20 votes

Is Kline right that Cauchy believed that continuous functions must be differentiable?

I do not believe Cauchy wrote anything explicit concerning the extent to which we can deduce differentiability from continuity. However, in many places it seems that Cauchy assumes continuity and then ...
Dave L Renfro's user avatar
15 votes
Accepted

Riemann's Contribution to Integration

The following is a slightly edited version of my 31 January 2003 sci.math post archived at google groups. Riemann [6] introduced his integral in his December 1853 Habilitationsschrift thesis. In his ...
Dave L Renfro's user avatar
14 votes
Accepted

What did Kurt Gödel mean by nonstandard analysis being "the analysis of the future"?

The quote is from the remark Gödel made after Robinson's talk at the Institute for Advanced Study in Princeton in March 1973. It is reproduced in the preface to the second edition of Robinson’s Non-...
Conifold's user avatar
  • 76k
12 votes
Accepted

When did "neighbourhood of a point" first appear in the history of Taylor series?

The terminology "neighbourhood of a point" (in German, "Umgebung einer bestimmten Stelle") with its current meaning, dates back at least to 1841 when Weierstrass wrote his Zur ...
user6530's user avatar
  • 3,870
11 votes
Accepted

What were the earliest “unpleasant” consequences of the Axiom of Choice (and its negation) to be deduced?

Vitali's construction is probably the first example "unpleasant" in the modern sense, but well-ordering of the continuum was unpleasant enough to some at the time of Zermelo's proof in 1904. Accepting ...
Conifold's user avatar
  • 76k
11 votes

Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?

Some reflections of J. Michael Steele (cf. The Cauchy-Schwarz Master Class. Cambridge University Press, 2004, pp. 10-12) on this matter: THE PACE OF SCIENCE -- THE DEVELOPMENT OF EXTENSIONS ...
José Hdz. Stgo.'s user avatar
11 votes
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Source of a Poincaré quote: "Logic sometimes makes monsters..."

McTutor most likely took the passage from Kline's Mathematical Thought From Ancient to Modern Times, v.3, p.973, they reproduced his translation verbatim. Kline references Poincare's essay Dans la ...
Conifold's user avatar
  • 76k
10 votes

Is Kline right that Cauchy believed that continuous functions must be differentiable?

Many mathematicians did believe that continuous functions are mostly differentiable (except for some isolated points perhaps) until Weierstrass made a splash by publishing an example of a nowhere ...
Conifold's user avatar
  • 76k
10 votes
Accepted

When was the first recorded occurence of irrational and imaginary number usage in number theory?

Irrational numbers were used by the ancient Greeks when they were discovered. The earliest texts did not survive but there are plenty of them in Euclid. Though they are not called numbers. The theory ...
Alexandre Eremenko's user avatar
9 votes
Accepted

Historical occurrences of mathematicians substituting terms for $x$ in the denominator of $\mathrm{d}y/\mathrm{d}x$?

Gibbs (1889, p. 140): $ \qquad \dfrac{d\,\log\mathrm V}{d\,\log p} = - \dfrac{d\,\log n}{d\,\log\lambda} $ Riemann (1868, p. 89): $ \qquad \dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=...
Francois Ziegler's user avatar
9 votes

Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?

Maybe it is interesting to note that the term "l’inégalité de Schwarz" was coined by Poincaré in an 1896 paper in Acta Mathematica 20, p. 73, and was used in the French and German literature for the ...
Jan Peter Schäfermeyer's user avatar
8 votes

What did Kurt Gödel mean by nonstandard analysis being "the analysis of the future"?

In the 1970s Gödel recommended Robinson for membership in the British Academy of Sciences. His recommendation was based in part on Robinson's framework for analysis with infinitesimals. From the ...
Mikhail Katz's user avatar
  • 5,772
8 votes

What did the ratio of two magnitudes mean to ancient Greek mathematicians?

Fowler's Ratio in Early Greek Mathematics is a standard reference on the subject, see also his book Mathematics Of Plato's Academy (both are freely available). Book V, Definition 3 of Euclid's ...
Conifold's user avatar
  • 76k
8 votes
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What was Lebesgue's original definition of a measurable set?

Lebesgue used what Caratheodory called the outer measure. Fermat's library has an annotated translation of Lebesgue's 1901 paper On a Generalization of the Definite Integral, he is much briefer than ...
Conifold's user avatar
  • 76k
8 votes

What brought about the need for real analysis and formal logic in recent years?

As I explained in my answer to your other question, mathematics was always done using ordinary (non-formalized) logic. Attempts to formalize logic begin with Aristotle. (This is called "formal logic")....
Alexandre Eremenko's user avatar
8 votes
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Did Cauchy ever deal with double or triple integrals?

Yes, he did, multiple times. Singular double integrals (1814) In Mémoire sur les intégrales définies (1814) Cauchy studied why switching the order of integration in a double integral can sometimes ...
Conifold's user avatar
  • 76k
8 votes

Historically, what led to the question of the validity of interchange of limit operations?

G. H. Hardy wrote in A Course in Pure Mathematics in Appendix II, section A note on double limit problems Let us consider some special instances. In $\S\ 213$ we proved that \begin{align*} \log(1+x)=...
Markus Scheuer's user avatar
8 votes
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When and why was the concept of "having a least upper bound" dubbed "completeness", as in Axiom of Completeness?

According to Burn, Irrational numbers in English language textbooks, 1890–1915: Constructions and postulates for the completeness of the real numbers, "completeness" was first used by ...
Conifold's user avatar
  • 76k
7 votes
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Who was the first to prove the mean value theorem?

As per your comment requesting details of the linked paper : Besenyei's paper begins with a history of the development of Rolle's Theorem into its general form. It follows this with the history of ...
nwr's user avatar
  • 6,859
7 votes
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Who first proved the "Cantor-Heine theorem" on uniform continuity?

An explicit definition of uniform continuity was first published by Heine in Über Trigonometrische Reihen (On Trigonometric Series), Journal für die Reine und Angewandte Mathematik, 71 (1870), pp. 353–...
Conifold's user avatar
  • 76k
7 votes

What was Lebesgue's original definition of a measurable set?

See : Thomas Hawkins, Lebesgue's theory of integration, AMS (1975), page 122, regarding Lebesgue's thesis "Intégrale, longueur, aire," published in 1902: $m(E)$ is a nonnegative measure on bounded ...
Mauro ALLEGRANZA's user avatar
7 votes
Accepted

Were there proofs of the Lebesgue Differentiation Theorem without using maximal functions?

The original proof appears in Lebesgue's Lecons sur l'integration et la recherche des fonctions primitives, Paris, 1904, freely available if you read French. He only considered continuous monotone ...
Conifold's user avatar
  • 76k
7 votes
Accepted

How did Peano prove his existence theorem without Ascoli's theorem?

Peano did do something else, and not quite right, apparently. In 1969 Kennedy published a note in the American Mathematical Monthly titled Is There an Elementary Proof of Peano's Existence Theorem, ...
Conifold's user avatar
  • 76k
7 votes
Accepted

When did the Notion of "Limit" Arise and for What Purpose?

Rigorous notion of limit for special cases arose in the work of Eudoxus and Archimedes, when determining the length of a circle, volume of the pyramid etc. (The work of Eudoxus did not survive, we ...
Alexandre Eremenko's user avatar
6 votes

Who gets credit for the real numbers?

The Archimedean property as it is called, was used as an axiom by Archimedes, and he credited Eudoxus of Cnidus, who predates Euclid; also see this. In Section 7: Stevin, Malet says: In fact Stevin ...
Peter Diehr's user avatar
6 votes

Who gets credit for the real numbers?

Many people get credit, because this was a long story beginning in the ancient Greece. Euclid has a theory of proportions (based on earlier research) which is equivalent to modern theory of real ...
Alexandre Eremenko's user avatar
6 votes

Conditionally convergent series

This is the second part of my "preliminary answer". Both parts together are a little more than a third of what I ultimately intend to post, but I don't know when I'll get around to doing so. ...
Dave L Renfro's user avatar
6 votes

Who attached Buniakovsky's name to the Cauchy-Schwarz inequality?

This is the elephant phenomenon that Russian mathematicians like to joke about. It is widely known that Russia is the homeland of the elephants ("Rossia--rodina slonov"). Similarly any important ...
Mikhail Katz's user avatar
  • 5,772
6 votes

What is history behind Smith-Volterra-Cantor sets?

See : David Bressoud, A radical approach to Lebesgue's theory of integration, Cambridge UP (2008), 4.1 The Smith-Volterra-Cantor Sets : Cantor would prove in 1883 that given any set S, its derived ...
Mauro ALLEGRANZA's user avatar
6 votes

Does anybody know the history of how Peter Gustav Lejeune Dirichlet came up with the “nowhere continuous” Dirichlet function?

Page 169 of his 1829 paper. It arises as the simplest example of a function for which his proof of Fourier’s theorem fails (because it’s not integrable): It would remain for us to consider the ...
Francois Ziegler's user avatar

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