New Podcast: Letters from Spain #1

New Podcast: Letters from Spain #1

I recently started on a new podcast project, Letters from Spain, where I hope to document my life in Madrid, and to reflect on some of the differences between Spain and the United States.

To listen to the first episode, click here.

Below is the text of the podcast:


Letter #1: Beginning

I am here to talk about Spain.

But I should begin with some reservations. Talking about other cultures is a dangerous enterprise. A major risk is exoticizing the culture—making it seem altogether unusual and even nonsensical. From there, it is a short step to dismissing the culture completely, treating it as an illogical accident of humankind, a bizarro land where nothing is as it should be. On the other extreme we may normalize the culture by focusing exclusively on the ways in which it is not so very different. This way we treat the other culture as we treat ourselves, which is partly good; however, this way we may fail to recognize how a culture is genuinely special.

This is only the beginning of our troubles. To talk about something, we must ourselves have a point of view; and that is formed, of course, by our own culture. For me that culture is American, specifically from New York, specifically from Westchester County, specifically from the town of Sleepy Hollow. For me, that is ‘normal,’ and this sense of normality shapes my perspective. I cannot help but compare Spain to this culture, my culture, and to see everything Spanish as, in a sense, a deviation. Is it possible to talk about a culture in purely objective terms? I doubt it; and even if it were possible, I doubt that it would be worth listening to. Culture is, among other things, a system of values, and you cannot understand it without having values of your own.

I am going on, listing difficulties, and yet there are still more risks. An obvious one is the use of stereotypes. Now, what is a stereotype? It is not merely a generalization, but a widely known and popularly believed generalization, usually with positive or negative ramifications. Each country has its share of stereotypes—the Spanish dance flamenco, go to bullfights, and sleep siestas, while Americans eat hamburgers and live in big houses. And so on. Now, some people say that stereotypes are problematic because they are generalizations. I don’t think that’s true. All knowledge consists of generalizations. And some generalizations are perfectly true. It is true, for example, that Spanish people tend to eat dinner later than Americans.

The problem with stereotypes, then, is not that they are generalizations, but that they are misapplied or untrue generalizations. Most Spanish people don’t like flamenco, or go to bullfights, or have time in the middle of the day for a nap. And, besides, these stereotypes are troublesome because they project a kind of fantasy version of Spain, where the people are living passionate, dangerous lives under the scorching Mediterranean sun. Don’t get me wrong, these things do exist in Spain, and they are interesting facets of Spanish culture. But to characterize the whole country that way is highly inaccurate, to say the least.

Considering all of these risks, then, what am I here to do? In this podcast, I hope to use my own experience in Spain to consider some of the subtler differences between life here and life back in the United States. To tell you something about myself, my name is Roy. I am a 28-year-old English teacher, living in Madrid. I decided to move here over four years ago, when I was working in Manhattan in an office job that, shall we say, did not fulfill my dreams of post-college life. I wanted an escape, to see the wider world, to go on an adventure; and Europe seemed to be the answer. I had been reading about European history for years. In undergrad, I studied cultural anthropology, and my advisor did his research in the south of Spain. Dreams of castles and philosophers’ graves beguiled me, and soon I found myself on a plane to Madrid.

You may ask, why Spain? Well, the answer is not very inspiring. Simply, Spain is one of the easiest countries to legally work in for Americans. It was an entirely opportunistic move. But, it was a fortunate one, since I became enamored of the country within months. My backstory explains my own bias. Like many people, I suspect, I came to Spain seeking an escape from the dreary world of American adulthood, and I found one. Thus, for me, Spain is tinged with a kind of rosy hue, as a place of refuge and adventure. I have lived here long enough for some of this to have worn off, but still I am predisposed to see all things Spanish as good. Still, I do hope I will avoid idealizing this country, since such a romanticized image would have little value. 

So in this podcast I want to explain what I have come to learn and appreciate about this place, and why I have chosen to stay year after year. And I will do this from an inescapably American perspective. The differences between Spanish and American culture goes far beyond flamenco and siestas, and I think these subtler differences have much to teach us. I hope do to this without either collapsing the differences between these two cultures, and without making Spain seem impossibly exotic. Let us see if I can thread the needle. 

Review: The History of Ancient Egypt

Review: The History of Ancient Egypt
The History of Ancient Egypt

The History of Ancient Egypt by Bob Brier

My rating: 5 of 5 stars

Ancient Egypt, like dinosaur bones and outer space, is one of those things which seem to attract universal curiosity. It certainly did in my case. I remember visiting the Egyptian section in the Met, as a young boy, and marveling over the mummies and the massive sarcophagi, the mysterious hieroglyphs and monumental statues.

There is something curiously foreign, even inhuman, about Egyptian artifacts. For one, they are old beyond anything we are accustomed to think about. To cite one oft-repeated fact, there is more time between construction the Great Pyramid and the life of Julius Caesar, than between Julius Caesar and our own time. Even ancient history seems like yesterday by comparison. Aside from mere time, Egypt’s culture is strikingly unlike our own. God-kings who marry their sisters to keep the blood-line pure, mummified bodies interred in graves full of gold, jackal-headed gods and hieroglyphic script—it is alien indeed.

Yet it is beautiful. Egyptian art is undoubtedly one of the great art traditions in the world—as anyone knows who has examined the bust of Nefertiti in Berlin, the seated scribe in Paris, or the statue of Hatshepsut in New York. It is a unified and coherent aesthetic, permeating everything from the smallest objects to the greatest temples, and lasting for thousands of years with only minor change. Even if it is enchanting, however, the art of Egypt also evokes this sensation of distance. Every image is so stylized, every human form is so rigid and unrealistic, every aesthetic choice pre-determined by tradition, that it is difficult to get a sense of real people behind these objects.

This sense of distance, of foreignness, of mystery, is what makes Egypt so exciting to study. (It is also why people talk about ancient aliens.) And Bob Brier is an ideal guide. I have never loved anything or anyone as much as Brier loves Egypt. This enthusiasm is infectious, and makes his series of lectures a real pleasure. He describes how he climbed into the Bent Pyramid—an early, failed attempt to create a pyramid—and how he traveled to the turquoise mines used by Egyptians themselves. He narrates Napoleon’s conquest of Egypt and explains how the Rosetta Stone was translated. He even describes in detail how to make a mummy—and he should know, since he made one himself.

Apart from these entertaining asides, Brier takes the listener through the whole history of Ancient Egypt, from prehistory to the death of Cleopatra. It is a fascinating story, and Brier is a wonderful storyteller. A lifelong resident of the Bronx, his verbal mannerisms may remind one—pleasantly or unpleasantly—of the man in the White House; yet he knows how to dramatize the relevant details enough to make them effortlessly stick in the memory. His love of a good story does lead him astray, at times. For my part, his two lectures on the Biblical stories, Joseph and Exodus, were somewhat too credulous of their veracity. He is similarly generous when it comes to Herodotus. And his theory of Tutankhamun’s murder has now been disproven.

Aside from these mild criticisms, I should note that the series does show its age. Recorded in 1999, lots has happened in the world of Egyptology since then, notably the advances in DNA and medical technology which allow us to know more about the lives of Egyptians. For example, we now know far more about Tutankhamun’s many physical ailments, and we also know that he was not the son of Nefertiti.

Nevertheless, these lectures remain a wonderful introduction to the times. I cannot emphasize enough how enjoyable they are. They convert you into an Egypt fanatic. Now I want teach myself hieroglyphics and to go to Egypt myself. It must be incredible to see all of this in person. For now, however, I will have to be content with Brier’s virtual tour and whatever museums I can visit.



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Review: On the Soul (Aristotle)

Review: On the Soul (Aristotle)
De Anima (On the Soul)

De Anima by Aristotle

My rating: 5 of 5 stars

As I have lately been making my way through Aristotle’s physical treatises, I have often observed that many of Aristotle’s errors stem from his tendency to see the physical world as analogous to a biological organism. So it is a pleasure to finally see Aristotle back on his home territory—living things. While Aristotle’s work in proto-physics and proto-chemistry is interesting mainly from a historical perspective, this work is interesting in its own right; in just a hundred pages, Aristotle manages to assemble a treatise on the fundamentals of life.

The first thing the modern student will notice is that Aristotle means something quite different by ‘soul’ than how we normally understand the word. The word ‘soul’ has come to mean an immaterial, specter-like wraith, the spiritual core of one’s personality—trapped, only temporarily, in a body; and this view has, over the years, caused problems for philosophers and theologians alike, for it remains to be explained how an immaterial spirit could move a material body, or how a material body could trap an immaterial spirit. Aristotle avoids these awkward questions. What he means is quite different.

Aristotle begins by observing that all forms of behavior, human or animal, require a body. Even supposedly ‘mental’ states, such as anger, love, and desire, all have concomitant physical manifestations: an angry man gets red in the face, a man in love stares at his beloved, and a man who desires alcohol tries to get it. From this, Aristotle quickly concludes that all the Pythagorean and Platonic talk of the transmigration of souls is silly; a soul needs a body, just as a body needs a soul. Furthermore, a specific soul doesn’t need just any body, but it needs its specific body. Soul and body are, in other words, codependent and inseparable. In Aristotle’s words, “each art must use its tools, each soul its body.”

This still leaves the question unanswered, what is a soul? Aristotle answers that the soul is the form of the body. Alright, what does that mean? Keep this in mind: when Aristotle says ‘form’, he is not merely talking about the geometrical shape of the object, but means something far more general: the form, or essence, of something is that by which it is what it is. Here’s an example: the form of a bowl is that which makes a bowl a bowl, as opposed to something else like, say, a plate or a cup. In this particular case, the form would seem to be the mere shape of the object; isn’t the thing that makes a bowl a bowl its shape? But consider that there is no such thing as a disembodied bowl; for a bowl to be a bowl, it must have a certain shape, be within a certain size range, and be embodied in a suitable material. All of these qualifications, the shape, size, and material, Aristotle would include in the ‘form’ of an object.

So the soul of living things is the quality (or qualities) that differentiate them from nonliving things. Now, the main difference between animate and inanimate objects is that animate objects possess capacities; therefore, the more capacities a living thing has, the more souls we must posit. This sounds funny, but it’s just a way of speaking. Plants, for Aristotle, are the simplest forms of living beings; they only possess the ‘vegetative soul’, which is what makes them grow and develop. Animals possess additional souls, such as that which allows them to sense, to desire, to imagine, and—in the case of humans—to think. The ‘soul’, then, is a particular type of form; it is a form which gives its recipient a certain type of capability. Plants are only capable of growth; animals are capable of growing, of moving, and of many other things.

Aristotle sums up his view in a memorable phrase: “From all this it is obvious that the affections of soul are enmattered formulable essences.” These capacities cannot be ‘enmattered’ in just anything, but must be embodied in suitable materials; plants are not made of just anything, but their capacities for growth always manifest themselves in the same types of material. Aristotle sums up this point with another memorable phrase: “soul is an actuality or formulable essence of something that possesses the potentiality of being besouled.”

So an oak tree is made of material with the potentiality of being ‘besouled’, i.e., turned into a living, growing oak tree. Conversely, a life-sized statue of an oak tree made of bronze would still not be an oak tree, even if it shared several aspects of its form with a real oak tree. It isn’t made of the right material, and thus cannot possess the vegetative soul.

I have given a somewhat laborious summary of this because I think it is a very attractive way of looking at living things. It avoids all talk of ‘ghosts in the machine’, and concentrates on what is observable. (I should note, however, that Aristotle thought that ‘mind’, which is the faculty of reason, is immaterial and immortal. Nobody’s perfect.)

I also find Aristotle metaphysical views attractive. True to his doctrine of the golden mean, he places equal emphasis on matter and form. He occupies an interesting middle-ground between the idealism of Plato and the materialism of Democritus. In order for a particular thing to be what it is, it must both have a certain form—which is embodied in, but not reducible to, its matter—and be made of the ‘right’ types of matter. Unlike Plato’s ideals, which reside in a different sphere of reality, existing as perfect essences devoid of matter, Aristotle’s forms are inherent in their objects, and thus are neither immaterial nor simply the matter itself

The treatise ceases to be as interesting as it progresses, but there are a few gems along the way. He moves on to an investigation of the five senses, and, while discussing sight, has a few things to say about light. Aristotle defines light as the quality by which something transparent is transparent; in other words, light is the thing that can be seen through transparent things. I suppose that’s a respectable operational definition. Aristotle also considers the idea that light travels absurd; nothing could go that fast:

Empedocles (and with him all others who used the same forms of expression) was wrong in speaking of light as ‘traveling’ or being at a given moment between the earth and its envelope, its movement being unobservable by us; that view is contrary both to the clear evidence of argument and to the observed facts; if the distance traversed were short, the movement might have been unobservable, but where the distance is from extreme East to extreme West, the draught upon our powers of belief is too great.

Aristotle also has a few interesting things to say about sense:

By a ‘sense’ is meant what has the power of receiving into itself the sensible forms of things without the matter. This must be conceived of as taking place in the way in which a piece of wax takes on the impress of a signet-ring without the iron or gold; we say that what produces the impression is a signet of bronze or gold, but its particular metallic constitution makes no difference: in a similar way the sense is affected by what is colored or flavored or sounding, but it is indifferent what in each case the substance is; what alone matters is what quality is has, i.e. in what ratio its constituents are combined.

So we don’t take in the matter of a bowl through our eyes, but only its form. All of our senses, then, are adapted for observing different aspects of the forms of objects. Thus, Aristotle concludes, all knowledge consists of forms; when we learn about the world, we are mentally reproducing the form of the world in our minds. As he says: “It follows that the soul is analogous to the hand; for as the hand is a tool of tools [i.e. the tool by which we use tools], so the mind is the form of forms [i.e. the form by which we apprehend forms].” (Notice how deftly Aristotle wields his division of everything into matter and form; he uses it to define souls, to define senses, and then to define knowledge. It is characteristic of him to make so much headway with such seemingly simple divisions.)

For a long time, I was perplexed that Aristotle was so influential. I was originally repulsed by his way of thinking, put off by his manner of viewing the world. His works struck me as alternately pedantic, wrongheaded, or obvious. How could he have exerted such a tremendous influence over the Western mind? Now, after reading through much more Aristotle, this is no longer perplexing to me; in fact, I often find myself thinking along his lines, viewing the world through his eyes. It takes, I believe, a lot of exposure in order to really develop a sympathy for Aristotle’s thought; but with its emphasis on balance, on growth, on potentiality, it succeeds in being a very aesthetically compelling (if often incorrect) way of viewing things.

This piece represents, to me, Aristotle at his best. It is a grand synthesis of philosophy and biology, probably not matched until William James’s psychological work. Unlike many gentlemanly philosophers who shut themselves in their studies, trying to explain human behavior purely through introspection, Aristotle’s biologically rooted way of seeing things combines careful observation—of humans and nonhumans alike—with philosophical speculation. It is a shame that only the logic-chopping side of Aristotle was embraced by the medievals, and not his empirical outlook.

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Review: On the Heavens (Aristotle)

Review: On the Heavens (Aristotle)

On the Heavens by Aristotle

My rating: 4 of 5 stars


This is quite a charming little book. In it, one can find the description of an entire way of viewing the natural world. Aristotle moves on from the abstract investigations of the Physics to more concrete questions: Is the earth a sphere or flat? What are the fundamental constituents of matter? Why do some things fall, and some things rise? Is the earth the center of everything? Aristotle’s answers, I’m afraid, have not stood the test of time; such, it appears, is the risk of all science—obsolescence.

The reader is immediately presented with a beautiful piece of Aristotelian reasoning. First, the good philosopher reminds us that “the perfect is naturally prior to the imperfect, and the circle is a perfect thing.” Circular motion, therefore, is more perfect than simple up-and-down motion like we see on earth; and since we do not find bodies whose natural motion is circular on earth, and since nature always strives towards perfection, it follows that there must be bodies not on earth which naturally move in a circular fashion. Again, since none of the earth-bound elements—fire, water, air, and earth—exhibit natural (i.e. unforced) circular motion, it follows that the heavenly bodies must be composed of something different; and this different substance (let us call it aether), since is exhibits the most perfect motion, must be itself perfect.

In Aristotle’s words:

… we may infer with confidence that there is something beyond the bodies that are about us on this earth, different and separate from them; and that the superior glory of its nature is proportionate to its distance from this world of ours.

Everything below the moon must be born and pass away; but the heavenly bodies abide forever in their circular course. Q.E.D.

In his physical investigations, it seems that Aristotle was not especially prescient. For example, he argues against “the Italian philosophers known as the Pythagoreans… At the centre, they say, is fire, and the earth is one of the stars, creating night and day by its circular motion about the centre.” Not so, says Aristotle; the earth is the center. He also argues against Democritus’s atomic theory, which posits the existence of several different types of fundamental particles, which are intermingled with “void,” or empty spaces in between them.

To be fair, Aristotle does think that the earth is round; he even includes an estimation of the earth’s circumference at 400,000 stadia, which is, apparently, somewhere around 40,000 miles. (The current-day estimate is about 24,000 miles.) Aristotle also thinks that “heavy” objects tend toward the earth’s surface; but puzzlingly (for the modern reader), he doesn’t think this has anything to do with the pull of the earth, but instead thinks it has something to do with earth’s position in the center of all things. In his words: “If one were to remove the earth to where the moon now is, the various fragments of earth would each move not towards it but to the place in which it now is.”

Then Aristotle launches into his investigation of the elements. As aforesaid, Aristotle posits four sublunary elements: earth, water, fire, and air. Earth is the heaviest, followed by water, and then air; and fire is the lightest. Aristotle believes that these elements have “natural” motions; they tend toward their proper place. Earth tries to go downward, towards the center of the planet. Fire tries to go upward, towards the stars. Aristotle contrasts this “natural” motion with “unnatural” or “violent” motion, which is motion from an outside source. I can, of course, pick up a piece of earth, thereby thwarting its natural tendency towards its proper place on the ground.

The elements naturally sort themselves into order: we have earth on the bottom, then water floating on top, then the air sitting on the water, and fire above the air. (Where all that fire is, I can’t say.) There are some obvious difficulties with this theory. For example, how can boats float? and birds fly? This leads Aristotle to a very tentative definition of buoyancy, with which he ends the book:

… since there are two factors, the force responsible for the downward motion of the heavy body and the disruption-resisting force of the continuous surface, there must be some ratio between the two. For in proportion as the force applied by the heavy thing towards disruption and division exceeds that which resides in the continuum, the quicker will it force its way down; only if the force of the heavy thing is the weaker, will it ride upon the surface.

The more one reads Aristotle, the more one grasps just how much his worldview was based on biology. The key word of his entire philosophy is entelechy, which simply means the realization of potential. We can see this clearly in his definition of motion: “The fulfillment of what exists potentially, in so far as it exists potentially, is motion.” That’s a mouthful, but think of it this way: the act of building a house can be thought of as the expression of the potential of a house; the physical house in progress is the partially actualized house, but the building itself is the potential qua potential.

It is easy to see how Aristotle might get interested in the expression of potentialities from investigating living things. For what is an egg but a potential chicken? What is a child but a potential man? This idea of fully realizing one’s potential is at the basis of his ethics and his physics; just as fire realizes its potential for moving upwards, so do citizens realize their potential through moderation. Aristotle’s intellectual method is also heavily marked by one who spent time investigating life; for it is the dreary task of a naturalist to catalogue and to categorize, to investigate the whole by looking at the parts.

While this mindset served him admirably in many domains, it misled him in the investigation inanimate matter. To say that chickens grow from eggs as an expression of potential is reasonable; but to attribute the downward motion of rocks as an expression of their potential sounds odd. It is as if you asked somebody why cars move, and they responded “because it is the nature of the vehicle”—which would explain exactly nothing. But it is difficult not to be impressed by Aristotle; for even if he reached the wrong conclusions, at least he was asking the right questions.

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Historic Hudson Homes: The Grange

Historic Hudson Homes: The Grange

This post is part of my series on Historic Hudson Homes:


I stumbled upon the Alexander Hamilton’s house completely by accident. I met a couple of friends from out of town, and in the process of exploring the Upper West Side we came upon it.

The discovery that one of the Founding Fathers’ houses was here, in Harlem, was astounding. The humble wooden house could hardly look more out of place nowadays. Set amidst the landscape of glass, steel, and concrete, the Federal style building looks as if it has been plopped down from another age. And in a way it was. The house has been moved from its original location not once, but twice.

When Hamilton had the house built, in the early 1800s, he was already in the twilight of his career. He had finished his epochal tenure as the Secretary of the Treasury—during which he constructed the nation’s financial system from the ground up—and had been reduced by a series of quarrels and scandals to an ordinary private citizen. To name just two watersheds in his life, he was publically exposed (and exposed himself) as an adulterer, and he also openly quarelled with John Adams, the presidential candidate of his political party, the Federalists. He was a brilliant statesman but a poor politician.

After this semi-voluntary retirement from the life of politics, Hamilton worked as a high-profile lawyer in New York City. Brilliant in everything he did, his career in law was no exception. To escape from the bustle of the city, Hamilton hired John McComb Jr.—the architect who designed NYC’s majestic City Hall, among other things—to build a rustic retreat for himself and his large family (he had eight children). At the time, Harlem was of course nothing like the city it is today. The area was still mostly farms and undeveloped woodland, an ideal spot to build a country house. 

Hamilton did not have much time to enjoy his little paradise, however. Just two years after it was completed in 1802, Hamilton was killed in a duel with Vice-President Aaron Burr—another high-profile NYC lawyer, and a long-time rival of Hamilton’s. The property stayed in the family for some decades hence, occupied by his long-surviving wife Elizabeth and their children. But eventually it was sold and, by the late 1800s, it was in foreclosure and doomed to destruction. The building only survived because of the intervention of the St. Luke’s Episcopal Church, which bought the house and had it relocated, to conform with the Manhattan street grid.

This purchase was not for purely disinterested historical reasons, however. The church modified the house so that it could be used for services—a purpose which it served for many years. Eventually a larger church building was constructed, enclosing the old home. Then, in the 1920s, a historical preservation society intervened, buying the property and partially restoring some of its original features and decoration. It was only in the 1960s that it was designated a National Historic Landmark. But it remained hemmed in by the many tall buildings that sprung up around it.

Finally, in 2006, the house was moved to a more open location, in St. Nicholas Park, which was still within Hamilton’s original property. The move was a minor feat of technical achievement, gently cradling the entire structure and shifting it, inch by inch, to its new home. Now it stands in a relatively open space, near a few churches, some brownstown apartments, a playground, a taxi service, and a funeral home. And this curious conjunction of the historical, the mundane, the holy, the profitable, and everything else, is what New York City is all about. 

For anyone who has visited Jefferson’s Monticello or Washington’s Mount Vernon, Hamilton’s little home may be something of a let down. Hamilton was never greatly wealthy in life; he owned no slaves—indeed, was part of the Abolition Society—and, though addicted to work, was scrupulously honest in his personal finances. In any case, much of the house’s original furnishings and character were lost in its two moves and its many renovations.

The visitor enters on the ground floor, what used to be the basement. There is a small gift shop near the entrance, with books by and about Hamilton. The rest of the space is a miniature museum. There is a time-lapse video of the epochal re-location of the house, and a short documentary that plays on a loop, giving an overview of Hamilton’s life. His accomplishments are also displayed in a series of informational panels and timelines. My favorite object on display was a mourning scarf—a scarf made and sold on the occasion of his untimely death, for people to honor his life. Amusingly, the scarf gives an incorrect account of Hamilton’s background, saying that he was “descended by his father of respectable Scottish ancestry, by his mother was reputably connected to New York.”

The Dining Room

The reality, as you may know, was very different. Hamilton’s father, James, was a complete failure in life, and abandoned Hamilton and his brother when they were very young. (James may not even have been Hamilton’s real father, anyway, as Chernow speculates.) Further, Hamilton’s mother was far from reputable—she abandoned her husband and had Hamilton out of wedlock—and, in any case, died when Alexander was very young, leaving him little in the way of property or connections. Hamilton was, in other words, both a bastard and an orphan, as his musical proudly proclaims. But I suppose it is not polite to mention that at a funeral.

The Salon

The house’s period rooms are all upstairs, on the first floor, accessible with a small wooden staircase from the basement. The hours of access are extremely limited (on weekdays from 12:00 to 1:00, and from 3:00 to 4:00), which makes visiting somewhat inconvenient. But it does not take much time to see everything. There are three main rooms to visit: the dining room, the salon, and Hamilton’s study (where he must have spent most of his time). The dining room features a long elegant table and is well lit with three tall windows. The salon is where I imagine the family spent much of their time, playing cards and receiving guests. Hamilton’s study is much as one would expect: decorated with maps and well-stocked with books. It must have been far messier and fuller when he was alive. He was an avid reader and inveterate scribbler.

Two notable paintings hang on the walls, one a full-length portrait of Hamilton himself, looking like a sophisticated dandy. There is also a portrait of George Washington, given as a gift to the Hamiltons from the Washingtons (the two men worked very closely together). The two original paintings are now in the Metropolitan Museum of Art, where they hang proudly in the American Wing; the paintings here are reproductions. Even more captivating is a life-sized bust of Hamilton, dressed like a Roman statesman and looking quite like one too. According to the guide, Hamilton’s wife considered the bust extremely lifelike, and even spoke to it after his death. It is a wonderful piece, bringing him to life with a winsome and wise smile. (Thomas Jefferson, Hamilton’s great rival, had a replica of this bust in his house in Monticello, placed opposite to his own, so the two would be forever in opposition.)

And that is about all there is to see. If anything, the house is charming because it shows how humble may be the life of an extraordinary historical figure.

But I should not end this post without mentioning another monument next door. Up the hill from Hamilton’s Grange is one of the most attractive college campuses I have ever seen: the North Campus of CUNY’s City College. The entire complex is built in a flamboyant neo-gothic style, designed by the architect George Browne Post (many of whose other important works have been demolished). Everything follows a unified aesthetic, from the elaborate iron grillwork on the gates to the little lamps flanking the entrances. The stonework on the buildings is particularly delightful, with enough subtle variation in color and texture to remain endlessly pleasant to examine. Every detail is well-done, from the wry gargoyles scattered throughout to the ornamented roofs and towers.

The centerpiece of the complex is Shepard Hall—named after a prominent NYC lawyer—which features a central building in the shape of a cathedral, and two curving wings on either side. Unfortunately I could not enter the building, as all the doors were locked. If I had gone inside, I would have seen the monumental Great Hall, a sort of central nave at the end of which is a mural by Edwin Blashfield, the artist who decorated the Library of Congress. In front of this hall is a statue of Alexander S. Webb, a general in the Civil War and the second president of the university. The statue makes him out to look like quite a swaggering military figure, and not exactly a bookish man.

(On the subject of statues, I ought, too, to mention the statue of Alexander Hamilton in front of the Broadway Spanish Baptist Church nearby. Here he looks rather like he does in the aforementioned painting, a heroic dandy.)

To me, the campus is a delightful and inspiring place. Indeed, the only American university campus of comparable beauty I know of is the University of Virginia, designed by Thomas Jefferson. Having gone to a rather unattractive university myself, I am very sensitive to the effect that architecture can have on the mood and atmosphere of an institute of higher learning. The strong verticality of the gothic buildings lifts one up and their historical form inspires a sense of participating in an honorable intellectual tradition.

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Review: Calculus (Kline)

Review: Calculus (Kline)

Calculus: An Intuitive and Physical Approach by Morris Kline

My rating: 5 of 5 stars

Morris Kline’s book, Mathematics for the Nonmathematician, is my favorite book on the discipline. Kline showed an amazing ability to explain mathematical concepts intuitively, and to situate them within a sensible human context. In his hands, math was not simply a series of equations or deductive proofs, but an integral aspect of our civilization: a crucial tool in our species’ attempt to understand and manipulate the world. The book changed my view of the subject.

So when I found that Kline had written a book on the calculus, I knew that I had to read it. Calculus represents the furthest I have ever gone with mathematics in my formal schooling. By the time that I graduated high school, I was a problem-solving machine—with so many rules of algebra, trigonometry, derivation, and integration memorized that I could breeze through simple exercises. Yet this was a merely mechanical understanding. I was like a well-trained dog, obeying orders without comprehension; and this was apparent whenever I had to do any problems that required deeper thinking.

In time I lost even this, leaving me feeling like any ordinary mathematical ignoramus. My remedial education has been slow and painful. This was my primary object in reading this book: to revive whatever atrophied mathematical skills lay dormant, and to at least recover the level of ability I had in high school. Kline’s text was perfect for this purpose. His educational philosophy suits me. Rather than explain the calculus using formal proofs, he first tries to shape the student’s intuition. He does this through a variety of examples, informal arguments, and graphic representation, allowing the learner to get a “feel” for the math before attempting a rigorous definition.

He justifies his procedure in the introduction:

Rigor undoubtedly refines the intuition but does not supplant it. . . . Before one can appreciate a precise formulation of a concept or theorem, he must know what idea is being formulated and what exceptions or pitfalls the wording is trying to avoid. Hence he must be able to call upon a wealth of experience acquired before tackling the rigorous formulation.

This rings true to my experience. In my first semester of university, when I thought that I was going to study chemistry, I took an introductory calculus course. It was divided into lectures with the professor and smaller “recitative” classes with a graduate student. In the lectures, the professor would inevitably take the class through long proofs, while the grad student would show us how to solve the problems in the recitatives. I inevitably found the professor’s proofs to be pointless, and soon decided to avoid them altogether, since they confused me rather than aided me. I got an A-minus in the class.

Though Kline forgoes the rigor one would expect in formal mathematics, this book is no breezy read. It is a proper textbook, designed to be used in a two-semester introductory course, complete with hundreds of exercises. And as fitting for such a purpose, this book is dry. Gone are the fascinating historical tidbits and gentle presentation of Kline’s book on popular mathematics. This book is meant for students of engineering and the sciences—students who need to know how to solve problems correctly, or planes will crash and buildings will collapse. But Kline is an excellent teacher in this context, too, and explains each concept clearly and concisely. It was often surprisingly easy to follow along.

The exercises are excellent as well, designed to progress in difficulty, and more importantly to encourage independent thinking. Rather than simply solving problems by rote, Kline encourages the student to apply the concepts creatively and in new contexts. Now, I admit that the sheer amount of exercises taxed my patience and interest. I wanted a refreshment, and Kline gave me a four-course meal. Still, I made sure to do at least a couple problems per section, to check whether I was actually understanding the basic idea. It helped immensely to have the solutions manual, which you can download from Dover’s website.

In the end, I am very glad to have read this book. Admittedly this tome did dominate my summer—as I plowed through its chapters for hours each day, trying to finish the book before the start of the next school year—and I undoubtedly tried to read it far too quickly. Yet even though I spent a huge portion of my time with this book scratching my head, getting questions wrong, it did help to restore a sense of intellectual confidence. Now I know for sure that I am still at least as smart as I was at age 18.

And the subject, if often tedious, is fascinating. Learning any branch of mathematics can be intensely satisfying. Each area interlocks with and builds upon the other, forming a marvelous theoretical edifice. And in the case of the calculus, this abstract structure contains the tools needed to analyze the concrete world—and that is the beauty of math.

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