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Abel, Henrik Neils – Zeno of Elea
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Neils Abel
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Abel, Henrik Neils
(born 1802)
Norwegian algebraist who proved the unsolvability of the general quintic (fifth degree) equation by radicals. He published this landmark result when he was 22, but had great difficulty in getting contemporary establishment mathematicians (including Gauss and Cauchy) to notice his work. Suffering from tuberculosis, he returned to Norway from Paris, but continued to publish papers in Crelle’s Journal. Just when the mathematical world was beginning to wake up to his achievements, when he was 26 years old, Abel succumbed to his illness.
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Archimedes
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Archimedes
(born 287 bce)
Greek geometer and applied mathematician who lived at Syracuse, a Greek settlement on the coast of Sicily. He is famous both for his profound mathematical work and for his ingenious mechanical inventions, which at the end of his life he used to defend his city against the Romans during the Second Punic War. When he discovered the hydrostatic principle – that a floating body of a given weight displaces a volume of liquid having an equal weight – he is reputed to have jumped from his bath and run naked through the streets shouting “Eureka!” (I have found it!) Archimedes is credited with many brilliant mathematical discoveries, especially in properties of geometrical figures, areas and volumes of conics, and principles of trigonometry. (He is thought to have known Heron's formula – several centuries before Heron.) He found areas of plane figures by the method of exhaustion, prefiguring modern integral calculus. He was killed by a Roman soldier during the sack of Syracuse.
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Aristotle
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Aristotle
(born 384 bce)
Greek philosopher and scientist who developed syllogistic logic as a formal scholastic discipline. He defined the syllogism as a “discourse in which certain things having been stated, something else follows of necessity from their being so.” Aristotle’s philosophy of the infinite, in which he held that only the potential infinite, and not any collection which was actually infinite, could be entertained by the reason, held great authority for 2,000 years, until Georg Cantor introduced his theory of transfinite sets in the 19th century. Aristotle wrote widely on every field of learning, interpreted and disagreed with the writings of his teacher Plato, and served as tutor to Alexander the Great.
Bourbaki, Nicolas
Although the myth of an actual French mathematician named Bourbaki was maintained for many years, it became generally known in 1952 that “Bourbaki” is the nom de plume of a private congress of young French mathematicians which formed in 1935, with the purpose of writing a comprehensive exposition of modern mathematics in many volumes, called Elements. Bourbaki’s original members included André Weil, Henri Cartan, Claude Chevalley, and Jean Dieudonné. Other mathematicians have been recruited into Bourbaki in the years since, including G. de Rham, A. Grothendieck, S. Lang, and R. Thom. A principle motivation for the activity of the Bourbaki group is the felt need to unify the entire corpus of mathematical work in the face of accelerating diversification and specialization of mathematics.
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Georg Cantor
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Cantor, Georg
(born 1845)
German mathematician and founder of modern set theory. Cantor’s concern with the set-theoretic nature of the real numbers began while he was working on certain properties of trigonometric series during the early 1870’s. Inspired by the work of his friend Dedekind, Cantor proved in 1873 that the rational numbers are countable, and later the same year proved that the real numbers themselves are uncountable. These proofs introduced important methods, including the notion of a one-to-one correspondence and the method of diagonalization. Cantor’s results were counterintuitive and broke with established dogma about the nature of infinity. Consequently, he became a controversial figure, opposed by some mathematicians (e.g., Kronecker) who used their influence to stifle Cantor’s career as much as possible. This opposition exacerbated Cantor’s predisposition to severe depression, and he ultimately died, of a heart attack, in a sanitorium.
Related article: Cantor's Theorem
Related article: Cantor Set
Related MiniText: Infinity -- You Can't Get There From Here...
Carroll, Lewis
See Charles Lutwidge Dodgson.
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René Descartes
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Descartes, René
French mathematician who is generally considered to have laid the foundations for modern mathematics. His greatest achievement was the invention of analytic geometry, in which the methods of algebra and those of geometry are used together. He is also a central figure in the history of modern philosophy; his treatises Meditations and Discourse on Method laid the groundwork both for modern rationalism and modern skepticism. Descartes did not actually use the rectangular coordinate system known as the Cartesian plane (this was developed by Leibniz and others), and he permitted only positive values for his variables. Nonetheless, his development of algebraic methods in geometry made possible an explosion of analytic discoveries by his successors, the most important of which was the discovery of the calculus less than a generation after Descartes’ death.
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Lewis Carroll
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Dodgson, Charles Lutwidge
(born 1832)
Oxford mathematician most famous for the books Alice in Wonderland and Through the Looking Glass, which he wrote under the pseudonym Lewis Carroll. However, he also wrote several mathematics textbooks, and delighted in inventing bizarre and humorous syllogisms for exercises in Aristotelian logic. These syllogisms commonly appear in introductory texts on logic to this day. He also formulated what is now called Carroll's Paradox, which shows the need for formal rules of inference in any system of logic.
Related article: Carroll's Paradox
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self portrait
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Escher, Maurits Cornelis
(born 1898)
Dutch graphic artist whose works exhibit many mathematical concepts, including tesselations, infinity, and polyhedra.
Related MiniText: Mathematical Art of M.C. Escher
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Fibonacci
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Fibonacci
(born 1170)
Medieval Italian mathematician, whose real name was Leonardo of Pisa. (“Fibonacci” is a nickname meaning “son of good nature.”) Although best known for the number-sequence which bears his name, his most important contribution to mathematics was introducing the Arabic numerals to Europe through his book Liber Abaci (pub. 1202), a treatise on algebraic methods of arithmetic, and the application of these methods in business. Fibonacci was a businessman and the son of a businessman, educated in North Africa, and it was on his extensive travels in the near east and Africa that he became familiar with the notation which made practical commerce so much easier, since it was a place notation and hence made algorithmic arithmetic calculations possible. It was in the Liber Abaci that the problem which leads to his famous sequence was posed and solved:How many pairs of rabbits will be produced in a year, beginning with a single pair, if in every month each pair bears a new pair which becomes productive from the second month on? The solution gives rise to the sequence beginning 1, 1, ..., and in which each succeeding term is given by adding the previous two.
Cf. Fibonacci sequence.
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Gottlob Frege
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Frege, Gottlob
(born 1848)
German philosopher and logician, and the founder of mathematical logic. He wrote Begriffsschrift (1879) and Grundlagen der Arithmetik (1884), in which he formulated the principles of the propositional calculus and the logical foundations of arithmetic, respectively. In the course of his work Frege developed other branches of logic, including the predicate calculus, second-order logic, and a version of naive set theory. He believed that arithmetic itself could be shown to be a branch of logic, and his chief concern was to determine whether proofs in arithmetic rest on logic alone, or need empirical facts for their support. Frege was a realist, as is demonstrated by his thesis that concepts are denotations of their extensions. The number 2, for example, is a concept denoting its extension – the class of all collections having two elements. This is contrasted with Bertrand Russell's conception of number as having no referent, but existing only as a contextual definition.
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Galileo
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Galilei, Galileo
(born 1564)
Italian mathematician, physicist, and astronomer who is generally associated with the birth of modern science. Such discoveries as that bodies of different size and weight fall at the same speed (i.e., that gravitational acceleration is independent of mass), overthrew two millenia of Aristotelian doctrines. They also greatly annoyed certain factions within the Church, which was only just coming to terms with the birth of science, and ultimately brought about a showdown with ecclesiastical authorities that effectively silenced Galileo at the end of his life. Galileo’s importance in the history of mathematics stems less from his purely mathematical work than from his promulgation of an outlook, especially in the sciences, that intellectual inquiry must not be fettered by prejudice if it is to bear fruit.
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Kurt Gödel
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Gödel, Kurt
(1906 – 1978)
The foremost logician of the 20th century. He is famous for many deep results in the foundations of mathematics, including the Godel Completeness Theorem, the Godel Incompleteness Theorem, and that the generalized continuum hypothesis and the axiom of choice are consistent with the other axioms of set theory. He moved from Austria to the Institute for Advanced Study at Princeton, New Jersey, in 1940. He was a lifelong friend of Albert Einstein, and in the 1950’s formulated versions of Einstein’s relativity theories that permitted the logical possibility of time travel.
Cf. logic
Related article: Gödel's Theorems
Hilbert’s Problems
ARTICLE
23 problems posed by Hilbert to the community of mathematicians at the Paris conference of 1900. See the article for details.
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Johannes Kepler
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Kepler, Johannes
(born 1571)
German astronomer and mathematician who discovered that planetary motion is elliptical. Early in his life, Kepler set about to prove that the universe obeyed Platonistic mathematical relationships; that for instance the planetary orbits were circular and at distances from the sun proportional to the Platonic solids. However, when his friend the astronomer Tycho Brahe died he bequeathed to Kepler his immense collection of astronomical observations. After decades of studying these observations, Kepler realized that his earlier surmises about planetary motion were naive, and he formulated his three laws of planetary motion, now known as Kepler's Laws. He did not have a unifying theory for these laws, however; this had to wait another generation, until Isaac Newton formulated his laws of gravity and motion.
Related article: Kepler's Laws
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Leibniz
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Leibniz, Gottfried Wilhelm
(1646 – 1716)
German mathematician and philosopher, and co-discoverer of calculus (independently of Newton). Leibniz sought to reduce the mechanics of human thought to a logical calculus, and in this respect foreshadowed modern interest in artificial intelligence and cognitive science. He also described the binary number system and laid the foundation of dynamics. Leibniz’s philosophy was characterized by the assertion that all true propositions are analytic, but that some propositions (which seem synthetic to us) could only be known to be analytic by God.
Peano, Giuseppe
(born 1858)
Italian mathematician best known for the Peano axioms, which define the natural numbers in terms of sets and formalize arithmetic in a logically rigorous way. Peano was primarily an analyst, however, and was the inventor of space-filling curves, that is surjective functions from the unit interval to the unit square.
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Pythagoras
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Pythagoras
(born 580 bce)
What we know of this ancient Greek mystic is mostly legend, for of the true facts of his life almost nothing is known. However, it is believed that he started a religious cult, the Pythagoreans, among whose beliefs was that number was the highest reality, and that all other aspects of reality could be understood in terms of integers and ratios of integers.
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Bertrand Russell
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Russell, Bertrand
(born 1872)
English philosopher and logician. Wrote Principia Mathematica (1913) with Alfred North Whitehead, an attempt to reduce all of mathematics to symbolic logic. This was perhaps the most important effort in the logicist program. Russell introduced type theory into the theory of sets and classes in an effort to avoid the kind of antinomy in the foundations of mathematics as was exemplified in the so-called Russell paradox. The goal of Russell’s system was to show that any true mathematical proposition can be established by logic alone, a goal which was severely compromised by Kurt Gödel’s proof of the Gödel Incompleteness Theorem in 1931.
Zeno of Elea
Greek philosopher famous for certain paradoxes, including the Paradox of the Arrow, the Paradox of the Tortoise and Achilles, and the Paradox of the Moving Rows. These constitute the earliest known arguments from infinity.
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