0 and 1 in Greek mathematics

Following up on an earlier post about zero in Greek mathematics and this timeline of zero, I want to say something more about the role of 0 (zero) and 1 (one) in ancient Greek thought. Unfortunately, some of the discussion on Greek mathematics out there is a bit like this:

0 and 1 as quantities

The ancient Greeks could obviously count, and they had bankers, so they understood credits and debts, and the idea of your bank account being empty. However, they had not reached the brilliant insight of Brahmagupta, around 628 AD, that you could multiply a debt (−) and a debt (−) to get a credit (+).

The ancient Greeks had three words for “one” (εἷς = heis, μία = mia, ἑν = hen), depending on gender. So, in the opening line of Plato’s Timaeus, Socrates counts: “One, two, three; but where, my dear Timaeus, is the fourth of those who were yesterday my guests … ? (εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων … ; )”

The Greeks had two words for “nothing” or “zero” (μηδέν = mēden and οὐδέν = ouden). So, in the Christian New Testament, in John 21:11, some fisherman count fish and get 153, but in Luke 5:5, Simon Peter says “Master, we toiled all night and took nothing (οὐδὲν)!”

0 and 1 in calculations

In ordinary (non-positional) Greek numerals, the Greeks used α = 1, ι = 10, and ρ = 100. There was no special symbol for zero. Greek mathematicians, such as Archimedes, wrote numbers out in words when stating a theorem.

Greek astronomers, who performed more complex calculations, used the Babylonian base-60 system. Sexagesimal “digits” from 1 to 59 were written in ordinary Greek numerals, with variations of ō for zero. The overbar was necessary to distinguish ō from the letter ο, which denoted the number 70 (since an overbar was a standard way of indicating abbreviations, it is likely that the symbol ō was an abbreviation for οὐδὲν).

Initially (around 100 AD) the overbar was quite fancy, and it became shorter and simpler over time, eventually disappearing altogether. Here it is in a French edition of Ptolemy’s Almagest of c. 150 AD:

In Greek-influenced Latin astronomical calculations, such as those used by Christians to calculate the date of Easter, “NULLA” or “N” was used for zero as a value. Such calculations date from the third century AD. Here (from Gallica) is part of a beautiful late example from around 700 AD (the calendar of St. Willibrord):

Outside of astronomy, zero does not seem to get mentioned much, although Aristotle, in his Physics (Book 4, Part 8) points out, as if it is a well-known fact, that “there is no ratio of zero (nothing) to a number (οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν),” i.e. that you cannot divide by zero. Here Aristotle may have been ahead of Brahmagupta, who thought that 0/0 = 0.

0 and 1 as formal numbers?

We now turn to the formal theory of numbers, in the Elements of Euclid and other works. This is mathematics in a surprisingly modern style, with formal proofs and (more or less) formal definitions. In book VII of the Elements (Definitions 1 & 2), Euclid defines the technical terms μονάς = monas (unit) and ἀριθμὸς = arithmos (number):

1. A monas (unit) is that by virtue of which each of the things that exist is called one (μονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται).
2. An arithmos (number) is a multitude composed of units (ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος).

So 1 is the monas (unit), and the technical definition of arithmos excludes 0 and 1, just as today the technical definition of natural number is taken by some mathematicians to exclude 0. However, in informal Greek language, 1 was still a number, and Greek mathematicians were not at all consistent about excluding 1. It remained a number for the purpose of doing arithmetic. Around 100 AD, for example, Nicomachus of Gerasa (in his Introduction to Arithmetic, Book 1, VIII, 9–12) discusses the powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512 = α, β, δ, η, ιϛ, λβ, ξδ, ρκη, σνϛ, φιβ) and notes that “it is the property of all these terms when they are added together successively to be equal to the next in the series, lacking a monas (συμβέβηκε δὲ πᾱ́σαις ταῖς ἐκθέσεσι συντεθειμέναις σωρηδὸν ἴσαις εἶναι τῷ μετ’ αὐτὰς παρὰ μονάδα).” In the same work (Book 1, XIX, 9), he provides a multiplication table for the numbers 1 through 10:

The issue here is that Euclid was aware of the fundamental theorem of arithmetic, i.e. that every positive integer can be decomposed into a bag (multiset) of prime factors, in no particular order, e.g. 60 = 2×2×3×5 = 2×2×5×3 = 2×5×2×3 = 5×2×2×3 = 5×2×3×2 = 2×5×3×2 = 2×3×5×2 = 2×3×2×5 = 3×2×2×5 = 3×2×5×2 = 3×5×2×2 = 5×3×2×2.

Euclid proves most of this theorem in propositions 30, 31 and 32 of his Book VII and proposition 14 of his Book IX. The number 0 is obviously excluded from consideration here, and the number 1 is special because it represents the empty bag (even today we recognise that 1 is a special case, because it is not a prime number, and it is not composed of prime factors either – although, as late as a century ago, there were mathematicians who called 1 prime, which causes all kinds of problems):

• If two numbers (arithmoi) by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers (ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει) – i.e. if a prime p divides ab, then it divides a or b or both
• Any composite number is measured by some prime number (ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – i.e. it has a prime factor
• Any number (arithmos) either is prime or is measured by some prime number (ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – this would not be true for 1
• If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it (ἐὰν ἐλάχιστος ἀριθμὸς ὑπὸ πρώτων ἀριθμῶν μετρῆται, ὑπ᾽ οὐδενὸς ἄλλου πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν ἐξ ἀρχῆς μετρούντων) – this is a partial expression of the uniqueness of prime factorisation

The special property of 1, the monas or unit, was sometimes expressed (e.g. by Nicomachus of Gerasa) by saying that it is the “beginning of arithmoi … but not itself an arithmos.” As we have already seen, nobody was consistent about this, and there was, of course, no problem in doing arithmetic with 1. Everybody agreed that 1 + 2 + 3 + 4 = 10. In modern mathematics, we would avoid problems by saying that natural numbers are produced using the successor function S, and distinguish that function from the number S(0) = 1.

The words monas and arithmos occur in other Greek writers, not always in the Euclidean technical sense. For example, in a discussion of causes and properties in the Phaedo (105c), Plato tells us that “if you ask what causes an arithmos to be odd, I shall not say oddness, but the monas (οὐδ᾽ ᾧ ἂν ἀριθμῷ τί ἐγγένηται περιττὸς ἔσται, οὐκ ἐρῶ ᾧ ἂν περιττότης, ἀλλ᾽ ᾧ ἂν μονάς).” Aristotle, in his Metaphysics, spends some time on the philosophical question of what the monas really is.

In general, the ancient Greeks seem to have had quite a sophisticated understanding of 0 and 1, though hampered by poor vocabulary and a lack of good symbols. Outside of applied mathematics and astronomy, they mostly worked with what we would call the multiplicative group of the positive rational numbers. What they were missing was any awareness of negative numbers as mathematical (not just financial) concepts. That had to wait until Brahmagupta, and when it came, 0 suddenly became a whole lot more interesting, because it eventually became possible to define more advanced mathematical concepts like fields.

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The history of zero: an infographic

Following up on an earlier post about Zero in Greek mathematics, here is a timeline for the use of zero in Europe (click to zoom). I have used images of, or quotes from, primary sources where possible (reliably dated Indian primary sources are much harder to find than Greek ones, unfortunately).

Chinese uses of zero are probably also derived from the Greeks, but Mayan uses are clearly independent.

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Modal logic, ethics, and obligation

Recently, I posted about necessary truth, the logic of belief, and epistemic logic. I would like to follow up on that one more time by discussing deontic logic, the logic of obligation and moral action. We can capture this concept using the 4 rules of D4 modal logic. The first 3 of these are the same as those I used for belief. I am replacing the previous modal operators with  Ⓞ  which is intended to be read as “it is obligatory that” (hence the O in the circle):

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

where  ~ Ⓞ ~P  is read as “~ P is not obligatory,” i.e. “P is permissible.” For those who prefer words rather than symbols:

• if P is any tautology, then P is obligatory
• if P and (P implies Q) are both obligatory, then Q is obligatory
• if P is obligatory, then it is obligatory that P is obligatory
• if P is obligatory, then P is permissible

For these rules as they stand, the only things that are obligatory are necessary truths like 2 + 2 = 4. This is because you can’t get an “ought” from an “is.” Apart from the first rule, there is no way of introducing a  Ⓞ  symbol out of nowhere. Consequently, if we are to reason about ethics and morality, we must begin with some deontic axioms that already contain the  Ⓞ   symbol. For people of faith, these deontic axioms may be given by God, as in the 10 Comandments, which include:

Ⓞ  you do not murder.
Ⓞ  you do not commit adultery.
Ⓞ  you do not steal.
Ⓞ  you do not bear false witness against your neighbor.

Immanuel Kant famously introduced the categorical imperative, a deontic axiom which Kant thought implied all the other moral rules, and thus provided the smallest possible set of deontic axioms:

Ⓞ  [you] act only according to that maxim whereby you can, at the same time, will that it should become a universal law.

Others have suggested the greatest happiness of the greatest number as a principle. Fyodor Dostoevsky, William James, and Ursula Le Guin are among those who have explained the problem with this:

“Tell me yourself, I challenge your answer. Imagine that you are creating a fabric of human destiny with the object of making men happy in the end, giving them peace and rest at last, but that it was essential and inevitable to torture to death only one tiny creature – that baby beating its breast with its fist, for instance – and to found that edifice on its unavenged tears, would you consent to be the architect on those conditions?” (Fyodor Dostoevsky, “The Grand Inquisitor,” in The Brothers Karamazov, 1880; 4.35 on Goodreads)

“Or if the hypothesis were offered us of a world in which Messrs. Fourier’s and Bellamy’s and Morris’s Utopias should all be outdone and millions kept permanently happy on the one simple condition that a certain lost soul on the far-off edge of things should lead a life of lonely torture, what except a specifical and independent sort of emotion can it be which would make us immediately feel, even though an impulse arose within us to clutch at the happiness so offered, how hideous a thing would be its enjoyment when deliberately accepted as the fruit of such a bargain?” (William James, “The Moral Philosopher and the Moral Life,” 1891)

“Some of them understand why, and some do not, but they all understand that their happiness, the beauty of their city, the tenderness of their friendships, the health of their children, the wisdom of their scholars, the skill of their makers, even the abundance of their harvest and the kindly weathers of their skies, depend wholly on this child’s abominable misery.” (Ursula K. Le Guin, “The Ones Who Walk Away from Omelas,” 1973; reprinted in The Wind’s Twelve Quarters, 1975; 4.05 on Goodreads)

The meaning of deontic statements can be described using Kripke semantics, which exploits the idea of possible worlds (i.e. alternate universes). To say that some statement is obligatory is to say that the statement would be true in better possible worlds (we write w₁ → w₂ to mean that w₂ is a better possible world than w₁).

In any given world v, the statement  Ⓞ P  is equivalent to :

• P  is true in all better worlds w_i (i.e. all those with v → w_i)

Likewise, in any given world v, the statement  ~ Ⓞ ~P  (P is permissible) is equivalent to:

• P  is true in at least one better world w_i (i.e. one with v → w_i)

The rules of deontic logic imply two conditions on these arrows between possible worlds:

• if  w₁ → w₂ → w₃  then  w₁ → w₃  (i.e. chains of arrows are treated like arrows too)
• in every world v there is at least one arrow  v → w  (i.e. chains of arrows don’t stop; this includes the case of  v → v)

A number of philosophers have suggested that deontic logic leads to paradoxes. In all cases that I have seen, these “paradoxes” have involved simple errors in the use of deontic logic – errors that become obvious when the deontic statements are translated into statements about possible worlds.

There are limitations to deontic logic, however. For example, if we say that it is obligatory not to steal, this means that, in all better possible worlds, nobody steals. If we also say that it is obligatory to punish thieves, this means that, in all better possible worlds, thieves are punished. However, if it is obligatory not to steal, better possible worlds have no thieves, so the two statements do not combine well.

Some people would, no doubt, suggest that fiction like that of Dostoevsky is a better tool than logic for exploring such issues. In cases where the writer is a genius, they are probably right.

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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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Modal logic, knowledge, and an old joke

Recently, I posted about necessary truth and about the logic of belief. I would like to follow up on that by discussing epistemic logic, the logic of knowledge. Knowledge is traditionally understood as justified true belief (more on that below), and we can capture the concept of knowledge using the 4 rules of S4 modal logic. These are in fact the same 4 rules that I used for for necessary truth, and the first 3 rules are the same as those I used for belief (the fourth rule adds the fact that knowledge is true).

Knowledge is specific to some person, and I am replacing the previous modal operators with  Ⓚ  which is intended to be read as “John knows” (hence the K in the circle):

• 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 John knows P
• if John knows both P and (P implies Q), then John knows Q
• if John knows P, then John knows that he knows P
• if John knows P, then P is true

Epistemic logic is useful for reasoning about, among other things, electronic commerce (see this paper of mine from 2000). How does a bank know that an account-holder is authorising a given transaction? Especially if deceptive fraudsters are around? Epistemic logic can highlight which of the bank’s decisions are truly justified. For this application, the first rule (which implies knowing all of mathematics) actually works, because both the bank’s computer and the account-holder’s device can do quite sophisticated arithmetic, and hence know all the mathematical facts relevant to the transaction they are engaged in.

But let’s get back to the idea of knowledge being justified true belief. In his Theaetetus, Plato has Theaetetus suggest exactly that:

“Oh yes, I remember now, Socrates, having heard someone make the distinction, but I had forgotten it. He said that knowledge was true opinion accompanied by reason [ἔφη δὲ τὴν μὲν μετὰ λόγου], but that unreasoning true opinion was outside of the sphere of knowledge; and matters of which there is not a rational explanation are unknowable – yes, that is what he called them – and those of which there is are knowable.” (Theaetetus, 201c)

Although he also uses essentially this same definition in other dialogues, Plato goes on to show that it isn’t entirely clear what kind of “justification” or “reason” is necessary to have true knowledge. In a brief 1963 paper entitled “Is Justified True Belief Knowledge?,” the philosopher Edmund Gettier famously took issue with the whole concept of justified true belief, and provided what seemed to be counterexamples.

My personal opinion, which I have argued elsewhere, is that “justified true belief” works fine as a definition of knowledge, as long as the justification is rigorous enough to exclude beliefs which are “accidentally correct.” For analysing things like electronic commerce, a sufficient level of rigour would involve the use of epistemic logic, as described above.

One of Gettier’s supposed counterexamples involves a proposition of the form  P ∨ Q  (P or Q) such that:

• Smith believes and knows  P ⇒ (P ∨ Q)
• Smith believes P
• P is false
• Q is true, and therefore so is  P ∨ Q

From these propositions we can use doxastic logic to infer that Smith believes the true statement  P ∨ Q,  but we cannot infer (using epistemic logic) that Smith knows  P ∨ Q. A famous old joke is perhaps relevant here:

“A physicist, a philosopher, and a mathematician are travelling through Scotland by train. Through the window, they observe a black sheep in a field. ‘Aha,’ says the physicist, ‘I see that Scottish sheep are black!’ The philosopher responds, ‘No! Some Scottish sheep are black!’ The mathematician, looking shocked, replies: ‘What are you guys saying? All we know is that at least one sheep in Scotland is black on at least one side.’”

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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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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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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.

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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.”

Piranesi by Susanna Clarke: 4½ stars

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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.

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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 the Trees Clap Their Hands by Virginia Stem Owens: 2½ stars

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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

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