Chapter XX (part 3)

Mathematics, or the science of magnitudes, is that system which studies the quantitative relations between things; logic, or the science of concepts, is that system which studies the qualitative (categorical) relations between things. Logic has been built up quite in the same way as mathematics. As with logic, so also with mathematics (at least the generally known mathematics of "finite" and "constant" quantities), both were deduced p. 247 by us from the observation of the phenomena of our world. Generalizing our observations, we gradually discovered those relations which we called the fundamental laws of the world. In logic, these fundamental laws are included in the axioms of Aristotle and of Bacon. A is A . ( That which was A will be A .) A is not Not-A . ( That which was Not-A will be Not-A .) Everything is either A or Not-A . Everything will be either A or Not-A . The logic of Aristotle and Bacon, developed and supplemented by their many followers, deals with concepts only . Logos, the word, is the object of logic. An idea, in order to become the object of logical reasoning, in order to be subjected to the laws of logic, must be expressed in a word. That which cannot be expressed in a word cannot enter into a logical system. More-over a word can enter into a logical system, can be subjected to logical laws, only as a concept . At the same time we know very well that not everything can be expressed in words . In our life and in our feelings there is much that cannot be expressed in concepts. Thus it is clear that even at the present moment, at the present stage of our development, not everything can be entirely logical for us. There are many things which in their substance are outside of logic altogether. This includes the entire region of feelings, emotions, religion. All art is just one entire illogicality; and as we shall presently see, mathematics , the most exact of sciences, is entirely illogical. If we compare the axioms of the logic of Aristotle and of Bacon with the axioms of mathematics as it is commonly known, we find between them complete similarity. The axioms of logic, A is A . A is not Not-A . p. 248 Everything is either A or Not-A . fully correspond to the fundamental axioms of mathematics, to the axioms of identity and difference. Every magnitude is equal to itself. The part is less than the whole. Two magnitudes, equal separately to a third, are equal to each other, etc . The similarity between the axioms of mathematics and those of logic extends very far, and this permits us to draw a conclusion about their similar origin. The laws of mathematics and of logic are the laws of the reflection of the phenomenal world in our receptivity and in our reasoning faculty. Just as the axioms of logic can deal with concepts only, and are related solely to them, so the axioms of mathematics apply to finite and constant magnitudes only, and are related solely to them. THESE AXIOMS ARE UNTRUE IN RELATION TO INFINITE AND VARIABLE MAGNITUDES, just as the axioms of logic are untrue even in relation to emotions, to symbols, to the musicality and the hidden meaning of words , to say nothing of those ideas which cannot be expressed in words. What does this mean? It means that the axioms of logic and of mathematics are deduced by us from the observation of phenomena , i.e., of the phenomenal world, and represent in themselves a certain conditional incorrectness , which is necessary for the knowledge of the unreal "subjective" world—in the true meaning of that word. ______ As has been said before, we have in reality two mathematics . One, the mathematics of finite and constant numbers , represents a quite artificial construction for the solution of problems based on conditional data.