**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 |

**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. 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, 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 20e and the page Infinitesimal analysis without the axiom of choice.

Cauchy's sum theorem

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