Transcendental law of homogeneity

Heuristic principle enunciated by Gottfried Wilhelm Leibniz

In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potentiarum et differentiarum, et de lege homogeneorum transcendentali.[1] Henk J. M. Bos describes it as the principle to the effect that in a sum involving infinitesimals of different orders, only the lowest-order term must be retained, and the remainder discarded.[2] Thus, if a {\displaystyle a} is finite and d x {\displaystyle dx} is infinitesimal, then one sets

a + d x = a . {\displaystyle a+dx=a.}

Similarly,

u d v + v d u + d u d v = u d v + v d u , {\displaystyle u\,dv+v\,du+du\,dv=u\,dv+v\,du,}

where the higher-order term du dv is discarded in accordance with the TLH. A recent study argues that Leibniz's TLH was a precursor of the standard part function over the hyperreals.[3]

See also

  • Law of continuity
  • Adequality

References

  1. ^ Leibniz Mathematische Schriften, (1863), edited by C. I. Gerhardt, volume V, pages 377–382)
  2. ^ Bos, Henk J. M. (1974), "Differentials, higher-order differentials and the derivative in the Leibnizian calculus", Archive for History of Exact Sciences, 14: 1–90, doi:10.1007/BF00327456, S2CID 120779114
  3. ^ Katz, Mikhail; Sherry, David (2012), "Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond", Erkenntnis, 78 (3): 571–625, arXiv:1205.0174, doi:10.1007/s10670-012-9370-y, S2CID 254471766
  • v
  • t
  • e
Gottfried Wilhelm Leibniz
Mathematics and
philosophy
Works
Category
  • v
  • t
  • e
History
Related branches
Formalizations
Individual concepts
Mathematicians
Textbooks


Stub icon

This article about the history of mathematics is a stub. You can help Wikipedia by expanding it.

  • v
  • t
  • e