The Mathematician's Fight with Tradition, Religion, Property, and the State
Birth of MathematicsHow Mathematics Came To Be
Many thousands of years ago, in the ancient civilization of Egypt, a group of people gathered to debate and discuss the Nile's flooding. The strength of the flood waters was so great that the land of inhabitants could actually sink forever into that river. Yet some people had to live by the river, because that was the main method of transportation for everyone. A social welfare scheme was developed: those who lost land from the flood would be compensated with an equal amount of land.
One main objection halted the whole project: "There's no way to perfectly measure the area of one piece of land so it's the same as another!" One person responded by inventing an entirely new field of mathematics -- geometry. Because of mathematics, the farmers who sacrificed their land to guarantee access to the river would not be without any form of social security. It was geometry and the first geometer that established some minimum level of social justice in the ancient, Egyptian civilization.
In Egypt, it was the need for guaranteed living standards that impelled great thinkers to create geometry. [*1] In the Middle East, it would be the desire to provide a fair inheritance to children that led to the creation of algebra. [*2] And all across Europe, it was the question of fair prices and just trade that brought about the modern field of probability and statistics. [*3] To many people, mathematics can feel cold and unnatural, sometimes awkward and sometimes lonely, but they probably don't know that mathematicians throughout history have been inspired to some of their greatest creations by completely social issues. Mathematical thinkers are not formed in the test tube of cold logic but in the ferment of the Social Struggle.
A New Way of Thinking for Civilization
One day, approximately 2,500 years ago on one of the many beaches of the Mediterranean Sea, a mathematician walked past a fishing boat with its nets full of fish. The net was hoisted in dry air, as the fish squirmed against the torturous suffocation. Without any hesitation, the mathematician asks, "Would you like to bet that I can guess how many fish are in the net?" The fishermen respond that if the mathematician can guess correctly, they will do anything he says. A careful calculation brought the number 153, and after counting each fish, it turned out that this stranger was correct. And his only wish was, "Return those fish alive back to the seas." [*4] This was Pythagoras.
It is due to Pythagoras that we owe the Pythagorean Theorem: A squared plus B squared equals C squared. To quote the mathematician Jacob Bronowski, speaking more than two thousand years after Pythagoras, "To this day, that remains the most important, single theorem in the whole of mathematics." [*5] Pythagoras stands out as one of the first important thinkers of mathematics, partly for his mathematical achievements and partly for the school of thought he established. Geometry, algebra, and probably were invented as methods to solve pressing problems of people and society, but the ideas of the Pythagoreans were not inspired by some worldly problem -- they were inspired by their reverence and near-spiritual worship of numbers. [*6]
The Pythagorean Theorem is an essential component of any mathematical education. It is drilled and redrilled into the heads of young students and proven and reproven by their instructors. But very few would know how to respond if you told them that Pythagoras had abolished all property among his followers, he prohibited ownership of ideas, and the killing of animals for consumption was regarded as the savaged, uncivilized habits of a mindless beast without a sense of either society or culture. Pythagoras was a student of the universe's lessons on mathematics, but he was a revolutionary teacher to a civilization that was just being born.
The First Revolutionary, the First Mathematician
In the sixth century BC, it was the time when Buddha, Confucius, and Lao Tzu were teaching their spiritual philosophies in the East, while in the West, the greatest mind to fill that role was Pythagoras. It was more than 2,300 years before Karl Marx, and Pythagoras had established a communal organization of property within the school of the Pythagoreans. [*7] After giving a lecture to two thousand listeners in Crotona, the crowd agreed to abolish property and that "each community should own all in common." One quote which is attributed to him is "A man should consider in a certain sense his brothers are part of him, just as my eyes are part of me; also my legs, my hands and other parts of me. For the relation of men to the social organism is the same as the members to the body." [*8]
Throughout the many, tiny, individual, Pythagorean societies of secret mathematicians, the rule of conduct was held to "not eat any creature endowed with a soul." [*9] Women and girls were taken in as pupils [*10] several thousand years before European universities would allow the same thing to happen. [*11] Not only physical property, but intellectual property, too, was communalized, "No Pythagorean took individual credit for a discovery..." [*12] Or, according to Uta C. Merzbach and Carl B. Boyer, "Knowledge and property were held in common..." [*13] The stories about Pythagoras were so mythical that some writers exaggerated his accomplishments, suggesting stories with "Pythagoras himself as operating an entire revolution in the form of government, not only in Croton, but likewise in the other Italian cities." [*14] But all was not well with the Pythagorean school. In the darkness of that utopian society, a beast was lurking: authority and government.
Authority and GovernmentThe Beginnings of the Pythagoreans
Pythagoras was the founder of the Pythagorean School, but it lasted for hundreds of years. Its organization was hierarchical and authoritarian. [*15] According to the historians of philosophy, George F. McLean and Patrick J. Aspell, "...the Pythagoreans used an authoritarian approach in deciding questions..." [*16] There were other secret societies with some political intrigues, but the Pythagoreans can be considered Authoritarian on two accounts: first, the group was "strictly sectarian, presenting only the Pythagorean line on any issue," and second, the group was "authoritarian, involving no discussion and embodying the ipse dixit ['he himself said it'] mentality." [*17] One historian, with multiple ancient sources, tells us that, "he was the only one who could speak; his followers were not allowed to." [*18]
This authoritarianism remained a permanent, fixed structure of the Pythagorean School's organization for its centuries of existence. Mathematical questions could immediately become political questions; doubting the mathematical legitemacy of a theorem could mean becoming outcast and politically ostracized by the group. There was one theorem that Pythagoras asserted as a fundamental principle, that all numbers could be represented as a ratio of two integers. In the language of modern mathematics, this statement is equivalent to: all numbers are Rational numbers. [*19] Belief in this was mandatory for all Pythagoreans, and doubting it was considering blasphemy. But then came along the mathematician Theaetetus.
The square root of 2. It's something that sounds so simple, and yet just a few brief lines by Theaetetus is enough to prove that it is absolutely an irrational number. [*20] An entire school of mathematical thought, full of brilliant minds, was founded with loyal belief that all numbers are a ratio of two integers -- and the whole system of thought crumbles because of just a few sentences by Theaetetus. And yet, this ground-breaking discovery was not the first. Theaetetus was no Pythagorean, but there were other Pythagoreans who may have stumbled upon an irrational number.
The Mathematical Outcast, Hippasus
There are three possible accusations against Hippasus of Metapontum, as well as three possible fates. According to one historian, the charges were "political insubordination, having headed a democratic movement against the conservative Pythagorean rule," "disclosure of the geometry of the pentagon or dodecahedron," or "the disclosure of a mathematical discovery of devastating significance for Pythagorean philosophy -- the existence of incommensurable magnitudes [irrational numbers]." [*21] The details of his punishment are also uncertain, as historian Kitty Ferguson tells us, "His punishment, from the gods or the Pythagoreans, depending on which story to believe, was drowning at sea, expulsion from the community, and/or the construction of a tomb to him as though he were dead." [*22] According to Charles Seife, in a book about the history of the number zero...
"...whatever Hippasus's true fate was, there is little doubt that he was reviled by his brothers. The secret he revealed shook the very foundations of the Pythagorean doctrine, but by considering the irrational an anomaly, the Pythagoreans could keep the irrationals from contaminating their view of the universe." [*23]Hippasus probably just wanted to be an honest mathematician, but his words were probably met with: "That means we have to question the whole system of communal property, vegetarianism, anti-sexism, ownerless ideas, and absolute obedience to the Pythagoras or whoever leads the school." It's not just about the square root of two -- it's about politics. Pythagoras in his own age revolted against the times of his day, and he brought about many interesting theorems with his rebellion. But while Pythagoras flourished, Hippasus perished. The authoritarian world built by Pythagoras could never give birth to another Pythagoras. Followers of Pythagoras may have been interesting and they may have produced their own theorems, but they were bound and chained. They were terrified by the new ideas. So mathematics went forward without them and they were left behind.
The End of Pythagoras
The fate of Hippasus concretes our image of Pythagoras as an authoritarian. Apollonius describes the Pythagoreans as having become completely corrupt and tyrannical, although there may be some exaggeration, "falsification, hired assassins, bribed judges, the expulsion of families and children, etc. were all put to use." [*24] The Pythagoreans supported aristocracy, which we know from Robert Browne: "The influence also of himself, and his followers, was sufficiently powerful to impose an aristocratic constitution in Croton and the neighboring states... The league which he established... constituted also a political bond of union, and its object was to propagate aristocratic principles." [*25]
Pythagoras never feared a democratic uprising against the authoritarian and aristocratic trends of his school, but he should have been worried about others who were similarly authoritarian and aristocratic. He refused the politician Cylon entrance into his society, so Cylon "commenced a violent struggle with the philosopher." [*26] According to Pythagorean historian Raymond Bernard, "Cylon ascended the tribune again and again. He demonstrated that the doctrines of Pythagoras were a crime against liberty and were undemocratic and against the rights of the masses." [*27] Cylon organized a mob against the Pythagorean school, which burned out the Pythagoreans, and his followers threw themselves onto the flames to make a bridge so that the old philosopher could escape. [*28]
Pythagoras went from city to city seeking refuge, and was denied again and again. At Locri, they gave him food and supplies, but refused him sanctuary, "their message to him as they turned him away was that they admired his wisdom but liked their present condition and way of life and did not wish to change." [*29] He finally found sanctuary at the temple in Metapontum -- ironic, because this is where Hippasus was from, the Pythagorean that Pythagoras had persecuted brutally, just now as he himself was escaping from persecution. According to legend, he spent the last forty days of his life starving himself to death and mourning for his friends. [*30] The absence of Pythagoras left an unfillable hole in mathematics and philosophy, two words which he is responsible for. [*31]
Followers of Pythagoras who Came After
Plato was among the philosophers in this ancient world who a mind for both philosophy and mathematics. He had received numerous invitations to be a tutor to King Dionysus II, and when he finally accepted, he found himself a prisoner of the king, "Plato was soon not only out of favor but in danger for his life." [*32] To quote a historian of Plato, "If Plato had been a virtual prisoner before, now he was in danger: Athenian rowers among the mercenaries told him some of their number were plotting to kill him..." [*33] This philosopher sent letters to friends begging for help, like Pythagoras escaping persecutors, and one person finally responded. Plato was "liberated only upon the intercession of Archytas." [*34]
It was Archytas, a Pythagorean, who had finally saved Plato from an oppressive tyrant. [*35] Unlike the books of Pythagoras, there are some existing fragments of the works of Archytas, and these provide some useful insight into the Pythagorean reasoning about aristocratic government. Here we can see the absolute obsession that the Pythagoreans had with numbers...
"The aristocratic constitution is founded on the subcontrary proportion, and is the justest, for this proportion attributes the greatest results to the greatest terms, and the smallest to the smallest. The democratic constitution is founded on the geometrical proportion, in which the results of the great and small are equal." [*36]The "subcontrary proportion" is is the Harmonic Mean, [*37] which is the equation (2 * a * b) / (a + b), while the equation for the Geometric Proportion is the square root of (a * b). Translated to sociology, Archytas believed that the value of the few, wealthy personalities of an aristocracy is greater than the value of many, small individuals of a democracy. The problem with this view is that aristocracy doesn't award position by merit but by birth or fortune, so the greatest of its citizens are not the greatest in terms of thinking or ability, and in fact, its best thinkers may be those in the lowest class. This is certainly the case with ancient Athenian society, where we read stories from Aristophanes about how slaves can outwit their masters. [*38]
Mathematicians Victimized by the State
The politics of the Pythagoreans was certainly a confusing mixture of aristocratic dominance and communal sharing. But the story of the Pythagoreans echoes throughout the story of mathematics. Archimedes, the ancient inventor and discoverer of numerous theorems, was slaughtered by a Roman imperial soldier. [*39] Mathematicians in China made numerous discoveries, but little is known even about them, since emperors have routinely burned their books. [*40] Robert Recorde, inventor of the equal sign, was sued by the state for criticizing the government and died in debtor's prison. [*41] [*42]
Marquis de Condorcet, who contributed to calculus and probability, was a representative and revolutionary during the French Revolution, and was eventually executed during the Reign of Terror in France. [*43] Jean-Victor Poncelet, contributor to the field of analytic geometry, did much of his research as a prisoner of the Russian tzar. [*44] Jean Leray, mathematician of topology, languished in a Nazi prisoner-of-war camp, [*45] and Otto Neugebauer, along with many other mathematicians, were forced to flee Europe to escape the hands of the Nazis. [*46] Everywhere, thinkers of mathematics have stood out among their peers as a threat to the established order, and everywhere, the government has persecuted them.
Class WarThe Dutch Astronomer and Council-Communist
In the Netherlands in 1873, an astronomer was born who would change the way we look at the universe. One historian of science describes his activities in 1925, telling us that he "founded his own astronomical laboratory (again, without a telescope), which gradually developed into an astrophysical institute." [*47] According to a publication by the United States Secretary of State, he is listed as one of several individuals responsible for establishing astrophysics "as a subdiscipline in Europe, [while] in the United States it was still a novelty." [*48]
In 1950, he was "awarded the Gold Medal of the Royal Astronomical Society in 1951 for his work on astrophysics and the structure of the Galaxy." When asked to give the Royal Society a lecture, he chose the history of astronomy as a topic, bewildering some of his listeners, "... it is by no means usual for a distinguished modern scientist not only to take an interest in the history of his science, but also to make significant original contributions to its study." [*49] From his well-known, 1951 book on the history of astronomy, he wrote, "...astronomy faces a host of new problems. Problems not as before purely astronomical, but problems of space and time, of universe and science, involving physics, mathematics and astronomy." [*50] He was admired by fellow scholars because he "...puts special emphasis on the role of quantification and the intellectual craving for beauty." [*51]
But this astronomer had many, many enemies. In 1916, he applied as a director of an observatory in the Netherlands, but he was yet cursed by the national government for having spent eight years working with the German Socialist movement, besides his frequent, anti-capitalist writings. On the opposite side, Vladimir Lenin and the Bolshevik Party, representing the established political authorities of the Soviet Union, condescended to write a pamphlet on why this astronomer and mathematician cannot be trusted. [*52] Lenin would pretend to shrug off the anti-Bolshevik criticism of this astronomer, "There have always been attacks on the 'dictatorship of leaders' in our Party." [*53] Persecuted by both Soviet and Capitalist governments, caught in the fire of global empires, this astronomer managed to change the way the whole world thought of the universe -- his name was Anton Pannekoek. To him, we owe the words...
"Capitalism, indeed, cannot be annihilated by a change in the commanding persons; but only by the abolition of commanding. The real freedom of the workers consists in their direct mastery over the means of production. The essence of the future free world community is not that the working masses get enough food, but they direct their work themselves, collectively. For the real content of their life is their productive work; the fundamental change is not a change in the passive realm of consumption, but in the active realm of production." [*54]Anton Pannekoek's dream was like a Pythagorean ideal without the authoritarianism or control. When he wanted to be an astronomer and was more qualified than any other individual, the national government directly intervened and withheld him from the position, just like Cylon leading the aristocracy to burn down the house of Pythagoras. Fortunately, the city council of Amsterdam was completely and thoroughly Socialist, and with Pannekoek being turned away from the role of astronomer, he was quickly offered the position of professor of astronomy, which he accepted. [*55]
According to one historian, Pannekoek "...began his studies at the age of fifteen as an amateur astronomer fascinated by the Milky Way," [*56] and only forty years later, he was the first person who had "measured the brightness of the Northern Milky Way based on photometry of photographs..." [*57] Despised by Leninist Bolsheviks and rejected by Dutch Nationalists, Pannekoek was still fascinated by the physics of the natural world around him just as much as he dreamed of a world without oppression or exploitation. One evening, he would calculate the orbits of objects in space, and the next, he would write, "The proletarian revolution is not simply the vanquishing of capitalist power. It is the rise of the whole working people out of dependence and ignorance into independence and clear consciousness of how to make their life." [*58]
The American Linguist and Anarchist
Pannekoek corrected one of the flaws of Pythagoras: authoritarianism and absolutism. Other mathematicians could make the same calculations and arrive at similar conclusions. Another similar mathematician can be given the indecisive description of "a rationalist philosopher or as the formal system theorist who opened the way to the artificially constructed 'languages' of theoretical computer science." [*59] Computer science in general would depend on his work, as his essential work "in automata theory in the late 1950's and '60s is quite well known in computer science. It was the basis for the Benjamin Franklin medal he received..." [*60] One historian summarizes him with: "... he brought theoretical linguistics into the ambit of information theory, computer science, AI, and computational psychology." [*61]
His name is Noam Chomsky, and he is usually known for his contributions to linguistics and mathematics or to Anarchism and radical anti-Capitalism, but rarely are both sides of this character examined together. To quote one of his articles...
"...international solidarity can take new and more constructive forms as the great majority of the people of the world come to understand that their interests are pretty much the same and can be advanced by working together. There is no more reason now than there has ever been to believe that we are constrained by mysterious and unknown social laws, not simply decisions made within institutions that are subject to human will -- human institutions, that have to face the test of legitimacy and, if they do not meet it, can be replaced by others that are more free and more just, as often in the past." [*62]The First Revolutionary Period of France
When the French Revolution broke out in 1789, that was the indicator for each peasant to take "...communal action against a large proprietor: a field worked by the lord's wage laborers, say, might be communally seized..." [*63] From one historian, "In France the peasants had seized the land, and they kept it." [*64] The spirit of the day was: "Revolutionary intellectuals proclaim the death of the old order and the birth of a new society; revolutionary peasants kill the tax collector and seize the land." [*65] But the revolution was immediately under threat from the day it was born, facing the armies from more than ten hostile nations: Great Britain and Russia, Spain and the Holy Roman Empire, Prussia and Portugal. Even the United States declared war on France, [*66] a sharp twist of fate when the United States would just be another British colony without French aid during the American Revolution. [*67]
Everything threatened the peasant armies and the new society with a basis of communalized land. Armies led by princes and presidents, by popes and parliaments, were marching to crush everything the French Revolution had accomplished. All seemed to be lost -- except, among the revolutionaries, there was a mathematician, Lazare Carnot, a contributor to the fields of geometry and calculus. As an elected representative, he was at the scene of one of the first battles of the French Revolution...
"Carnot, in agony at the disorder, rallied the soldiers, formed them anew on the plain, -- cashiered, in the sight of the whole army, the general who had allowed himself to be beaten by disobeying his orders, and seizing the musket of a grenadier, he marched at the head of the columns in the costume of a Representative of the People... The campaign of seventeen months, conducted by Carnot, and during which the troops of the Republic never laid down their arms, was one of the most successful and glorious that France can boast." [*68]The peasant revolution could now thrive and grow, because of Lazare Carnot, but when the revolution was completely dismantled by Emperor Napoleon, Carnot was among the few to offer any resistance to the personality cult surrounding Napoleon, suffering exile for it. [*69] There were other mathematicians in France at this time: Gaspard Monge known for descriptive geometry, [*70] Joseph-Louis Lagrange known for celestial mechanics and calculus, [*71] Pierre Simon Laplace known for probability, [*72] Adrien Marie Legendre known for geometry and number theory, [*73] Nicolas Condorcet known for calculus and probability, [*74] as well as the above-mentioned Carnot. Many of them were brought up accepting religious wafers on their tongues and always praising the nation as students at the military academies, but when the Revolution came, not one of them sided with the old, decaying order. [*75]
ReligionThe New Mathematicians and Skepticism
One day in Italy in the early 1800's, a mathematician walked down the street enjoying the beautiful scenery and fresh air. But something caught this man's attention. Stopping his walk, he took a moment to examine a water pump on the side of the road. A small sign indicated that it was erected for the love of "God and Country," but upon examining the device, the mathematician discovered that "every time that a wayfarer pumped some water for himself, he pumped a larger quantity into the owner's house." Having told this story to a friend, he remarked, "There is only one thing which I hate more than piety, and that is patriotism." This was Charles Babbage, inventor of the programmable computer. [*76] The moral to the story is thorough: religion and nation have pretended to satisfy human thirst in order to deceive and control the common people.
A young child of age three has his will in the power of a medievil-minded court system. Both of his parents were dead and his father was an Atheist who had organized meetings to fight for women's equality and the right to birth control. [*77] As an adult, this child would later write, "My father wished my brother and me to be brought up as free-thinkers, and appointed two free-thinkers as our guardians. The Court of Chancery, however, at the request of my grandparents, set aside the will..." [*78] He was then forced into a Christian education, by orders of a British court, in the year of 1875. "I think perhaps the Court of Chancery might have regretted that since. It does not seem to have done as much good as they hoped," he could joke as an adult, then known as a well-respected philosopher and mathematician. [*79] This was the childhood of Bertrand Russell.
The contributions of Bertrand Russell to mathematics are vast and deep, covering the areas of set theory, logic, linguistics, and computer science. But if you were to ask him his opinion on politics, there be an endless fountain of opinion, and he would probably tell you, "Either war or civilization must end..." [*80] Throughout his life, he would be known as an enemy of religion and its influence, becoming well-known for the essay "Why I Am Not A Christian," and being admired everywhere for his skepticism of traditional religion. To quote one of his writings, "Cruel persecutions have been commoner in Christendom than anywhere else. What appears to justify persecution is dogmatic belief. Kindliness and tolerance only prevail in proportion as dogmatic belief decays." [*81]
A young student sits before the advisory board of the University of Cambridge. It's supposed to be a day of celebration and joy, as this young, brilliant mind is being asked to join the school's program for graduate students. However, in order to enroll, the student is required to take a test affirming absolute loyalty and devotion to the Church of England as organized by King Henry the Eighth. This was an unacceptable request and he refused. So he was denied the right to be a student, because "his religion was personal, and he was no friend of any organized church, certainly not the Church of England." [*82]
This math student then transferred to University College London, where he would become the first professor of mathematics, at the early age of only twenty-two -- and even then, he would leave that college, too, over a dispute on religious discrimination. [*83] His name was Augustus De Morgan, and his contributions to complex logic, relation logic, and probability logic were all essential to modern mathematics. [*84] Modern computers depend upon the mathematical discoveries of De Morgan; but he was the kind of mathematician who valued his own dignity and independence more than submitting to the domination of church authorities.
The Trial Against the Millenia's Greatest Mathematician
Pythagoras worshiped numbers. Even numbers were considered female and odd numbers were considered male; one is the number of reason and four is the number of justice. [*85] Numbers were associated with mysticism and mythology, but prayer was nothing more than solving geometry problems. Pythagoras' struggles against his enemies were not focused on his veneration for geometric patterns so much as his support for aristocratic and authoritarian organization. After two thousand years, mathematicians are still making statements against the established, religious norm. The only real change that can be observed is that mathematicians tend to be far more suspicious of myth and superstition than Pythagoras.
Emperor Napoleon, always beaming with pride about the accomplishments of France, would occasionally read the writings of French mathematicians, if they were famous. Among them, there were the writings of Pierre Simon Laplace, author of one of the first books accurately describing the motion of the solar system. [*86] It was an impressive accomplishment at such an early period of time, but Napoleon was disappointed. The emperor complained, "Monsieur Laplace, they tell me you have written this large book on the system of the universe, and have never even mentioned its Creator." The mathematician responded to the emperor, "I have no need for that hypothesis." [*87]
But what is it about Babbage, Russell, De Morgan, and Laplace that they all have in common? There is only one thing we can point to: they are all mathematicians who came after Galileo Galilei. His crime was defending the theory that the earth is spinning and rotates around the sun, known then as the Copernican Theory. To cite the inquisition records...
"We have three Inquisition documents testifying to the fact that Galileo was admonished not to hold or defend the Copernican theory. All three are connected with the name Bellarmine, who was to carry out the admonition under the explicit direction of the Pope..." [*88]The documents by Bellarmine which admonished Galileo were never presented at the trial, they were "discovered" more than a decade after the trial, and they were all unsigned; they were obvious forgeries, though some historians consider them to be "incomplete drafts." [*89] From another historian: "The Pope thereupon ordered Galileo to face the Inquisition. Galileo was interrogated by the Inquisition four times and was threatened with torture." [*90] When the mathematician made a discovery, the emperor did not complain -- the emperor acted: "Threatened with torture on June 21, 1632, Galileo agreed with the position of the church and declared the earth to be motionless with the sun moving around it." [*91] After being forced to publicly recant his statements about the solar system, he quietly said, "And yet it [the Earth] moves!"
Galileo and Company
Galileo, a professor of mathematics at the age of 69, was condemned to house arrest for the remaining period of his life, "forced to recite prayers every day and wasn't supposed to be allowed visitors, though neither of these were well enforced." [*92] In 1642, he died, suffering from blindness, insomnia, and hernia, and still a prisoner to the pope. [*93] To quote mathematician Jacob Bronowski, "The effect of the trial and the imprisonment was to put a total stop to the scientific tradition in the Mediterranean. From now on, the scientific revolution moved to Northern Europe." [*94] Thus we have Babbage, De Morgan, Laplace, and Russell, all mathematicians who thrived by being a great distance away from the pope and religious control.
It was not merely torture, but also death that Galileo was threatened with, as the last heretic who discredited the church, Giordano Bruno, was burned alive for refusing to withdraw his statements. [*95] Bruno had been accused of heresy, but his contributions to astrophysics within the 1500's were amazing: "When Giordano Bruno came to view unlimited Euclidean space as natural space filled with an incalculable number of solar systems, the idea arose of the infinity of the spatial universe." [*96] His geometric models led him to "arrive mentally, at least, at nearly limitless magnitudes... extremes not only of immensity but also of tininess... He had begun, in other words, to ask the questions to which only calculus would provide answers." [*97] To quote George Foote, biographer of infidels and heretics...
"The Venetian Council transferred him to Rome, where be languished for seven years in a pestiferous dungeon, and was repeatedly tortured, according to the hellish code of the Inquisition. At length, on February 10th, 1600, he was led out to the Church of Santa Maria, and sentenced to be burnt alive, or, as the Holy Church hypocritically phrased it, to be punished 'as mercifully as possible, and without effusion of blood.' Haughtily raising his bead, he exclaimed: 'You are more afraid to pronounce my sentence than I to receive it.' He was allowed a week's grace for recantation, but without avail; and on the 17th of February, 1600, he was burnt to death on the Field of Flowers." [*98]Mathematicians Victimized by Religion
Galileo and Bruno were not the first to be persecuted by a church. Religious persecution is as old as religion. Hypatia of Alexandra, born in 370 BC, is another name from this history of religious abuse. She was Platonist, or a follower of Plato, but she also considered herself a follower of Pythagorean ideas, both far in contrast to the Christian society all around her. Hypatia was the first female mathematician in the history of the world, as well as a professor of mathematics, and "on a warm March day in AD 414, a crowd of Christian zealots seized her, stripped her, and proceeded to scrape her flesh from her bones using sharp shells. Next, they cut up her body and burned the pieces." [*99]
Hypatia was well known for her commentaries on mathematical and astronomical works that were both contemporary and ancient, such as Apollonius and Diophantus. Some scholars have suggested that we wouldn't even have the books of Diophantus today without Hypatia, because only she could explain their value to the people around her. [*100] Working with Synesius, together they invented the astrolabe, a device used to predict the positions of stars, which can be used for navigational purposes. [*101] To quote one historian who succinctly describes Hypatia's place in the history of science and mathematics...
"Since there were to be no significant advances in mathematics, astronomy or physics anywhere in the West for another 1,000 years, Hypatia has come to symbolize the end of ancient science... after Hypatia came only the chaos and barbarism of the Dark Ages." [*102]Hypatia and Bruno, Galileo and De Morgan, Russell and Laplace, but there are so many others, too. Anaxagoras, Greek astronomer and mathematician, was imprisoned because he stated that "the sun was not a deity but a huge red-hot stone as big as the whole of Peloponnesus [Southern Greece]". [*103] The Bernoulli family, comprised of at least eight mathematicians and physicists, provided tremendous contributions to calculus, physics, and probability, and they almost met the same fate as Bruno at the hand of the Spanish Inquisition in 1583. [*104] Bernard Bolzano, one of the essential developers of real topology within mathematics, was a professor who criticized nationalism, antisemitism, and religious fanaticism, and "Owing to repeated controversies with the authorities he was called an 'atheist,' lost his job and was put on trial in 1820." [*105]
In 557 AD, Justinian I, a Christian saint, banned and evicted the mathematical and philosophical schools in Greece. [*106] In 1476, the mathematician and astronomer Regiomontanus died from poisoning, having been invited by the Catholic Church to reform the calendar, but actually being murdered for criticizing the Ptolemaic world view, which held that the earth was the center of the universe. [*107] Petrus Ramus, mathematician and professor of mathematics, was slaughtered by fanatic mobs of the Catholic Church for the same reason in the St. Bartholomew's Day Massacre in 1572. [*108] Mathematicians everywhere have a devotion to truth that isn't quite like that of any other professional -- and naturally, this always means a hostile confrontation with the forces of religion.
The Mathematician's RevolutionA New Type of Scientist
In 1938, one of the greatest physicists of the twentieth century told his sister about the United States, "The truth is that nothing but money counts here," and to another friend, "Everything here is rushed and ruthless, a real dance around the golden calf, a wild and ugly dance." [*109] In May of 1949, he would shock many in the scientific community by writing an article about Socialism: "I am convinced there is only one way to eliminate these grave evils, namely through the establishment of a socialist economy, accompanied by an educational system which would be oriented toward social goals." Capitalism was described as "the real source of the evil." [*110]
This scientist would let the world know that there was one thing worse than Capitalism: "War constitutes the most formidable obstacle to the growth of international cooperation, especially in its effect upon culture." [*111] The strain of Pythagoras could still be felt in this mathematician, too, "So I am living without fats, without meat, without fish, but am feeling quite well this way. It almost seems to me that man was not born to be a carnivore," [*112] and eventually telling us, "Nothing will benefit human health and increase the chances for survival of life on Earth as much as the evolution to a vegetarian diet." [*113] Pacifist and Anti-War, Socialist and Anti-Capitalist, Democratic and Anti-Soviet, Vegetarian and Anti-Animal Abuse -- this was Albert Einstein.
Pythagoras never really died. He simply took on new shapes and forms. Einstein is best known for his work in the area for his Theory of General Relatively, publishing the gravitational equations in 1915 with the comment that it marked "a true triumph of the methods of the general differential calculus founded by Gauss, Riemann, Christoffel, Ricci..." [*114] Unlike Pythagoras, Einstein was not authoritarian. He was asked to be the president of Israel in 1952, but turned down the offer, saying, "It is true that many a rebel has in the end become a figure of responsibility, but I cannot bring myself to do so." [*115] Mathematics can advance in the least expected ways sometimes. But there are other ways in which it has fallen so far, far behind.
A New Type of Mathematician
There is another famous mathematician from the twentieth century whose story needs to be told. From 1940 to 1942, he worked secretly at a government program in Beltchey Park given the nickname "Ultra." The project was decrypting the secret messages of the Nazis. [*116] Because of his work "...when Hitler sent an order to his troops, Churchill and Roosevelt received the same information, at the same time," [*117] and according to one historian, he "was doing more than perhaps any other individual during the entire war in ensuring the Third Reich's ultimate defeat." [*118] Much earlier, in 1936, with the publication of the paper "On Computable Numbers," this mathematician had invented the entire field of computer science, [*119] with his total field of study "ranging from the fundamental theory of computability to the question of what might constitute true artificial intelligence." [*120]
In 1952, he was arrested and jailed by the police, with twelve charges against him, all generally of the same type with one of them reading: "...being a male person, committed an act of gross indecency with Arnold Murray, a male person." He was charged with being gay, classified by the government intelligence agencies as a "security risk," and sentenced to chemical castration. [*121] [*122] While abroad in Greece, he dipped an apple in cyanide and then ate it, dying shortly afterward. [*123] It has been suggested that he chose this way of committing suicide because it would "allow those who wished to do so to believe it an accident." [*124] Both the history of mathematics and social justice are incomplete without his name -- Alan Turing.
Archytas was the ancient Pythagorean who freed Plato from the hands of a tyrant, but were there really no Pythagoreans left when Turing was arrested? Where was Archytas at a time like this? Turing was only 42 years old at the time of his death; like the loss of Hypatia and Galileo, what kind of Dark Age can we expect now because of Christian ideas that dominated the British lawbooks? Again and again, the obscure mathematician is persecuted, tortured, and abused by the authorities, perpetually victimized for being unafraid of the truth. Some historians believe that the stories of Hypatia, Galileo, and Turing prevent future authorities from committing such unwarranted cruelty. But the clear-minded thinking of the mathematician finds the same condemnation from the same, impossible authority.
During the French Revolution, Carnot organized a peasant rebellion that brushed aside the armies of the British King as easily as it swept away the navies of the American President. Where was Carnot at a time like this? The social struggle constantly molds and remolds itself, generation after generation, era after era. Even today, there is still the internal conflict of society, with a mathematician focused at the center of the controversy.
A New Type of Philosopher
This mathematician is most well-known as a cryptologist. Just barely a teenager in the late 1980's, he wrote a program called Sycophant, which allowed him and a group of hackers known as the International Subversives to break into computer systems of the US military. [*125] The list of computers they broke into "read like a Who's Who of the American military-industrial complex." [*126] Even though his university studies in mathematics and physics were "mixed but generally mediocre," [*127] he worked with human rights workers who told him "tales of abuse from repressive regimes such as East Timor, Russia, Kosovo, Guatemala, Iraq, Sudan, and the Democratic Republic of the Congo." To help the human rights workers, he co-invented the Rubberhose Deniable Encryption System: it was designed so that those being tortured for keys to encrypted data can give up a fake password that looks like it works when it doesn't, giving a fake information instead of the real information. [*128]
Although he was an open source developer, [*129] his primary role was in distribution of highly classified documents to the general public. There is one meeting with a CIA espionage agent that made him known all over the world. This CIA agent had worked with an anti-Castro, anti-Communist group responsible for terrorist attacks killing hundreds of people. After several more meetings, the CIA agent reported that the cryptologist had raped her and he was quickly detained -- his name is Julian Assange. [*130]
Twitter entries from the woman and photographs of her with Assange occurring only days after the alleged rape have discredited her allegations. After the alleged rape, the alleged victim introduced Assange to another woman at a political convention, who made the same allegations against Assange. [*131] Like Galileo, he faced forged testimony, and like Galileo, it looks as though the remainder of his days may be spent under house arrest at the Ecuadorian Embassy in London. [*132]
But what is it that has brought about so much attention on this one mathematician who leaked secret documents to the public? There is no way to answer without mentioning Wikileaks. Torture manuals from Guantanamo Bay expose "among other matters, systematic methods to prevent prisoners meeting with the Red Cross and the use of extreme psychological stress as torture." [*133] There are also secret torture camps operated by the US army in Iraq using "beatings, burnings, and lashings," [*134] and of greater interest, many of those inmates were never even suspected of terrorism, being detained because they were journalists. [*135] There was also a leak of 400,000 documents implicating the US military in "torture, summary executions and war crimes" involving over 15,000 previously unreported deaths. [*136] This is why Assange is so desperately wanted by authorities. He has exposed their system of justice for the hoax it is, like Galileo shattering the Ptolemaic world view of the Catholic Church.
Julian Assange was not the first and he will not be the last. From Pythagoras to Einstein, from Carnot to Pannekoek, from Hypatia to Galileo, from De Morgan to Russell, from Hippasus to Turing, the social struggle constantly turns within mathematics as it does throughout civilization. Mathematicians will always try to solve problems, whether these are problems of mind and theory or people and society. To quote the mathematician Jacob Bronowski, "The intellectual commitment and the emotional commitment working as one has made the ascent of man." [*137]
*1. "A History of Mathematics," by Uta C. Merzbach and Carl B. Boyer, Third Edition, published by John Wiley & Sons, Inc., Hoboken, New Jersey, 2011, page 8.