The Philosophy of Mathematics

From the beginning of CHAPTER I. FIRST PRINCIPLES OF THE INFINITESIMAL METHOD - THE METHOD OF EXHAUSTION. The student of mathematics, on passing from the lower branches of the science to the infinitesimal analysis, finds himself in a strange and almost wholly foreign department of thought. He has not risen, by easy and gradual steps, from a lower into a higher, purer, and more beautiful region of scientific truth. On the contrary, he is painfully impressed with the conviction,...
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From the beginning of CHAPTER I. FIRST PRINCIPLES OF THE INFINITESIMAL METHOD - THE METHOD OF EXHAUSTION.
The student of mathematics, on passing from the lower branches of the science to the infinitesimal analysis, finds himself in a strange and almost wholly foreign department of thought. He has not risen, by easy and gradual steps, from a lower into a higher, purer, and more beautiful region of scientific truth. On the contrary, he is painfully impressed with the conviction, that the continuity of the science has been broken, and its unity destroyed, by the influx of principles which arc as unintelligible as they are novel. He finds himself surrounded by enigmas and obscurities, which only serve to perplex his understanding and darken his aspirations after knowledge. That clearness of evidence, which is the boast of the mathematics, and which has hitherto cheered and stimulated his exertions, forsakes him as soon as he enters on the study of the infinitesimal calculus, and the darkness of doubt settles on his path. If, indeed, he does not abandon the study in disgust or despair, as thousand* have done, he pursues it for the sake of a diploma or a degree, or from some less worthy motive than the love of science' He certainly derives from it comparatively little advantage in the cultivation of his intellectual powers; because the dark and unintelligible processes he is required to perform scarcely demand a natural exercise of them.
These disadvantages of the study are due, for the most part, to the manner in which the calculus is usually taught. In most elementary works on the differential calculus, the first principles of the science are not set forth at all, or else so imperfectly defined as to mislead the student from the clear path of mathematical science into a region of clouds and darkness. I have frequently made the experiment with some of the best of such works. I have more than once put them into the hands of a class of from ninety to a hundred students, among whom there Mere mathematical minds of no ordinary power, and required them to tell me what are the first principles of the infinitesimal method or calculus. Yet, after having read and mastered the first chapter, which, of course, contained a discussion of "First Principles," not one of them had acquired the least notion of what those principles are. Not one of them could even name the first principles of the science, much less define them. In this respect, the most capable and diligent members of the class were on a perfect level with the most stupid and indolent. Indeed, if the authors of the books themselves knew what the first principles of the calculus are, they were very careful not to unveil their knowledge.