|
|
|
|
|
|
10. Augustin Cauchy
Augustin Cauchy was an incredibly prodigious, prolific and inventive mathematician. Home-schooled, he awed famous mathematicians at an early age, and in contrast to other great mathematicians, like Gauss and Newton, he was almost over-eager to publish; and in his day his fame surpassed that of even Gauss and has continued to grow even to this day. Cauchy did important work in analysis, algebra and number theory. One of his important contributions was the ``theory of substitutions'' (permutation group theory). Cauchy's research also included convergence of infinite series, differential equations, determinants, and probability. He invented the calculus of residues. Although he was one of the first great mathematicians to focus on abstract mathematics (another was Leonard Euler), he also made important contributions to mathematical physics, e.g. the theory of elasticity. He was the first to prove Fermat's conjecture that every positive integer can be expressed as the sum of k k-gon numbers for any k, and also refined Euler's results in discrete topology. Another of Cauchy's contributions was his insistence on rigorous proofs. 9. Kurt Godel
 Kurt Gödel was an Austrian-American mathematician, physicist, logician, and philosopher. One of the most significant logicians of all time, Gödel's work has had immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were attempting to use logic and set theory to understand the foundations of mathematics. Godel proved this was impossible, a ground-breaking result whose implications are still felt to this day. Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna. The more famous incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known as Gödel numbering, which codes formal expressions as natural numbers. Gödel's incompleteness theorems struck a fatal blow to David Hilbert's program towards a universal mathematical formalism which was based on Principia Mathematica. He also showed, along with Paul Cohen, that the continuum hypothesis cannot be disproved from the accepted axioms of set theory, if those axioms are consistent. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic. 8. Srinivasa Ramanujan
 Unlike most mathematicians who insisted on rigorous proofs to their results, Ramanujan often omitted proofs. While still in India, he recorded the bulk of his results in four notebooks of loose leaf paper. These results were mostly written up without any derivations. This is probably the origin of the misperception that Ramanujan was unable to prove his results and simply thought up the final result directly. Mathematician Bruce C. Berndt, in his review of these notebooks and Ramanujan's work, says that Ramanujan most certainly was able to make the proofs of most of his results, but chose not to. This style of working may have been for several reasons. Since paper was very expensive, Ramanujan would do most of his work and perhaps his proofs on slate, and then transfer just the results to paper. Using a slate was common for mathematics students in India at the time. He was also quite likely to have been influenced by the style of G. S. Carr's book, which stated results without proofs. Finally, it is possible that Ramanujan considered his workings to be for his personal interest alone; and therefore only recorded the results. As for his place in the world of Mathematics, Bruce C. Berndt said this: “Paul Erdos has passed on to us G. H. Hardy's personal ratings of mathematicians. Suppose that we rate mathematicians on the basis of pure talent on a scale from 0 to 100, Hardy gave himself a score of 25, J.E. Littlewood 30, David Hilbert 80 and Ramanujan 100.'" 7. Henri Poincare
 Henri Poincaré was a clumsy and frail man, and supposedly flunked an IQ test, but he was one of the most creative mathematicians ever, and surely the greatest mathematician of the ``intuitionist'' style. He produced a large amount of brilliant work in almost all areas of mathematics, but also found time to become a famous popular writer of philosophy. His masterpieces include combinatorial (or algebraic) topology, the theory of differential equations, foundations of homology, the theory of periodic orbits, and the discovery of automorphic functions (a unifying foundation for the trigonometric and elliptic functions). He even anticipated modern chaos theory. He posed ``Poincare's conjecture,'' which for an entire century was one of the most famous unsolved problems in mathematic. Recently Poincare's conjecture was settled and the first Million Dollar math prize in history is likely to be awarded.
As with all the greatest mathematicians, Poincaré was interested in physics. He made revolutionary advances in fluid dynamics and celestial motions. And, much of Poincare’s work on the Lorentz Transformation was instrumental to Einstein’s Theory of Relativity, so much so that it is often claimed that Einstein plagiarized his theory from Poincare and Lorentz. Despite being a brilliant researcher, Poincaré was resistant to contributions from mathematicians like Georg Cantor and saw mathematical work in economics and finance as peripheral, and as such, he ranks lower on our list than his body would work would normally suggest. 6. George Cantor
 Georg Cantor single-handedly created modern set theory, defining cardinal numbers, well-ordering, ordinal numbers, and discovering the Theory of Transfinite Numbers. He defined equality between cardinal numbers based on the existence of a bijection, and was the first to demonstrate that the real numbers have a higher cardinal number than the integers, in effect proving that not all infinities are the same size!
Although there are infinitely many distinct transfinite numbers, Cantor conjectured that C, the cardinality of the reals, was the second smallest transfinite number. This ``Continuum Hypothesis'' was included in Hilbert's famous List of Problems, and was finally resolved many years later by Kurt Godel and Paul Cohen: Cantor's Continuum Hypothesis is an ``Undecidable Statement'' of Set Theory. A genius far ahead of his time, Cantor's revolutionary set theory attracted vehement opposition from many of the leading mathematicians and thinkers of his day. Poincare call it a "grave disease". Kronecker, who would make it his personal mission to ruin Cantor, said he was a "charlatan" and "corrupter of youth". And, writing decades after Cantor's death, Wittgenstein lamented that mathematics is "ridden through and through with the pernicious idioms of set theory," and dismissed Cantor’s theory as "utter nonsense" that is "laughable" and "wrong". His ideas even angered theologians. Despite this, Cantor's invention of modern set theory is now considered one of the most important achievements in modern mathematics. In addition to his most famous achievements, Cantor also made advances in number theory and trigonometric series. He gave the modern definition of irrational numbers, and anticipated the theory of fractals. But despite his great genius, a few supporters, Cantor suffered from poverty, and even malnourishment, during World War I. He died alone on January 6, 1918 in the sanatorium where he had spent the final year of his life. His genius only recognized after his death. As Hilbert famously declared: "No one shall expel us from the Paradise that Cantor has created."
|
|
|
> |
