The beauty of Fjordman's thinking is that he is clear, concise, meticulous and thorough ....I like mathematics :)
In his essay, Fjordman posits that the revelation of scientific discovery was at the onset for the glory of G-d. I still believe that. "Many of the scholars who created modern science, including Galileo and Newton, believed that they were honoring God by studying his Creation."
FJORDMAN: MATHEMATICS AND RELIGION
My good friend Ohmyrus, whose website I publish essays at every now and then, is an ethnic Chinese man and a Christian. He believes that secularism promotes a short term attitude towards life since for secular people their time horizon is their own lifetime. Religious people, on the other hand, have their eyes on eternity. Many European medieval cathedrals, for instance Cologne Cathedral in Germany, took centuries to complete. The people who built them literally moved mountains of stones, even though they would in many cases not see the completed result themselves. They did this because they looked to the hereafter. Ohmyrus believes that some of the same principles applied to science as well.
Neither Roman, Egyptian, Chinese nor Indian civilization created the Scientific Revolution; they all stagnated after making initial gains in knowledge. This is because the natural human tendency is to want immediate results. If the research does not yield reasonably quick benefits, interest wanes. Yet you needed a critical mass of accumulated knowledge before the Scientific Revolution could be ignited. The Bible commands mankind to subdue the Earth, but in order to do so, men need to understand how the world works. In addition to this, the Bible portrays God as a Creator who made the universe work according to rational laws. Since God's laws are immutable, it remains for us to discover them. Many of the scholars who created modern science, including Galileo and Newton, believed that they were honoring God by studying his Creation. They saw science as a religious duty. Ohmyrus writes:
“Robert Boyle (1627-1691) in his last will and testament urged his colleagues at the Royal Society of London that ‘they and all other Searchers into Physical Truths may thereby add to the Glory of God and to the Comfort of Mankind.’ René Descartes (1596-1650) said that rational laws must exist because God is perfect and therefore acts in a manner as constant and immutable as possible except for miracles which occur rarely. Other scientists during the Age of Enlightenment that also shared this view of a rational Creator God who created the universe according to rational laws were Newton, Kepler and even Galileo….Thus you have a group of people eager to discover what these scientific laws are in order to glorify God even though they may not yield any immediate benefits. Thus scientific discoveries can accumulate for years, decades and even centuries without any practical use for them. Eventually, of course these scientific discoveries yielded new inventions and other benefits. This permitted the eventual breakthrough which became the Scientific Revolution. Today, science has a momentum of its own and does not need religious motivation to sustain research. But that is why scientific revolution took place in Christendom and not elsewhere.”
Since we live in a more secular age, many observers will doubtlessly dismiss this explanation out of hand. But after having studied the astronomical achievements of the Babylonians and the Mayas, I feel quite certain that you cannot understand why they did what they did and put so much effort into astronomical observations unless you understand their religious world view as well. It is quite likely that the same principle applies to European Christians.
Joel Mokyr, professor at the Department of Economics at Northwestern University, writes about innovation and economic history in his book The Gifts of Athena: Historical Origins of the Knowledge Economy. Was there a link between the so-called Scientific Revolution of seventeenth century Europe and the Industrial Revolution that followed some generations later? Some scholars have questioned this connection, but Mokyr argues that the missing link between the two was what he terms the Industrial Enlightenment. There was a new mentality, and the spillover effects of this were as important as the actual knowledge generated by it.
The Industrial Enlightenment’s debt to the Scientific Revolution consisted of scientific method, scientific mentality and scientific culture. Scientific method meant more accurate measurements, controlled experiment and an insistence on reproducibility. Increasingly precise barometers, thermometers, clocks and other instruments helped unlock previously unknown natural phenomena. There was a mental shift away from reliance on past authority to empirical verifications of facts, with an emphasis on experiment and refinements of the experimental method. Important was also scientific mentality, the concept that the world was orderly and that natural phenomena could be predicted and described mathematically.
According to Mokyr, “The early seventeenth century witnessed the work of Kepler and Galileo that explicitly tried to integrate mathematics with natural philosophy, a slow and arduous process, but one that eventually changed the way all useful knowledge was gathered and analyzed. Once the natural world became intelligible, it could be tamed: because technology at base involves the manipulation of nature and the physical environment, the metaphysical assumptions under which people engaged in production operate, are ultimately of crucial importance. The Industrial Enlightenment learned from the natural philosophers - especially from Newton, who stated it explicitly in the famous opening pages of Book Three of the Principia - that the phenomena produced by nature and the artificial works of mankind were subject to the same laws. That view squarely contradicted orthodox Aristotelianism. The growing belief in the rationality of nature and the existence of knowable natural laws that govern the universe, the archetypical Enlightenment belief, led to a growing use of mathematics in pure science as well as in engineering and technology.”
The most widely cited consequence of the Scientific Revolution was the increasing use of mathematics in natural philosophy and eventually in technical communications. By making mathematics accessible not only to mathematicians but to instrument makers, engineers, designers and artillery officers it became a tool of communication.
The belief that the world could be described mathematically was culture-specific and not at all self-evident. Most civilizations in the past did not share this view, but it is true that you can find elements of it among some of the ancient Greeks. Galileo mentioned Archimedes frequently, and leading figures during the Scientific Revolution were well aware of his importance as a pioneer in the use of mathematics in the treatment of physical problems.
According to Victor J. Katz in A History of Mathematics, second edition, “Archimedes was the first mathematician to derive quantitative results from the creation of mathematical models of physical problems on earth. In particular, Archimedes is responsible for the first proof of the law of the lever and its application to finding centers of gravity as well as the first proof of the basic principle of hydrostatics and some of its important applications.”
The principle of the lever was known before this, but no-one that we know of had created a mathematical model for it before Archimedes in the third century BC. In contrast, while the Romans used mathematics in building roads, aqueducts etc. this was not terribly sophisticated. They added virtually nothing to the Greek achievement in science and mathematics.
As Katz says, “It has long been recognized that there was no ‘Roman mathematics.’ Of course, there were original mathematicians who worked in the Roman empire, primarily in Alexandria, but all of them were part of the continuing Greek tradition. There is no record, however, of any mathematicians who lived and worked at the center of the empire or who wrote in Latin. The great orator Cicero even admitted that the Romans were not interested in mathematics….It appears that the mathematics used in surveying, in architecture, or in the other activities necessary to administer the empire, all taken from earlier discoveries, was sufficient to solve whatever problems arose. With no need for more and no official encouragement of those whose intellectual curiosity ran toward that particular domain, the Roman Empire of the West survived for 500 years without making any contributions to the world’s store of mathematical knowledge.” The extensive use of hydraulic construction in the Roman Empire and in medieval times led to innovations such as aqueducts and the waterwheel, but this added little to Archimedean mathematical theory. Leonardo da Vinci brought some updates and may have aided the notions of pressure of Galileo’s disciple Torricelli. Blaise Pascal formulated the law of isotropic pressure and persuaded his brother-in-law to verify the altitude-dependence of barometric pressure. The Oxford Guide to the History of Physics and Astronomy explains:
“Daniel Bernoulli based his Hydrodynamica of 1738 on Leibnizian vis viva (kinetic energy) conservation, thus obtaining the relation between wall pressure, velocity, and height (Bernoulli’s law). His word ‘hydrodynamica’ expressed the synthesis between conceptions of hydrostatics and hydraulics. Only after suitable extensions of Newtonian dynamics and differential calculus did ‘hydrodynamics’ come to mean a general theory of fluid motion. In 1744 and 1752, Jean Le Rond d’Alembert published treatises in which he applied his general principle of dynamics to fluid motion and established the paradoxical lack of resistance to the motion of a solid through a perfect fluid. Probably motivated by this breakthrough, in 1755 the Swiss geometer Leonhard Euler obtained the partial-differential equation for the motion of a perfect fluid by equating the forces acting on a fluid element to the product of its acceleration and mass. He also showed how to derive Bernoulli’s law from this equation. In his Mécanique analitique of 1788, Joseph Louis Lagrange solved Euler’s equation for simple cases of two-dimensional fluid motion and proved a few important theorems….The French masters of late eighteenth-century hydraulics, Jean-Charles Borda, Charles Bossut, and Pierre-Louis-Georges Du Buat, combined experiment, global balance of momentum or vis viva, and physical intuition.”
The scope of this semi-empirical approach grew with the later work of French, British and German engineers. Hydrodynamics progressed during the nineteenth century, but despite the practical orientation of some theorists it still didn’t always meet hydraulic and other engineering needs. The full development of hydrodynamics happened in the twentieth century; its growth as a mathematical discipline closely mirrored that of science in general.
The ancient Greek mathematician Pythagoras lived in the decades before 500 BC. Pythagoras himself left no written record, so the mathematical doctrines of his school can only be surmised from the works of later writers known as the Pythagoreans. The Pythagoreans believed that the Earth is a sphere, the most perfect solid, although the shadow of the Earth cast on the Moon during an eclipse added observational credibility to this theory. Pythagoras and his followers were fascinated with music and studied the properties of vibrating strings and musical harmonies. They held the belief that similar mathematical harmonies could be found in the universe in general. While the modern study of waves and acoustics began with the Scientific Revolution from the seventeenth century AD onwards, because of his great interest in music Pythagoras is sometimes considered the founder of acoustics in the European tradition. The Pythagorean doctrine was that “numbers are the substance of all things.”
According to Victor J. Katz, “What the Pythagoreans meant by this was not only that all known objects have a number, or can be ordered and counted, but also that numbers are at the basis of all physical phenomena. For example, a constellation in the heavens could be characterized by both the number of stars that compose it and its geometrical form, which itself could be thought of as represented by a number. The motions of the planets could be expressed in terms of ratios of numbers. Musical harmonies depended on numerical ratios: Two plucked strings with ratio of length 2:1 give an octave, with ratio 3:2 give a fifth, and with ratio 4:3 give a fourth. Out of these intervals an entire musical scale can be created. Finally, the fact that triangles whose sides are in the ratio of 3:4:5 are right-angled established a connection of number with angle. Given the Pythagoreans’ interest in number as a fundamental principle of the cosmos, it is only natural that they studied the properties of positive integers, what we would call the elements of the theory of numbers.”
Plato followed the general principles of the Pythagoreans. Reputedly, nobody could enter Plato’s Academy in Athens who did not know geometry. Geometry literally means “Earth measurement” and initially originated through activities of obvious practical importance, in the surveying techniques and building projects of the ancient Egyptians and others. However, Greek mathematicians elaborated geometry into a level of abstract sophistication far beyond anything the Egyptians had ever done. Plato emphasized mathematics at his Academy and in The Republic for the education of the philosopher-kings, the rulers of his ideal state. His most important theoretical treatise on physics and cosmology is the Timaeus, written around 360 BC. Plato believed that the universe is the rational creation of a Divine Craftsman, or Demiurge. This concept was further elaborated by later Neoplatonists such as Plotinus.
Johannes Kepler revived Pythagorean number mysticism in the early seventeenth century, but now with an added Christian perspective. Scholar James Evans writes:
“The revival of the Pythagorean approach to nature was an aspect of Renaissance Neoplatonism. Kepler went on to become the most outstanding mathematical astronomer of his generation. His greatest gifts were inexhaustible patience, great calculating ability, and a relentless drive to understand. But his motives for astronomical research always involved a quest for higher knowledge. Everywhere, he sought for connections between apparently disparate realms of thought. He wanted to know God’s plan for the cosmos….At another stage he revived the Pythagorean doctrine of the harmony of the spheres, in an attempt to associate the speeds of the planets with musical notes. When William Gilbert discovered that the Earth is a giant magnet, Kepler seized on that idea, too, and sought in magnetism an explanation of the motive force that the Sun exerts on the planets. Kepler’s approach to nature, combining elements of animism, mathematical mysticism, and physical reasoning, was not at all unusual for the Renaissance. What made Kepler’s work different was that he also became a skillful technical astronomer and that he judged his own cosmological speculations very strictly: if he was right, then everything should work out in numerical detail.”
While leading scholars during the Scientific Revolution such as Galileo, Kepler and Newton were indeed inspired by the mathematics of the ancient Greeks, their Christian world view made the connection between mathematics and the natural world even more powerful and explicit. Isaac Newton spent a great deal of time looking for hidden codes in the Bible, and undoubtedly believed that he was studying both of God’s Books: The Bible and the Book of Nature. Nothing similar happened in East Asia, or indeed in any other civilization.
According to scholar John North, “Unlike Platonic and Aristotelian thought, Chinese thought was not overtly philosophical, but rather, it was historical. Joseph Needham, a well-known authority on the history of science in China, has suggested that the reason for this is that Chinese religion had no lawgiver in human guise, so that the Chinese did not naturally think in terms of laws of nature.” Naturally occurring regularities and phenomena could be observed, of course, but the Chinese did not generally deduct universal mathematical laws from them, as “Many Chinese writers betrayed a belief that, while broad analogies are to be found in the world, reality is essentially too subtle to be encoded in general principles.”
Joel Mokyr cites the verdict of a ninth century Arab author that “the curious thing is that the Greeks are interested in theory but do not bother about practice, whereas the Chinese are very interested in practice and do not bother much about the theory.” Science and technology are today frequently seen as nearly synonymous terms, but the close connection between the two, where new technologies are often (though by no means always, even today) the result of scientific advances, is a modern, Western invention. The Chinese were traditionally far better at creating great, practical technology with a limited theoretical basis than they were at developing a mathematical-scientific world view, equivalent to what happened in Europe.
It is possible that this difference was partly rooted in cultural factors. According to Mokyr, in Europe, the physical world was viewed as orderly, whereas “the Chinese employ words like thien fa (laws of heaven), yet, as Needham insisted, these are laws without a lawgiver. In that sense, of course, the Chinese may have been closer to a twentieth century way of thinking about nature than to the thinking of Kepler and Newton.”
Muslims have a notion of a Creator God which is superficially similar to that of Jews and Christians, but upon closer inspection, the Allah of the Koran is not quite like the God of the Bible. Islam is a fatalistic religion where the emphasis is on the absolute sovereignty of Allah. Stanley Jaki, a priest and physicist, explains that it was the highly influential philosopher al-Ghazali in the eleventh century in particular who “denounced natural laws, the very objective of science, as a blasphemous constraint upon the free will of Allah.” Relatively early in its history, therefore, science in the Islamic world was deprived of the philosophical foundation it needed in order to flourish. Consequently, Professor Jaki observes, “the improvements brought by Muslim scientists to the Greek scientific corpus were never substantial.”
As Robert Spencer states in his book Religion of Peace?: Why Christianity Is and Islam Isn't,
“Muslims believe that Allah's hand is unfettered - he can do anything. The Qur'an explicitly refutes the Judeo-Christian view of God as a God of reason when it says: ‘The Jews say: Allah's hand is fettered. Their hands are fettered and they are accursed for saying so’ (5:64). In other words, it is heresy to say that God operates by certain natural laws that we can understand through reason. This argument was played out throughout Islamic history. Muslim theologians argued during the long controversy with the Mu'tazilite sect, which exalted human reason, that Allah was not bound to govern the universe according to consistent and observable laws. ‘He cannot be questioned concerning what He does’ (Qur’an 21:23). Accordingly, observations of the physical world had no value; there was no reason to expect that any pattern to its workings would be consistent, or even discernable. If Allah could not be counted on to be consistent, why waste time observing the order of things? It could change tomorrow.”
The leading medieval Jewish philosopher Moses Maimonides, who lived under Islamic rule and was well familiar with Islamic thinking, explains Islamic cosmology in this way: “Human intellect does not perceive any reason why a body should be in a certain place instead of being in another. In the same manner they say that reason admits the possibility that an existing being should be larger or smaller than it really is, or that it should be different in form and position from what it really is; e.g., a man might have the height of a mountain, might have several heads, and fly in the air; or an elephant might be as small as an insect, or an insect as huge as an elephant. This method of admitting possibilities is applied to the whole Universe. Whenever they affirm that a thing belongs to this class of admitted possibilities, they say that it can have this form and that it is also possible that it be found differently, and that the one form is not more possible than the other; but they do not ask whether the reality confirms their assumption....[They say] fire causes heat, water causes cold, in accordance with a certain habit; but it is logically not impossible that a deviation from this habit should occur, namely, that fire should cause cold, move downward, and still be fire; that the water should cause heat, move upward, and still be water. On this foundation their whole [intellectual] fabric is constructed.”
The American author and sociologist of religion Rodney Stark agrees that Islam does not have “a conception of God appropriate to underwrite the rise of science...Allah is not presented as a lawful creator but is conceived of as an extremely active God who intrudes in the world as he deems it appropriate. This prompted the formation of a major theological bloc within Islam that condemns all efforts to formulate natural laws as blasphemy in that they deny Allah's freedom to act.”
Rodney Stark puts great, perhaps too great, emphasis on the Christian commitment to reason in the historic development of capitalism and science in Europe. According to him, “Christian faith in reason was influenced by Greek philosophy. But the more important fact is that Greek philosophy had little impact on Greek religions. Those remained typical mystery cults, in which ambiguity and logical contradictions were taken as hallmarks of sacred origins. Similar assumptions concerning the fundamental inexplicability of the gods and the intellectual superiority of introspection dominated all of the other major world religions.”
While I personally think Stark focuses too much on a single causative factor, it is true that European scholars, starting with Scholastic natural philosophers during the Middle Ages, on the whole believed the world to be orderly. The Scientific Revolution was ignited, however, only when the organized body of scholars from the universities and eventually the scientific societies combined mathematics with a more systematic use of the experimental method.
The world view of senior European scientists during the early modern era could be described as “God meets geometry,” the idea that the universe could be described mathematically and rationally. Muslims shared the concept of a single Creator, but their version of God wasn’t helpful in this regard. The Koran is deeply inconsistent. The notion that Allah is incomprehensible and provides no correlation between cause and effect had a negative impact on the development of empirical sciences in the Islamic world. In contrast, for Jews and Christians, God has created the universe according to a certain logic, which can be described. Kepler firmly believed the Solar System was created according to God’s plan, which he attempted to unlock. Sir Isaac Newton was passionately interested in religion and wrote extensively about it. Even Albert Einstein, who was certainly not an orthodox, religious Jew, still retained some residue of the idea that the universe was created according to a logic which is, to a certain extent, comprehensible and accessible to human reason: “I believe in Spinoza’s God, who reveals Himself in the lawful harmony of the world, not in a God who concerns Himself with the fate and the doings of mankind.”
Andrew Robinson asks a timely question in his book The Story of Measurement: “What are we to make of the fact that nature, physical reality, can be explained in terms of mathematics formulated by human beings? Do numbers and their interrelationships have a real existence ‘out there’, independent of the mind, which we humans ‘discover’, or are they pure inventions of the mind, which we impose on reality?”
The Hungarian American Nobel laureate Eugene Paul Wigner (1902-1995) was born in Budapest into a family of non-practicing Jews. He studied together with John von Neumann in Hungary and later became a good friend of another Hungarian Jewish physicist, Leó Szilárd. Wigner attended colloquia at the University of Berlin with prominent scientists such as Albert Einstein, Max Planck, Werner Heisenberg, Max von Laue and Walther Nernst. He became David Hilbert’s assistant at Göttingen in 1927, but after the Nazis came to power in Germany he became a naturalized a citizen of the United States in 1937. His sister married the great English theoretical physicist Paul Dirac. Wigner worked on the Manhattan Project during WWII, and made substantial contributions to the development of quantum mechanics.
Wigner once gave a lecture entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences. According to him, “mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections….The enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and there is no rational explanation for it.”
As Andrew Robinson comments, “Ever since Galileo presented the numerical values he found in his experiments as proof that the laws of motion he had deduced were not his, but nature’s, leading scientists have pondered this profound question - without coming to any conclusion. The physicist Heinrich Hertz remarked that, ‘One cannot escape the feeling that these mathematical formulae have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.’ Einstein, epigrammatic as ever, said: ‘As far as the propositions of mathematics refer to reality, they are not certain; and so far as they are certain, they do not refer to reality.’ But later, he distinctly modified this view: ‘Experience remains, of course, the sole criterion of the physical utility of a mathematical construction. But the creative principle resides in mathematics. In a certain sense, therefore, I hold it true that pure thought can grasp reality, as the ancients dreamed.’”
The ancient Greeks made great advances for their time in science and in mathematics. The Romans after them contributed virtually nothing to science or to mathematics. The correlation between science and mathematics is apparently quite strong. Up until the Italian Renaissance, Europeans assimilated some external influences via the Middle East, prominent among them the Hindu numeral system with the zero which we use today as well as some advances in algebra made by Muslim scholars. However, after that, from about the fourteenth to the twentieth century, Europe outperformed all other civilizations in the world in mathematics. By that I don't just mean to say that Europeans outperformed all other cultures individually, but combined. It is possible to argue that European global leadership was stronger in mathematics than in any other scholarly discipline. Perhaps the simplest explanation for why the Scientific Revolution happened in Europe is because the language of nature is written in mathematics, as Galileo famously said, and Europeans did more than any other civilization to develop - or discover - the vocabulary of this language.
This brings us to the next question: Does mathematics have an independent existence in nature or does the human mind invent it? The answer potentially has huge philosophical implications. The people who created modern science lived predominantly in Europe, an overwhelmingly Christian continent with an important Jewish minority. They apparently had an advantage when they assumed the universe to be designed by a rational Creator. I admit this is a challenging dilemma for those of us who are not religious: Why can nature apparently be described mathematically and rationally if it has not been designed by a rational Creator? As a non-religious man, this is the only religious argument that I find difficult to answer.