The noted blogger Fjordman is filing this report via Gates of Vienna.
For a complete Fjordman blogography, see The Fjordman Files. There is also a multi-index listing here.
Charles Murray’s book Human Accomplishment includes rankings of influential personalities in Western, Chinese and Indian philosophy. The men at the top — Aristotle, Confucius and Sankara, respectively — are there because in some sense they defined what it meant to be Western, Chinese or Indian. The same is not true of artists, no matter how great they are.
Who does Murray personally consider to be the most accomplished individual who ever lived? “ Aristotle. He more or less invented logic, which was of pivotal importance in human history (and no other civilization ever came up with it independently). He wrote the essay on ethics (‘Nicomachean Ethics’) that to my mind contains the bedrock truths about the nature of living a satisfying human life. He made huge contributions to aesthetics, political theory, methods of classification and scientific observation. Who else even comes close?”
Georges Lemaître (1894-1966), a Belgian priest, introduced his primeval “cosmic egg” in the 1920s. In 1948 the American Ralph Alpher (1921-2007) together with the Ukrainian-born American physicist George Gamow (1904-1968) outlined a theory of how the first elements formed in the early universe (Big Bang nucleosynthesis). Later that year, collaborating with scientist Robert Herman (1914-1997), an American-born son of a Russian Jewish immigrant just like Alpher himself, Alpher predicted the existence of a cosmic background radiation resulting from the Big Bang. Two individuals who didn’t realize what they had found in the 1960s stumbled across evidence of this radiation and received the 1978 Nobel Prize in Physics for the discovery. The contributions of Alpher and Herman were overlooked. A Nobel Prize cannot be shared by more than three individuals, nor can it be awarded posthumously.
This Big Bang model was modified in the 1980s with the introduction of the concept of an early period of cosmic inflation by the American cosmologist Alan Guth (born 1947) and the Russian-born physicist Andrei Linde (born 1948). Alpher’s contributions have unquestionably helped shape the way we currently look at the universe, yet he is not mentioned in Murray’s Human Accomplishment and is entirely overlooked in a number of other works on scientific history. By contrast, Aristotle’s physical ideas have all been discredited centuries ago, yet most educated people have heard of Aristotle. Does that mean that Aristotle is overrated and is primarily famous for being wrong? Not quite so. His biological works have stood the test of time, and his personal contributions to the development of scientific logic are profound.
Aristotle’s six works on logic are known collectively as the Organon, which means “tool” or “instrument.” This reflects the awareness that logic was not a science in itself but a tool for rationally analyzing the world. The vocabulary of logic, syllogism, types of logical fallacy, the elements of deductive reasoning and a long list of terms for analyzing propositions date back to Aristotle. The power of his logic was so great that the importance of logic overrode empiricism for centuries. The balance was restored when it was combined with experiment. Francis Bacon published his Novum Organum (“The New Organon”) in 1620 with the Baconian method. During the Scientific Revolution, supporters of the experimental method frequently criticized what they considered blind adherence to Aristotelian philosophy.
Perhaps the most impressive aspect of Aristotle’s work is the sheer scale of it, and how he extended his investigations to include all natural phenomena. There was no known equivalent to him in ancient Egypt, Mesopotamia, India, China or Mesoamerica. He enjoyed an exceptional prestige throughout the Middle Ages in the Middle East and Europe, so much so that even his errors prevailed into the modern era, especially in physics and astronomy. Many theologians and natural philosophers simply referred to him as “the Philosopher.” Yet his tremendous influence resulted not merely from intellectual subservience on the part of medieval scholars, but also “from the overwhelming explanatory power of Aristotle’s philosophical and scientific system. Aristotle prevailed through persuasion, not coercion.”
The nature of mathematical proof is related to Aristotelian logic, but mathematical logic predates it in time. The development of mathematical proof started among the ancient Greeks in the sixth century BC with Thales and Pythagoras. Other ancient civilizations practiced mathematics in an intuitive and experimental manner to solve practical problems. It was the Greeks who came to insist that geometric statements should be established by careful deductive reasoning rather than by trial and error. The logico-deductive method can provide conclusions/new knowledge by means of reasoning step by step from established knowledge. In his excellent book A History of Mathematics, Second Edition, Victor J. Katz states:
“Aristotle believed that logical arguments should be built out of syllogisms, where ‘a syllogism is discourse in which, certain things stated, something other than what is stated follows of necessity from their being so.’ In other words, a syllogism consists of certain statements that are taken as true and certain other statements that are then necessarily true. For example, the argument ‘if all monkeys are primates, and all primates are mammals, then it follows that all monkeys are mammals’ exemplifies one type of syllogism, while the argument ‘if all Catholics are Christians and no Christians are Moslem, then it follows that no Catholic is Moslem’ exemplifies a second type. After clarifying the principles of dealing with syllogisms, Aristotle notes that syllogistic reasoning enables one to use ‘old knowledge’ to impart new.”
Aristotle distinguished between the basic truths that are peculiar to each particular science and the ones that are common to all. The former are often called postulates, the latter as axioms. Yet “Although Aristotle emphasized the use of syllogisms as the building blocks of logical arguments, Greek mathematicians apparently never used them. They used other forms, as have most mathematicians down to the present. Why Aristotle insisted on syllogisms is not clear. The basic forms of argument actually used in mathematical proof were analyzed in some detail in the third century B.C.E. by the Stoics, of whom the most prominent was Chrysippus (280-206 B.C.E.). This form of logic is based on propositions, statements that can be either true or false, rather than on the Aristotelian syllogisms.”
Aristotle was a member of Plato’s Academy for twenty years. When he returned to Athens in 335 BC he founded a rival school in another Athenian gymnasium, the Lyceum. Athens had by then acquired educational leadership within the Greek world. One of the visitors there was Zeno (ca. 335-263 BC), who was born in Citium in Cyprus and founded the influential Stoic school of philosophy. Men should face the world with “stoic calm.” The Stoics “looked upon the passions as essentially irrational, and demanded their complete extirpation. They envisaged life as a battle against the passions, in which the latter had to be completely annihilated. Hence their ethical views end in a rigorous and unbalanced asceticism.”
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According to Lindberg, “Zeno of Citium arrived in Athens about 312 and subsequently began to teach in the stoa poikile (painted colonnade) in a corner of the Athenian agora, thus founding a school of what came to be called ‘Stoic’ philosophy. Epicurus, an Athenian citizen born on the island of Samos, returned to Athens about 307, purchased a house and garden, and there founded a school of ‘Epicurean’ philosophy that survived into the Christian era. The Academy, the Lyceum, the Stoa, and the Garden of Epicurus — the four most prominent schools in Athens — all developed institutional identities that enabled them to survive their founders. The Academy and the Lyceum seem to have had continuous existence until the beginning of the first century B.C. (perhaps until the sack of Athens by the Roman general Sulla in 86 B.C.). It is often claimed that the Academy survived until it was closed by the Emperor Justinian in A.D. 529. The truth seems to be that Neoplatonists (so-called because of their departure from or reinterpretation of various Platonic doctrines) refounded the Academy in the fifth century A.D. and managed to keep it alive until about 560 or later.”
The Megarians were followers of Euclid of Megara (ca. 430-360 BC), a pupil of Socrates. They were interested in logical puzzles and influenced Stoic logic. Many Greek mathematicians followed the forms of argument delineated by Stoic philosophers such as Chrysippus of Soli, who was a systematizer of Stoic philosophy and in Antiquity was considered a logician comparable in stature to Aristotle himself. His works have mostly been lost, but he aided the popularity that Stoicism enjoyed during Hellenistic and Roman times, with prominent adherents such as the Roman Emperor Marcus Aurelius (AD 121-180).
Parmenides was one of the first philosophers to use an extended argument for his views rather than merely proposing a vision of reality, yet he never formulated his principles in a systematic manner. Zeno of Elea’s arguments, or “Zeno’s Paradoxes,” could establish a claim by showing that its opposite leads to absurd consequences. This line of argumentation is known as reductio ad absurdum. It is likely that Zeno consciously used it in a systematic way.
As author Marvin J. Greenberg says, “The orderly development of theorems with proofs about abstract entities became characteristic of Greek mathematics and was entirely new. This was the first major revolution in the history of mathematics. How this revolution came about is not well understood by historians. Among Greek philosophers, dialectics, the art of arguing well, which originated in Parmenides’ Eleatic school of philosophy, played an important role. And undoubtedly proofs were an outgrowth of the need to convince others in a debate.”
The Greeks’ basic political organization was the polis or city-state. Its government could be democratic or monarchical but was usually ruled by law, and thus its citizens were motivated to learn the skills of argument and debate. The Sophists were teachers of rhetoric, prepared to teach anything anyone was ready to pay for. Plato criticized them for this as his teacher Socrates did not accept payment, but the Sophists did contribute to a culture of argumentation. This political atmosphere favored the development of proof in mathematics. To contrast between (mere) persuasion and demonstration, Aristotle defined a logic proceeding from self-evident primary premises via valid deductions to incontrovertible conclusions.
As G.E.R Lloyd puts it in The Ambitions of Curiosity, the development of scientific logic and mathematical proof may seem natural and inevitable to us today, “But when we reflect that neither the Chinese nor any other ancient mathematical tradition did so, there would appear to be more to it than mere intellectual attractiveness. What more may be answered in part, I suggest, by the negative models provided by the styles of argument cultivated in those other peculiarly Greek institutions of the law-courts (dikasteria) and political assemblies. It was dissatisfaction with the merely persuasive arguments used there that led some philosophers and mathematicians to develop their alternative.”
The concept of mathematical proof reached its highest state of development in Greek geometry. Among Euclid’s most important predecessors was the astronomer Hippocrates of Chios (ca. 470-410 BC), who compiled the first significant Greek work on the elements of geometry. Archytas of Tarentum (ca. 400-350 BC) was a Greek mathematician, political leader, Pythagorean philosopher and musical theorist from southern Italy and a friend of Plato. Euclid may have borrowed from his work in his own treatment of number theory.
Although Euclid gave the work an overarching structure and added some material of his own, he is first and foremost famous for creating a brilliant synthesis of the work of others. Almost nothing is known about him personally, but he lived in the early Hellenistic period. It is assumed that he taught at the Museum and Library at Alexandria, founded around 300 BC.
As Katz states, “The most important mathematical text of Greek times, and probably of all time, the Elements of Euclid, written about 2300 years ago, has appeared in more editions than any work other than the Bible….Biographies of many famous mathematicians indicate that Euclid’s work provided their initial introduction into mathematics, that it in fact exited them and motivated them to become mathematicians. It provided them with a model of how ‘pure mathematics’ should be written, with well-thought-out axioms, precise definitions, carefully stated theorems, and logically coherent proofs. Although there were earlier version of Elements before that of Euclid, his is the only one to survive, perhaps because it was the first one written after both the foundations of proportion theory and the theory of irrationals had been developed in Plato’s school and the careful distinctions always to be made between number and magnitude had been propounded by Aristotle. It was therefore both ‘complete’ and well organized.”
There were some notable logicians and natural philosophers during the Middle Ages; a few in the Middle East, more in Europe but almost nobody of note in Byzantium. The next revolution in logic came in Western Europe. Gottfried Wilhelm Leibniz in his 1666 publication De Arte Combinatoria first proposed the idea of an algebra of logic, but later developments happened along somewhat different lines. Author Marvin Jay Greenberg writes:
“George Boole and Augustus de Morgan began to carry out his idea. Boolean algebra is now the foundation for computer arithmetic and is very important in pure mathematics. In 1879 Gottlob Frege brought quantifiers into logic, introducing what is now known as the predicate calculus, but with terrible notation. Most of the currently used notation and methods of mathematical logic stem from the society of logicians founded in the 1880s by Giuseppe Peano along with Mario Pieri. They emphasized the importance of a formal symbolic language for mathematics to remove the ambiguities of natural languages, to make mathematics utterly precise, and to permit the mathematical study of entire mathematical theories. Many years later, this formalization also enabled the programming of computers to do mathematics. The discovery and validation of non-Euclidean geometries, together with Georg Cantor’s invention of set theory and Karl Weierstrass’ rigorous presentation of analysis, caused mathematicians to study axiomatics seriously for the first time. It was not until 1889 that axioms for the arithmetic of natural numbers were satisfactorily formulated — by Peano, based on Richard Dedekind’s set-theoretic development using the successor function (and influenced by earlier algebraic work of Hermann Grassmann).”
Hermann Grassmann (1809-1877) was a German polymath educated at the University of Berlin. He was more famous in his day as a scholar of Sanskrit and other Indo-European languages than as a mathematician. His basic idea of a general calculus of vectors was first published in 1844, followed by a reworked version in 1862. According to author John Derbyshire, “He defined such concepts as linear dependence and independence, dimension, basis, subspace, and projection. He in fact went much further, working out ways to multiply vectors and express changes of basis, thus inventing the modern concept of ‘an algebra’ in a much more general way than Hamilton with his quaternions. All this was done in a strongly algebraic style, emphasizing the entirely abstract nature of these new mathematical objects and introducing geometrical ideas as merely applications of them.”
Grassmann’s work was studied by Giuseppe Peano and by the Frenchman Élie Cartan (1869-1951), the son of a poor blacksmith. Cartan showed unusual ability and was able to obtain state funds for his education. He lectured at the Universities of Montpellier, Lyon and Nancy before moving to Paris and did valuable work on Lie algebras and group theory. His son Henri Cartan (1904-2008) became a distinguished scholar in his own right and a member of the group of mathematicians writing under the collective pseudonym Nicolas Bourbaki.
The German scholar Gottlob Frege (1848-1925) was one of the founders of modern mathematical logic and analytic philosophy. He had a major influence on the prominent Austrian and British philosopher Ludwig Wittgenstein (1889-1951), born in Vienna in the Habsburg Empire to a family with a Jewish background who practiced Christianity. His father was an industrialist and patron of the arts. Composers Johannes Brahms, Richard Strauss, Gustav Mahler and the Catalan conductor Pablo Casals (1876-1973) frequented the family, and Wittgenstein senior collected works of artists like the French sculptor Auguste Rodin (1840-1917) and the Austrian painter Gustav Klimt (1862-1918). In 1911 Ludwig Wittgenstein went to Cambridge to study with Bertrand Russell. His philosophical influences include the Danish theologian Søren Kierkegaard and Arthur Schopenhauer. His major works are Tractatus Logico-Philosophicus from 1922 and Philosophical Investigations from 1953. Wittgenstein spent many years at Cambridge University in England where he eventually died.
The gifted Italian mathematician Giuseppe Peano (1858-1932) was born and raised in the Piedmont region of northern Italy, an area justly renowned for its fine wines. He spent most of his career teaching at the University of Turin where he became a lecturer of infinitesimal calculus in 1884 and a professor in 1890. He was above all a pioneer in the development of a symbolic logic and the use of the axiomatic method, stressing the necessity of rigor.
Alfred North Whitehead (1861-1947) was an English mathematician and philosopher who wrote a Universal Algebra (1898) based on Grassmann, Boole and Hamilton. Bertrand Russell (1872-1970) was a prominent and controversial British philosopher, author and social reformist who made valuable contributions to the development of mathematical logic. In 1950 he was awarded the Nobel Prize in Literature. Part of Peano’s logic notation was adopted by Russell and Whitehead in their Principia Mathematica (1910-13), which had a complicated but precise symbolism. Like Hilbert’s approach, however, it failed in its ultimate purpose.
According to Marvin Jay Greenberg, “The most influential foundational works in logic in the early twentieth century were the Principia Mathematica of Bertrand Russell and A. N. Whitehead; the work of David Hilbert with his associates Wilhelm Ackermann, Paul Bernays, and John von Neumann; and the contributions of Thoralf Skolem. By formalizing all rules of reasoning and axioms in a purely symbolic language, mathematicians were able to study entire branches of their subject, such as Peano arithmetic and elementary geometry and Zermelo-Fraenkel set theory. They were then able to prove theorems about those branches — theorems that are called metamathematical because they are about mathematical theories, not about numbers or geometric figures or sets. The most important metamathematical theorems are the completeness and incompleteness theorems of Kurt Gödel from the early 1930s, which revolutionized our thinking about the nature of mathematics. Also vitally important in the 1930s were the equivalent determinations of the class of effectively computable number-theoretic functions by Alan Turing, Alonzo Church, Emil Post, and Gödel.”
The German-born Jewish mathematician Abraham Fraenkel (1891-1965) was a Zionist who, after leaving Kiel, taught at the Hebrew University of Jerusalem from 1929. Building on the work of German scholar Ernst Zermelo (1871-1953) from 1908 he helped to create axiomatic set theory in 1922. Thoralf Skolem (1887-1963), a Norwegian mathematician and professor at the University of Oslo, made further contributions in the field of mathematical logic and set theory, as did the German mathematician Wilhelm Friedrich Ackermann (1896-1962). Paul Bernays [pdf] (1888-1977) from Switzerland collaborated with David Hilbert and made significant contributions to the development of mathematical logic and the philosophy of mathematics.
The English mathematician and logician Augustus De Morgan (1806-1871) published his Formal Logic in 1847. Fellow Englishman George Boole (1815-1864) became the inventor of Boolean logic, the basis of modern digital computer logic. Emil Post (1897-1954) was an American logician associated with Columbia University in New York City. Another prominent mathematician and logician in the United States, Alonzo Church (1903-1995) of Princeton University, was one of the founders of theoretical computer science along with the great English mathematician, cryptanalyst and computer scientist Alan Turing (1912-1954).
Felix Klein (1849-1925), born in Düsseldorf, Germany to a Prussian family, was active in many branches of mathematics and a highly influential teacher, yet he is best known for his work on the connections between geometry and group theory. After 1870, during the period of German unification under the leadership of the “Iron Chancellor” Otto von Bismarck, he collaborated with the Norwegian mathematician Marius Sophus Lie, who introduced him to the group concept which had been pioneered by mathematicians such as the Norwegian Niels Henrik Abel and the Frenchman Évariste Galois. “Klein’s synthesis of geometry as the study of the properties of a space that are invariant under a given group of transformations, known as the Erlanger Programm (1872), profoundly influenced mathematical development” and gave a unified approach to geometry which is now the standard accepted view.
Emmy Noether (1882-1935) was the daughter of a Jewish mathematician. She came from the town of Erlangen in Bavaria and took classes at the university there. In 1915 she was invited by Hilbert and Klein to the University of Göttingen. Often regarded as the most important woman in the history of mathematics, she made many contributions to abstract algebra. Noether’s Theorem, proved by her in 1915, “establishes a quite remarkable connection between the symmetries of a physical system and its conserved quantities, a connection that has acquired particular importance for the entire body of modern physics because of the emphasis all contemporary theories put on both symmetries and conservation laws.”
The Hungarian János Bolyai (1802-1860), the Russian Nikolai Lobachevsky (1792-1856) and above all the brilliant German mathematician Bernhard Riemann (1826-1866) in the mid-nineteenth century founded non-Euclidean geometry, which we will deal with later when looking at the general theory of relativity. Following the development non-Euclidean geometry in late nineteenth century Europe came a renewed interest in Euclid’s geometry.
The most successful attempt to set up a complete set of axioms from which Euclidean geometry could be derived was David Hilbert (1862-1943). He rose to prominence after Felix Klein called him to Göttingen, the preeminent university for mathematics in Germany and perhaps the world prior to the rise of the Nazis. He was rivaled only by the French mathematical physicist Henri Poincaré as the leading mathematician during his lifetime. He made outstanding contributions to many mathematical fields; the infinite-dimensional spaces used in quantum mechanics, Hilbert spaces, are named after him. His treatise Grundlagen der Geometrie (Foundations of Geometry) from 1899 “provided brand new important insights into the foundations of geometry.” There were many axiom schemes developed in this period to clarify various areas of mathematics. Hilbert’s work was the culmination of this process where the ideas of Aristotle and Euclid were reconfirmed as the model for pure mathematics.
At the International Congress of Mathematicians in Paris in 1900, Hilbert challenged his fellow scholars with a list of twenty-three unsolved problems in mathematics that turned out to be some of the most important of the twentieth century. Many of his problems have now been solved. In the year 2000, the Clay Mathematics Institute in the United States offered a million-dollar prize to anyone who can solve seven problems considered to be among the most challenging for the new century. One of them, the Poincaré conjecture, has been solved.
The Russian Jewish mathematician Grigori Perelman was born in 1966 in Saint Petersburg. In 2002-2003 the former child prodigy published on the Internet the solution to the Poincaré conjecture. He was aided by earlier progress made by fine mathematicians such as Richard Hamilton (born 1943) and William Thurston (born 1946) in the USA but made the final breakthrough in solitude. In 2006, Perelman was awarded the prestigious Fields Medal but declined to accept it. In 2010 it was announced that he had met the criteria to receive the first Clay Millennium Prize Problems award of one million US $. He is reported to have said that “I’m not interested in money or fame. I don’t want to be on display like an animal in a zoo.”
Hilbert’s axiomatic approach was criticized by Luitzen Egbertus Jan Brouwer (1881-1966), a Dutch mathematician and philosopher from the University of Amsterdam who contributed substantially to the emerging field of topology and founded the philosophy of intuitionism in opposition to Hilbert’s formalistic ideas. As writer Roger G. Newton states in From Clockwork to Crapshoot: A History of Physics, “In Hilbert’s publicly stated view, the success of intuitionism would destroy mathematics. The intuitionist movement has now lost momentum, but Hilbert’s axiomatic program was dealt a mortal blow from another direction, by the work of the Austrian mathematician Kurt Gödel (1906-1978). Gödel’s famous theorem states that in every consistent and sufficiently strong formal axiomatic system there would necessarily arise statements that could be neither proved nor disproved within that system.”
Kurt Gödel was born in Austria-Hungary. Although a non-Jew he found the situation unacceptable after the Anschluss in 1938, when Austria became a part of Nazi Germany, and relocated to the USA. While sometimes mentally unstable he was a brilliant logician. He made his deepest impact on mathematics in 1931 at the University of Vienna. According to the Stanford Encyclopedia of Philosophy online he “ founded the modern, metamathematical era in mathematical logic. His Incompleteness Theorems, among the most significant achievements in logic since, perhaps, those of Aristotle, are among the handful of landmark theorems in twentieth century mathematics. His work touched every field of mathematical logic, if it was not in most cases their original stimulus. In his philosophical work Gödel formulated and defended mathematical Platonism, involving the view that mathematics is a descriptive science, and that the concept of mathematical truth is an objective one.”