Arithmetic and Combinatorics: Kant and His Contemporaries

By Gottfried Martin; Judy Wubnig | Go to book overview

CHAPTER THREE
The Axioms of Arithmetic

THE AXIOMS OF ADDITION were published by Johann Schultz in 1789-92 in the Prüfung der Kantischen Critik der Reinen Vernunft, in 1790 in the Anfangsgründe der reinen Mathesis, and again in 1805 in Kurzer Lehrbegriff der Mathematik ( 2d ed., 1820). From then on they appeared in numerous works whose authors can be called Kantians in either a narrower or a wider sense. Kiesewetter and Zimmermann were actually students of Kant's. The first statement of the axioms given from outside the small circle of students occurs in System der Elemente der allgemeinen Grössenlehre by Friedrich Wilhelm August Murhard ( 1729- 1853) in 1798. Martin Ohm ( 1792- 1872) and Jakob Fries then followed. The axioms were taken over into mathematics proper by Hermann Grassmann, Sir William Rowan Hamilton ( 1805- 1865) and Hermann Hankel. In this way the axioms of arithmetic gradually appeared in specifically mathematical textbooks, for example, those of Ernst Schröder ( 1841- 1902), Gottlob Frege, Hermann Schubert ( 1848- 1911), Otto Stolz ( 1842- 1905), Richard Baltzer ( 1818- 1887), and Carl Anton Bretschneider ( 1808- 1878). Peano gave the final statement in 1889 in his Arithmetices principia.

After giving an account of these two groups, I will try to show that there is a direct line from Kant to modern mathematics, basically through Ohm and Grassmann, although it is not really possible to establish this conclusively in view of the great number of works of the Kantians in which the axioms had already appeared. At the end of this chapter I will examine the question of whether the axioms stem from Kant or Schultz.

To show unequivocally Kant's importance as a productive mathematician I take the commutative law of multiplication as an example, so that the originality and importance of Kant's procedure will become clear.

First, here is Schultz's account in the Anfangsgründe, which is dated 1790 (although reference is made in it to the Prüfung, which bears the date 1791 on its title page.) He says:

-34-

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Arithmetic and Combinatorics: Kant and His Contemporaries
Table of contents

Table of contents

  • Title Page iii
  • Contents v
  • Foreword vii
  • Translator's Preface xi
  • Preface [1938] xv
  • Introduction xvii
  • Part I 1
  • Chapter One - The Axiomatics and Logic of Mathematics 3
  • Chapter Two - The Analytic Principles 11
  • Chapter Three - The Axioms of Arithmetic 34
  • Part II 51
  • Chapter Four - Problems About Classes of Numbers 53
  • Chapter Five - Combinatorics and the Idea of A Systematic Ontology 59
  • Chapter Six - Synthetic Judgment in Arithmetic 85
  • Appendix - Examination of Kant's "Critique of Pure Reason," Part I, Section 4 129
  • Notes 141
  • Bibliography 165
  • Name Index 185
  • Subject Index 191
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