Archive for February, 2018

Susskind & Friedman: Quantum Mechanics

February 22, 2018

Quantum Mechanics
The Theoretical Minimum
Leonard Susskind and Art Friedman
(Basic, 2014)
xx + 364 p.

Books on quantum mechanics tend to come to two main varieties: introductions for non-scientists, which normally focus on the conceptual underpinnings of the subject and avoid mathematics, and technical books written for advanced undergraduates or higher. This book, however, doesn’t quite fall in either camp. It spends a good deal of time carefully exploring the conceptual foundations, but it also does contain enough mathematics — all of it fairly gentle, but pertinent — so that the reader is not only told, but also is able to see, how certain of the most famous predictions of quantum mechanics follow from those conceptual foundations.

Susskind and Friedman begin with a simple quantum system, a single quantum spin, and use it to lay out the unusual logic of quantum states, emphasizing how it differs from the logic of classical physics. They discuss both time independent and time dependent quantum mechanics, emphasizing the value of the former for deducing the energy states of a system and of the latter for deducing how the system evolves in time. About one-third of the book is devoted to an exploration of entanglement, traditionally one of the strangest aspects of the quantum world, but they take some pains to argue that entanglement does not imply any sort of non-locality, as is sometimes claimed. Later sections of the book transition to the topic of wavefunctions and particles, and creep right up to the edge of quantum field theory, so as to peer over for a moment. At the end, they give a nice treatment of the quantum mechanical harmonic oscillator, which is one of the simplest but most important quantum systems.

Susskind is one of the best-known physicists of his generation. It is not all that common, I think, for such an eminent scientist to have a passion for teaching his subject to beginners, so I very much admire what he is doing here. The development of the subject is clear, with intermediate steps worked, and the significance of conclusions are emphasized. The book has a welcoming tone, and the enthusiasm of the authors is evident. They do not refrain from an occasional joke. The book is, apparently, derived from an online course which Susskind has given at Stanford. (Friedman is one of his students, and it is unclear to me exactly what his role has been in writing the book.) The book has been typeset with LaTeX, as is right and just.

It’s a little hard to say who the target audience is. It would be accessible, certainly, to an interested reader who had been trained in, for instance, engineering. The mathematics required doesn’t extend much beyond complex numbers, basic calculus, and linear algebra, and even these are given quick explanations in the text. I could see it being a very good read for an ambitious high school student or beginning undergraduate who has an interest in the subject. Or, for that matter, the target might be me: a trained physicist who has been out of academia for a while and would enjoy a trip down memory lane.

The book is part of a series, in fact, that goes under the title “The Theoretical Minimum”. It was preceded by a book on classical mechanics (to which this volume makes occasional reference), and has now been succeeded by a recent book on special relativity and classical field theory. I’ve not read either of those, but I had enough fun with this one that I might.

Newman on some great authors

February 15, 2018

In The Idea of a University, John Henry Newman discussed prospects for the eventual development of an organic body of Catholic literature in English to rival the Protestant literature that has formed our language to date. In the process, he made reference to a number of the greatest writers of the Western European tradition, including our English ones, and offered brief comments or judgments upon them. Some of these are quite entertaining, others surprising (Pascal!). I reproduce several of them here, without further comment from me.


Swift and Addison: “the most native and natural of our writers”

Voltaire: “an open scoffer at every thing sacred, venerable, or high-minded”

Pascal: “does not approve himself to a Catholic judgment”

Ariosto: “is allowed on all hands to occupy the first rank of Literature” but is guilty of “coarse sensuality”

Boccaccio: “the first of Italian prose-writers”

Shakespeare: “There is in Shakespeare neither contempt of religion nor scepticism, and he upholds the broad laws of moral and divine truth with the consistency and severity of an Æschylus, Sophocles, or Pindar. There is no mistaking in his works on which side lies the right; Satan is not made a hero, nor Cain a victim, but pride is pride, and vice is vice, and, whatever indulgence he may allow himself in light thoughts or unseemly words, yet his admiration is reserved for sanctity and truth. From the second chief fault of Literature, as indeed my last words imply, he is not so free; but, often as he may offend against modesty, he is clear of a worse charge, sensuality, and hardly a passage can be instanced in all that he has written to seduce the imagination or to excite the passions.”

Pope: “a rival to Shakespeare, if not in genius, at least in copiousness and variety”, “he was actually a Catholic, though personally an unsatisfactory one.”

Johnson: holds “the special title of moralist in English Literature”

Locke: “scarcely an honour to us in the standard of truth, grave and manly as he is”

Bacon: “deserves by his writings to be called the most orthodox of Protestant philosophers”

Hobbes, Hume, and Bentham: “simply a disgrace”

Old English Daniel

February 13, 2018

Translated from the Old English by Craig Williamson
(Penn, 2016) [c.850]
20 p.

Continuing my reading of the poems in the Junius Manuscipt, I come to this narrative poem based on episodes in the Book of Daniel, Chapters 1-5. At roughly 700 lines, it is somewhat longer than Exodus, but it seems to me a much more straightforward poem.

The poet tells the story of the sack of Jerusalem by King Nebuchadnezzar and the captivity of the Hebrew people, Daniel among them. Daniel gains favour with the king as an able interpreter of dreams. Meanwhile the king is drunk on hubris, and we hear the story of the burning fiery furnace, and of the three Hebrew men, Hananiah, Azariah, and Mishael, who are protected from the flames by an angel of God. The king has a second dream, and again Daniel interprets, foretelling the seven-years madness of the king. Finally we hear the tale of Belshazzar’s feast, with the strange handwriting on the wall and Daniel summoned to read it.

I noted, in comments on earlier poems in this manuscript, that the poets sought to cast the Biblical characters as warriors, mighty in their fury and dauntless, even when this was without any real warrant in the Biblical text. Interestingly, that does not happen in this poem. Instead, the poet sees Daniel as a different kind of hero: a sage, a man of knowledge who sees into mysteries. I do not know if this was a character whom the poet’s audience would have recognized as familiar.

The poem is odd, structurally. The first dream of Nebuchadnezzar is passed over rapidly, but the second is presented in great detail. Even more oddly, the final panel of the poem ends abruptly, after Daniel is summoned but before he interprets the handwriting — he is given just enough time to berate the revelers for their impiety. And of course those of us who teach catechism classes to youngsters are surprised to read a poem about Daniel that never mentions the lion’s den (though the explanation is straightforward enough: that occurs in Daniel, Chapter 6). In his introductory notes, Craig Williamson notes these peculiar features of the poem and points to a lively scholarly debate about them.

My favourite part of the poem is the song of praise which the three young men sing from the centre of the fiery furnace. The imagery is beautiful and the effect is resplendent; let me write out the whole thing:

The three bold surviviors, wise in mind,
Said to their Creator with a single voice:
“Gracious Father, let the full beauty
Of the world’s crafts, each created wonder,
The heavens and angels, the bright clear waters,
Each of your beings in its own degree,
Everything above in its glory and grandeur,
Praise your power and worship you.
Let the sun and moon, the stars in heaven,
The planets parading in the night sky,
The waters of earth and air, the dew and rain,
Praise and glorify you. Let all souls sing,
Exalting the name of almighty God.
Let burning fire and bright summer,
Night and day, land and sea,
Light and darkness, heat and cold,
Frost and dew, rain and river,
Spring-snap and winter-wonder,
Cloud-drift and snow-drift,
All weathers, all seasons, glorify God.
Let all creatures in the curve of creation
Extol your blessings, eternal Lord —
Lightning-flash and thunder-clap,
Earth-hills and summer-spills,
Salt-waves and spring-surges,
The deep thrum of whales singing,
The high drift of birds winging,
Water-flow and wind-blow,
Cattle in the field, beasts in the wild.
Let the children of men celebrate your love,
Bring you the best of their hearts’ hymns.
Let the people of Israel, your faithful servants,
Praise you, proclaiming your glory revealed
In the wealth of the world, in bright nature’s
Bountiful being, in each creatures’ song.
Your hands hold each heart’s virtue,
Each mind’s making, each soul’s yearning.
We three children of God speak out
With a singular voice rising from the flames —
Hananiah, Azariah, and Mishael together.
We glorify God in the sanctity of our hearts.
We bless you forever, Lord of all nations,
Almighty Father, true Son of the Creator,
Savior of souls, Healer of hearts,
Holy Spirit, and all-knowing God.
We celebrate your vision in our way of seeing,
Your powerful truth in our best way of being.
You reign supreme in the realm of heaven,
Higher than the sun-road over the world-roof.
You are a Poet making, a Creator shaping,
The Holy Word weaving itself in the world
Moment by moment from beginning to end —
All light, all life, the soul of our seeking,
The way of our walking in every bright land.”

Admitting all of the necessary caveats about reading in translation, I hear in these lines a joyful lyricism that I’ve not encountered before in Anglo-Saxon poetry — albeit on limited exposure. At any rate, I like it.

The fourth and final poem in the Junius Manuscript is based on the New Testament. Christ and Satan will be our matter next time.

Another circle canon

February 9, 2018

A few days ago I posted some videos showing the scores of music notated on a circular stave. Here’s another, and it’s even better. Salve Radix was set by Richard Sampson to celebrate the birth of Mary Tudor to Henry VIII and Catharine of Aragon. The piece is for four voices, 2 in the countertenor and 2 in the bass, with each pair singing in canon at an interval of a fifth. (It says “canon at the fourth” in the video, but to me it looks like a fifth. Mind, I am a chimpanzee in such matters and happy to be corrected.)

I’m not sure why these are notated in a circular pattern; once around and it’s done. Perhaps it’s just because it’s pretty on the page. But the intricacy of the musical construction is still delightful to behold. My thanks to Jordan Alexander Key for making the video.

Macaulay: For the Children’s Sake

February 6, 2018

For the Children’s Sake
Foundations of Education for Home and School
Susan Schaeffer Macaulay
(Crossway, 1984)
x + 165 p.

I’ve had a middling, half-dormant interest in educational theory since I was myself a student, but becoming a parent, which includes becoming responsible for the education of new persons, not to mention becoming a de facto teacher in many respects, raises the issues afresh. Actually, it’s that the practical decisions about what school one’s children should attend become pressing, and so one begins to evaluate educational practices, and resorts to theory only in an effort to think things through clearly and consistently. It is true that people have been arguing about how best to educate the young for 3000 years, and nobody has settled the main questions yet (such as “What is the point of education?” and “Why are Teachers’ Colleges not carpet-bombed?”), but, still, perhaps the effort is not entirely worthless.

This little book introduces us, at one remove, to the thought and practices of Charlotte Mason (1842-1923), an English educator who lies outside the mainstream tradition, but whose ideas have, in the past few decades, become fairly influential in the homeschool movement in North America, largely because of this book, or so I surmise.

Mason’s starting point is disarmingly simple: children are people, and should be treated as such. Teaching is a personal encounter; so is learning. As such, each child should be allowed and encouraged to develop at his or her own rate and in his or her own way, not made to comply with an set of impersonal objectives and milestones. Writes Mason,

“We must know something about the material we are to work upon if the education we offer is not to be scrappy and superficial. We must have some measure of a child’s requirements, not based on his uses to society, nor upon the standard of the world he lives in, but upon his own capacity and needs.” (from Toward a Philosophy of Education)

This basic commitment explains why she is a marginal figure in the world of public education, for it is difficult to give a mass education model this personal touch (though, to be fair, Mason was herself a schoolteacher and developed her ideas in that context). It is equally clear, and for the same reasons, why homeschoolers have picked her up, for this approach is eminently suitable to their situation.

Mason also believed that since formal education is a preparation for life after formal education, education should be like life. It should be a matter of enjoyment and interest. Children should be encouraged to be motivated by factors intrinsic to education, like curiosity and a love of learning, not factors extrinsic, like grades. We should aim to foster a loving, joyful environment for learning, in which the pleasure of learning is taught by example.

Since students are people, part of their education consists in developing those stable habits of thought and action that will enable them to be successful students and people: the virtues. In this Charlotte Mason is consistent with the classical tradition, for which moral formation is at least as important as intellectual formation. She sought especially to encourage the scholarly virtues in her students: attention, concentration, self-control, and truthfulness.

Since children are persons and their education is our responsibility, we owe them, Mason reasons, the best we have. They should therefore be invited to experience and enjoy our best books, our finest art, our most beautiful music, and so forth. She had a word for educational material that condescends to the child’s intelligence, moral judgment, or aesthetic sensibility: “twaddle”. It’s helpful to have a word, because schools are full of the stuff. We can safely assume, with little risk of error, that all politically-motivated educational materials are twaddle. We want to avoid it:

“The person rises to understand, master, and enjoy whatever he is surrounded with in language, ideas, literature, and in appreciation of beauty. If you share with children the very best, carefully chosen to meet their needs, they will amaze everyone.”

This is actually true. In our home I’ve seen it especially with the music that the kids like. It is my practice not to play pop music at home or in the car, so they do not have much exposure to it; we listen to classical music. At the same time, they are members of a children’s choir in which they sing good sacred music: Mozart, Schubert, Handel, and Gregorian chant. There is no condescension to “children’s tastes”, and they rise to the occasion. My six-year-old son sings Latin motets to himself while building Lego. Our three-year-old’s favourite music is Vivaldi’s Gloria; he sings it in his bed at night. Children will feast on what we feed them, but they are, at first, poor judges of quality. Much of teaching consists in supplying a steady diet of good quality nourishment for their minds and hearts.

And not only should children be given the best we have to give them, but they should be invited to experience and enjoy it on their terms, not ours, taking from it what they find, not what we think they should find. In real life, when we read a novel, we all do so to engage with the story we are reading, but how many of us would persist if, upon finishing each chapter, we had to answer a series of questions about it? Is it wise, then, to ask students to do this? Mason thought not, and therefore counselled against reading comprehension tests. Instead, she had her students do “narrations”, in which they would re-tell, in their own words and after their own manner, a story they had read. When you stop to think about it, this is a brilliant and beautiful idea, for everyone loves to talk about something they enjoyed reading, and, more to the point, narrating a story requires a much more thorough and nuanced and personal engagement with a book than does answering a set of specific questions. Try it.

In fact, narration is pretty much exactly what I’ve been doing on this blog all these years; I can speak from experience: it’s rewarding. Since reading this book (some months ago now), I’ve also been having my daughter give narrations of some of the things she’s been reading, and she, too, responds wonderfully to the challenge. I’m a believer.

Since we want children to engage personally with what they read and learn, another of Mason’s recommendations is that they be given real books to read, whole and complete, rather than compilations of short excerpts from longer books, because doing so puts them into sustained contact with another person — the author — with whom they then begin to develop a relationship.

Indeed, the development of relationships — with God, with the natural world, and with other people — is a key organizing principle for a Charlotte Mason-style education. A relationship with God is best developed through experiencing a faith lived joyfully, with prayer and devotion, in the home and school. A child’s understanding of the natural world she thought best fostered by direct contact with nature, through nature walks, in which close observation and full sensory immersion are encouraged. (In many Charlotte Mason homeschools, it seems that such nature walks take the place of a science textbook, at least for younger students.) Relationships with others are cultivated, as we have already said, through books, historical studies (taking care to try to understand the complexity and foreignness of the past, not interpreting everything within contemporary frameworks or judging by contemporary standards), and interactions with the teacher and with other students. And children must also get to know themselves, which was one reason why Mason believed that children’s lives should have plenty of time and space for unstructured imaginative play; I agree with her heartily on that.


Susan Schaeffer Macaulay’s purpose in this book was to rescue Charlotte Mason’s ideas from the obscurity into which they had fallen, and to present them as providing a compelling educational philosophy for today. She writes well, both about the principles and ideas of Charlotte Mason, and about her own education, which was consonant with Mason’s approach on many points. She has a gentle, thoughtful authorial voice, and, unlike many authors of popular books on education, is not garrulous. (Incidentally, I was surprised to discover, mid-way through the book, that she is the daughter of Francis Schaeffer, the guru of intellectual-leaning Evangelical Christians of a certain vintage.) Although the book is about Charlotte Mason’s approach to education, direct quotes from her are rare, and so it is a little unclear to me how much of the book’s content derives from her, and how much is Macaulay’s interpretation and elaboration.

In a book on education Roger Scruton argued that the purpose of education is not principally to benefit the student who receives it, but to ensure that the culture to which that student belongs is received and perpetuated. I found the impersonal slant of this view jarring, even as I could see the point he was making. The much more personal approach to education proposed in this book is not really inconsistent with Scruton’s concerns though; Macaulay does not propose that there be no core curriculum, or that students, though encouraged to encounter books with their own native intelligence and feeling, somehow create value in things simply by liking them. Rather, she proposes a curated education, in which children encounter what, in the judgment of their teacher, is the richest and most worthy material, whether it be literature, art, music, or what have you. In fact it seems a perfect vehicle for passing on and truly appropriating a cultural tradition, which was what Scruton was advocating.

Circular musical notation

February 3, 2018

For no very evident reason I’ve recently come across a few pieces of music for which the score has a peculiar feature: the music is notated in a circle, rather than on the usual linear staves.

The first is a piece by Baude Cordier, a fifteenth-century French composer, which has been preserved in the famous Chantilly Codex. It is entitled Tout par compas suy composes (“From a compass I am composed“), and the title gives a clue to the rationale for the odd notation: this is a circle canon, in which the voices wrap around and repeat, just like a circle, with the two discantus voices singing in canon. This excellent video animates the score so that we can follow along more readily:

This is not the only eccentric score from Cordier; he also notated a love song, Belle, Bone, Sage, in such a way that the staves form a heart.

The second example comes from George Crumb, a 20th century composer. Crumb is rather famous for his unorthodox musical scores. (His score for Black Angels is an extreme example.) Today though we will look at his circular score for The Magic Circle of Infinity, a playful piece for piano. It sounds a bit like a mad music-box, and I can imagine a mad little ballerina spinning on top of it as the score spins.

Another fun score by Crumb is his Spiral Galaxy, which spirals.

Thanks to Jordan Alexander Key for making these videos. His YouTube channel is a good one.