Chapter 10
The nineteenth century: making it true
Berkeley's objection stood unanswered for about ninety years. This chapter is the answer, and it took four countries and most of the century.
The pressure came from heat
Joseph Fourier's Théorie analytique de la chaleur (1822) claimed that any function has a trigonometric series. It was not fully true, and it was not properly provable, and it is the single most productive wrong-sounding statement of the century. Forcing it to make sense produced Dirichlet's definition of function, Riemann's integral, Cantor's set theory, and Lebesgue's measure (Source 383, O'Connor). The whole of modern analysis is downstream of a problem about heat.
There is an institutional cause too. The École Polytechnique was founded by the French revolutionary government to train engineers, and rigorous calculus is partly a by-product of the Revolution's need for them (Source 360, Grabiner, p. 189).
Fourier himself was arrested twice during the Terror, went to Egypt with Napoleon, and governed a province. He also believed heat was good for you and kept his rooms unbearably hot (Source 383, O'Connor).
Bolzano, silenced
Bernard Bolzano published Rein analytischer Beweis in Prague in 1817. The title is the whole argument: "Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation". The word doing the work is rein, purely: no pictures, no motion, no geometry (Source 363, Bolzano).
Guess before you read on
A curve starts below the axis and ends above it, so it has to cross somewhere. Every mathematician before 1817 agreed, and so does everybody you will ever ask. Bolzano called his paper Purely analytic proof, and the word doing the work is purely. Name what he said was wrong with the standard argument.
I have a guess
It is a picture, not a proof. Using a fact about geometry to establish a fact about numbers imports applied mathematics into pure, and he would not have it (Source 363, Bolzano; Source 387, Ruch).
If "it obviously crosses" felt like enough, you are in the company of everyone who came before him. The discomfort you are feeling now is the whole of the nineteenth century, and this chapter is what it produced.
His objection is one a fifteen-year-old feels immediately. Everybody thought the intermediate value theorem was obviously fine: a curve that starts below the axis and ends above it must cross it. Bolzano says that is a picture, not a proof (Source 363, Bolzano), and that using a fact about geometry to prove a fact about numbers is importing applied mathematics into pure (Source 387, Ruch). That distinction is worth a whole lesson.
He got the IVT, a working definition of continuity, and the Bolzano-Weierstrass theorem, all before Cauchy.
Then the state destroyed his career. A Catholic priest appointed to a chair of religious studies created to inoculate students against the French Revolution, he preached pacifism, social justice, and the absurdity of national hatred, until the Emperor personally fired him in December 1819 or January 1820 and forbade him to publish (Source 370, Morscher). His mail was opened. He was tried by his own Church, told to recant, and refused (Source 371, O'Connor). He kept writing for twenty-nine more years, mostly into a drawer.
Around 1830 he constructed a continuous, nowhere-differentiable function, roughly forty years before Weierstrass. Be precise about what he proved: he showed the non-differentiability points are dense, with a flawed continuity argument, and he did not complete the theorem (Source 384, O'Connor; Source 386, Johansson). The manuscript sat in the National Library in Vienna until Martin Jašek found it in 1920, a hundred years after Bolzano was silenced and two years after the Habsburg empire that silenced him ceased to exist. Bohemia had become Czechoslovakia, and Czech mathematicians went looking for their own forgotten genius (Source 386, Johansson). Six weeks after Jašek's announcement, Rychlík and Jarník independently proved the theorem Bolzano could not finish. The Functionenlehre was printed in 1930.
Cauchy, and a fight that is still running
Augustin-Louis Cauchy's Cours d'Analyse (1821) and Résumé (1823) are where the modern definitions arrive. He defined the derivative by a limit, gave the first real definition of the definite integral as a limit of sums, and proved the fundamental theorem in something close to the modern way.
He also explained the integral sign to his students by telling them it is a letter S that has been stretched, S for somme (Source 362, Cauchy). Two centuries later students still draw it without being told why.
His examples of a limit are charmingly old-fashioned: an irrational number is the limit of fractions approximating it, and "in geometry, the surface of the circle is the limit toward which the surfaces of inscribed polygons converge as the number of their sides grows" (Source 361, Cauchy, p. 4). The most modern definition in the book is illustrated with Archimedes.
And here is the live dispute, which you should teach as a dispute.
Cauchy's 1821 definition of continuity is stated in infinitesimal language, in his own words: "an infinitely small increment of the variable always produces an infinitely small increment of the function itself" (Source 361, Cauchy, p. 35). No epsilons. And his 1823 preface states his program as reconciling rigor "with the simplicity that results from the direct consideration of infinitely small quantities" (Source 362, Cauchy).
On page 131 of the Cours he states, with complete confidence, that a convergent series of continuous functions has a continuous sum. That is false as stated, because it needs uniform convergence, which nobody had yet. Niels Henrik Abel published the counterexample in 1826, citing the page number:
Dans l'ouvrage cité de M. Cauchy on trouve (page 131) le théorème suivant... "In the work of M. Cauchy cited above one finds, on page 131, the following theorem..." (Source 364, Abel, p. 71, footnote. English translated here, not quoted from a published translation.)
His counterexample is You can watch the error being caught in real time, in print, on facing pages. Abel was 23 or 24, he corrected the most famous analyst in Europe in a footnote, and in the main text he called the book indispensable. It is a masterclass in how to disagree (Source 364, Abel).
The dispute: was Cauchy simply wrong, or was he reasoning coherently with infinitesimals in a framework that later got vindicated? Judith Grabiner, Umberto Bottazzini, Craig Fraser, and Gert Schubring say the theorem is false. Detlef Laugwitz, Mikhail Katz, and others argue for a coherent infinitesimal reading (Source 367, Bascelli; Source 368, Borovik). Both sides were read in full for this book and the question is not settled. One of the papers is titled as a direct riposte to Grabiner's article title, thirty years later (Source 368, Borovik). Historians of mathematics subtweet each other in journal titles.
Cauchy the man: born in Paris five weeks after the storming of the Bastille, and on the losing side of every subsequent French revolution. He followed Charles X into exile in 1830 rather than swear a loyalty oath, giving up his salary, his chairs, and his country (Source 372, O'Connor).
The Galois myth does not survive the documents. The version students meet, a genius destroyed by a jealous establishment with his life's work binned by Cauchy, is not what they show. The lost manuscript was the February 1830 Grand Prize submission, held by Fourier, who died that April. Cauchy's documented 1829 role was to referee it and advise resubmission (Source 380, O'Connor). What the documents show is worse and more ordinary: a bad-tempered teenager, a referee who died, a second referee who wanted clearer writing, and a pointless duel. That is a better lesson than the myth.
Abel, and a quotation that is a translation of a translation
Abel's "divergent series are the invention of the devil" is not what Abel wrote, and the phrase has been traced back to its source (Source 490, Abel).
The letter is to Bernt Holmboe, from Berlin, not Paris, dated 16 January 1826 (the 1902 editors argue it was written on the 14th and dated when sent). The original Dano-Norwegian, at p. 16:
Divergente Rækker ere i det Hele noget Fandensskab, og det er en Skam at man vover at grunde nogen Demonstration derpaa.
Translated directly: "Divergent series are, on the whole, a piece of devilry, and it is a disgrace that anyone should dare to found any demonstration upon them."
The French translation in the same 1902 volume, by P. G. la Chesnais, reads "une invention du diable". "Invention of the devil" is la Chesnais's phrase, not Abel's (noget Fandensskab means "some devilry"), and the English version drops Abel's hedge i det Hele, "on the whole" (Source 490, Abel).
The human framing is good too: Abel is 23, on a traveling scholarship, writing home to his old schoolmaster. In the same run of letters he is being fitted for new clothes so as to be complet when he reaches Paris, going to the opera, and playing cards. The sentence that ended up on a thousand lecture-room walls was a young man's aside in a chatty letter, and he softened it immediately (Source 490, Abel).
Abel died on 6 April 1829, aged 26, of tuberculosis, two days before a letter arrived from Crelle announcing a professorship in Berlin (Source 364, Abel).
His line about Gauss is also worth having: "He is like the fox, who effaces his tracks in the sand with his tail" (Source 385, O'Connor), on Gauss's habit of removing every trace of how he found things.
Weierstrass
Karl Weierstrass taught gymnasium for years, including gymnastics and handwriting, published almost nothing for over a decade, and then produced one paper that made him famous across Europe within months (Source 373, O'Connor). Sources say he became a professor at 39, though he was 40 or 41, and give his teaching span as 14 years with some variation. He is the strongest available answer to a student who thinks it is too late for them.
The ε-δ definition reached its final form through his Berlin lectures, many of which were published only through students' notes, which is why "who wrote it first" is genuinely tangled (Source 373, O'Connor). He built on Bolzano and Cauchy. The ε is literally the "e" of erreur, and the δ of différence (Source 360, Grabiner). The symbol of mathematical precision started life as the symbol of being wrong.
The monster function, 1872. Continuous everywhere, differentiable nowhere. Get three details right:
- Weierstrass presented it on 18 July 1872 and never published it. Paul du Bois-Reymond published it, in 1875 (Source 386, Johansson).
- He opened by giving the credit to Riemann (c. 1861) (Source 386, Johansson).
- Charles Cellérier is a third independent discoverer (c. 1860, published 1890) (Source 386, Johansson).
Hermite's reaction is verified from the primary source, Correspondance vol. II p. 318, letter 374, Paris, 20 May 1893:
Je me détourne avec effroi et horreur de cette plaie lamentable des fonctions continues qui n'ont point de dérivées. "I turn away with dread and horror from this lamentable plague of continuous functions that have no derivatives." (Source 365, Hermite)
The standard English version drops the word "continuous", which destroys the point. And the better line, in the same letter, is the one nobody quotes: "L'Analyse retire d'une main ce qu'elle donne de l'autre", "Analysis takes back with one hand what it gives with the other" (Source 365, Hermite). Hermite wrote it at 70, two years before he died, to a friend who had one year to live, and in the same letter he cheerfully says that when it is thundery he does nothing but daydream or sleep.
The ending is the best part. Sixty years later, Banach and Mazurkiewicz proved that almost every continuous function is nowhere differentiable (Source 388, Kesavan). Hermite was not turning away from a plague. He was turning away from the general case.
And there is an internal irony worth showing a class: the standard proof that Weierstrass's monster is continuous uses the Weierstrass M-test and the theorem that a uniform limit of continuous functions is continuous. The tool that tames the monster is the repair of the exact error Cauchy made in 1821 (Source 389, Calder).
Riemann, Dedekind, Cantor, Lebesgue
Bernhard Riemann defined his integral in his 1854 Habilitationsschrift on trigonometric series, because he could not even state his question until he had said what an integral is (Source 391, Riemann). He was 27. It was not published until 1867, after he died at 39. His definition uses arbitrary partitions and arbitrary tags, and it ends:
If it does not have this property, then ∫f(x)dx has no meaning. (S391, Riemann's Werke; translated from the German here, so treat the wording as a rendering rather than a set text)
What students learn as "the Riemann integral" is usually Darboux's 1875 reformulation with upper and lower sums (Source 391, Riemann, and the notation file).
At the same Habilitation, the candidate proposes three topics and the examiner picks. Gauss, against custom, picked the third, the one on the foundations of geometry that Riemann had prepared least. The result is the mathematics Einstein needed sixty years later (Source 375, O'Connor). A job interview produced the toolkit of general relativity.
Richard Dedekind gives the best classroom hook in the whole century. A young professor about to teach calculus for the first time realizes he cannot prove the most basic fact about limits without drawing a picture, is embarrassed, and resolves to fix it. He fixed it on a specific Wednesday:
I succeeded Nov. 24, 1858, and a few days afterward I communicated the results of my meditations to my dear friend Durège with whom I had a long and lively discussion. (Source 366, Dedekind, Beman's translation, p. 2)
Then he did not publish for fourteen years, because "the presentation did not seem altogether simple, and further, the theory itself had little promise" (Source 366, Dedekind).
Charles Méray published a construction of the reals in 1869, before both Dedekind (1872) and Cantor (Source 378, O'Connor). He is usually skipped, and he is a subtler "who got robbed" case than Bolzano: nobody suppressed him. He published in French, in a normal journal, in a country whose mathematical culture had no appetite for the question. He was just in the wrong place (Source 378, O'Connor).
Georg Cantor built the actual infinite into mathematics and called infinitesimals "the cholera-bacilli of mathematics" (Source 369, Bell). His 1888 manifesto sentence is the fault line running to constructivism: whether we can decide something is irrelevant to whether it is true (Source 382, O'Connor). Kronecker, and later Brouwer and Bishop, thought exactly the opposite. The romantic story that Kronecker single-handedly drove Cantor mad does not survive checking: MacTutor dates the first breakdown to 1884 and presents the illness as recurrent and independent, while documenting the professional obstruction as real (Source 376, O'Connor).
Henri Lebesgue's 1902 thesis Intégrale, longueur, aire answered the question the Riemann integral could not: what do you do with a function that is discontinuous everywhere? His coin analogy, that Riemann counts coins in the order you pull them out of your pocket while Lebesgue sorts them into denominations first, reaches us in paraphrase. Nobody here traced the exact words back to Lebesgue, so take the analogy as his idea rather than as his sentence (Source 377, O'Connor). Two founders of modern French analysis, Lebesgue and Borel, fell out permanently over war work in 1914 to 1918 (Source 377, O'Connor). Mathematics is done by people who stop speaking to each other.
The coda: Berkeley answered, 232 years later
In 1966 Abraham Robinson published Non-standard Analysis, giving infinitesimals a rigorous foundation through model theory. Berkeley's ghosts got a birth certificate (Source 381, O'Connor; Source 432, OConnor and Robertson).
The fix that makes them legal is a single extra idea: the standard part. Take the ratio, then round to the nearest real number. H. Jerome Keisler's 1976 textbook puts the payoff in one line:
Leibniz was on the right track, but 300 years too soon! (Source 440, Keisler)
Errett Bishop attacked it on constructivist grounds (Source 390, Katz). His 1977 review is quoted all over the internet, and the review itself was blocked on every attempt to reach it here, so the sentences you will meet carry no source anyone has checked (Source 395, Bishop; Source 390, Katz). The argument is the nineteenth century's again in twentieth-century dress: Kronecker told Cantor mathematics must be built from the integers; Bishop told Robinson a proof must carry numerical content. Same objection, ninety years apart, aimed at the two things nineteenth-century rigor made respectable: the actual infinite and the infinitely small (Source 390, Katz).
And the loop closes on Chapter 2. Archimedes had both methods in 250 BCE. He found his answers with infinitesimals and published them with something like exhaustion, because he did not trust the first method in print. Two thousand two hundred years later, that is still what most mathematicians do (Source 440, Keisler).
One question before you go
Weierstrass presented a function in 1872 that is continuous everywhere. What can it not do, anywhere at all?
Show the answer
Have a derivative. Continuous everywhere, differentiable nowhere. Hermite turned away "with dread and horror from this lamentable plague of continuous functions that have no derivatives", and the standard English version drops the word "continuous", which destroys the point (Source 365, Hermite).
Derivatives 1.6 trains you to read a derivative off the shape of a graph, and this function is the edge of that skill. Sixty years later Banach and Mazurkiewicz proved almost every continuous function is one of these, so the well-behaved curves in your course are the rare case (Source 388, Kesavan).