Great Currents of Mathematical Thought, Vol. II: Mathematics in the Arts and Sciences (Dover Phoenix Editions)

By F. Le Lionnais

Years in the making, this feast of mathematical ideas comprises works by 50 eminent French scholars.?The first book of the two-volume set embraces "Mathematics: Concepts and Development," with several... This description may be from another edition of this product.

• Format: Hardcover
• Language: English
• ISBN: 0486495795
• ISBN13: 9780486495798
• Release Date: March 2004
• Publisher: Dover Publications
• Length: 272 Pages
• Weight: 1.05 lbs.
• Dimensions: 8.8" x 0.8" x 5.8"

Customer Reviews

Fun

This book is quite amusing, albeit rather too French. I shall comment only on the editor's own contribution, which concerns beauty in mathematics. To the outsider it may seem that "Beauty often appears at feasts where only utility or truth have been invited." But the outsider is wrong to take beauty for an illicit mistress, for our engagement is no secret: "it is most often alluring esthetic satisfactions which have motivated modern mathematicians to cultivate their cherished study with such ardor" (p. 121). This proposition is easily proved by enumerating mathematical geniuses who made great discoveries at a very early age, for "how can one conceive of the attraction that mathematical truths could exercise upon such very young children if not by the incitement of their imagination and of their desire to play joined to the exercise of their mind?" (p. 148). Beauty comes "under two grand banners": "classicism, all elegant sobriety, and romanticism, delighting in striking effects and aspiring to passion" (p. 123). Classicism is "austerity," "mastery over diversity." It "intrigues us especially when we are expecting a certain disorder. This bounty is all the more delightful in that to some it seems unmerited, to others won after a mighty struggle." (p. 124). An example is Bernoulli's work on the logarithmic spiral, which revealed it to be "equal to its caustics by reflection and refraction, to its evolute, and to numerous other derived or conjugate curves. For this reason Bernoulli requested that on his tombstone be engraved a logarithmic spiral, above the followinf inscription: eadem numero mutata resurgo. 'This marvelous spiral', he wrote, 'gives me such overwhelming pleasure that I can scarcely satisfy my desire to contemplate it.'" (p. 126). We are susceptible to being misled by this type of beauty. For example, we are impressed to learn that the exponential function, "like a phoenix rising again from its own ashes, is its own derivative," but our excitement fades "as soon as we realise that it is not very surprising for the differential equation y=y' to have a solution" (pp. 126-127). "This impression of finality often plays a large role in the esthetic enjoyment which the sciences can provide us. There is, to be sure, no finality in mathematics ... but the very frailty of our intelligence engenders these illusions which so stir our emotions." (p. 127). "Without as yet leaving the empire of classical beauty, we can add the delightful Ionian slenderness to the rigid simplicity of the Doric order." (p. 127). E.g., quadratic reciprocity, Euler's formula. As for methods rather than results, "a method earns the epithet of classic when it permits the attainment of powerful effects by moderate means" (p. 136). E.g., Riemann surfaces, classifications of conics. The discussion of romantic beauty is decidedly less satisfying. It is simply identified with disturbing entities (e.g., i, quantities greater than infinity, non-differentiable curves,