21

"Counterexamples" to Cauchy's theorem were "discovered" as soon as he proved it. Of course Cauchy knew all these "counterexamples" but he insisted that his theorem and its proof is correct until his death. In particular, Fourier's book on heat is full of these counterexamples. Fourier tries to clarify the matter by defining a continuous function as one "...


21

An excellent source for the early history of smooth non-analytic functions is the paper Gerald G. Bilodeau. The origin and early development of non-analytic infinitely differentiable functions, Arch. Hist. Exact Sci., 27 (1982), 115-135. MR84g:26017. Also, Dave Renfro wrote an excellent historical Essay on nowhere analytic $C^\infty$ functions in two ...


18

Zeno (around 500 BC) raised this paradox to argue against the notion of "plurality", arguing that a belief in the existence of many things rather than only one leads to absurd conclusions: If there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited. See section 2.2 of Zeno's paradoxes in the ...


14

This question is based on a misunderstanding. The statement that $\pi$ is constant has precise meaning: $\pi$ is a ratio of the length of circumference to the length of diameter. The statement that it is constant means that it is the same for all circles. (This statement is independent of the representation of this ratio with digits). Contrary to what many ...


13

I'll add to Alexandre's answer that with the way Cauchy thought about continuum and defined convergence, the "counterexamples" did not converge to the limits we ascribe to them today. To Cauchy "points" are "moving points" so he would consider $x=1-1/n$ with variable $n$ going to $\infty$ a "point" infinitesimally close to $1$. But $(1-1/n)^n$ does not ...


13

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-Standard Analysis (1974). Here is the full text of the remark (boldface mine): "I would like to point out a fact that was not explicitly mentioned by ...


12

Aristotle gave the first systematic rebuttal of Zeno, in particular he wrote in Physics: "…a line cannot be composed of points, the line being continuous and the point indivisible". According to Aristotle, a line can be composed only of smaller, indefinitely divisible lines, and not of points without magnitude. This was the mainstream view until the "...


12

The first rigorous integration theory in due to Eudoxus and Archimedes. It is called the method of exhaustion, and it allowed Archimedes to find the volumes of the balls, pyramids, cones, areas of segments of parabolas etc. (By the way, it was proved in 20th century that one cannot find the volume of a pyramid by pure geometric methods, some form of integral ...


12

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 thesis he also gave an example, correctly verified, of a Riemann integrable function whose discontinuities form a dense set. Riemann's thesis wasn't widely known ...


11

Georg Cantor discovered it. You can see at least : The Early Development of Set Theory : in late 1873, came a surprising discovery that fully opened the realm of the transfinite. In correspondence with Dedekind, Cantor asked the question whether the infinite sets $\mathbb N$ of the natural numbers and $\mathbb R$ of real numbers can be placed in one-to-...


11

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 that the continuum can be dissociated into points (contra Aristotle) was recent and hard enough, that it could be well-ordered strained credulity further. ...


10

One answer to your question could be that the separation actually came pretty late. Wikipedia claims that "Earlier authors held the result to be intuitively obvious, and requiring no proof.", so until continuity was formalized by Bolzano and Cauchy, I believe it doesn't make sense to find evidence. So we need to look for people who read Bolzano or Cauchy, ...


10

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 Augustin-Louis Cauchy (1789-1857) published his famous inequality in 1821 in the second of two notes on the theory of inequalities that formed the final part of ...


10

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 of proportions of Eudoxus-Euclid is equivalent to the theory of real numbers. Euclid's book X contains some very complicated theory of some irrational numbers. ...


9

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 integral inequality until well into the 20th century. https://archive.org/stream/actamathematica20upps#page/73/mode/1up The term "Cauchy-Schwarz inequality" was ...


9

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}=4\dfrac{d\smash[t]{\dfrac{dy}{d(x+\alpha t)}}}{d(x-\alpha t)} $ Hamilton (1843, p. 362): $ \quad\ \dfrac d{d\log t}\,(1+t)^{-m}\,\mathrm T_m $ Jacobi (1841, p....


8

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 introduces derivatives. From what I can determine, this has led historians to "read Cauchy" in different ways according to the embedded contextual ...


8

Fair warning: this answer doesn't completely answer the question, but I think it may answer the question as well as it is possible to do. The article which Hankel wrote (published 1971) that is generally credited with "rediscovering" Bolzano's work was an article in section 1 Theil 90 (Gregorius - Grezin) of Allgemeine Encyclopädie der ...


8

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 viewpoint of Gödel's realism, the significance of Robinson's work was to extend further the ordered number system, beyond the natural, integer, rational, and real ...


8

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"). At the later time, the idea arose to use mathematics to formalize logic. One of the early proponents of this idea was Leibniz, and it achieved further ...


8

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 lead to different results. This led him to introduce the notion of "singular integral". For $K = \frac{z}{x^2 + z^2}$ he showed that $\int_0^1\int_0^1\frac{\...


8

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)=x-\frac{1}{2}x^2+\frac{1}{3}x^3-\cdots \end{align*} where $-1<x\leq 1$, by integrating the equation \begin{align*} 1/(1+t)=1-t+t^2-\cdots \end{align*} ...


7

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 the hypotheses and claims that this results in a true theorem, providing a proof thereof. The existence of a correct result (modulo modified hypotheses) could ...


7

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 differentiable continuous function in 1875. But it was not so much for the rigor reason implied by Kline, and repeated in older history books, as for the difference ...


7

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 various generalisations of Rolle's Theorem equivalent to MVT and attributed to the likes of Cauchy, Bonnet, Serret, Dini, and Harnack. Looking at Besenyei's ...


7

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 sets $E$, such that: (1) $m(E) \ne 0$ for some set $E$; (2) $m(E + a) = m(E)$ for every real number $a$; (3) if $E_n$ are ...


7

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 the subsequent elaborations: "Let us consider a set of points of $(a, b)$; one can enclose in an infinite number of ways these points in an enumerably infinite ...


7

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, where he wrote: "In 1886 Giuseppe Peano stated [8] that the initial value problem: $y' = f(x,y),\, y(a) = b$, has a solution on the sole condition that $f$ is ...


7

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 know about it from Euclid and ancient historians of mathematics). The argument to find these limits was called the "method of exhaustion". It is ...


6

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 mathematical result ultimately must have a Russian source and since Buniakovski translated some of Cauchy's work into Russian, he was a natural choice for ...


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