Role of Mathematics in the Rise of Science

Role of Mathematics in the Rise of Science

SALOMON BOCHNER
Copyright Date: 1966
Pages: 400
https://www.jstor.org/stable/j.ctt7ztnfp
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  • Book Info
    Role of Mathematics in the Rise of Science
    Book Description:

    The central theme of these essays is the nature and role of mathematics, its growth and spread, and its involvement with ever-wider areas of knowledge. The author attempts to determine the decisive and creative aspects of the abstractness" of mathematics which have made it the dominant intellectual force that it is. He frequently confronts the mathematics and physics of today with the mathematics and physics of the Greeks, which, however renowned, was not yet capable of this abstractness.

    Originally published in 1981.

    ThePrinceton Legacy Libraryuses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

    eISBN: 978-1-4008-5282-6
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. PREFACE
    (pp. v-vi)
    S. B.
  3. Table of Contents
    (pp. vii-2)
  4. INTRODUCTION
    (pp. 3-10)

    The essays of this collection are all concerned with the role of mathematics in the rise and unfolding of Western intellectuality, with the sources and manifestations of the clarity and the mystique of mathematics, and with its ubiquity, universality, and indispensability.

    We will frequently confront the mathematics of today with the mathematics of the Greeks; and in such a confrontation it is pertinent to take the entire mathematical development since a.d. 1600 as one unit. Therefore, “modern mathematics” will mean for us, invariably, mathematics since 1600, and not since some later date, even though, for good reasons, the mathematics of...

  5. PART I. ESSAYS
    • CHAPTER 1 FROM MYTH TO MATHEMATICS TO KNOWLEDGE
      (pp. 13-58)

      What indeed is mathematics? This question, if asked in earnest, has no answer, not a satisfactory one; only part answers and observations can be attempted.

      A neat little answer, a citizen’s description of mathematics in capsule form, is preserved in the writings of a Chureh Father of the 3rd century a.d.; Anatolius of Alexandria, bishop of Laodicea, reports that a certain (unnamed) “jokester,” using words of Homer which had been intended for something entirely different, put it thus: ¹

      Small at her birth, but rising every hour,

      While scarce the skies her horrid [mighty] head can bound,

      She stalks on...

    • CHAPTER 2 HOW HISTORY OF SCIENCE DIFFERS FROM OTHER HISTORY
      (pp. 59-130)

      There are differences of consequence, we contend, between history of science and other history, mainly general (that is, politico-social) history, but also history of literature, and even history of philosophy proper. The differences are differences of degree only, but they are significant. In detail, we will be mainly concerned with history of basic developments in mathematics and physics, but our observations will usually apply to larger areas of history, although the scope of the larger area may vary with the context. We assert as follows.

      (1) History of science is younger, considerably younger than general history, and in historiographic analyses...

    • CHAPTER 3 REVOLUTIONS IN PHYSICS AND CRISES IN MATHEMATICS
      (pp. 131-142)

      We will now deal with two topics which, although separable, are closely connected with each other. The first and larger part of the chapter is concerned with the conception of a revolution in physics, as recently blueprinted in a provocative book by Thomas Kuhn.¹ We will make observations which are seemingly in conflict with those of Kuhn, but I really intend to amplify and qualify some of Kuhn’s theses rather than to dissent from them, and my approach is somewhat different anyhow. After that, we will make some observations on revolutions in physics as far as the underlying mathematics is...

    • CHAPTER 4 ARISTOTLE’S PHYSICS AND TODAY’S PHYSICS
      (pp. 143-178)

      In making some observations on the nature of the physics of the Greeks, especially of the physics of Aristotle, our aim will be to point up similarities and dissimilarities, analogies and discrepancies, between the physics of antiquity and the physics of today. The aim is not to find out how much of Greek physics survives in ours, but to verify that to a certain extent the similarities in the modes of general thinking, ancient and modern, also reflect themselves in parallelisms between the doctrines of physics which these modes of thinking have produced.

      Greek physics, while being observational and perhaps...

    • CHAPTER 5 THE ROLE OF MATHEMATICS IN THE RISE OF MECHANICS
      (pp. 179-208)

      Mechanics, as a part of physics, is very mathematical, in imagery, content, and consequences. We will make some observations on the mathematical structure of mechanics during some phases of its development, and we will be concerned with the inward mathematical texture rather than with the outward mathematical setting. But we will not go essentially beyond the beginning of the 19th century. After that time, mechanics and paradigms from mechanics began to penetrate into many other areas of theoretical physics, so that observations on the mathematical nature of mechanics after the early 19th century would have to take large parts of...

    • CHAPTER 6 THE SIGNIFICANCE OF SOME BASIC MATHEMATICAL CONCEPTIONS FOR PHYSICS
      (pp. 209-254)

      The present chapter supplements and continues the preceding one, and we will again make observations on the relations between mathematics and physics on the level of abstractions and conceptualizations; in the preceding chapter we emphasized mechanics, and at present we will be more concerned with physics in general.

      The Greeks created rational philosophy and a general conception of rational philosophy, and they created a general conception of mathematics and a general conception of physics. Their mathematics is justly renowned; the Greeks imparted to it from the beginning a certain metaphysical accent which has become a distinguishing mark of mathematics in...

    • CHAPTER 7 THE ESSENCE OF MATHEMATICS
      (pp. 255-274)

      Mathematics is frequently encountered in association and interaction with Astronomy, Physics, and other branches of Natural Science, and it also has deep-rooted affinities to what are called Humanities nowadays. Actually, it is a realm of knowledge entirely by itself, and one of considerable scope too; the word “mathematics” stems from a root which means Learnable Knowledge as such. Mathematical knowledge is commonly deemed to have a high degree of validity, binding onHomo sapiensirrespective of cultural conditioning and predilection, although it can be argued that in the past cultural settings have affected its development noticeably.

      Even as far as...

    • CHAPTER 8 THE ESSENCE OF ANALYSIS
      (pp. 275-300)

      Analysis is one of three main divisions of mathematics, the other two being (i) Geometry and Topology and (ii) Algebra and Arithmetic. In extent, Analysis is the largest; it comprises subdivisions which are nearly autonomous and which are easier to describe than the whole division.

      Greek mathematics had a geometrical tenor, as the works of Euclid, Archimedes, Apollonius of Perga, and Pappus testify. It did initiate some durable topics of analysis, but the organized creation of analysis began only in “modern” times around a.d. 1600. Analysis then came into being in stages, growing up in intimacy with Mechanics and Theoretical...

  6. PART II. BIOGRAPHICAL SKETCHES
    (pp. 301-372)

    Aeschylus (525–456 b.c.), Attic Tragedian, the first, and still one of the greatest, tragedians in the “West” of whom entire plays survive. Aeschylus derived from landed aristocracy, and, although a lifelong poet by avocation, he led, as customary with Athenians of his generation, a crowded citizen’s life in the service of the polis. For instance, tradition has it, and there is no reason to doubt it, that he was present at the battles of Marathon, Artemisium, Salamis, and Plataea in the years 490–479.

    Of his few surviving works, his last was his trilogyOresteia(Agamemnon, Choeophoroi, Eumenides) which...

  7. INDEX
    (pp. 373-386)