Logic as the Science of the Pure Concept

by Benedetto Croce

The conception of a mathematical science of nature is at variance with the thesis that recognizes the ineliminable historical foundation of the natural sciences and the consequences which follow from it. It is claimed that this mathematical science, in expressing the ideal and end of the natural sciences, would express also their true nature, which is not empirical but abstract, not synthetic but analytic, not inductive but deductive. The mathematical conception of the natural sciences would imply perfect mechanism, the reduction of all phenomena to quantity without quality, the representation of each phenomenon by means of a mathematical formula, which should be its adequate definition.

Various definitions of mathematics.

But the nature of mathematics cannot be considered a mystery in our time. Mathematics (as has lately been said with a subtlety equal to its truth) is a science "in which it can never be known what we are talking about, nor whether what we are talking about be true" These affirmations are made one after the other by all mathematicians who are conscious of their own methods. In what sense can a process that merits such a description be called a science? A science that states no sort of truth does not belong to the theoretic spirit, since it is not even poetry; and a science which is not related to anything is not even an empirical science, which is always related to a definite group of representations. For this reason, others incline to consider mathematics sometimes as language, sometimes as logic. But mathematics is neither language in general nor any special language; it is not language in the universal sense, co-extensive with expression and with art; nor is it a historically given language, which would be a contingent fact; nor a class of languages (phonetic, pictorial, or musical language, etc.), which would be an approximate and empirical definition, inapplicable in a function like mathematics, which expresses its own original nature. It is not logic, because there is only one logic, and thought thinks always as thought. If it is maintained, on the other hand, that the human spirit has also a special logic, which is that of mathematicizing, a return is made to the problem to be solved, namely, what is mathematicizing? that is to say, this logic, which is not the logic of thought, because it does not give truth, and is not the logic of the empirical sciences, because it does not depend upon representations.

Mathematical process.

Any sort of arithmetical operation can serve as an example of mathematical process. Let us take the multiplication: 4×4 = 16. The sign = (equals) indicates identity: 4×4 is identical with 16, as it is identical with an infinite number of such formulæ, since there can be infinite definitions of every number. What do we learn from such an equivalence concerning the reality, phenomenal or absolute, to which the human mind aspires? Nothing at all. But we learn how to substitute 16 for 8×2, for 9+7, for 21-5, for 32÷2, for 4², for √256, and so on. One or the other substitution is of service, according to circumstances. When, for instance, some one promises to pay us 4 lire daily, and we wish to know the total amount of lire, that is to say, the object that we shall have at our disposal after four days, we shall carry out the operation 4×4=16. Again, when we have 32 lire to divide into equal parts between ourselves and another, we shall have recourse to the formula: 32÷2 = 16. Mathematics as Mathematics does not know, but establishes formulæ of equality; it does not subserve knowing, but counting and calculating what is already known.

Apriority of mathematical principles.

For counting and calculating Mathematics requires formulæ, and to 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.

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.

and not intuitible.