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. [Sidenote: _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. [Sidenote: _Apriority of mathematical principles._]