MATHEMATICS AND THE QUESTIONOF COS~UC MIND (WITH Omsrt ESSAYS). CasSius Jackson Keyser, Adrain Professor Emeritus of Mathematics, Columbia University. Scripta Mathematica. Yeshiva College, Amsterdam Avenue and 186th Street, New York City, 1935. v 121 pp. 12.5 X 19 cm. $0.75.
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This bonk, Number Two of the Scripta Mathematica Library, consists of six essays, reprinted, with some alterations, from articles that have recently appeared in various journals. The author discusses the meaning of mathematics and its relation t o nature and life for "intellectual non-specialisis," for those "who like to think, not for the multitude of those who are satisfied with being merely told." To read it understandingly does not require a mathematician's knowledge of mathematics. I n the first essay, "The Meaning of Mathematics," distinction is made hetween the nature of mathematics and the nature of mathematicsl applications. Considering mathematics from the viewpoints (1) of an "intellectual enterprise" aimed t o establish hypothetical propositions, (2) of a "body of knowledge," and (3) of a "type of thinking,'' a comprehensive definition of mathematics is reached, based on the concept of a "hypothetical doctrinal function" as developed from the "type of thinking" viewpoint. "An ideal exposition of any branch of mathematics always is a well-wrought postulational discourse, a logically ordered presentation of both the product or result and the processes of postulational thinking." Considering mathematics and science to embrace the whole realm of man's knowledge, "as is expressible by propositions," science for the "Actual world," mathematics for the "IogicaUy Possible world," the range of possible applications of mathematics coincides with the potential scope of science, and this scope embraces all the subject matters, material or mental, physical or psychical, organic or inorganic, of the Actual World." Mathematics and science are differentiated thus. "Scientific truth resides in categoricals. refers t o their content, and admits of no test hut obsemtionMathematical truth resides only in hypotheticals, refers to their forms, and owns no test save deduction " I n "The Bearings of Mathematics" reference is made to a "certain rich manifold of light-giving relations connecting mathematics with those great human interests and human concerns in which there is, properly, no question of establishing mathematical propositions or of making mathematical applications." The bearings are carried by "the relations which tie it as one among the cognate expression-Forms of a given Culture to the other Forms thereof." such as philosophy, poetry, painting, etc. As hearings the author discusses, among others, ideals as conformed to the mathematical concepts of limit, relations, change, invariance, logical criticism, inlinity. The relatively brief "Mathematics and the Question of the Cosmic Mind" states that since "the universe has headily more and more revealed itself as being, not a chaos, hut a cosmos, as being, that is, a realm or locus of humanly discoverable and humanly understandable logical relationships, order, and law-we may say with almost perfect confidence that our Universe is, essentially and ultimately, a realm of Mind." "Mitigating the Tragedy of our Modern Culture" defines this tragedy as the fact that many individuals "who by their native endowments of intellect were in their vouth well oualified for the
appeals of other interests, have chosen, or have been constrained, to devote their energies t o other pursuits.-It is these men and women who keenly feel the pain of being told in their mature ,-ears, and of having to believe, that their conceptions of the world they inhabit-their conceptions of the essential natures and interrelations of Space, Time, Matter, and Mid-are as grotesquely crude, inadequate, and false as those of a peasant." The means of mitigating this tragedy is the development of "the a r t of so presenting and so expounding scientific and mathematical ideas, methods, and achievements, together with their philosophic implications and spiritual significance, as to engage
the interest and reach the understanding of intellectual laymen." In the essay "On the Study of Legal Science" the author contends that law, being a "variant function of Variables," can be studied by the mathematical method. If law is to be a branch of science i t must deal with natural phenomena. "The science of law will consist of categorical propositions setting forth the distinctive behavior of the judiciary together, of course, with the stimuli calling it forth and the circumstances conditioning it." The concluding essay, "William Benjamin Smith," is a eulogy of the life and work of the deceased educator. Any "intellectual non-specialist" in mathematics will be amply repaid by reading this entertaining text and obtaining the viewpoint of an authority on "sheer mathematics." Cnas. G. E l c n ~ r ~
POPOET'SQunNTIr~TlveANALYSIS Revised by Murray J. Rice and Warm P. Cor1eIyou. Alfxd University. Third edition. P. Blakiston's Son & Co., Inc., Philadelphia, 1935. xxii 555 P P 75 figs. 23 X 15 cm. $4.00.
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The well-known text on elementary quantitative analysis by the late Professor Popoff of the University of Iowa, which some years ago passed through two editions, now appears in the third edition as a very extensive revision a t the hands of two chemistry professon at Alfred University. This book was a pioneer ariginally in combining in one volume theory, practice, complete diiections for laboratory procedure, and problems, and as such paved the way for several later texts of similar type. The original purposes and methods are retained in this revision but much of the text has heen rewritten and the whole has been rearranged into two books. Book I is intended to supply material for an elementary course and Book I1 the material for an advanced course which might he called "Special Topics and Special Methods in Analytical Chemistry," although the instructor has the option of using any of this material as he wishes. Within Book I the subject matter is divided into sharp classifications: elementary theory, rules of computation, dictionaries (the actual term and method used) of experimental rules and regulations, of instruments and apparatus, of reagents and materials, and of unit ooerations. followed hv the snecific methods. Book I1 is simihrly divided for advanced material, including besides the usual rravimetrh: and volumetric methods, clear and r\lensivc treatment of elrctrodcporition, putmtiomtrric and condurrimrrrir analyses, silicate, gas, and iron and steel analyses. This book is, therefore, quite complete for the usual onesemester, or year courses in quantitative analysis. While the material is organized in an unusually clear-cut fashion, the success of its use will depend on the genius and foresight exerted by the instructor to the end that the average sophomore student will not be Lost in the wanderins between Books I and 11. The is placed wherever the operation can be suspended to a later period. I t must be concluded that this revisioshas been consummated with great earnestness and effort, judiciously based on an established hook and name, but almost uniquely the creation of Rice and Cortelyou. The work must compete with excellent late texts and others in process of publication. I n general the choice of material, the modern treatment, and the inclusion in one volume of all necessary aspects of the subject, embracing even questions and problems, seem adequate. The mechanics of the published volume add much t o the general merit: a heavy washable cover, strong binding, freedom from error, and commendable typography, especially the use of bold-faced type for equations and formulas which causes them to stand out like friendly lighthouses to save the student from shipwreck on the dark and rock-bound coasts of quantitative analysis. The text begins with a welkhosen qnotation-not from Aristotle, Liebig. or Fresenius, but from Edgar A. Guest. I t ends in the last line of the index with the words-"as areducing agent." As such for
