German history: All that characterizing German speculative science, experimental science, and historical science

The German mind is powerfully mathematical, but it is only mathematical. This formulaic exposition sums up everything we have said about the characteristics of the speculative, experimental and historical sciences in Germany. If this exclusive mathematical mind is the mark of The German intelligence, what is the source of that chaotic confusion, that deep gloom, which is so often revealed in the works produced by it?

Finally, what is clearer, what is more closely arranged than geometry? Observe this concert-goer. His ear was very sensitive and experienced, and distinguished with astonishing accuracy the slightest difference in pitch or timbre. It recognizes the most intricate chords. Harmony and melody contain no mystery to it. At the end of the concert, the man began to leave.

From beginning to end, he expertly conducted the orchestra through the intricacies of the music. Now he moved on carefully, pausing slightly to collide with people and objects. But don't be surprised. He is blind. In exactly the same fashion, German science is neither dim nor chaotic when it comes to things that come from the mathematical mind. The German deputy mathematicians Weierstrasse and Schwarz brought their inquiry into admirable order. In it, they are overly concerned with clarity.

But, leaving the proper domain of deductive method, The German reason wandered blindly into the realm where only the intuitive mind could see. So it actually mimics the blind. Where the usual channel for people is to direct themselves by sight, blind people resort to the only senses at their command, namely hearing and touch. In this fashion, German science, deprived of the eye of common sense and the eye of the intuitive mind, tried to follow the geometrical method where such an eye might be indispensable.

But this approach does not give it the eye it needs. Just as there are two kinds of mind, the intuitive mind and the mathematical mind, each of them contributes to the structure of science what is unique to it, so that without one the work of the other can never be complete; Similarly, there are two kinds of order: mathematical order and natural order. Each of these orders, when used in its proper place, is a source of inspiration.

But if one imposes a natural order on material subject to the judgment of the mathematical mind, one immediately falls into error. If one were to ask mathematical methods to elucidate what is subordinate to the intuitive mind, one would still be in a deep gloom. Following the mathematical method means never coming up with any statement that cannot be proved by the aid of a previously established statement. To follow the natural way means to bring one truth together with other truths which affect things similar in nature, and to separate judgments involving dissimilar things.

Within geometry itself it is sometimes necessary to consider the natural method. In fact, it is possible to conceive of several different permutations for the same set of theorems without at least losing the precision that constitutes the whole method of geometry. In this case, the intuitive mind will enlighten the mathematician as to which of these propositions is the most natural, and thus the best. This is a basic task, and one that is often completely ignored by mathematicians who are merely mathematicians.

Inspired by Descartes and PASCAL, the Logic of the Priory of Porroyal has denounced such mathematicians. Among the faults it accuses him of having is this: please take no notice of the true natural order. This is where the greatest weakness of mathematicians can be found. They imagine that there are few other ways of observing, other than that the first proposition should be sufficient to prove the ones that follow. Therefore, there is no trouble with the law of the true method, which always exists at the beginning of the simplest and most general things, in order to proceed next to the most complex and special things.

They mix everything up and deal with lines and planes, triangles and squares in a disorderly manner, to the point of graphically proving a single line and causing an infinite number of other inversions that undermine this fine science. Euclid's The Original, Hacha is full of this shortcoming. It is not surprising that German mathematicians were largely responsible for this failure. But, for fear of being too specialized, we would like to show here, with a few examples, that the exclusive pursuit of algebraic accuracy has often led mathematicians on the other side of the Rhine to despise in the most absolute degree the order that natural kinship can impose in the problems they deal with.

However, for our present purposes, we have to include too many and too specific details. When the mathematical mind is deprived of the aid of the intuitive mind and claims to be self-sufficient, it is not only unable to arrange mathematical theories according to the natural order, but also unable to recognize the kinship that exists between the various sciences. It ignores the fundamental connections that link mathematics to the rest of human knowledge.