Galois’s 200th birthday

Évariste Galois was a French mathematician who died under mysterious circumstances after a duel when he was only 20 years old!
Even though he was so young, he is considered the father of modern algebra and he is the founder of Galois theory.
He was the first to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, the first to use the term group in mathematics in the modern way and so on.


In 1828, he attempted the entrance exam to École Polytechnique, the most prestigious institution for mathematics in France at the time, without the usual preparation in mathematics, and failed for lack of explanations on the oral examination. In that same year, he entered the École Normale (then known as l’École préparatoire), a far inferior institution for mathematical studies at that time, where he found some professors sympathetic to him.

In the following year, Galois’ first paper, on continued fractions,[3] was published. It was at around the same time that he began making fundamental discoveries in the theory of polynomial equations. He submitted two papers on this topic to the Academy of Sciences. Augustin Louis Cauchy refereed these papers, but refused to accept them for publication for reasons that still remain unclear. However, in spite of many claims to the contrary, it appears that Cauchy recognized the importance of Galois’ work, and that he merely suggested combining the two papers into one in order to enter it in the competition for the Academy’s Grand Prize in Mathematics. Cauchy, a highly eminent mathematician of the time, considered Galois’ work to be a likely winner.

On July 28, 1829, Galois’ father committed suicide after a bitter political dispute with the village priest. A couple of days later, Galois made his second and last attempt at entering the Polytechnique, and failed yet again. It is undisputed that Galois was more than qualified; however, accounts differ on why he failed. The legend holds that he thought the exercise proposed to him by the examiner to be of no interest, and, in exasperation, threw at the examiner’s head the rag used to erase the blackboard. More plausible accounts state that Galois made too many logical leaps and baffled the incompetent examiner, evoking the student’s rage. The recent death of his father may have also influenced his behavior.

Having been denied admission to the Polytechnique, Galois took the Baccalaureate examinations in order to enter the École Normale. He passed, receiving his degree on December 29, 1829. His examiner in mathematics reported, “This pupil is sometimes obscure in expressing his ideas, but he is intelligent and shows a remarkable spirit of research.”

In April 1829 Galois had his first mathematics paper published on continued fractions in the Annales de mathématiques. On 25 May and 1 June he submitted articles on the algebraic solution of equations to the Académie des Sciences. Cauchy was appointed as referee of Galois’ paper.

Galois sent Cauchy further work on the theory of equations, but then learned from Bulletin de Férussac of a posthumous article by Abel which overlapped with a part of his work. Galois then took Cauchy‘s advice and submitted a new article On the condition that an equation be soluble by radicals in February 1830. The paper was sent to Fourier, the secretary of the Paris Academy, to be considered for the Grand Prize in mathematics. Fourier died in April 1830 and Galois’ paper was never subsequently found and so never considered for the prize.

Galois, after reading Abel and Jacobi‘s work, worked on the theory of elliptic functions and abelian integrals. With support from Jacques Sturm, he published three papers in Bulletin de Férussac in April 1830. However, he learnt in June that the prize of the Academy would be awarded the Prize jointly to Abel (posthumously) and to Jacobi, his own work never having been considered.

There is a movie that is based on his life

In 1830, during the revolution, Galois was expelled from school for publicly criticizing the director of his school for failing to support the Revolution. Based on the suggestion of a friend Galois wrote a new paper on his research though. “Sur les conditions de re’solubilite’ des e’quations par radicaux,” is the only finished article on his theory of the solution of equations. Unfortunately, the Academy returned his paper stating that he needed to write a fuller explanation. Shortly after being expelled from school, Galois was arrested for political offenses and spent most of the last year and a half of his life in prison. He did write a scratchy and hastily written account of his researches which he entrusted to his friend August Chevalier. This account was written the night before his death and has been preserved. He was killed in a dual on May 31, 1832. The first full and clear presentation of Galois theory was given in 1870 by Camille Jordan in a book.

A famous quote from Galois:

“Unfortunately what is little recognized is that the most worthwhile scientific books are those in which the author clearly indicates what he does not know; for an author most hurts his readers by concealing his difficulties.”

More on his work in subsequent post




Hilbert’s 23 problems

Found this over here and sharing it over here


Hilbert’s address of 1900 to the International Congress of Mathematicians in Paris is perhaps the most influential speech ever given to mathematicians, given by a mathematician, or given about mathematics. In it, Hilbert outlined 23 major mathematical problems to be studied in the coming century. Some are broad, such as the axiomatization of physics (problem 6) and might never be considered completed. Others, such as problem 3, were much more specific and solved quickly. Some were resolved contrary to Hilbert’s expectations, as the continuum hypothesis (problem 1).

Hilbert’s address was more than a collection of problems. It outlined his philosophy of mathematics and proposed problems important to his philosophy.

Although almost a century old, Hilbert’s address is still important and should be read (at least in part) by anyone interested in pursuing research in mathematics.

In 1974 a symposium was held at Northern Illinois University on the Mathematical developments arising from Hilbert problems. A major mathematician discussed progress on each problem and how work on the problem has influenced mathematics. Also, 23 new problems of importance were described. The two-volume proceedings of the symposium was edited by Felix Browder and published by the American mathematical Society in 1976. See also Irving Kaplansky’s Hilbert’s problems, University of Chicago, Chicago, 1977.

There is also a collection on Hilbert’s Problems, edited by P. S. Alexandrov, Nauka, Moscow, 1969, in Russian, which has been translated into German.

Types of Infinities

One surely would have heard the word infinity many times in his life, but what most of the people dont know is that there are different kinds of infinities.

This post tries to throw some light on this.

You can see that there are infinite types of infinity via Cantor’s theorem which shows that given a set A, its power set P(A) is strictly larger in terms of infinite size (the technical term is “cardinality”).

A very nice introduction to the many different notions of infinity in mathematics is Rudy Rucker’s book: Infinity and the Mind. Unlike many other popularizations, this is written by someone who did a Ph.D. on the topic. Moreover, Rucker has gone to great lengths to make the presentation faithful to the mathematics but still accessible to an educated layperson. Below is an excerpt on the Alephs.

Now coming to the concept of countable and uncountable infinities, here’s an explanation that i have found really interesting:

Suppose no one every taught you the names for ordinary numbers. Then suppose that you and I agreed that we would trade one bushel of corn for each of my sheep. But there’s a problem, we don’t know how to count the bushels or the sheep! So what do we do?

We form a “bijection” between the two sets. That’s just fancy language for saying you pair things up by putting one bushel next to each of the sheep. When we’re done we swap. We’ve just proved that the number of sheep is the same as the number of bushels without actually counting.

We can try doing the same thing with infinite sets. So suppose you have the set of positive numbers and I have the set of rational numbers and you want to trade me one positive number for each of my rationals. Can you do so in a way that gets all of my rational numbers?

Perhaps surprisingly the answer is yes! You make the rational numbers into a big square grid with the numerator and denominators as the two coordinates. Then you start placing your “bushels” along diagonals of increasing size, see wikipedia.

This says that the rational numbers are “countable” that is you can find a clever way to count them off in the above fashion.

The remarkable fact is that for the real numbers there’s no way at all to count them off in this way. No matter how clever you are you won’t be able to scam me out of all of my real numbers by placing a natural number next to each of them. The proof of that is Cantor’s clever “diagonal argument.”
For those interested in the origin of this above discussion can go here:

More on this in my next post.

Ludwig Wittgenstein – A memoir

Ludwig Wittgenstein‘s father was Karl Wittgenstein who
was Jewish while his mother was a Roman Catholic. Ludwig was baptised into the
Catholic Church. His parents were both very musical and Ludwig was brought up in
a home which was always filled with music, Brahms being a frequent guest.
Ludwig’s parents had eight children who were all highly gifted both artistically
and intellectually. There were three girls, Gretl, Hermine, and Helene, and five
boys Hans, Kurt, Rudolf, Paul, and Ludwig. The family were wealthy
industrialists having made a fortune in the steel industry and, being one of the
wealthiest families in Austria, they were able to provide the best possible
education for their children.

Perhaps at this stage we should make some comments on Ludwig’s
brothers and sisters, for it will help to understand something of Ludwig’s
lifestyle as he grew up and also what he went through. Three of the boys, Hans,
Kurt, and Rudolf, all committed suicide later in their lives. Paul was a
talented pianist who lost an arm during World War I. Ravel composed Concerto
for the Left Hand
for him. Gretl had her portrait painted by Gustav Klimt,
the great Austrian Art Nouveau painter. Hermine wrote an important article on
Wittgenstein which is published in [<a href="javascript:ref(' R Rhees (ed.), Recollections of Wittgenstein (Oxford, 1984).’,16)”>16]
and from which we give some quotes.

Ludwig was the youngest of the children and he was educated at
home until he was fourteen years of age. He showed an interest in mechanical
things as he grew up and when he was ten years old he made a working sewing
machine. In 1903 Wittgenstein began three years of schooling at the Realschule
in Linz, Austria, which specialised in mathematics and natural science. Coming
from a cultured background into a school filled with working class children gave
Wittgenstein a difficult and unhappy time. He did not understand his fellow
pupils and to them he seemed [<a href="javascript:ref(' R Rhees (ed.), Recollections of Wittgenstein (Oxford, 1984).’,16)”>16]:-

… like a being from another world.

How could they be expected to understand the frail shy boy who
spoke with a stammer, and whose father was one of the richest men in Austria?
The school enhanced Wittgenstein’s love of technology, however, and made him
decide to study engineering at university. In 1906 he went to Berlin where began
his studies in mechanical engineering at the Technische Hochschule in
Charlottenburg. Intending to study for his doctorate in engineering,
Wittgenstein went to England in 1908 and registered as a research student in an
engineering laboratory of the University of Manchester.

His first project involved the study of the behaviour of kites
in the upper atmosphere of the earth. He moved from this to further study of
aeronautical research, this time examining the design of a propeller with a
small jet engine on the end of each blade. At this stage Wittgenstein was much
more practically minded than one might suppose, given his later highly
theoretical work, and he not only studied the theoretical design of the
propeller but he actually built and tested it.

The tests of the propeller were successful but, needing to
understand more mathematics for his research, he began a study which soon
involved him in the foundations of mathematics. Russell had published his
Principles of Mathematics in 1903 and Wittgenstein turned to this work as
he sought a better understanding of foundations of his subject. He became so
interested in Russell‘s work that he decided that he
wanted to learn more. Wittgenstein travelled to Jena to ask Frege‘s advice and was told that he
should study under Russell.

Wittgenstein left his aeronautical research in Manchester in
1911 to study mathematical logic with Russell in Trinity College, Cambridge.
Russell was not one to be easily
impressed by a student, but he was certainly very impressed by Wittgenstein. Russell wrote that teaching
Wittgenstein was:-

… one of the most exciting intellectual adventures
[of my life]. … [Wittgenstein had] fire and
penetration and intellectual purity to a quite extraordinary degree. …
[He] soon knew all that I had to teach.

Russell also wrote [<a href="javascript:ref(' D F Pears, Wittgenstein (London, 1971).’,14)”>14]:-

His disposition is that of an artist, intuitive and moody.
He says every morning he begins his work with hope, and every evening he ends in

By 1912 Russell had become convinced that
Wittgenstein possessed a genius which should be directed towards mathematical
philosophy. He therefore persuaded Wittgenstein to give up any ideas that he
still had to resume his applied mathematical work on aeronautics.

The first paper that Wittgenstein presented was to the
Cambridge Philosophical Society in 1912. Entitled What is
[<a href="javascript:ref(' B McGuinness, Wittgenstein: A Life: Young Ludwig, 1889-1921 (1988).’,12)”>12]:-

… [it] shows that from the very beginning
Wittgenstein recognised the importance of understanding the nature of
philosophical problems and of reflecting on the appropriate methods for
approaching them.

During this period at Cambridge, Wittgenstein continued to work
on the foundations of mathematics and also on mathematical logic. He suffered
depression, however, and threatened suicide on a number of occasions. He found
Cambridge a less than ideal place to work since he felt that the academics there
were merely trying to be clever in their discussions while their ideas lacked
depth. When he told Russell that he wanted to leave
Cambridge and go to Norway, Russell tried to dissuade him [<a href="javascript:ref(' D F Pears, Wittgenstein (London, 1971).’,14)”>14]:-

I said it would be dark, and he said he hated daylight. I
said it would be lonely, and he said he prostituted his mind talking to
intelligent people. I said he was mad, and he said God preserve him from sanity.
(God certainly will.)

Despite Russell‘s attempts to stop him,
Wittgenstein went to Skjolden in Norway and this proved an extremely fruitful
period during which lived in isolation working on his ideas on logic and
language that would form the basis of his great work the Tractatus
It was also a period when he continued to suffer
depression. His letters to Hermine spoke of his mental torment (see [<a href="javascript:ref(' R Rhees (ed.), Recollections of Wittgenstein (Oxford, 1984).’,16)”>16])
and she wrote that during this time he lived:-

… in a heightened state of intellectual intensity, which
verged on the pathological.

When World War I broke out in 1914 Wittgenstein immediately
travelled from Skjolden to Vienna to join the Austrian army. He was keen to
enlist since he wanted to face death [<a href="javascript:ref(' R Rhees (ed.), Recollections of Wittgenstein (Oxford, 1984).’,16)”>16]:-

Now I should have the chance to be a decent human being, for
I’m standing eye to eye with death.

He served first on a ship then in an artillery workshop but he
found his fellow soldiers very difficult as they subjected him to cruelty. In
1916 he was sent as a member of a howitzer regiment to the Russian front where
he gained many distinctions for bravery. In 1918 he was sent to north Italy in
an artillery regiment and he was there at the end of the war, becoming a
prisoner of the Italians in Cassino. During these four years of active service
Wittgenstein had written his great work in logic, the Tractatus, and the
manuscript was found in his rucksack when he was taken prisoner. He was allowed
to send the manuscript to Russell while he was held in a prison
camp in Italy.

Having written what he believed was his final word on
philosophy, Wittgenstein’s intention was now to give up his study of the
subject. Released from detention in 1919, he gave away the family fortune he had
inherited and, in the following year, trained as a primary school teacher in
Austria. He was trained in the methods of the Austrian School Reform Movement,
which believed that the main aim of a teacher was to arouse a child’s curiosity
and to help the child develop as an independent thinker. The Movement rejected
the method of teaching which encouraged children to simply learn to repeat
facts. But although Wittgenstein was a strong believer in these principles and
tried with great enthusiasm to provide the children that he taught in the
mountain village of Wiener Neustadt with the best possible education, there were
factors working against his success. Perhaps the biggest difficulty that
Wittgenstein faced was that giving away the family fortune did nothing to enable
someone with his highly privileged background to fit into the culture of the
children of farmers who he taught.

During this period Wittgenstein was again desperately unhappy
and came close to committing suicide on a number of occasions. The thought that
he was appreciated by the children kept him at his task, but he found
difficulties in keeping relations between himself and the other teachers on a
friendly basis. Eventually, feeling largely that he had failed as a primary
school teacher, Wittgenstein gave up in 1925. He still did not feel that he
wanted to return to an academic life so he worked at a number of different jobs.
First he worked as a gardener’s assistant in the Hüsseldorf monastery near
Vienna, living in the tool-shed for three months. Then he worked as an architect
for two years occupied in the design and construction of a mansion house for his
sister Gretl near Vienna.

Although Wittgenstein had not wished to return to academic life
during this period he was not completely isolated from the study of mathematical
logic, the foundations of mathematics, and philosophy. He met with Ramsey, who was making a special study
of the Tractatus and had travelled from Cambridge to Austria on several
occasions to have discussions with him, and he also met with philosophers from
the Vienna Circle. There have been many theories put forward to explain why he
returned to academic life, but at the heart of it must be that in the
discussions he had, he came to see problems with the Tractatus.

In 1929 Wittgenstein returned to Cambridge where he submitted
the Tractatus as his doctoral thesis. This work considers the
relationship of language to the world. Words, Wittgenstein argued, were
representations of objects and combining words led to propositions which were
statements about reality, or as he says, pictures of reality. Such statements,
of course, may picture a reality which is true or false. Conversely, the world
as presented by Wittgenstein in the Tractatus, consists of facts. These
facts can be broken down into states of affair, which in turn can be broken down
into combinations of objects. This is essentially an atomic theory with the
world built from simple objects. He argues that there is a bijection (one-one
correspondence) between language and the world.

In the Preface to Philosophical Investigations written
sixteen years after he returned to Cambridge, Wittgenstein wrote:-

… since beginning to occupy myself with philosophy again,
sixteen years ago, I have been forced to recognise grave mistakes in what I
wrote in that first book. I was helped to realise these mistakes – to a degree
which I myself am hardly able to estimate – by the criticism which my ideas
encountered from Frank Ramsey, with whom I discussed
them in innumerable conversations during the last two years of his life.

However, it was not until 1953, two years after Wittgenstein’s
death, that this second great work Philosophical Investigations was
published. In this work Wittgenstein studied [<a href="javascript:ref(' B McGuinness, Wittgenstein: A Life: Young Ludwig, 1889-1921 (1988).’,12)”>12]:-

… the philosophy of language and philosophical psychology.
… the form of the book is quite unique. … we first get a part of
distinct, numbered remarks, varying in length from one line to several
paragraphs, and a second part of fourteen sections, half a page to thirty-six
pages long … instead of presenting arguments and clearly stated conclusions,
these remarks reflect on a wide range of topics without ever producing a clear
final statement on any of them.

How does his approach in the Philosophical
differ from that in the Tractatus? He is still
concerned with language, but in his later thinking words are not unvarying
representations of objects, but rather are diverse. He draws an analogy between
words and tools in a tool-box:-

… there is a hammer, pliers, a saw, a screw-driver, a
ruler, a glue-pot, nails and screws. The function of words are as diverse as the
functions of these objects.

It was not that a word had a meaning, rather it had a use.
Another illustration that he gives is an analogy between words and pieces in a
chess game. The meaning of a chess piece is not determined by its physical
appearance, rather it is determined by the rules of chess. Similarly the meaning
of a word is its use governed by rules.

After the award of his doctorate, Wittgenstein was appointed a
lecturer at Cambridge and he was made a fellow of Trinity College. In the
following years Wittgenstein lectured there on logic, language, and the
philosophy of mathematics. He was appointed to the chair in philosophy at
Cambridge in 1939. Malcolm, a student of Wittgenstein, writes in [<a href="javascript:ref(' N Malcolm, Ludwig Wittgenstein : a memoir (London, 1958).’,10)”>10]
about Wittgenstein’s lectures which he attended in 1939:-

His lectures were given without preparation and without
notes. He told me once that he tried to lecture from notes but was disgusted
with the result; the thoughts that came out were ‘stale’, or, as he put it to
another friend, the words looked like ‘corpses’ when he began to read them. In
the methods that he came to use, his only preparation for the lecture, as he
told me, was to spend a few minutes before the class met, recollecting the
course that the inquiry had taken at the previous meeting. At the beginning of
the lecture he would give a brief summary of this and then he would start from
there, trying to advance the investigation with fresh thoughts. …
[W]hat occurred in these class meetings was largely new

G H von Wright was a pupil of Wittgenstein at Cambridge. He
writes [<a href="javascript:ref(' N Malcolm, Ludwig Wittgenstein : a memoir (London, 1958).’,10)”>10]:-

Wittgenstein thought that his influence as a teacher was, on
the whole, harmful to the development of independent minds in his disciples. I
am afraid that he was right. And I believe that I can partly understand why it
should be so. Because of the depth and originality of his thinking, it is very
difficult to understand Wittgenstein’s ideas and even more difficult t
incorporate them into one’s own thinking. At the same time the magic of his
personality and style was most inviting and persuasive. to learn from
Wittgenstein without coming to adopt his forms of expression and catchwords and
even to imitate his tone of voice, his mien and gestures was almost

There is a suggestion here that Wittgenstein would never have
fitted in as the leader of a large group of students and researchers. Although
he did have students who went on produce important work, yet remain true to his
way of thinking, Wittgenstein always seemed an isolated figure. He seemed to
understand the reasons for this when he wrote:-

Am I the only one who cannot found a school or can a
philosopher never do this? I cannot found a school because I do not really want
to be imitated. Not at any rate by those who publish articles in philosophy

Wittgenstein remained at Cambridge until he resigned in 1947
except for the period of World War II during which he worked as a hospital
porter in Guy’s Hospital in London. He also spent time working as a laboratory
assistant in the Royal Victoria Infirmary before returning to his duties at
Cambridge in 1944. After three years back at Cambridge he retired and moved to
an isolated cottage on the west coast of Ireland. His health deteriorated and in
1949 cancer was diagnosed. Wittgenstein did not seem unhappy at the diagnosis
since he said that he did not wish to live any longer. He continued to work on
his ideas until a few days before his death, the power and depth of his
intellect being undiminished by illness.

McGinn, in [<a href="javascript:ref(' B McGuinness, Wittgenstein: A Life: Young Ludwig, 1889-1921 (1988).’,12)”>12],
gives a fair estimate of Wittgenstein:-

The power and originality of his thought show a unique
philosophical mind and many would be happy to call him a genius.

Wittgenstein was never happy with his own writings and as a
result only the one major work, the Tractatus, was published during his
life. A wealth of material from his lectures and notes has subsequently been
published. That his ideas are found difficult is something that he was well
aware of and he felt that in some way he did not fit into the world in which he
lived. Let us end with a quote from his own writing about why ideas are found

Why is philosophy so complicated? It ought to be entirely
simple. Philosophy unties the knots in our thinking that we have, in a senseless
way, put there. To do this it must make movements that are just as complicated
as these knots. Although the result of philosophy is simple, its method cannot
be if it is to succeed. The complexity of philosophy is not a complexity of its
subject matter, but of our knotted understanding.