Paul Erdős: The Wandering Genius Who Loved Only Numbers

 

On 26 March 1913, Paul Erdős was born: one of the most prolific and influential mathematicians of the 20th century, a legendary figure not only for his academic contributions, but also for his eccentric lifestyle and his unique vision of mathematics as a collaborative and universal activity.

Born in Budapest, Hungary, to a Jewish family, Erdős lived a nomadic existence, traveling incessantly between conferences, universities and colleagues' homes, with a suitcase and a few personal belongings, dedicating his life entirely to mathematics.

He is known for his work in fields such as number theory, combinatorics, graph theory and probability. He published over 1,500 papers before his death in 1996, collaborating with more than 500 co-authors, a record that gave rise to the famous "Erdős number", an index measuring the collaborative distance between a mathematician and Erdős himself through joint publications. Among his most famous results are the Erdős-Kac theorem (also known as the fundamental theorem of probabilistic number theory), his contributions to Ramsey theory, and his work on prime numbers. His ability to pose deep and challenging problems often opened up new avenues of research.

Erdős’s legacy goes beyond his theorems. He embodied an approach to mathematics as a collective, almost mystical enterprise: for him, mathematical problems were part of an ideal “Book,” written by some kind of transcendent entity, which mathematicians were tasked with discovering. His recurring phrase, “my brain is open,” reflected his willingness to collaborate with anyone, regardless of age, nationality, or prestige. This spirit inspired a culture of openness and sharing in the mathematical community. Erdős was also a symbol of cultural resilience. Growing up in a time of political turmoil and persecution of Jews in Europe, he transformed personal hardship into tireless creative energy. His minimalist lifestyle – he owned little, lived as a guest of others, and donated much of his earnings in prizes for solving mathematical problems – reflected an almost monastic dedication to knowledge.

Morally, Erdős represents an ideal of intellectual altruism. He did not seek personal fame or material wealth, but the advancement of knowledge for the common good. His generosity manifested itself in the monetary prizes he offered for solving problems he proposed, a way of stimulating the curiosity and talent of others. Furthermore, his refusal to conform to social conventions – he never married and did not have a permanent home – can be seen as a rebellion against materialism and a call to pursue what truly matters. Erdős was also deeply human: he loved to joke, invented terms such as "epsilons" for calling children (referring to the small mathematical quantity ε), and had an inclusive ethic that made him a friend to all. His death from a heart attack, while attending a conference in Warsaw on 20 September 1996, was emblematic: he died doing what he loved, surrounded by the community he had helped build.

Paul Erdős left a legacy that intertwines mathematical brilliance, a collaborative cultural approach, and a moral example of selfless dedication. His life demonstrates that true greatness lies not only in achievements, but in the way one lives and inspires others. To this day, his name is synonymous with creativity, connection, and a passion for discovery.

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Image: Erdős with Terence Tao (ten-year-old ), 1985. Via Wikimedia Commons 

References and further reading

Paul Erdős- Biography 

PAUL ERDŐS 

N Is a Number: A Portrait of Paul Erdös 

The Man Who Loved Only Numbers 

Emmy Noether: The Queen Of Mathematics Who Revolutionized Physics

Born on 23 March 1882, Emmy Noether, considered by Einstein the most important woman in history of mathematics, had a great passion for theoretical physics and abstract algebra.

She made many relevant contributions to the field of mathematics.

In 1915, Noether accepted an invitation from Hilbert and Klein to move to Göttingen, the "Mecca of Mathematics."

Many of the faculty did not want her there, but she worked hard and soon was given a job as a lecturer.

Even though she still was not paid for her efforts, for the first time, Noether was teaching under her own name.

Three years later, she began receiving a small salary for her work.

There she became a world-class algebraist who attracted students and younger colleagues who themselves became leading mathematicians.

In fact, during her time at the University of Göttingen, Emmy accumulated a small following of students known as "Noether's boys".

In 1933, Hitler and the Nazis came into power in Germany.

Noether moved to the USA, where Bryn Mawr College offered her a position teaching. There she taught until her death in 1935, being an inspiring teacher for her students.

"In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began. In the realm of algebra, in which the most gifted mathematicians have been busy for centuries, she discovered methods which have proved of enormous importance in the development of the present-day younger generation of mathematicians. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature."

- Albert Einstein, Princeton University, May 1,1935 (Source)

Emmy discovered the symmetries of a physical system are inextricably linked to physical quantities that are conserved, such as energy. These ideas became known as Noether’s theorem, the foundations of modern physics.

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References

Celebrate the mathematics of Emmy Noether

Emmy Noether, Mentors & Colleagues 

Emmy Noether

Image: Portrait of Emmy Noether, before 1910 (Public Domain)

Author: unknown

Image Source 

La Matematica È il Segreto Nascosto per Capire il Mondo

Pubblico un interessante video da TED Ideas worth spreading, in cui Roger Antonsen (logico, matematico e informatico norvegese) spiega come un piccolo cambio di prospettiva può rivelare schemi, numeri e formule quali passaggi verso l'empatia e la comprensione.

Il filmato è sottotitolato in lingua italiana, ma, se preferite la lettura, più avanti trovate la traduzione della TED translator Silvia Fornasiero.

Teoria dei Numeri: La Regina della Matematica

Fonte dell'immagine

Si dice che Carl Friedrich Gauss, uno dei più grandi matematici di sempre, avesse affermato:

"La matematica è la regina delle scienze e la teoria dei numeri è la regina della matematica".

Le proprietà dei numeri primi giocano un ruolo cruciale nella teoria dei numeri, che studia le proprietà dei numeri interi; una domanda intrigante è come essi siano distribuiti all'interno degli altri numeri interi.
Nel 19° secolo ci sono stati indubbi progressi nella risposta a questa domanda con la dimostrazione del Teorema dei numeri primi, ma abbiamo anche visto Bernhard Riemann proporre quello che molti considerano il più grande problema irrisolto in matematica - l'Ipotesi di Riemann.

La teoria dei numeri è un'antica disciplina. Le prime formulazioni dei problemi della teoria dei numeri, e le soluzioni di alcuni di essi, risalgono, infatti, a Pitagora e alla sua scuola e sono enunciate negli Elementi di Euclide.
Euclide ha dimostrato l'infinità dei numeri primi con il metodo della reductio ad absurdum, un metodo di dimostrazione che procede con la formulazione di una proposizione che poi si risolve in una contraddizione, dimostrando così che la proposizione è falsa.
Un altro problema è, sin dai tempi più antichi, quello della risoluzione di equazioni a coefficienti interi: Pitagora risolveva equazioni quadratiche legate ai triangoli rettangoli, Euclide utilizzava equazioni lineari per calcolare il massimo comun divisore di due numeri interi e Archimede studiava equazioni quadratiche, note oggi come equazioni di Pell. 

Il Problema dei Tre Raggi

Il Problema dei Tre Raggi è l'apprezzabile lavoro svolto da Tamar Barabi, una studentessa israeliana del primo anno di scuola superiore. ll risultato dei suoi sforzi è stato pubblicato su Pi in the Sky* (Numero 20, 2017), alle pagine 26 e 27.

L'intraprendente fanciulla racconta come un bel dì, dopo aver risolto un esercizio di geometria, si rese conto che la risoluzione poteva esserne facilitata applicando un semplice teorema...che scoprì in seguito, con somma meraviglia, non essere stato ancora formulato!
Consultò, infatti, il suo insegnante e alcuni suoi parenti impegnati nel campo della matematica fuori del suo paese di origine, e alla fine- d'accordo con i suoi genitori- si mise in contatto con un professore universitario e altri esperti. 
Appurato che il teorema non era stato ancora formulato, nonostante la sua evidente semplicità e valenza logica, decise di elaborarne la formulazione con relativa dimostrazione, un po' aiutata in questo dal suo insegnante e da suo padre, insegnante pure lui.

Ecco a voi il teorema, in inglese.