Four ways

Following my review of the book Piranesi by Susanna Clarke, I wanted to say something about different ways of seeking knowledge. I see four fundamental options, which I list below, and illustrate graphically above (click to zoom).

P & P (agreement / synthesis)

I use the formula P & P to reflect the situation where different ways of thinking – such as Science, Art, and Religion – are all telling the same story, and therefore form part of a grand cultural synthesis. This was a characteristic of medieval thought in Europe, where Art frequently told religious stories, and Thomas Aquinas had integrated Religion with the best available Science of his day. Perhaps the pinnacle of the medieval approach is the poetry of Dante Alighieri (depicted above), where Religion and Science are combined together with poetic Art. But that was 700 years ago, of course.

P & Q (complementarity)

I use P & Q to reflect the situation where Science, Art, Religion, etc. are seen as complementary but incommensurable. They all produce their own kind of “truth” (P versus Q). I can study the stars, but independently of that, I can also see them as beautiful. For the case of Science and Religion, Stephen Jay Gould has called this approach non-overlapping magisteria.

The problem with this approach is a kind of fragmentation of life. Art is distinguished from Technology in ways that the ancient Greeks would have found bizarre. Increasingly, people seem to be fighting against this situation.

P > ~P (over-riding)

I use P > ~P to reflect the situation where Science, Art, Religion, etc. are seen as contradictory (P versus not P), but one source of “truth” is seen as superior to, and thus over-riding, the others. This includes the case of religious people who do not believe that observation of the universe can produce valid truth. It also includes scientism, or the belief that Science trumps everything else (a doomed approach, because the foundations of Science are themselves not scientific; they are philosophical and mathematical). I have illustrated this option with the depiction of Isaac Newton by William Blake. This was not intended to be a positive depiction; around about the same time Blake famously wrote “May God us keep / From Single Vision and Newton’s sleep.

The novel Piranesi touches on the problems of scientism: “It is a statue of a man kneeling on his plinth; a sword lies at his side, its blade broken in five pieces. Roundabout lie other broken pieces, the remains of a sphere. The man has used his sword to shatter the sphere because he wanted to understand it, but now he finds that he has destroyed both sphere and sword. This puzzles him, but at the same time part of him refuses to accept that the sphere is broken and worthless. He has picked up some of the fragments and stares at them intently in the hope that they will eventually bring him new knowledge.

P & ~P (contradiction / chaos)

Finally, I use P & ~P to reflect the situation where Science, Art, Religion, etc. are seen as contradictory (P versus not P) but the contradiction is embraced. Your “truth” may be completely contradictory to my “truth,” but that’s OK. The result of this is a kind of postmodernist chaos that seems to me fundamentally unstable. Indeed, former adherents of this approach seem now to be moving towards a new single dominant metanarrative.

So those are four ways of seeking knowledge. Can we indeed live with contradiction? Can the problems of complementarity be resolved? Or is it possible to construct some new synthesis of Science, Art, Religion, and other ways of seeking knowledge? The novel Piranesi raises some interesting questions, but gives no answers, of course.

Artwork from a Florentine artist, Ryan N. McFarlane/U.S. Navy, Auguste Rodin, William Blake, and Ivan Ayvazovsky.


Piranesi: a book review


Piranesi (2020) by Susanna Clarke

I have been reading a fabulous new book called Piranesi by Susanna Clarke, the author of Jonathan Strange & Mr Norrell. The title of her new novel is drawn from the Italian artist Giovanni Battista Piranesi, and it takes place within an enormous and magical flooded House that is reminiscent of some of Piranesi’s art. “The Beauty of the House is immeasurable; its Kindness infinite,” Susanna Clarke writes. Adding to the enjoyment of this wonderful novel has been a series of podcasts by Joy Marie Clarkson (starting here).


The Prisons – A Wide Hall with Lanterns by Giovanni Battista Piranesi (1745)

There are multiple references to the Narnia stories of C.S. Lewis. One example is the similarity of the Albatross scene to the one in The Voyage of the Dawn Treader. Another is the way that “Valentine Andrew Ketterley” of “an old Dorsetshire family” (Part 4) suggests Uncle Andrew Ketterley from The Magician’s Nephew: “The Ketterleys are, however, a very old family. An old Dorsetshire family ….”

Working through this novel, I’ve been repeatedly struck with a strange sense of déjà vu. Either Susanna Clarke and I read the same books, or she is revealing to me something that, in an inarticulate way, I already knew. Or possibly both. That said, some of the echoes I see to other books are, no doubt, coincidence.


Some fan art of mine, prompted by the novella Rain Through Her Fingers by Rabia Gale, which is set in a flooded city that Piranesi reminds me of

I am reviewing the novel here on ScientificGems because it has a lot to say about Science, Knowledge, and how to relate to the World: “I realised that the search for the Knowledge has encouraged us to think of the House as if it were a sort of riddle to be unravelled, a text to be interpreted, and that if ever we discover the Knowledge, then it will be as if the Value has been wrested from the House and all that remains will be mere scenery. The sight of the One-Hundred-and-Ninety-Second Western Hall in the Moonlight made me see how ridiculous that is. The House is valuable because it is the House. It is enough in and of Itself. It is not the means to an end.” (Part 2). This recalls something that C.S. Lewis wrote in The Abolition of Man: “For magic and applied science alike the problem is how to subdue reality to the wishes of men …” Indeed, Susanna Clarke makes us ask “is Science truly our friend?”

More specifically, Susanna Clarke argues against Reductionist views of the world, and the need to approach the objects of study with Love: “It is a statue of a man kneeling on his plinth; a sword lies at his side, its blade broken in five pieces. Roundabout lie other broken pieces, the remains of a sphere. The man has used his sword to shatter the sphere because he wanted to understand it, but now he finds that he has destroyed both sphere and sword. This puzzles him, but at the same time part of him refuses to accept that the sphere is broken and worthless. He has picked up some of the fragments and stares at them intently in the hope that they will eventually bring him new knowledge.” (Part 7)

One may count the petals of a violet, for example, and grind it up to extract the ionones and anthocyanins responsible for odour and colour. But something has been lost in so doing, and the resulting description does not exhaust everything that can be said about the flower. This problem is amplified for those who do not themselves experience the flower, but rely on descriptions by others.

The novel also references Plato and the importance of universals: “You make it sound as if the Statue was somehow inferior to the thing itself. I do not see that that is the case at all. I would argue that the Statue is superior to the thing itself, the Statue being perfect, eternal and not subject to decay.” (Part 6). As Lewis would say: “It’s all in Plato, all in Plato: bless me, what do they teach them at these schools!

Expanding on a statement by Tertullian (c. 160–225), Galileo famously said: “[Science] is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth.” (Galileo, Il Saggiatore, 1623, tr. Stillman Drake)

This is true, of course, but the House does not speak to us only in mathematical language.


Plato in the Musei Capitolini, Rome (photo: Marie-Lan Nguyen)

There is much more to be said about this wonderful novel. It concludes with a repetition of the words: “The Beauty of the House is immeasurable; its Kindness infinite.” There is a whole philosophy of Science there.

Goodreads rates the novel as 4.3 out of 5, and reviews of the novel are mostly glowing. The Guardian calls it an “elegant and singular novel” while the LA Review of Books says “a work of intellectual intensity.” It made the top ten fantasy novel list for the 2021 Locus Awards (although it did not win). I’m giving it four and a half stars. And let me say to my readers: “may your Paths be safe … your Floors unbroken and may the House fill your eyes with Beauty.

4.5 stars
Piranesi by Susanna Clarke: 4½ stars


The Klein Bottle

I have been thinking about the famous Klein bottle (above). To quote a limerick by Leo Moser:

A mathematician named Klein
Thought the Möbius band was divine.
Said he: “If you glue
The edges of two,
You’ll get a weird bottle like mine.”

Just for fun, here is a Game of Life glider on a Klein bottle. The top and bottom of the square are considered to be joined, so as to form a tube. The ends of the tube (vertical sides of the square) are also joined, but with a reversal of orientation (a manoeuvre not really possible in three dimensions). The glider changes colour as it changes orientation.

Video produced using the R animation package.


And the Trees Clap Their Hands: a book review


And the Trees Clap Their Hands: Faith, Perception, and the New Physics by Virginia Stem Owens (1983, republished 2005, 148 pages)

I recently read And the Trees Clap Their Hands: Faith, Perception, and the New Physics by Virginia Stem Owens. This is a book that addresses the important question “What does it all mean?” with regards to science – what does science really tell us about the world, and how should we respond to that? How can we make sense of it all in a human way?

Virginia Stem Owens was born in 1941 and became a pastor’s wife in the Presbyterian Church (USA), also gaining an MA in English literature from the University of Kansas and a Master of Arts in Religion from the Iliff School of Theology in Denver. She has written numerous books.

And the Trees Clap Their Hands was an enjoyable read, but Owens’ lack of scientific experience is responsible for several flaws in the book. I was a little disappointed at the lack of footnotes and at some glaring errors of fact. For example (p. 92), Owens confuses turbulence (a phenomenon of liquids and gases in the “Old Physics”) with Brownian motion (a microscopic phenomenon resulting from the existence of atoms). I also felt that she skipped over some important things, while not getting others quite right. I should point out, too, that the “New Physics” of the subtitle (relativity and quantum theory) is now roughly a century old. On the other hand, Owens’ writing is lyrically beautiful:

The body I am today came yesterday in a crate of avocados from California. India spins in my tea-drenched blood this morning. Minerals dissolved for millennia in a subterranean aquifer irrigate my interior, passing through the portals of my cell walls, which are themselves filigrees of chemical construction. I am really only a river of dissolute stones, the wash of world-water dammed for a melting moment in the space I call my body, some of it ceaselessly brimming over the spillway and flowing on down drains, into other tributaries, catching in some other body’s pond, until one day the whole structure cracks and buckles, giving up in one great gush its reservoir of mineraled water.” (p. 126)


What does it all mean? Answers have come from Plato, Galileo, and the Bible, to mention just three sources (bust of Plato photographed by Marie-Lan Nguyen)

Relativity and Time

Since Owens mentions relativity several times, I was surprised to see no mention of spacetime. As Hermann Minkowski wrote in 1923, “The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.

Relativity implies that there is no absolute “present moment” in the Universe, and hence that, of the three theories of time illustrated below, only the Block Universe (Eternalism) can be correct. The philosophical and religious implications of this are huge, and would have been worth discussing.


Three views of time: in Presentism, only the Present exists; in the Growing Block Universe, the past exists as well; and in the Block Universe, also called Eternalism, the universe forms a four-dimensional spacetime “block” in which the future is already written (image © Anthony Dekker)

Quantum Theory and Observation

One of the key aspects of the century-old “New Physics” is that both matter and light exist as waves and particles. The true reality is a combination of two seemingly contradictory perspectives (this has been used by other writers as an analogy for the relationship between e.g. determinism and free will). The wave function of a particle changes over time according to the Schrödinger equation. It can also undergo wave function collapse, changing from something fuzzy and spread out to something far more definite. How and why the latter phenomenon occurs remains quite mysterious.

One of the classic experiments exploring this involves firing electrons at a detector screen through a double slit:


The double slit experiment (credit: NekoJaNekoJa and Johannes Kalliauer)

Being fuzzy waves, electrons go through both slits at the same time, undergoing interference effects characteristic of waves. They are then detected as particles, with comparatively precise locations:


Individual electrons being detected by a screen after passing through a double slit (credit: Thierry Dugnolle)

Physicists used the word “observation” for the electron being detected by the screen (thus having its wave function collapse). Owens takes it for granted that this means “observation by a human mind” and that the human race therefore, in a sense, creates the universe by observing it. However, this use of the word “observation” is not what most physicists mean (indeed, the universe fairly obviously existed before there were any people). It is, in fact, not clear exactly what constitutes a wave-function-collapsing “observation,” but recent work with quantum computing suggests that it happens even when nobody’s looking (and even when you don’t want it to).

Along the lines of Gary Zukav (whom she cites), Owens tries to build a semi-pantheistic philosophy on top of this – something that is not actually justified by the physics. She also makes a big thing of “the impossibility of isolating the observer from the world” (p. 85), which is not actually a huge problem in the physical sciences, if you know what you’re doing. It’s more of a problem with animal behaviour (as in the famous example below) and an enormous problem in psychology and anthropology.


Konrad Lorenz interacting with geese in the 1930s (Are you my mother?)

Religion and God

Owens is writing from an explicitly Christian (Presbyterian) perspective, which doesn’t quite sit comfortably with the New Age Zukavian material in this book (there also appears to be some influence from Owen Barfield’s Saving the Appearances: A Study in Idolatry). And while Owens highlights the issue of nonlocality in quantum theory, she does not explore how this might relate to an omnipresent God “behind the scenes.” There are some beautifully written spiritual reflections, but the connection of the religious material to the scientific is somewhat tenuous. Owens seems to want “and all the trees of the field shall clap their hands” to be more than a metaphor, but the physics doesn’t really help with that. In addition, there seems to be some theological confusion regarding the doctrine of the Incarnation.

Goodreads gives this book a score of 3.9. In spite of the beautiful writing and genuine sense of wonder, I can’t go nearly that high (side issue: Goodreads somehow has a cover image with the wrong title!).

* * and a half
And the Trees Clap Their Hands by Virginia Stem Owens: 2½ stars


Planet Narnia: a book review


Planet Narnia: The Seven Heavens in the Imagination of C. S. Lewis (2008) by Michael Ward

More than a decade ago, on a blog that no longer exists, I reviewed Planet Narnia: The Seven Heavens in the Imagination of C. S. Lewis by Michael Ward (written a few years before Ward converted to Catholicism). Ward’s thesis was that C. S. Lewis wrote The Chronicles of Narnia based on a secret plan linking the seven novels to the seven classical planets (Sun, Moon, Mercury, Venus, Mars, Jupiter, and Saturn). This plan was allegedly so secret that Lewis shared it with none of his friends.

People are still reading Planet Narnia, so it’s worthwhile re-stating my opinions, especially since I’ve done some recent analyses that are relevant.

While Ward’s book has been widely praised, not everybody has agreed with his thesis. In an interview, Lewis’s stepson Douglas Gresham stated, “A very nice man and a friend of mine, Michael Ward, has recently written and published a book all about how the Narnian Chronicles are all based on the seven planets of the medieval astronomical system. I like Michael enormously, but I think his book is nonsense.

Devin Brown is one scholar who is critical. Brenton Dickieson is another. In fact, in a letter to a child, Lewis himself is quite clear that there was no plan: “The series was not planned beforehand as she thinks. When I wrote The Lion I did not know I was going to write any more. Then I wrote P. Caspian as a sequel and still didn’t think there would be any more, and when I had done The Voyage I felt quite sure it would be the last, but I found I was wrong.

There is something terribly seductive about correspondence theories like Ward’s. Plato, for example, argued that the atoms of the “four elements” (Earth, Air, Fire, and Water) corresponded to four of the Platonic Solids. Earth atoms were cubes, because cubes can be stacked to form shapes. Water atoms were icosahedra, because water flows and icosahedra are the roundest of the Platonic Solids. Fire atoms were tetrahedra, because fire burns, and that’s obviously a result of the sharp points on fire atoms. And by now we’re basically just assuming the hypothesis, so we simply state that air atoms are octahedra. Plato was a great philosopher, but that isn’t how reasoning is supposed to work.

In a similar vein to Justin Barrett’s “Some Planets in Narnia: A Quantitative Investigation of the Planet Narnia Thesis” (Seven, vol. 27, Jan 2010), I have explored the frequencies of 20 words or word groups associated with the seven planets (I allow for variations in word endings). The 20 bar charts show the frequencies, adjusted for the total word count of each book (but the small white numbers show actual counts). The bars list the seven Narnia books in order of internal chronology (i.e. The Magician’s Nephew first). The stars mark the book that Michael Ward thinks is associated with the relevant planet; the star is black if the relevant bar is indeed the highest.

One might add the exclamation “By Jove!,” which occurs only twice in the supposedly “Jovial” The Lion, the Witch and the Wardrobe, but 5 times in Prince Caspian and 4 times in The Silver Chair. Looking at the chart, we start well with two “hits” linking the Sun with The Voyage of the Dawn Treader, but then there are only four more hits (and most of those can be explained purely on the basis of plot). All up, 6 hits out of 20, when we would have expected 20/7, or just under 3. This is not statistically significant.

It is also worth noting that we know how Lewis wrote when he was making connections to the planets. For example, Lewis’s Perelandra is based on Venus, and Lewis throws in 7 references to the metal associated with her: “coppery” (3×), “copper-coloured” (3×), and “coppery-green” (1×). There are no such references in Out of the Silent Planet, and just one in That Hideous Strength (and that is in a reference to Venus: “I have long known that this house is deeply under her influence. There is even copper in the soil. Also – the earth-Venus will be specially active here at present.”). However, there is only one mention of copper in The Magician’s Nephew, which Ward claims is also linked to Venus (“The feathers shone chestnut colour and copper colour.”). That is to say, an important Lewisian reference to Venus is not present in any significant way in The Magician’s Nephew.

Now Ward gets around these and other problems by claiming that Lewis didn’t always follow the plan: “Nevertheless, for all its apparent ungraciousness, we can bear in mind that Lewis was unlikely to have been perfectly successful in carrying out his own plan” (p. 233). But if Lewis didn’t follow the plan, one questions what kind of “plan” it was.


Hope, Love, and Faith (photo: Anthony Dekker)

In his story “The Honour of Israel Gow,” G. K. Chesterton writes: “I only suggested that because you said one could not plausibly connect snuff with clockwork or candles with bright stones. Ten false philosophies will fit the universe; ten false theories will fit Glengyle Castle. But we want the real explanation of the castle and the universe.” In that spirit, I offer an alternate theory (the core of which was developed collaboratively) which I also blogged about more than a decade ago. Not that I think that my theory is necessarily right, just that it’s a better theory than Ward’s, and therefore casts doubt on his proposal. A theory should, after all, explain the facts better than any alternative theory.

And my theory is this: that the seven Narnia stories are linked to the Seven Virtues: Love, Faith, Hope, Prudence, Temperance, Fortitude, and Justice. This fits what we know about composition. Of course, if you thought you were writing just one Christian children’s book, it would be about Love. Of course the next two books written would be about Faith and Hope. Of course the four “cardinal virtues” would come last. So how does this work?

I get 11 “hits.” That’s 11 out of 25, because I discarded 5 options while producing the chart, but the match is still extremely significant, with p < 0.04%. Shields are a common Christian symbol of Faith (Ephesians 6:16) and they are mentioned especially often in Prince Caspian, which I associate with that virtue (and see also the line “We don’t forget. I believe in the High King Peter and the rest that reigned at Cair Paravel, as firmly as I believe in Aslan himself.”).

Anchors are a common Christian symbol of Hope (Hebrews 6:19) and they are mentioned especially often in The Voyage of the Dawn Treader (and see also “But Reepicheep here has an even higher hope… I expect to find Aslan’s own country. It is always from the east, across the sea, that the great Lion comes to us.”). The Horse and His Boy is full of bravery in the face of fear, i.e. of Fortitude (e.g. “And now at last, brave girl though she was, her heart quailed. Supposing the others weren’t there! Supposing the ghouls were! But she stuck out her chin (and a little bit of her tongue too) and went straight towards them.”).

The Silver Chair has Puddleglum to demonstrate Prudence, and the final judgement in The Last Battle demonstrates Justice. It is really only for The Magician’s Nephew that I have failed to make my case, but there one can take Uncle Andrew and Jadis as examples of the absence of Temperance (as in “… he thinks he can do anything he likes to get anything he wants.”).


Hope and anchors have a long association. This flag was embroidered by Jane, Lady Franklin for one of many expeditions searching for her lost husband (photo credit)

To quote Chesterton again: “Ten false philosophies will fit the universe; ten false theories will fit Glengyle Castle.” Another theory for the Narnia books that has been suggested to me is that they correspond to the seven liberal arts, with the first three books written corresponding to the Trivium (Grammar, Logic, and Rhetoric, in that order) and the last four books written corresponding to the Quadrivium (Arithmetic, Geometry, Music, and Astronomy).

We can count words as before, except that the category “long words” (corresponding to Rhetoric) includes all words of 11 or more letters, such as “crestfallen,” “unmitigated,” or “waterspouts.” We get 8 “hits” this way (2 more than for Ward’s theory). Allowing for discarded options, this is statistically significant, with p < 2%. The pairings of The Lion, the Witch and the Wardrobe with Grammar and Prince Caspian with Logic are rather unconvincing, but I think that this theory is still better than that of Michael Ward.

One can even find characteristic colours in the books, with the words “green,” “white,” “blue,” and “black” occurring particularly often in The Magician’s Nephew (52), The Lion, the Witch and the Wardrobe (59), The Voyage of the Dawn Treader (26), and The Silver Chair (39), respectively.

Chesterton (or, rather, Father Brown) concludes with what is really the fundamental principle of science: “But we want the real explanation of the castle and the universe.” And I don’t think that Michael Ward has the real explanation of the Narnia books. Indeed, even if one assumes that the Narnia stories follow a plan, there are better candidates for a plan than the one that Ward suggests.

Goodreads rates Planet Narnia 4.3 out of 5, because (judging by the comments) people largely seem to believe Ward’s argument, which I find so unconvincing. However, I can really only give his book two stars:

* *
Planet Narnia by Michael Ward: 2 stars


Perelandra: a book review


Perelandra (1943) by C. S. Lewis (1996 cover by Kinuko Y. Craft)

Having blogged about Out of the Silent Planet and That Hideous Strength, the first and last novels of the “Space Trilogy” or “Cosmic Trilogy” by C. S. Lewis, I should also mention Perelandra, the middle volume.

While Out of the Silent Planet is science fiction, Perelandra is better described as religious fantasy (with portions of what could be called supernatural horror). However, in a 1962 discussion with Kingsley Amis and Brian Aldiss, Lewis states that “The starting point of the second novel, Perelandra, was my mental picture of the floating islands. The whole of the rest of my labors in a sense consisted of building up a world in which floating islands could exist. And then, of course, the story about an averted fall developed. This is because, as you know, having got your people to this exciting country, something must happen.” When Aldiss responds “But I am surprised that you put it this way round. I would have thought that you constructed Perelandra for the didactic purpose,” Lewis replies “Yes, everyone thinks that. They are quite wrong.

The basic idea of the floating islands of vegetation on the ocean of Perelandra (what we call Venus) may have came from the novel Last and First Men by Olaf Stapledon (XII§4), after mankind has chosen to exterminate the native civilisation of Venus (something that Stapledon seems to approve of, but which Lewis explicitly criticised in Out of the Silent Planet): “Man now busied himself in preparing his new home. Many kinds of plant life, derived from the terrestrial stock, but bred for the Venerian environment, now began to swarm on the islands and in the sea. For so restricted was the land surface, that great areas of ocean had to be given over to specially designed marine plants, which now formed immense floating continents of vegetable matter.

In Chapter 3 of Perelandra, when Elwin Ransom first arrives on Venus, there are some wonderful descriptive passages, which go far, far beyond Stapledon’s bald statement: “It seems that he must have remained lying on his face, doing nothing and thinking nothing for a very long time. When he next began to take any notice of his surroundings he was, at all events, well rested. His first discovery was that he lay on a dry surface, which on examination turned out to consist of something very like heather, except for the colour which was coppery. Burrowing idly with his fingers he found something friable like dry soil, but very little of it, for almost at once he came upon a base of tough interlocked fibres. Then he rolled round on his back, and in doing so discovered the extreme resilience of the surface on which he lay. It was something much more than the pliancy of the heather-like vegetation, and felt more as if the whole floating island beneath that vegetation were a kind of mattress. He turned and looked ‘inland’ – if that is the right word – and for one instant what he saw looked very like a country. He was looking up a long lonely valley with a copper-coloured floor bordered on each side by gentle slopes clothed in a kind of many-coloured forest. But even as he took this in, it became a long copper-coloured ridge with the forest sloping down on each side of it. Of course he ought to have been prepared for this, but he says that it gave him an almost sickening shock. The thing had looked, in that first glance, so like a real country that he had forgotten it was floating – an island if you like, with hills and valleys, but hills and valleys which changed places every minute so that only a cinematograph could make a contour map of it.

It’s a great pity that Venus is nothing like that at all.


Lewis wrote Perelandra while a Fellow at Magdalen College, Oxford.

Much of the rest of the book consists of conversations about theology and moral philosophy between Ransom, the scientist Weston (who first appeared in Out of the Silent Planet), and Tinidril, one of the two “people” on Venus. Tinidril corresponds to Eve in the Bible, so that we get a sort of alternate history of the “Temptation of Eve.” Weston is possessed by a “Force” that turns out to be Satan or a demon. As a former academic myself, it is interesting to see Lewis’s ascending hierarchy of potential moral failings:

“RANSOM: ‘Does that mean in plainer language that the things the Force wants you to do are what ordinary people call diabolical?’
WESTON: ‘My dear Ransom, I wish you would not keep relapsing on to the popular level. The two things are only moments in the single, unique reality. The world leaps forward through great men and greatness always transcends mere moralism. When the leap has been made our “diabolism” as you would call it becomes the morality of the next stage; but while we are making it, we are called criminals, heretics, blasphemers. …’
‘How far does it go? Would you still obey the Life-Force if you found it prompting you to murder me?’
‘Yes.’
‘Or to sell England to the Germans?’
‘Yes.’
‘Or to print lies as serious research in a scientific periodical?’
‘Yes.’
‘God help you!’ said Ransom.

The chart below shows a chapter-by-chapter frequency analysis of various names and words in the book (some obvious synonyms were also used in counting words, and characters mentioned but not appearing are included). There is also a chapter-by-chapter polarity (sentiment) analysis at the bottom of the chart.

When one considers the theological subject matter, the conflict with the Un-man, and the underground scenes towards the end, the novel is a little reminiscent of Dante’s Inferno, but the final chapters are far more like the Paradiso, and (in a letter) Lewis himself tells us that some of the conversations between Ransom and Tinidril draw on Matilda in the Purgatorio. Aspects of the conflict between Ransom and Weston recall the interaction between Frodo and Gollum in The Lord of the Rings, while other aspects of Perelandra are like visiting the Elves. It is not a simple book.

Goodreads rates the novel as the best of Lewis’s trilogy, giving it 3.99 out of 5. I’m giving it five stars, but readers not interested in theology or moral philosophy would no doubt rate it lower.

5 stars
Perelandra by C. S. Lewis: 5 stars


Out of the Silent Planet: a book review

Since Mars is on my mind right now…


Out of the Silent Planet (1938) by C. S. Lewis (1996 cover by Kinuko Craft)

C. S. Lewis is famous for the Narnia novels, but more than a decade before they were written, he published Out of the Silent Planet. Lewis was to write two sequels (I have blogged about the last volume, That Hideous Strength). However, Out of the Silent Planet is essentially science fiction, while the sequels are better described as fantasy.

Considered as a work of science fiction, Out of the Silent Planet was a pioneering novel. It followed (and was influenced by) novels by H. G. Wells, David Lindsay, and Olaf Stapledon, but preceded the work of Brian Aldiss, Arthur C. Clarke, and other British science fiction authors of the “Golden Age.” Lewis’s novel was one of the first to sound a cautious note about human colonisation of other planets. In his 1958 essay “Religion and Rocketry,” Lewis describes the rather sad results that human colonisation could have: “We know what our race does to strangers. Man destroys or enslaves every species he can. Civilized man murders, enslaves, cheats, and corrupts savage man. Even inanimate nature he turns into dust bowls and slag-heaps. There are individuals who don’t. But they are not the sort who are likely to be our pioneers in space.

Lewis’s word hnau (sentient being) was borrowed by James Blish in his alien-contact novel A Case of Conscience. In Out of the Silent Planet, Lewis brings colonisation to a halt after three deaths; Ursula K. Le Guin’s novel The Word for World Is Forest is one of many where exploitative colonisation runs its course. Several ideas in the 2009 film Avatar can be traced back to classic novels that were, in turn, ultimately inspired by Lewis.

In Out of the Silent Planet, Lewis introduces not one, but four, sentient species living in harmony with each other: the hrossa (farmers and poets), the séroni (scientists and philosophers), the pfifltriggi (artists, miners, and engineers), and the eldila (angels, essentially).

The scientist Weston actually recommends exterminating the inhabitants of the planet to make room for humans, claiming “Your tribal life with its stone-age weapons and beehive huts, its primitive coracles and elementary social structure, has nothing to compare with our civilization – with our science, medicine and law, our armies, our architecture, our commerce, and our transport system which is rapidly annihilating space and time. Our right to supersede you is the right of the higher over the lower. Life … is greater than any system of morality; her claims are absolute. It is not by tribal taboos and copy-book maxims that she has pursued her relentless march from the amœba to man and from man to civilization.” (this was an idea seriously suggested in the 1920s and 1930s, but one which was perhaps seen for what it was after World War II). The businessman Dick Devine simply wants to exploit the inhabitants for profit. Only the hero, Elwin Ransom, is interested in them for who they are.


Lewis wrote Out of the Silent Planet while a Fellow at Magdalen College, Oxford.

The science in Out of the Silent Planet is surprisingly good, considering that Lewis’s area of expertise was English literature. No doubt Lewis ran his ideas past the scientists at Magdalen College. There are some clangers in Lewis’s discussion of gravity onboard the spacecraft, and he no doubt knew that the theory of Martian canals was obsolete (although Robert A. Heinlein and Ray Bradbury were still using the idea of canals in the 1940s and 1950s). Lewis does an excellent job of explaining the impact of low gravity on Martian life, of pointing out that space is not “dark and cold,” and of describing the need for oxygen on the Martian surface.

He learned from the sorn that he was right in thinking they were near the limits of the breathable. Already on the mountain fringe that borders the harandra and walls the handramit, or in the narrow depression along which their road led them, the air is of Himalayan rarity, ill breathing for a hross, and a few hundred feet higher, on the harandra proper, the true surface of the planet, it admits no life.

Bringing in a specialist linguist (Elwin Ransom) to decipher the Martian language is a neat trick (and one repeated in Poul Anderson’s The High Crusade). Lewis’s friend J. R. R. Tolkien may have been an inspiration for this character.


Lewis’s naming of the planets. Out of the Silent Planet is set on Malacandra (Mars).

Lewis’s special interest was Mediaeval and Renaissance literature. In Out of the Silent Planet he is taking the cosmology of Dante’s great trilogy (see my discussion of the Inferno, Purgatorio, and Paradiso) and putting it through a Copernican Revolution, while retaining the Christian worldview. Naïve correspondences with Dante should be resisted, however, since Lewis was not originally intending to write a trilogy. If one absolutely must draw links, Out of the Silent Planet would probably correspond to the Purgatorio, with its emphasis on permissible vs wrongful desires.

Goodreads rates this ground-breaking novel 3.92. I’m giving it four and a half stars.

4.5 stars
Out of the Silent Planet by C. S. Lewis: 4½ stars


2 + 2 = 4 and mathematical models

One of the strangest aspects of 2020 was a number of people arguing that 2 + 2 = 4 wasn’t necessarily true, and that it might be the case that 2 + 2 = 5. In fact, 2 + 2 = 4 is not only true throughout the universe, it is true in every possible alternate universe as well. A great many silly arguments were made by people trying to defend 2 + 2 = 5. But there is a deeper response here that relates to how applied mathematics works, and a few people have been trying to express that.

In general, applied mathematics questions have the form A → B, where A is some real-world situation, and B our desired but unknown answer. We abstract our real-world situation A to the mathematical object X, so that X → Y is the mathematical analogue of our real-world A → B. Mathematical questions have clearly-defined answers (although finding them is not always easy). We can then take our mathematical answer Y and reverse-translate it to the real world, giving something that we claim is a good approximation to B.

In the mathematical world of X and Y, everything is crisp and clear (and 2 + 2 = 4). In the real world, things are messy and ill-defined. Furthermore, in going from A to X we introduce simplifications and approximations which may mean that the mathematical answer Y is not “fit for purpose.”

For example, our real-world question might be “what is the distance from Melbourne to Sydney?” There are three ambiguities here: What is “Melbourne”? What is “Sydney”? And what is “distance”?

The conventional location marker for the city of Melbourne, Australia is the Old Melbourne General Post Office (now a shopping centre, at 37°48′49″S, 144°57′48″E). Likewise, the conventional location marker for the city of Sydney is the Sydney General Post Office (at 33°52′4″S, 151°12′27″E). Let’s use those coordinates. But what is “distance”? If “distance” means “as the crow flies,” then a simple answer might be to find the great-circle distance on a sphere approximating the Earth (let’s use the equatorial radius of 6,378.137 km). This distance can be calculated fairly easily as 714.2 km, which might be a close enough answer for many purposes.

A better mathematical model might be distance on the WGS reference ellipsoid model of the Earth. This gives the slightly lower value 713.8 km (according to Google Earth or raster::pointDistance).

Alternatively, “distance” might mean “by road,” in which case we need a computer representation of Australia’s road network. For this question, Google Maps reports a distance of 878 km via the M31. The expected travel time by car (9 hours and 7 minutes when I looked) might be even more useful.

It may therefore be the case that out of various mathematical answers Y, many are not “what you wanted.” But that failure to abstract correctly does not invalidate the mathematical truths involved in X → Y. In particular, it does not invalidate 2 + 2 = 4. It just means that you picked up the beautiful crystal knife of mathematics and cut yourself with it.

As Korzybski liked to say, “the map is not the territory.”


No, it is not true that 2 + 2 = 5

The year 2020 was an unusual year. One of the strangest aspects was a number of people arguing that 2 + 2 = 4 wasn’t necessarily true, and that it might be the case that 2 + 2 = 5. This is an idea that had been mentioned by George Orwell in his dystopic novel 1984:

All rulers in all ages have tried to impose a false view of the world upon their followers, but they could not afford to encourage any illusion that tended to impair military efficiency. So long as defeat meant the loss of independence, or some other result generally held to be undesirable, the precautions against defeat had to be serious. Physical facts could not be ignored. In philosophy, or religion, or ethics, or politics, two and two might make five, but when one was designing a gun or an aeroplane they had to make four. Inefficient nations were always conquered sooner or later, and the struggle for efficiency was inimical to illusions.

In fact, 2 + 2 = 4 is true regardless of your culture or your skin colour (although you might represent the fact using an alternate set of symbols, like  +  = ). If there are aliens out there, it is true for them too. What’s more, 2 + 2 = 4 is true in every possible alternate universe as well as in this one.

Some defenders of 2 + 2 = 5 have appealed to modular arithmetic, and (to take one example) modulo 3 we have 2 = { −1, 2, 5, … } and 1 = { −2, 1, 4, … }, using overbars to distinguish congruence classes from integers (in order to be precise). Consequently, 2 + 2 = 4 = 1. However, we never get 2 + 2 = 5 in such systems, other than in the trivial case where all integers are equivalent (the proof of this is actually quite straightforward). We certainly do not get 2 + 2 = 5.


Vesper Flights: a book review and reflection


Vesper Flights by Helen Macdonald

I have been waiting eagerly for a copy of Vesper Flights by Helen Macdonald. If you have read my review of her H is for Hawk, you will understand why. Early reviews of the new book were also positive – “powerful essays,” said The Guardian; “soul-stirring,” said USA Today; “a beautiful and generous book,” said npr. The Goodreads community gave it 4.2 out of 5.

I was fortunate enough to get a copy of Vesper Flights for Christmas. It is a collection of 41 essays, and Helen Macdonald writes “I hope that this book works a little like a Wunderkammer. It is full of strange things and it is concerned with the quality of wonder.” Many of the essays have an autobiographical component. Several moved me to tears.


A Wunderkammer painted by Domenico Remps around 1695 (click images in this review to zoom)

The essays in the collection are:

  1. Nests – a reflection on bird’s nests
  2. Nothing Like a Pig – coming face-to-face with a wild boar
  3. Inspector Calls – a beautifully written and touching account of an autistic boy meeting a parrot
  4. Field Guides – a visit to Australia, and praise for field guides

The hairpin banksia gets a mention in essay #4

  1. Tekels Park – reminiscences of a childhood spent among nature in Tekels Park
  2. High-Rise – a wonderful account of the surprising amount of life that can be found in the night-time sky
  3. The Human Flock – about migration
  4. The Student’s Tale – about a refugee
  5. Ants – about nuptial flights in ants

A winged queen ant (photo credit)

  1. Symptomatic – about migraines and impending doom
  2. Sex, Death, Mushrooms – “Many toxic fungi closely resemble edible ones, and differentiating each from each requires careful examination, dogged determination and often the inspection of spores stained and measured under a microscope slide.
  3. Winter Woods – walking through woods in the winter
  4. Eclipse – an eclipse is an emotional experience
  5. In Her Orbit – with Nathalie Cabrol in the Atacama Desert, site of the now-defunct Carrera Solar Atacama (this chapter is based on a New York Times article)

San Pedro de Atacama, Chile (photo credit)

  1. Hares
  2. Lost, But Catching Up
  3. Swan Uppingswan upping on the Thames as social commentary
  4. Nestboxes – are they for the birds, for us, or both?
  5. Deer in the Headlights – this essay highlights the problem of deer-vehicle collisions (the UK gets about 1 per thousand people per year); Australia has a kangaroo-vehicle collision problem of similar magnitude, but that issue is perhaps viewed a little differently
  6. The Falcon and the Tower – about urban peregrine falcons, specifically in Dublin (see also this short documentary film)

The towers of the decommissioned Poolbeg Generating Station in Dublin, with a magnification of the western (leftmost) tower. These towers, around 207 m high, are home to the peregrine falcons described in essay #20 (photo credit)

  1. Vesper Flights – the central and title essay, based on a New York Times article, is about swifts
  2. In Spight of Prisons – all about glow-worms, Lampyris noctiluca
  3. Sun Birds and Cashmere Spheresgolden orioles and bearded reedlings
  4. The Observatory – “a swan had come towards me and offered me strange
    companionship at a time when I thought loneliness was all I could feel.
  5. WickenWicken and other fens, which I imagine inspired the home of Puddleglum in the Narnia stories

A hide at Wicken Fen (photo credit)

  1. Storm
  2. Murmurations – “Words to accompany Sarah Wood’s 2015 film Murmuration x 10
  3. A Cuckoo in the House – about cuckoos and the man who inspired the character ‘M.’ Yes, that ‘M.’
  4. The Arrow-Stork – the arrow-stork and the study of bird & animal migration
  5. Ashes – on tree diseases
  6. A Handful of Corn – as a famous song says: “Come feed the little birds, show them you care, and you’ll be glad if you do; their young ones are hungry, their nests are so bare, all it takes is tuppence from you.

  1. Berries
  2. Cherry Stones
  3. Birds, Tabled – a fascinating exploration of the morality of bird-watching versus bird-keeping and the class conflicts involved (a number of reviewers online have taken issue with this chapter, specifically)
  4. Hiding
  5. Eulogy
  6. Rescue – a beautiful account of bird rescue and wildlife rehabilitation
  7. Goats
  8. Dispatches from the Valleys – a heavily autobiographical chapter, raising all kinds of spiritual questions (but not really answering them)
  9. The Numinous Ordinary – “I kept trying to find the right words to describe certain experiences and failing. My secular lexicon didn’t capture what they were like. You’ve probably had such experiences yourself – times in which the world stutters, turns and fills with unexpected meaning.
  10. What Animals Taught Me – “When I was a child I’d assumed animals were just like me. Later I thought I could escape myself by pretending I was an animal. Both were founded on the same mistake. For the deepest lesson animals have taught me is how easily and unconsciously we see other lives as mirrors of our own.

Not surprisingly, about half the chapters in this book are about birds, in some way or other:

At its best, this book is as good as the superb H is for Hawk, but is not consistently so (indeed, it scarcely could be). While some of the chapters are truly wonderful, others have a moralistic tone that I thought was a little more heavy-handed than it needed to be, and which became a little repetitive after a while. In the last chapter Helen Macdonald offers a corrective: “These days I take emotional solace from knowing that animals are not like me, that their lives are not about us at all.” Or, as C.S. Lewis once put it:

Come out, look back, and then you will see … this astonishing cataract of bears, babies, and bananas: this immoderate deluge of atoms, orchids, oranges, cancers, canaries, fleas, gases, tornadoes and toads. How could you ever have thought this was the ultimate reality? How could you ever have thought that it was merely a stage-set for the moral drama of men and women? She is herself. Offer her neither worship nor contempt. Meet her and know her. If we are immortal, and if she is doomed (as the scientists tell us) to run down and die, we shall miss this half-shy and half-flamboyant creature, this ogress, this hoyden, this incorrigible fairy, this dumb witch.

Helen Macdonald has a genuine talent for showing the reader what she saw, and the reader of a book like this will feel appropriate things in that situation. Perhaps the more moralistic tone is the inevitable, and possibly appropriate, nature of an essay written for a newspaper or magazine. The fact that this book is a collection of such essays would then explain why it feels a little repetitive at times.

My recommendation: buy this book, but only read a few chapters each week. And think about them.

* * * *
Vesper Flights by Helen Macdonald: 4 stars