The taming of this jungle by reduction to the discrete has been a principal task, if not the principal task, of mathematics. It teems with such exotic and intractable entities as incommensurable lines, horn angles, space curves, one-sided surfaces. By contrast, the continuous is a jungle, a labyrinth. 1Ĥ recommended it as a paragon of intelligibility, a realm within which reason can be realized to its fullest extent. Synonyms for “discrete” include separate, distinct, detached, disjunct. It is a curious fact that, while “continuity” and “discreteness” are antonyms, “continence” and “discreetness” are synonyms. This same root yields the verb “discern”-to recognize as distinct or separate-and the cognate “discreet”-to show discernment, hence “well-behaved”. 3 The word “discrete” derives from a Latin root meaning “to separate”. This was surely the result of (continued) exposure to the phrase “space-time continuum” with which popular accounts of relativity theory have been peppered. 2 The term “continuum” has become a buzzword, especially popular with science fiction writers (e,g. Synonyms for “continuous” include: connected, entire, unbroken, uninterrupted. The simplicity of the principles governing discreteness has The word “continuous” derives from a Latin root meaning “to hang together” or “to cohere” this same root gives us the nouns “continent”-an expanse of land unbroken by sea-and “continence”-self-restraint in the sense of “holding oneself together”. Populated by units lacking intrinsic qualities and so wholly indistinguishable from one another, in the dominion of the discrete difference is manifested through plurality alone. The discrete is a model of tidiness in which quality is reduced to quantity and over which the concept of number reigns supreme. The realm of the continuous is traditionally associated with intuition, that of the discrete, with reason. Continuity connotes unity discreteness, plurality. Opposed to continuity is discreteness: to be discrete 3 is to be separated, like the scattered pebbles on a beach or the leaves on a tree. A continuous entity-a continuum 2 -has no “gaps”. To be continuous1 is to constitute an unbroken or uninterrupted whole, like the ocean or the sky. We are all familiar with the idea of continuity. William Blake The homeland, friends, is a continuous act As the world is continuous. To see a World in a Grain of Sand And a Heaven in a Wild Flower, Hold Infinity in the palm of your hand And Eternity in an hour. Introduction Continuous as the stars that shine And twinkle on the milky way, They stretched in never-ending line Along the margin of a bay: Ten thousand saw I at a glance, Tossing their heads in sprightly dance. Forster, and that is what I have tried to do here. Still, the thread itself, if tangled in places, has been luminous. In writing the book I have found it necessary to thread my way through a wealth of sources, both philosophical and mathematical and it is inevitable that a number of topics have not received the attention they deserve. So the first part of the book is largely philosophical, while the second is almost exclusively mathematical. First, to trace the historical development of the concepts of the continuous and the infinitesimal and second, to describe the ways in which these two concepts are treated in contemporary mathematics. Bell Dedicated, once again to mt dear wife Mimi Rather, following Cohen’s lead, the Marburg philosophers sought to clarify Leibniz’s principle of continuity and to exploit it in making sense of infinitesimals and related concepts.John L. The Marburg school neither clung to the traditional approach of logically and metaphysically dubious infinitesimals nor Whiggishly subscribed to the new orthodoxy of the “great triumvirate” of Cantor, Dedekind, and Weierstrass that declared infinitesimals conceptus nongrati in mathematical discourse. Our main thesis is that Marburg neo-Kantian philosophy formulated a sophisticated position toward the problems raised by the concepts of limits and infinitesimals. Among its various currents, the Marburg school (Cohen, Natorp, Cassirer, and others) was the one most interested in matters scientific and mathematical.
The dominant current of philosophy in Germany at the time was neo-Kantianism. Some of the protagonists of the said revolution were Cauchy, Cantor, Dedekind, and Weierstrass.
We seek to elucidate the philosophical context in which one of the most important conceptual transformations of modern mathematics took place, namely, the so-called revolution in rigor in infinitesimal calculus and mathematical analysis.
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