View source for Law of continuity

{{Short description|Principle that whatever succeeds for the finite also succeeds for the infinite}}
{{redirects|Principle of continuity||Dedekind cut}}

The '''law of continuity''' is a heuristic principle introduced by [[Gottfried Wilhelm Leibniz|Gottfried 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".<ref>Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2011) [https://doi.org/10.1007%2Fs10699-011-9223-1 A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography]. [[Foundations of Science]]. {{doi|10.1007/s10699-011-9223-1}} See [https://arxiv.org/abs/1104.0375 arxiv]</ref> Kepler used the law of continuity to calculate the area of the circle by representing it 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 [[infinitesimal]]s, laying the groundwork for [[infinitesimal calculus]]. The [[transfer principle]] provides a mathematical implementation of the law of continuity in the context of the [[hyperreal number]]s.

A related law of continuity concerning [[intersection number]]s in [[geometry]] was promoted by [[Jean-Victor Poncelet]] in his "Traité des propriétés projectives des figures".<ref>Poncelet, Jean Victor. [https://archive.org/details/traitdespropri01poncuoft Traité des propriétés projectives des figures]: T. 1. Ouvrage utile à ceux qui s' occupent des applications de la géométrie descriptive et d'opérations géométriques sur le terrain." (1865), pp. 13–14</ref><ref>Fulton, William. Introduction to intersection theory in algebraic geometry. No. 54. American Mathematical Soc., 1984, p. 1</ref>

==Leibniz's formulation==
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'').<ref>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.</ref>

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."<ref>Leibniz, Gottfried Wilhelm, and Leroy E. Loemker. Philosophical Papers and Letters. 2d ed. Dordrecht: D. Reidel, 1970, p. 544 </ref> 

The law of continuity became important to Leibniz's justification and conceptualization of the infinitesimal calculus.

==See also==
*[[Transcendental law of homogeneity]]
*[[Uniformitarianism]]
*[[As above, so below]]

== References ==
{{reflist}}

{{Gottfried Wilhelm Leibniz}}
{{Infinitesimals}}

[[Category:Nonstandard analysis]]
[[Category:Gottfried Wilhelm Leibniz]]
[[Category:Infinity]]
[[Category:History of calculus]]
[[Category:Mathematics of infinitesimals]]
[[Category:Metaphysical principles]]


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