They are, then and now, incomplete developments aimed at convincing one's that some results are true. Now is the corpus of arguments used by our fathers "consistent"? This question does not make real sense, because "arguments" are not results, and are not "true or false", either individually or in groups. The corpus of results is certainly a corpus of true results, so is consistent, and was certainly recognized as such even by Berkeley (to my knowledge, the first serious contradiction involving results of calculusĬame 150 years after the founding period with Cauchy's theorem that a limit of continuous functions is continuous,Ĭombined with counter-examples from Fourier's theory, so is completely out of our scope). What we do have is a corpus of results from the founders of calculus (say Pascal, Descartes, Fermat, Newton, Leibniz), and a corpus of arguments they used to justify them. "free of contradictions", which was as clear then as now), but in the meaning of "way of arguing". The problem does not lie in the ambiguous meaning of "consistent" (which just means The question is not precise enough to get a definite answer, but not for the reason most people say in commentaries. Obviously, a modern mathematician would ask for either a concrete mathematically defined model of both $R^+$ and the valuation map - and the two mainstream such models are listed above, with perhaps more yet to come under category 3 - or at least to set it up in the form of calculus of propositions, with rigorous rules of inference albeit w/o a fixed interpretation of objects. after Euler!) can find a support of this point of view. Maybe, a devoted scholar of Leibnizz, Euler, etc. That is, accept that a non-0 infinitesimal is not equal to the real number 0, it just has the value 0. This is I believe the only sound way to view the medieval controversies around infinitesimals. For instance the value of a non-0 infinitesimal is 0, the value of its inverse is $\infty$, but $0\cdot \infty=1$ makes little sense in $R$. It occurs that the evaluation map cannot be a homomorphism, it always lacks something. For instance,ġ) $R^+$ consists of all convergent infinite real sequences and the valuation map is theĢ) $R^+$ is a nonstandard extension of $R$ and the valuation map is the ``standard part'' map ($\infty$ for infinitely large objects)ģ) Nilpotent or any other applicable exotics. $R^+ \to R\cup \$ - call it the valuation map. That is, one considers an extension - call it $R^+$ - of the true reals R and a map There is a related thread at the history SE: Ĭoming back to the B.Berkeley critics, there is a common denominator of all known getarounds, both the two mainstream ones (Wstrass and NSA) and exotic ones like the SDG interpretation. Thus he was using the term "inconsistent" in much the same sense it is used in modern logic. Berkeley claimed that calculus was based on an inconsistency that can be expressed in modern notation as $(dx\not=0)\wedge(dx=0)$. Was the early calculus consistent as far as most practitioners were concerned, as Vickers contended, or was it a most inconsistent way of arguing, as did Berkeley and Dunham? On the other hand, Peter Vickers in 2007 challenged "The ubiquitous assertion that the early calculus of Newton and Leibniz was an inconsistent theory" at (soon to appear in book form at Oxford University Press), and concluded that this only holds in a limited sense and "can only be imputed to a small minority of the relevant community". none of this mattered if the foundations were rotten". Although the results of the calculus seemed to be valid. Dunham concludes: "Bishop Berkeley had made his point. This passage is quoted by William Dunham in 2004. George Berkeley wrote in 1734 with reference to the early calculus that such a method is "a most inconsistent way of arguing, and such as would not be allowed in Divinity". Rather the question is of its CONSISTENCY. This question does NOT concern the RIGOR, or lack thereof, of the early calculus.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |