Two suggestions by Rota and Feynman on how to do science and math

Giancarlo Rota (1932-1999) was an Italian-born, then naturalized American, mathematician and philosopher. As happens to outstanding thinkers, in his late years he was often invited to give a speech on more or less everything he wanted. At the Rotafest organized in his honour at MIT he spelled out a list of suggestions for scientists1:

A speaker should try to give his audience something they can take home. But what? I have been collecting some random bits of advice that I keep repeating to myself, do’s and don’ts of which I have been and will always be guilty. […] The advice we give others is the advice that we ourselves need.

There would be a lot to say on both the content of those suggestions and wheter the same pieces of advice that are valid for masters are also of great interest for all scientists. But for a change, let me cut this short and just present the two among them that I found the most interesting (one of which is in fact by the physicist Richard Feynman), leaving to the readers all extra-thinking.

5. Every mathematician has only a few tricks. A long time ago an older and well known number theorist made some disparaging remarks about Paul Erdös’ work. You admire Erdös contributions to mathematics as much as I do, and I felt annoyed when  the older mathematician flatly and definitively stated that all of Erdös work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten.  But on reading the proofs of Hilbert’s striking and deep theorems in  invariant theory, it was surprising to verify that Hilbert’s proofs relied  on the same few tricks. Even Hilbert had only a few tricks!

7. Use the Feynman method. Richard Feynman was fond of giving the following advice on how to be a genius. You have to keep a dozen of your favorite problems constantly present in your mind, although by and large they will lay in a dormant state. Every time you hear or read a new trick or a new result, test it against each of your twelve problems to see whether it helps. Every once in a while there will be a hit, and people will say: “How did he do it? He must be a genius!”


1. This and following citations are taken from: Ten lessons I wish I had been thaught (presented at MIT on April 20, 1996, on the occasion of the Rotafest), in Giancarlo Rota, Indiscrete Thoughts,  Modern Birkhäuser Classics (1997), pages 195–203.

Why do we do Science? Galileo’s answer, according to Brecht

Why do we do Science? The following is an extract from Bertolt Brecht’s play Leben des Galilei (Life of Galileo), roughly from the 1940s. Galileo Galilei, after having discovered the Satellites of Jupiter, was forced to silence by an Inquisition decree, since his findings were against the official position of the Church. Here Galileo speaks with a young pupil, both a monk and a scientist, who believes that the decree is false but nevertheless has to be followed, since it aims at preserving the equilibrium in the life of the common people. Galileo’s answer is a condemnation of the raison d’État and a passionate apology of science, as both an instrument to improve people’s life conditions, and a compelling temptation to human curiosity. Apart from that, the whole extract is an excellent piece of play.

Young Monk: Mr. Galilei, I couldn’t sleep for three nights. I did not know how to reconcile the decree, that I read, with the satellites of Jupiter, that I saw. Hence, today I decided to come to you after saying Mass.

Galileo: What for? To tell me that Jupiter has no satellites?

Young Monk: No. That I succeeded in understanding the wiseness of the decree. It showed me the reason why research without boundaries is dangerous for humanity, thus I decided to quit astronomy. But I thought it was appropriate to explain you some reasons why even an astronomer could prevent himself from working on that theory. […]

Let me tell you about myself. I grew up in the countryside, son of peasants. They are simple people. They know everything about olive trees, but little about everything else. When I look at the phases of Venus, I see my parents in front of me, sitting with my sister next to the fireplace, eating their humble dinner. I can see the beam above their heads, that the smoke has turned black, and I can see their old workers hands, and their small spoons. They’re not happy, but even in their unhappiness there is a secret order, marked by cyclic events: the cleaning of the floor in the house, the seasons in the fields,  the tax collection. Adversities come upon them with regularity. The back of my father did not become crooked at once, but year after year by working in the fields, and the birth labours that turned my mother into an asexual being also came at regular intervals. They take the strength to climb a stony path despite being soaked with sweat, to give birth to children, and even to eat, from the feeling of Continuity and Necessity that is given to them by the land that each year becomes green again, by their small church and the readings of the Bible, which they listen to each Sunday. It is reassuring to them that the eye of God lies upon them, inquiring and almost anxious. That the theater of the world is built around them, so that each actor can properly play his role, whether big or small. What would they say, if I told them they just lie on a heap of rocks, that continuously rotates into the empty space like any other planet, just one among many, and not even so important! Then what would all their patience, their misery be for? And the Holy Scriptures – that declare as necessary all their sweat, their patience, their hunger, their submission, what are they good for, if they are discovered to be full of errors? I can see their gaze becoming gloomy, their spoon falling on the fireplace, I can see how they would feel betrayed and cheated. They would say: “So, there is no eye lying upon us, we alone have to take care of ourselves, ignorant, old and worn as we are? No one gave as a role to play, but the one to be miserable in this tiny planet, that is not even autonomous? There is no meaning for our misery, hunger is just not eating, and not a test of strength. Fatigue is just bending your back and carrying weights, not a credit.” Do you now understand why I see in the decree of the Holy Congregation an act of motherly charity, of  great goodness?

Galileo:  Goodness! Maybe you mean that, since there ‘s nothing left, the wine has been drunk and their lips are dry, all they are left with is kissing the soutane? But why is there nothing left? Why is the order of this nation just the order of an empty chest, and the only thing that is necessary is to work oneself to death? Between hills flourishing with grapes and luxuriant cornfields! Your peasants are paying for the wars that are conducted by the representatives of the mild Jesus in Spain and Germany. Do you know why is the Earth in the center of the Universe? So that the Chair of Peter could be! That’s what all is about! You are right, this has nothing to do with planets, it has to do with peasants. And please don’t come and tell me about the beauty of natural phenomena like oldness! Do you know how does the oyster produce the pearl?  Wounded by the presence of an external body which puts in danger its life, for instance a grain of sand, the oyster covers it with mother-of-pearl.  It almost dies during the process. The pearl can go the hell, I care more about the health of the oyster. Virtues do not only arise in misery, my dear. If your people were healthy and prosperous, they could develop the virtues of health and prosperity. But now the exhausting work in the fields just generates the virtues of exhaustingness, and I refuse it. My dear friend, my new water pumps can do more wonders than their absurd, superhuman work.  […]

But we cannot develop machines for pumping the water from rivers, if we cannot study the biggest machine that lies in front of us: the celestial bodies. The sum of the angles of a triangle cannot change according to the wishes of the Papal Court. And I cannot compute the trajectories of the flying bodies, so that also the flight of witches on broomsticks are explained.

Young Monk: And don’t you believe that the truth, if it is actually truth, will impose itself, even without us?

Galileo: No, no, no. The truth will impose itself, only if we will. The victory of the reason can only be the victory of those who reason. It sounds like if to you, the peasants are just like the musk on their huts! How can someone think that the sum of the angles of a triangle could be in conflict with their interests? But if they do not move up and learn to think, even the best irrigation system cannot help them. For God’s sake, I can see the divine patience of your people, but where is their divine fury?

Young Monk: They are tired.

Galileo (passing a manuscript to him): Son, aren’t you a physicist? Here are written the reason why the oceans have high and low tides. But you are not allowed to read it, do you understand me? Ah, you are already reading it? So, are you also a physicist? (The Young Monk is absorbed in the manuscript) An apple from the tree of  knowledge! He immediately bites it. He will be condemned forever, but he couldn’t help but bite it. Sometimes I think that I would be happy to be buried in a prison 20 meters under, where I could see no light, if I could know what light really is. And the worst is that what I know, I have to tell the others. Like a lover, or a drunkard, like a betrayer. It is a real vice, and it will ruin me. How much time will I resist speaking to walls, that’s the real question.

Young Monk (pointing at a line in the manuscript): There is a sentence I cannot understand.

Galileo: I will explain it to you, I will explain it to you.

On April, 25th, La7 is broadcasting the monologue ITIS Galileo by Marco Paolini in prime time. If you understand Italian, don’t miss it.

On The Copernican Revolution by T.S. Kuhn

I have always found hard to imagine how concepts and ideas we are now accustomed to, once were not part of everyone’s life. Some of those thoughts appear to me so central and grounding, that I believe their fall would imply the collapse of most of our everyday credo, and a world were they are absent would need to be built with totally different bricks. Those thoughts are not always related to science, or not only to exact science at least. For one thing, you could think about the Earth being round — ok, this is pretty much exact science. But also, say, the importance of subconscious in everyday behaviour. The late-roman, early-christian sense of pietas and respect for the sufferings of people.

The book The Copernican Revolution by T. S. Kuhn is the story of how an idea – one that today we could not even fantasize to be true, but back in the Middle Ages could not even be fantasized to be false – crumbled, and in a slow domino effect brought together with it the whole concept of the universe in its broadest sense. This idea placed a motionless Earth in the center of a huge rotating Sphere, where all stars were set, and which also sealed the (hence finite) Universe. The Sun itself lied on this Sphere, but, alone among the stars, was moving of a circular motion on the Sphere itself.

This post is devoted to present Kuhn’s book. After this short introduction, in the next three sections I summarize his main arguments, with few comments of mine (underlined). The last section is devoted to some other comments of mine.  (This is a good point for a huge disclaimer. I did not have any formal education in history or philosophy of science, and I have neither the time nor the patience to read sources. So I have to accept that all those sources say exactly what Kuhn claims they say, except for the extracts that actually appear in the text, and that I can hence judge myself.)

The “central Earth” model mentioned above (with minor and less minor modifications) has old roots, dating back to Anaximander (6th century BC) and having its most representative spokesman in Ptolemy (2nd century AD) and his treatise Almagesto (more details on it come in the next section). During the centuries, those roots interwove with everyday physics (to be read: physics dealing with events we see every day, like rocks being thrown and objects hitting each other), with philosophy, moral, and religion. Those ideas nourished each other, condensing in a coherent, solid backbone, which lied behind educated people everyday actions, as well as reflected the meaning they gave to life and Universe. As we shall see, replacing the central Earth universe with a new one required dissembling this infrastructure, and this turned out to be a quite hard and slow process.

The fundamentals of Ptolemy’s model

Back to Ptolemy’s model. So, Earth is motionless and lies in the center of the Universe, while all stars are in a huge rotating Sphere, far away from the Earth, which is called Celestial Sphere. As for predicting the apparent movement of the stars, Ptolemy’s model works pretty well – it is used even nowadays for navigating without GPRS. Today we know that those stars are actually far away from each other. But since they are far away also from Earth, a rough measure (the one that was possible back then) of their behaviour seem to imply that they all rotate at the same speed. As for the Sun, it cannot be imagined as rotating together with other stars – it is too close to the Earth when compared to them. Hence Ptolemy, while still setting it on the Celestial Sphere, endowed it with an extra revolution movement, which again roughly approximates its real behavior.

One immediately notes that this simple and quite elegant model completely ignores the planets, the reason being that their behaviour, even from a qualitative point of view, is way less predictable and regular than the one of the stars. For instance, an observer lying on the Earth usually sees each planet going east, but sometimes one of them happens to go west for a short period, restarting its normal motion towards east shortly afterwords (this is called the apparent retrograde motion).

The motion of Mars in 2003 as seen from the Earth (image taken from Wikipedia).

The motion of Mars in 2003 as seen from the Earth (image taken from Wikipedia).

Hence, early astronomers were challenged with finding a model that could be used to predict the behaviour of the planets. The Celestial Sphere being widely accepted as truthful, most systems tried to settle this open issue by inserting the planets in orbits around the Earth somewhere between the Earth itself and the external Sphere.

They all deviated from the tradition as little as possible, and mostly designed for the planets trajectories obtained as the composition of circles. One example is the following: draw a circular orbit O1 around the Earth. Now fix a point P on this orbit, and draw another circular orbit O2 around P, and put your planet, say Mars, somewhere on O2. Now let P move around O1, O2 move together with P, and Mars orbit on O2. For suitable rotational speeds of O1 and O2, an observer from the Earth will see Mars mostly going in one direction, and from time to time proceeding backwards, hence experiencing the apparent retrograde motion (O1 is called deferent, while O2 is known as epycicle – see the picture below).

The motion of a planet lying on an epicycle, and its apparent motion as seen from the Earth

Hence such a model can, at least qualitatively, predict the behaviour of a planet. For obtaining accurate quantitative predictions, one needs to complicate things a little bit – adding orbits O3, O4, etc. centered on points of O2, O3, etc., and using other tricks. Ptolemy was not the first to introduce such observations, but was the first that embedded them in a coherent framework, which provided much more accurate predictions than any other system before.

This is essentially the general model of Universe in which educated people lived, up to the end of the Middle Ages. But now that they had the astronomical infrastructure, they needed to fill it with a cosmology, that is, with a system of physical and metaphysical laws that reflected everyday experiences and the moral credo, and that moreover was coherent with Ptolemy’s model. The most complete and coherent “system” they had at hand was the one by Aristotle (4th century BC), which lived way before Ptolemy, but shared with him the general idea of a Celestial Sphere universe. Aristotle’s work dealt with optics, motion, biology, medicine, as well as with politics and ethics. As for religion, Scholars from the middle ages strove to find a coherence between Aristotle and Ptolemy’s universe, and the Catholic Dogma, mostly coming from the Bible and the early Christian thinkers. As an example Kuhn reports the following passage by Thomas Aquinas (Summa totius Theologiae), that tries to justify the following quotation from the Bible (Genesis, I.6): “And God said, Let there be a firmament in the midst of the waters, and let it divide the waters from the waters”:

As, however, this theory can be shown false by solid reasons, it cannot be held to be the sense of Holy Scripture. It should rather be considered that Moses was speaking to ignorant people, and that out of condescension to their weakness he put before them only such things as are apparent to sense. Now even the most uneducated can perceive by their senses that earth and water are corporeal, whereas it is not evident to all that air also is corporeal. […] Moses, then, while he expressly mentions water and earth, makes no express mention of air by name, to avoid setting something beyond their knowledge.

With modifications and “reinterpretations” like the one above, a rather coherent and self-contained cosmology was achieved. I will not even try to give details on how this complex interplay of science, moral, religion, and everyday experience realized. Instead, I will sketch a simplifying example, which I hope makes the point: humans are God’s beloved creation, and Earth was created for humans. That is why Earth is in the center of the universe, and that is why it is round, for round is the most perfect of shapes, and God is perfect. For symmetric reasons, every object is attracted to the center, hence a stone that I drop will fall towards the ground, since it is attracted to the center of the Earth, which is at the same time the center of the universe. God lives in the heavens, which are above us, hence the skies are made of a divine material, which is incorruptible, while things on Earth get old and tear apart, for we are sons of God but also sinners. That is why you and I grow old and die, while the Sun will keep appearing at the same, predictable time, in the same place, with the same shape.

The long life of Ptolemy’s Universe and its slow death

The years of decadence that followed the collapse of the Roman Empire dissipated most European knowledge, including the one on the works by Ptolemy and Aristotle. Together with others, those were to be slowly rediscovered in the XXII centuries, and immediately became an essential reading for any scientist of the period (even though the concept of scientist is much more recent — a term that better suits the period is natural philosopher). After a while, scholars became unsatisfied with the accuracy of its predictions and some theoretical incongruences of the model, and proposed minor modifications here and there to present better systems, with a serious general improvement nor of the model, neither of its accuracy. For a long time, no serious scholar made a real attempt to questionize the fundamentals of this universe – the central Earth. Let us present the main reasons for this.

First, the authority that Aristotle and Ptolemy had among the scholars was incommensurable. Ipse dixit (“He himself said so”) was a common way to back up an opinion by claiming that Aristotle himself expressed that idea. Centuries of decadence in science made scholars think of greek wisdom as a limitless source of knowledge, and they hardly dare to question the very base of their arguments. For a second reason, let us consider again the example at the end of the previous section. The cosmology that lies behind it is kept together by a tense equilibrium. If you remove one piece, then all the building starts trembling and may fall down. If Earth is not the center of the universe, does it mean that humans are by no means special to God ? And moreover, why do stones keep falling towards the ground ? A model willing to replace the Ptolemaic was confronted with many questions like those. This was clearly unfair, for the same question were answered in the Ptolemaic model after centuries of discussions, tries and errors, experiments, while a brand new system was required to have them ready altogether. Also, the equilibrium reached between the Holy Scripture and the scientific status-quo risked to be dismantled by any substantial modification of the system, and any reformer would have had the fierce opposition of the Church.

So many scholars, even criticizing the work by Aristotle and Ptolemy, did not dare to take the big leap and remove the Earth from the center of the Universe. A peculiar case is that of Nicola D’Oresme, who basically state what today we know as Galilean relativity, hence showing that in principle one could not determine if the Earth was moving or the Sun, but did not turn this possibility into reality.

Even though he was one of the most prominent astronomers of his era, Copernicus himself waited several decades before having his manifesto The Revolunitionibus Orbium Caelestium printed in 1543. His model put a motionless Sun at the center of the universe, with all the planets, including the Earth, orbiting in circular orbits around it. He kept the other stars in a circular sphere, which again was far beyond all planets and sealed the Universe, but he now assumed the sphere to be fixed. His model also provides a number of other minor modification to the state-of-the-art system, but none as substantial as the ones just mentioned, hence I will not write more about it.

Copernicus’ model, as appeared in his treatise De revolutionibus orbium coelestium (taken from Wikipedia)

One may ask what made Copernicus finally take the hazardous decision of divulgating his theory: he firmly believed that his system, if compared to Ptolemy’s, was simpler and more elegant. Several observations on the behaviour of the planets which were justified in Ptolemaic model using ad-hoc hypotheses, followed in Copernicus’ from the very general structure of the system. For one thing, the apparent retrograde motion of planets can be justified qualitatively without the use of deferents and epicycles, as it comes with the model (though when one needs precise quantitative predictions, one has to add epicycles and use similar tricks, making Copernicus’ system basically as complicated as Ptolemy’s). There is another, more philosophical reason: Copernicus was highly influenced by the neoplatonic doctrine which saw in the Sun the source of life and energy, and almost identified it with God. Take for instance the following extract by Liber de Sole by Marsilio Ficino, a Neoplatonist from Florence:

Nothing reveals the nature of the Good more fully than the light [of the Sun]. First, light is the most brilliant and clearest of sensible objects. Second, […]. Fourth, the heat which accompanies it fosters and nourishes all things is the universal generator and mover […] The Sun can signify God himself to you, and who shall dare to say the Sun is false!

To our eyes, this text has nothing to do with science. Surprisingly, Copernicus’ also proposed philosophical justifications of a similar kind for supporting many assumptions in his work. For instance, here is the one for the centrality of the Sun:

In the middle of all sits Sun enthroned. In this most beautiful temple could we place this luminary in any better position from which he call illuminated the whole at once ? He is rightly called the Lamp, the Mind, the Ruler of the Universe; Hermes Trismegistus names him the Visible God, Sophocles’ Electra calls him the All-seeing. So the Sun sits as upon a royal throne ruling his children the planets which circle around him.

Hence, the neoplatonic philosophy allowed Copernicus to consider a universe whose center was the Sun, providing the necessary philosophical background for a new cosmology. Such motivations sound extremely awkward to our present sensibility. Nowadays people working in science, even those that study the fundamental laws of nature, do not usually aim at embedding their theories in a more general philosophical setting. There are multiple reasons for this. Let me mention two: research is very specialized, and people working in a specific area do not have the interest, let alone the competence, to define a metaphysical framework that backs up their theories or experimental results. Moreover, we are now much more humble regarding the goal of our studies. We do not aim at describing reality itself, but just a (possibly accurate) model of it. Centuries of research showed us that each theory, even the most succesful (eg Newton’s mechanics), is doomed to be replaced at some point by a more advanced one, which dealt with a wider class of phenomena, or with the same phenomena in a more accurate way. Hence, there is little reason for finding a philosophical explanation for a theory, when the one thing we know about it is that it is false!

Even though most astronomers appreciated Copernicus’ work, for all the prejudices mentioned above and because Copernicus’ predictions were not much more precise than Ptolemy’s, the central Sun hypothesis was either ignored or considered to be a pure technical one, i.e. an assumption made for mathematical convenience and not representing the real behaviour of the universe. Hence, for most people working in the field, the Earth remains where the Bible and Ptolemy placed it, ie at the center of the universe. Needless to say, outside the scientific world, people hardly get to know Copernicus’ model. In the long run, this was an advantage: being not so considered, Copernicus’ idea was sheltered from the troublesome attentions of censorship.

As a beginner of a revolution, Copernicus was hardly revolutionary at all. Trained as a Catholic priest, he did move the Earth from the center of the Universe, but kept unchanged almost everything else: the Universe was still finite, still was it sealed by the huge Celestial Sphere. In a really medieval fashion, he felt he needed philosophical arguments (the neoplatonic importance of the Sun) to back up his theory. As we already mentioned, his predictions were not much better than Ptolemy’s. Still many sea changes were required to transform his universe into Newton’s.

The first one was provided by Kepler, who aimed at computing more accurate predictions of the orbits of the planet. Starting from Copernicus’ model (hence with the Sun at the center of the Universe), after extensive measure and failed experiments, he ceased to try to describe the orbits via composition of circles. Instead, he set each planet on an ellipsis, with the Sun on one of the foci. Also, he managed to fit experimental data with elliptic orbits by assuming that the imaginary line that connects a planet to the Sun sweeps out equal areas during equal intervals of time. (I am stopping here, for an introduction to Kepler’s laws on planetary motion is not important to us: the interested reader may consult e.g. Wikipedia). This brought him to extremely accurate predictions of the behaviour of the planets, thus spreading in the astronomic community a renewed interest in the hypothesis of the central Sun. In fact, even if describing all Kepler’s observations with a central Earth system was in principle possible, this model would have needed to be too complicated to be considered a serious alternative. Kepler knew that Copernicus did not have the braveness or the intuition to have the revolution he started completed. Copernicus still assumed that the Earth played a major role among all planets, and endowed it with features that other planets did not have (for instance, in Copernicus’ systems all planes of orbits intersected in the center of Earth’s orbit). One of Kepler’s main achievement was to understand that the Earth was just another planet, hence he deprived it of any special role. But Kepler still had some traits which clearly distinguish him from our concept of a modern scientist: for instance, he still needed to back up his theory with some neoplatonic conception of the Sun. For him, the Sun is

a fountain of light, rich in fruitful heat, most fair, limpid, and pure to the sight, […], called king of the planets for his motion, heart of the world for his power, its eye for his beauty, and which alone we should judge worthy of the Most High God […] Who would hesitate to confer the votes of the celestial motions on him who already has been administrating all other movements and changes by the benefit of the light which is entirely in his possession ? […] By the highest right we return to the Sun, who alone appears, by virtue of his dignity and power, suited for this motive duty and worthy to become the home of God himself, not to say the first mover.

He imagined that, because of this, the Sun was emanating a force called anima motrix that attracted planets towards it. When digging into some other aspects of his work that were to be soon forgotten, this neoplatonic background is even more evident: for instance, he related the size of the orbits to the platonic solids, and hence to the number of those the number of planets.

Kepler’s Platonic solids model of the solar system (taken from Wikipedia)

Kepler’s work marked a clear point for the central Earth model, but it was thanks to the experimental evidence and the communicative verve of Galileo that the astronomic battle was finally won. Galileo’s most important contributions were obtained pointing the telescope towards the stars. Looking at the Sun and at the Moon, he unveiled their irregular surfaces, thus demolishing the idea that the heavens were immutable. Looking at Jupiter, he discovered his four satellites, hence showing that there existed celestial bodies that were not orbiting around the Earth. (There are a lot of nice sources on Galileo’s life and work. One may look at this or, for a less scientific though enjoyable point of view, at this)

Building up a new universe

After Kepler and Galileo’s work, there was no doubt that from an astronomical point of view the Copernicus’ system (or, better said, its evolved version due to Kepler) was clearly better than Ptolemy’s. Thus, the only arguments against the central Earth hypothesis could be non-astronomical. Most of them regarded the coherence of the Universe, posing question similar to the ones mentioned in the previous section. The physics and the moral that were appropriate for Ptolemy’s universe were clearly unsatisfactory for the new one, and no one had good substitutes for those. Little by little, these questions were to be answered coherently with the new system. Contributions to this started with Kepler and Galileo, proceeded with Descartes and Hooke, and have their main representative in Newton. I will just focus on the work of the latter, as his answers are the most definitive and relevant.

The great intuition of Newton (and Hooke, that had it roughly at the same time) was to understand that the physical law that kept the Earth around the Sun was of the same kind of the one that make objects on Earth fall towards its surface. One may be surprised by the fact that an idea that no one had for thousands of years, came to mind to two people in the same years. This can be easily explained. So far, scientists started from the assumption that the heavens were of some different matter from the Earth, thus they have to be subjected to different rules. But Kepler had shown that Earth was just another planet, while Galileo had observed that the Moon and the Sun were as irregular as the Earth. Hence, there was no reason anymore for assuming different physics laws for what was on Earth and what was not. Following his qualitative intuition, Newton was also able to provide a mathematical formula showing how this attraction realizes (again, go to Wikipedia for details…). He exploited it to give a theoretical explanation of Kepler’s orbit, thus embedding the astronomic system in the physical setting natural philosophers were longing for. The new concept of gravity he introduced required some time to be accepted by his contemporaries (which dislike an “attraction” between objects that could not be explained through purely mechanics arguments), but then turned out to be of extraordinary importance. Newton himself tried hard to find a purely mechanical explanation to this law, but in the end he convinced himself (and his fellows natural philosophers) that the gravity was an intrinsic property of the objects, and that it could not be explained otherwise. This is another fundamental brick he lied for the development of modern science: the fact that every scientific hypothesis or deduction must be justified within experimental science itself, and one needs not to appeal to some external justification. Hence, if no scientific reason for the existence of gravity can be found, one should not look for a metaphysical one. The following passage from his “Principia Mathematica” clearly illustrate this:

I explained so far the phenomena happening in the sky and in the sea via gravity, but I never explained the cause of gravity. For this force arises uniquely from some cause, that penetrates till the center of the Sun and of planets […] His action extends everywhere till endless distances […] Actually, I was not able to deduce the grounds of those properties from phenomena yet, and I feign no hypothesis (“Hypothesis non fingo”). Any assumption that cannot be deduced from phenomena is to be called “hypothesis”; and in experimental philosophy there is no place for hypothesis, whether metaphysical, or physical, or on occult features, or mechanic. In this philosophy propositions must be deduced from phenomena, and made general by induction. […] It is sufficient that gravity exists de facto, and behaves according to the rules we exposed, and explained the movements of celestial bodies and of our sea.

Newton’s contribution brought the new universe to a higher level of self-coherence and conformity with natural phenomena than Ptolemy’s had ever been. Not only his results, but his very approach was an example for all scientists to come, who were to discover other fundamental laws that provided explanation for more and more events (this is a super-interesting topic, but definitively out of the scope of this post). Even the relativistic and quantum mechanics, that were to replace Newton’s in the XXth century, can be seen as a product of the more modern concept of science that Copernicus’ universe produced.

Some considerations

Kuhn wrote The Copernican Revolution starting from some material he produced for a course on history of Science at Harvard. Those lectures were intended for students in humanities with a general interest in science. This is clearly reflected by the style of the book: no assumption on the reader’s previous knowledge is made, every new concept is defined and explained, often with examples. Kuhn gently guides you through the technical parts of Copernicus and Ptolemy’s work (though he does not provide full details), and confines some more difficult parts to a technical appendix. Hence the book is really enjoyable whatever is your background. His elegant prose is coupled with a very scientific approach to the matter: his reasoning never lacks in coherence or asks the reader for a leap of faith. Not only it provides you with brilliant ideas and concepts, but the way you are introduced to those is very stimulating: if you dedicate to this book the time it deserves, you’ll be rewarded with tons of nice intuitions and a modified attitude towards science, whether you previously were into it or not. I would say that this book is even more enjoyable if you’re not into science, hence you’re not used to look at it as an integrating part of your life and society.

This book can be also interpreted as a prequel to Kuhn’s manifesto The structure of Scientific Revolutions, where he exposes his general theory on how science evolves (about which I will write some other time). In the short summary I gave of The Copernican Revolution, I did not mention the general observations Kuhn makes starting from the “case study” given by the Copernican Revolution. Though they are many, witty, and point the reader’s attention towards Change in Science, in particular how difficult thinking out of the box is. Even scientist that produced a new idea, often were too much soaked in the status quo to realize what the furthest consequences of their innovation could have been. I believe this is a lesson of incommensurable value for everyone working (not only) in science.