Einstein, the searcher

by Alexander Moszkowski

The conversation touched on famous expressions, words carved in stone, in particular a saying of Kant which seeks to fix the foundation and the limits of knowledge. "Every science of Nature," the great philosopher of Königsberg had said, "contains just as much Truth as it contains mathematics." And since, ultimately. Nature includes everything—for a demarcation between physical and mental science no longer seems possible—then, if we follow Kant, we should have to regard mathematics as the sole measure of science.

It is certainly not yet possible to enter into a discussion on this point with historians, medical or legal practitioners. They would be justified in refusing it, since, in their subjects, "truth" is not the sole factor, and because we cannot see at present how the conception of a comprehensive mathematical truth is to find a place in them. But when we question a physicist on this point, who unceasingly uses mathematics as his chief instrument, we should surely expect him to answer with an unconditional affirmative. At least, I should not have been surprised if Einstein had answered in this way, and if he had indeed claimed its validity for every branch of science.

But Einstein considered this quotation to be true only conditionally, in that he accepted it as a principle, but did not regard it as universal. That is, he does not recognize mathematics as the only test of truth.

"The sovereignty of mathematics," said Einstein, "is based on very simple assumptions; it is rooted in the conception of magnitude itself. Its dominant position is due to the fact that it gives us much more delicate means of distinguishing between infinitely varied possibilities than any other method of thought that expresses itself in language and is restricted to the use of words. The greater the field taken into consideration, the clearer does this become; but even in such a narrow range as 1 to 100, an estimate such as 27 is incomparably more exact than can be expressed in words in any other way. If we think of a series of sensations, ranging from pleasure to pain, or from sweet to bitter, we find that words leave us in an uncertain, confused state, and we do not succeed in fixing on a point of the series with the same precision as we above fixed on the 27 out of the 100. But when the theory of magnitude plays a part in the question, as, for example, in a series of tones, whose vibrations exhibit a mathematical sequence, we immediately attain a much higher order of precision by using numbers...."

That is why there is a sort of scientific pleasure in the sequence of tones, so my thoughts ran on. Leibniz remarks that "Music is the pleasure of the human soul, which arises from counting without knowing that it is counting." Here Pythagoras' "Number is the essence of all things" is verified. As soon as we arrive at the stage at which we feel the psychological essence of number, we fall into a sort of ecstasy, because, in our subconscious minds, we experience not only the pleasure of sense but also the underlying truth.

Einstein resumed: "Kant's remark is correct in the sense that it sets up two things in clear contradiction to one another. On the one hand, he has in view the fruits of knowledge of ordinary life, in which our ordinary perceptions and experiences are intermingled and cannot be disentangled by inductive methods and deductive considerations. Opposed to these, and to be regarded of higher rank, are the properly scientific constructions—that is, such in which we find a neat differentiation of connected thoughts that are based on regular foundations and that form the links of a chain of deduction. Whenever our science succeeds in detaching this logically ordered knowledge from its sense-sources, it has a mathematical character, and the amount of truth contained in it will accordingly be determined by Kant's criterion. But Kant demands too much when he asks us to apply this scale to all attainable knowledge of science. It would seem advisable to draw limitations if his remark is to serve as a regulative measure. A great part of biological science will in future still be obliged to make its way independently of purely mathematical considerations."

"Your reflections, Professor, would then also apply to the saying of Galilei: The book of Nature lies open before us, but is written in letters other than those of our alphabet; its characters are composed of triangles, quadrilaterals, circles, and spheres."

"With all due honour to the beauty of this observation, I cannot refrain from doubting its universal validity. If we were to accept it unconditionally we should have to regard the paths of all research as purely mathematical, and this would exclude certain very important possibilities, above all, certain forms of intuition that have shown themselves to be extremely fruitful. Thus, according to Galilei's interpretation, the book of Nature would have been illegible for Goethe, for his spirit was entirely non-mathematical, indeed anti-mathematical. But he possessed a particular form of intuition that expressed itself as a feeling which put him into direct contact with Nature, with the result that he obtained a clearer vision than many an exact investigator."

"Do you then consider intuitive gifts to be separable at all in form and in kind?"

"It would be pedantic to seek to establish a fundamental difference, even if we may regard the non-mathematical intuition of Goethe as a very striking case. Moreover, as I have often emphasized, all great achievements of science start from intuitive knowledge, namely, in axioms, from which deductions are then made. It is possible to arrive at such axioms only if we gain a true survey of thought-complexes that are not yet logically ordered; so that, in general, intuition is the necessary condition for the discovery of such axioms. And it cannot be denied that, in the great majority of minds with a mathematical tendency, this intuition exhibits itself as a characteristic of their creative power."