The 20th Century continued the trend of the 19th
towards increasing generalization and abstraction in mathematics, in
which the notion of axioms as “self-evident truths” was largely
discarded in favour of an emphasis on such logical concepts as
consistency and completeness.
It also saw mathematics become a major profession, involving thousands of new Ph.D.s each year and jobs in both teaching and industry, and the development of hundreds of specialized areas and fields of study, such as group theory, knot theory, sheaf theory, topology, graph theory, functional analysis, singularity theory, catastrophe theory, chaos theory, model theory, category theory, game theory, complexity theory and many more.
The eccentric British mathematician G.H. Hardy and his young Indian protégé Srinivasa Ramanujan, were just two of the great mathematicians of the early 20th Century who applied themselves in earnest to solving problems of the previous century, such as the Riemann hypothesis. Although they came close, they too were defeated by that most intractable of problems, but Hardy is credited with reforming British mathematics, which had sunk to something of a low ebb at that time, and Ramanujan proved himself to be one of the most brilliant (if somewhat undisciplined and unstable) minds of the century.
Others followed techniques dating back millennia but taken to a 20th Century level of complexity. In 1904, Johann Gustav Hermes completed his construction of a regular polygon with 65,537 sides (216 + 1), using just a compass and straight edge as Euclid would have done, a feat that took him over ten years.
The early 20th Century also saw the beginnings of the rise of the field of mathematical logic, building on the earlier advances of Gottlob Frege, which came to fruition in the hands of Giuseppe Peano, L.E.J. Brouwer, David Hilbert and, particularly, Bertrand Russell and A.N. Whitehead, whose monumental joint work the “Principia Mathematica” was so influential in mathematical and philosophical logicism.
The century began with a historic convention at the Sorbonne in Paris
in the summer of 1900 which is largely remembered for a lecture by the
young German mathematician David Hilbert
in which he set out what he saw as the 23 greatest unsolved
mathematical problems of the day. These “Hilbert problems” effectively
set the agenda for 20th Century mathematics, and laid down the gauntlet
for generations of mathematicians to come. Of these original 23
problems, 10 have now been solved, 7 are partially solved, and 2 (the
Riemann hypothesis and the Kronecker-Weber theorem on abelian
extensions) are still open, with the remaining 4 being too loosely
formulated to be stated as solved or not.
Hilbert was himself a brilliant mathematician, responsible for several theorems and some entirely new mathematical concepts, as well as overseeing the development of what amounted to a whole new style of abstract mathematical thinking. Hilbert's approach signalled the shift to the modern axiomatic method, where axioms are not taken to be self-evident truths. He was unfailingly optimistic about the future of mathematics, famously declaring in a 1930 radio interview “We must know. We will know!”, and was a well-loved leader of the mathematical community during the first part of the century.
However, the Austrian Kurt Gödel was soon to put some very severe constraints on what could and could not be solved, and turned mathematics on its head with his famous incompleteness theorem, which proved the unthinkable - that there could be solutions to mathematical problems which were true but which could never be proved.
Alan Turing, perhaps best known for his war-time work in breaking the German enigma code, spent his pre-war years trying to clarify and simplify Gödel’s rather abstract proof. His methods led to some conclusions that were perhaps even more devastating than Gödel’s, including the idea that there was no way of telling beforehand which problems were provable and which unprovable. But, as a spin-off, his work also led to the development of computers and the first considerations of such concepts as artificial intelligence.
With the gradual and wilful destruction of the mathematics community of Germany and Austria by the anti-Jewish Nazi regime in the 1930 and 1940s, the focus of world mathematics moved to America, particularly to the Institute for Advanced Study in Princeton, which attempted to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. Many of the brightest European mathematicians, including Hermann Weyl, John von Neumann, Kurt Gödel and Albert Einstein, fled the Nazis to this safe haven.
John von Neumann is considered one of the foremost mathematicians in
modern history, another mathematical child prodigy who went on to make
major contributions to a vast range of fields. In addition to his
physical work in quantum theory and his role in the Manhattan Project
and the development of nuclear physics and the hydrogen bomb, he is
particularly remembered as a pioneer of game theory, and particularly
for his design model for a stored-program digital computer that uses a
processing unit and a separate storage structure to hold both
instructions and data, a general architecture that most electronic
computers follow even today.
André Weil was another refugee from the war in Europe, after narrowly avoiding death on a couple of occasions. His theorems, which allowed connections to be made between number theory, algebra, geometry and topology, are considered among the greatest achievements of modern mathematics. He was also responsible for setting up a group of French mathematicians who, under the secret nom-de-plume of Nicolas Bourbaki, wrote many influential books on the mathematics of the 20th Century.
Perhaps the greatest heir to Weil’s legacy was Alexander Grothendieck, a charismatic and beloved figure in 20th Century French mathematics. Grothendieck was a structuralist, interested in the hidden structures beneath all mathematics, and in the 1950s he created a powerful new language which enabled mathematical structures to be seen in a new way, thus allowing new solutions in number theory, geometry, even in fundamental physics. His “theory of schemes” allowed certain of Weil's number theory conjectures to be solved, and his “theory of topoi” is highly relevant to mathematical logic. In addition, he gave an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve. Although, after the 1960s, Grothendieck all but abandoned mathematics for radical politics, his achievements in algebraic geometry have fundamentally transformed the mathematical landscape, perhaps no less than those of Cantor, Gödel and Hilbert, and he is considered by some to be one of the dominant figures of the whole of 20th Century mathematics.
Paul Erdös was another inspired but distinctly non-establishment figure of 20th Century mathematics. The immensely prolific and famously eccentric Hungarian mathematician worked with hundreds of different collaborators on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. As a humorous tribute, an "Erdös number" is given to mathematicians according to their collaborative proximity to him. He was also known for offering small prizes for solutions to various unresolved problems (such as the Erdös conjecture on arithmetic progressions), some of which are still active after his death.
The field of complex dynamics (which is defined by the iteration of
functions on complex number spaces) was developed by two Frenchmen,
Pierre Fatou and Gaston Julia, early in the 20th Century. But it only
really gained much attention in the 1970s and 1980s with the beautiful
computer plottings of Julia sets and, particularly, of the Mandelbrot
sets of yet another French mathematician, Benoît Mandelbrot. Julia and
Mandelbrot fractals are closely related, and it was Mandelbrot who
coined the term fractal, and who became known as the father of fractal
geometry.
The Mandelbrot set involves repeated iterations of complex quadratic polynomial equations of the form zn+1 = zn2 + c, (where z is a number in the complex plane of the form x + iy). The iterations produce a form of feedback based on recursion, in which smaller parts exhibit approximate reduced-size copies of the whole, and which are infinitely complex (so that, however much one zooms in and magifies a part, it exhibits just as much complexity).
Paul Cohen is an example of a second generation Jewish immigrant who followed the American dream to fame and success. His work rocked the mathematical world in the 1960s, when he proved that Cantor's continuum hypothesis about the possible sizes of infinite sets (one of Hilbert’s original 23 problems) could be both true AND not true, and that there were effectively two completely separate but valid mathematical worlds, one in which the continuum hypothesis was true and one where it was not. Since this result, all modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.
Another of Hilbert’s problems was finally resolved in 1970, when the young Russian Yuri Matiyasevich finally proved that Hilbert’s tenth problem was impossible, i.e. that there is no general method for determining when polynomial equations have a solution in whole numbers. In arriving at his proof, Matiyasevich built on decades of work by the American mathematician Julia Robinson, in a great show of internationalism at the height of the Cold War.
In additon to complex dynamics, another field that benefitted greatly from the advent of the electronic computer, and particulary from its ability to carry out a huge number of repeated iterations of simple mathematical formulas which would be impractical to do by hand, was chaos theory. Chaos theory tells us that some systems seem to exhibit random behaviour even though they are not random at all, and conversely some systems may have roughly predictable behaviour but are fundamentally unpredictable in any detail. The possible behaviours that a chaotic system may have can also be mapped graphically, and it was discovered that these mappings, known as "strange attractors", are fractal in nature (the more you zoom in, the more detail can be seen, although the overall pattern remains the same).
An early pioneer in modern chaos theory was Edward Lorenz, whose interest in chaos came about accidentally through his work on weather prediction. Lorenz's discovery came in 1961, when a computer model he had been running was actually saved using three-digit numbers rather than the six digits he had been working with, and this tiny rounding error produced dramatically different results. He discovered that small changes in initial conditions can produce large changes in the long-term outcome - a phenomenon he described by the term “butterfly effect” - and he demonstrated this with his Lorenz attractor, a fractal structure corresponding to the behaviour of the Lorenz oscillator (a 3-dimensional dynamical system that exhibits chaotic flow).
1976 saw a proof of the four colour theorem by Kenneth Appel and
Wolfgang Haken, the first major theorem to be proved using a computer.
The four colour conjecture was first proposed in 1852 by Francis Guthrie
(a student of Augustus De Morgan), and states that, in any given
separation of a plane into contiguous regions (called a “map”) the
regions can be coloured using at most four colours so that no two
adjacent regions have the same colour. One proof was given by Alfred
Kempe in 1879, but it was shown to be incorrect by Percy Heawood in 1890
in proving the five colour theorem. The eventual proof that only four
colours suffice turned out to be significantly harder. Appel and Haken’s
solution required some 1,200 hours of computer time to examine around
1,500 configurations.
Also in the 1970s, origami became recognized as a serious mathematical method, in some cases more powerful than Euclidean geometry. In 1936, Margherita Piazzola Beloch had shown how a length of paper could be folded to give the cube root of its length, but it was not until 1980 that an origami method was used to solve the "doubling the cube" problem which had defeated ancient Greek geometers. An origami proof of the equally intractible "trisecting the angle" problem followed in 1986. The Japanese origami expert Kazuo Haga has at least three mathematical theorems to his name, and his unconventional folding techniques have demonstrated many unexpected geometrical results.
The British mathematician Andrew Wiles finally proved Fermat’s Last Theorem for ALL numbers in 1995, some 350 years after Fermat’s initial posing. It was an achievement Wiles had set his sights on early in life and pursued doggedly for many years. In reality, though, it was a joint effort of several steps involving many mathematicans over several years, including Goro Shimura, Yutaka Taniyama, Gerhard Frey, Jean-Pierre Serre and Ken Ribet, with Wiles providing the links and the final synthesis and, specifically, the final proof of the Taniyama-Shimura Conjecture for semi-stable elliptic curves. The proof itself is over 100 pages long.
The most recent of the great conjectures to be proved was the Poincaré Conjecture, which was solved in 2002 (over 100 years after Poincaré first posed it) by the eccentric and reclusive Russian mathematician Grigori Perelman. However, Perelman, who lives a frugal life with his mother in a suburb of St. Petersburg, turned down the $1 million prize, claiming that "if the proof is correct then no other recognition is needed". The conjecture, now a theorem, states that, if a loop in connected, finite boundaryless 3-dimensional space can be continuously tightened to a point, in the same way as a loop drawn on a 2-dimensional sphere can, then the space is a three-dimensional sphere. Perelman provided an elegant but extremely complex solution involving the ways in which 3-dimensional shapes can be “wrapped up” in even higher dimensions. Perelman has also made landmark contributions to Riemannian geometry and geometric topology.
John Nash, the American economist and mathematician whose battle against paranoid schizophrenia has recently been popularized by the Hollywood movie “A Beautiful Mind”, did some important work in game theory, differential geometry and partial differential equations which have provided insight into the forces that govern chance and events inside complex systems in daily life, such as in market
economics, computing, artificial intelligence, accounting and military theory.
The Englishman John Horton Conway established the rules for the so-called "Game of Life" in 1970, an early example of a "cellular automaton" in which patterns of cells evolve and grow in a grid, which became extremely popular among computer scientists. He has made important contributions to many branches of pure mathematics, such as game theory, group theory, number theory and geometry, and has also come up with some wonderful-sounding concepts like surreal numbers, the grand antiprism and monstrous moonshine, as well as mathematical games such as Sprouts, Philosopher's Football and the Soma Cube.
Other mathematics-based recreational puzzles became even more popular among the general public, including Rubik's Cube (1974) and Sudoku (1980), both of which developed into full-blown crazes on a scale only previously seen with the 19th Century fads of Tangrams (1817) and the Fifteen puzzle (1879). In their turn, they generated attention from serious mathematicians interested in exploring the theoretical limits and underpinnings of the games.
Computers continue to aid in the identification of phenomena such as Mersenne primes numbers (a prime number that is one less than a power of two - see the section on 17th Century Mathematics). In 1952, an early computer known as SWAC identified 2257-1 as the 13th Mersenne prime number, the first new one to be found in 75 years, before going on to identify several more even larger.
With the advent of the Internet in the 1990s, the Great Internet
Mersenne Prime Search (GIMPS), a collaborative project of volunteers who
use freely available computer software to search for Mersenne primes,
has led to another leap in the discovery rate. Currently, the 13 largest
Mersenne primes were all discovered in this way, and the largest (the
45th Mersenne prime number and also the largest known prime number of
any kind) was discovered in 2009 and contains nearly 13 million digits.
The search also continues for ever more accurate computer approximations
for the irrational number π, with the current record standing at over 5 trillion decimal places.
The P versus NP problem, introduced in 1971 by the American-Canadian Stephen Cook, is a major unsolved problem in computer science and the burgeoning field of complexity theory, and is another of the Clay Mathematics Institute's million dollar Millennium Prize problems. At its simplest, it asks whether every problem whose solution can be efficiently checked by a computer can also be efficiently solved by a computer (or put another way, whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure). The solution to this simple enough sounding problem, usually known as Cook's Theorem or the Cook-Levin Theorem, has eluded mathematicians and computer scientists for 40 years. A possible solution by Vinay Deolalikar in 2010, claiming to prove that P is not equal to NP (and thus such insolulable-but-easily-checked problems do exist), has attracted much attention but has not as yet been fully accepted by the computer science community.
HARDY AND RAMANUJAN
The eccentric British mathematician G.H. Hardy is known for his
achievements in number theory and mathematical analysis. But he is
perhaps even better known for his adoption and mentoring of the
self-taught Indian mathematical genius, Srinivasa Ramanujan.
Hardy himself was a prodigy from a young age, and stories are told about how he would write numbers up to millions at just two years of age, and how he would amuse himself in church by factorizing the hymn numbers. He graduated with honours from Cambridge University, where he was to spend most of the rest of his academic career.
Hardy is sometimes credited with reforming British mathematics in the early 20th Century by bringing a Continental rigour to it, more characteristic of the French, Swiss and German mathematics he so much admired, rather than British mathematics. He introduced into Britain a new tradition of pure mathematics (as opposed to the traditional British forte of applied mathematics in the shadow of Newton), and he proudly declared that nothing he had ever done had any commercial or military usefulness (he was also an outspoken pacifist).
Just before the First World War, Hardy (who was given to flamboyant gestures) made mathematical headlines when he claimed to have proved the Riemann Hypothesis. In fact, he was able to prove that there were infinitely many zeroes on the critical line, but was not able to prove that there did not exist other zeroes that were NOT on the line (or even infinitely many off the line, given the nature of infinity).
Meanwhile, in 1913, Srinivasa Ramanujan, a 23-year old shipping clerk from Madras, India, wrote to Hardy (and other academics at Cambridge), claiming, among other things, to have devised a formula that calculated the number of primes up to a hundred million with generally no error. The self-taught and obsessive Ramanujan had managed to prove all of Riemann’s results and more with almost no knowledge of developments in the Western world and no formal tuition. He claimed that most of his ideas came to him in dreams.
Hardy was only one to recognize Ramanujan's genius, and brought him to Cambridge University, and was his friend and mentor for many years. The two collaborated on many mathematical problems, although the Riemann Hypothesis continued to defy even their joint efforts.
A common anecdote about Ramanujan during this time relates how Hardy
arrived at Ramanujan's house in a cab numbered 1729, a number he claimed
to be totally uninteresting. Ramanujan is said to have stated on the
spot that, on the contrary, it was actually a very interesting number
mathematically, being the smallest number representable in two different
ways as a sum of two cubes. Such numbers are now sometimes referred to
as "taxicab numbers".
It is estimated that Ramanujan conjectured or proved over 3,000 theorems, identities and equations, including properties of highly composite numbers, the partition function and its asymptotics and mock theta functions. He also carried out major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Among his other achievements, Ramanujan identified several efficient and rapidly converging infinite series for the calculation of the value of π, some of which could compute 8 additional decimal places of π with each term in the series. These series (and variations on them) have become the basis for the fastest algorithms used by modern computers to compute π to ever increasing levels of accuracy (currently to about 5 trillion decimal places).
Eventually, though, the frustrated Ramanujan spiralled into depression and illness, even attempting suicide at one time. After a period in a sanatorium and a brief return to his family in India, he died in 1920 at the tragically young age of 32. Some of his original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired vast amounts of further research and have have found applications in fields as diverse as crystallography and string theory.
Hardy lived on for some 27 years after Ramanujan’s death, to the ripe old age of 70. When asked in an interview what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, and even called their collaboration "the one romantic incident in my life". However, Hardy too became depressed later in life and attempted suicide by an overdose at one point. Some have blamed the Riemann Hypothesis for Ramanujan and Hardy's instabilities, giving it something of the reputation of a curse
RUSSELL AND WHITEHEAD
Bertrand Russell and Alfred North Whitehead were British mathematicians, logicians and philosophers, who were in the vanguard
of the British revolt against Continental idealism in the early 20th
Century and, between them, they made important contributions in the
fields of mathematical logic and set theory.
Whitehead was the elder of the two and came from a more pure mathematics background. He became Russell’s tutor at Trinity College, Cambridge in the 1890s, and then collaborated with his more celebrated ex-student in the first decade of the 20th Century on their monumental work, the “Principia Mathematica”. After the First World War, though, much of which Russell spent in prison due to his pacifist activities, the collaboration petered out, and Whitehead’s academic career remained ever after in the shadow of that of the more flamboyant Russell. He emigrated to the United States in the 1920s, and spent the rest of his life there.
Russell was born into a wealthy family of the British aristocracy, although his parents were extremely liberal and radical for the times. His parents died when Russell was quite young and he was largely brought up by his staunchly Victorian (although quite progressive) grandmother. His adolescence was very lonely and he suffered from bouts of depression, later claiming that it was only his love of mathematics that kept him from suicide. He studied mathematics and philosophy at Cambridge University under G.E. Moore and A.N. Whitehead, where he developed into an innovative philosopher, a prolific writer on many subjects, a committed atheist and an inspired mathematician and logician. Today, he is considered one of the founders of analytic philosophy, but he wrote on almost every major area of philosophy, particularly metaphysics, ethics, epistemology, the philosophy of mathematics and the philosophy of language.
Russell was a committed and high-profile political activist throughout his long life. He was a prominent anti-war activist during both the First and Second World Wars, championed free trade
and anti-imperialism, and later became a strident campaiger for nuclear
disarmament and socialism, and against Adolf Hitler, Soviet
totalitarianism and the USA’s involvement in the Vietnam War.
Russell's mathematics was greatly influenced by the set theory and logicism Gottlob Frege had developed in the wake of Cantor's
groundbreaking early work on sets. In his 1903 "The Principles of
Mathematics", though, he identified what has come to be known as
Russell's Paradox (a set containing sets that are not members of
themselves), which showed that Frege's naive set theory could in fact
lead to contradictions. The paradox is sometimes illustrated by this
simplistic example: "If a barber shaves all and only those men in the
village who do not shave themselves, does he shave himself?"
The paradox seemed to imply that the very foundations of the whole of mathematics could no longer be trusted, and that, even in mathematics, the truth could never be known absolutely (Gödel's and Turing's later work would only make this worse). Russell's criticism was enough to rock Frege’s confidence in the entire edifice of logicism, and he was gracious enough to admit this openly in a hastily written appendix to Volume II of his "Basic Laws of Arithmetic".
But Russell's magnum opus was the monolithic “Principia Mathematica”, published in three volumes in 1910, 1912 and 1913. The first volume was co-written by Whitehead, although the later two were almost all Russell’s work. The aspiration of this ambitious work was nothing less than an attempt to derive all of mathematics from purely logical axioms, while avoiding the kinds of paradoxes and contradictions found in Frege’s earlier work on set theory. Russell achieved this by employing a theory or system of "types”, whereby each mathematical entity is assigned to a type within a hierarchy of types, so that objects of a given type are built exclusively from objects of preceding types lower in the hierarchy, thus preventing loops. Each set of elements, then, is of a different type than each of its elements, so that one can not speak of the "set of all sets" and similar constructs, which lead to paradoxes.
However, the “Principia" required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the “axiom of infinity” (which guarantees the existence of at least one infinite set, namely the set of all natural numbers), the “axiom of choice” (which ensures that, given any collection of “bins”, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins, and that there is no "rule" for which object to pick from each) and Russell’s own “axiom of reducibility” (which states that any propositional truth function can be expressed by a formally equivalent predicative truth function).
During the ten years or so that Russell and Whitehead spent on the "Principia", draft after draft was begun and abandoned as Russell constantly re-thought his basic premises. Russell and his wife Alys even moved in with the Whiteheads in order to expedite the work, although his own marriage suffered as Russell became infatuated with Whitehead's young wife, Evelyn. Eventually, Whitehead insisted on publication of the work, even if it was not (and might never be) complete, although they were forced to publish it at their own expense as no commercial publishers would touch it.
Some idea of the scope and comprehensiveness of the “Principia” can
be gleaned from the fact that it takes over 360 pages to prove
definitively that 1 + 1 = 2. Today, it is widely considered to be one of
the most important and seminal works in logic since Aristotle's
"Organon". It seemed remarkably successful and resilient in its
ambitious aims, and soon gained world fame for Russell and Whitehead.
Indeed, it was only Gödel's 1931 incompleteness theorem that finally
showed that the “Principia” could not be both consistent and complete.
Russell was awarded the Order of Merit in 1949 and the Nobel Prize in Literature in the following year. His fame continued to grow, even outside of academic circles, and he became something of a household name in later life, although largely as a result of his philosophical contributions and his political and social activism, which he continued until the end of his long life. He died of influenza in his beloved Wales at the grand old age of 97
HILBERT
David Hilbert was a great leader and spokesperson for the discipline
of mathematics in the early 20th Century. But he was an extremely
important and respected mathematician in his own right.
Like so many great German mathematicians before him, Hilbert was another product of the University of Göttingen, at that time the mathematical centre of the world, and he spent most of his working life there. His formative years, though, were spent at the University of Königsberg, where he developed an intense and fruitful scientific exchange with fellow mathematicians Hermann Minkowski and Adolf Hurwitz.
Sociable, democratic and well-loved both as a student and as a teacher, and often seen as bucking the trend of the formal and elitist system of German mathematics, Hilbert’s mathematical genius nevertheless spoke for itself. He has many mathematical terms named after him, including Hilbert space (an infinite dimensional Euclidean space), Hilbert curves, the Hilbert classification and the Hilbert inequality, as well as several theorems, and he gradually established himself as the most famous mathematician of his time.
His pithy enumeration of the 23 most important open mathematical questions at the 1900 Paris conference of the International Congress of Mathematicians at the Sorbonne set the stage for almost the whole of 20th Century mathematics. The details of some of these individual problems are highly technical; some are very precise, while some are quite vague and subject to interpretation; several problems have now already been solved, or at least partially solved, while some may be forever unresolvable as stated; some relate to rather abstruse backwaters of mathematical thought, while some deal
with more mainstream and well-known issues such as the Riemann
hypothesis, the continuum hypothesis, group theory, theories of
quadratic forms, real algebraic curves, etc.
As a young man, Hilbert began by pulling together all of the may
strands of number theory and abstract algebra, before changing field
completely to pursue studies in integral equations, where he
revolutionized the then current practices. In the early 1890s, he
developed continuous fractal space-filling curves in multiple
dimensions, building on earlier work by Guiseppe Peano. As early as
1899, he proposed a whole new formal set of geometrical axioms, known as
Hilbert's axioms, to substitute the traditional axioms of Euclid.
But perhaps his greatest legacy is his work on equations, often referred to as his finiteness theorem. He showed that although there were an infinite number of possible equations, it was nevertheless possible to split them up into a finite number of types of equations which could then be used, almost like a set of building blocks, to produce all the other equations.
Interestingly, though, Hilbert could not actually construct this finite set of equations, just prove that it must exist (sometimes referred to as an existence proof, rather than constructive proof). At the time, some critics passed this off as mere theology or smoke-and-mirrors, but it effectively marked the beginnings of a whole new style of abstract mathematics.
This use of an existence proof rather than constructive proof was
also implicit in his development, during the first decade of the 20th
Century, of the mathematical concept of what came to be known as Hilbert
space. Hilbert space is a generalization of the notion of Euclidean
space which extends the methods of vector algebra and calculus to spaces
with any finite (or even infinite) number of dimensions. Hilbert space
provided the basis for important contributions to the mathematics of
physics over the following decades, and may still offer one of the best mathematical formulations of quantum mechanics.
Hilbert was unfailingly optimistic about the future of mathematics, never doubting that his 23 problems would soon be solved. In fact, he went so far as to claim that there are absolutely no unsolvable problems - a famous quote of his (dating from 1930, and also engraved on his tombstone) proclaimed, “We must know! We will know!” - and he was convinced that the whole of mathematics could, and ultimately would, be put on unshakable logical foundations. Another of his rallying cries was “in mathematics there is no ignorabimus”, a reference to the traditional position on the limits of scientific knowledge.
Unlike Russell, Hilbert’s formalism was premised on the idea that the ultimate base of mathematics lies, not in logic itself, but in a simpler system of pre-logical symbols which can be collected together in strings or axioms and manipulated according to a set of “rules of inference”. His ambitious program to find a complete and consistent set of axioms for all of mathematics (which became known as Hilbert’s Program), received a severe set-back, however, with the incompleteness theorems of Kurt Gödel in the early 1930s. Nevertheless, Hilbert's work had started logic on a course of clarification, and the need to understand Gödel's work then led to the development of recursion theory and mathematical logic as an autonomous discipline in the 1930s, and later provided the basis for theoretical computer science.
For a time, Hilbert bravely spoke out against the Nazi repression of his Jewish mathematician friends in Germany and Austria in the mid 1930s. But, after mass evictions, several suicides, many deaths in concentration camps, and even direct assassinations, he too eventually lapsed into silence, and could only watch as one of the greatest mathematical centres of all time was systematically destroyed. By the time of his death in 1943, little remained of the great mathematics community at Göttingen, and Hilbert was buried in relative obscurity, his funeral attended by fewer than dozen people and hardly reported in the press.
GÖDEL
Kurt Gödel grew up a rather strange, sickly child in Vienna. From an
early age his parents took to referring to him as “Herr Varum”, Mr Why,
for his insatiable curiosity. At the University of Vienna, Gödel first
studied number theory, but soon turned his attention to mathematical
logic, which was to consume him for most of the rest of his life. As a
young man, he was, like Hilbert,
optimistic and convinced that mathematics could be made whole again,
and would recover from the uncertainties introduced by the work of Cantor and Riemann.
Between the wars, Gödel joined in the cafe discussions of a group of intense intellectuals and philosophers known as the Vienna Circle, which included logical positivists such as Moritz Schlick, Hans Hahn and Rudolf Carnap, who rejected metaphysics as meaningless and sought to codify all knowledge in a single standard language of science.
Although Gödel did not necessarily share the positivistic philosophical outlook of the Vienna Circle, it was in this enviroment that Gödel pursued his dream of solving the second, and perhaps most overarching, of Hilbert’s 23 problems, which sought to find a logical foundation for all of mathematics. The ideas he came up with would revolutionize mathematics, as he effectively proved, mathematically and philosophically, that Hilbert’s (and his own) optimism was unfounded and that such a foundation was just not possible.
His first achievement, which actually served to advance Hilbert's Program, was his completeness theorem, which showed that all valid statements in Freges's "first order logic" can be proved from a set of simple axioms. However, he then turned his attention to "second order logic", i.e a logic powerful enough to support arithmetic and more complex mathematical theories (essentially, one able to accept sets as values of variables).
Gödel’s incompleteness theorem (technically "incompleteness theorems", plural, as there were actually two separate theorems, although they are usually spoken of together) of 1931 showed that, within any logical system for mathematics (or at least in any system that is powerful and complex enough to be able to describe the arithmetic of the natural numbers, and therefore to be interesting to most mathematicians), there will be some statements about numbers which are true but which can NEVER be proved. This was enough to prompt John von Neumann to comment that "it's all over".
His approach began with the plain language assertion such as “this
statement cannot be proved”, a version of the ancient “liar paradox”,
and a statement which itself must be either true or false. If the
statement is false, then that means that the statement can be proved,
suggesting that it is actually true, thus generating a contradiction.
For this to have implications in mathematics, though, Gödel needed to
convert the statement into a "formal language" (i.e. a pure statement of
arithmetic). He did this using a clever code based on prime numbers,
where strings of primes play the roles of natural numbers, operators,
grammatical rules and all the other requirements of a formal language.
The resulting mathematical statement therefore appears, like its natural
language equivalent, to be true but unprovable, and must therefore
remain undecided.
The incompleteness theorem - surely a mathematician’s worst nightmare - led to something of a crisis in the mathematical community, raising the spectre of a problem which may turn out to be true but is still unprovable, something which had not been even considered in the whole two millennia plus history of mathematics. Gödel effectively put paid, at a stroke, to the ambitions of mathematicians like Bertrand Russell and David Hilbert who sought to find a complete and consistent set of axioms for all of mathematics. His work PROVED that any system of logic or numbers that mathematicians ever come up with will always rest on at least a few unprovable assumptions. His conclusions also imply that not all mathematical questions are even computable, and that it is impossible, even in principle, to create a machine or computer that will be able to do all that a human mind can do.
Unfortunately, the theorems also led to a personal crisis for Gödel.
In the mid 1930s, he suffered a series of mental breakdowns and spent
some significant time in a sanatorium. Nevertheless, he threw himself
into the same problem that had destroyed the mental well-being of Georg Cantor
during the previous century, the continuum hypothesis. In fact, he made
an important step in the resolution of that notoriously difficult
problem (by proving that the the axiom of choice is independence from
finite type theory), without which Paul Cohen would probably never have been able to come to his definitive solution. Like Cantor and others after him, though, Gödel too suffered a gradual deterioration in his mental and physical health.
He was only kept afloat at all by the love of his life, Adele Numbursky. Together, they witnessed the virtual destruction of the German and Austrian mathematics community by the Nazi regime. Eventually, along with many other eminent European mathematicians and scholars, Gödel fled the Nazis to the safety of Princeton in the USA, where he became a close friend of fellow exile Albert Einstein, contributing some demonstrations of paradoxical solutions to Einstein's field equations in general relativity (including his celebrated Gödel metric of 1949).
But, even in the USA, he was not able to escape his demons, and was dogged by depression and paranoia, suffering several more nervous breakdowns. Eventually, he would only eat food that had been tested by his wife Adele, and, when Adele herself was hospitalized in 1977, Gödel simply refused to eat and starved himself to death.
Gödel’s legacy is ambivalent. Although he is recognized as one of the great logicians of all time, many were just not prepared to accept the almost nihilistic consequences of his conclusions, and his explosion of the traditional formalist view of mathematics. Worse news was still to come, though, as the mathematical community (including, as we will see, Alan Turing) struggled to come to grips with Gödel’s findings
TURING
The British mathematician Alan Turing is perhaps most famous for his
war-time work at the British code-breaking centre at Bletchley Park
where his work led to the breaking of the German enigma code (according
to some, shortening the Second World War at a stroke, and potentially
saving thousands of lives). But he was also responsible for making Gödel’s
already devastating incompleteness theorem even more bleak and
discouraging, and it is mainly on this - and the development of computer
science that his work gave rise to - that Turing’s mathematical legacy
rests.
Despite attending an expensive private school which strongly emphasized the classics rather than the sciences, Turing showed early signs of the genius which was to become more prominent later, solving advanced problems as a teenager without having even studied elementary calculus, and immersing himself in the complex mathematics of Albert Einstein's work. He became a confirmed atheist after the death of his close friend and fellow Cambridge student Christopher Morcom, and throughout his life he was an accomplished and committed long-distance runner.
In the years following the publication of Gödel’s incompleteness theorem, Turing desperately wanted to clarify and simplify Gödel’s rather abstract and abstruse theorem, and to make it more concrete. But his solution - which was published in 1936 and which, he later claimed, had come to him in a vision - effectively involved the invention of something that has come to shape the entire modern world, the computer.
During the 1930s, Turing recast incompleteness in terms of computers
(or, more specifically, a theoretical device that manipulates symbols,
known as a Turing machine), replacing Gödel's
universal arithmetic-based formal language with this formal and simple
device. He first proved that such a machine would be capable of
performing any conceivable mathematical computation if it were
representable as an algorithm. He then went on to show that, even for
such a logical machine, essentially driven by arithmetic, there would
always be some problems they would never be able to solve, and that a
machine fed such a problem would never stop trying to solve it, but
would never succeed (known as the “halting problem”).
In the process, he also proved that there was no way of telling beforehand which problems were the unprovable ones, thus providing a negative proof to the so-called Entscheidungsproblem or “decision problem“, posed by David Hilbert in 1928. This was a further slap in the face for a mathematics community still reeling from Gödel’s crushing incompleteness theorem.
After the war, Turing continued the work he had begun, and worked on the development of early computers such as ACE (Automatic Computing Engine) and the Manchester Mark 1. Although the computer he developed was a very basic and limited machine by modern standards, Turing clearly saw its potential, and dreamed that one day computers would be more than machines, capable of learning, thinking and communicating. He was the first to develop ideas for a chess-playing computer program, and saw mastery in the game as one of the goals that designers of intelligent machines should strive for.
Indeed, he was the first to address the problem of artificial
intelligence, and proposed an experiment now known as the Turing Test in
an attempt to define a standard for a machine to be called
"intelligent". By this test, a computer could be said to "think" if it
could fool a human interrogator into thinking that the conversation was
with a human. This showed remarkable foresight at a time long before the
Internet, when the only available computers were the size of a room and
less powerful than a modern pocket calculator.
Turing’s personal philosophy was to be free from hypocrisy, compromise and deceit. He was, for example, a homosexual at a time when it was both illegal and even dangerous, yet he never hid it nor made it an issue. Unlike Gödel (who strongly believed in the power of intuition, and who was convinced that the human mind was capable of going beyond the limitations of the systems he described), Turing clearly felt a certain affinity with computers and, to some extent, he saw them as embodying this admirable absence of lies or hypocrisy.
After the war, he was kept under surveillance as a potential security risk by the authorities and eventually, in 1952, he was arrested, charged and found guilty of engaging in a homosexual act. As a result, he was chemically castrated by an injection of the female hormone estrogen, which caused him to grow breasts and also affected his mind. In 1954, Turing was found dead, having committed suicide with cyanide
WEIL
André Weil was a very influential French mathematician around the
middle of the 20th Century. Born into a properous Jewish family in
Paris, he was brother to the well-known philosopher and writer Simone
Weil, and both were child prodigies. He was passionately addicted to
mathematics by the age of ten, but he also loved to travel and study
languages (by the age of sixteen he had read the "Bhagavad Gita" in the
original Sanskrit).
He studied (and later taught) in Paris, Rome, Göttingen and elsewhere, as well as at the Aligarh Muslim University in Uttar Pradesh, India, were he further explored what would become a life-long interest in Hinduism and Sanskrit literature.
Even as a young man, Weil made substantial contributions in many areas of mathematics, and was particularly animated by the idea of discovering profound connections between algebraic geometry and number theory. His fascination with Diophantine equations led to his first substantial piece of mathematical research on the theory of algebraic curves. During the 1930s, he introduced the adele ring, a topological ring in algebraic number theory and topological algebra, which is built on the field of rational numbers.
It was also at this time that he became a founding member, and the de
facto early leader, of the so-called Bourbaki group of French
mathematicians. This influential group published many textbooks on
advanced 20th Century mathematics under the assumed name of Nicolas
Bourbaki, in an attempt to give a unified description of all mathematics
founded on set theory. Bourbaki has the distinction of having been
refused membership of the American Mathematical Society for being
non-existent (although he was a member of the Mathematical Society of
France!)
When the Second World War broke out, Weil, a committed conscientious objector, fled to Finland, where he was mistakenly arrested as a possible spy. Having made his way back to France, he was again arrested and imprisoned as for refusing to report for military service. In his trial, he cited the Bhagavad Gita to justify his stand, arguing that his true dharma was the pursuit of mathematics, not assisting in the war effort, however just the cause. Given the choice of five more years in prison or joining a French combat unit, though, he chose the latter, an especially lucky decision given that the prison was blown up shortly afterwards.
But it was in 1940, in a prison near Rouen, that Weil did the work that really made his reputation (although his full proofs had to wait until 1948, and even more rigorous proofs were supplied by Pierre Deligne in 1973). Building on the prescient work of his countryman Évariste Galois in the previous century, Weil picked up the idea of using geometry to analyze equations, and developed algebraic geometry, a whole new language for understanding solutions to equations.
The Weil conjectures on local zeta-functions effectively proved the
Riemann hypothesis for curves over finite fields, by counting the number
of points on algebraic varieties over finite fields. In the process, he
introduced for the first time the notion of an abstract algebraic
variety and thereby laid the foundations for abstract algebraic geometry
and the modern theory of abelian varieties, as well as the theory of
modular forms, automorphic functions and automorphic representations.
His work on algebraic curves has influenced a wide variety of areas,
including some outside of mathematics, such as elementary particle
physics and string theory.
In 1941, Weil and his wife took the opportunity to sail for the United States, where they spent the rest of the War and the rest of their lives. In the late 1950s, Weil formulated another important conjecture, this time on Tamagawa numbers, which remained resistant to proof until 1989. He was instrumental in the formulation of the so-called Shimura-Taniyama-Weil conjecture on elliptic curves which was used by Andrew Wiles as a link in the proof of Fermat’s Last Theorem. He also developed the Weil representation, an infinite-dimensional linear representation of theta functions which gave a contemporary framework for understanding the classical theory of quadratic forms.
Over his lifetime, Weil received many honorary memberships, including the London Mathematical Society, the Royal Society of London, the French Academy of Sciences and the American National Academy of Sciences. He remained active as professor emeritus at the Institute for Advanced Studies at Princeton until a few years before his death
COHEN
Paul Cohen was one of a new generation of American mathematician
inspired by the influx of European exiles over the War years. He himself
was a second generation Jewish immigrant, but he was dauntingly
intelligent and extremely ambitious. By sheer intelligence and force of
will, he went on to garner for himself fame, riches and the top
mathematical prizes.
He was educated at New York, Brooklyn and the University of Chicago, before working his way up to a professorship at Stanford University. He went on to win the prestigious Fields Medal in mathematics, as well as the National Medal of Science and the Bôcher Memorial Prize in mathematical analysis. His mathematical interests were very broad, ranging from mathematical analysis and differential equations to mathematical logic and number theory.
In the early 1960s, he earnestly applied himself to the first of Hilbert’s 23 list of open problems, Cantor’s continuum hypothesis, whether or not there exists a set of numbers of numbers bigger than the set of all natural (or whole) numbers but smaller than the set of real (or decimal) numbers. Cantor was convinced that the answer was “no” but was not able to prove it satisfactorily, and neither was anyone else who had applied themselves to the problem since.
Some progress had been made since Cantor.
The Zermelo-Fraenkel set theory, as modified by the Axiom of Choice
(commonly abbreviated together as ZFC), developed between about 1908 and
1922, had become accepted as the standard form of axiomatic set theory
and the most common foundation of mathematics.
Kurt Gödel had demonstrated in 1940 that the continuum hypothesis is consistent with ZFC (more specifically, that the continuum hypothesis cannot be disproved from the standard Zermelo-Fraenkel set theory, even if the axiom of choice is adopted. Cohen’s task, then, was to show that the continuum hypothesis was independent of ZFC (or not), and specifically to prove the independence of the axiom of choice.
Cohen’s extraordinary and daring conclusion, arrived at using a new technique he developed himself called "forcing", was that both answers could be true, i.e. that the continuum hypothesis and the axiom of choice were completely independent from ZFC set theory. Thus, there could be two different, internally consistent mathematics, one where the continuum hypothesis was true, and there was no such set of numbers, one where and the hypothesis was false and a set of numbers did exist. The proof seemed to be correct, but Cohen’s methods (particularly his new technique of “forcing”) were so new that no-one was really quite sure until Gödel finally gave his stamp of approval in 1963.
His findings were as revolutionary as Gödel’s own. Since that time, mathematicians have built up two different mathematical worlds, one in which the continuum hypothesis applies and one in which it does not, and modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.
Cohen’s paradigm-changing proof brought him fame, riches and mathematical prizes galore, and he became a top professor at Stanford and Princeton. Flushed with success, he decided to tackle the Holy Grail of modern mathematics, Hilbert’s eighth problem, the Riemann hypothesis. However, he ended up spending the last 40 years of his life, until his death in 2007, on the problem, still with no resolution (although his approach has given new hope to others, including his brilliant student, Peter Sarnak).
ROBINSON AND MATIYASEVICH
In a field almost completely dominated by men, Julia Robinson was one
of the very few women to have made a serious impact on mathematics -
others who merit mention are Sophie Germain and Sofia Kovaleskaya in the
19th Century, and Alicia Stout and Emmy Noether in the 20th - and she
became the first women to be elected as president of the American
Mathematical Society.
Brought up in the deserts of Arizona, Robinson was a shy and sickly child but showed an innate love for, and facility with, numbers from an early age. She had to overcome many obstacles and to fight to be allowed to continue studying mathematics, but she persevered, obtained her PhD at Berkeley and married a mathematician, her Berkeley professor, Raphael Robinson.
She spent most of her career pursuing computability and “decision problems”, questions in formal systems with “yes” or “no” answers, depending on the values of some input parameters. Her particular passion was Hilbert’s tenth problem, and she applied herself to it obsessively. The problem was to ascertain whether there was any way of telling whether or not any particular Diophantine equation (a polynomial equation whose variables can only be integers) had whole number solutions. The growing belief was that no such universal method was possible, but it seemed very difficult to actually prove that it would NEVER be possible to come up with such a method.
Throughout the 1950s and 1960s, Robinson, along with her colleagues Martin Davis and Hilary Putnam, doggedly pursued the problem, and eventually developed what became known as the Robinson hypothesis, which suggested that, in order to show that no such method existed, all that was needed was to construct one equation whose solution was a very specific set of numbers, one which grew exponentially.
The problem had obsessed Robinson for over twenty years and she confessed to a desperate desire to see its solution before she died, whoever might achieve it. In order to progress further, though, she needed input from the young Russian mathematician, Yuri Matiyasevich.
Born and educated in Leningrad (St. Petersburg), Matiyasevich had already distinguished himself as a mathematical prodigy, and won numerous prizes in mathematics. He turned to Hilbert’s tenth problem as the subject of his doctoral thesis at Leningrad State University, and began to correspond with Robinson about her progress, and to search for a way forward.
After pursuing the problem during the late 1960s, Matiyasevich finally discovered the final missing piece of the jigsaw in 1970, when he was just 22 years old. He saw how he could capture the famous Fibonacci sequence of numbers using the equations that were at the heart of Hilbert’s tenth problem, and so, building on Robinson’s earlier work, it was finally proved that it is in fact impossible to devise a process by which it can be determined in a finite number of operations whether Diophantine equations are solvable in rational integers.
In a poignant example of the internationalism of mathematics at the
height of the Cold War, Matiyasevich freely acknowledged his debt to
Robinson’s work, and the two went on to work together on other problems
until Robinson’s death in 1984.
Among, his other achievements, Matiyasevich and his colleague Boris Stechkin also developed an interesting “visual sieve” for prime numbers, which effectively “crosses out” all the composite numbers, leaving only the primes. He has a theorem on recursively enumerable sets named after him, as well as a polynomial related to the colourings of triangulation of spheres. He is head of the Laboratory of Mathematical Logic at the St. Petersburg Department of the Steklov Institute of Mathematics of Russian Academy of Sciences, and is a member of several mathematical societies and boards.
It also saw mathematics become a major profession, involving thousands of new Ph.D.s each year and jobs in both teaching and industry, and the development of hundreds of specialized areas and fields of study, such as group theory, knot theory, sheaf theory, topology, graph theory, functional analysis, singularity theory, catastrophe theory, chaos theory, model theory, category theory, game theory, complexity theory and many more.
The eccentric British mathematician G.H. Hardy and his young Indian protégé Srinivasa Ramanujan, were just two of the great mathematicians of the early 20th Century who applied themselves in earnest to solving problems of the previous century, such as the Riemann hypothesis. Although they came close, they too were defeated by that most intractable of problems, but Hardy is credited with reforming British mathematics, which had sunk to something of a low ebb at that time, and Ramanujan proved himself to be one of the most brilliant (if somewhat undisciplined and unstable) minds of the century.
Others followed techniques dating back millennia but taken to a 20th Century level of complexity. In 1904, Johann Gustav Hermes completed his construction of a regular polygon with 65,537 sides (216 + 1), using just a compass and straight edge as Euclid would have done, a feat that took him over ten years.
The early 20th Century also saw the beginnings of the rise of the field of mathematical logic, building on the earlier advances of Gottlob Frege, which came to fruition in the hands of Giuseppe Peano, L.E.J. Brouwer, David Hilbert and, particularly, Bertrand Russell and A.N. Whitehead, whose monumental joint work the “Principia Mathematica” was so influential in mathematical and philosophical logicism.
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 Part of the transcript of Hilbert’s 1900 Paris lecture, in which he set out his 23 problems |
Hilbert was himself a brilliant mathematician, responsible for several theorems and some entirely new mathematical concepts, as well as overseeing the development of what amounted to a whole new style of abstract mathematical thinking. Hilbert's approach signalled the shift to the modern axiomatic method, where axioms are not taken to be self-evident truths. He was unfailingly optimistic about the future of mathematics, famously declaring in a 1930 radio interview “We must know. We will know!”, and was a well-loved leader of the mathematical community during the first part of the century.
However, the Austrian Kurt Gödel was soon to put some very severe constraints on what could and could not be solved, and turned mathematics on its head with his famous incompleteness theorem, which proved the unthinkable - that there could be solutions to mathematical problems which were true but which could never be proved.
Alan Turing, perhaps best known for his war-time work in breaking the German enigma code, spent his pre-war years trying to clarify and simplify Gödel’s rather abstract proof. His methods led to some conclusions that were perhaps even more devastating than Gödel’s, including the idea that there was no way of telling beforehand which problems were provable and which unprovable. But, as a spin-off, his work also led to the development of computers and the first considerations of such concepts as artificial intelligence.
With the gradual and wilful destruction of the mathematics community of Germany and Austria by the anti-Jewish Nazi regime in the 1930 and 1940s, the focus of world mathematics moved to America, particularly to the Institute for Advanced Study in Princeton, which attempted to reproduce the collegiate atmosphere of the old European universities in rural New Jersey. Many of the brightest European mathematicians, including Hermann Weyl, John von Neumann, Kurt Gödel and Albert Einstein, fled the Nazis to this safe haven.
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Von Neumann’s computer architecture design |
André Weil was another refugee from the war in Europe, after narrowly avoiding death on a couple of occasions. His theorems, which allowed connections to be made between number theory, algebra, geometry and topology, are considered among the greatest achievements of modern mathematics. He was also responsible for setting up a group of French mathematicians who, under the secret nom-de-plume of Nicolas Bourbaki, wrote many influential books on the mathematics of the 20th Century.
Perhaps the greatest heir to Weil’s legacy was Alexander Grothendieck, a charismatic and beloved figure in 20th Century French mathematics. Grothendieck was a structuralist, interested in the hidden structures beneath all mathematics, and in the 1950s he created a powerful new language which enabled mathematical structures to be seen in a new way, thus allowing new solutions in number theory, geometry, even in fundamental physics. His “theory of schemes” allowed certain of Weil's number theory conjectures to be solved, and his “theory of topoi” is highly relevant to mathematical logic. In addition, he gave an algebraic proof of the Riemann-Roch theorem, and provided an algebraic definition of the fundamental group of a curve. Although, after the 1960s, Grothendieck all but abandoned mathematics for radical politics, his achievements in algebraic geometry have fundamentally transformed the mathematical landscape, perhaps no less than those of Cantor, Gödel and Hilbert, and he is considered by some to be one of the dominant figures of the whole of 20th Century mathematics.
Paul Erdös was another inspired but distinctly non-establishment figure of 20th Century mathematics. The immensely prolific and famously eccentric Hungarian mathematician worked with hundreds of different collaborators on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory. As a humorous tribute, an "Erdös number" is given to mathematicians according to their collaborative proximity to him. He was also known for offering small prizes for solutions to various unresolved problems (such as the Erdös conjecture on arithmetic progressions), some of which are still active after his death.
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The Mandelbrot set, the most famous example of a fractal |
The Mandelbrot set involves repeated iterations of complex quadratic polynomial equations of the form zn+1 = zn2 + c, (where z is a number in the complex plane of the form x + iy). The iterations produce a form of feedback based on recursion, in which smaller parts exhibit approximate reduced-size copies of the whole, and which are infinitely complex (so that, however much one zooms in and magifies a part, it exhibits just as much complexity).
Paul Cohen is an example of a second generation Jewish immigrant who followed the American dream to fame and success. His work rocked the mathematical world in the 1960s, when he proved that Cantor's continuum hypothesis about the possible sizes of infinite sets (one of Hilbert’s original 23 problems) could be both true AND not true, and that there were effectively two completely separate but valid mathematical worlds, one in which the continuum hypothesis was true and one where it was not. Since this result, all modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.
Another of Hilbert’s problems was finally resolved in 1970, when the young Russian Yuri Matiyasevich finally proved that Hilbert’s tenth problem was impossible, i.e. that there is no general method for determining when polynomial equations have a solution in whole numbers. In arriving at his proof, Matiyasevich built on decades of work by the American mathematician Julia Robinson, in a great show of internationalism at the height of the Cold War.
In additon to complex dynamics, another field that benefitted greatly from the advent of the electronic computer, and particulary from its ability to carry out a huge number of repeated iterations of simple mathematical formulas which would be impractical to do by hand, was chaos theory. Chaos theory tells us that some systems seem to exhibit random behaviour even though they are not random at all, and conversely some systems may have roughly predictable behaviour but are fundamentally unpredictable in any detail. The possible behaviours that a chaotic system may have can also be mapped graphically, and it was discovered that these mappings, known as "strange attractors", are fractal in nature (the more you zoom in, the more detail can be seen, although the overall pattern remains the same).
An early pioneer in modern chaos theory was Edward Lorenz, whose interest in chaos came about accidentally through his work on weather prediction. Lorenz's discovery came in 1961, when a computer model he had been running was actually saved using three-digit numbers rather than the six digits he had been working with, and this tiny rounding error produced dramatically different results. He discovered that small changes in initial conditions can produce large changes in the long-term outcome - a phenomenon he described by the term “butterfly effect” - and he demonstrated this with his Lorenz attractor, a fractal structure corresponding to the behaviour of the Lorenz oscillator (a 3-dimensional dynamical system that exhibits chaotic flow).
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Example of a four-colour map |
Also in the 1970s, origami became recognized as a serious mathematical method, in some cases more powerful than Euclidean geometry. In 1936, Margherita Piazzola Beloch had shown how a length of paper could be folded to give the cube root of its length, but it was not until 1980 that an origami method was used to solve the "doubling the cube" problem which had defeated ancient Greek geometers. An origami proof of the equally intractible "trisecting the angle" problem followed in 1986. The Japanese origami expert Kazuo Haga has at least three mathematical theorems to his name, and his unconventional folding techniques have demonstrated many unexpected geometrical results.
The British mathematician Andrew Wiles finally proved Fermat’s Last Theorem for ALL numbers in 1995, some 350 years after Fermat’s initial posing. It was an achievement Wiles had set his sights on early in life and pursued doggedly for many years. In reality, though, it was a joint effort of several steps involving many mathematicans over several years, including Goro Shimura, Yutaka Taniyama, Gerhard Frey, Jean-Pierre Serre and Ken Ribet, with Wiles providing the links and the final synthesis and, specifically, the final proof of the Taniyama-Shimura Conjecture for semi-stable elliptic curves. The proof itself is over 100 pages long.
The most recent of the great conjectures to be proved was the Poincaré Conjecture, which was solved in 2002 (over 100 years after Poincaré first posed it) by the eccentric and reclusive Russian mathematician Grigori Perelman. However, Perelman, who lives a frugal life with his mother in a suburb of St. Petersburg, turned down the $1 million prize, claiming that "if the proof is correct then no other recognition is needed". The conjecture, now a theorem, states that, if a loop in connected, finite boundaryless 3-dimensional space can be continuously tightened to a point, in the same way as a loop drawn on a 2-dimensional sphere can, then the space is a three-dimensional sphere. Perelman provided an elegant but extremely complex solution involving the ways in which 3-dimensional shapes can be “wrapped up” in even higher dimensions. Perelman has also made landmark contributions to Riemannian geometry and geometric topology.
John Nash, the American economist and mathematician whose battle against paranoid schizophrenia has recently been popularized by the Hollywood movie “A Beautiful Mind”, did some important work in game theory, differential geometry and partial differential equations which have provided insight into the forces that govern chance and events inside complex systems in daily life, such as in market
economics, computing, artificial intelligence, accounting and military theory.The Englishman John Horton Conway established the rules for the so-called "Game of Life" in 1970, an early example of a "cellular automaton" in which patterns of cells evolve and grow in a grid, which became extremely popular among computer scientists. He has made important contributions to many branches of pure mathematics, such as game theory, group theory, number theory and geometry, and has also come up with some wonderful-sounding concepts like surreal numbers, the grand antiprism and monstrous moonshine, as well as mathematical games such as Sprouts, Philosopher's Football and the Soma Cube.
Other mathematics-based recreational puzzles became even more popular among the general public, including Rubik's Cube (1974) and Sudoku (1980), both of which developed into full-blown crazes on a scale only previously seen with the 19th Century fads of Tangrams (1817) and the Fifteen puzzle (1879). In their turn, they generated attention from serious mathematicians interested in exploring the theoretical limits and underpinnings of the games.
Computers continue to aid in the identification of phenomena such as Mersenne primes numbers (a prime number that is one less than a power of two - see the section on 17th Century Mathematics). In 1952, an early computer known as SWAC identified 2257-1 as the 13th Mersenne prime number, the first new one to be found in 75 years, before going on to identify several more even larger.
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Approximations for π |
The P versus NP problem, introduced in 1971 by the American-Canadian Stephen Cook, is a major unsolved problem in computer science and the burgeoning field of complexity theory, and is another of the Clay Mathematics Institute's million dollar Millennium Prize problems. At its simplest, it asks whether every problem whose solution can be efficiently checked by a computer can also be efficiently solved by a computer (or put another way, whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure). The solution to this simple enough sounding problem, usually known as Cook's Theorem or the Cook-Levin Theorem, has eluded mathematicians and computer scientists for 40 years. A possible solution by Vinay Deolalikar in 2010, claiming to prove that P is not equal to NP (and thus such insolulable-but-easily-checked problems do exist), has attracted much attention but has not as yet been fully accepted by the computer science community.
HARDY AND RAMANUJAN
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G.H. Hardy (1877-1947) and Srinivasa Ramanujan (1887-1920) |
Hardy himself was a prodigy from a young age, and stories are told about how he would write numbers up to millions at just two years of age, and how he would amuse himself in church by factorizing the hymn numbers. He graduated with honours from Cambridge University, where he was to spend most of the rest of his academic career.
Hardy is sometimes credited with reforming British mathematics in the early 20th Century by bringing a Continental rigour to it, more characteristic of the French, Swiss and German mathematics he so much admired, rather than British mathematics. He introduced into Britain a new tradition of pure mathematics (as opposed to the traditional British forte of applied mathematics in the shadow of Newton), and he proudly declared that nothing he had ever done had any commercial or military usefulness (he was also an outspoken pacifist).
Just before the First World War, Hardy (who was given to flamboyant gestures) made mathematical headlines when he claimed to have proved the Riemann Hypothesis. In fact, he was able to prove that there were infinitely many zeroes on the critical line, but was not able to prove that there did not exist other zeroes that were NOT on the line (or even infinitely many off the line, given the nature of infinity).
Meanwhile, in 1913, Srinivasa Ramanujan, a 23-year old shipping clerk from Madras, India, wrote to Hardy (and other academics at Cambridge), claiming, among other things, to have devised a formula that calculated the number of primes up to a hundred million with generally no error. The self-taught and obsessive Ramanujan had managed to prove all of Riemann’s results and more with almost no knowledge of developments in the Western world and no formal tuition. He claimed that most of his ideas came to him in dreams.
Hardy was only one to recognize Ramanujan's genius, and brought him to Cambridge University, and was his friend and mentor for many years. The two collaborated on many mathematical problems, although the Riemann Hypothesis continued to defy even their joint efforts.
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Hardy-Ramanujan "taxicab numbers" |
It is estimated that Ramanujan conjectured or proved over 3,000 theorems, identities and equations, including properties of highly composite numbers, the partition function and its asymptotics and mock theta functions. He also carried out major investigations in the areas of gamma functions, modular forms, divergent series, hypergeometric series and prime number theory.
Among his other achievements, Ramanujan identified several efficient and rapidly converging infinite series for the calculation of the value of π, some of which could compute 8 additional decimal places of π with each term in the series. These series (and variations on them) have become the basis for the fastest algorithms used by modern computers to compute π to ever increasing levels of accuracy (currently to about 5 trillion decimal places).
Eventually, though, the frustrated Ramanujan spiralled into depression and illness, even attempting suicide at one time. After a period in a sanatorium and a brief return to his family in India, he died in 1920 at the tragically young age of 32. Some of his original and highly unconventional results, such as the Ramanujan prime and the Ramanujan theta function, have inspired vast amounts of further research and have have found applications in fields as diverse as crystallography and string theory.
Hardy lived on for some 27 years after Ramanujan’s death, to the ripe old age of 70. When asked in an interview what his greatest contribution to mathematics was, Hardy unhesitatingly replied that it was the discovery of Ramanujan, and even called their collaboration "the one romantic incident in my life". However, Hardy too became depressed later in life and attempted suicide by an overdose at one point. Some have blamed the Riemann Hypothesis for Ramanujan and Hardy's instabilities, giving it something of the reputation of a curse
RUSSELL AND WHITEHEAD
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Bertrand Russell (1872-1970) and A.N. Whitehead (1861-1947) |
of the British revolt against Continental idealism in the early 20th
Century and, between them, they made important contributions in the
fields of mathematical logic and set theory.Whitehead was the elder of the two and came from a more pure mathematics background. He became Russell’s tutor at Trinity College, Cambridge in the 1890s, and then collaborated with his more celebrated ex-student in the first decade of the 20th Century on their monumental work, the “Principia Mathematica”. After the First World War, though, much of which Russell spent in prison due to his pacifist activities, the collaboration petered out, and Whitehead’s academic career remained ever after in the shadow of that of the more flamboyant Russell. He emigrated to the United States in the 1920s, and spent the rest of his life there.
Russell was born into a wealthy family of the British aristocracy, although his parents were extremely liberal and radical for the times. His parents died when Russell was quite young and he was largely brought up by his staunchly Victorian (although quite progressive) grandmother. His adolescence was very lonely and he suffered from bouts of depression, later claiming that it was only his love of mathematics that kept him from suicide. He studied mathematics and philosophy at Cambridge University under G.E. Moore and A.N. Whitehead, where he developed into an innovative philosopher, a prolific writer on many subjects, a committed atheist and an inspired mathematician and logician. Today, he is considered one of the founders of analytic philosophy, but he wrote on almost every major area of philosophy, particularly metaphysics, ethics, epistemology, the philosophy of mathematics and the philosophy of language.
Russell was a committed and high-profile political activist throughout his long life. He was a prominent anti-war activist during both the First and Second World Wars, championed free trade
and anti-imperialism, and later became a strident campaiger for nuclear
disarmament and socialism, and against Adolf Hitler, Soviet
totalitarianism and the USA’s involvement in the Vietnam War.
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Russell’s Paradox |
The paradox seemed to imply that the very foundations of the whole of mathematics could no longer be trusted, and that, even in mathematics, the truth could never be known absolutely (Gödel's and Turing's later work would only make this worse). Russell's criticism was enough to rock Frege’s confidence in the entire edifice of logicism, and he was gracious enough to admit this openly in a hastily written appendix to Volume II of his "Basic Laws of Arithmetic".
But Russell's magnum opus was the monolithic “Principia Mathematica”, published in three volumes in 1910, 1912 and 1913. The first volume was co-written by Whitehead, although the later two were almost all Russell’s work. The aspiration of this ambitious work was nothing less than an attempt to derive all of mathematics from purely logical axioms, while avoiding the kinds of paradoxes and contradictions found in Frege’s earlier work on set theory. Russell achieved this by employing a theory or system of "types”, whereby each mathematical entity is assigned to a type within a hierarchy of types, so that objects of a given type are built exclusively from objects of preceding types lower in the hierarchy, thus preventing loops. Each set of elements, then, is of a different type than each of its elements, so that one can not speak of the "set of all sets" and similar constructs, which lead to paradoxes.
However, the “Principia" required, in addition to the basic axioms of type theory, three further axioms that seemed to not be true as mere matters of logic, namely the “axiom of infinity” (which guarantees the existence of at least one infinite set, namely the set of all natural numbers), the “axiom of choice” (which ensures that, given any collection of “bins”, each containing at least one object, it is possible to make a selection of exactly one object from each bin, even if there are infinitely many bins, and that there is no "rule" for which object to pick from each) and Russell’s own “axiom of reducibility” (which states that any propositional truth function can be expressed by a formally equivalent predicative truth function).
During the ten years or so that Russell and Whitehead spent on the "Principia", draft after draft was begun and abandoned as Russell constantly re-thought his basic premises. Russell and his wife Alys even moved in with the Whiteheads in order to expedite the work, although his own marriage suffered as Russell became infatuated with Whitehead's young wife, Evelyn. Eventually, Whitehead insisted on publication of the work, even if it was not (and might never be) complete, although they were forced to publish it at their own expense as no commercial publishers would touch it.
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A small part of the long proof that 1+1 =2 in the “Principia Mathematica” |
Russell was awarded the Order of Merit in 1949 and the Nobel Prize in Literature in the following year. His fame continued to grow, even outside of academic circles, and he became something of a household name in later life, although largely as a result of his philosophical contributions and his political and social activism, which he continued until the end of his long life. He died of influenza in his beloved Wales at the grand old age of 97
HILBERT
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David Hilbert (1862-1943) |
Like so many great German mathematicians before him, Hilbert was another product of the University of Göttingen, at that time the mathematical centre of the world, and he spent most of his working life there. His formative years, though, were spent at the University of Königsberg, where he developed an intense and fruitful scientific exchange with fellow mathematicians Hermann Minkowski and Adolf Hurwitz.
Sociable, democratic and well-loved both as a student and as a teacher, and often seen as bucking the trend of the formal and elitist system of German mathematics, Hilbert’s mathematical genius nevertheless spoke for itself. He has many mathematical terms named after him, including Hilbert space (an infinite dimensional Euclidean space), Hilbert curves, the Hilbert classification and the Hilbert inequality, as well as several theorems, and he gradually established himself as the most famous mathematician of his time.
His pithy enumeration of the 23 most important open mathematical questions at the 1900 Paris conference of the International Congress of Mathematicians at the Sorbonne set the stage for almost the whole of 20th Century mathematics. The details of some of these individual problems are highly technical; some are very precise, while some are quite vague and subject to interpretation; several problems have now already been solved, or at least partially solved, while some may be forever unresolvable as stated; some relate to rather abstruse backwaters of mathematical thought, while some deal
with more mainstream and well-known issues such as the Riemann
hypothesis, the continuum hypothesis, group theory, theories of
quadratic forms, real algebraic curves, etc.
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Hilbert’s algorithm for space-filling curves |
But perhaps his greatest legacy is his work on equations, often referred to as his finiteness theorem. He showed that although there were an infinite number of possible equations, it was nevertheless possible to split them up into a finite number of types of equations which could then be used, almost like a set of building blocks, to produce all the other equations.
Interestingly, though, Hilbert could not actually construct this finite set of equations, just prove that it must exist (sometimes referred to as an existence proof, rather than constructive proof). At the time, some critics passed this off as mere theology or smoke-and-mirrors, but it effectively marked the beginnings of a whole new style of abstract mathematics.
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Among other things, Hilbert space can be used to study the harmonics of vibrating strings |
one of the best mathematical formulations of quantum mechanics.Hilbert was unfailingly optimistic about the future of mathematics, never doubting that his 23 problems would soon be solved. In fact, he went so far as to claim that there are absolutely no unsolvable problems - a famous quote of his (dating from 1930, and also engraved on his tombstone) proclaimed, “We must know! We will know!” - and he was convinced that the whole of mathematics could, and ultimately would, be put on unshakable logical foundations. Another of his rallying cries was “in mathematics there is no ignorabimus”, a reference to the traditional position on the limits of scientific knowledge.
Unlike Russell, Hilbert’s formalism was premised on the idea that the ultimate base of mathematics lies, not in logic itself, but in a simpler system of pre-logical symbols which can be collected together in strings or axioms and manipulated according to a set of “rules of inference”. His ambitious program to find a complete and consistent set of axioms for all of mathematics (which became known as Hilbert’s Program), received a severe set-back, however, with the incompleteness theorems of Kurt Gödel in the early 1930s. Nevertheless, Hilbert's work had started logic on a course of clarification, and the need to understand Gödel's work then led to the development of recursion theory and mathematical logic as an autonomous discipline in the 1930s, and later provided the basis for theoretical computer science.
For a time, Hilbert bravely spoke out against the Nazi repression of his Jewish mathematician friends in Germany and Austria in the mid 1930s. But, after mass evictions, several suicides, many deaths in concentration camps, and even direct assassinations, he too eventually lapsed into silence, and could only watch as one of the greatest mathematical centres of all time was systematically destroyed. By the time of his death in 1943, little remained of the great mathematics community at Göttingen, and Hilbert was buried in relative obscurity, his funeral attended by fewer than dozen people and hardly reported in the press.
GÖDEL
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Kurt Gödel (1906-1978) |
Between the wars, Gödel joined in the cafe discussions of a group of intense intellectuals and philosophers known as the Vienna Circle, which included logical positivists such as Moritz Schlick, Hans Hahn and Rudolf Carnap, who rejected metaphysics as meaningless and sought to codify all knowledge in a single standard language of science.
Although Gödel did not necessarily share the positivistic philosophical outlook of the Vienna Circle, it was in this enviroment that Gödel pursued his dream of solving the second, and perhaps most overarching, of Hilbert’s 23 problems, which sought to find a logical foundation for all of mathematics. The ideas he came up with would revolutionize mathematics, as he effectively proved, mathematically and philosophically, that Hilbert’s (and his own) optimism was unfounded and that such a foundation was just not possible.
His first achievement, which actually served to advance Hilbert's Program, was his completeness theorem, which showed that all valid statements in Freges's "first order logic" can be proved from a set of simple axioms. However, he then turned his attention to "second order logic", i.e a logic powerful enough to support arithmetic and more complex mathematical theories (essentially, one able to accept sets as values of variables).
Gödel’s incompleteness theorem (technically "incompleteness theorems", plural, as there were actually two separate theorems, although they are usually spoken of together) of 1931 showed that, within any logical system for mathematics (or at least in any system that is powerful and complex enough to be able to describe the arithmetic of the natural numbers, and therefore to be interesting to most mathematicians), there will be some statements about numbers which are true but which can NEVER be proved. This was enough to prompt John von Neumann to comment that "it's all over".
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Gödel’s Incompleteness Theorem |
The incompleteness theorem - surely a mathematician’s worst nightmare - led to something of a crisis in the mathematical community, raising the spectre of a problem which may turn out to be true but is still unprovable, something which had not been even considered in the whole two millennia plus history of mathematics. Gödel effectively put paid, at a stroke, to the ambitions of mathematicians like Bertrand Russell and David Hilbert who sought to find a complete and consistent set of axioms for all of mathematics. His work PROVED that any system of logic or numbers that mathematicians ever come up with will always rest on at least a few unprovable assumptions. His conclusions also imply that not all mathematical questions are even computable, and that it is impossible, even in principle, to create a machine or computer that will be able to do all that a human mind can do.
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Representation of the Gödel Metric, an exact solution to Einstein's field equations |
He was only kept afloat at all by the love of his life, Adele Numbursky. Together, they witnessed the virtual destruction of the German and Austrian mathematics community by the Nazi regime. Eventually, along with many other eminent European mathematicians and scholars, Gödel fled the Nazis to the safety of Princeton in the USA, where he became a close friend of fellow exile Albert Einstein, contributing some demonstrations of paradoxical solutions to Einstein's field equations in general relativity (including his celebrated Gödel metric of 1949).
But, even in the USA, he was not able to escape his demons, and was dogged by depression and paranoia, suffering several more nervous breakdowns. Eventually, he would only eat food that had been tested by his wife Adele, and, when Adele herself was hospitalized in 1977, Gödel simply refused to eat and starved himself to death.
Gödel’s legacy is ambivalent. Although he is recognized as one of the great logicians of all time, many were just not prepared to accept the almost nihilistic consequences of his conclusions, and his explosion of the traditional formalist view of mathematics. Worse news was still to come, though, as the mathematical community (including, as we will see, Alan Turing) struggled to come to grips with Gödel’s findings
TURING
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Alan Turing (1912-1954) |
Despite attending an expensive private school which strongly emphasized the classics rather than the sciences, Turing showed early signs of the genius which was to become more prominent later, solving advanced problems as a teenager without having even studied elementary calculus, and immersing himself in the complex mathematics of Albert Einstein's work. He became a confirmed atheist after the death of his close friend and fellow Cambridge student Christopher Morcom, and throughout his life he was an accomplished and committed long-distance runner.
In the years following the publication of Gödel’s incompleteness theorem, Turing desperately wanted to clarify and simplify Gödel’s rather abstract and abstruse theorem, and to make it more concrete. But his solution - which was published in 1936 and which, he later claimed, had come to him in a vision - effectively involved the invention of something that has come to shape the entire modern world, the computer.
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Representation of a Turing Machine |
In the process, he also proved that there was no way of telling beforehand which problems were the unprovable ones, thus providing a negative proof to the so-called Entscheidungsproblem or “decision problem“, posed by David Hilbert in 1928. This was a further slap in the face for a mathematics community still reeling from Gödel’s crushing incompleteness theorem.
After the war, Turing continued the work he had begun, and worked on the development of early computers such as ACE (Automatic Computing Engine) and the Manchester Mark 1. Although the computer he developed was a very basic and limited machine by modern standards, Turing clearly saw its potential, and dreamed that one day computers would be more than machines, capable of learning, thinking and communicating. He was the first to develop ideas for a chess-playing computer program, and saw mastery in the game as one of the goals that designers of intelligent machines should strive for.
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Turing test |
Turing’s personal philosophy was to be free from hypocrisy, compromise and deceit. He was, for example, a homosexual at a time when it was both illegal and even dangerous, yet he never hid it nor made it an issue. Unlike Gödel (who strongly believed in the power of intuition, and who was convinced that the human mind was capable of going beyond the limitations of the systems he described), Turing clearly felt a certain affinity with computers and, to some extent, he saw them as embodying this admirable absence of lies or hypocrisy.
After the war, he was kept under surveillance as a potential security risk by the authorities and eventually, in 1952, he was arrested, charged and found guilty of engaging in a homosexual act. As a result, he was chemically castrated by an injection of the female hormone estrogen, which caused him to grow breasts and also affected his mind. In 1954, Turing was found dead, having committed suicide with cyanide
WEIL
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André Weil (1906-1998) |
He studied (and later taught) in Paris, Rome, Göttingen and elsewhere, as well as at the Aligarh Muslim University in Uttar Pradesh, India, were he further explored what would become a life-long interest in Hinduism and Sanskrit literature.
Even as a young man, Weil made substantial contributions in many areas of mathematics, and was particularly animated by the idea of discovering profound connections between algebraic geometry and number theory. His fascination with Diophantine equations led to his first substantial piece of mathematical research on the theory of algebraic curves. During the 1930s, he introduced the adele ring, a topological ring in algebraic number theory and topological algebra, which is built on the field of rational numbers.
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Weil was an early leader of the Bourbaki group who published many influential textbooks on modern mathematics |
When the Second World War broke out, Weil, a committed conscientious objector, fled to Finland, where he was mistakenly arrested as a possible spy. Having made his way back to France, he was again arrested and imprisoned as for refusing to report for military service. In his trial, he cited the Bhagavad Gita to justify his stand, arguing that his true dharma was the pursuit of mathematics, not assisting in the war effort, however just the cause. Given the choice of five more years in prison or joining a French combat unit, though, he chose the latter, an especially lucky decision given that the prison was blown up shortly afterwards.
But it was in 1940, in a prison near Rouen, that Weil did the work that really made his reputation (although his full proofs had to wait until 1948, and even more rigorous proofs were supplied by Pierre Deligne in 1973). Building on the prescient work of his countryman Évariste Galois in the previous century, Weil picked up the idea of using geometry to analyze equations, and developed algebraic geometry, a whole new language for understanding solutions to equations.
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An illustration of the "cycle évanescent" or "vanishing cycle" described in Deligne's proof of the Weil conjectures |
In 1941, Weil and his wife took the opportunity to sail for the United States, where they spent the rest of the War and the rest of their lives. In the late 1950s, Weil formulated another important conjecture, this time on Tamagawa numbers, which remained resistant to proof until 1989. He was instrumental in the formulation of the so-called Shimura-Taniyama-Weil conjecture on elliptic curves which was used by Andrew Wiles as a link in the proof of Fermat’s Last Theorem. He also developed the Weil representation, an infinite-dimensional linear representation of theta functions which gave a contemporary framework for understanding the classical theory of quadratic forms.
Over his lifetime, Weil received many honorary memberships, including the London Mathematical Society, the Royal Society of London, the French Academy of Sciences and the American National Academy of Sciences. He remained active as professor emeritus at the Institute for Advanced Studies at Princeton until a few years before his death
COHEN
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Paul Cohen (1934-2007) |
He was educated at New York, Brooklyn and the University of Chicago, before working his way up to a professorship at Stanford University. He went on to win the prestigious Fields Medal in mathematics, as well as the National Medal of Science and the Bôcher Memorial Prize in mathematical analysis. His mathematical interests were very broad, ranging from mathematical analysis and differential equations to mathematical logic and number theory.
In the early 1960s, he earnestly applied himself to the first of Hilbert’s 23 list of open problems, Cantor’s continuum hypothesis, whether or not there exists a set of numbers of numbers bigger than the set of all natural (or whole) numbers but smaller than the set of real (or decimal) numbers. Cantor was convinced that the answer was “no” but was not able to prove it satisfactorily, and neither was anyone else who had applied themselves to the problem since.
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One of several alternative formulations of the Zermelo-Fraenkel Axioms and Axiom of Choice |
Kurt Gödel had demonstrated in 1940 that the continuum hypothesis is consistent with ZFC (more specifically, that the continuum hypothesis cannot be disproved from the standard Zermelo-Fraenkel set theory, even if the axiom of choice is adopted. Cohen’s task, then, was to show that the continuum hypothesis was independent of ZFC (or not), and specifically to prove the independence of the axiom of choice.
Cohen’s extraordinary and daring conclusion, arrived at using a new technique he developed himself called "forcing", was that both answers could be true, i.e. that the continuum hypothesis and the axiom of choice were completely independent from ZFC set theory. Thus, there could be two different, internally consistent mathematics, one where the continuum hypothesis was true, and there was no such set of numbers, one where and the hypothesis was false and a set of numbers did exist. The proof seemed to be correct, but Cohen’s methods (particularly his new technique of “forcing”) were so new that no-one was really quite sure until Gödel finally gave his stamp of approval in 1963.
His findings were as revolutionary as Gödel’s own. Since that time, mathematicians have built up two different mathematical worlds, one in which the continuum hypothesis applies and one in which it does not, and modern mathematical proofs must insert a statement declaring whether or not the result depends on the continuum hypothesis.
Cohen’s paradigm-changing proof brought him fame, riches and mathematical prizes galore, and he became a top professor at Stanford and Princeton. Flushed with success, he decided to tackle the Holy Grail of modern mathematics, Hilbert’s eighth problem, the Riemann hypothesis. However, he ended up spending the last 40 years of his life, until his death in 2007, on the problem, still with no resolution (although his approach has given new hope to others, including his brilliant student, Peter Sarnak).
ROBINSON AND MATIYASEVICH
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Julia Robinson (1919-1985) and Yuri Matiyasevich (1947- ) |
Brought up in the deserts of Arizona, Robinson was a shy and sickly child but showed an innate love for, and facility with, numbers from an early age. She had to overcome many obstacles and to fight to be allowed to continue studying mathematics, but she persevered, obtained her PhD at Berkeley and married a mathematician, her Berkeley professor, Raphael Robinson.
She spent most of her career pursuing computability and “decision problems”, questions in formal systems with “yes” or “no” answers, depending on the values of some input parameters. Her particular passion was Hilbert’s tenth problem, and she applied herself to it obsessively. The problem was to ascertain whether there was any way of telling whether or not any particular Diophantine equation (a polynomial equation whose variables can only be integers) had whole number solutions. The growing belief was that no such universal method was possible, but it seemed very difficult to actually prove that it would NEVER be possible to come up with such a method.
Throughout the 1950s and 1960s, Robinson, along with her colleagues Martin Davis and Hilary Putnam, doggedly pursued the problem, and eventually developed what became known as the Robinson hypothesis, which suggested that, in order to show that no such method existed, all that was needed was to construct one equation whose solution was a very specific set of numbers, one which grew exponentially.
The problem had obsessed Robinson for over twenty years and she confessed to a desperate desire to see its solution before she died, whoever might achieve it. In order to progress further, though, she needed input from the young Russian mathematician, Yuri Matiyasevich.
Born and educated in Leningrad (St. Petersburg), Matiyasevich had already distinguished himself as a mathematical prodigy, and won numerous prizes in mathematics. He turned to Hilbert’s tenth problem as the subject of his doctoral thesis at Leningrad State University, and began to correspond with Robinson about her progress, and to search for a way forward.
After pursuing the problem during the late 1960s, Matiyasevich finally discovered the final missing piece of the jigsaw in 1970, when he was just 22 years old. He saw how he could capture the famous Fibonacci sequence of numbers using the equations that were at the heart of Hilbert’s tenth problem, and so, building on Robinson’s earlier work, it was finally proved that it is in fact impossible to devise a process by which it can be determined in a finite number of operations whether Diophantine equations are solvable in rational integers.
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Matiyasevich-Stechkin visual sieve for prime numbers |
Among, his other achievements, Matiyasevich and his colleague Boris Stechkin also developed an interesting “visual sieve” for prime numbers, which effectively “crosses out” all the composite numbers, leaving only the primes. He has a theorem on recursively enumerable sets named after him, as well as a polynomial related to the colourings of triangulation of spheres. He is head of the Laboratory of Mathematical Logic at the St. Petersburg Department of the Steklov Institute of Mathematics of Russian Academy of Sciences, and is a member of several mathematical societies and boards.
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