The basic timeline/lineage of real analysis

Classical → Medieval → Early Renaissance mathematics (500 BCE–1600 CE): A few major achievements include the axiomatization of geometry; the discovery of irrational numbers, Archimedes’ proof of the area formula for a circle around 250 BCE; Omar Khayyam’s geometric solution of the cubic equation around 1100 CE; the algebraic solution (in radicals) of the cubic equation by del Ferro/Tartaglia/Cardano around 1540 CE. Awareness of the fact that there were line segments whose lengths could not be represented as ratios of integers generally led mathematicians to view geometry as a separate (and more general) field from number theory.

Birth of Analytic Geometry (early 1600’s) Rene Descartes (1596–1650) and Pierre de Fermat (1601–1665) normalize the practice of allowing each point on a geometric line to symbolize a real number. Similarly a point in the plane could be represented as an ordered pair of numbers (hence the term Cartesian plane). This philosophical shift allowed the techniques of algebra/number theory to use to solve geometric problems. Descartes posed the tangent line problem and wrote: I dare say that this is not only the most useful and the most general problem in geometry that I know, but even that I have ever desired to know, and he worked on solving it using algebra.

This laid the groundwork for the development of the calculus in the mid 1600’s, which is probably the most important advancement in math history. The term calculus is meant to emphasize its use as a computational tool in applied sciences, especially for computing tangent lines and areas bounded by curves.

Ren´eDescartes (1789–1857) Pierre de Fermat (1815–1897) Now vs. then: mathematical culture

Modern (20th/21st century) Late 17th/18th century Mathematical statements are purely abstract Mathematics is inseparable from the study of statements, irrespective of their potential physics, astronomy, and other empirical fields application to physical phenomena Mathematical statements may be justified by a All mathematical statements should be combination of intuitive/plausibility arguments, justified by precise, careful definitions and plus empirical evidence detailed, logical proof Much faith in the power of symbolic Awareness of “counterexamples,” skepticism of manipulation to solve deep problems “symbol-pushing” as a method of argument A function is an expression that can be clearly A function is an arbitrary correspondence explained by a combination of algebraic between inputs and outputs operations and geometric descriptions Infinitesimals

The early calculus was developed on the basis of an “infinitely small quantity” or an “infinitesimal.” Other terms and notations for infinitesimals Differentials fluxions, fluents dx, dy, dz o˙,y ˙ In computations of derivatives and integrals, infinitesimals were used simultaneously as both: a nonzero quantity (so they can be divided out of expressions), and as a negligible quantity, i.e. an amount so close to zero it can be safely ignored.

Despite this seeming inconsistency, early authors of the calculus did not generally try to give a formal definition of an infinitesimal quantity, and instead resorted to vague descriptions involving “continuous motion” or “evanescent increments.” Newton’s Description of Infinitesimals (emphasis mine)

Perhaps it may be objected, that there is no ultimate proportion of evanescent quantities; because the proportion, before the quantities have vanished, is not the ultimate, and when they are vanished, is none. But by the same argument, it may be alleged, that a body arriving at a certain place, and there stopping, has no ultimate velocity; because the velocity, before the body comes to that place, is not its ultimate velocity; when it has arrived, it has none. But the answer is easy; for by the ultimate velocity is meant that with which the body is moved, neither before it arrives at its last place and the motion ceases, nor after, but at the very instant it arrives; that is, that velocity with which the body arrives at its last place, and with which the motion ceases. And in like manner, by the ultimate ratio of evanescent quantities is to be understood the ratio of the quantities not before they vanish, nor afterwards, but with which they vanish. In like manner the first ratio of nascent quantities is that with which they begin to be. And the first or last sum is that with which they begin and cease to be(or to be augmented or diminished). There is a limit which the velocity at the end of the motion may obtain, but not exceed. This is the ultimate velocity. And there is the like limit in all quantities and proportions that begin and cease to be. And since such limits are certain and definite, to determine the same is a problem strictly geometrical. But whatever is geometrical we may be allowed to use in determining and demonstrating any other thing that is likewise geometrical. –Isaac Newton The power of infinitesimals

The calculus developed rapidly due in part to its immense computational power and empirical effectiveness, and its developers’ faith in algebraic manipulation. The “free-wheeling calculus” reached its pinnacle in the time of Leonhard Euler (1707–1783).

Euler was possibly the most prolific mathematician of all time, and made vast contributions to calculus, number theory, physics, including developing the study of transcendental functions like exponentials, logarithms, trig functions, etc...

He also freely manipulated infinitesimals and “infinitely large numbers” in his mathematics to prove absurdities, like 1 1 − 1 + 1 − 1 + 1 − 1 + ... = 2 and 1 + 2 + 3 + 4 + 5 + ... = −1. An Excerpt on Euler From Burton’s History of Mathematics Criticisms of calculus and infinitesimals

Mathematicians, and especially geometricians, abstracting a quantity from a material, make for themselves of it a kind of kingdom which is as free as possible... In a word then, there are mathematicians who, in their kingdom of abstraction, assume indivisible substances which are without parts, without length, and without width... but not, however, the physicists for whom, working in the field of matter, no such thing is permitted. –Pierre Gassendi (1592–1655)

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities? –Bishop George Berkeley (The Analyst 1734)

Another criticism: infinitesimals violate the “Principle of Archimedes”: for every  > 0 no matter how small and every K > 0 no matter how large, there exists a positive integer N such that N > K. Euler’s Institutiones Calculi Differentialis (1755) which tried to develop calculus operations as a natural extension of algebra; treated infinite series more or less as long polynomials; completely ignored questions of convergence.

Joseph Louis Lagrange’s Theorie des fonctions analytiques (1797) which tried to circumvent infinitesimals and limits altogether by assuming that EVERY function admits Taylor series representations. Derivatives are defined simply by taking the coefficients from the Taylor series representation. He wrote that his calculus was redeemed from all considerations of the infinitely small, from vanishing quantities, from limits and from fluxions, and reduced to algebraic analysis of finite quantities.

Jean-le-Rond d’Alembert, in the 1750’s, advocated for the limit as the underlying notion of the calculus, defining it thus: One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude however small, though the first magnitude may never exceed the magnitude it approaches.

Early (Failed?) Attempts at Foundations

Colin Maclaurin’s Treatise on Fluxions (1742) a massive tome that was a direct response to Berkeley’s criticisms, but apparently inadequate in addressing problems with infinitesimals as a foundational concept. Joseph Louis Lagrange’s Theorie des fonctions analytiques (1797) which tried to circumvent infinitesimals and limits altogether by assuming that EVERY function admits Taylor series representations. Derivatives are defined simply by taking the coefficients from the Taylor series representation. He wrote that his calculus was redeemed from all considerations of the infinitely small, from vanishing quantities, from limits and from fluxions, and reduced to algebraic analysis of finite quantities.

Jean-le-Rond d’Alembert, in the 1750’s, advocated for the limit as the underlying notion of the calculus, defining it thus: One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude however small, though the first magnitude may never exceed the magnitude it approaches.

Early (Failed?) Attempts at Foundations

Colin Maclaurin’s Treatise on Fluxions (1742) a massive tome that was a direct response to Berkeley’s criticisms, but apparently inadequate in addressing problems with infinitesimals as a foundational concept.

Euler’s Institutiones Calculi Differentialis (1755) which tried to develop calculus operations as a natural extension of algebra; treated infinite series more or less as long polynomials; completely ignored questions of convergence. Jean-le-Rond d’Alembert, in the 1750’s, advocated for the limit as the underlying notion of the calculus, defining it thus: One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude however small, though the first magnitude may never exceed the magnitude it approaches.

Early (Failed?) Attempts at Foundations

Colin Maclaurin’s Treatise on Fluxions (1742) a massive tome that was a direct response to Berkeley’s criticisms, but apparently inadequate in addressing problems with infinitesimals as a foundational concept.

Euler’s Institutiones Calculi Differentialis (1755) which tried to develop calculus operations as a natural extension of algebra; treated infinite series more or less as long polynomials; completely ignored questions of convergence.

Joseph Louis Lagrange’s Theorie des fonctions analytiques (1797) which tried to circumvent infinitesimals and limits altogether by assuming that EVERY function admits Taylor series representations. Derivatives are defined simply by taking the coefficients from the Taylor series representation. He wrote that his calculus was redeemed from all considerations of the infinitely small, from vanishing quantities, from limits and from fluxions, and reduced to algebraic analysis of finite quantities. Early (Failed?) Attempts at Foundations

Colin Maclaurin’s Treatise on Fluxions (1742) a massive tome that was a direct response to Berkeley’s criticisms, but apparently inadequate in addressing problems with infinitesimals as a foundational concept.

Euler’s Institutiones Calculi Differentialis (1755) which tried to develop calculus operations as a natural extension of algebra; treated infinite series more or less as long polynomials; completely ignored questions of convergence.

Joseph Louis Lagrange’s Theorie des fonctions analytiques (1797) which tried to circumvent infinitesimals and limits altogether by assuming that EVERY function admits Taylor series representations. Derivatives are defined simply by taking the coefficients from the Taylor series representation. He wrote that his calculus was redeemed from all considerations of the infinitely small, from vanishing quantities, from limits and from fluxions, and reduced to algebraic analysis of finite quantities.

Jean-le-Rond d’Alembert, in the 1750’s, advocated for the limit as the underlying notion of the calculus, defining it thus: One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude however small, though the first magnitude may never exceed the magnitude it approaches. he claimed that EVERY periodic function admitted such a representation, including certain specific discontinuous ones;

he gave a practical means of computing the coefficients An and Bn, and the computations yielded physically verifiable solutions.

So Fourier’s series required mathematicians to expand their notions of what constitutes a “function.”

It turns out the problem of WHICH periodic functions admit Fourier series representations depends on subtleties about the continuity points of a function, and its solution requires careful analysis (inspiring Dirichlet 1829, Riemann in the 1850’s, Cantor and others).

Also, it was evident to mathematicians of the era that some Fourier series could not be differentiated term-by-term, in contrast to the familiar Taylor series.

Foundational Crisis: Fourier Series and “Hard Questions” in Calculus In 1822, Joseph Fourier published a paper describing how to build functions which are solutions to a certain partial differential equation, called the heat equation, which arises in thermodynamics. The functions were described as infinite series of trigonometric functions, i.e. functions of the form

∞ X (An cos nx + Bn sin nx). n=0

His work was highly controversial because he gave a practical means of computing the coefficients An and Bn, and the computations yielded physically verifiable solutions.

So Fourier’s series required mathematicians to expand their notions of what constitutes a “function.”

It turns out the problem of WHICH periodic functions admit Fourier series representations depends on subtleties about the continuity points of a function, and its solution requires careful analysis (inspiring Dirichlet 1829, Riemann in the 1850’s, Cantor and others).

Also, it was evident to mathematicians of the era that some Fourier series could not be differentiated term-by-term, in contrast to the familiar Taylor series.

Foundational Crisis: Fourier Series and “Hard Questions” in Calculus In 1822, Joseph Fourier published a paper describing how to build functions which are solutions to a certain partial differential equation, called the heat equation, which arises in thermodynamics. The functions were described as infinite series of trigonometric functions, i.e. functions of the form

∞ X (An cos nx + Bn sin nx). n=0

His work was highly controversial because he claimed that EVERY periodic function admitted such a representation, including certain specific discontinuous ones; So Fourier’s series required mathematicians to expand their notions of what constitutes a “function.”

It turns out the problem of WHICH periodic functions admit Fourier series representations depends on subtleties about the continuity points of a function, and its solution requires careful analysis (inspiring Dirichlet 1829, Riemann in the 1850’s, Cantor and others).

Also, it was evident to mathematicians of the era that some Fourier series could not be differentiated term-by-term, in contrast to the familiar Taylor series.

Foundational Crisis: Fourier Series and “Hard Questions” in Calculus In 1822, Joseph Fourier published a paper describing how to build functions which are solutions to a certain partial differential equation, called the heat equation, which arises in thermodynamics. The functions were described as infinite series of trigonometric functions, i.e. functions of the form

∞ X (An cos nx + Bn sin nx). n=0

His work was highly controversial because he claimed that EVERY periodic function admitted such a representation, including certain specific discontinuous ones;

he gave a practical means of computing the coefficients An and Bn, and the computations yielded physically verifiable solutions. Also, it was evident to mathematicians of the era that some Fourier series could not be differentiated term-by-term, in contrast to the familiar Taylor series.

Foundational Crisis: Fourier Series and “Hard Questions” in Calculus In 1822, Joseph Fourier published a paper describing how to build functions which are solutions to a certain partial differential equation, called the heat equation, which arises in thermodynamics. The functions were described as infinite series of trigonometric functions, i.e. functions of the form

∞ X (An cos nx + Bn sin nx). n=0

His work was highly controversial because he claimed that EVERY periodic function admitted such a representation, including certain specific discontinuous ones;

he gave a practical means of computing the coefficients An and Bn, and the computations yielded physically verifiable solutions.

So Fourier’s series required mathematicians to expand their notions of what constitutes a “function.”

It turns out the problem of WHICH periodic functions admit Fourier series representations depends on subtleties about the continuity points of a function, and its solution requires careful analysis (inspiring Dirichlet 1829, Riemann in the 1850’s, Cantor and others). Foundational Crisis: Fourier Series and “Hard Questions” in Calculus In 1822, Joseph Fourier published a paper describing how to build functions which are solutions to a certain partial differential equation, called the heat equation, which arises in thermodynamics. The functions were described as infinite series of trigonometric functions, i.e. functions of the form

∞ X (An cos nx + Bn sin nx). n=0

His work was highly controversial because he claimed that EVERY periodic function admitted such a representation, including certain specific discontinuous ones;

he gave a practical means of computing the coefficients An and Bn, and the computations yielded physically verifiable solutions.

So Fourier’s series required mathematicians to expand their notions of what constitutes a “function.”

It turns out the problem of WHICH periodic functions admit Fourier series representations depends on subtleties about the continuity points of a function, and its solution requires careful analysis (inspiring Dirichlet 1829, Riemann in the 1850’s, Cantor and others).

Also, it was evident to mathematicians of the era that some Fourier series could not be differentiated term-by-term, in contrast to the familiar Taylor series. Early analysis: limits and continuity

Augustin-Louis Cauchy (1789–1857) – First to describe limits in terms of “arbitrary” and “sufficient” closeness. First to give rigorous proofs of theorems in calculus using arguments with numerical estimates, the  (epsilon for “erreur”) and δ (delta for “diff´erence”)proofs. Even though he was uniquely rigorous for his era, he still did not entirely avoid reference to infinitesimals, e.g. his definition of continuity: The function f (x) is continuous within given limits if between these limits an infinitely small increment i in the variable x always produces an infinitely small increment, f (x + i) − f (x), in the function itself.

Bernard Bolzano (1781–1848) – a Bohemian priest who was dismissed from his post as professor of religion at University of Prague for heresy; his contributions to early analysis (three major papers written in German) became known posthumously. He gave an essentially correct definition of continuity and was the first to attempt to prove the Intermediate Value Theorem. As a lemma, he stated the Bolzano-Weierstrass Theorem. He was also first to identify the Least Upper Bound property. In his career, Cauchy wrote two or three different proofs of the Mean Value Theorem. By modern standards, all of them were incorrect.

Cauchy also made the following famous error: he wrote that if (fn) is a sequence of continuous functions and limn→∞ fn(x) = f (x) for every x in the domain, then f must be continuous also. This is not true– more on this later in the class!

Clairaut’s theorem, which says that mixed partial derivatives are equal for functions which are twice continuously differentiable, was assumed by Nicolaus Bernoulli in 1721; and incorrectly proven by Euler, Clairaut (1740), Lagrange (1797), and Cauchy (1823). A working proof did not appear until 1873.

The legality of differentiating a series of functions term-by-term was assumed in early treatments of the calculus, but is not legal in general, and not even for Fourier series.

Other Errors in Analysis

Early practitioners of the calculus including Euler and Gauss assumed the Intermediate Value Theorem without justification. Cauchy also made the following famous error: he wrote that if (fn) is a sequence of continuous functions and limn→∞ fn(x) = f (x) for every x in the domain, then f must be continuous also. This is not true– more on this later in the class!

Clairaut’s theorem, which says that mixed partial derivatives are equal for functions which are twice continuously differentiable, was assumed by Nicolaus Bernoulli in 1721; and incorrectly proven by Euler, Clairaut (1740), Lagrange (1797), and Cauchy (1823). A working proof did not appear until 1873.

The legality of differentiating a series of functions term-by-term was assumed in early treatments of the calculus, but is not legal in general, and not even for Fourier series.

Other Errors in Analysis

Early practitioners of the calculus including Euler and Gauss assumed the Intermediate Value Theorem without justification.

In his career, Cauchy wrote two or three different proofs of the Mean Value Theorem. By modern standards, all of them were incorrect. Clairaut’s theorem, which says that mixed partial derivatives are equal for functions which are twice continuously differentiable, was assumed by Nicolaus Bernoulli in 1721; and incorrectly proven by Euler, Clairaut (1740), Lagrange (1797), and Cauchy (1823). A working proof did not appear until 1873.

The legality of differentiating a series of functions term-by-term was assumed in early treatments of the calculus, but is not legal in general, and not even for Fourier series.

Other Errors in Analysis

Early practitioners of the calculus including Euler and Gauss assumed the Intermediate Value Theorem without justification.

In his career, Cauchy wrote two or three different proofs of the Mean Value Theorem. By modern standards, all of them were incorrect.

Cauchy also made the following famous error: he wrote that if (fn) is a sequence of continuous functions and limn→∞ fn(x) = f (x) for every x in the domain, then f must be continuous also. This is not true– more on this later in the class! The legality of differentiating a series of functions term-by-term was assumed in early treatments of the calculus, but is not legal in general, and not even for Fourier series.

Other Errors in Analysis

Early practitioners of the calculus including Euler and Gauss assumed the Intermediate Value Theorem without justification.

In his career, Cauchy wrote two or three different proofs of the Mean Value Theorem. By modern standards, all of them were incorrect.

Cauchy also made the following famous error: he wrote that if (fn) is a sequence of continuous functions and limn→∞ fn(x) = f (x) for every x in the domain, then f must be continuous also. This is not true– more on this later in the class!

Clairaut’s theorem, which says that mixed partial derivatives are equal for functions which are twice continuously differentiable, was assumed by Nicolaus Bernoulli in 1721; and incorrectly proven by Euler, Clairaut (1740), Lagrange (1797), and Cauchy (1823). A working proof did not appear until 1873. Other Errors in Analysis

Early practitioners of the calculus including Euler and Gauss assumed the Intermediate Value Theorem without justification.

In his career, Cauchy wrote two or three different proofs of the Mean Value Theorem. By modern standards, all of them were incorrect.

Cauchy also made the following famous error: he wrote that if (fn) is a sequence of continuous functions and limn→∞ fn(x) = f (x) for every x in the domain, then f must be continuous also. This is not true– more on this later in the class!

Clairaut’s theorem, which says that mixed partial derivatives are equal for functions which are twice continuously differentiable, was assumed by Nicolaus Bernoulli in 1721; and incorrectly proven by Euler, Clairaut (1740), Lagrange (1797), and Cauchy (1823). A working proof did not appear until 1873.

The legality of differentiating a series of functions term-by-term was assumed in early treatments of the calculus, but is not legal in general, and not even for Fourier series. Modern analysis: Weierstrass and Riemann

Karl Weierstrass (1815–1897) “The Father of Modern Analysis” – First to write the formal, modern -δ definition of a limit as a basis for the calculus. Provided rigorous proofs for many theorems of Euler and other pillars of early calculus. He was a big influence on many visionaries in “mathematical foundations” leading into the 20th century, like Georg Cantor, Richard Dedekind, and David Hilbert. He also mentored the first woman to receive a Ph.D. in mathematics, Sonya Kovalevsky (1850–1891) whom he described as having “the gift of intuitive genius.”

Weierstrass was a purveyor of pathological examples in the calculus. He shocked the mathematical world in 1872 when he constructed a function which is continuous at every point, but differentiable at no point. It can be constructed as a Fourier series– we will look at it next week.

Bernhard Riemann (1826–1866) – First to give a rigorous definition of an integral which is independent from the definition of the derivative. Gave new criteria for existence of Fourier series representation. Major Takeaways

Analysis is hard, it involves deep ideas, and requires practice and trial and error.

Keep studying!!