Diophantus of Alexandria

Diophantus (+/- 250 AD) of Alexandria has been called 'the father of algebra' and an entire branch of mathematics has been named after him, the study of Diophantine Equations. The most famous problem in this field 'Diophantine Equations' is Fermat's 'Last Theorem'. Fermat was reading Diophantus' comments on the Pythagoran theorem when he conjectured that for an exponent n > 2, the equation \[ x^n + y^n = z^n \] has no integer solutions. This theorem was considered the hardest open problem in mathematics until solved by Andrew Wiles in 1994. Diophantus work was lost to the Western world for thousand years.

Anyway, I thought about Diophantus when I came across this beautiful equation which has an infinite number of integer solutions \[ x^3 + y^3 + z^3 = x^2 +y^2 + z^2.\]

Two mystery mathematicians.

#maths

Facts about two mathematicians.

#1.

His collected works appear in five volumes: the first contains 62 papers which (...) ; the second contains 107 of the 147 papers (...); the third includes 89 of the 180 papers (...); the fourth contains 98 of the 232 papers he published (...).

#2.

(...) school reports began to describe him as singular, bizarre, original and closed. - (...) took the examination of the École Polytechnique but failed. - This is the only student who has answered me poorly, he knows absolutely nothing. I was told that this student has an extraordinary capacity for mathematics. This astonishes me greatly, for, after his examination, I believed him to have but little intelligence. - (Later in life he was sent to prison twice.)

I am sure that you remember at least one of them, I am not sure about the other though. Do you recognize the persons already? - One of them is recognized as one of the greatest minds that ever lived. The other has been criticized for the lack of depth in his work.

Who are they?
- Mystery person #1.
- Mystery person #2.

What is the origin of mathematics ?

If radio astronomers would discover a signal containing a repeating sequence of prime numbers then they would claim to have found extra-terrestrial intelligent life. Why? Because they consider mathematics as universal throughout the entire universe to which only intelligent life forms have access.

Mathematics is universal. That basically means that mathematics is discovered and not created. We use, for example, $\pi$ as the ratio between circumference and diameter of a circle, that ratio is the same everywhere in the universe. Not the symbol $pi$, the decimal number system and so forth.

We take the existence of mathematics for granted, we don't question when and how it was created. Where does that vast body of mathematics come from? Was it created with the Big Bang? If so, than the ( mathematical ) models of physicists that explain their Big Bang theory look naively simple.

It's rather vague to discuss if there was mathematics before the Big Bang. But -if- there was a point from which everything was created than that creation must have included -all- of mathematics. If we don't accept that than we are saying that we are the most intelligent life form in the universe, because mathematics is created by us and not discovered by us.

What -is- the origin of mathematics?

Britain´s greatest code breaker / Alan Turing

Alan Turing ( represents the team that ) decisively changed the course of World War II. He is among the greatest scientists of the 20th century, if not all times. Alan Turing is the inventor of computers, programming and artificial intelligence. All the computers in operation today, including the billions of smartphones are in fact Turing Machines, the computer Turing invented conceptuallly. His thoughts were revolutionary. Computers in his days, were people. People, computing.

The British of today are working hard on clearing their conscience on how they treated Alan Turing. In 2009 Gordon Brown said that "he is sorry for the "appalling" way World War II code-breaker Alan Turing was treated for being gay." And now there is this documentary called "Britain's greatest codebreaker" based on the biography of Turing and sessions Turing had with a psychiatrist in Manchester, Franz Greenbaum.

In my opinion Alan Turing could only have flourished in Britain because of his eccentricity. What would have happened with Turing if he had to work at a patent-office like Einstein? Asa Briggs, one of the codebreakers at Bletchley Park mentioned that Turing often came to work with his pajamas under his jacket. That described the culture in that group, I suppose. One of the most important things in Turings private life was the loss of his best friend, the love of his. He never got over it although in Bletchley Park he became friends with a woman, the only woman codebreaker on the team. He even proposed to marry her but Turing called it off, because he wanted to live an honest life. That was his first mistake...

His second mistake was going to the police and accusing a male prostitute of stealing 50 pounds. The police did not care about the 50 pound robbery at all, they could book a professor for gross indecency. Turing now lived in a world where, it seems, nothing out of the ordinary was accepted. Society really wanted him. His sentence was that he could choose prison or enforced body change. He chose the body change, which was an experimental chemical castration. His security privileges were removed as well, meaning he could not continue to work for the UK Government Communications Headquarters. The drug, a synthetic version of female hormones, had a catastrophic effect on his body.

Alan Turing sadly ended his life in 1954.

See also:
- Alan Turing documentary

Rene Descartes

In the book 'Problem Solving and Number Theory' I read

The law of quadratic reciprocity was discovered for the first time, in a complex form, by L. Euler who published it in his paper entitled “Novae demonstrationes circa divisores numerorum formae $xx + nyy$.

Was there no notation for exponents in the time of the great Euler? I did not know how to formulate a query for Google, so I asked a question at Math/StackExchange.

I found out that Descartes introduced the notation for $x^2$. Descartes is famous for his quote "Cogito ergo sum", "Je pense donc je suis" or in plain English "I think, therefore I am". Descartes was my favourite mathematician in secondary school because he invented analytic geometry, one of the milestones in the development of mathematics. Before Descartes geometry was done strictly in the Euclidean way, by compass and ruler. Thanks to Descartes' quadrant and coordinates, geometric shapes like lines and circles could be respresented by algebraic equations. They become objects to do calculations with.

So far about Descartes.

Also, thanks to kind repliers I discovered the bookset "A History of Mathematical Notations, vols 1 and 2., by Florian Cajori". A real gem if you like the history of mathematics. Because the copyright expired one is allowed to freely download the original version. See the comments below the original question at StackExchange for a link to the full version of the book.

Link:
- Question at StackExchange

The difference between mathematics and numerology

Numerologist is a four-letter word among mathematicians ( I can imagine ). Is numerology the same as number theory? Not really, the difference between numerology and number theory, as a branch of mathematics, can best be explained using an example.

Both numerologists and mathematicians study identities like $$2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 = 666.$$ A mathematician would notice that this is the sum of the first seven primes and rewrite the sum as $$\sum_{k=1}^{7} p_k = 666.$$ The right-hand side sum is a number with equal digits, so the mathematician might look at other sums of consecutive primes and verify if these sums have similar patterns. Basically a mathematician is interested in anything that might lead to the formulation of a theorem, a proposition to prove mathematically.

A numerologist would immediately notice that $666$ is the number of "The Beast", the representation of evil in the Christian belief system. Then, a numerologist might consider prime numbers divine, since they are the building blocks of all integers, and might try to formulate some law of good and bad represented in the sum-formula. Numerologists also believe that future events can be predicted so they will be extra alert to that.

I believe that it was until the Middle Ages that there was no real distinction between the profession of numerologist and mathematician. Newton has been called the last Alchemist, perhaps he was the last mathematician / numerologist as well. ( History, I am afraid, is not my strongest point. )

Fermat on arithmeticians

Thanks to Sol Robeson ( in PI ) we call mathematicians who lost it "numerologists".

In 1657, Fermat challenged William Brouncker, of Castle Lynn in Ireland, and John Wallis to find integral solutions to the equations $$x^2 − 151y^2 = 1$$ and $$x^2 − 313y^2 = −1.$$ He ( Fermat ) cautioned them not to submit rational solutions because even the lowest type of arithmetician could devise such answers.

"An Introduction to Diophantine Equations, A Problem-Based Approach, Andreescu, Andrica & Cucurezeanu, Springer 2010"

Considering that Fermat used the qualification the lowest type of arithmetician there must have been a ranking in the computational branch those days. Until at least WW2, a computer, was the job description of someone who did "computational work" in banking, insurance, trading, logistics and what have you. Jobs like that exist even now, think of the actuarial sciences, but most of them if not all require a degree in mathematics. I am not sure but I suppose that in Fermat's days there must have been people responsible for the basic addition and multiplication type of calculations. Fermat called them "arithmeticians, of the lowest kind".

I am speculating of course. Fermat could have been a terrible arrogant man looking down on the working class. Considering that he was not a mathematician himself but that he wrote, on his own initiative, letters to the great minds of his time says at least something of his self-image.

Link: My previous post on Fermat

Littlewood's advice to Ingham

My advice about studying mathematics is to go deep, to slowly let it sink in and revise, revise and revise. I.e. imagine a student who just reached the mandatory 360 (540) points: if he would not pass a subset of all his previous exams -today- wouldn't the degree be a joke? Anyway, I am all for a final exam covering a subset of all previous exams. - Yet, I don't think students will like the idea. But why not? Once the math is 'in' you would only have to maintain it during the lifetime of the course.

Hardy (l), Littlewood (r)

Ingham

An advice given by Littlewood to Ingham ( both were giants in Number Theory while Littlewood, together with Hardy of course, got Ramanujan to Cambridge ) is the following:

Albert Ingham was educated at Stafford Grammar School, and from there he won a scholarship to Trinity College, Cambridge, in December 1917. After spending a few months in the army towards the end of World War I, he began his studies in January 1919. An outstanding undergraduate career saw him awarded distinction in the Mathematical Tripos and win a Smith's prize and the highest honours. In 1922 he was elected to a fellowship at Trinity for a dissertation on the zeta function and his next four years were occupied only with research, a few months of which were spent at Göttingen. During this time Ingham was greatly influenced by Littlewood who gave him the advice to:-

... work at a hard problem: you may not solve it but you'll solve another one.

An advice I have taken long ago. When you generally work on harder problems than those presented to you in TMA's and/or exams, your TMA questions tend to become trivial. This is one possible method to score high marks for your TMA's. Other known proven recipe's which I do not advise are:
- ask someone with a degree 'to help you' ;
- target a student in the forums and offer him help ( not knowing he is going to do all the helping himself );
- post portions of your questions in Dr Math type of forums, mathematicians are very helpful;

If you have a blog ( in mathematics, or any other field ) and people start acting friendly then they are working towards asking you
- to 'exchange' some OU course books;
- to 'compare' answers on a TMA.
Make sure you always say NO, you don't owe them anything. Friendly visitors come in many disguises.

Paul Erdös - documentary

Some people consider Paul Erdös as the most prolific mathematician who ever lived. Born in Hungary in 1913, Erdös wrote and co-authored over 1,500 papers and pioneered several fields in theoretical mathematics. "N is a Number: A Portrait of Paul Erdös" is a documentary about his life and work. - It is now available for on line viewing on Internet. - N is a Number. Portrait of Paul Erdös.

Paul Erdös 1913 - 1996

Don't expect anything about his (other) addiction in this documentary. In an article by Paul Hoffman published in November 1987, Atlantic Monthly profiled Erdös and discussed his Benzedrine habit. Erdös liked the article, "...except for one thing...You shouldn't have mentioned the stuff about Benzedrine. It's not that you got it wrong. It's just that I don't want kids who are thinking about going into mathematics to think that they have to take drugs to succeed."

Time to start debunking Euler ?

Some names are only to be spoken of with respect.

Apple has been around a lot longer than Google. Apple created the i-Tunes software and related store where you can buy songs for only $0.99 and they created the i-Phone gadget. If you are a fan of that gadget I am sure you bought at least four of them in the last three years although the first one is still working it doesn't look that good compared to the i-Phone 4. Besides that Apple created billions of cash for Steve Jobs who has no intention to share it with the poor like Bill Gates from Microsoft does. - Google on the other hand created free software like gmail, google docs and google maps, a free feature we take for granted. Oh, and they created Google Search of course. Since a while, also thanks to Google, we have digital access to the books of Leonhard Euler ( and all the other great mathematicians of course ).

Have a look at the page where Euler defines the number Pi, here at Google Books. ( If you don't read Latin, why not use Google Translate? ) I checked the first 120 decimals of Euler's Pi with those from Mathematica 8 (2010) . They are exactly the same.

When I looked at Pi in 120 decimals I knew immediately that Euler did not make these calculations by himself. He just could not. OK, he was a genius but he did not have more -time- than a mere humanoid like us. How many hours ( if not days, i.e. check double-check ) would it have cost to do that calculation? It's not just -that- calculation. It's about the sheer size and depth of his legacy. Although Euler was still an active mathematician when he died at age 76, it is beyond the capabilities of one human being to create that in one lifetime.

Euler must have used students and computing staff to write his books. We know that he was blind during the last years of his life but even in that period he continued to publish. Perhaps 'Euler' was merely a brand. I don't know. Who was 'Euler'?

Perhaps the comparison of Euler with Ramanujan would be somewhat like a comparison of Rembrandt with van Gogh. We know for sure that everything we read in Ramanujan's notebooks is written by Ramanujan himself. I am not sure that everything in the books of Euler is written by Euler himself, the same is true for Rembrandt and are the cause of the fact that so many Rembrandt's turned out to be work of his students.

Before Pythagoras

An interesting exhibition in New York - "Before Pythagoras: The Culture of Old Babylonian Mathematics November 12 - December 17, 2010"

Visit the website of Before Pythagoras

Nicolas Bourbaki

From Preface to the Third Edition of Galois Theory by Ian Stewart

Nicolas Bourbaki is the pseudonym of a group of mathematicians — mostly French, mostly young — who tidied up the mathematics of the mid-20th century in a lengthy series of books. Their guiding principle was never to prove a theorem if it could be deduced as a special case of a more general theorem. To study planar geometry, work in n dimensions and then let n = 2.