Review of Bledsoe's The Philosophy of Mathematics. [From The Nation 4.148 (30 April 1868): 355-356.] ³⁸

The Philosophy of Mathematics, with Special Reference to the Elements of Geometry and the Infinitesimal Method. By Professor A. T. Bledsoe, A.M., LL.D., late of the University of Virginia. (Philadelphia: J. B. Lippincott &Co. 1868.)

This work is mainly devoted to the rationale of the various infinitesimal methods, and to a succinct and interesting discussion of the essential logic of these methods. We have discovered nothing of value essentially original in the philosophical exposition of these methods,

unless in a realistic and uncompromising application of the “method of limits” to their philosophy.

One English author, Mr. Todhunter, who ventures to ignore the question whether a variable actually reaches its limits or not, appears to be regarded by the author as an arch-heretic, because he purposely omits from the definition of a limit the words “that value which the function never actually attains.” Realism would seem quite out of place in matters so technical and abstract; and to us it seems quite unimportant in the “method of limits” to consider whether an actual infinity of steps are continuously passed over in “passing” to the limit of a ratio, or whether the limits of value in the ratios of infinitesimals he only substituted (or understood to be substituted) for their actual ratios. Either would fully justify that neglect of infinitesimal terms in sums and differences which the method explains.

The author is undoubtedly correct in regarding the “method of limits” as the only complete rationale of the exactness of the infinitesimal calculus, but he appears to us in his discussion of this method to be more a metaphysician than a mathematician, more concerned and interested in collateral considerations on the theory than in the practice of mathematical research; and we especially dissent from his views on the causes of the difficulties which students find in the common presentations of fundamental mathematical principles. The “axioms” of mathematics are properly only axioms of an art, and only claim to be the simplest, direct, practical maxims for mathematical deductions, not necessarily the most obvious of truths philosophically considered; though they are the most extensive and useful in the art of mathematical investigation. This is especially true of modern mathematics, which lack and forego the rigor of the ancient geometry, because they do not so much aim at a clear philosophical comprehension of what we already know as at a trustworthy and expeditious method of discovering what we are still ignorant of. No mathematician at all familiar with the infinitesimal calculus can doubt of its rigorous exactness in dealing with abstract hypothetical problems; even though he can render no clear account of its logic. As an abstract instrument of scientific research it may be better known through its technical axioms than in its philosophical foundations.

But our author appears to have chiefly in view in his work the difficulties of the mathematical teacher, the difficulties of making the principles of the calculus understood by the student. That he greatly over-estimates the philosophical difficulties to which his work is mainly devoted we attribute to his own admirable idiosyncrasy. He clearly knows and recounts his own difficulties rather than those of students in general. Simple algebra, which involves none of these difficulties, is a stumbling-block to the very sort of minds for which he has written his book. He knows much better what he is talking about than what he is talking to. The main difficulty is not metaphysical. It does not so much consist in a lack of the evidences, for the reception of which the human reason is especially apt, as in that want of disposition and attention to which the human will, since the fall, is especially prone. We would not be understood, however, as depreciating the importance of clear philosophical views in the study of an art either in mathematics or in politics, and we believe that this most interesting account of the controversy which has engaged the attention of metaphysicians since the origin of modern mathematical analysis will be instructive as well as interesting to most mathematical readers.