the place to criticize them, because substantially it has already
been done in the course of the exposition of our theories; and what is
left would reduce itself to a criticism of minute errors, which finds a
more suitable place in reviews dealing with books of the day than in
philosophic treatises. So true is it that those classificatory systems
pass with the day that witnessed their birth.
[Sidenote: _Coincidence of that problem with the search for the
categories, when understood in a strictly philosophic sense._]
We are concerned only to demonstrate more clearly that the demand
inherent in such attempts is identical with that which leads to the
establishing of a doctrine of the categories or a philosophic system.
It is indeed possible to discover now and then in the demands for a
classification of the sciences, two demands, the one limited, the other
wider. The first takes the form of a demand for a classification of the
forms of knowledge, as in the Baconian system, and in the others which
repeat the type. Here the sciences are divided according to the three
faculties, memory (natural and civil history), imagination (narrative,
dramatic and parabolical poetry), and reason (theology, philosophy of
nature and philosophy of man). The other tends to a classification
not according to gnoseological forms alone, but according to objects,
according to all the real principles of being, as in the system of
Comte and in those derived from it. Now a classification of the first
kind coincides with researches relating to the forms of the theoretic
spirit, and the problems that it exposes cannot be solved save by
penetrating into the problems of these forms. Otherwise it is not
possible to say if, for example, the Baconian classification be exact
or no, and if not, where it should be corrected. But in passing to
the other form of classification, according to objects or to the real
principles of being, we pass from the sea to the ocean, because that
coincides with the entire philosophic system. The classification of
Comte, for example, is his positivism itself, and it is not possible to
accept or refute or evaluate the one, without accepting or refuting or
submitting to examination the other. There are people who ingenuously
believe that they can understand things by representing them on a
sheet of paper, in the form of a genealogical tree or of a table rich
in graphic signs of inclusion and exclusion. But when we seriously
engage upon the work, we perceive that in order to draw up the tree and
construct the table, it is above all things needful to have understood
them. The pen falls from the hand and the head is obliged to bend
itself in meditation, when it does not prefer to abandon the dangerous
game and amuse itself in other ways.
[Sidenote: _Forms of knowledge and literary-didactic forms._]
And this is just the occasion to make clear the distinction that
we have on several occasions employed, between forms of knowledge
and literary or didactic forms of knowledge, between the orders of
knowledge and books. The arrangement of books is not always determined
solely by the demand for the strict treatment of a determinate problem;
very frequently, its motive is supplied by the practical need of having
certain different pieces of knowledge collected together, in order not
to be obliged to go and search for them in several places, that is to
say, in their true places. Thus, side by side with scientific treatises
properly so-called, are to be found scholastic compilations and
manuals. Such are Geographies, Pedagogies, juridical or philological
Encyclopædias, Natural Histories, and so on. Authors, even outside
strictly scholastic limits, used formerly to consider it convenient
sometimes to isolate, sometimes to unite certain orders of knowledge,
and to baptize the mutilation or mixture with a particular name. It is
evident that when dealing with these hybrid compilations and formations
the philosopher and the historian of the sciences, who seek not books,
but ideas, must carry out a series of analyses and syntheses, of
disassociations and associations, without allowing themselves to be
seduced by the authority of the writers or by the solidity of these
mixtures, which have become traditional.
[Sidenote: _Prejudices arising from these last._]