Law of Continuity

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The Law of Continuity is a heuristic principle introduced by Leibniz based on earlier work by Nicholas of Cusa and Johannes Kepler. It is the principle that "whatever succeeds for the finite, also succeeds for the infinite".[1] Kepler used it to calculate the area of the circle by representing the latter as an infinite-sided polygon with infinitesimal sides, and adding the areas of infinitely many triangles with infinitesimal bases. Leibniz used the principle to extend concepts such as arithmetic operations, from ordinary numbers to infinitesimals, laying the groundwork for infinitesimal calculus. A mathematical implementation of the law of continuity is provided by the transfer principle in the context of the hyperreal numbers.

Leibniz's formulation[edit]

Leibniz expressed the law in the following terms in 1701:

In any supposed continuous transition, ending in any terminus, it is permissible to institute a general reasoning, in which the final terminus may also be included (Cum Prodiisset).[2]

He means here that, in a continuous phenomenon of any kind (be it motion in space or change in a mathematical function) the maxima and minima can be included as valid points. Leibniz provided several examples. He asserted that, for any quantities A and B such that A is greater than B where “A is continually diminished, until A becomes equal to B,” then “it is permissible to include under a general reasoning” both the prior cases where A > B and the final “terminus” where A = B. [3]

In a 1702 letter to French mathematician Pierre Varignon subtitled “Justification of the Infinitesimal Calculus by that of Ordinary Algebra," Leibniz adequately summed up the true meaning of his law, stating that "the rules of the finite are found to succeed in the infinite."[4] In the example above, he refers to the fact that, when A = B, the difference between the two is infinitely small; the Law of Continuity assures Leibniz that an infinitely small difference can be treated as an equality and hence compared to finite differences (or inequalities). Thus Leibniz asserts that the infinite and the finite can be treated alike.

The Law of Continuity became important to Leibniz's justification and conceptualization of the infinitesimal calculus, although the manuscript (see above) in which he develops the continuity-based version of calculus was never published during his lifetime.

See also[edit]


  1. ^ Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science. doi:10.1007/s10699-011-9223-1 [1] See arxiv
  2. ^ Child, J. M. (ed.): The early mathematical manuscripts of Leibniz. Translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes by J. M. Child. Chicago-London: The Open Court Publishing Co., 1920.
  3. ^ Leibniz, Gottfried Wilhelm, J. M. Child, and C. I. Gerhardt. The Early Mathematical Manuscripts of Leibniz; translated from the Latin texts published by Carl Immanuel Gerhardt with critical and historical notes. Chicago: Open court Pub. Co., 1920, p. 147.
  4. ^ Leibniz, Gottfried Wilhelm, and Leroy E. Loemker. Philosophical Papers and Letters. 2d ed. Dordrecht: D. Reidel, 1970, p. 544