merely a humbler variation

A man who sets out to justify his existence and his activities has to distinguish two different questions. The first is whether the work which he does is worth doing; and the second is why he does it, whatever its value may be. The first question is often very difficult, and the answer very discouraging, but most people will find the second easy enough even then. Their answers, if they are honest, will usually take one or other of two forms; and the second form is a merely a humbler variation of the first, which is the only answer we need consider seriously.

If intellectual curiosity, professional pride, and ambition are the dominant incentives to research, then assuredly no one has a fairer chance of satisfying them than a mathematician. His subject is the most curious of all—there is none in which truth plays such odd pranks. It has the most elaborate and the most fascinating technique, and gives unrivalled openings for the display of sheer professional skill. Finally, as history proves abundantly, mathematical achievement, whatever its intrinsic worth, is the most enduring of all.

G. H. Hardy - A Mathematician's Apology.

The proof of Fermat's Last Theorem

Professor Andrew Wiles in 1993 presented his proof to the public for the first time at a conference in Cambridge. In August 1993 however, it turned out that the proof contained a gap. In desperation, Andrew Wiles tried to fill in this gap, but found out that the error he had made was a very fundamental one. According to Wiles, the crucial idea for circumventing, rather than closing this gap, came to him on 19 September 1994. Together with his former student Richard Taylor, he published a second paper which circumvented the gap and thus completed the proof. Both papers were published in 1995 in a special volume of the Annals of Mathematics.

The Most Famous Problem in Mathematics (Fermat's Last Theorem)

Observations on Diophantus  had been written by Fermat in the margin next to Diophantus' Problem 8 in Book II, which asks 'Given a number which is a square, write it as a sum of two other squares.' Fermat's note said (in Latin) that:

On the other hand, it is impossible for a cube to be written as a sum
of two cubes or a fourth power to be written as a sum of two fourth
powers or, in general, for any number which is a power greater
than the second to be written as a sum of two like powers. I have
a truly marvellous demonstration of this proposition which this
margin is too narrow to contain.
In algebraic terms, Diophantus' problem asks for rational numbers
x, y, z satisfying the equation

x^2 + y^2 = Z^2.

This turns out to be a fairly easy task. What Fermat's marginal comment asserts is that if n is a natural number greater than 2, then the equation

x^n + y^n = z^n

has no rational solutions.

The Weierstrass Function July 18 1871 - II

The Gödel Incompleteness

At the turn of the century, the world-famous German mathematician David Hilbert proposed a programme for the development of all of mathematics within the strict formalization of the axiomatic method. According to Hilbert's belief, all of mathematics could be regarded as the formal, logical manipulation of symbols based on prescribed axioms. (This would mean that, in principle, a computer could be programmed to 'do all of mathematics'.) But in 1930, with two startling and totally unexpected theorems, the young Austrian mathematician Kurt Gödel demonstrated that Hilbert's programme could not possibly succeed.

Niccolò Tartaglia

Apart from his solution of the cubic equation, Tartaglia is remembered for other contribution to science. It was he who discovered that a projectile should fired at 45 degrees to achieve maximum range. His conclusion was based on incorrect theory, however, as is clear from Tartaglia's diagrams of trajectories, Tartaglia's Italian translation of the Elements was the first printed translation of Euclid in a modern language.

Bourbaki

J’apprends que Grothendieck n’est plus membre de Bourbaki. Je le regrette beaucoup, ainsi que les circonstances qui ont amené cette décision...Ce qui importait, c’est une opposition systématique, plus ou moins explicitée selon les uns ou les autres, contre son point de vue mathématique, ou plutôt son emploi par Bourbaki....C’est un scandale que Bourbaki, non seulement ne soit pas à la tête du mouvement functorial, mais encore n’y soit même pas à la queue. . . . Si certains membres fondateurs (e.g., Weil) désirent revenir sur leur décision de ne pas influencer Bourbaki dans la direction qu’il désire prendre, qu’ils le disent explicite- ment....Si Bourbaki refuse, non pas de se mettre dans le nouveau mouvement, mais d’en prendre la tête, alors les traités visant à la redaction des éléments des mathématiques (et pas seulment à ceux de la géométrie algébrique) seront rédigés par d’autres, qui s’inspireront non pas de l’esprit de Bourbaki 1960, mais de son esprit 1939. Ce serait dommage.

The Weierstrass Function July 18 1871 - I

Funzione continua ma non derivabile in nessun punto.

harmony and regularity

... In effect, if one extends these functions by allowing complex values for the arguments, then there arises a harmony and regularity which without it would remain hidden.

B. Riemann, 1851

plague of functions

I turn away in fright and horror from this lamentable plague of functions that do not have derivatives.

C. Hermite, 1893

Rolle’s theorem

Rolle’s theorem is a simple but important result, familiar to anyone who has moved just beyond elementary calculus into the beginnings of analysis. Essentially it tells us that if a differentiable function has equal values at a and b, then somewhere between those two points it must have a local maximum or a local minimum.

A more formal statement of the theorem, typical of those given in modern textbooks, is as follows.

Let f be a function that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b). If f(a) = f(b), then there exists a point c in (a, b) for which f ́(c) = 0.

Karl Weierstrass 2

........But Weierstrass did not leave Berlin. He wrote to Kova-

levskaya about Kronecker:

"I deeply regret that such a gifted person, possessing

such indisputable scientific merits, can also be so envious

and petty in his vanity."

Karl Weierstrass 1

Karl Weierstrass, Professor of Mathematics at Berlin

University, was famous as a great scientist, Leo Koenigsberger

being one of his students.

Weierstrass was a giant of thinking and has left an

indelible mark on mathematics. His name is familiar to

everyone who studied the theory of complex variables.

Proceeding from the theory of real numbers he had developed

he provided a logical substantiation for mathematical

analysis, while his theory of analytic functions is very

significant.

The name Weierstrass is deeply ingrained in many sections

and theorems of mathematics: the Bolzano-Weierstrass

theorem, the Weierstrass theory of elliptic functions; the

study of sufficient conditions for the maximum of an integral

(in the calculus of variations); the geodesic lines and minimal

surfaces in differential geometry; the theory of elementary

divisors in linear algebra; the application of series in the

theory of analytic functions (in 1841, when Weierstrass was

twenty-six, he knew of the theorem Laurent published in

1843); the theory of analytic continuation.