Modal logic, necessity, and science fiction

A necessary truth is one that is true in all possible universes. We can capture the concept of necessary truth with the 4 rules of S4 modal logic (where □ is read “necessarily”):

• if P is any tautology, then  □ P
• if  □ P  and  □ (P ⇒ Q)  then  □ Q
• if  □ P  then  □ □ P
• if  □ P  then  P

For those who prefer words rather than symbols:

• if P is any tautology, then P is necessarily true
• if P and (P implies Q) are both necessarily true, then Q is necessarily true
• if P is necessarily true, then it is necessarily true that P is necessarily true
• if P is necessarily true, then P is true (in our universe, among others)

The first rule implies that the truths of mathematics and logic (□ 2 + 2 = 4, etc.) are necessary truths (they must obviously be so, since one cannot consistently imagine an alternate universe where they are false). The second rule implies that the necessary truths include all logical consequences of necessary truths. The last two rules imply that  □ P  is equivalent to  □ □ P,  □ □ □ P,  etc. In other words, there is only one level of “necessary” that needs to be considered.

As it stands, these rules only allow us to infer the truths of mathematics and logic (such as  □ 2 + 2 = 4). One must add other necessary axioms to get more necessary truths than that. A Christian or Muslim might, for example, add “Necessarily, God exists,” and spend time exploring the logical consequences of that.

Countless things that are true in our universe are not necessarily true, such as “Water freezes at 0°C” or “Trees are green” or “Bill Clinton was President of the United States in the year 2000.”

For historical truths like the latter, it’s obvious that they are contingent on events, rather than being necessary. There is a substantial body of “alternate history” fiction which explores alternatives for such contingent truths, such as these four novels (pictured above):

• Fatherland (Robert Harris, 1992): a detective story set in a universe where Hitler won the war; it is the week leading up to his 75^th birthday (3.99 on Goodreads)
• The Peshawar Lancers (S.M. Stirling, 2002): European civilisation is destroyed by the impact of comet fragments in 1878; a new Kiplingesque Anglo-Indian steampunk civilisation arises (3.86 on Goodreads)
• SS-GB (Len Deighton, 1978): Hitler defeats Britain in 1941; British police face moral dilemmas cooperating with the SS (3.74 on Goodreads)
• Romanitas (Sophia McDougall, 2005): the Roman Empire is alive and well in present-day London; slaves are still crucified (3.24 on Goodreads; first of a trilogy)

Three plant pigments: green beech, brown kelp, and red gracilaria algae (cropped from photographs by Simon Burchell, Stef Maruch, and Eric Moody)

The truths of biology are just as contingent as the truths of history. Trees are (mostly) green, but even on our own planet, brown and red are viable alternative colours for plants. From an evolutionary perspective, Stephen Jay Gould expresses the contingency this way:

“any replay of the tape [of life] would lead evolution down a pathway radically different from the road actually taken.” (Stephen Jay Gould, Wonderful Life: The Burgess Shale and the Nature of History, 1989)

(some of his colleagues would take issue with the word “radically,” but still accept the word “different”). From a Christian point of view, the contingency of biology follows from the doctrine of the “Free Creation” of God, independently of any beliefs about evolution. To quote Protestant theologian Louis Berkhof:

“God determines voluntarily what and whom He will create, and the times, places, and circumstances, of their lives.” (Louis Berkhof, Systematic Theology, Part I, VII, D.1.c)

The Catholic Church shares the same view, as none other than Thomas Aquinas makes clear (using the terminology of necessary truth):

“It seems that whatever God wills He wills necessarily. For everything eternal is necessary. But whatever God wills, He wills from eternity, for otherwise His will would be mutable. Therefore whatever He wills, He wills necessarily. … On the contrary, The Apostle says (Ephesians 1:11): ‘Who works all things according to the counsel of His will.’ Now, what we work according to the counsel of the will, we do not will necessarily. Therefore God does not will necessarily whatever He wills.” (Summa Theologiae, Part I, 19.3)

Having taken this line, one might ask why mathematical truths are necessary rather than contingent. The astronomer Johannes Kepler resolves this problem this by telling us that they are not created:

“Geometry existed before the Creation, is co-eternal with the mind of God.” (Johannes Kepler, Harmonices Mundi)

In fiction, alternative biologies are normally explored in the context of some other planet, because alternate earths are pretty much logically equivalent to other planets. Here are four examples of fictional biology:

• Out of the Silent Planet (C.S. Lewis, 1938): written from a Christian perspective, this novel has three intelligent humanoid alien species living on the planet Mars (3.92 on Goodreads; see also my book review)
• The Mote in God’s Eye (Larry Niven and Jerry Pournelle, 1974): this novel is one of the best alien-contact novels ever written (4.07 on Goodreads)
• the xenomorph from the film Aliens (1986)
• the Klingon character Worf from the TV series Star Trek: The Next Generation (1987–1994)

The truths of physics are contingent as well; our universe could have been set up to run on different rules. Science fiction authors often tweak the laws of physics slightly in order to make the plot work (most frequently, to allow interstellar travel). Fantasy authors invent alternate universes which differ from ours far more dramatically:

• Dune (Frank Herbert, 1965): faster-than-light travel is a feature of the plot; it follows that interstellar navigation requires looking into the future (4.25 on Goodreads; see also my book review)
• Great North Road (Peter F. Hamilton, 2012): “Stargate” style portals are a key feature of this novel (4.07 on Goodreads)
• The Many-Coloured Land (Julian May, 1981): a science fiction incorporating psychic powers (4.07 on Goodreads; first of a series)
• Magician (Raymond E. Feist, 1982): a classic fantasy novel which explores some of the internal logic of magic along the way (4.31 on Goodreads; first of a series)

Because mathematical truths are necessary truths, they are potentially common ground with intelligent aliens. This is one theme in the book (later film) Contact:

“‘No, look at it this way,’ she said smiling. ‘This is a beacon. It’s an announcement signal. It’s designed to attract our attention. We get strange patterns of pulses from quasars and pulsars and radio galaxies and God-knows-what. But prime numbers are very specific, very artificial. No even number is prime, for example. It’s hard to imagine some radiating plasma or exploding galaxy sending out a regular set of mathematical signals like this. The prime numbers are to attract our attention.’” (Carl Sagan, Contact, 1985; 4.14 on Goodreads)

Of course, Carl Sagan or his editor should have realised that 2 is prime. Even intelligent beings can make mistakes.

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In this post series: logic of necessary truth, logic of belief, logic of knowledge, logic of obligation

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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.”

Photo: Katie Lee Osborne

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.

Perelandra by C. S. Lewis: 5 stars

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Fairy tale retellings

Little Red Riding Hood, as depicted by Gustave Doré (1883)

A few years ago, I blogged about fairy tales. “About once every hundred years some wiseacre gets up and tries to banish the fairy tale,” C.S. Lewis wrote in 1952, and Richard Dawkins had done exactly that.

Fairy tales are stories that have stood the test of time, and that means they have power. That power can be harnessed to teach science to children, but I don’t want to talk about that today; I want to talk about fairy tale retellings, which have become popular again in recent years.

It seems that Einstein did not say “If you want your children to be intelligent, read them fairy tales” – but fairy tales do develop the imagination and speak to the human heart. And retellings keep fairy tales fresh.

In the Forests of Serre (2003) by Patricia A. McKillip

Fairy tales are generally classified as fantasy, and most retold fairy tales fall within that genre too. Among my favourites are the dream-like novels of Patricia A. McKillip, including In the Forests of Serre (2003), which incorporates Slavic tales of Baba Yaga and the Firebird. In fact, pretty much everything that Patricia A. McKillip has written is superb.

“This Mortal Mountain” (1967), a novelette by Roger_Zelazny, collected in The Doors of His Face, the Lamps of His Mouth (1971) and This Mortal Mountain (2009)

Fairy tales can be retold as science fiction too. After all, “any sufficiently advanced technology is indistinguishable from magic.” In “This Mortal Mountain” (1967), Roger Zelazny mashes together Sleeping Beauty (or “Doornroosje” as I first learned to call it) with Dante’s Purgatorio, in a story of mountain-climbing on a distant planet: “‘A forty-mile-high mountain,’ I finally said, ‘is not a mountain. It is a world all by itself, which some dumb deity forgot to throw into orbit.’ … I looked back at the gray and lavender slopes and followed them upward once more again, until all color drained away, until the silhouette was black and jagged and the top still nowhere in sight, until my eyes stung and burned behind their protective glasses; and I saw clouds bumping up against that invincible outline, like icebergs in the sky, and I heard the howling of the retreating winds which had essayed to measure its grandeur with swiftness and, of course, had failed.”

The spell described in this novelette is purely technological, but yet the story reduces me to tears every time I read it: “The planes of her pale, high cheeks, wide forehead, small chin corresponded in an unsettling fashion with certain simple theorems which comprise the geometry of my heart.”

The Lunar Chronicles, which I have not read, are a series of young adult science fiction fairy tale retellings, so the science fiction spin still exists.

Faerie Tale (1988) by Raymond E. Feist

Many fairy tales were originally intended to be scary. The terror of walking through a wolf-infested forest armed with, at most, a knife for protection is something that is difficult to imagine today, when Canis lupus is so much less common in the wild than it used to be. Deliberately swimming in shark-infested waters is perhaps the closest modern equivalent. Added to the wolves, bears, trolls, and giants, fairy tales also frequently have supernatural threats. In Faerie Tale (1988), Raymond E. Feist retells some Irish mythology as the straight horror it was perhaps once meant to be.

The Mountain of the Wolf (2016) by Elisabeth Grace Foley

Fairy tales can also be retold with great success as Westerns. As with science fiction retellings, the frontier elements of danger and of the unknown help to set the scene. A particularly good example is The Mountain of the Wolf (2016), in which Elisabeth Grace Foley retells Little Red Riding Hood (or “Roodkapje” as I first learned to call it), but with a believable motivation for Red Riding Hood’s presence in the danger zone (I grew up with a Dutch children’s game that acted out the story; Red Riding Hood’s motivation in the original tale always struck me as confused).

The Reluctant Godfather (2017) by Allison Tebo

Finally, fairy tales can be twisted. The outcome may be altered; the hero may become the villain; the beautiful dragon may be rescued from a ravening princess. This can become very dark, bordering on horror, or it may be light comic fantasy. And amusing recent example of the latter is The Reluctant Godfather (2017), a retelling of Cinderella by Allison Tebo in which the fairy godmother is (a) male and (b) totally uninterested in helping Cinderella out. In the movie world, Hoodwinked! is a well-known example of the twisted fairy tale in its comic form.

So there you have it. How do you take your fairy tales: black, or with cream and sugar?

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What is Magic?

I have been reading some interesting fantasy novels recently by Rabia Gale: novels which blur the line between magic and technology (four stars for Mourning Cloak, by the way). That prompted me to ask: where does that line actually fall? Arthur C. Clarke once said, “any sufficiently advanced technology is indistinguishable from magic.” So is there a line at all? What is magic, exactly?

One might perhaps define magic as the manipulation of the world using forces other than those commonly recognised. Exactly what that involves depends on where one thinks magic comes from.

(A): One might consider magical abilities to be divine gifts. God gives Moses the ability to carve a path through the Red Sea, for example. In practice, this category of activity is not normally called “magic,” and is more likely to go under the name of “miracle.” Generally, it is understood as divine manipulation of the world through a person, rather than by that person.

(B): The opposite scenario is where the magical gifts come from some darker power. In Navajo mythology, for example, witches gain their power by committing intrinsically evil acts, such as murder or incest. Obviously, magic of this kind must be avoided. Indeed, the Navajo prefer not even to speak of it. However, some form of this kind of magic is often used by the “bad guys” in fantasy.

(C): A third option is that some individuals are simply born with magical abilities. This is distinguished from case (A) in that there is normally an entire class of magical people or beings, and these people or beings have some form of free will regarding the use of their abilities. In the works of J.R.R. Tolkien, the Maiar (angels) and the Valar (archangels) fall into this category, as do the elves. Free will implies that some of these entities turn to evil: in the works of Tolkien, this includes Melkor or Morgoth (one of the Valar), and Saruman, Sauron, and the Balrogs (all Maiar of various kinds).

This approach to magic is a staple of young adult (YA) literature, in part because it encourages adolescents to think about the talents that they may have been born with, and how those talents will (or should) influence the trajectory of their lives. One thinks of Ged in A Wizard of Earthsea, for example, or of Pug in Feist’s Magician. Indeed, it is not only young adults who find this theme compelling.

(D): The fourth approach to magic is that it is a totally natural part of the universe. Anyone can learn to manipulate the universe with magic, just as one can learn electrochemistry or thermodynamics. Approached in this way, magic simply becomes a kind of fictional science.

There is also a combination of (C) and (D) where magic requires a combination of study and natural talent. This naturally leads to a “school for mages” novel, which can be a little dull (in my opinion, at least), unless the author finds a way of extracting the protagonist from the school (as in A Wizard of Earthsea), or of creatively subverting the whole idea of the school (as in The Bards of Bone Plain).

Human magic in the works of J.R.R. Tolkien seems in most cases to be a combination of (B) and (D). When human beings attempt to seize what comes naturally to elves, the outcome seems to inevitably be one of darkness and shadow, with the Nazgûl (Ringwraiths) being the extreme example. There is a parable here which applies also to technology, much like Michael Crichton’s Jurassic Park and his other cautionary tales. Any sufficiently misguided technology, it seems, is indistinguishable from black magic.

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Superman is boring

Superman, as orginally described, was invulnerable. Having a hero with superpowers that are too strong makes for a boring story, because a good story needs conflict. There are several ways of handling that, of course.

1. Kryptonite weakens my powers

According to some accounts, kryptonite was invented specifically to make Superman less invulnerable and boring (Paul Fairchild explains why this was a bad decision). Kryptonite, of one kind or another, is a classic solution to the problem of an overly strong superhero which, to some extent, has been used by multiple authors. It can be overused, however. If your superhero is always weak, why have such a character at all? A better variation of this approach is for the protagonist to carry his or her own metaphorical kryptonite inside, as some kind of “fatal flaw.”

2. My powers come at a heavy cost

This is one of the easiest ways for an author to ensure that his or her character does not overuse their superpowers. These superpowers may cause pain, coma, physical harm, or other damage that enforces a break between uses of the superpowers. For example, the psychic Greg Mandel in Peter F. Hamilton’s Mindstar Rising and its two sequels suffers severe headaches when his powers are used to excess. Variations of this approach are used in a number of fantasy novels.

3. My powers disturb or frighten me

A good example of this option is Doctor Who, in the eponymous TV series, who often needs to be talked into taking action. The advantage of this approach is that it produces a great deal of interesting dialogue on why the superpowers are disturbing or frightening.

4. I am still learning to use my powers

This option is particularly common in young adult fiction. It allows the author to have an attempted use of powers either succeed or fail at any point; but this makes sense with a young protagonist. The young magician Pug in Raymond E. Feist’s Riftwar Saga is a good example. So is Luke Skywalker in the original Star Wars movie trilogy. To some extent, Aragorn in The Lord of the Rings can be viewed as having a combination of (3) and (4). But, however the author does it, I think that some limitation on superpowers is essential for a story to remain interesting. What do you think?

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On children’s novels

Having previously mentioned children’s fantasy literature, I thought I might follow up the theme a little more. In one kind of children’s novel, a group of child protagonists resolve some serious dilemma with little or no adult assistance (generally, the plot involves adventure of some kind). Such a story prompts the child reader to ask: “How would I handle a problem like that?” Of course, such a story also requires a reason for the lack of adult help. There appear to be four main possibilities, which I list below. Can my readers think of any others?

1: Complicit Adults

One common reason for the children to act independently is that the adults are complicit in an evil that the protagonists struggle against. A classic example is The White Mountains by John Christopher (1967). Thirteen-year-old Will is looking forward to his coming-of-age ceremony, until he discovers that this involves a mind-control device being implanted in his brain by the aliens who control the Earth. This kind of novel inevitably focuses on escape – in Will’s case, a dramatic journey on foot from England to the “White Mountains” of the title.

The White Mountains by John Christopher (1967).

2: The Desert Island

Another option involves stranding the protagonists in a location without helpful adults. The journey to this location can be undertaken through either physical or magical means, and the focus of the novel is on achieving some goal(s) and then returning home. A famous example is The Wonderful Wizard of Oz by L. Frank Baum (1900).

The Wonderful Wizard of Oz by L. Frank Baum (1900).

The “desert island” option can be combined with option #1 – that is, there are adults on the island, but they are pirates (or evil in some other way). An example would be the planet Camazotz in A Wrinkle in Time by Madeleine L’Engle (1963).

A Wrinkle in Time by Madeleine L’Engle (1963).

3: The Unbelievable Story

Another way of ruling out adult help is by making the dilemma completely unbelievable, as in Something Wicked This Way Comes by Ray Bradbury (1962). “‘The Chief of Police,’ Will said. ‘He’d listen to us—’ ‘Yeah,’ said Jim. ‘He’d wake just long enough to send for the butterfly net. Hell, William, hell, even I don’t believe what’s happened the last twenty-four hours.’”

Something Wicked This Way Comes by Ray Bradbury (1962).

A somewhat less satisfactory variation of this option is simply to make the adults too busy to pay attention. Alternatively, the “unbelievable story” option can be combined with the first two. For example, The Lion, the Witch and the Wardrobe by C.S. Lewis (1950) combines elements of all three options.

The Lion, the Witch and the Wardrobe by C.S. Lewis (1950).

4: The All-Out War

Finally, the dilemma can involve a conflict which is so serious that it forces the recruitment of what are essentially child soldiers. The City of Gold and Lead (John Christopher, 1968) and The Pool of Fire (John Christopher, 1968), the sequels to The White Mountains, are two examples.

The City of Gold and Lead by John Christopher (1968).

Other examples of this option include Ender’s Game (Orson Scott Card, 1985) and Tomorrow, When the War Began (John Marsden, 1993).

Ender’s Game by Orson Scott Card (1985).

Of course, there are many other kinds of children’s literature, and protagonists may also be adults, talking animals, hobbits, or other creatures. But exploring those is out of scope for this post, which is already an odd one for a science blog.

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On fairy tales

“About once every hundred years some wiseacre gets up and tries to banish the fairy tale,” C.S. Lewis wrote in 1952. The wiseacre of our time seems to be Richard Dawkins who, two years ago, told the world that fairy tales could be harmful because they “inculcate a view of the world which includes supernaturalism” (he had said similar things in 2008). In a later clarification, he added that fairy tales could “be wonderful” and that they “are part of childhood, they are stretching the imagination of children” – provided some helpful adult emphasises that “Do frogs turn into princes? No they don’t.”

But many scientists grew up with, and were inspired by, fantasy literature. For example, Jane Goodall tells of growing up with the novel The Story of Doctor Dolittle (as I did!). In fact, many science students and professional scientists avidly read fantasy literature even as adults (as they should). The booksthatmakeyoudumb website lists, among the top 10 novels read at CalTech and MIT, Harry Potter, Dune, and The Lord of the Rings. And Alice in Wonderland was written by a mathematician.

This is a science blog, so I have a strong emphasis on scientific truth, which tells us many important ecological and physiological facts about, for example, frogs. Without science, we’d all still be struggling subsistence farmers. But there is actually more than scientific truth out there.

There is also mathematical truth. Are the links in this frog network all equivalent? Yes, they are – but that is decided by mathematical proof, not by scientific experiment. It is in fact a purely abstract mathematical question – the background picture of the frog is actually irrelevant.

And there is ethical truth. Is it OK to eat frog’s legs? Science does not give us the answer to this (although logic can help us decide if our answer is consistent with our other beliefs), but fantasy literature often helps us to explore such ethical questions. Tolkien’s The Lord of the Rings is one superb example. Would you “snare an orc with a falsehood”? Would you attempt to take the One Ring and “go forth to victory”?

There is metaphorical truth. A frog may, in spite of what Dawkins says, be a handsome prince – there’s more to the universe than can be seen at first glance. Or, as Antoine de Saint-Exupéry put it, “What is essential is invisible to the eye.” Children often learn this important fact from fairy tales.

And there is even religious and philosophical truth. Does the frog-goddess Heqet exist, for example? Does the universe exist? Is there a spoon? The methods of philosophy are different from the methods of science, and some amateur philosophers simply state their beliefs without actually justifying them, but philosophy is actually very important. Science itself is based on certain philosophical beliefs about reality.

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