19
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 ...
- 3,037
13
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 ...
- 3,037
12
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-...
- 73.2k
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 ...
- 73.2k
11
votes
Accepted
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 ...
- 73.2k
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 ...
- 73.2k
10
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
...
- 1,787
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 ...
- 47.5k
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 ...
- 1,364
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}=...
- 7,654
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 ...
- 4,962
8
votes
Accepted
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 ...
- 73.2k
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")....
- 47.5k
8
votes
Accepted
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 ...
- 73.2k
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)=...
- 251
7
votes
Accepted
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–...
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7
votes
Accepted
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 ...
- 6,689
7
votes
$\epsilon$-$\delta$ definition of continuity
Regarding the use of $\epsilon$ and $\delta$ in the context of the definition of continuity of functions, we can say that they "were in the air" since Cauchy.
See Augustin-Louis Cauchy's ...
- 14.3k
7
votes
Why did mathematicians not see that $f_n(x)=x^n$ is a counterexample to Cauchy's "theorem" about limits of continuous functions?
Cauchy's formulation of the theorem in 1821 is ambiguous and at any rate in a 1853 article Cauchy seems to state that it was incorrect as stated. On the other hand, in the 1853 article Cauchy modifies ...
- 4,962
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 ...
- 14.3k
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 ...
- 73.2k
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, ...
- 73.2k
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 ...
- 47.5k
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 ...
- 4,962
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 ...
- 690
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 ...
- 47.5k
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 ...
- 14.3k
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 ...
- 7,654
6
votes
What brought about the need for real analysis and formal logic in recent years?
We didn't seem to have a "proof theory" where we all agreed what constituted a proof or what was considered a correct / incorrect proof.
Yes we did. (The Greeks theorized proof by contradiction, ...
- 7,654
6
votes
How did Newton and Leibniz interpret the integral?
It was known even before Newton and Leibniz that areas under curves can be found by inverting the "computation of derivatives" (drawing tangents). In explicit geometric form this "fundamental theorem ...
- 73.2k
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