The English mathematician G.H. Hardy famously said, "... Math is more like a young man's game than any other art or science. Here, how correct is his understanding of "young people"?
Admittedly, proving a mathematical theorem requires a great deal of creativity, the ability to reacquaint the problem, and thinking in ways that no one else has thought of.
However, it also requires a lot of experience and knowledge. After all, if you don't understand a problem, you can't prove it. Many unconfirmed conjectures are based on a mountain of concepts that often takes years to reach.
Based on Wikipedia, I've laid out a timeline of mathematical developments from 1501 to 2015, tracing 250 major events in the field of mathematics: new proofs of theorems, the publication of important work, or the germination of core mathematical concepts.
Here are some of these events:
In 1540, the 18-year-old Lodovico Ferrari solved the quadratic equation.
In 1799, at the age of 22, Carl Friedrich Gauss proved the fundamental theorem of algebra (every polynomial equation has a solution in a complex number).
In 1925, at the age of 24, Werner Heisenberg, Jordan, and Born established matrix representations of quantum mechanics.
In 2004, 29-year-oldSecion Tao and Ben Green proved the Green–Tao theorem.
In 1522, the 30-year-old Adam Ries explained the use of Arabic numerals and their advantages over Roman numerals.
In 2003, 37-year-old Grigori Perelman proved the Poincaré conjecture.
In 2003, 41-year-old Andrew Wiles proved part of Taniyama's conjecture, thus proving Fermat's Last Theorem.
In 1929, the 47-year-old Emmy Noether first introduced a general representation theory of groups and algebras.
In 2013, the 58-year-old Zhang Yitang proved that there are infinite pairs of gaps for finite prime numbers.
In 1618, the 68-year-old John Napier first mentioned the base e of the natural logarithm in a work on logarithm.
Author draws
The age of the great mathematicians
Data Summary:
Their average age is 37 years;
The median age is slightly lower, at 35 years;
25% of mathematicians in mathematical memorabilia have made important mathematical achievements under their 30s;
42% of mathematicians make significant results between the ages of 30 and 39;
33% of mathematicians make significant achievements in their 40s or older;
The youngest of these is 18 years old, and in 1540 Lodovico Ferrari derived a general solution to the quadratic equation;
The oldest was 73 years old, and in 1825 Adrien-Marie Legendre and Peter Gustav Lejeune Dirichlet proved Fermat's Last Theorem for the n=5 case.
From 20 to 70 years old
I made an interactive chart, The Age of the Great Mathematicians, which readers can click here or below for detailed information (using the slider to investigate the achievements of mathematicians of all ages). Some of the highlights are worth noting:
Between the ages of 20 and 29
In 1832, the French mathematician Variste Galois, at the age of 21, proposed the general conditions for the solvability of algebraic equations, so he basically established group theory and Galois theory, and he was also the first to use the mathematical term "group" to represent a set of permutations, along with Nils Abel as the founder of modern group theory. But tragically and legendically, he was killed in a duel shortly after he came up with these theories.
In 1913, the 26-year-old Indian mathematician Srinivasa Ramanujan wrote a letter to Hardy with a long list of unproven theorems in which Ramanujan pleaded with Hardy to help him lift him out of poverty. Of course, Ramanujan's discovery in the letter must have predated him before he was 26 years old.
Between the ages of 30 and 39
In 2008, Marian Mirzahani, a mathematician in Tehran, Iran, proved a long-unresolved conjecture at the age of 31: William Thurston proposed that seismic map flows on Teichmüller space were all traversal systems. Six years later, she was awarded the Fields Medal in recognition of her "outstanding contribution to dynamics and geometry under Riemann surfaces and their modular spaces." On July 14, 2017, Mirzahani died of breast cancer at the age of 40.
In 1837, the 32-year-old German mathematician Gustav Lejeune Dirichlet developed analytic number theory in a paper on the presence of primes in a given arithmetic series. This is Dirichlet's fourth job on the math timeline, with the first three jobs taking place when he was 20, 27 and 32.
In 1915. Albert Eins, a 36-year-old theoretical physicist, published his theory of general relativity, a decade ago when he was a clerk at the Swiss Patent Office.
Between the ages of 40 and 49
In 1993, after several years of secret research on Fermat's Last Theorem, Andrew Wiles, 40, announced that he had proved Fermat's Last Theorem. It is well known that during the review process, there was an error in the certificate, but in the following year, the error was corrected. Wiles expanded his work at the age of 46, completing all of Taniyama Shimura's conjectures.
In 1918, at the age of 41, G.H. Hardy and Srinivasa Ramanujan developed an asymptotic formula for the division function. Perhaps his collaboration with young creative genius Ramanujan is one of the reasons he laments that mathematics is a "young man's game."
Over 50 years old
In 2013, Chinese-American mathematician Zhang Yitang first demonstrated the existence of infinite pairs of prime numbers with finite gaps, thus making a qualitative breakthrough in the number theory problem of twin prime number conjecture. In 1991, Zhang Yitang did not get a recommendation letter from his supervisor after obtaining his doctorate, the academic road was bumpy, he relied on miscellaneous work for a while to make a living, he worked as an accountant for several years, and worked in a fast food restaurant in the subway. (Source)
In 1722, the French mathematician Abraham de Moivre linked complex numbers to trigonometry and proposed Deimovor's formula; at the age of 66, a normal distribution was introduced to approximate the binomial distribution.
There is a humorous anecdote about the elderly mathematician:
It is often said that Deimovor, who had always been interested in the numbers, had predicted that he would need to sleep 15 minutes more each day than the day before, and that he would die when the total amount of sleep reached 24 hours, on November 27, 1754, which was also the time of his death. (Source)
Data and methods
Most of the data is obtained from the mathematical timeline on Wikipedia, starting from "modern" (16th century). It would be great to start with archimedes and Hypatia in ancient Greek times, but the historical record is not precise enough, so here it starts with "modern".
I've also expanded the data available on the Wikipedia timeline with other notable results, such as the work of mathematical physicists like Albert Einstein, Bohr, and Heisenberg.
Surprisingly, some of the recent big events in the mathematical community also have some data collection problems. I'd like to include proof of Hao Huang's 2019 sensitivity conjecture about Boolean functions, but I can't find his date of birth. Extrapolating from his educational experience, he may be in his 30s.
Similarly, I want to include Joan Taylor, an amateur mathematician who, along with Joshua Socolar, discovered the Socolar-Taylor tile collage that solved the problem raised by Roger Penrose. But I can't find her date of birth, but on her website, she says she started thinking about it in 1990, 20 years before the results published in 2011.
Final thoughts
Statistically, while most of the remarkable achievements are made by mathematicians between the ages of 20 and 40, the age range is still wide. As we get older, mathematicians tend to produce in the book and summary genres, and of course there are still some examples of groundbreaking innovative proofs made by old mathematicians.
Every mathematician's story is different. Among them are prodigies who died young, prodigious scholars covering various fields, and people who insist on delving into a problem for 20 years.
For us math amateurs, we may not want to change the world, but just enjoy it, knowing that our brains are thinking all the time and can continue to enjoy the joy of mathematics, which is enough to be happy.
By Afiq Hatta
Translation: Nuor
Reviewer: xux