Reviews for Let's think about mathematics

The average rating for Let's think about mathematics based on 2 reviews is 3 stars.

Review # 1 was written on 2014-05-21 00:00:00 Arnaud Perucca Felix Klein traveled to America in 1893 in order to convene the world congress of mathematicians held in conjunction with the world fair of that year in Chicago. Aside: Klein, along with Georg Cantor, was instrumental in organizing the official international congresses of mathematicians (at which the Fields medal is awarded), the first of which took place in Zürich in 1897 and which have recurred every four years since except for interruptions during the first and second world wars. While touring the newly founded University of Chicago in Evanston, he gave a series of lectures (in English) as a synopsis of the current state of the field, which - in a period when international communication was less convenient than it has become for us - must have keenly interested his American audience, since he enjoyed the reputation of being the greatest living geometer of his generation in Germany. The Evanston lectures remain highly topical for us today as a window into what had become of the Erlanger Programm that Klein himself had announced at the start of his career two decades before. At just 90 pages (plus an appendix, about which a little below), these none-too-technical lectures should make for required reading to every student of mathematics and its history. A telegraphic summary of topics covered can be gathered by reproducing the table of contents: I. Clebsch II. Sophus Lie III. Sophus Lie IV. On the Real Shape of Algebraic Curves and Surfaces V. Theory of Functions and Geometry VI. On the Mathematical Character of Space-Intuition, and the Relation of Pure Mathematics to the Applied Sciences VII. The Transcendency of the Numbers e and π VIII. Ideal Numbers IX. The Solution of Higher Algebraic Equations X. On Some Recent Advances in Hyperelliptic and Abelian Functions XI. The Most Recent Researches in Non-Euclidean Geometry These lectures are to be perused not to gain a mastery of the material (in any case, Klein's terminology is outdated) but to win an appreciation of the distinguished author's informed and wise perspectives on the field of mathematics as it then existed. In as much as every reader will fasten upon what appeals to his idiosyncratic sensibilities, it would be pointless to seek to reduce Klein's remarks to a synthetic thesis and so this reviewer will confine himself to offering a handful of observations on what most strikes him. 1) Right off the bat Klein serves wholesome food for thought: Among mathematicians in general, three main categories may be distinguished; and perhaps the names logicians, formalists, and intuitionists may serve to characterize them. (1) The word logician is here used, of course, without reference to the mathematical logic of Boole, Peirce, etc.; it is only intended to indicate that the main strength of the men belonging to this class lies in their logical and critical power, in their ability to give strict definitions, and to derive rigid deductions therefrom. The great and wholesome influence exerted in Germany by Weierstrass in this direction is well known. (2) The formalists among the mathematicians excel mainly in the skilful formal treatment of a given question, in devising for it an 'algorithm'. Gordan, or let us say Cayley and Sylvester, must be ranged in this group. (3) To the intuitionists, finally, belong those who lay particular stress on geometrical intuition (Anschauung), not in pure geometry only, but in all branches of mathematics. What Benjamin Peirce has called 'geometrizing a mathematical question' seems to express the same idea. (pp. 1-2) 2) The following remarks underscore why the Erlanger Programm matters to Klein: Sphere-geometry has been treated in two ways that must be carefully distinguished. In one method, which we may call the elementary sphere-geometry, only the five co-ordinates a : b : c : d : e are used, while in the other, the higher, or Lie's, sphere-geometry, the quantity r is introduced. In this latter system, a sphere has six homogeneous co-ordinates, a, b, c, d, e, r, connected by the equation (1). From a higher point of view the distinction between these two sphere-geometries, as well as their individual character, is best brought out by considering the group belonging to each. Indeed, every system of geometry is characterized by its group, in the meaning explained in my Erlangen Programm; i.e. every system of geometry deals only with such relations of space as remain unchanged by the transformations of its group. In the elementary sphere-geometry the group is formed by all the linear substitutions of the five quantities a, b, c, d, e, that leave unchanged the homogeneous equation of the second degree b2 + c2 + d2 - ae = 0. This gives ∞25-15 = ∞10 substitutions. By adopting this definition we obtain point-transformations of a simple character. The geometrical meaning of equation (2) is that the radius is zero. Every sphere of vanishing radius, i.e. every point, is therefore transformed into a point. Moreover, as the polar 2bbʹ + 2ccʹ + 2ddʹ - aeʹ - aʹ e = 0 remains likewise unchanged in the transformation, it follows that orthogonal spheres are transformed into orthogonal spheres. Thus the group of the elementary sphere-geometry is characterized as the conformal group, well known as that of the transformation by inversion (or reciprocal radii) and through its applications in mathematical physics. (pp. 10-11) Here we see that Klein is motivated by concrete problems such as form the subject of Lie's investigations and not merely by a drive to abstraction and systematicity - as most everyone these days is for the most part! 3) The following comments on naïve versus refined intuition are welcome: In my address before the Congress of Mathematics at Chicago I referred to the distinction between what I called the naïve and the refined intuition. It is the latter that we find in Euclid; he carefully develops his system on the basis of well-formulated axioms, is fully conscious of the necessity of exact proofs, clearly distinguishes between the commensurable and incommensurable, and so forth. The naïve intuition, on the other hand, was especially active during the period of the genesis of the differential and integral calculus. Thus we see that Newton assumes without hesitation the existence, in every case, of a velocity in a moving point, without troubling himself with the inquiry whether there might not be continuous functions having no derivative. At the present time we are wont to build up the infinitesimal calculus on a purely analytical basis, and this shows that we are living in a critical period similar to that of Euclid. It is my private conviction, although I may perhaps not be able to fully substantiate it with complete proofs, that Euclid's period also must have been preceded by a 'naïve' stage of development. (p. 38) Could mathematics prosper again more vigorously were we to cultivate a rescuscitated naïveté - hence, one less naïve than what went before, but an intuition that still strives to recover a greater naïveté than we have now instead of committing itself entirely to the refinement of the analytical? Vgl. Novalis: Je unwissender man von Natur ist, desto mehr Kapazität für das Wissen. Jede neue Erkenntnis macht einen viel tiefern, lebendigern Eindruck. Man bemerkt dieses deutlich beim Eintritt in eine Wissenschaft. Daher verliert man durch zu vieles Studieren an Kapazität. Es ist eine der ersten Unwissenheit entgegengesetzte Unwissenheit. Jene ist Unwissenheit aus Mangel, diese aus Überfluss der Erkenntnisse. Letztere pflegt die Symptome des Skeptizismus zu haben. Es ist aber ein unechter Skeptizismus, aus indirekter Schwäche unsers Erkenntnisvermögens. Man ist nicht im Stande, die Masse zu durchdringen und sie in bestimmter Gestalt vollkommen zu beleben: die plastische Kraft reicht nicht zu. So wird der Erfindungsgeist junger Köpfe und der Schwärmer sowie der glückliche Griff des geistvollen Anfängers oder Laien leicht erklärbar. [Blütenstaub Nr. 90] Friedrich Nietzsche: Reife des Mannes: das heißt den Ernst wiedergefunden haben, den man als Kind hatte, beim Spiel. [Sprüche und Zwischenspiele Nr. 94, in Jenseits von Gut und Böse] The appendix (pp. 91-100) offers an overview of the development of mathematics at German universities during the nineteenth century. Klein tosses out a few pointed remarks on the Americans who went to study there we ought to pay attention to today but don't. He recommends that they would profit more from their stay in Europe if they were first to outfit themselves with a greater degree of intellectual maturity by standing for a degree at an American university before going overseas to polish their expertise in Germany. Somebody seems to have listened; for that became the practice among American physicists in the 1920's and 1930's who were to make signal contributions to the emerging quantum theory (van Vleck, Oppenheimer etc.). Conclusion: the most important take-away from these Evanston lectures would be the reflection that in Klein's day there reigned a balance between the concrete and the abstract that we have since lost, to our detriment. Take a glance, for instance, at papers appearing these days on higher category theory. They are crammed with all kinds of fancy formalism in the language of homological algebra and numerous commutative diagrams etc. but scarcely ever, if it all, does one perform an actual computation of something in a specific case. Back on the day (the 1950's) it was de rigueur in certain circles to criticize the then-new sheaf theory for amounting to an empty formalism and for diverting everyone from working out relevant results that would advance the state of knowledge, say in the analysis of partial differential equations. To an extent, this must have been true, but in ensuing decades the sheaf-theoretic perspective does seem to have established its credentials in leading to greater understanding of problems outside itself proper (albeit it does demand a substantial effort at building up preparatory formalism in order to get going on real research problems), i.e., concerning what were traditional goals of research prior to its advent. Is the same the case again today? Perhaps, but it's difficult to convince oneself of this unreservedly. Certainly it would be healthier if one were to check out new formal and abstract findings by verifying their usefulness when applied to concrete problems whose solution is already known or by proving something novel and non-trivial with their aid - as was expected as a matter of course among Klein's contemporaries but is not anymore among us. Another reflection. Perhaps Klein's Erlanger Programm, vital as it was in its heyday has outlived its usefulness. Einstein's general theory of relativity accustoms us to think in terms of Riemannian spaces having no special symmetry anymore (apart from the trivial issue that one can reimpose symmetry as a means of arriving at special solutions, as Killing does) - was this a mistake? The great advantage of Klein's approach consists in the circumstance that the symmetry groups he considers, being finite-dimensional in every case, are simple enough to be amenable to our spatial intuition and thereby to yield formulae that can be computed with and evaluated. Can we ever get back to a situation like this, in the context of present-day, radically expanded knowledge? Would new and better concepts and principles bring about such a favorable turn of events? Let us leave the reader with ruminations along these or possibly other lines about the mathematics itself, prompted by Klein's fine Evanston lectures. As for the state of the field of mathematics, there is a little more to say about current practices in light of Klein's remarks. Probably because overproduction of PhD's was not so great a problem in Klein's era as it has become for us, those who did follow an academic track were expected to undertake more original research for the doctorate (which, if granted, gave permission to teach at the university as a privat dozent) and would have dared to put themselves forward for a tenured academic post only once, after having engaged in a protracted period of reflection as a budding young scholar, they could formulate a blueprint for a promising research program, as did Klein himself in 1872 (see our review here of his Eintrittsrede at the University of Erlangen). Everyone nowadays thinks himself entitled to be a professor for the mere effort of having demonstrated the capacity to crank out a sufficient number of minutely detailed papers at a fast enough pace devoted solely to advancing other people's research agendas, without containing very much in the way of a novel point of view of one's own. If one were to pause to develop one's own ideas from scratch or even to gain perspective through a deeper study of existing literature, productivity would suffer! But one does not attain to the stature of a Felix Klein overnight, nor during a rushed post-doctoral fellowship or two. Intellectual standards were higher during the late nineteenth and early twentieth centuries (prior to the expansion of university faculties that ensued after the second world war); there prevailed a greater degree of self-selection and only those who knew themselves possessed of a gift for original research would have had the temerity to pursue an academic career. For the formative [gestaltende] intellectual power of the mind is quite rarer than and altogether qualitatively distinct from a far-more-readily acquired technical facility - what we have forgotten in the aftermath of the Cold War when there was a pressing need for the contributions of the many who can indeed be trained to attain proficiency at the latter. The desperate straights into which the contemporary system of grant funding has fallen, which while paying lip-service to the former in practice places an all-but exclusive premium on the latter, are doubtless responsible for arresting progress at the frontiers of knowledge. The funding agencies are well satisfied to record a spate of technically competent papers being published and neglect to note that any progress they promote is more specious than real, in that - for all the profusion of minor theorems receiving proof - at the conceptual level, less serious thought than ever is being attempted or carried out. The paradox flows from a decision to optimize the wrong objective function!

Review # 2 was written on 2014-09-24 00:00:00 Patrick Brugger A collection of twelve short stories, all by different authors.