Σάββατο 23 Σεπτεμβρίου 2017

Euler, the Beethoven of Mathematics - OpenMind

Euler, the Beethoven of Mathematics - OpenMind

Euler, the Beethoven of Mathematics

In the educational studies of every scientist, there are a
few individual names that seem to emerge from course to course. But
above those of Newton, Galileo, or Einstein, there is
one name that probably surpasses them all as the first to appear—once
children master the four basic arithmetic operations, their approach to
logic begins with set theory and its Venn diagrams. But these are but
one particular case of those invented by a mathematician whose name
designates constants, functions, equations, laws, theorems, and almost
any other type of mathematical entity: Euler.

The Swiss Leonhard Euler (15 April 1707 – 18
September 1783) was one of the greatest intellectual supermen in the
history of mankind. The numbers serve to demonstrate his incredible
mental superpowers—over his 76 years of life he published more than 800
works, totalling some 30,000 pages. It has been estimated that almost a third of all the science and mathematics written in the eighteenth century bears his signature. After his death, his obituary required 56 pages to list all his publications.

But even the numbers fall short in describing a prodigious mind whose
talent manifested itself in some anecdotes. Perhaps the best known is
that he was able to recite Virgil’s Aeneid from beginning to end, detailing in what line every page of the edition he owned began and ended.

Superhuman computing power

Memory was not the only ability in which his brain seemed to
anticipate our current machines—his computing power was also superhuman.
He spent the last 17 years of his life almost totally blind,
due to a cataract in his left eye and a degenerative lesion in the
right one, whose origin varies according to the versions. But if this
disease affected his output, it was only to increase it; “in this way I
will have fewer distractions,” he once said. At one stage he was writing
an average of one work a week and joked about his enormous production,
claiming that his pencil outperformed him in intelligence. Like a
Beethoven unable to hear his music, Euler could hardly see his calculations,
but in his head he counted tables of lunar movements with such clarity
that an apprentice tailor could serve as a secretary without the need
for mathematical training.

Leonhard Euler portrayed by Jakob Emanuel Handmann circa 1756. Source: Deutsches Museum, Munich
Leonhard Euler portrayed by Jakob Emanuel Handmann circa 1756. Source: Deutsches Museum, Munich
On one occasion, two students disagreed over the result of the sum of
17 terms in a series, as the results of the two operations differed in
the fiftieth decimal place. Without a pencil or slate, Euler computed
the correct result in his mind in a few seconds. The anecdote was
referred to by his contemporary and colleague, the Frenchman Nicolas de
Condorcet, who at Euler’s death wrote a lengthy eulogy to “one of the
greatest and most extraordinary men that Nature ever produced.”

Curiously, that genius might have been lost to mathematics if Euler
had followed in the footsteps of his father to serve as pastor of the
Reformed Church, as planned. The advice of the mathematician Johann
Bernoulli, a friend of the family, was key in directing Euler’s
footsteps definitively towards mathematics and science.

Precocious in his studies and in his career, he soon began to stand out, which led him to travel to occupy prestigious positions in the Academies of Saint Petersburg and Berlin.
The most prolific mathematician in history was not only the main
founder of what we now know as classical mathematics, exploring a wide
variety of fields and introducing much of the notation used today, but
he also explored other disciplines such as astronomy, optics,
engineering, magnetism, ballistics, navigation, shipbuilding, philosophy
and music. It is said that his musical theory did not triumph because
it was too advanced in mathematical computations for musicians, and too
musical for mathematicians.

A Gift for Dissemination

Euler was also endowed with a gift for dissemination, without having
dedicated himself professionally to teaching. Proof of this is the
publication that was a best seller in its time—Letters to a German Princess, On Different Subjects in Physics and of Philosophy,
a work in three volumes that began to be published in 1768 and that
collects the letters written by Euler to his pupil, Friederike Charlotte
of Brandenburg-Schwedt, princess of Anhalt-Dessau and niece of the King
of Prussia Federico the Great.

The famous Goldbach Conjecture first appeared in a letter addressed to Euler by Christian Goldbach. Source: Departament of Mathematics and Statistics - Dalhousie University
The famous Goldbach Conjecture first appeared in a letter addressed to Euler by Christian Goldbach. Source: Departament of Mathematics and Statistics – Dalhousie University
In fact, Euler’s correspondence is also a treasure trove—the famous
Goldbach Conjecture, one of the oldest yet unsolved mathematical
problems, first appeared in 1742 in a letter addressed to Euler by the
German mathematician Christian Goldbach, his friend since they met each
other at the St. Petersburg Academy.

It was in this Russian city that, on September 18, 1783, Euler was
calculating the ascent of hot air balloons—which at that time were
causing a furore in Europe—and argued over dinner with his colleague
Anders Johan Lexell about the orbit of the newly discovered planet
Uranus. As Condorcet wrote, it was later, while drinking tea and playing
with his grandson, when “all of a sudden the pipe that he was smoking
slipped from his hand and he ceased to calculate and live.”

Javier Yanes

Παρασκευή 7 Ιουλίου 2017

Women in Maths - Δημοσιεύσεις

Women in Maths - Δημοσιεύσεις

Voisin (Professor at Collège de France, member of the
Académie des Sciences, Paris, recipient of CNRS Gold medal
2016):``... I would not say that I chose math as a career; I got
interested, so I started, then I continued and it was a sort of
addiction. I never really 'thought' of doing this, it's like
this was simply obvious and also the easiest way. How can I say;
once I started seriously doing maths, there was no alternative. I
got used to it, I had to do this. Since I started, I never wanted to
do something different. I would even say I find it more and more
interesting over time...

The fact is that my family did
not care so much, because I come from a very large family: I have
eight sisters and three brothers. My parents were very happy if we
were independent and earned money. I left my family's home when I
was 17, I got a scholarship and, starting from this point, I
never had to ask money from my parents. I should say that when I
was a child I had some contacts with maths, especially geometry, but
my parents did not care so much about our future careers; if I
had been a teacher in high school they would have been happy.

...What is hard are the moments when you lack inspiration to formulate
new ideas, new problems. Also sometimes it happened that I did
some research which was unsuccessful. It is important to be able to stop
something which does not work, not to spend too much time and
energy on an idea that you drive by force. You need to change. I
always found travelling very useful for this, because if you are alone
you tend to stay stuck on a subject, while if you travel you
get some distance and you can try something new, a new subject,
your mind has a new drive, a new energy.

...I had excellent
working conditions, because I had no teaching, I could teach
only when I wanted to, and in high level courses. I had a CNRS
position, so I was able to work at home, no time and energy
wasted in public transportation. Life was made very easy by my CNRS
position; and you know the French system of child care, so I had no
excuse for not working full-time. I should mention that what made
my life so very easy is that my husband is also a
mathematician, so not only the every day schedule is much softer, but
we understood both that we needed time for us. At the weekends,
I used to work in the morning and he in the afternoon. That
was nice, we both agreed that we should do things this way.

like very much the moment I start a new research, I like
very much the moment I have something in my mind: sometimes it
is barely an idea, sometimes it's just the beginning of
something. But there is this quality of the dream, and the fact
that your mind works alone, you do not need to force it.

I also
like to give talks; this is a bit different, but I like it very much. I
have to challenge myself to discuss, because I am what in
French we call ’introvertie’. There is a lot of introversion
in our work, because we are contemplating something. But there
is also a part of our work that is different, discussing,
giving talks, attending conferences, which is also nice. Still, for
me, the very nice part of my job is when I work on something new by

The bad part....there is some bad part, some
suffering, when you are trying to do something which is
difficult. There are some moments when you spend much energy,
and moments in which the dynamics of research is a little
lost. You don’t feel you are inside of mathematics. But I am afraid this
is especially bad for my family....''

This is a short extract from an interview with Claire by the EWM. For the full interview (worth reading!), please see

Φωτογραφία του χρήστη Women in Maths.

Κυριακή 2 Απριλίου 2017

Yves Meyer, Wavelet Expert, Wins Abel Prize | Quanta Magazine

Yves Meyer, Wavelet Expert, Wins Abel Prize | Quanta Magazine

How the French Mathematician Sophie Germain Paved the Way for Women in Science and Almost Saved Gauss’s Life – Brain Pickings

How the French Mathematician Sophie Germain Paved the Way for Women in Science and Almost Saved Gauss’s Life – Brain Pickings

How the French Mathematician Sophie Germain Paved the Way for Women in Science and Almost Saved Gauss’s Life

A century after the trailblazing French mathematician Émilie du Châtelet popularized Newton and paved the path for women in science, and a few decades before the word “scientist” was coined for the Scottish mathematician Mary Somerville, Sophie Germain
(April 1, 1776–June 27, 1831) gave herself an education using her
father’s books and became a brilliant mathematician, physicist, and
astronomer, who pioneered elasticity theory and made significant
contributions to number theory.

In lieu of a formal education, unavailable to women until more than a century later,
Germain supplemented her reading and her natural gift for science by
exchanging letters with some of the era’s most prominent mathematicians.
Among her famous correspondents was Carl Friedrich Gauss, considered by
many scholars the greatest mathematician who ever lived. Writing under
the male pseudonym M. LeBlanc — “fearing the ridicule attached to a
female scientist,” as she herself later explained — Germain began
sharing with Gauss some of her theorem proofs in response to his magnum
opus Disquisitiones Arithmeticae.

Sophie Germain

Their correspondence began in 1804, at the peak of the French
occupation of Prussia. In 1806, Germain received news that Napoleon’s
troops were about to enter Gauss’s Prussian hometown of Brunswick.
Terrified that her faraway mentor might suffer the fate of Archimedes,
who was killed when Roman forces conquered Syracuse after a two-year
siege, she called on a family friend — the French military chief M.
Pernety — to find Gauss in Brunswick and ensure his safety. Pernety
tasked one of his battalion commanders with traveling two hundred miles
to the occupied Brunswick in order to carry out the rescue mission.

But Gauss, it turned out, was unscathed by the war. In a letter from
November 27 of 1806, included in the altogether fascinating Sophie Germain: An Essay in the History of the Theory of Elasticity (public library), the somewhat irate battalion commander reports to his chief:

Just arrived in this town and have bruised myself with
your errand. I have asked several persons for the address of Gauss, at
whose residence I was to gather some news on your and Sophie Germain’s
behalf. M. Gauss replied that he did not have the honor of knowing you
or Mlle. Germain… After I had spoken of the different points contained
in your order, he seemed a little confused and asked me to convey his
thanks for your consideration on his behalf.
Carl Friedrich Gauss (Portrait by Jensen)

Upon receiving the comforting if somewhat comical news, Germain felt
obliged to write to Gauss and clear his confusion about his would-be
savior’s identity. After coming out as the woman behind the M. LeBlanc
persona in a letter from February 20 of 1807, she tells Gauss:

The appreciation I owe you for the encouragement you have
given me, in showing me that you count me among the lovers of sublime
arithmetic whose mysteries you have developed, was my particular
motivation for finding out news of you at a time when the troubles of
the war caused me to fear for your safety; and I have learned with
complete satisfaction that you have remained in your house as
undisturbed as circumstances would permit. I hope, however, that these
events will not keep you too long from your astronomical and especially
your arithmetical researches, because this part of science has a
particular attraction for me, and I always admire with new pleasure the
linkages between truths exposed in your book.
Gauss responds a few weeks later:


Your letter … was for me the source of as much pleasure as surprise.
How pleasant and heartwarming to acquire a friend so flattering and
precious. The lively interest that you have taken in me during this war
deserves the most sincere appreciation. Your letter to General Pernety
would have been most useful to me, if I had needed special protection on
the part of the French government.

Happily, the events and consequences of war have not affected me so
much up until now, although I am convinced that they will have a large
influence on the future course of my life. But how I can describe my
astonishment and admiration on seeing my esteemed correspondent M.
LeBlanc metamorphosed into this celebrated person, yielding a copy so
brilliant it is hard to believe? The taste for the abstract sciences in
general and, above all, for the mysteries of numbers, is very rare: this
is not surprising, since the charms of this sublime science in all
their beauty reveal themselves only to those who have the courage to
fathom them. But when a woman, because of her sex, our customs and
prejudices, encounters infinitely more obstacles than men in
familiarizing herself with their knotty problems, yet overcomes these
fetters and penetrates that which is most hidden, she doubtless has the
most noble courage, extraordinary talent, and superior genius. Nothing
could prove me in a more flattering and less equivocal way that the
attractions of that science, which have added so much joy to my life,
are not chimerical, than the favor with which you have honored it.

The scientific notes which your letters are so richly filled have
given me a thousand pleasures. I have studied them with attention, and I
admire the ease with which you penetrate all branches of arithmetic,
and the wisdom with which you generalize and perfect. I ask you to take
it as proof of my attention if I dare to add a remark to your last
With this, Gauss extends the gift of constructive criticism on some
mathematical solutions Germain had shared with him — the same gift which
trailblazing feminist Margaret Fuller bestowed upon Thoreau,
which shaped his career. Although Gauss eventually disengaged from the
exchange, choosing to focus on his scientific work rather than on
correspondence, he remained an admirer of Germain’s genius. He advocated
for the University of Gottingen to award her a posthumous honorary
degree, for she had accomplished, despite being a woman and therefore
ineligible for actually attending the University, “something worthwhile
in the most rigorous and abstract of sciences.”

She was never awarded the degree.

Red fish pond in front of the girls’ school named after Germain

After the end of their correspondence, Germain heard that the Paris Academy of Sciences had announced a prix extraordinaire
— a gold medal valued at 3,000 francs, roughly $600 then or about
$11,000 now — awarded to whoever could explain an exciting new physical
phenomenon scientists had found in the vibration of thin elastic
surfaces. The winning contestant would have to “give the mathematical
theory of the vibration of an elastic surface and to compare the theory
to experimental evidence.”

The problem appeared so difficult that it discouraged all other
mathematicians except Germain and the esteemed Denis Poisson from
tackling it. But Poisson was elected to the Academy shortly after the
award was announced and therefore had to withdraw from competing. Only
Germain remained willing to brave the problem. She began work on it in
1809 and submitted her paper in the autumn of 1811. Despite being the
only entrant, she lost — the jurors ruled that her proofs were

Germain persisted — because no solution had been accepted, the
Academy extended the competition by two years, and she submitted a new
paper, anonymously, in 1813. It was again rejected. She decided to try a
third time and shared her thinking with Poisson, hoping he would
contribute some useful insight. Instead, he borrowed heavily from her
ideas and published his own work on elasticity, giving Germain no
credit. Since he was the editor of the Academy’s journal, his paper was
accepted and printed in 1814.

Still, Germain persisted. On January 8, 1816, she submitted a third
paper under her own name. Her solution was still imperfect, but the
jurors decided that it was as good as it gets given the complexity of
the problem and awarded her the prize, which made her the first woman to
win an accolade from the Paris Academy of Sciences.

But even with the prize in tow, Germain was not allowed to attend
lectures at the Academy — the only women permitted to audit were the
wives of members. She decided to self-publish her winning essay, in
large part in order to expose Poisson’s theft and point out errors in
his proof. She went on to do foundational mathematical work on
elasticity, as well as work in philosophy and psychology a century
before the latter was a formal discipline. Like Rachel Carson,
Germain continued to work as she was dying of breast cancer. A paper
she published shortly before her terminal diagnosis precipitated the
discovery the laws of movement and equilibrium of elastic solids.

Her unusual life and enduring scientific legacy are discussed in great detail in the biography Sophie Germain. Complement it with the stories of how Ada Lovelace became the world’s first computer programmer, how physicist Lise Meitner discovered nuclear fission, was denied the Nobel Prize, but led the way for women in science anyway, and how Harvard’s unsung 19th-century female astronomers revolutionized our understanding of the universe decades before women could vote.

Mathematician Emmy Noether Should Be Your Hero | Smart News | Smithsonian

Mathematician Emmy Noether Should Be Your Hero | Smart News | Smithsonian