Reappraisal of the procedures of Cauchy's infinitesimal analysis
|Pioneer||Journal where reappraisal appeared||Title of article||Link to article containing reappraisal|
|A. L. Cauchy||Perspectives on Science||Cauchy's continuum||11b|
|A. L. Cauchy||Foundations of Science||Who gave you the Cauchy-Weierstrass tale? The dual history of rigorous calculus||12b|
|A. L. Cauchy||Foundations of Science||A Cauchy-Dirac delta function||13g|
|A. L. Cauchy||Mat. Stud.||Cauchy, infinitesimals and ghosts of departed quantifiers||17a|
|A. L. Cauchy||Foundations of Science||Controversies in the foundations of analysis: Comments on Schubring's Conflicts||17e|
|A. L. Cauchy||Foundations of Science||Cauchy's infinitesimals, his sum theorem, and foundational paradigms||18e|
|A. L. Cauchy||Antiquitates Mathematicae||19th century real analysis, forward and backward||19a|
|A. L. Cauchy||Real Analysis Exchange||Cauchy's work on integral geometry, centers of curvature, and other applications of infinitesimals||20b|
|A. L. Cauchy||British Journal for the History of Mathematics||Continuity between Cauchy and Bolzano: Issues of antecedents and priority||20a|
|A. L. Cauchy||Mathematics Today||A two-track tour of Cauchy's Cours||21f|
|A. L. Cauchy||Antiquitates Mathematicae||Historical infinitesimalists and modern historiography of infinitesimals||22a|
Some misconceptions about Cauchy's analysis
Misconception 1. George Berkeley detected inconsistencies in the foundations of the infinitesimal calculus.
Such an assumption is incorrect, as we argued in the leading philosophy journal Erkenntnis here: 13f. See the perceptive review by Marcelle Guillaume for MathSciNet at this link.
Misconception 2. Cauchy for his part introduced differentials in a completely finitistic manner, indicating that at least on this score he distanced himself from infinitesimals.
Cauchy indeed gave such a definition of differentials, but his definition is a rather tautological one: Cauchy set dx equal to any nonzero real number and defined dy to be f'(x)dx. On the other hand, Cauchy's infinitesimals are involved in the definition of the derivative f'(x).
Misconception 3. Cauchy systematically used the epsilon-delta definition of limit.
In fact, Cauchy's definition of limit follows Lacroix, who was a very broad mathematician and used a wide variety of foundational approaches, including infinitesimals, and not merely limits. Cauchy's definition of limit was based on a primitive notion of variable quantity. Cauchy never gave an epsilon-delta definition of limit. While some of the proofs he gave in Cours d'Analyse did use incipient forms of epsilon-delta arguments, they all lack the trademark modern aspect of exhibiting an explicit dependence of delta on epsilon.
Misconception 4. Cauchy defined infinitesimals as a particular type of sequence.
In most cases, Cauchy says that such a sequence becomes an infinitesimal, implying that a certain change in nature is taking place. What the nature of such a change may be is not explicitly clarified by Cauchy. However, from the fact that he presents a series of theorems on asymptotic behavior of infinitesimals, it is reasonable to assume, as do some scholars including Laugwitz, that two sequences are identified if they differ only in a finite number of terms. The claim that Cauchy defined an infinitesimal as a variable or a function tending to zero, or as a null sequence, is found persistently in the literature but is incorrect, since in most cases Cauchy writes that a sequence becomes an infinitesimal rather than an infinitesimal being a sequence.
Misconception 5. Completeness of the real numbers entails that there is no room there to fit in infinitesimals in rigorous real analysis.
This may be true but that's because a Dedekind-complete continuum is necessarily Archimedean. There is no evidence that Cauchy would have accepted the latter property. The syntactic approach to analysis with infinitesimals enables infinitesimals to be found within the ordinary real line; see Infinitesimal analysis without the axiom of choice.
Misconception 6. Cauchy introduced limits, a rival approach to infinitesimals.
While Cauchy followed Lacroix' approach to limits, the true opposition is not between infinitesimals and limits, but rather between infinitesimals and epsilon-delta definitions as eventually developed by Weierstrass. The concept of limit itself is present in both approaches. In the modern infinitesimal approach, the limit is defined via the standard part, or shadow. Cauchy was also aware of the fact that to compute the limit one needs to discard higher-order terms in the infinitesimal.
Misconception 7. Once rigorous foundations were introduced for analysis, there is no more need for infinitesimals which were thereby superseded.
The mathematical developments at the end of the 19th century not so much superseded infinitesimals, as failed in formalize them as part of the foundational developments that took place at that time. As far as I know Cantor was the one to have claimed infinitesimals to be contradictory; I am not aware of any such claims by either Dedekind or Weierstrass.
Misconception 8. The real numbers of Cantor and Dedekind were accepted as the foundation of modern analysis and infinitesimals were excluded from real analysis.
This is a largely incorrect claim, as documented in the long article by P. Ehrlich in 2006, who documents an uninterrupted chain of work on non-Archimedean systems by mathematicians such as Thomae, Stolz, du Bois-Reymond, Veronese, Borel, and others.
Misconception 9. Few mathematicians are still willing to spend time on the controversial infinitesimals.
Today infinitesimals are no longer controversial since Abraham Robinson provided a rigorous foundation for them, and Terry Tao makes routine use of them in his research. Moreover, they are a more efficient tool in teaching calculus than non-infinitesimal epsilon-delta methods, as argued in the article 17h.
Misconception 10. Cauchy's concept of infinitesimal is just an abbreviation for variables having limit to zero.
For Cauchy the primitive notion was not that of limit but rather that of a variable quantity, and he defined both infinitesimals and limits in terms of variable quantities.
Misconception 11. It is not worth bringing infinitesimal to the subject again so as to get students confused.
On the contrary, the students prefer infinitesimal definitions over epsilon-delta ones by a large margin, and so do those of their teachers who are familiar with the approach combining infinitesimals and epsilon-delta; see article 17h.
Misconception 12. Euler's and Leibniz's infinitesimals were constant, but Cauchy choose to define them as variables of a specific kind.
This is a dubious interpretation of Cauchy's infinitesimals, as I argued. In many articles and books Cauchy uses infinitesimals in ways very similar to Leibniz and Euler; for details see the article 20b.
Misconception 13. A mathematical implementation of infinitesimals was achieved by Abraham Robinson in 1961, this led to a re-evaluation of Cauchy's infinitesimal, but seems did not make much sense.
To whom did this not make much sense? I would argue that the Robinson-Laugwitz reading of Cauchy's infinitesimals in general and the sum theorem in particular "does not make sense" only to historians who learned only the epsilon-delta approach (due to Weierstrass and Dini, not Cauchy) as undergraduates, and are unwilling to learn new approaches to analysis. It turns out that infinitesimal analysis does not depend on assuming the axiom of choice and can be done without it; see the article 21e and the page Infinitesimal analysis without the axiom of choice.
Cauchy's sum theorem
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