Lecture 4: The Renaissance
Modern science, born in the Middle Ages, came to maturity in the Renaissance. The fundamental ideas due to Grosseteste, Buridan and Oresme, spread throughout Europe and profoundly influenced the succeeding centuries. Leonardo da Vinci was notable among those who gathered together the ideas of his predecessors, often without acknowledgement, and embodied them in his writings.
So far, the concepts describing motion, though sound, were largely qualitative. It was the great achievement of the Renaissance natural philosophers to make them more precise, to express them numerically, and to describe their temporal variation by means of differential equations. This was achieved through studies of the motions of the heavens, particularly of the moon and the planets. The first step was made by Copernicus (1473-1543) who boldly put the sun in the centre of the solar system, thus making possible a dynamical theory of planetary motions, and also stirring up a theological hornet's nest.
It is essential for the establishment of a scientific theory that its consequences agree with experimental measurements, and it was Tycho Brahe (1546-1601) who made very careful measurements of the motions of the planets that enabled Kepler (1571-1631) to establish the planetary orbits and his laws of planetary motion.
Meanwhile Galileo (1564-1642) was working on the fundamental dynamical concepts of mass, velocity, acceleration and momentum, expressing them in precise mathematical form and seeing how they were related for projectiles and falling bodies. He also plunged into the theological debate on the relation of the new scientific ideas to the teaching of Holy Scripture.
Finally Newton (1642-1727) put in place the cornerstone by postulating his laws of motion and developing the differential calculus. This, together with his theory of universal gravitation, enabled him to derive Kepler's laws and to calculate precisely the motions of the moon and of the planets. This achievement established modern science and gave it lasting prestige to the extent that it became the paradigm of intellectual accomplishment. It also led to the idea of the world as a vast clockwork that only needed to be set in motion by God and that thenceforth it would to go on for ever without further action by Him.
Since the earliest times man has observed the motions of the heavenly bodies and noted their regularities. The stars have the same uniform circular motion as if they are all fixed to a gigantic sphere that rotates once every day. There are exceptions to this: the planets, or wanderers, have more complicated motions relative to the background of the fixed stars, and the comets come and go at intervals. The sun and the moon have yearly and monthly motions that are still familiar even to modern city-dwellers.
The ancients from the Babylonians onwards were concerned mainly to describe these motions in a way that would permit accurate predictions. This was necessary to keep track of the seasons so that they would know when to plant their crops. It was natural to take the earth as the centre and to describe all the motions of the heavenly bodies as seen from the earth. Ptolemy in ancient times imagined the planets to be attached to spheres rolling on spheres so that their motions could be described by a complicated system of cycles and epicycles. These gave the motions of the planets quite well, but as the measurements of the motions became more accurate more and more epicycles had to be introduced to fit them. Furthermore, Ptolemy's system was purely a description, that gave no understanding of why the motions take place as they do.
Copernicus realised that it is altogether simpler to put the sun in the centre and to assume that the earth and planets move around it. He proposed this as a description of the solar system as it really is, and not just as a convenient calculational tool. This theory immediately encounters the difficulty that if the earth is moving the stars should appear differently with the seasons. The only way out of this, which later was proved to be correct, is to assume that the stars are at such immense distances away that the motion of the earth makes no appreciable difference to their appearances. There was however at that time no way of proving this, and so it remained a powerful argument against the heliocentric theory. There are other difficulties: if the earth is moving round the sun it must be moving very rapidly, so why do we not feel a strong wind, why does everything not fly off the earth, and why do not falling bodies fall behind?
Another type of difficulty is that a moving earth seems to be contrary to Holy Scripture. Thus in the book of Joshua we read that during a battle the sun stood still, which implies that normally the sun is moving and the earth is stationary. This difficulty can be avoided by saying that the Copernican scheme is only intended as a calculational convenience, and not as an account of reality. Although Copernicus himself regarded these motions as real, to avoid trouble the Lutheran theologian Osiander wrote an anonymous Preface to Copernicus' book saying that it was not an account of the reality of things but was only intended to facilitate practical calculations. This was readily accepted by scientists used to the old way of thinking and also as a way of avoiding trouble with the ecclesiastical authorities. Nevertheless it is profoundly unsatisfactory since if generally adopted it denies that science gives us real knowledge about the world. Scientists never really believe that, whatever they may say.
Osiander's attempt to avoid trouble failed. Luther himself indignantly rejected the idea that the sun is at rest, and Melanchthon, although appreciating the astronomical advantages of the Copernican system also argued against the earth's motion. It was therefore required that astronomical theories should not only agree with the facts, but also with the Scriptures taken literally. This was to lead to much trouble later on.
4.3. Tycho Brahe.
It very often happens that important scientific advances are made possible only by more accurate measurements. It is interesting to ask why scientists often devote their lives to increasing the precision of measurements. If we can understand what we know already, why go to the trouble? If the results agree with what we know, we have wasted our time, and if they do not, then we give ourselves a lot more trouble trying to understand them. The reason lies in the passion of the scientists to know more about the world, and this knowledge can only come from the comparison of our theories with the most accurate experimental data.
Tycho Brahe devoted thirty-five years of his life to the task of making the most accurate measurements of the positions of the stars and planets that were possible at the time, before the invention of the telescope. He erected huge instruments at his Observatory and made a large number of observations. These were inherited by Kepler who used them to obtain the orbits of the planets to unrivalled accuracy.
Brahe also proposed an alternative to the Copernican system in which the earth remained stationary while all the other heavenly bodies moved relative to it in such a way that the positions relative to the earth remained the same as in the Copernican system. He did this in order to maintain the agreement with observation and to avoid theological trouble. A third reason was the desire to avoid conflict with Aristotelian cosmology, which put a stationary earth in the centre. His system did not command general assent, and furthermore his discovery of a new star in 1572 was a serious blow to the Aristotelian system, which maintained the immutability of the heavenly bodies.
The life of Kepler shows clearly the importance of our fundamental beliefs on all our work. Kepler followed Plato in his appreciation of the value of numbers. He went so far as to say that our knowledge of numbers is of the same kind as God's. He also passionately believed in the order of the universe, and his life's work was to bring order, exact numerical order, into our knowledge of the heavens.
He worked with Tycho Brahe and soon realised the value of the immense treasure that Brahe had accumulated by his lifetime of measurements. He saw that he could use them to calculate the orbits of the planets, which of course, following Aristotle, he believed to be circular, as befits perfect incorruptible celestial matter. He tried to compute the orbit of Mars, and indeed found that it is nearly circular, but however hard he worked he could not make it fit Tycho's measurements. He found a circular orbit that agreed with the measurements to about ten minutes of arc, but not to about two, which was the accuracy of the measurements. Most people would have said that this was good enough, and gone on to do something else. But this was not good enough for Kepler, who believed that the fit must be exact, within the uncertainties of the measurements. So he toiled on for years and years, until he finally realised that he could never get a circle to fit; it must be some sort of oval.
Kepler was not concerned simply with geometry; he wanted to understand the origin of planetary motions. He asked himself: 'If the sun is indeed the origin and source of planetary motions, then how does this fact manifest itself in the motions of the planets themselves?' He found that Mars moves a little faster when it is nearer to the sun and a little slower when it is further from the sun. After five years work he discovered that the radius vector from the sun to the planet sweeps out equal areas in equal times.
This does not by itself determine the shape of the orbit, but after three more years work he found that the orbit is an ellipse. This was a watershed in the history of science. It finally destroyed Aristotelian physics and opened the way to Newton's theory of planetary orbits, which showed that they are indeed ellipses. This breakthrough was made possible by Kepler's belief in the order of nature, in a mathematically and numerically exact order, that followed from his belief in the rationality of the world created by God.
Kepler did not stop there. He continued his work on the planetary orbits, and established a relation between the period of the planet and its mean distance from the sun: the square of the former is proportional to the cube of the latter. These striking regularities strengthened Kepler's belief in the order in nature.
It took him over twenty years to establish his three laws by a most arduous analysis of Brahe's measurements. In his early work Mysterium Cosmographicum in 1596 he exclaimed: 'Oh! that we could see the day when both sets of figures agree with each other'. When he had discovered his third law, twenty-two years later, he was able to say in a reprint of his book: 'We have lived to see this day after 22 years and rejoice in it.'
Kepler thought about the dynamics of the solar system and was influenced by Gilbert's ideas on magnetism. He suggested that every material body is attracted to every other material body. He was however not able to specify the form of this attraction, or to use it to obtain his three laws of planetary motion. That was the achievement of Newton. Before that could be accomplished, however, dynamics had to be made a quantitative science, and this was the work of Galileo.
Despite the pioneering work of Buridan and Oresme, Copernicus and Kepler, the prevailing view of the world in Renaissance times was still that of Aristotle. It was still a world of purpose that can be understood by general philosophical reasoning. It is primarily due to the work of Galileo that this was replaced by the more modern view that the world is a mechanism that we can understand through the use of mathematics. He said, in effect, that purpose has no place in physics, and that bodies move following mathematical laws that we must discover by experiment. We must abandon the absurdly optimistic idea that we can somehow intuit the essences of things, and from that intuition deduce their properties. We no longer speculate about the causes of motion; instead we devote ourselves to the humble yet practicable task of describing motion as accurately as possible using the language of mathematics.
Kepler realised the importance of numerical accuracy in describing how the planets move, and established the laws of planetary motion. Galileo did the same for motions on the surface of the earth. He studied how balls roll down an inclined plane, and how projectiles move through the air, and was able to express his results in simple laws connecting positions, velocities and times.
The fundamental ideas of dynamics were established qualitatively by Buridan and his successors, and subsequently there was much discussion about the motions of falling bodies and of projectiles, in particular about the relationships between the distance fallen, the time taken and the velocity acquired; is there some simple relationship such as the velocity being proportional to the time taken or to the distance fallen? What are the trajectories of projectiles; are they simple geometrical curves? Concepts such as momentum and energy were only refined to their present precision by centuries of effort, and the first step was to answer these fundamental questions about motion.
Although his work had the effect of discrediting Aristotelian physics, Galileo remained under his influence in many ways. He failed to appreciate the importance of Kepler's work, and continued to believe that the orbits of the planets are circular.
Galileo realised the importance of accurate measurements, but was in a more difficult situation than Kepler. Long times, such as the periods of rotation of the planets, can be measured quite accurately by primitive means, but it is far more difficult to measure accurately the much shorter time taken by a body to fall a measured distance. Galileo used his own pulse to measure the period of swing of the lamp in the cathedral of Pisa, and found that it is independent of the amplitude. For a falling body a more accurate measure is needed, and he used a thin jet of water coming from a large jar, weighing the amount that came out during the fall. He further increased the accuracy by allowing the ball to roll down an inclined plane instead of falling freely, for then the time is much longer and so easier to measure.
By such measurements he showed that the velocity acquired is proportional to the time taken and that the distance traversed is proportional to the square of the time taken. This was established for balls rolling down a plane inclined at various angles, and by extrapolation he deduced that it also applies when the plane is vertical, corresponding to free fall. From his results he obtained a rough estimate of what we now call the acceleration due to gravity.
The work of Galileo shows very clearly many of the essential features of modern scientific investigation. He believed that the relationships between distance, time and velocity are expressible by simple formulae, and made careful experiments to see which possibility is correct. He knew that his measurements were subject to some experimental uncertainty, so that the results confirmed his relationships only approximately. He was able, like any good experimentalist, to see through his imperfect results to those corresponding to perfectly accurate measurements in a world where there is no friction, air resistance or other disturbing effects.
Galileo also studied the motion of projectiles, and found that their trajectories are parabolae, so that the range is a maximum when the angle of elevation of the gun is 45°. The famous story of his dropping two weights from the top of the leaning tower of Pisa is apocryphal.
As we saw with Kepler, the advance of science often depends on the precision of the measurements. Those of Brahe were as accurate as possible by direct sighting. The next step, the invention of the telescope, was due to Galileo. Lenses had been known for centuries and were used for spectacles and by Leewenhoek to look at very small things. Galileo realised that they could be put together and used to magnify the heavens. He made the first telescope, and immediately made a series of critical discoveries.
He saw the sunspots, an imperfection unexpected in a perfect Aristotelian sphere, and the mountains on the moon. He discovered several of the satellites of the planet Jupiter, and found that they revolved around it. This was just like a miniature solar system, and gave support to the Copemican idea of the heliocentric solar system. He saw the phases of Venus, showing that Venus moves on an orbit around the sun. He realised that if the earth is at rest, the celestial sphere must rotate with a high velocity, so that it is much easier to suppose that it is the earth that is rotating.
All these and many other arguments convinced Galileo that Copernicus was correct, and he vigorously propagated his beliefs. This inevitably brought him into conflict with the scientists of the time, namely the Aristotelian establishment. It is not easy for us to realise the strength and still more the psychologically compelling power of the Aristotelian system. It was a well thought-out, carefully integrated system that offered a plausible and detailed account of the whole of knowledge. Its concepts and world view had been used by the scholastic theologians in the construction of their vast and impressive theological systems. They were well-equipped to answer Galileo's arguments, which were persuasive rather than demonstrative. Galileo could not prove that the earth moves round the sun; he could only argue by analogy from what he considered reasonable. His argument from the tides, which he considered the strongest, is in fact false. There is also a very strong argument against the heliocentric system from the absence of stellar parallax, and to this Galileo could only reply that the stars must be very far away. It was not until 1725 that the discovery of stellar aberration by Bradley provided evidence for the heliocentric theory, and precise measurements by Bessel and Henderson in 1840 detected stellar parallax. It was unfortunate for Galileo that he did not invent the Foucault pendulum, which easily shows the rotation of the earth. Thus to the Aristotelians Galileo seemed to be advocating, on very shaky grounds, a revolution in thought that struck at the heart of a comprehensive world view that had stood for two thousand years. No wonder the Aristotelian establishment was outraged by his discoveries, and refused to look through his telescope.
Galileo was not only a great scientist, but also a brilliant writer and populariser. He wrote not only in Latin, but also in the vernacular language, so his popular writings had wide appeal. He was an enthusiastic advocate of the Copernican system, scorning the subterfuge of Osiander, and demolishing the arguments of his opponents with ill-concealed sarcasm.
The Aristotelian scientists realised that they could make trouble for Galileo by saying that the Copernican theory is contrary to Scripture. In the book of Joshua, for instance, it is said that during the battle the sun stood still, implying that normally it is in motion. Galileo was a pious Catholic, and did not believe that his work was in any way contrary to the teachings of the Church. He therefore, with his customary vigour, defended his work on theological grounds. He adopted the view, originally due to Augustine, that the Scriptures tell us how to go to heaven, not how the heavens go. Thus if there appears to be any disagreement between science and the scriptures, it must be either that the science is wrong or we have misinterpreted scripture. Convinced of the truth of the Copernican system, he therefore suggested that when the Bible speaks of the sun standing still this is just adopting our ordinary everyday language and is not intended to imply anything about the real nature of the motion of the earth in relation to the sun.
Although this is all perfectly reasonable, and indeed correct, the Church authorities were not used to being lectured to by a layman on theology. Furthermore, Galileo's ideas were certainly unsettling at a time of great theological upheaval. They knew that Galileo had not proved that the earth moves round the sun, so they suggested to Galileo that he refrain from further polemics until he had definite proof of the correctness of the Copernican system.
Although he had promised to remain quiet, Galileo caused further trouble a few years later by writing a dialogue between a defender and an opponent of the heliocentric system, putting some arguments due to the Pope in the mouth of the simple-minded opponent. Until that time, the Church had been very sympathetic to Galileo, and he was friendly with many high dignitaries including the Pope himself. But this new action could not be ignored. Galileo was therefore summoned before the tribunal, and required to renounce his errors and sentenced to detention in his own villa. This was a serious mistake that has cost the Church dearly, and has now been formally revoked.
In any discussion among physicists about the achievements of the great physicists of the past, there is never any question about who was the greatest. Although he modestly remarked that he was standing on the shoulders of giants, and that he was like a boy playing by the seashore, now and then picking up a smoother pebble or a prettier shell than ordinary, while the great ocean of truth lay undiscovered before him, a better summary of his achievements is to be found inscribed below his bust in Trinity College Cambridge: The marble image of a mind supreme, travelling through strange seas of thought alone.
His predecessors Copernicus, Brahe, Kepler and Galileo, were indeed giants, and they laid the secure foundations of modern physics. They established the laws of planetary motion and of motions on the earth, but there was no vision of the whole, no way of calculating all motions with precision. That was provided by Newton, and in so doing he founded modern science.
The essential nature of modern science, distinguishing it from the achievements of the past, is that it provides the laws governing the behaviour of matter and the way to apply these laws to calculate what will happen in any situation. Newton did all this for dynamics; it was left to others to do the same for electromagnetism and for atoms and nuclei.
The first step was to formulate the three laws of motion. They were not entirely new, in the sense that some aspects were glimpsed in the work of his predecessors, but he put them all together in concise form. The first, that every body continues in its state of rest or rectilinear motion until acted upon by a force, was already implicit qualitatively in the work of Buridan. The second law says that the acceleration of a body due to a force is proportional to the force divided by the mass of the body, and the third that to every action there is an equal and opposite reaction. To extract the momentous consequences of these apparently simple laws requires mathematical techniques that were not available. Newton provided them by inventing the differential calculus.
These calculations cannot be done unless we know the forces between bodies, expressed mathematically. Newton took the idea of Kepler and others that all masses attract each other, and postulated that the force between two masses is proportional to the product of their masses and inversely proportional to the square of the distance between them.
Taken together, these three achievements established the science of, dynamics. We are familiar with the story of the apple that gave him the idea that the apple falling to the earth follows the same law as the moon. Why do we say that the moon is falling towards the earth, when it remains, at a constant distance? This is because according to the First Law it naturally tends to go in a straight line. It is only the gravitational attraction of the earth that pulls it towards the earth so that its orbit is a circle around the earth. It is falling all the time towards the earth.
To check this idea, Newton calculated the rate of fall, but the result was only approximately correct. Since only exact agreement is acceptable he concluded that his idea was wrong, and put the calculations aside. Later on he read about a new determination of the radius of the earth, and realised that this might be the explanation of his previous result. He put the new number in his calculation, and it fitted perfectly within the uncertainties.
The importance of this result can hardly be over-estimated. Newton had shown that his dynamics applies equally to the motions of the celestial and terrestrial bodies. This is quite contrary to Aristotle, who held that they have quite different natures, the one being incorruptible and the other corruptible. It was the Christian doctrine of creation by God that destroyed this distinction, and thus made Newton's work possible.
Newton's First Law is also directly contrary to Aristotle, who required the continuing action of the mover to sustain motion. Newton's Law, however, says that a body continues in its state of motion or rest without the action of a mover. A force is required not to sustain motion, but to change it.
Newton was of a very retiring nature and seldom published his results, knowing very well that this would only involve him in tiresome controversies with stupid people. He heard speculations about the form of the law of force that would give the observed elliptical orbits, and was asked what he thought. He replied that he had shown that it is the inverse square law. Eventually he was persuaded to write up his work, and in eighteen months he wrote his Principia, the most influential book in the whole history of science. While many people have praised it, very few, even among scientists, have ever read that formidable tome. Quite recently, a very eminent theoretical physicist, Chandrasekhar, has worked carefully through the Principia, and he said that it is almost unbelievable that any man could have written so profound a book in such a short time.
With the work of Newton, modern science came to maturity. The laws of motion were given in the form of differential equations, and the mathematical techniques needed to solve them established. In any situation, it is simply necessary to specify the initial conditions and then obtain solutions of the equations of motion that satisfy these conditions. This gives the complete subsequent motion of the system for all time. Kepler's laws and Galileo's laws can be obtained in this way from Newton's laws of motion, together with the law of gravitational attraction that specifies the forces. This problem, that exercised the giants of physics for centuries, is now a relatively simple exercise for an undergraduate.
Newton's Principia is difficult partly because it is written using geometrical ideas, which were more familiar to his contemporaries. In the following centuries his work was extended and applied to a wide range of problems by Lagrange, Laplace and many other physicists.
This concept of strict laws of nature that could be used to predict all future behaviour had a great influence on contemporary thought. It makes the world into a great machine, created in the beginning and set into motion by God, but afterwards going on inexorably in accord with Newton's laws. After He has created the world, God can no longer have any influence on it; he has been rendered redundant. This philosophy is known as deism, and gives rise to many theological problems. It effectively denies the possibility of free actions, and therefore of responsibility and free will. We are all machines, automata that behave strictly in accord with Newton's laws. In the words of Whitehead, it is just the hurrying of material, endlessly, meaninglessly. As we shall see, this is an illegitimate extrapolation of Newton's work.
Newton realised these consequences of his work, in particular the implication that the world is eternal. This in turn suggested the possibility of other earths with human inhabitants. Knowing that Halley had run into trouble for apparently believing the eternity of the world, Newton emphasised the need for God to intervene periodically to correct His mechanism. He believed this to be necessary to maintain the planets in their orbits in spite of the perturbing influences of the other planets. Thus he was able to show that all remains in the hands of God. Subsequently, however, Laplace proved that the solar system is stable against such perturbations, so that the periodic intervention by God is unnecessary. The problem of God's action in a determined mechanistic world is still a live issue today.
4.7. The Scientific Attitude.
With the work of Newton, physics came to maturity and achieved a form that has remained the same ever since. This is therefore an appropriate point to describe the scientific attitude, and this can be done by quotations from scientists.
'Learn by trying to understand simple things in terms of other ideas, -- always honestly and directly. What keeps the clouds up, why can't I see stars in the daytime, why do colours appear in oily water, what makes the lines on the surface of water being poured from a pitcher, why does a hanging lamp swing back and forth -- and all the innumerable little things you see all around you. Then when you have learned what an explanation really is, you can then go on to more subtle questions.' (Feynman, 357).
The scientist is 'a permanent child, finding ever more ingenious ways of taking the world apart to see what was inside' (Feynman, 77).
'Science is a way to teach how something gets to be known, what is not known, to what extent things are known (for nothing is known absolutely), how to handle doubt and uncertainty, what the rules of evidence are, how to think about things so that judgment can be made, how to distinguish truth from fraud, and from show' (Feynman, 285).
'Our imagination is stretched to the utmost not, as in fiction, to imagine things which are not really there, but just to comprehend those things which are there' (Feynman, 325).
'If you are a scientist you believe that it is good to find out how the world works; that it is good to find out what the realities are; that it is good to turn over to mankind at large the greatest possible power to control the world . . . It is not possible to be a scientist unless you believe that the knowledge of the world, and the power which this gives, is a thing which is of intrinsic value to humanity and that you are using it to help the spread of knowledge, and are willing to take the consequences. (Oppenheimer, quoted Feynman, 209).
4.8. The Three Ages of Science.
The historical development of science can be divided into three stages in two different ways, within science itself and also against the background of the beliefs of society.
Within science, we can distinguish three distinct views of the world, as an organism, as a mechanism and as a mathematical formalism. The ancient Greeks, anxious to preserve human values, viewed the world as an organism, but this was a failure. In the Renaissance, when science came to maturity with Newton, the world became a mechanism, following precise mathematical equations. If we know the initial conditions, these enable us to calculate all subsequent motions, and the results are in precise agreement with our measurements. Finally, in the present century, mechanistic physics proved inadequate, and now for much of our work, we rely mainly on the mathematical equations, although physical understanding is still essential.
The relation between science and the society in which it lives also passes through three stages. In the first stage, most of the beliefs about the material world prevented or hindered the growth of science. Eventually science struggled into existence in the High Middle Ages when the Christian beliefs about the world provided the necessary conditions for its birth. It came to maturity in the Renaissance, and at that time most of the pioneer scientists were believing Christians who saw their work as showing forth the glory of God. At this point science became an autonomous self-sustaining enterprise that develops in accord with its own internal criteria, although it was still within a broadly Christian society. It needed the material support of society, and the relations between science and society were not always smooth. There have been from the beginning tensions between the scientists on the one hand and the theologians, the Church and the State authorities on the other, and among scientists between Christians and secularists. These can be a stimulating and healthy tensions, but they often degenerate into mutual incomprehension and hostility. Examples abound: Luther and Melanchthon attacked the Copernican theory, and Galileo was punished for his views. Nowadays the tension continues: many theologians are apprehensive of science, scientists like Hawking and Dawkins attack and pour scorn on theology, and there are continual battles with Government agencies to obtain sufficient support for scientific research.
In the present century we have moved into the third stage, when science more and more frequently finds itself in a society that is no longer broadly Christian, but indifferent or anti-Christian, either in its ideological roots or through the authority of the State. Science has spread rapidly over the world to non-Christian societies, as people become aware of its technological applications. But there is an important distinction; technology is easily exported and readily welcomed, but science is extremely difficult to export. Schools and universities all over the world teach science, but in most cases this has proved rather difficult, and it is not at all easy to establish really fruitful research programmes. Twice in Europe science has come under totalitarian regimes that have treated science as a slave or as a god, in Nazi Germany and in Soviet Russia. In both cases the effect on science has been disastrous, and science withered. Can science survive in a post-Christian society?
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