Review: Treatise on Light

Review: Treatise on Light
Treatise on Light (Illustrated Edition)

Treatise on Light by Christiaan Huygens

My rating: 4 of 5 stars

But sound, as I have said above, only travels 180 toises in the same time of one second: hence the velocity of light is more than six hundred thousand times greater than that of sound.

This little treatise is included in volume 34 of the Great Books of the Western World, which I used to read Newton’s Principia and his Opticks. In this edition the Treatise comes out to about 50 pages, so I decided it was worth combing through. Christiaan Huygens is one of the relatively lesser known figures of the scientific revolution. But even a brief acquaintance with his life and work is enough to convince one that he was a thinker of gigantic proportion, in a league with Descartes and Leibniz. His work in mechanics prefigured Newton’s laws, and his detailed understanding of the physics of pendulums (building from Galileo’s work) allowed him to invent the pendulum clock. His knowledge of optics also improved the technology of telescope lenses, which in turn allowed him to describe the rings of Saturn and discover the first of Saturn’s moons, Titan.

Apart from all this, Huygens was the progenitor of the wave theory of light. This is in contrast with the corpuscular theory of light (in which light is conceived of as little particles), put forward 14 years later in Isaac Newton’s Opticks. Newton’s theory quickly became more popular, partially because of its inherent strength, and partially because it was Isaac Newton who proposed it. But Huygens’s wave theory was revived and seemingly confirmed in the 19th century by Thomas Young and Augustin-Jean Fresnel.

Essentially, Huygens’s idea was to use sound as an analogy for light. Just as sound consists of longitudinal waves (vibrating in the direction they travel) propagated by air, so light must consist of much faster waves propagated by some other, finer medium, which Huygens calls the ether. He conceives of a luminous object, such as a burning coal, as emitting circular waves at every point in its surface, spreading in every direction throughout a space.

Like Newton, Huygens was aware of Ole Rømer’s calculation of the speed of light. It had long been debated whether light is instantaneous or merely moves very quickly. Aristotle rejected the second option, thinking it inconceivable that something could move so fast. Little progress had been made since then, because making a determination of light’s speed presents serious challenges: not only is light several orders of magnitude faster than anything in our experience, but since light is the fastest thing there is, and the bearer of our information, we have nothing to measure it against.

This changed once astronomers began measuring the movement of the Jovian moons. Specifically, the moon Io is eclipsed by Jupiter every 42.5 hours; but as Rømer measured this cycle at different points in the year, he noticed that it varied somewhat. Realizing that this likely wasn’t due to the moon’s orbit itself, he hypothesized that it was caused by the varying distance of Earth to Jupiter, and he used this as the basis for the first roughly accurate calculation of the speed of light. Newton and Huygens both accepted the principle and refined the results.

Huygens gets through his wave theory, reflection, and refraction fairly quickly; and in fact the bulk of this book is dedicated to an analysis of Icelandic spar—or, as Huygens calls it, “The Strange Refraction of Icelandic Crystal.” This is a type of crystal that is distinctive for its birefringence, which means that it refracts light of different polarizations at different angles, causing a kind of double image to appear through the crystal. Huygens delves into a detailed geometrical analysis of the crystal, which I admit I could not follow in the least; nevertheless, the defining property of polarization eludes him, since to understand it one must conceive of light as a transverse, not a longitudinal, wave (that is, unlike a sound wave, which cannot be polarized). In the end, he leaves this puzzling property of the crystal for future scientists, but not without laying the groundwork of observation and theory that we still rely upon.

All together, this little treatise is a deeply impressive work of science: combining sophisticated mathematical modeling with careful experimentation to reach surprising new conclusions. Huygens illustrates perfectly the rare mix of gifts that a scientist must have in order to be successful: a sharp logical mind, careful attention to detail, and a creative imagination. The world is full of those with only one or two of these qualities—brilliant mathematicians with no interest in the real world, obsessive recorders and cataloguers with no imagination, brilliant artists with no gift for logic—but it takes the combination to make a scientist of the caliber of Huygens.

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Review: A Very Short Introduction to Galileo

Review: A Very Short Introduction to Galileo
Galileo: A Very Short Introduction

Galileo: A Very Short Introduction by Stillman Drake

My rating: 4 of 5 stars

There is not a single effect in Nature, not even the least that exists, such that the most ingenious theorists can ever arrive at a complete understanding of it.

One of the most impressive aspects of the Very Short Introduction series is the range of creative freedom allowed to its writers. (Either that, or its flexibility in repurposing older writings; presumably a version of this book was published before the VSI series even got off the ground, since its author died in 1993.) This is a good example: For in lieu of an introduction, Stillman Drake, one of the leading scholars of the Italian scientist, has given us a novel analysis of Galileo’s trial by the Inquisition.

Admittedly, in order to contextualize the trial, Drake must cover all of Galileo’s life and thought. But Drake’s focus on the trial means that many things one would expect from an introduction—for example, an explanation of Galileo’s lasting contributions to science—are only touched upon, in order to make space for what Drake believed was the crux of the conflict: Galileo’s philosophy of science.

Galileo Galilei was tried in 1633 for failing to obey the church’s edict that forbade the adoption, defense, or teaching of the Copernican view. And it seems that he has been on trial ever since. The Catholic scientist’s battle with the Catholic Church has been transformed into the archetypical battle between religion and science, with Galileo bravely championing the independence of human reason from ancient dogma. This naturally elevated Galileo to the status of intellectual heroe; but more recently Galileo has been criticized for falling short of this ideal. Historian of science, Alexandre Kojève, famously claimed that Galileo hadn’t actually performed the experiments he cited as arguments, but that his new science was mainly based on thought experiments. And Arthur Koestler, in his popular history of astronomy, criticized Galileo for failing to incorporate Kepler’s new insights. Perhaps Galileo was not, after all, any better than the scholastics he criticized?

Drake has played a significant role in pushing back against these arguments. First, he used the newly discovered working papers of Galileo to demonstrate that, indeed, he had performed careful experiments in developing his new scheme of mechanics. Drake also points out that Galileo’s Dialogue Concerning the Two Chief World Systems was intended for popular audiences, and so it would be unreasonable to expect Galileo to incorporate Kepler’s elliptical orbits. Finally, Drake draws a hard line between Galileo’s science and the medieval theories of motion that have been said to presage Galileo’s theories. Those theories, he observes, were concerned with the metaphysical cause of motion; whereas Galileo abandoned the search for causes, and inaugurated the use of careful measurements and numerical predictions in science.

Thus, Drake argues that Galileo never saw himself as an enemy of the Church; to the contrary, he saw himself as fighting for its preservation. What Galileo opposed was the alignment of Church dogma with one very particular interpretation of scripture, which Galileo believed would put the church in danger of being discredited in the future. Galileo attributed this mistaken policy to a group of malicious professors of philosophy, who, in the attempt to buttress their outdated methods, used Biblical passages to make their views seem orthodox. This was historically new. Saint Augustine, for example, considered the opinions of natural philosophers entirely irrelevant to the truth of the Catholic faith, and left the matter to experts. It was only in Galileo’s day (during the Counter-Reformation) that scientific theories became a matter of official church policy.

Drake’s conclusion is that Galileo’s trial was not so much a conflict between science and religion (for the two had co-existed for many centuries), but between science and philosophy: the former concerned with measurement and prediction, the latter concerned with causes. And Drake notes that many contemporary criticisms of Galileo—leaving many loose-ends in his system, for example—mirror the contemporary criticisms of his work. The trial goes on.

Personally I found this book fascinating and extremely lucid. However, I am not sure it exactly fulfills its promise as an introduction to Galileo. I think that someone entirely new to Galileo’s work, or to the history and philosophy of science, may not get as much out of this work. Luckily, most of Galileo’s own writings (translated by Drake) are already very accessible and enjoyable.

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Review: Newton’s Principia

Review: Newton’s Principia
The Principia

The Principia by Isaac Newton

My rating: 5 of 5 stars

It is shown in the Scholium of Prop. 22, Book II, that at the height of 200 miles above the earth the air is more rare than it is at the surface of the earth in the ratio of 30 to 0.0000000000003998, or as 75,000,000,000,000 to 1, nearly.

Marking this book as “read” is as much an act of surrender as an accomplishment. Newton’s reputation for difficulty is well-deserved; this is not a reader-friendly book. Even those with a strong background in science and mathematics will, I suspect, need some aid. The historian of mathematics Colin Pask relied on several secondary sources to work his way through the Principia in order to write his excellent popular guide. (Texts by S. Chandrasekhar, J. Bruce Brackenridge, and Dana Densmore are among the more notable vade mecums for Newton’s proofs.) Gary Rubenstein, a math teacher, takes over an hour to explain a single one of Newton’s proofs in a series of videos (and he had to rely on Brackenridge to do so).

It is not that Newton’s ideas are inherently obscure—though mastering them is not easy—but that Newton’s presentation of his work is terse, dense, incomplete (from omitting steps), and at times cryptic. Part of this was a consequence of his personality: he was a reclusive man and was anxious to avoid public controversies. He says so much himself: In the introduction to Book III, Newton mentions that he had composed a popular version, but discarded it in order to “prevent the disputes” that would arise from a wide readership. Unsurprisingly, when you take material that is intrinsically complex and then render it opaque to the public, the result is not a book that anyone can casually pick up and understand.

The good news is that you do not have to. Newton himself did not advise readers, even mathematically skilled readers, to work their way through every problem. This would be enormously time-consuming. Indeed, Newton recommended his readers to peruse only the first few sections of Book I before moving on directly to Book III, leaving most of the book completely untouched. And this is not bad advice. As Ted said in his review, the average reader could gain much from this book by simply skipping the proofs and calculations, and stopping to read anything that looked interesting. And guides to the Principia are certainly not wanting. Besides the three mentioned above, there is the guide written by Newton scholar I. Bernard Cohen, published as a part of his translation. I initially tried to rely on this guide; but I found that, despite its interest, it is mainly geared towards historians of science; so I switched to Colin Pask’s Magnificent Principia, which does an excellent job in revealing the importance of Newton’s work to modern science.

So much for the book’s difficulty; on to the book itself.

Isaac Newton’s Philosophiæ Naturalis Principia Matematica is one of the most influential scientific works in history, rivaled only by Darwin’s On the Origin of Species. Quite simply, it set the groundwork for physics as we know it. The publication of the Principia, in 1687, completed the revolution in science that began with Copernicus’s publication of De revolutionibus orbium coelestium over one hundred years earlier. Copernicus deliberately modeled his work on Ptolemy’s Almagest, mirroring the structure and style of the Alexandrian Greek’s text. Yet it is Newton’s book that can most properly be compared to Ptolemy’s. For both the Englishman and the Greek used mathematical ingenuity to draw together the work of generations of illustrious predecessors into a single, grand, unified theory of the heavens.

The progression from Copernicus to Newton is a case study in the history of science. Copernicus realized that setting the earth in motion around the sun, rather than the reverse, would solve several puzzling features of the heavens—most conspicuously, why the orbits of the planets seem related to the sun’s movement. Yet Copernicus lacked the physics to explain how a movable earth was possible; in the Aristotelian physics that held sway, there was nothing to explain why people would not fly off of a rotating earth. Furthermore, Copernicus was held back by the mathematical prejudices of the day—namely, the belief in perfect circles.

Johannes Kepler made a great stride forward by replacing circles with ellipses; this led to the discovery of his three laws, whose strength finally made the Copernican system more efficient than its predecessor (which Copernicus’s own version was not). Yet Kepler was able to provide no account of the force that would lead to his elliptical orbits. He hypothesized a sort of magnetic force that would sweep the planets along from a rotating sun, but he could not show why such a force would cause such orbits. Galileo, meanwhile, set to work on the new physics. He showed that objects accelerate downward with a velocity proportional to the square of the distance; and he argued that different objects fall at different speeds due to air resistance, and that acceleration due to gravity would be the same for all objects in a vacuum. But Galileo had no thought of extending his new physics to the heavenly bodies.

By Newton’s day, the evidence against the old Ptolemaic system was overwhelming. Much of this was observational. Galileo observed craters and mountains on the moon; dark spots on the sun; the moons of Jupiter; and the phases of Venus. All of these data, in one way or another, contradicted the old Aristotelian cosmology and Ptolemaic astronomy. Tycho Brahe observed a new star in the sky (caused by a supernova) in 1572, which confuted the idea that the heavens were unchanging; and observations of Haley’s comet in 1682 confirmed that the comet was not somewhere in earth’s atmosphere, but in the supposedly unchanging heavens.

In short, the old system was becoming unsustainable; and yet, nobody could explain the mechanism of the new Copernican picture. The notion that the planets’ orbits were caused by an inverse-square law was suspected by many, including Edmond Haley, Christopher Wren, and Robert Hooke. But it took a mathematician of Newton’s caliber to prove it.

But before Newton published his Principia, another towering intellect put forward a new system of the world: René Descartes. Some thirty years before Newton’s masterpiece saw the light of day, Descartes published his Principia Philosophiæ. Here, Descartes summarized and systemized his skeptical philosophy. He also put forward a new mechanistic system of physics, in which the planets are borne along by cosmic vortexes that swirl around each other. Importantly, however, Descartes’s system was entirely qualitative; he provided no equations of motion.

Though Descartes’s hypothesis has no validity, it had a profound effect on Newton, as it provided him with a rival. The very title of Newton’s book seems to allude to Descartes’s: while the French philosopher provides principles, Newton provides mathematical principles—a crucial difference. Almost all of Newton’s Book II (on air resistance) can be seen as a detailed refutation of Descartes’s work; and Newton begins his famous General Scholium with the sentence: “The hypothesis of vortices is pressed with many difficulties.”

In order to secure his everlasting reputation, Newton had to do several things: First, to show that elliptical orbits, obeying Kepler’s law of equal areas in equal times, result from an inverse-square force. Next, to show that this force is proportional to the mass. Finally, to show that it is this very same force that causes terrestrial objects to fall to earth, obeying Galileo’s theorems. The result is Universal Gravity, a force that pervades the universe, causing the planets to rotate and apples to drop with the same mathematical certainty. This universal causation effectively completes the puzzle left by Copernicus: how the earth could rotate around the sun without everything flying off into space.

The Principia is in a league of its own because Newton does not simply do that, but so much more. The book is stuffed with brilliance; and it is exhausting even to list Newton’s accomplishments. Most obviously, there are Newton’s laws of motion, which are still taught to students all over the world. Newton provides the conceptual basis for the calculus; and though he does not explicitly use calculus in the book, a mathematically sophisticated reader could have surmised that Newton was using a new technique. Crucially, Newton derives Kepler’s three laws from his inverse-square law; and he proves that Kepler’s equation has no algebraic solution, and provides computational tools.

Considering the mass of the sun in comparison with the planets, Newton could have left his system as a series of two-body problems, with the sun determining the orbital motions of all the planets, and the planets determining the motions of their moons. This would have been reasonably accurate. But Newton realized that, if gravity is truly universal, all the planets must exert a force on one another; and this leads him to the invention of perturbation theory, which allows him, for example, to calculate the disturbance in Saturn’s orbit caused by proximity to Jupiter. While he is at it, Newton calculates the relative sizes and densities of the planets, as well as calculates where the center of gravity between the gas giants and the sun must lie. Newton also realized that gravitational effects of the sun and moon are what cause terrestrial tides, and calculated their relative effects (though, as Pask notes, Newton fudges some numbers).

Leaving little to posterity, Newton realized that the spinning of a planet would cause a distortion in its sphericity, making it marginally wider than it is tall. Newton then realized that this slight distortion would cause tidal locking in the case of the moon, which is why the same side of the moon always faces the earth. The slight deformity of the earth is also what causes the procession of the equinoxes (the very slow shift in the location of the equinoctial sunrises in relation to the zodiac). This shift was known at least since Ptolemy, who gave an estimate (too slow) of the rate of change, but was unable to provide any explanation for this phenomenon.

The evidence mustered against Descartes’s theory is formidable. Newton describes experiments in which he dropped pendulums in troughs of water, to test the effects of drag. He also performed experiments by dropping objects from the top of St. Paul’s Cathedral. What is more, Newton used mathematical arguments to show that objects rotating in a vortex obey a periodicity law that is proportional to the square of the distance, and not, as in Kepler’s Third Law, to the 3/2 power. Most convincing of all, Newton analyzes the motion of comets, showing that they would have to travel straight through several different vortices, in the direction contrary to the spinning fluid, in order to describe the orbits that we observe—a manifest absurdity. While he is on the subject of comets, Newton hypothesizes (correctly) that the tail of comets is caused by gas released in proximity to the sun; and he also hypothesizes (intriguingly) that this gas is what brings water to earth.

This is only the roughest of lists. Omitted, for example, are some of the mathematical advances Newton makes in the course of his argument. Even so, I think that the reader can appreciate the scope and depth of Newton’s accomplishment. As Pask notes, between the covers of a single book Newton presents work that, nowadays, would be spread out over hundreds of papers by thousands of authors. The result is a triumph of science. Newton not only solves the longstanding puzzle of the orbits of the planets, but shows how his theory unexpectedly accounts for a range of hitherto separate and inexplicable phenomena: the tides, the procession of the equinoxes, the orbit of the moon, the behavior of pendulums, the appearance of comets. In this Newton demonstrated what was to become the hallmark of modern science: to unify as many different phenomena as possible under a single explanatory scheme.

Besides setting the groundwork for dynamics, which would be developed and refined by Euler, d’Alembert, Lagrange, Laplace, and Hamilton in the coming generations, Newton also provides a model of science that remains inspiring to practitioners in any field. Newton himself attempts to enunciate his principles, in his famous Rules of Reasoning. Yet his emphasis on inductivism—generalizing from the data—does not do justice to the extraordinary amount of imagination required to frame suitable hypotheses. In any case, it is clear that Newton’s success was owed to the application of sophisticated mathematical models, carefully tested against collections of physical measurements, in order to unify the greatest possible number of phenomena. And this was to become a model for other intellectual disciples to aspire to, for good and for ill.

A striking consequence of this model is that its ultimate causal mechanism is a mathematical rule rather than a philosophical principle. The planets orbit the sun because of gravity, whose equations accurately predict their motions; but what gravity is, why it exists, and how it can affect distant objects, is left completely mysterious. This is the origin of Newton’s famous “I frame no hypothesis” comment, in which he explicitly restricts himself to the prediction of observable events rather than speculation on hidden causes (though he was not averse to speculation when the mood struck him). Depending on your point of view, this shift in emphasis either made science more rational or more superficial; but there is little doubt that it made science more effective.

Though this book is too often impenetrable, I still recommend that you give it a try. Few books are so exalting and so humbling. Here is on display the furthest reaches of the power of the human intellect to probe the universe we live in, and to find hidden regularities in the apparent chaos of experience.

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Review: Epitome of Copernican Astronomy & Harmonies of the World

Review: Epitome of Copernican Astronomy & Harmonies of the World

Epitome of Copernican Astronomy and Harmonies of the WorldEpitome of Copernican Astronomy and Harmonies of the World by Johannes Kepler

My rating: 3 of 5 stars

The Earth sings MI, FA, MI so that you may infer even from the syllables that in this our domicile MIsery and FAmine obtain.

Thomas Kuhn switched from studying physics to the history of science when, after teaching a course on outdated scientific models, he discovered that his notion of scientific progress was completely mistaken. As I plow through these old classics in my lackadaisical fashion, I am coming to the same conclusion. For I have discovered that the much-maligned Ptolemy produced a monument of observation and mathematical analysis, and that Copernicus’s revolutionary work relied heavily on this older model and was arguably less convincing. Now I discover that Johannes Kepler, one of the heroes of modern science, was also something of a crackpot.

The mythical image of the ideal scientist, patiently observing, cataloguing, calculating—a person solely concerned with the empirical facts—could not be further removed from Kepler. Few people in history had such a fecund and overactive imagination. Every new observation suggested a dozen theories to his feverish mind, not all of them testable. When Galileo published his Siderius Nuncius, for example, announcing the presence of moons orbiting Jupiter, Kepler immediately concluded that there must be life on Jupiter—and, why not, on all the other planets. Kepler even has a claim of being the first science-fiction writer, with his book Somnium, describing how the earth would appear to inhabitants of the moon (though Lucian of Samothrace, writing in the 2nd Century AD, seems to have priority with his fantastical novella, A True Story). This imaginative book, by the way, may have contributed to the accusations that Kepler’s mother was a witch.

In reading Kepler, I was constantly reminded of a remark by Bertrand Russell: “The first effect of emancipation from the Church was not to make men think rationally, but to open their minds to every sort of antique nonsense.” Similarly, the decline in Aristotle’s metaphysics did not prompt Kepler to reject metaphysical thinking altogether, but rather to speculate with wild abandon. But Kepler’s speculations differed from the ancients’ in two important respects: First, even when his theories are not testable, they are mathematical in nature. Gone are the verbal categories of Aristotle; and in comes the modern notion that nature is the manifestation of numerical harmonies. Second, whenever Kepler’s theories are testable, he tested them, and thoroughly. And he had ample data with which to test his speculations, since he was bequeathed the voluminous observations of his former mentor, Tycho Brahe.

At its worst, Kepler’s method resulted in meaningless numerical coincidences that explained nothing. As many a statistician has learned, if you crunch enough numbers and enough variables, you will eventually stumble upon a serendipitous correlation. This aptly describes Kepler’s use of the five Platonic solids to explain planetary orbits; by trying many combinations, Kepler found that he could create an arrangement of these regular solids, nested within one another, that mostly corresponded with the size of the planets’ orbits. But what does this explain? And how does this help calculation? The answer to both of these questions is negative; the solution merely appeals to Kepler’s sense of mathematical elegance, and reinforced his religious conviction that God must have arranged the world harmoniously.

Another famous example of this is Kepler’s notion of the “harmonies of the world.” By playing with the numbers of the perihelion, aphelion, orbital lengths, and so forth, Kepler assigns a melodic range to each of the planets. Mercury, having the most elongated orbit, has the biggest range; while Venus’s orbit, which most approximates a perfect circle, only produces a single note. Jupiter and Saturn are the basses, of course, while Mars is the tenor, Earth and Venus the altos, and Mercury the soprano. He then suggests (though vaguely) that there are beings on the sun, capable of sensing this heavenly music. (The composer Laurie Spiegel created a piece in which she recreates this music; it is not exactly Bach.) Once more, we naturally ask: What would all this speculation on music and harmonies explain? And once more, the answer is nothing.

Kepler’s writing is full of this sort of thing—torturous explorations of ratios, data, figures, which strike the modern mind as ravings rather than reasoning. But the fact remains that Kepler was one of the great scientific geniuses of history. He was writing in a sort of interim period between the fall of Aristotelian science and the rise of Newtonian physics, a time when the mind of Europe was completely untethered to any recognizable paradigm, free to luxuriate in speculation. Most people in such circumstances would produce nothing but nonsense; but Kepler managed to invent astrophysics.

What gives Kepler a claim to this title was his conception of a scientific law (though he did not put it as such). Astronomers from Ptolemy to Copernicus used schemes to predict planetary movements; but there was no one underlying principle which could explain everything. Kepler’s relentless search for numerical coincidences led him to statements that unified observations of all the planets. These are now known as Kepler’s Laws.

The first of these was the seemingly simple but revolutionary insight that planets orbit in ellipses, with the sun at one of the foci. It is commonly said that previous astronomers preferred circles for petty metaphysical reasons, seeing them as perfect. But there were other reasons, too. Most obviously, the mathematics of shapes inscribed in circles was well-understood; this was the basis of trigonometry.

Yet the use of circles to track orbits that, in reality, are not circular, created some problems. Thus in the Ptolemaic system the astronomer used one circle (the eccentric) for the distance, and another, overlapping circle (the equant) for the speed. When these were combined with the epicycles (used to explain retrogression) the resultant orbits, though composed of perfect circles, were anything but circular. Kepler’s use of ellipses obviated the need for all these circles, reducing a complicated machinery into a single shape. It was this innovation that made the Copernican system so much more efficient than the Ptolemaic one. As Owen Gingerich, a Copernican scholar, has said: “What passes today as the ‘Copernican System’ is in detail the Keplerian system.”

Yet the use of ellipses, by itself, would not have been so useful were it not for Kepler’s Second Law: that planets sweep out equal areas in equal times of their orbits. For when a planet is closest to the sun (at perihelion) it is moving its fastest; and when it is furthest (at aphelion) it is slowest; and this creates a constant ratio (which is the result of the conserved angular momentum of each planet). Ironically, of the two, Ptolemy was closer than Copernicus to this insight, since Ptolemy’s much-maligned equant (the imaginary point around which a planet travels at a constant speed) is a close approximation of the Second Law. Even so, I think that Kepler moved far beyond all previous astronomy with these insights, jumping from observed and analyzed regularities to general principles.

Kepler’s Third Law seemed to have excited the astronomer the most, since he even includes the exact date at which he made the realization: “… on the 8th of March in this year One Thousand Six Hundred and Eighteen but unfelicitously submitted to calculation and rejected as false, finally, summoned back on the 15th of May, with a fresh assault undertaken, outfought the darkness of my mind.” This law states that, for every planet, the ratio of the orbital period squared to the orbital size cubed, is constant. (For the orbital size Kepler used half the major axis of the ellipse.)

While it is no doubt striking that this ratio is almost the same for every planet (this is because the planet’s mass is negligible compared with the sun’s), it is difficult to completely sympathize with Kepler’s excitement, since the resultant law is not useful for predicting orbits, and its significance was only explained much later by Newton as a derivable conclusion from his equations. Kepler, being the man he was, used this mathematical constant to fuel his metaphysical speculations.

However much, then, that Kepler’s theories may strike us nowadays as baseless, crackpot theorizing, he must be given a commanding place in the history of science. The reason I cannot rate this collection any higher is that Kepler is extremely tiresome to read. In his more lucid moments, his imaginative energy is charming. But much of the book consists of whole paragraphs of ratio after ratio, shape after shape, number after number, and so it is easy to get lost or bored. Since I have a decent grasp of music theory, I thought I might be able to get something out of his Harmonies of the World, but I found even that section mostly opaque, swirling in obscure and impenetrable reasoning.

The great irony, then, is that Kepler’s writings can strike the modern-day reader as far less “scientific” than Ptolemy’s; but perhaps we should expect such ironies from a man who helped to inaugurate modern science, but who made his living casting horoscopes.

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Review: On the Revolutions of Heavenly Spheres

Review: On the Revolutions of Heavenly Spheres

On the Revolutions of Heavenly SpheresOn the Revolutions of Heavenly Spheres by Nicholas Copernicus

My rating: 4 of 5 stars

And though all these things are difficult, almost inconceivable, and quite contrary to the opinion of the multitude, nevertheless in what follows we will with God’s help make them clearer than day—at least for those who are not ignorant of the art of mathematics.

The Copernican Revolution has become the prime exemplar of all the great transformations in our knowledge of the world—a symbol of scientific advance, the paradigmatic clash of reason and religion, a shining illustration of how cold logic can beat out old prejudices. Yet reading this groundbreaking book immediately after attempting Ptolemy’s Almagest—the Bible of geocentric astronomy—reveals far more similarities than differences. Otto Neugebauer was correct in calling Copernicus’s system an ingenious modification of Hellenistic astronomy, for it must be read against the background of Ptolemy in order to grasp its significance.

The most famous section of De revolutionibus was, ironically, not even written by Copernicus, but by the presumptuous Andreas Osiander, a Lutheran theologian who was overseeing the publication of the book, and who included a short preface without consulting or informing Copernicus. Knowing that Copernicus’s hypothesis could prove controversial (Luther considered it heretical), Osiander attempted to minimize its danger by asserting that it was merely a way of calculating celestial positions and did not represent physical reality: “for it is not necessary that the hypotheses should be true, or even probable; but it is enough if they provide a calculate which fits the observations.”

Though this assertion obviously contradicts the body of the work (in which Copernicus argues at length for the reality of the earth’s movement), and though Copernicus and his friends were outraged by the insertion, it did help to shield the book from censure. And arguably Osiander was being a good and true Popperian—believing that science is concerned with making accurate predictions, not in giving us “the truth.” In any case, Osiander was no doubt correct in this assertion: “For it is sufficiently clear that this art is absolutely and profoundly ignorant of the causes of the apparent irregular movements.” Neither Ptolemy nor Copernicus had any coherent explanation of what caused the orbits of the planets, which would not come until Newton.

After this little interpolation, Copernicus himself wastes no time in proclaiming the mobility of the earth. In retrospect, it is remarkable that it took such a long stretch of history for the heliocentric idea to emerge. For it instantly explains many phenomena which, in the Ptolemaic system, are completely baffling. Why do the inner planets (Venus and Mercury) move within a fixed distance of the sun? Why does the perigee (the closest point in the orbit) of the outer planets (Mars, Jupiter, Saturn) occur when they are at opposition (i.e., when they are opposite in the sky from the sun), and why does their apogee (the farthest point) occur when they are in conjunction (when they are hidden behind the sun)? And why do the planets sometimes appear to move backwards relative to the fixed stars?

But putting the earth in orbit between Venus and Mars neatly and instantly explains all of these mysteries. Mercury and Venus always appear a fixed distance from the sun because they are orbiting within the earth’s orbital circle, and thus from our position appear to go back and forth around the sun. Mars, Jupiter, and Saturn, by contrast, can appear at any longitudinal distance from the sun because their orbits are outsider ours; but if Mars’ orbit were tracked from Jupiter, for example, it would, like Venus and Mercury, appear to go back and forth around the sun. Also note that Mars will appear to go “backwards” from earth when earth overtakes the red planet, due to our planet’s shorter orbital period. And since Mars will be closest when it is on the same side of the sun as earth (opposition from the sun), and furthest when it is on far side of the sun (conjunction with the sun), this also explains the apogee and perigee positions of the outer planets.

This allows Copernicus to collapse five circles—one for each of the planets, which were needed in the Ptolemaic system to account for these anomalies—into one circle: namely, the earth’s orbit. The advantages are palpable.

Nevertheless, while I think the benefits of putting the planets in orbit around the sun are obvious, perhaps even to a traditionalist, it is not obvious why Copernicus should put the earth in motion around the sun rather than the reverse. Indeed, this is exactly what the eminent astronomer Tycho Brahe did, several generations later. For it makes no observational difference whether the sun or the earth is in motion. And in the Aristotelian physics of the time, the former solution makes a great deal more sense, since the heavens were supposed to be constituted of lightest elements and the earth of the heaviest elements. So how could the heavy earth move so quickly? What is more, there is no concept of inertia in Aristotelian physics, and so no explanation for why people would not fly off the earth if it were in rapid motion.

Copernicus takes a brief stab at answering these obvious counterarguments, even offering a primitive notion of inertia: “As a matter of fact, when a ship floats on over a tranquil sea, all the things outside seem to the voyagers to be moving in a movement which is the image of their own, and they think on the contrary that they themselves and all the things with them are at rest.” Even so, it is obvious that such a brief example does not suffice to refute the entire Aristotelian system. Clearly, a whole new concept of physics was needed if the earth was to be in motion, one which did not arrive until Isaac Newton, born nearly two hundred years after Copernicus. It took a certain amount of boldness, or obtuseness, for Copernicus to proclaim the earth’s motion without at all being able to explain how the heaviest object in the universe—or so they believed—could hurtle through space.

In structure and content, De revolutionibus follows the Algamest pretty closely: beginning with mathematical preliminaries, onward to the orbits of the sun (or, in this case the earth), the moon, and the planets—with plenty of tables to aid calculation—as well as a description of his astronomical instruments and a chart of star locations, and finally ending with deviations in celestial latitude (how far the planets deviate north and south from the ecliptic in their orbits). Copernicus was even more wedded than Ptolemy to the belief that celestial objects travel in perfect circles, which leads him to repudiate Ptolemy’s use of the equant (the point around which a planet moves at a constant speed). The use of the equant upset Copernicus’s sense of elegance, you see, since its center is different from the actual orbit’s center, thus requiring two overlapping circles.

Copernicus’s own solution was an epicyclet, which revolves twice westward (clockwise, from the celestial north pole) for each rotation eastward on the deferent. And so, ironically, though Ptolemy is sometimes mocked for using epicycles, Copernicus followed the same path. I also find it amusing that the combined effect of these circular motions, in both Ptolemy and Copernicus, added up to a non-circular orbit; clearly nature had different notions of elegance than these astronomers. In any case, it would have to wait until Kepler until it was realized that the planets actually follow an ellipse.

Perhaps the greatest irony is that Copernicus’s book is not any easier to use than Ptolemy’s as a recipe book for planetary positions. Now, it is far beyond my powers to even attempt such a calculation. But in his Very Short Introduction to Copernicus (which I recommend), Owen Gingerich takes the reader through the steps to calculation the position of Mars on Copernicus’s birthday: February 19, 1473. To do this you needed the radix, which is a root position of the planet recorded at a specified time; and you also need the planet’s orbital speed (the time needed for one complete orbit, in this case 687 days). The year must be converted into sexigesimal (base 60) system, and then converted in elapsed Egyptian years (which lack a leap year), in order to calculate the time elapsed since the date of the radix’s position (in this case is January 1st, 1 AD). Then this sexigesimal number can be looked up in Copernicus’s tables; but this only gives us the location of Mars with respect to the sun. To find out where it will appear in the sky, we also need the location of earth, which is another tedious process. You get the idea.

I read the bulk of this book while I was on vacation in rural Canada. Faced with the choice between relaxation or self-torture, I naturally chose the latter. While most of my time was spent scratching my head and helplessly scratching the page with a pencil, the experience was enough to show me—as if I needed more demonstration after Ptolemy—that astronomy is not for the faint of heart, but requires intelligence, patience, and care.

There was one advantage to reading the book on vacation. For it is the only time of year when I am in a place without light pollution. The stars, normally hiding behind street lights and apartment buildings, shone in the hundreds. I would have seen even more were it not for the waxing moon. But this did give me the opportunity to get out an old telescope—bought as a birthday present for a cousin, over a decade ago—and examine the moon’s pitted surface. It is humbling to think that even such basic technology was years ahead of Copernicus’s time.

Looking at the brilliant grey circle, surrounded by a halo of white light, I felt connected to the generations of curious souls who looked at the same moon and the same stars, searching for answers. So Copernicus did not, in other words, entirely spoil my vacation.

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Review: Sidereal Messanger

Review: Sidereal Messanger

Sidereus Nuncius, or The Sidereal MessengerSidereus Nuncius, or The Sidereal Messenger by Galileo Galilei

My rating: 4 of 5 stars

A most excellent a kind service has been performed by those who defend from envy the great deeds of excellent men and have taken it upon themselves to preserve from oblivion and ruin names deserving of immortality.

This book (more of a pamphlet, really) is proof that you do not need to write many pages to make a lasting contribution to science. For it was in this little book that Galileo set forth his observations made through his newly improved telescope. In 50-odd pages, with some accompanying diagrams and etchings, Galileo quickly asserts the roughness of the Moon’s surface, avers the existence of many more stars than can be seen with the naked eye, and—the grand climax—announces the existence of the moons of Jupiter. Suddenly the universe seemed far bigger, and stranger, than it had before.

The actual text of Siderius Nuncius does not make for exciting reading. To establish his credibility, Galileo includes a blow-by-blow account of his observations of the moons of Jupiter, charting their nightly appearance. The section on our Moon is admittedly more compelling, as Galileo describes the irregularities he observed as the sun passed over its surface. Even so, this edition is immeasurably improved by the substantial commentary provided by Albert van Helden, who gives us the necessary historical background to understand why it was so controversial, and charts the aftermath of the publication.

Though Galileo is sometimes mistakenly credited with inventing the telescope, spyglasses were widely available at the time; what Galileo did was improve his telescope far beyond the magnification commonly available. The result was that, for a significant span of time, Galileo was the only person on the planet with the technology to closely and accurately observe the heavens. The advantage was not lost on him, and he made sure that he published before he got scooped. In another shrewd move, he named the newly-discovered moons of Jupiter after the Grand Duke Cosimo II and his brothers, for which they were known as the Medician Stars (back then, the term “star” meant any celestial object). This earned him patronage and protection.

Galileo’s findings were controversial because none of them aligned with the predictions of Aristotelian physics and Ptolemaic astronomy. According to the accepted view, the heavens were pure and incorruptible, devoid of change or imperfection. Thus it was jarring to find the moon’s surface bumpy, scarred, and mountainous, just like Earth’s. Even more troublesome were the Galilean moons. In the orthodox view the Earth was the only center of orbit; and one of the strongest objections against Copernicus’s system was that it included two centers, the Sun and the Earth (for the Moon). Galileo’s finding of an additional center of orbit meant that this objection ceased to carry any weight, since in any case we must posit multiple centers. Understandably there was a lot of skepticism at first, with some scholars doubting the efficacy of Galileo’s new instrument. But as other telescopes caught up with Galileo’s, and new anomalies were added to the mix—the phases of Venus and the odd shape of Saturn—his observations achieved widespread acceptance.

Though philosophers and historians of science often emphasize the advance of theory, I find this text a compelling example of the power of pure observation. For Galileo’s breakthrough relied, not on any new theory, but on new technology, extending the reach of his senses. He had no optical theory to guide him as he tinkered with his telescope, relying instead on simple trial-and-error. And though theory plays a role in any observation, some of Galileo’s findings—such as that the Milky Way is made of many small stars clustered together—are as close to simple acts of vision as possible. Even if Copernicus’s theory was not available as an alternative paradigm, it seems likely to me that advances in the power of telescopes would have thrown the old worldview into a crisis. This goes to show that observational technology is integral to scientific progress.

It is also curious to note the moral dimension of Galileo’s discovery. Now, the Ptolemaic system is commonly lambasted as narcissistically anthropocentric, placing humans at the center of it all. Yet it is worth pointing out that, in the Ptolemaic system, the heavens are regarded as pure and perfect, and everything below the moon as corruptible and imperfect (from which we get the term “sublunary”). Indeed, Dante placed the circles of paradise on the moon and the planets. So arguably, by making Earth the equal of the other planets, the new astronomy actually raised the dignity of our humble abode. In any case, I think that it is simplistic to characterize the switch from geocentricity to heliocentricity as a tale of declining hubris. The medieval Christians were hardly swollen with pride by their cosmic importance.

As you can see, this is a fascinating little volume that amply rewards the little time spent reading it. Van Helden has done a terrific job in making this scientific classic accessible.

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Review: Almagest

Review: Almagest

The Almagest: Introduction to the Mathematics of the HeavensThe Almagest: Introduction to the Mathematics of the Heavens by Ptolemy

… it is not fitting even to judge what is simple in itself in heavenly things on the basis of things that seem to be simple among us.

In my abysmal ignorance, I had for years assumed that tracking the orbits of the sun and planets would be straightforward. All you needed was a starting location, a direction, and the daily speed—and, with some simple arithmetic and a bit of graph paper, it would be clear as day. Attempting to read Ptolemy has revealed the magnitude of my error. Charting the heavenly bodies is a deviously complicated affair; and Ptolemy’s solution must rank as one of the greatest intellectual accomplishments of antiquity—fully comparable with the great scientific achievements of European Enlightenment. Indeed, Otto Neugebauer, the preeminent scholar of ancient astronomy, went so far as to say:

One can perfectly well understand the ‘Principia’ without much knowledge of earlier astronomy but one cannot read a single chapter in Copernicus or Kepler without a thorough knowledge of Ptolemy’s “Almagest”. Up to Newton all astronomy consists in modifications, however ingenious, of Hellenistic astronomy.

With more hope than sense, I cracked open my copy of The Great Books of the Western World, which has a full translation of the Almagest in the 16th volume. Immediately repulsed by the text, I then acquired a students’ edition of the book published by the Green Lion Press. This proved to be an excellent choice. Through introductions, preliminaries, footnotes, and appendices—not to mention generous omissions—this edition attempts to make Ptolemy accessible to a diligent college student. Even so, for someone with my background to attain a thorough knowledge of this text, he would still require months of dedicated study with a teacher as a guide. For the text is difficult in numerous ways.

Most obviously, this book is full of mathematical proofs and calculations, which are not exactly my strong suit. Ptolemy’s mathematical language—relying on the Greek geometrical method—will be unfamiliar to students who have not read some Euclid; and even if it is familiar, it proves cumbrous for the sorts of calculations demanded by the subject. To make matters worse, Ptolemy employs the sexagesimal system (based on multiples of 60) for fractions; so his numbers all must be converted into our decimals for calculation. What is more, even the names of the months Ptolemy uses are different, bearing their Egyptian names (Thoth, Phaöphi, Athur, etc.), since Ptolemy was an Alexandrian Greek. Yet even if we put these technical obstacles to the side, we are left with Ptolemy’s oddly infelicitous prose, which the translator describes thus:

In general, there is a sort of opacity, even awkwardness, to Ptolemy’s writing, especially when he is providing a larger frame for a topic or presenting a philosophical discussion.

Thus, even in the non-technical parts of the book, Ptolemy’s writing tends to be headache-inducing. All this combines for form an unremitting slog. So since my interest in this book was amateurish, I skimmed and skipped liberally. Yet this text is so rich that, even proceeding in such a dilettantish fashion, I managed to learn a great deal.

Ptolemy’s Almagest, like Euclid’s Elements, proved so comprehensive and conclusive when it was published that it rendered nearly all subsequent astronomical work obsolete or superfluous. For this reason, we know little about Ptolemy’s predecessors, since there was little point in preserving their work after Ptolemy summed it up in such magnificent fashion. As a result it is unclear how much of this book is original and how much is simply adapted. As Ptolemy himself admits, he owes a substantial debt to the astronomer Hipparchus, who lived around 200 years earlier. Yet it seems that Ptolemy originated the novel way of accounting for the planets’ position and speed, which he puts forth in later books.

Ptolemy begins by explaining the method by which he will measure chords; this leads him to construct one of the most precise trigonometric tables from antiquity. Later, Ptolemy goes on to produce several proofs of spherical trigonometry, which allows him to measure distances on the inside of a sphere, making this book an important source for Greek trigonometry as well as astronomy. Ptolemy also employs Menelaus’ Theorem, which also uses the fixed proportions of a triangle to establish ratios. From this I see that triangles are marvelously useful shapes, since they are the only shape which is rigid—that is, the angles cannot be altered without also changing the ratio of the sides, and vice versa. This is also, by the way, what makes triangles such strong structural components.

Ptolemy gets down to business in analyzing the sun’s motion. This is tricky for several reasons. For one, the sun does not travel parallel to the “fixed stars” (so called because the stars do not position change relative to one another), but rather at an angle, which Ptolemy calculates to be around 23 degrees. We now know this is due to earth’s axial tilt, but for Ptolemy is was the obliquity of the ecliptic. Also, the angle that the sun travels through the sky is determined by one’s latitude; this also determines the seasonal shifts in day-length; and during these shifts, the sun rises on different points on the horizon. To add to these already daunting variables, the sun also shifts in speed during the course of the year. And finally, Ptolemy had to factor in that the procession of the equinoxes—the ecliptic’s gradual westward motion from year to year.

The planets turn out to be even more complex. For they all exhibit anomalies in their orbits which entail further complications. Venus, for example, not only speeds up and slows down, but also seems to go forwards and backwards along its orbit. This leads Ptolemy to the adoption of epicylces—little circles which travel along the greater circle, called the “deferent,” of the planet’s orbit. But to preserve the circular motion of the deferent, Ptolemy must place the center (called the “eccentric”) away from earth. Then, Ptolemy introduces another imaginary circle, around which the planet travels with constant velocity: and the center of this is called the “equant.” Thus the planet’s motion was circular around one point (the eccentric) and constant around another (the equant), neither of which coincide with earth. In addition to all this, the orbit of Venus is not exactly parallel with the sun’s orbit, but tilted, and its tilt wobbles throughout the year. For Ptolemy to account for all this using only the most primitive instruments and without the use of calculus or analytic geometry is an extraordinary feat of patience, vision, and drudgery.

Even after writing all this, I am not giving a fair picture of the scope of Ptolemy’s achievement. This book also includes an extensive star catalogue, with the location and brightness of over one thousand stars. He argues strongly for earth’s sphericity and even offers a calculation of earth’s diameter (which was 28% too small). Ptolemy also calculates the distance from the earth to the moon, using the lunar parallax (the difference in the moon’s appearance when seen from different positions on earth), which comes out the quite accurate figure of 59 earth radii. And all of this is set forth in dry, sometimes baffling prose, accompanied by pages of proofs and tables. One can see why later generations of astronomers thought there was little to add to Ptolemy’s achievement, and why Arabic translators dubbed it “the greatest” (from which we get the English name).

A direct acquaintance with Ptolemy belies his popular image as a metaphysical pseudo-scientist, foolishly clinging to a geocentric model, using ad-hoc epicycles to account for deviations in his theories. To the contrary, Ptolemy scarcely ever touches on metaphysical or philosophical arguments, preferring to stay in the precise world of figures and proofs. And if science consists in predicting phenomena, then Ptolemy’s system was clearly the best scientific theory around for its range and accuracy. Indeed, a waggish philosopher might dismiss the whole question of whether the sun or the earth was at the “center” as entirely metaphysical (is it falsifiable?). Certainly it was not mere prejudice that kept Ptolemy’s system alive.

Admittedly, Ptolemy does occasionally include airy metaphysical statements:

We propose to demonstrate that, just as for the sun and moon, all the apparent anomalistic motions of the five planets are produced through uniform, circular motions; these are proper to the nature of what is divine, but foreign to disorder and variability.

Yet notions of perfection seem hard to justify, even within Ptolemy’s own theory. For the combined motion of the deferent and the epicycle do not make a circle, but a wavy shape called an epitrochoid. And the complex world of interlocking, overlapping, slanted circles—centered on imaginary points, riddled with deviations and anomalies—hardly fits the stereotypical image of an orderly Ptolemaic world.

It must be said that Ptolemy’s system, however comprehensive, does leave some questions tantalizingly unanswered. For example, why do Mercury and Venus stay within a definite distance from the sun, and travel along at the same average speed as the sun? And why are the anomalies of the “outer planets” (Mars, Jupiter, Saturn) sometimes related to the sun’s motion, and sometimes not? All this is very easy to explain in a heliocentric model, but rather baffling in a geocentric one; and Ptolemy does not even attempt an explanation. Even so, I think any reader of this volume must come to the conclusion that this is a massive achievement—and a lasting testament to the heights of brilliance and obscurity that a single mind can reach.

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Review: The New Organon

Review: The New Organon

The New OrganonThe New Organon by Francis Bacon

My rating: 4 of 5 stars

Since I’ve lately read Aristotle’s original, I thought I’d go ahead and read Bacon’s New Organon. The title more or less says it all. For this book is an attempt to recast the method of the sciences in a better mold. Whereas Aristotle spends pages and pages enumerating the various types of syllogisms, Bacon dismisses it all with one wave of the hand—away with such scholarly nonsense! Because Aristotle is so single-mindedly deductive, his scientific research came to naught; or, as Bacon puts it, “Aristotle, who made his natural philosophy a mere bond servant to his logic, thereby [rendered] it contentious and well-nigh useless.”

What is needed is not deduction—which draws trivial conclusions form absurd premises—but induction. More specifically, what is needed is a great deal of experiments, the results of which the careful scientist can sort into air-tight conclusions. Down with the syllogism; up with experiment. Down with the schoolmen; up with the scientists.

In my (admittedly snotty) review of Bacon’s Essays, I remarked that he would have done better to have written a work entirely in aphorisms. Little did I know that Bacon did just that, and it is this book. Whatever Bacon’s defects were as a politician or a philosopher, Bacon is the undisputed master of the pithy, punchy maxim. In fact, his writing style can be almost sickening, so dense is it with aphorism, so rich is it with metaphor, so replete is it with compressed thought.

In the first part of his New Organon all of the defects of Bacon’s style are absent, and all of his strengths are present in full force. Indeed, if this work consisted of only the first part, it would have merited five stars, for it is a tour de force. Bacon systematically goes through all of the errors the human mind is prone to when investigating nature, leaving no stone unturned and no vices unexamined, damning them all in epigram after epigram. The reader hardly has time to catch his breath from one astonishing insight, when Bacon is on to another.

Among these insights are, of course, Bacon’s famous four idols. We have the Idol of the Tribe, which consist of the errors humans are wont to make by virtue of their humanity. For our eyes, our ears, and our very minds distort reality in a systematic way—something earlier philosophers had, so far as I know, neglected to account for. We have then the Idols of the Cave, which are the foibles of the individual person, over and above the common limitations of our species. Of these may include certain pet theories, preferences, accidents of background, peculiarities of taste. And then finally we have the Idols of the Market Place, which are caused by the deceptive nature of language and words, as well as the Idols of the Theater, which consists of the various dogmas present in the universities and schools.

Bacon also displays a remarkable insight into psychology. He points out that humans are pattern-seeking animals, which leads us to sometimes see patterns which aren’t there: “The human understanding is of its own nature prone to suppose the existence of more order and regularity in the world than it finds.” Bacon also draws the distinction, made so memorable in Isaiah Berlin’s essay, between foxes and hedgehogs: “… some minds are stronger and apter to mark the differences of things, others to mark their resemblances.” Bacon also notes, in terms no psychologist could fault, a description of confirmation bias:

The human understanding when it has once adopted an opinion (either as being the received opinion or as being agreeable to itself) draws all things else to support and agree with it. And though there be a greater number and weight of instances to be found on the other side, or else by some distinction sets aside and rejects, in order that by this great and pernicious predetermination the authority of its former conclusions may remain inviolate.

Part two, on the other hand, is a tedious, rambling affair, which makes the patient reader almost forget the greatness of the first half. Here, Bacon moves on from condemning the errors of others to setting up his own system. In his opinion, scientific enquiry is a simple matter of tabulation: make a table of every situation in which a given phenomenon is always found, and then make a table of every situation in which a given phenomenon is never found; finally, make a table of every situation in which said phenomenon is sometimes found, shake well, and out comes your answer.

The modern reader will not recognize the scientific method in this process. For we now know that Bacon’s induction is not sufficient. (Though, he does use his method to draw an accurate conclusion about the nature of heat: “Heat is a motion, expansive, restrained, and acting in its strife upon the smaller particles of bodies.”) What Bacon describes is more or less what we’d now call ‘natural history’, a gathering up of facts and a noting of regularities. But the scientific method proper requires the framing of hypotheses. The hypothesis is key, because it determines what facts need to be collected, and what relationship those facts will have with the theory in question. Otherwise, the buzzing world of facts is too lush and fecund to tabulate; there are simply too many facts. Furthermore, Bacon makes the somewhat naïve—though excusable, I think—assumption that a fact is simply a fact, whereas we now know that facts are basically meaningless unless contextualized; and, in science, it is the theory in question which contextualizes said facts.

The importance of hypotheses also makes deduction far more important than Bacon acknowledges. For the aspiring experimentalist must often go through a long chain of deductive reasoning before he can determine what experiment should be performed in order to test a theory. In short, science relies on both deductive and inductive methods, and the relationship of theory to data is far more intertwined than Bacon apparently thinks. (As a side note, I’d also like to point out that Bacon wasn’t much of a scientist himself; he brings up the Copernican view of the heliocentric solar system many times, only to dismiss it as ridiculous, and also seems curiously unaware of the other scientific advances of his day.)

In a review of David Hume’s Enquiry Concerning the Principles of Morals, I somewhat impertinently remarked that the English love examples—or, to use a more English word, instances. I hope not to offend any English readers, but Bacon confirms me in this prejudice—for the vast bulk of this work is a tedious enumeration of twenty-seven (yes, that’s almost thirty) types of ‘instances’ to be found in nature. Needless to say, this long and dry list of the different sorts of instances makes for both dull reading and bad philosophy, for I doubt any scientist in the history of the world ever made progress by sorting his results into one of Bacon’s categories.

So the brilliant, brash, and brazen beginning of this book fizzles out into pedantry that, ironically enough, rivals even Aristotle’s original Organon. So, to repeat myself, the title of this book more or less says it all.

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Review: The Structure of Scientific Revolutions

Review: The Structure of Scientific Revolutions

The Structure of Scientific RevolutionsThe Structure of Scientific Revolutions by Thomas S. Kuhn

My rating: 5 of 5 stars

Observation and experience can and must drastically restrict the range of admissible scientific belief, else there would be no science. But they cannot alone determine a particular body of such belief. An apparently arbitrary element, compounded of personal and historical accident, is always a formative ingredient of the beliefs espoused by a given scientific community at a given time.

This is one of those wonderfully rich classics, touching on many disparate fields and putting forward ideas that have become permanent fixtures of our mental furniture. Kuhn synthesizes insights from history, sociology, psychology, and philosophy into a novel conception of science—one which, despite seemingly nobody agreeing with it, has become remarkably influential. Indeed, this book made such an impact that the contemporary reader may have difficulty seeing why it was so controversial in the first place.

Kuhn’s fundamental conception is of the paradigm. A paradigm is a research program that defines a discipline, perhaps briefly, perhaps for centuries. This is a not only a dominant theory, but a set of experimental methodologies, ontological commitments, and shared assumptions about standards of evidence and explanation. These paradigms usually trace their existence to a breakthrough work, such as Newton’s Principia or Lavoisier’s Elements; and they persist until the research program is thrown into crisis through stubborn anomalies (phenomena that cannot be accounted for within the theory). At this point a new paradigm may arise and replace the old one, such as the switch from Newton’s to Einstein’s system.

Though Kuhn is often spoken of as responding to Popper, I believe his book is really aimed at undermining the old positivistic conception of science: where science consists of a body of verified statements, and discoveries and innovations cause this body of statements to gradually grow. What this view leaves out is the interconnection and interdependence between these beliefs, and the reciprocal relationship between theory and observation. Our background orients our vision, telling us where to look and what to look for; and we naturally do our best to integrate a new phenomenon into our preexisting web of beliefs. Thus we may extend, refine, and elaborate our vision of the world without undermining any of our fundamental theories. This is what Kuhn describes as “normal science.”

During a period of “normal science” it may be true that scientific knowledge gradually accumulates. But when the dominant paradigm reaches a crisis, and the community finds itself unable to accommodate certain persistent observations, a new paradigm may take over. This cannot be described as a mere quantitative increase in knowledge, but is a qualitative shift in vision. New terms are introduced, older ones redefined; previous discoveries are reinterpreted and given a new meaning; and in general the web of connections between facts and theories is expanded and rearranged. This is Kuhn’s famous “paradigm shift.” And since the new paradigm so reorients our vision, it will be impossible to directly compare it with the older one; it will be as if practitioners from the two paradigms speak different languages or inhabit different worlds.

This scandalized some, and delighted others, and for the same reason: that Kuhn seemed to be arguing that scientific knowledge is socially solipsistic. That is to say that scientific “truth” was only true because it was given credence by the scientific community. Thus no paradigm can be said to be objectively “better” than another, and science cannot be said to really “advance.” Science was reduced to a series of fashionable ideas.

Scientists were understandably peeved by the notion, and social scientists concomitantly delighted, since it meant their discipline was at the crux of scientific knowledge. But Kuhn repeatedly denied being a relativist, and I think the text bears him out. It must be said, however, that Kuhn does not guard against this relativistic interpretation of his work as much as, in retrospect, he should have. I believe this was because Kuhn’s primary aim was to undermine the positivistic, gradualist account of science—which was fairly universally held in the past—and not to replace it with a fully worked-out theory of scientific progress himself. (And this is ironic since Kuhn himself argues that an old paradigm is never abandoned until a new paradigm takes its place.)

Though Kuhn does say a good deal about this, I think he could have emphasized more strongly the ways that paradigms contribute positively to reliable scientific knowledge. For we simply cannot look on the world as neutral observers; and even if we could, we would not be any the wiser for it. The very process of learning involves limiting possibilities. This is literally what happens to our brains as we grow up: the confused mass of neural connections is pruned, leaving only the ones which have proven useful in our environment. If our brains did not quickly and efficiently analyze environmental stimuli into familiar categories, we could hardly survive a day. The world would be a swirling, jumbled chaos.

Reducing ambiguities is so important to our survival that I think one of the primary functions of human culture is to further eliminate possibilities. For humans, being born with considerable behavioral flexibility, must learn to become inflexible, so to speak, in order to live effectively in a group. All communication presupposes a large degree of agreement within members of a community; and since we are born lacking this, we must be taught fairly rigid sets of assumptions in order to create the necessary accord. In science this process is performed in a much more formalized way, but nevertheless its end is the same: to allow communication and cooperation via a shared language and a shared view of the world.

Yet this is no argument for epistemological relativism, any more than the existence of incompatible moral systems is an argument for moral relativism. While people commonly call themselves cultural relativists when it comes to morals, few people are really willing to argue that, say, unprovoked violence is morally praiseworthy in certain situations. What people mean by calling themselves relativists is that they are pluralists: they acknowledge that incompatible social arrangements can nevertheless be equally ethical. Whether a society has private property or holds everything in common, whether it is monogamous or polygamous, whether burping is considered polite or rude—these may vary, and yet create coherent, mutually incompatible, ethical systems. Furthermore, acknowledging the possibility of equally valid ethical systems also does not rule out the possibility of moral progress, as any given ethical system may contain flaws (such as refusing to respect certain categories of people) that can be corrected over time.

I believe that Kuhn would argue that scientific cultures may be thought of in the same pluralistic way: paradigms can be improved, and incompatible paradigms can nevertheless both have some validity. Acknowledging this does not force one to abandon the concept of “knowledge,” any more than acknowledging cultural differences in etiquette forces one to abandon the concept of “politeness.”

Thus accepting Kuhn’s position does not force one to embrace epistemological relativism—or, at least not the strong variety, which reduces knowledge merely to widespread belief. I would go further, and argue that Kuhn’s account of science—or at least elements of his account—can be made to articulate even with the system of his reputed nemesis, Karl Popper. For both conceptions have the scientist beginning, not with observations and facts, but with certain arbitrary assumptions and expectations. This may sound unpromising; but these assumptions and expectations, by orienting our vision, allow us to realize when we are mistaken, and to revise our theories. The Baconian inductivist or the logical positivist, by beginning with an raw mass of data, has little idea how to make sense of it and thus no basis upon which to judge whether an observation is anomalous or not.

This is not where the resemblance ends. According to both Kuhn and Popper (though the former is describing while the second is prescribing), when we are revising our theories we should if possible modify or discard the least fundamental part, while leaving the underlying paradigm unchanged. This is Kuhn’s “normal science.” So when irregularities were observed in Uranus’ orbit, the scientists could have either discarded Newton’s theories (fundamental to the discipline) or the theory that Uranus was the furthest planet in the solar system (a superficial fact); obviously the latter was preferable, and this led to the discovery of Neptune. Science could not survive if scientists too willingly overturned the discoveries and theories of their discipline. A certain amount of stubbornness is a virtue in learning.

Obviously, the two thinkers also disagree about much. One issue is whether two paradigms can be directly compared or definitively tested. Popper envisions conclusive experiments whose outcome can unambiguously decide whether one paradigm or another is to be preferred. There are some difficulties to this view, however, which Kuhn points out. One is that different paradigms may attach very different importance to certain phenomena. Thus for Galileo (to use Kuhn’s example) a pendulum is a prime exemplar of motion, while to an Aristotelian a pendulum is a highly complex secondary phenomenon, unfit to demonstrate the fundamental properties of motion. Another difficulty in comparing theories is that terms may be defined differently. Einstein said that massive objects bend space, but Newtonian space is not a thing at all and so cannot be bent.

Granting the difficulties of comparing different paradigms, I nevertheless think that Kuhn is mistaken in his insistence that they are as separate as two languages. I believe his argument rests, in part, on his conceiving of a paradigm as beginning with definitions of fundamental terms (such as “space” or “time”) which are circular (such as “time is that measured by clocks,” etc.); so that comparing two paradigms would be like comparing Euclidian and non-Euclidian geometry to see which is more “true,” though both are equally true to their own axioms (while mutually incompatible). Yet such terms in science do not merely define, but denote phenomena in our experience. Thus (to continue the example) while Euclidian and non-Euclidian geometries may both be equally valid according to their premises, they may not be equally valid according to how they describe our experience.

Kuhn’s response to this would be, I believe, that we cannot have neutral experiences, but all our observations are already theory-laden. While this is true, it is also true that theory does not totally determine our vision; and clever experimenters can often, I believe, devise tests that can differentiate between paradigms to most practitioners’ satisfaction. Nevertheless, as both Kuhn and Popper would admit, the decision to abandon one theory for another can never be a wholly rational affair, since there is no way of telling whether the old paradigm could, with sufficient ingenuity, be made to accommodate the anomalous data; and in any case a strange phenomena can always be tabled as a perplexing but unimportant deviation for future researchers to tackle. This is how an Aristotelian would view Galileo’s pendulum, I believe.

Yet this fact—that there can be no objective, fool-proof criteria for switching paradigms—is no reason to despair. We are not prophets; every decision we take involves risk that it will not pan out; and in this respect science is no different. What makes science special is not that it is purely rational or wholly objective, but that our guesses are systematically checked against our experience and debated within a community of dedicated inquirers. All knowledge contains an imaginative and thus an arbitrary element; but this does not mean that anything goes. To use a comparison, a painter working on a portrait will have to make innumerable little decisions during her work; and yet—provided the painter is working within a tradition that values literal realism—her work will be judged, not for the taste displayed, but for the perceived accuracy. Just so, science is not different from other cultural realms in lacking arbitrary elements, but in the shared values that determine how the final result is judged.

I think that Kuhn would assent to this; and I think it was only the widespread belief that science was as objective, asocial, and unimaginative as a camera taking a photograph that led him to emphasize the social and arbitrary aspects of science so strongly. This is why, contrary to his expectations, so many people read his work as advocating total relativism.

It should be said, however, that Kuhn’s position does alter how we normally think of “truth.” In this I also find him strikingly close to his reputed nemesis, Popper. For here is the Austrian philosopher on the quest for truth:

Science never pursues the illusory aim of making its answers final, or even probable. Its advance is, rather, towards the infinite yet attainable aim of ever discovering new, deeper, and more general problems, and of subjecting its ever tentative answers to ever renewed and ever more rigorous tests.

And here is what his American counterpart has to say:

Later scientific theories are better than earlier ones for solving puzzles in the often quite different environments to which they are applied. That is not a relativist’s position, and it displays the sense in which I am a convinced believer in scientific progress.

Here is another juxtaposition. Popper says:

Science is not a system of certain, or well-established, statements; nor is it a system which steadily advances towards a state of finality. Our science is not knowledge (episteme): it can never claim to have attained truth, or even a substitute for it, such as probability. … We do not know: we can only guess. And our guesses are guided by the unscientific, the metaphysical (though biologically explicable) faith in laws, in regularities which we can uncover—discover.

And Kuhn:

One often hears that successive theories grow ever closer to, or approximate more and more closely to, the truth… Perhaps there is some other way of salvaging the notion of ‘truth’ for application to whole theories, but this one will not do. There is, I think, no theory-independent way to reconstruct phrases like ‘really there’; the notion of a match between the ontology of a theory and its ‘real’ counterpart in nature now seems to me illusive in principle.

Though there are important differences, to me it is striking how similar their accounts of scientific progress are: the ever-increasing expansion of problems, or puzzles, that the scientist may investigate. And both thinkers are careful to point out that this expansion cannot be understood as an approach towards an ultimate “true” explanation of everything, and I think their reasons for saying so are related. For since Popper begins with theories, and Kuhn with paradigms—both of which stem from the imagination of scientists—their accounts of knowledge can never be wholly “objective,” but must contain an aforementioned arbitrary element. This necessarily leaves open the possibility that an incompatible theory may yet do an equal or better job in making sense of an observation, or that a heretofore undiscovered phenomenon may violate the theory. And this being so, we can never say that we have reached an “ultimate” explanation, where our theory can be taken as a perfect mirror of reality.

I do not think this notion jeopardizes the scientific enterprise. To the contrary, I think that science is distinguished from older, metaphysical sorts of enquiry in that it is always open-ended, and makes no claim to possessing absolute “truth.” It is this very corrigibility of science that is its strength.

This review has already gone on for far too long, and much of it has been spent riding my own hobby-horse without evaluating the book. Yet I think it is a testament to Kuhn’s work that it is still so rich and suggestive, even after many of its insights have been absorbed into the culture. Though I have tried to defend Kuhn from accusations of relativism or undermining science, anyone must admit that this book has many flaws. One is Kuhn’s firm line between “normal” science and paradigm shifts. In his model, the first consists of mere puzzle-solving while the second involves a radical break with the past. But I think experience does not bear out this hard dichotomy; discoveries and innovations may be revolutionary to different degrees, which I think undermines Kuhn’s picture of science evolving as a punctuated equilibrium.

Another weakness of Kuhn’s work is that it does not do justice to the way that empirical discoveries may cause unanticipated theoretical revolutions. In his model, major theoretical innovations are the products of brilliant practitioners who see the field in a new way. But this does not accurately describe what happened when, say, DNA was discovered. Watson and Crick worked within the known chemical paradigm, and operated like proper Popperians in brainstorming and eliminating possibilities based on the evidence. And yet the discovery of DNA’s double helix, while not overturning any major theoretical paradigms, nevertheless had such far-reaching implications that it caused a revolution in the field. Kuhn has little to say about events like this, which shows that his model is overly simplistic.

I must end here, after thrashing about ineffectually in multiple disciples in which I am not even the rankest amateur. What I hoped to re-capture in this review was the intellectual excitement I felt while reading this little volume. In somewhat dry (though not technical) academic prose, Kuhn caused a revolution that still forceful enough to make me dizzy.

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Review: The Logic of Scientific Discovery

Review: The Logic of Scientific Discovery

The Logic of Scientific DiscoveryThe Logic of Scientific Discovery by Karl R. Popper

My rating: 4 of 5 stars

We do not know: we can only guess.

Karl Popper originally wrote Logik der Forchung (The Logic of Research) in 1934. This original version—published in haste to secure an academic position and escape the threat of Nazism (Popper was of Jewish descent)—was heavily condensed at the publisher’s request; and because of this, and because it remained untranslated from the German, the book did not receive the attention it deserved. This had to wait until 1959, when Popper finally released a revised and expanded English translation. Yet this condensation and subsequent expansion have left their mark on the book. Popper makes his most famous point within the first few dozen pages; and much of the rest of the book is given over to dead controversies, criticisms and rejoinders, technical appendices, and extended footnotes. It does not make for the most graceful reading experience.

This hardly matters, however, since it is here that Popper put forward what has arguably become the most famous concept in the philosophy of science: falsification.

This term is widely used; but its original justification is not, I believe, widely understood. Popper’s doctrine must be understood as a response to inductivism. Now, in 1620 Francis Bacon released his brilliant Novum Organum. Its title alludes to Aristotle’s Organon, a collection of logical treatises, mainly focusing on how to make accurate deductions. This Aristotelian method—dominated by syllogisms: deriving conclusions from given premises—dominated the study of nature for millennia, with precious little to show for it. Francis Bacon hoped to change all that with his new doctrine of induction. Instead of beginning with premises (‘All men are mortal’), and reasoning to conclusions (‘Socrates is mortal’), the investigator must begin with experiences (‘Socrates died,’ ‘Plato died,’ etc.) and then generalize a conclusion (‘All men are mortal’). This was how science was to proceed: from the specific to the general.

This seemed all fine and dandy until, in 1738, David Hume published his Treatise of Human Nature, in which he explained his infamous ‘problem of induction.’ Here is the idea. If you see one, two, three… a dozen… a thousand… a million white swans, and not a single black one, it is still illogical to conclude “All swans are white.” Even if you investigated every swan in the world but one, and they all proved white, you still could not conclude with certainty that the last one would be white. Aside from modus tollens (concluding from a negative specific to a negative general), here is no logically justifiable way to proceed from the specific to the general. To this argument, many are tempted to respond: “But we know from experience that induction works. We generalize all the time.” Yet this is to use induction to prove that induction works, which is paradoxical. Hume’s problem of induction has proven to be a stumbling block for philosophers ever since.

In the early parts of the 20th century, the doctrine of logical positivism arose in the philosophical world, particularly in the ‘Vienna Circle’. This had many proponents and many forms, but the basic idea, as explained by A.J. Ayer, is the following. The meaning of a sentence is equivalent to its verification; and verification is performed through experience. Thus the sentence “The cat is on the mat” can be verified by looking at the mat; it is a meaningful utterance. But the sentence “The world is composed of mind” cannot be verified by any experience; it is meaningless. Using this doctrine the positivists hoped to eliminate all metaphysics. Unfortunately, however, the doctrine also eliminates human knowledge, since, as Hume showed, generalizations can never be verified. No experience corresponds, for example, to the statement: “Gravitation is proportional to the product of mass and the inverse square of distance,” since this is an unlimitedly general statement, and experiences are always particular.

Karl Popper’s falsificationism is meant to solve this problem. First, it is important to note that Popper is not, like the positivists, proposing a criterion of ‘meaning’. That is to say that, for Popper, unfalsifiable statements can still be meaningful; they just do not tell us anything about the world. Indeed, he continually notes how metaphysical ideas (such as Kepler’s idea that circles are more ‘perfect’ than other shapes) have inspired and guided scientists. This is itself an important distinction because it prevents him from falling into the same paradox as the positivists. For if only the statements with empirical content have meaning, then the statement “only the statements with empirical content have meaning” is itself meaningless. Popper, for his part, regarded himself as the enemy of linguistic philosophy and considered the problem of epistemology quite distinct from language analysis.

To return to falsification, Popper’s fundamental insight is that verification and falsification are not symmetrical. While no general statement can be proved using a specific instance, a general statement can indeed be disproved with a specific instance. A thousand white swans does not prove all swans are white; but one black swan disproves it. (This is the aforementioned modus tollens.) All this may seem trivial; but as Popper realized, this changes the nature of scientific knowledge as we know it. For science, then, is far from what Bacon imagined it to be—a carefully sifted catalogue of experiences, a collection of well-founded generalizations—and is rather a collection of theories which spring up, as it were, from the imagination of the scientist in the hopes of uniting several observed phenomena under one hypothesis. Or to put it more bluntly: a good scientific theory is a guess that does not prove wrong.

With his central doctrine established, Popper goes on to the technicalities. He discusses what composes the ‘range’ or ‘scope’ of a theory, and how some theories can be said to encompass others. He provides an admirable justification for Occam’s Razor—the preference for simpler over more complex explanations—since theories with fewer parameters are more easily falsified and thus, in his view, more informative. The biggest section is given over to probability. I admit that I had some difficulty following his argument at times, but the gist of his point is that probability must be interpreted ‘objectively,’ as frequency distributions, rather than ‘subjectively,’ as degrees of certainty, in order to be falsifiable; and also that the statistical results of experiments must be reproducible in order to avoid the possibility of statistical flukes.

All this leads up to a strangely combative section on quantum mechanics. Popper apparently was in the same camp as Einstein, and was put off by Heisenberg’s uncertainty principle. Like Einstein, Popper was a realist and did not like the idea that a particle’s properties could be actually undetermined; he wanted to see the uncertainty of quantum mechanics as a byproduct of measurement or of ‘hidden variables’—not as representing something real about the universe. And like Einstein (though less famously) Popper proposed an experiment to decide the issue. The original experiment, as described in this book, was soon shown to be flawed; but a revised experiment was finally conducted in 1999, after Popper’s death. Though the experiment agreed with Popper’s prediction (showing that measuring an entangled photon does not affect its pair), it had no bearing on Heisenberg’s uncertainty principle, which restricts arbitrarily precise measurements on a single particle, not a pair of particles.

Incidentally, it is difficult to see why Popper is so uncomfortable with the uncertainty principle. Given his own dogma of falsifiability, the belief that nature is inherently deterministic (and that probabilistic theories are simply the result of a lack of our own knowledge) should be discarded as metaphysical. This is just one example of how Popper’s personality was out of harmony with his own doctrines. An advocate of the open society, he was famously authoritarian in his private life, which led to his own alienation. This is neither here nor there, but it is an interesting comment on the human animal.

Popper’s doctrine, like all great ideas, has proven both influential and controversial. For my part I think falsification a huge advance over Bacon’s induction or the positivists’ verification. And despite the complications, I think that falsifiability is a crucial test to distinguish, not only science from pseudo-science, but all dependable knowledge from myth. For both pseudo-science and myth generally distinguish themselves by admirably fitting the data set, but resisting falsification. Freud’s theories, for example, can accommodate themselves to any set of facts we throw at them; likewise for intelligent design, belief in supernatural beings, or conspiracy theories. All of these seem to explain everything—and in a way they do, since they fit the observable data—but really explain nothing, since they can accommodate any new observation.

There are some difficulties with falsification, of course. The first is observation. For what we observe, or even what we count as an ‘observation’, is colored by our background beliefs. Whether to regard a dot in the sky as a plane, a UFO, or an angel is shaped by the beliefs we already hold; thus it is possible to disregard observations that run counter to our theories, rather than falsifying the theories. What is more, theories never exist in isolation, but in an entire context of beliefs; so if one prediction is definitively falsified, it can still be unclear what we must change in our interconnected edifice of theories. Further, it is rare for experimental predictions to agree exactly with results; usually they are approximately correct. But where do we draw the line between falsification and approximate correctness? And last, if we formulate a theory which withstands test after test, predicting their results with extreme accuracy time and again, must we still regard the theory as a provisional guess?

To give Popper credit, he responds to all of these points in this work, though perhaps not with enough discussion. But all these criticisms belie the fact that so much of the philosophy of science written after Popper has taken his work as a starting point, either attempting to amplify, modify, or (dare I say it?) falsify his claims. For my part, though I was often bored by the dry style and baffled by the technical explanations, I found myself admiring Popper’s careful methodology: responding to criticisms, making fine distinctions, building up his system piece by piece. Here is a philosopher deeply committed to the ideal of rational argument and deeply engaged with understanding the world. I am excited to read more.

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