“Transcendental law of homogeneity”的意思、由来-开放百科全书

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Transcendental law of homogeneity

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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 is finite and is infinitesimal, then one sets

Similarly,

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]

References

1. ^Leibniz Mathematische Schriften, (1863), edited by C. I. Gerhardt, volume V, pages 377–382)
2. ^{{citation | last1 = Bos | first1 = Henk J. M. | author1-link = Henk J. M. Bos | arxiv = | doi = 10.1007/BF00327456 | issue = | journal = Archive for History of Exact Sciences | pages = 1–90 | title = Differentials, higher-order differentials and the derivative in the Leibnizian calculus | volume = 14 | year = 1974}}
3. ^{{citation | last1 = Katz | first1 = Mikhail | author1-link = Mikhail Katz | last2 = Sherry | first2 = David | author2-link = | arxiv = 1205.0174 | doi = 10.1007/s10670-012-9370-y | issue = | journal = Erkenntnis | pages = | title = Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, and Their Foes from Berkeley to Russell and Beyond | volume = | year = 2012}}

2 : History of calculus|Gottfried Leibniz

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