Nicholas Bourbaki
Nicholas Bourbaki was not a person but a pseudonym chosen for a secretive math group founded by nine great mathematicians in France in the mid-1930s. Henri Cartan, André Weil, Szolem Mandelbrojt, Frenchmen Claude Chevalley, and Jean Dieudonne were some of the active founders who collectively wrote many academic papers under the Bourbaki pseudonym to represent the essence of a “contemporary mathematician.”
“The Genius Mathematician That Never Lived”. 2021. Medium. https://madiha-7874.medium.com/the-genius-mathematician-that-never-lived-46879f8e13bb.
When observing the recent past of mathematical history, you encounter the name Nicolas Bourbaki quite often. The reason for this is quite simple, actually. Nicolas Bourbaki, born 1935, might be the author of the most math books in history. More importantly, he has written books ranging from mathematical history to algebraic topology, and even calculus. Where it becomes mysterious, however, is that not a single person has seen him. That is because no one by the name of Nicolas Bourbaki has ever lived.
Ali. “The Secret Math Society: Nicolas Bourbaki”. 2022. Medium. https://ali.medium.com/the-secret-math-society-nicolas-bourbaki-59c05f1a67f3.
Marcus du Sautoy argues that mathematics is the driving force behind modern science: mathematical secrets of Nicholas Bourbaki.
Hemanth. “The Story Of The Rockstar Mathematician Who Never Lived”. 2022. Medium. https://medium.com/street-science/the-story-of-the-rockstar-mathematician-who-never-lived-ac523b64a5b4.
The story begins at the prestigious school: École normale supérieure (ENS) in France. In the early 1900s, a student named André Weil had attended the school. During his time there, a new professor had presented a certain “theorem of Bourbaki.” Later on, it turned out that this new professor was just an upperclassman who was pulling a prank. Somehow, the narrative and the humour of this prank stuck with Weil.
Georg Cantor
The mathematician Georg Cantor was born in St. Petersburg, Russia, on March third, 1845. His family moved to Germany when he was 11. As a youth, he excelled in maths and was an outstanding violinist. He went on to study maths at the University of Berlin under some of the great’s of the day, including Leopold Kronecker, Karl Weierstrass. Later, his research led him to consider infinity, not just some abstract concept, but a new type of number, a transfinite number.
What is something that is trivial to a mathematician but would amaze a layperson?
One of the most beautiful mathematical facts that are (currently) trivial to a mathematician, but would amaze a layperson is this:
“Some infinities are bigger than others”.
If anyone tells you that this is obvious, they are lying. Proving this is my favorite lecture when I teach Discrete Math I.
This was proven in the 1870s by Georg Cantor. A number of mathematicians didn’t accept Cantor’s proof, at least for a while.
Cantor essentially showed that the cardinality of the set of Real Numbers is greater than that of the Natural Numbers. He did this by showing that no matter how you tried to pair up the Reals with the Naturals, there would always be some Reals that were not used.
Ever weirder, given any infinite set, we can create a greater infinite set. This leaves us with this beautiful statement:
“Given an infinite set, there is an infinite set with a greater order of infinity.”
Incredible.
In 1900, the great mathematician, David Hilbert (1862 – 1943), gave a widely regarded talk about some of the unsolved problems in mathematics. During this talk he mentioned Cantor’s proof that the set of Reals is a higher order of infinity than the set of Natural Numbers. He then said:
“Nothing will remove us from the paradise that Cantor has created for us.”
Farage, Tim. “What Is Something That Is Trivial To A Mathematician But Would Amaze A Layperson?” 2022. Quora. https://qr.ae/pv5F0m.
“A Lonely Mathematician At The Edge Of Infinity: Georg Cantor”. 2021. Medium. https://mathladyhazel.medium.com/a-lonely-mathematician-at-the-edge-of-infinity-georg-cantor-62d36729761.
The German mathematician Georg Cantor pioneered set theory and gave us our modern understanding of the mathematics of infinity.
Farage, Tim. “How Is It Determined That One Infinite Set Is “Larger” Than Another?”. 2023. Quora. https://qr.ae/przHnm.
One of the most beautiful mathematical facts that are (currently) trivial to a mathematician, but would amaze a layperson is this: “Some infinities are bigger than others”. Ever weirder, given any infinite set, we can create a greater infinite set. This leaves us with this beautiful statement: “Given an infinite set, there is an infinite set with a greater order of infinity.” Incredible.
Marcus du Sautoy argues that mathematics is the driving force behind modern science: Georg Cantor, numbers and infinity.
Nagrath, Aditya. “Georg Cantor’s Controversial Career”. 2022. Medium. https://anagrath.medium.com/georg-cantors-controversial-career-c1d2fffe7464.
Georg Cantor was a German mathematician who would eventually be known as a pioneer of “new mathematics.” His work was controversial because it disrupted previously settled matters regarding integer sets. Though it took years for the world to recognize his genius, Cantor’s brilliant methods are now a foundation for modern mathematics. Read on to learn more about this contentious genius.
Paul Erdös
Paul Erdős (1913 – 1996) was one of the most productive mathematicians in history. Born in Hungary, he solved countless problems in graph theory, number theory, combinatorics, analysis, probability, and other parts of mathematics. During his life, Erdös published around 1,500 papers and collaborated with more than 500 other mathematicians. In fact, he spent most of his life living out of a suitcase, travelling to seminars, and visiting colleagues!
Labh, Sunny. “Paul Erdős, The Mathematician Of Highest Intellectual Calibre”. 2022. Medium. https://piggsboson.medium.com/paul-erd%C5%91s-the-mathematician-of-highest-intellectual-calibre-825bf30fdf0.
I’ve talked about several mathematicians in my blogs from Maryam Mirzakhani to Terrence tao, from Gottfried Leibniz to Leonhard Euler. I particularly take interest in speaking about mathematicians because they perceive the world and the universe around them in ways that no one else can. Mathematical reasoning unlocks several other areas of your brain and enhances your overall thinking mechanism. This story, however, is about the mathematician who had one of the most incomprehensible personalities of all. His way of living his life and doing mathematics was unique, as mentioned by all his collaborators. I’ve written a separate story about the working ethics of the Hungarian mathematician Paul Erdős, if you haven’t read it you can read it here.
Nagrath, Aditya. “Paul Erdös: Collaborative Mathematics”. 2021. Medium. https://anagrath.medium.com/paul-erd%C3%B6s-collaborative-mathematics-37a04a0d2f3d.
“Paul Erdös – Timeline Of Mathematics – Mathigon”. 2021. Mathigon. https://mathigon.org/timeline/erdos.
Tovey, Craig Aaron. “Who Was The Most Genius Mathematician That You Had Ever Worked With, And What Was The Situation That Made You Think So?”. 2022. Quora. https://qr.ae/pvbEGy.
Paul Erdos. Other mathematicians I have known could, like Erdos, solve problems instantly. Erdos could do more. He would instantly alter your problem to make it a good research question. The first time I talked with him, I asked him two questions. The first was about prime numbers of a certain form. As soon as I finished speaking he answered that my guess was correct and followed as a corollary from a theorem proved 20 years previously. He immediately continued by proposing a generalization that might be true and would be worth exploring. My second question was about Ramsey numbers (part of graph theory). I had conjectured that R(k,m) ≤ R(k−1,m+1) for k>m, which intuitively seemed very plausible to me. Erdos immediately said “That is too hard. Try to prove R(3k,k) ≤ R(2k,2k). That would be very interesting.”
I visualize what Erdos could do as if what is known or readily proved in mathematics is the interior of a ball. Points on the ball’s boundary are the good research problems at the edge of the unknown. Points far from the ball are too difficult for us now. If you gave Erdos a problem inside the ball, he would project it outwards to get a harder version on the boundary that was worthy of research. If you gave Erdos a problem well outside the ball, he would project it inwards onto the ball to give a you a version that you might have a chance of solving. I have not worked with any other mathematician who could do that instantly.
Hilda Geiringer
Hilda Geiringer was the first female lecturer in applied mathematics in history. But not only applied mathematics, but the scope of Hilda’s acquaintance was also much more comprehensive. This Jewish woman was also skilled in applying her knowledge in various fields of science. Geiringer was one of the pioneers of applied mathematics in the twentieth century. At that time, applied mathematics was trying to establish its own uniqueness apart from mathematics. Where Hilda was the most significant contributor, she has also mastered her mathematics in probability and reproduction. Many parts of today’s engineering and science stand based on many of her works. Geiringer took her work more as an emotion than as a profession.
“Hilda Geiringer: The Woman Who Reshaped Mathematics”. 2021. Medium. https://medium.com/@samazgor/hilda-geiringer-the-woman-who-reshaped-mathematics-538b1f1073a.
“Hilda Geiringer – Biography”. 2021. Maths History. https://mathshistory.st-andrews.ac.uk/Biographies/Geiringer/.
“Hilda Geiringer – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Hilda_Geiringer.
G. H. Hardy
Godfrey Harold Hardy (1877–1947) was an English mathematician best remembered for his work in analytic number theory and mathematical analysis. He almost single-handedly injected rigour (then a feature limited to continental mathematics) into mainstream British mathematics. But when asked (in an interview with Paul Erdős) to single out his greatest contribution to mathematics, he unreservedly opted for his discovery of Srinivasa Ramanujan, whom he elevated from a clerical position in Madras (modern day Chennai, India) to the elite mathematical stature the genius of Ramanujan richly deserved.
“G. H. Hardy — Rigorous & Shy”. 2020. Medium. https://www.cantorsparadise.com/g-h-hardy-rigorous-shy-8f8afdd689e9.
“BBC Radio 4 Extra – A Brief History Of Mathematics, Hardy And Ramanujan”. 2021. bbc.co.uk. https://www.bbc.co.uk/programmes/b00ss1j4.
Marcus du Sautoy argues that mathematics is the driving force behind modern science: GH Hardy and the prime number ‘menace’.
“G. H. Hardy: Mathematics As Art”. 2021. Medium. https://anagrath.medium.com/g-h-hardy-mathematics-as-art-17ba137dd14a.
Sophiya Vasilyevna Kovalevskaya
The first woman to obtain a doctorate (in the modern sense) in mathematics, the first woman appointed to a full professorship in northern Europe and one of the first women to work for a scientific journal as an editor.
“Sofia Kovalevskaya”. 2023. mathwomen.agnesscott.org. https://mathwomen.agnesscott.org/women/kova.htm.
“Sofia Kovalevskaya – Biography”. 2023. Maths History. https://mathshistory.st-andrews.ac.uk/Biographies/Kovalevskaya/.
“Sofia KOVALEVSKY”. 2023. scientificwomen.net. https://scientificwomen.net/women/kovalevsky-sofia-50.
“Sofya Kovalevskaya – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Sofya_Kovalevskaya.
Maryam Mirzakhani
Maryam Mirzakhani, (born May 3, 1977, Tehrān, Iran—died July 14, 2017, Palo Alto, California, U.S.), Iranian mathematician who became (2014) the first woman and the first Iranian to be awarded a Fields Medal. The citation for her award recognized “her outstanding contributions to the dynamics and geometry of Riemann surfaces and their moduli spaces.”
The Fields Medal, often described as the mathematician’s Nobel Prize, is given every four years to no more than four mathematicians, all of whom are 40 or younger. She was named for her work on complex geometry and dynamic systems.
Ms. Mirzakhani, of Iran, received the award in Seoul, South Korea, in 2014. “This is a great honor. I will be happy if it encourages young female scientists and mathematicians,” she said at the time. “I am sure there will be many more women winning this kind of award in coming years.”
Ms. Mirzakhani — known for taking the difficult, complicated path to solve mathematical problems — studied the symmetry of curved surfaces and other theoretical concepts known as “pure mathematics.”
“Maryam Mirzakhani: First Woman to Win Math’s Nobel Prize | Elephant Learning”. 2023. Elephantlearning.Com. https://www.elephantlearning.com/post/maryam-mirzakhani-first-woman-to-win-maths-nobel-prize.
“Maryam Mirzakhani – Wikipedia”. 2023. en.wikipedia.org. https://en.wikipedia.org/wiki/Maryam_Mirzakhani.
University, Stanford. 2017. “Maryam Mirzakhani, Mathematician And Fields Medal Winner, Dies At Stanford | Stanford News”. Stanford News. https://news.stanford.edu/2017/07/15/maryam-mirzakhani-stanford-mathematician-and-fields-medal-winner-dies/.
“Maryam MIRZAKHANI”. 2023. scientificwomen.net. https://scientificwomen.net/women/mirzakhani-maryam-69.
“A Tenacious Explorer Of Abstract Surfaces | Quanta Magazine”. 2014. Quanta Magazine. https://www.quantamagazine.org/maryam-mirzakhani-is-first-woman-fields-medalist-20140812/.
“Maryam Mirzakhani – Biography”. 2023. Maths History. https://mathshistory.st-andrews.ac.uk/Biographies/Mirzakhani/.
“What Were Maryam Mirzakhani’s Greatest Contributions To Mathematics?”. 2023. Quora. https://www.quora.com/What-were-Maryam-Mirzakhanis-greatest-contributions-to-Mathematics/.
Emmy Noether
“Emmy Noether – Wikipedia”. 2020. en.wikipedia.org. https://en.wikipedia.org/wiki/Emmy_Noether.
Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She discovered Noether’s theorem, which is fundamental in mathematical physics. She was described by Pavel Alexandrov, Albert Einstein, Jean Dieudonné, Hermann Weyl and Norbert Wiener as the most important woman in the history of mathematics. As one of the leading mathematicians of her time, she developed some theories of rings, fields, and algebras. In physics, Noether’s theorem explains the connection between symmetry and conservation laws.
“Five Fast Facts About Mathematician Emmy Noether”. 2021. energy.gov. https://www.energy.gov/articles/five-fast-facts-about-mathematician-emmy-noether.
Scientists, Top, and List Scientists. 2021. “Emmy Noether – Biography, Facts And Pictures”. famousscientists.org. https://www.famousscientists.org/emmy-noether/.
“The Most Important Woman In 20th-Century Mathematics: Emmy Noether”. 2021. Medium. https://anagrath.medium.com/?p=1e3ad8ce499e.
Who did Albert Einstein regard as the most important woman in mathematics? That honor goes to Emmy Noether, a prolific mathematician, and physicist working in the early 20th-century. Besides working in a male-dominated field and fighting for recognition at every turn, Emmy also had to contend with the turmoil and chaos of the Nazi takeover in Germany. As a Jewish woman in academia, she was a high-profile target for the Nazi party. Although she successfully escaped Germany and came to America, many of her colleagues and family members were not so lucky. Emmy Noether’s life is an empowering and illuminating example of someone who loves what they do so much that they persevere despite nearly insurmountable obstacles.
Pincha, Yash. “She Who Changed Physics Forever”. 2022. Medium. https://medium.com/quantafy/she-who-changed-physics-forever-c508a46c5bf7.
Noether’s theorem, postulated by Amalie Emmy Noether, is by far, one of the most beautiful mathematical ideas out there. It doesn’t receive nearly as much credit as it deserves.
What is the most beautiful theorem in physics?
In my opinion, Noether’s theorem.
Not only is it the most beautiful theorem but it’s also one of the most important. It’s utilized in classical mechanics, classical field theory, quantum mechanics, and quantum field theory. It was proved by Emmy Noether, one of the most badass women ever.
The theorem states, “if a system has continuous symmetry, then there is a corresponding quantity whose values are conserved over a period of time.” In other words, Noether’s theorem establishes a link between continuous symmetries and conservation laws.
[Mathematical Proof]
In modern physics, you will find symmetries everywhere. They let you obtain conservation laws without the need of specifying the law as an axiom.
Some symmetries and their conservation laws include:
- Time-invariance →→ Energy
- Rotational symmetry →→ Angular momentum
- Phase shifts →→ Conservation of charge
Dwight Neuenschwander’s book Emmy Noether’s Wonderful Theorem goes into far more detail.
Elseweifi, Amr. “What Is The Most Beautiful Theorem In Physics?”. 2022. Quora. https://qr.ae/pvbERD.
Srinivasa Ramanujan
An Indian mathematician with least resource and great intelligence who lived on earth for very short period of time. Ramanujan was born on 22 Dec, 1887 in Tamil Brahmin Iyengar family, in Erode. He grew up in very religious thoughts, beliefs, but despite that passion towards mathematics was flourished since childhoods days. He was the prodigy, he was able to understand and solve mathematics well above his age. At age of 16 he got his mind at A Synopsis of Elementary Results in Pure and Applied Mathematics, a collection of 5000+ theorems, which lead him to develop Bernoulli Numbers independently. From then he did mathematics of his own, and independently rediscovers many known mathematics facts. Some of which he sent to GH Hardy, for which Hardy was impressed and his professional mathematician career started.
“Ramanujan”. 2020. Medium. https://medium.com/maths163/ramanujan-48b6c56c199d.
“BBC Radio 4 Extra – A Brief History Of Mathematics, Hardy And Ramanujan”. 2021. bbc.co.uk. https://www.bbc.co.uk/programmes/b00ss1j4.
Marcus du Sautoy argues that mathematics is the driving force behind modern science: GH Hardy and the prime number ‘menace’.
Sharma, Ashwin. “Lessons From A Self-Taught Mathematical Genius Who Succeeded Against All Odds”. 2022. Medium. https://medium.com/@ash.sharma3_14/lessons-from-a-self-taught-mathematical-genius-who-succeeded-against-all-odds-62fa50057279.
Srinivasa Ramanuja’s life is the most romantic story in science. His story is one of succeeding against all odds. A self-taught mathematical genius living in extreme poverty in South India became one of the greatest mathematicians of the 20th century. Despite being a two-time college dropout, he revolutionized the field of number theory and became the first Indian to be elected fellow of the Royal Society in England in 1918. His life was tragically cut short at the age of 32. His short, but brilliant life, is a testament to living life at the very frontiers of human discovery. Intertwined in Ramanujan’s incredible biography are three of the most important lessons we can learn from his life.
Ramanujan was a self-taught Indian mathematician who travelled to England to work with professor G H Hardy after sending him a letter describing some of his remarkable ideas. In this video we take a look at that letter and at Hardy’s initial response.
Bernhard Riemann
Bernhard Riemann (1826 – 1866) was a German mathematician working in the fields of analysis and number theory. He came up with the first rigorous definition of integration, studied differential geometry which laid the foundation for general relativity, and made groundbreaking discoveries regarding the distribution of prime numbers.
“Bernhard Riemann – Timeline Of Mathematics – Mathigon”. 2022. Mathigon. https://mathigon.org/timeline/riemann.
Mancuso, Olivia. “Bernhard Riemann: Shy, Brilliant, Revolutionary”. 2022. Medium. https://anagrath.medium.com/bernhard-riemann-shy-brilliant-revolutionary-773226ae3002.
Bernhard Riemann (1826–1866) was a shy German mathematician known for his fear of public speaking and the brilliant journey to overcome it. He became a mathematics professor after presenting a habilitation thesis on multidimensional spaces. His contributions to mathematics include Riemannian Geometry, which profoundly impacted Einstein’s theory of relativity. Read on to learn about this prolific German mathematician!
“The Distribution Of Primes – Divisibility And Primes – Mathigon”. 2022. Mathigon. https://mathigon.org/course/divisibility/distribution-of-primes#riemann.
When only 15 years old, the German mathematician Carl Friedrich Gauss ad a groundbreaking new idea: he counted the number of primes up to a certain point, and showed the results in a chart (see The Riemann Hypothesis on the The Distribution of Primes page for the interactive chart). Along the x-axis you can see all integers. Whenever there is a prime, the Prime Counting Function increases by one. As we zoom out, the blue line becomes very smooth. Gauss noticed that the shape of this function looks very similar to the function x/log(x). He predicted that the two functions are always “approximately similar”, and this was proven in 1896. However, there is still a significant error between the actual number of primes, and Gauss’s approximation. In 1859, the mathematician Bernhard Riemann discovered an approximation that looked much better, but he wasn’t able to prove that it would always work. His idea became known as the Riemann Hypothesis.
“Bernhard Riemann – Wikipedia”. 2020. en.wikipedia.org. https://en.wikipedia.org/wiki/Bernhard_Riemann.
Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as one of the most influential papers in analytic number theory. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.
“Bernhard Riemann – Biography, Facts And Pictures”. 2022. famousscientists.org. https://www.famousscientists.org/bernhard-riemann/.
Bernhard Riemann made profound, far-sighted discoveries with lasting consequences for mathematics and our understanding of space, gravity, and time. Riemannian geometry completely reformed the field of geometry and became the mathematical foundation of Einstein’s general theory of relativity. Finding a proof or disproof of the Riemann hypothesis continues to be the greatest, deepest, unsolved problem in number theory – the search for a solution has become the holy grail of mathematics. Another of Riemann’s innovations, Riemann surfaces, made a strong link between topology and complex function theory. Riemann was the first person to rigorously define the integral.
John von Neumann
John von Neumann (1903 – 1957) was a Hungarian-American mathematician, physicist and computer scientist. He made important contributions to pure mathematics, was a pioneer of quantum mechanics, and developed concepts like game theory, cellular automata, self-replicating machines, and linear programming.
During World War II, von Neumann was a key member of the Manhattan Project, working on the development of the hydrogen bomb. He later consulted for the Atomic Energy Commission and the US Air Force.
“John Von Neumann: The Colorful Mathematician Who Helped Design The Atomic Bomb”. 2022. Medium. https://anagrath.medium.com/john-von-neumann-the-colorful-mathematician-who-helped-design-the-atomic-bomb-f36576272881.
John von Neumann was a Hungarian child prodigy turned famous mathematician known for memorizing entire books, computing complex equations in his head, and having an animated personality.
He lived an eccentric lifestyle filled with frequent parties and loud music — but also was responsible for many critical military contributions in World War II. Von Neumann was truly a fascinating mathematician whose impressive resume could only be topped by his “off-color” sense of humor and zest for life.
“John Von Neumann – Wikipedia”. 2022. en.wikipedia.org. https://en.wikipedia.org/wiki/John_von_Neumann.
John von Neumann (December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. Von Neumann was regarded as perhaps the mathematician with the widest coverage of the subject in his time and was said to have been “the last representative of the great mathematicians who were equally at home in pure and applied mathematics”. He integrated pure and applied sciences.
“John Von Neumann – Biography, Facts And Pictures”. 2022. famousscientists.org. https://www.famousscientists.org/john-von-neumann/.
John Von Neumann was a polymath and pioneer of the application of operator theory to quantum mechanics, in the development of functional analysis. Along with fellow physicists Edward Teller and Stanislaw Ulam, von Neumann worked out key steps in the nuclear physics involving thermonuclear reactions and the hydrogen bomb.
Von Neumann wrote 150 published papers in his life; 60 in pure mathematics, 20 in physics, and 60 in applied mathematics.
His last work, published in 1958 “The Computer and the Brain”, explores the analogies between computing machines and the living human brain.
Thorbjørnsen, Gylve. “Who Is The Smartest Person Who Has Ever Lived? Was It Plato, Aristotle, Newton, Einstein Or Somebody Else? – Quora”. 2022. quora.com. https://qr.ae/pvHQr9.
This guy, to me at least seems like a genius. He was smart as hell, really knowledgeable and it was just purely insane.
He was one of the worlds most prolific mathematicians. His research helps in so many fields. He was also known to be able to recite entire books out of memory that he had read years beforehand.
I’m too lazy to write them in myself so here is quotes from him and about him from himself, and some extremely intelligent gents 😉
Mathematical Mysteries 