# Great contributions to mathematics by older mathematicians [closed]

It is often said that mathematicians hit their prime in their twenties, and some even say that no great mathematics is created after that age, or that older mathematicians have their best days behind them.

I don't think this is true. Gauss discovered his Theorema Egregium, a central result in differential geometry, in his fifties. Andrew Wiles proved Fermat's Last Theorem in his thirties.

Post many examples of great mathematics created over the age of 30, the older the better. Bonus points for mathematicians whose best work happened over the age of 30. I will define great mathematics to be something of great significance, such as proving a deep theorem, developing far-reaching general theory, or anything else of great impact to mathematics.

Addendum: Please also include a brief explanation of why the mathematical result posted is significant.

(Many say that 30 isn't that old, but I'm casting the net wide to get more examples. Age-30 examples would also help to debunk the "peak at twenties" myth. I do ask for examples to be as old as possible, so the lower bound isn't that important.)

I know that mathematicians can produce a lot of work throughout their lives - Euler is a great example - but here I want to focus on mathematics of great significance or impact.

Help me break this misconception!

## closed as too broad by Etienne, user147263, AlexR, user642796Dec 13 '14 at 9:27

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I proved my favorite theorem when I was 35. ;) – Cheerful Parsnip Dec 9 '14 at 14:05
• Strongly related, but not quite duplicate. Also, nearly the same question on MO. – apnorton Dec 9 '14 at 15:12
• I do not like the title of this question. – Ali Caglayan Dec 9 '14 at 18:04
• I would consider "older" to be $\geq 40$. Otherwise, the Fields Medal has missed its mark on "younger" researchers. – user155861 Dec 9 '14 at 23:44
• Fourier published his theory of heat when he was 55. – Pedro Tamaroff Dec 10 '14 at 0:42

The Weierstrass approximation theorem was published in 1885, when Karl Weierstrass was 70. As T.W. Körner writes in his Fourier analysis book (pg. 294):

Fejér discovered his theorem at the age of 19, Weierstrass published this theorem at the age of 70. With time the reader may come to appreciate why many mathematicians regard the second circumstance as even more romantic and heart warming than the first.

This 'misconception' is not proven or disproven by quoting specific counter examples, but by examining a fair number of mathematical discoveries and determining a trend.

# 17 equations that changed the world

I will be going by this list based on this book by Ian Stewart of the 17 equations that changed the world, I will be taking the earliest possible date that seems to make sense, because to disprove something that seems to be the most sensible approach. Additionally some of these equations aren't purely mathematical, but that's the nature of mathematics that per the OP is most influential, had the greatest impact.

Please understand, I am not judging whether these discoveries are rightly attributed, I simply followed the list!

• The logarithm and its identities - John Napier aged 64 published his work Mirifici Logarithmorum Canonis Descriptio (can't seem to find anything about when he actually first formulated the basis)

• The fundamental theorem of calculus - Newton aged 23 and Leibniz aged 28

• Newton's universal law of gravitation - Newton aged 44

• The origin of complex numbers - Hammilton aged 38

• Euler's formula for polyhedra - Euler aged 43

• The normal distribution - Quetelet aged 39 published his work Sur l'homme et le développement de ses facultés, ou Essai de physique sociale

• The wave equation - Bournoulli aged 38 published Hydrodynamica and Jean D'Alembert aged 29

• The Fourier transform - Fourier aged 43

• The Navier-Stokes equations - Navier aged 37

• Maxwell's equations - Maxwell aged 30

• Second law of thermodynamics - Too messy attribution

• Einstein's theory of relativity - Special Relativity aged 26

• The Schrödinger equation - Schrodinger aged 40

• Shannon's information theory - Shannon aged around 31

• The logistic model for population growth - Messy attribution and discovery

• The Black–Scholes model - Black aged 35 and Scholes aged 32

# Conclusion

Historically speaking the greatest mathematical discoveries were made between ages 25 and 45 (average of 35-37). As life expectancy post child years were similar to ours we do not need to normalize for that. It is however possible that in more recent years ages have gone down, leading to the stated believe, however I will not further examine this as it is hard to examine this objectively within the scope of the stated question. Either way, an age of 35 on average still can be considered fairly young, although it is clearly older than the questioned twenties the OP was skeptical about.

# Sources

I grabbed all the various years from Wikipedia and did the calculations myself.

Louis de Branges proved Bieberbach's Conjecture at age 53.

(I found the picture when researching his birthdate)

Grigori Perelman was 36 when he released his proof of Thurston's geometrization conjecture.

• Since when is 36 old? – MathematicsStudent1122 Sep 3 '16 at 9:31
• @MathematicsStudent1122 Read the question, not just the title. – Spenser Sep 3 '16 at 15:55

I think not only proper examples would help you to break this misconception but also the time itself. Everything is changing. Some fifty years ago there were no computers in people's life and people were dying earlier. Now it is the opposite. The education system also has changed with the advances in mathematics. The things that are learned nowadays are getting more and more advanced and it also requires alot of time to come to the very deep of the problem. The proof of Fermat's last theorem is a supporter of what is said here. His corrected version of the proof was published on 1995 and he was born in 1953. There is a documentary movie which shows how much effort he put into this problem but in this context more importantly when he decided to prove the theorem..

Kolmogorov–Arnold–Moser theory : http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Arnold%E2%80%93Moser_theorem - Kolmogorov was 50 at that time.

EDIT: I decided to edit my answer and add P.S.Novikov here (I only mentioned him in the comment earlier). He was a consistent achiever throughout all his life, but here I want to mention his (negative) solution of word problem in group theory. He proved the result at the age of 54.

I think that Leonard Euler "prime" years were when he was getting older until he died at 76. Not only did age not stop him but neither did blindness.

Serre is nearly 90 years old, and he still does mathematics. Of course, not to the same degree of magnitude as he used to do because of much poorer health, but he is still a member of the living mathematical community and we greatly benefit from him.

• I would even say that for some theorists, becoming blind even increases productivity, because there is less potential for distraction. Especially people with a higher sensoric sensitity (many introverts) can often be distracted by the slightest optical shock. I sometimes frighten when my wife suddenly moves (e.g. her fingers or head), after having been static for a few minutes :D – phresnel Dec 11 '14 at 9:45
• To substantiate your assessment of "prime years", do you know of specific pieces of great mathematics that Euler (or Serre) contributed in old age? – Herng Yi Dec 11 '14 at 12:14

Born in 1860, he began to develop the theory of dislocations in crystals that was later to become important in the understanding of the behaviour of ductile materials... when he was 45.

Alfred Lotka

Born in 1880, one of his earliest publications, proposed a solution to Ronald Ross's second malaria model (Lotka was 32). He published a thorough five-part analysis and extension of both Ross's malaria models... when he was 43.

Augustin-Louis Cauchy

Born in 1789, he presented a dozen papers on celestial mechanics to the Academy... when he was 51.

Phyllis Nicolson

Grace Hopper

Not a mathematician, but belongs to a related field (computer science). Born in 1906, she achieved the inaugural Computer Sciences Man of the Year award from the Data Processing Management Association when she was 63. She is also the mother of Cobol programming language and the term "bug" for an error in a computer system/software.

Weierstrass proved his famous Approximation Theorem at age 70.

Tibor Rado introduced "Busy Beaver" functions at age 67.

Freeman Dyson recently published an important contribution to the iterated prisoner's dilemma at age 89.

I don't know about Mathematics in particular, but one observation that may provide an explanation is that at least in some fields of research today, it seems the longer you stay in academia, the more the focus of your work shifts from actual research to administration and "politics".

As a PhD student and maybe post-doc, you might just be peaking at the sweet spot between having already acquired enough knowledge/insight to make meaningful contributions and not yet being drawn into the swamp of academic reality (lecturing, writing grant proposals, networking, etc.). For a lot of older researchers, at least in my experience, their research contribution mainly consists in supervising new generations of young researchers -- which is no small feat, by all means, but might explain why their personal research output declines.

So perhaps the reason is not so much the degradation of intellectual power, but simply given by the pragmatics of not having as much time to dedicate to research any more as when you're young.

Take this answer with a grain of salt, though, as I'm broadly generalizing. A lot of counter-examples have already been given in this thread by yourself and others.

• Downvoters, care to comment? – Thomas Dec 10 '14 at 11:09
• This is not an answer. – JiK Dec 10 '14 at 15:26
• More specifically, OP asked for examples of great mathematics done by people over 30, not an explanation of why people think academics peak in their 20s – Jemmy Dec 10 '14 at 16:58
• @Jeremy Since he posted this as "answer" (I guess it would have been to long for a comment), you are right. But he makes a valid point that is clearly related to the question. IMHO real discussion and insight can not arise if we follow the path of just strictly answering the question and do not allow any other related thoughts. – Thekwasti Dec 11 '14 at 9:21
• @Thekwasti Related, probably but this is a Q&A site, not a discussion forum and extended discussion is explicitly discouraged. If you want to talk about this further, please post on meta. – Jemmy Dec 11 '14 at 15:26

George Marsaglia was still active and making significant contributions to the art of pseudo-random number generators with long periods at the age of 86, just shortly before he died.

Maybe these were not "great" contributions since I could actually understand them :-)