The Lost World by Michael Crichton: a book review

The Lost World, by Michael Crichton (1995)

Recently, because this is the season for extra reading, I re-read The Lost World by Michael Crichton (which was made into a 1997 film). This novel turns 25 years old in September. Its main plot needs no explanation, of course. Just like Jurassic Park, there’s action, there’s excitement, and there’s dinosaurs chasing people.

As with all Crichton novels, there are technical and scientific themes that do not make it into the film. I had forgotten, for example, that the original mobile laboratory is solar powered: “He wants them light, I build them light. He wants them strong, I build them strong – light and strong both, why not, it’s just impossible, what he’s asking for, but with enough titanium and honeycarbon composite, we’re doing it anyway. He wants it off petroleum base, and off the grid, and we do that too. … The Explorer with the black photovoltaic panels on the roof and hood, the inside crammed with glowing electronic equipment. Just looking at the Explorer gave them a sense of adventure…” (pages 64 & 94)

Velociraptor skeletal cast at the Dinosaur Journey museum in Colorado (original photo by Jens Lallensack)

Another theme, naturally, is the changing scientific view of dinosaurs, and indeed other things, over time (in fact, the book and film are already out-of-date in some respects): “Back in the 1840s, when Richard Owen first described giant bones in England, he named them Dinosauria: terrible lizards. That was still the most accurate description of these creatures, Malcolm thought. … the Victorians made them fat, lethargic, and dumb – big dopes from the past. This perception was elaborated, so that by the early twentieth century, dinosaurs had become so weak that they could not support their own weight. … That view didn’t change until the 1960s, when a few renegade scientists, led by John Ostrom, began to imagine quick, agile, hotblooded dinosaurs. Because these scientists had the temerity to question dogma, they were brutally criticized for years, … But in the last decade, a growing interest in social behavior had led to still another view. Dinosaurs were now seen as caring creatures, living in groups, raising their little babies.” (page 83)

Tortuga Islands, Costa Rica (original photo by “rigocr”) – is this the mysterious Isla Sorna?

As with many Crichton novels, scientific hubris is a major theme. Other themes include the education of children (both dinosaur children and human children), information systems design, the theories of Stuart Kauffman about self-organisation and evolution, and the importance of what is now called the complex systems view.

Overall, this is a good solid action novel, with several scientific and philosophical themes to think about. Goodreads rates it 3.78. I’m giving it only 3½ stars, in part because it’s a little too much like Jurassic Park. But it’s certainly well worth a read.

The Lost World, by Michael Crichton: 3½ stars

Chemistry can be beautiful: the classic flame test

The flame test occasionally comes up in classic detective fiction: “He snapped off the lights, and we were left with only the sodium flame. In that green, sick glare a face floated close to mine – a corpse-face – livid, waxen, stamped with decay…” (Dorothy L. Sayers & Robert Eustace, The Documents in the Case)

Spectral lines in the image are taken from Kramida, A., Ralchenko, Yu., Reader, J., and NIST ASD Team (2019). NIST Atomic Spectra Database (ver. 5.7.1), [Online]. National Institute of Standards and Technology, Gaithersburg, MD. Photographs in the image are public domain, from Wikimedia Commons.

Dune by Frank Herbert: a book review

Dune by Frank Herbert (1965)

I recently re-read the classic 1965 novel Dune by Frank Herbert. This is Frank Herbert’s best book, and one of the best science fiction novels ever written. It won the Hugo Award in 1966 (jointly with Roger Zelazny’s This Immortal) and won the inaugural Nebula Award. It became a quite terrible 1984 film and a somewhat better miniseries.

Parts of the novel are reminiscent of the work of Cordwainer Smith, notably the idea of a desert planet producing spice, and the idea that navigating a faster-than-light ship requires a guild of unusual navigators who can see into the future. However, most of the novel was so original that it became a huge hit when it first appeared. Themes that are particularly notable are those of planetary ecology, intergalactic politics, and unusual human skills.

I have always been moved by Herbert’s idea of a symbolic ecological language that can “arm the mind to manipulate an entire landscape” (Appendix 1), and the idea of making ecological literacy a key part of education:

At a chalkboard against the far wall stood a woman in a yellow wraparound, a projecto-stylus in one hand. The board was filled with designs – circles, wedges and curves, snake tracks and squares, flowing arcs split by parallel lines. The woman pointed to the designs one after the other as fast as she could move the stylus, and the children chanted in rhythm with her moving hand.
Paul listened, hearing the voices grow dimmer behind as he moved deeper into the sietch with Harah.
‘Tree,’ the children chanted. ‘Tree, grass, dune, wind, mountain, hill, fire, lightning, rock, rocks, dust, sand, heat, shelter, heat, full, winter, cold, empty, erosion, summer, cavern, day, tension, moon, night, caprock, sandtide, slope, planting, binder. …’
” (Chapter 22)

The unusual ecology of the desert planet Arrakis encourages us, of course, to think more deeply about our own planet (and Arrakis was apparently inspired by the Oregon Dunes here on Earth).

Also fascinating is the idea that the human race has turned away from computers and the Internet, and gone back to training human minds to remember, calculate, and think:

‘Once men turned their thinking over to machines in the hope that this would set them free. But that only permitted other men with machines to enslave them.’
‘Thou shalt not make a machine in the likeness of a man’s mind,’ Paul quoted. […]
‘The Great Revolt took away a crutch,’ she said. ‘It forced human minds to develop. Schools were started to train human talents.’
” (Chapter 1)

The most obvious theme, and the source of the novel’s action, is the galaxy-wide intrigue between the noble House Corrino, House Atreides, and House Harkonnen; the resulting warfare between them; and the resistance of the desert Fremen to occupation (inspired by Lawrence of Arabia):

Paul took two deep breaths. ‘She said a thing.’ He closed his eyes, calling up the words, and when he spoke his voice unconsciously took on some of the old woman’s tone: ‘ “You, Paul Atreides, descendant of kings, son of a Duke, you must learn to rule. It’s something none of your ancestors learned”.’ Paul opened his eyes, said: ‘That made me angry and I said my father rules an entire planet. And she said, “He’s losing it.” And I said my father was getting a richer planet. And she said. “He’ll lose that one, too.” And I wanted to run and warn my father, but she said he’d already been warned – by you, by Mother, by many people.’” (Chapter 2)

Goodreads rates this classic science fiction novel 4.2. I’m giving it 4½ stars (but be aware that the sequels are not nearly as good).

Dune by Frank Herbert: 4½ stars

Skylark DuQuesne by E. E. Smith: a book review

Skylark DuQuesne by E. E. “Doc” Smith (serialised 1965)

I recently re-read E. E. “Doc” Smith’s Skylark DuQuesne, the final story of Smith’s Skylark series. Smith, of course, is famous for the Lensman series, which is a bit annoying in places, but which is still full of all kinds of interesting ideas. This book is another matter. It’s just bad. Now poor writing may forgivable in “space opera,” and age may play a factor here too – the novel was serialised beginning in June 1965, when Smith was aged 75, and was published as a book in 1966 (Smith died during the serialisation). This novel has so many flaws, in fact, that I can only mention some of them.

To begin with, the sexual titillation for teenage boys is just over the top. Is there any reason why the characters need to be naked quite so often? Or for one of them to be a stripper? It’s a trifle creepy, to be frank.

The mathematics depicted in the book is also disappointing: “‘Hold it!’ Seaton snapped, half an hour later. ‘Back up – there! This integral here. Limits zero to pi over two. You’re limiting the thing to a large but definitely limited volume of your generalized N-dimensional space. I think it should be between zero and infinity—and while we’re at it let’s scrap half of the third determinant in that no-space-no-time complex. Let’s see what happens if we substitute the gamma function here and the chi there and the xi there and the omicron down there in the corner.’” (Chapter 24: DuQuesne and Sleemet)

Smith is describing simple high-school calculus (integrating a function on a single real variable). Even by the standards of the time (never mind the future!) there was a lot more mathematics out there. Robert Heinlein had no trouble getting that fact across in 1952 in The Rolling Stones / Space Family Stone: “Their father reached up to the spindles on the wall, took down a book spool, and inserted it into his study projector. He spun the selector, stopped with a page displayed on the wall screen. It was a condensed chart of the fields of mathematics invented thus far by the human mind. ‘Let’s see you find your way around that page.’ The twins blinked at it. In the upper left-hand corner of the chart they spotted the names of subjects they had studied; the rest of the array was unknown territory; in most cases they did not even recognize the names of the subjects.” (Chapter IV: Aspects of Domestic Engineering).

And hinting at new, future, mathematics is quite possible too. Isaac Asimov did it in 1942 in the first part of Foundation: “‘Good. Add to this the known probability of Imperial assassination, viceregal revolt, the contemporary recurrence of periods of economic depression, the declining rate of planetary explorations, the…’ He proceeded. As each item was mentioned, new symbols sprang to life at his touch, and melted into the basic function which expanded and changed. Gaal stopped him only once. ‘I don’t see the validity of that set-transformation.’ Seldon repeated it more slowly. Gaal said, ‘”But that is done by way of a forbidden sociooperation.’ ‘Good. You are quick, but not yet quick enough. It is not forbidden in this connection. Let me do it by expansions.’ The procedure was much longer and at its end, Gaal said, humbly, ‘Yes, I see now.’” (Chapter 4)

In contrast, Smith’s novel seems to have the goal of making his teenage readers feel good about what they know, rather than encouraging them to grow (and that applies to both intellectual and moral growth).

A PDP-8 computer of 1965

A related problem (common to Smith’s novels, and indeed to much early science fiction) is the failure to imagine how computers might be used. The novel assumes powerful computers (“brains”) which can both sense and influence the physical world. Yet manual information processing is still the order of the day: “Tammon was poring over a computed graph, measuring its various characteristics with vernier calipers, a filar microscope, and an integrating planimeter, when Mergon and Luloy came swinging hand in hand into his laboratory” (Chapter 9: Among the Jelmi)

Finally, the antagonist Marc DuQuesne (the name is a Genesis 4:15 reference, since the surname is pronounced duːˈkeɪn) is a rather unpleasant kind of Nietzschean Übermensch, and the protagonist (Richard Seaton) is not much better. Julian May, in her excellent Saga of Pliocene Exile (and even better Galactic Milieu Series) apparently based her character Marc Remillard in part on Smith’s Marc DuQuesne. But Marc Remillard repents of his crimes, and atones for them, and is actually interesting to read about. Smith’s novel finishes with DuQuesne as arrogant, as unrepentant, and as banal as ever.

Goodreads rates Smith’s novel 3.8, and some old-school science fiction fans still seem to enjoy it. It was even nominated for the Hugo Award for Best Novel, back in 1966, although it can hardly be compared to the other nominees – Dune and This Immortal (tied winners), The Squares of the City, and The Moon Is a Harsh Mistress (which was re-nominated, and won, in 1967). I give Smith’s novel just one star – but if you are nevertheless intrigued, it is now public domain in Canada and is online there.

Skylark DuQuesne by E. E. “Doc” Smith: 1 star

Scientific alignment

I was thinking recently about the alignment (in the Dungeons & Dragons sense) of fictional scientists (see diagram above).

I was brought up on the Famous Five children’s stories by Enid Blyton. Perennially popular, even though flawed in certain ways, these novels star a rather grumpy scientist called Quentin (who had more than a little to do with my own desire to become a scientist). Quentin is certainly altruistic:

‘These two men were parachuted down on to the island, to try and find out my secret,’ said her father. ‘I’ll tell you what my experiments are for, George—they are to find a way of replacing all coal, coke and oil—an idea to give the world all the heat and power it wants, and to do away with mines and miners.’
‘Good gracious!’ said George. ‘It would be one of the most wonderful things the world has ever known.’
‘Yes,’ said her father. ‘And I should give it to the whole world—it shall not be in the power of any one country, or collection of men. It shall be a gift to the whole of mankind—but, George, there are men who want my secret for themselves, so that they may make colossal fortunes out of it.’
” (Enid Blyton, Five On Kirrin Island Again, 1947)

However, Quentin works for no organisation (barring some government consulting work) and draws no regular salary. He is clearly Chaotic Good.

Long before Quentin, Victor Frankenstein in Frankenstein (Mary Shelley, 1818) created his famous monster out of selfishness and hubris. However, he also desires to make things right, so Frankenstein seems to me Chaotic Neutral.

On the other hand, the experiments of Doctor Moreau in The Island of Doctor Moreau (H. G. Wells, 1896) mark him as Chaotic Evil. The same is true of the scientist Rotwang in the movie Metropolis (1927), who is the prototype of the evil “mad scientist” of many later films – in contrast to good “mad scientists” like Emmett “Doc” Brown in the Back to the Future movies (1985, 1989, 1990).

In all cases, however, there seems to be a bias towards portraying scientists as Chaotic. This is a little strange, because the organisational structures, processes, and rules governing science in the real world are better described as “ordered” or Lawful (in the Dungeons & Dragons sense). Perhaps chaotic characters are just more fun?

Not that everyone follows all the rules and procedures of course. When I take the What is your Scientific Alignment? test, my personal alignment comes out as Neutral Good.

Complexity and Randomness revisited

I have posted before (post 1 and post 2) about order, complexity, and randomness. The image above shows the spectrum from organised order to random disorder, with structured complexity somewhere in between. The three textual examples below illustrate the same idea.

Regular Complex Random
AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA AAAAAAAAAA … It was the best of times, it was the worst of times, it was the age of wisdom, it was the age of foolishness, it was the epoch of belief, it was the epoch of incredulity, it was the season of Light, it was the season of Darkness, it was the spring of hope, it was the winter of despair, … ShrfT e6IJ5 eRU5s nNcat qnI8N m-cm5 seZ6v 5GeYc w2jpg Vp5Lx V4fR7 hhoc- 81ZHi 5qntn ErQ2- uv3UE MnFpy rLD0Y DI3GW p23UF FQwl1 BgP36 RK6Gb 6lpzR nV03H W5X3z 2f1u8 OgpXy tY-6H HkwEU s0xLN 9W8H …

These three examples, and many intermediate cases, can be distinguished by the amount of information they contain. The leading way of measuring that information is with Kolmogorov complexity. The Kolmogorov complexity of a block of text is the length of the shortest program producing that text. Kolmogorov complexity is difficult to calculate in practice, but an approximation is the size of the compressed file produced by good compression software, such as 7-Zip. The chart below shows the number of bytes (a byte is 8 bits) for a compressed version of A Tale of Two Cities, a block of the letter ‘A’ repeated to the same length, and a block of random characters of the same length:

The random characters are chosen to have 64 possible options, which need 6 bits to describe, so a compression to about 75% of the original size is as expected. The novel by Dickens compresses to 31% of its original size.

But does this chart show information? Grassberger notes that Kolmogorov complexity is essentially just a measure of randomness. On this definition, random-number generators would be the best source of new information – but that’s not what most people mean by “information.”

An improvement is to introduce an equivalence relation “means the same.” We write X ≈ Y if X and Y have the same meaning. In particular, versions of A Tale of Two Cities with different capitalisation have the same meaning. Likewise, all meaningless random sequences have the same meaning. The complexity of a block of text is then the length of the shortest program producing something with the same meaning as that text (i.e. the complexity of X is the length of the shortest program producing some Y with X ≈ Y).

In particular, the complexity of a specific block of random text is the length of the shortest program producing random text (my R program for random text is 263 bytes), and we can approximate the complexity of A Tale of Two Cities by compressing an uppercase version of the novel. This definition of complexity starts to look a lot more like what we normally mean by “information.” The novel contains a large amount of information, while random sequences or “AAAAA…” contain almost none:

Those who hold that information satisfies the rule ex nihilo nihil fit can thus be reassured that random-number generators cannot create new information out of nothing. However, if we combine random-number generators with a selection procedure which filters out anything that “means the same” as meaningless sequences, we can indeed create new information, as genetic algorithms and genetic programming have demonstrated – although Stuart Kauffman and others believe that the evolution of biological complexity also requires additional principles, like self-organisation.

Something is going on with the primes…

The chart below illustrates the Erdős–Kac theorem. This relates to the number of distinct prime factors of large numbers (integer sequence A001221 in the On-Line Encyclopedia):

Number No. of prime factors No. of distinct prime factors
1 0 0
2 (prime) 1 1
3 (prime) 1 1
4 = 2×2 2 1
5 (prime) 1 1
6 = 2×3 2 2
7 (prime) 1 1
8 = 2×2×2 3 1
9 = 3×3 2 1
10 = 2×5 2 2
11 (prime) 1 1
12 = 2×2×3 3 2
13 (prime) 1 1
14 = 2×7 2 2
15 = 3×5 2 2
16 = 2×2×2×2 4 1

The Erdős–Kac theorem says that, for large numbers n, the number of distinct prime factors of numbers near n approaches a normal distribution with mean and variance log(log(n)), where the logarithms are to the base e. That seems to be saying that prime numbers are (in some sense) randomly distributed, which is very odd indeed.

In the chart, the observed mean of 3.32 is close to log(log(109)) = 3.03, although the observed variance of 1.36 is smaller. The sample in the chart includes 17 numbers with 8 distinct factors, including 1,000,081,530 = 2×3×3×5×7×19×29×43×67 (9 factors, 8 of which are distinct).

The Erdős–Kac theorem led to an episode where, following the death of Paul Erdős in 1996, Carl Pomerance spoke about the theorem at a conference session in honour of Erdős in 1997. Quoting Albert Einstein (“God does not play dice with the universe”), Pomerance went on to say that he would like to think that Erdős and [Mark] Kac replied “Maybe so, but something is going on with the primes.” The quote is now widely misattributed to Erdős himself.

Complexity in medicine: some thoughts

I have been thinking recently about medicine and complexity, as a result of several conversations over many years. In particular, the Cynefin framework developed by Dave Snowden (see diagram below) seems a useful lens to use (this thought is not original to me – see among others, the articles “The Cynefin framework: applying an understanding of complexity to medicine” by Ben Gray and “Cynefin as reference framework to facilitate insight and decision-making in complex contexts of biomedical research” by Gerd Kemperman). I will also refer to two case studies from the book Five Patients by Michael Crichton, which is still quite relevant, in spite of being written in 1969.

The Cynefin framework developed by Dave Snowden. The central dark area is that of Disorder/Confusion, where it is not clear which of the four quadrants apply (image: Dave Snowden).

The Cynefin framework divides problems into four quadrants: Obvious, Complicated, Complex, and Chaotic. In addition, the domain of Disorder/Confusion reflects problems where there is no clarity about which of the other domains apply. In medicine, this reflects cases where multiple factors are at work – potentially, multiple chronic conditions as well as one or more acute ones. These conditions can exist in all four quadrants. Ben Gray gives the example of a child with a broken arm linked to both a vitamin deficiency and an abusive home environment. Several quite different interventions may be required.

The Obvious Quadrant

The quadrant of the Obvious applies to conditions with clear cause and effect, where there is a single right answer. According to Dave Snowden, the appropriate response is to sense what is going on, categorise the situation as one on a standard list, and then to respond in the way that people have been trained to do. This response may be trivial (a band-aid, say), or it may involve enormous professional skill. In medicine, much of nursing falls in this quadrant, as does much of surgery.

Michael Crichton’s Five Patients discuses the case of Peter Luchesi, a man admitted to Massachusetts General Hospital during 1969 with a crushed arm and nearly severed hand, as the result of an industrial accident:

Three inches above the left wrist the forearm had been mashed. Bones stuck out at all angles; reddish areas of muscle with silver fascial coats were exposed in many places. The entire arm about the injury was badly swollen, but the hand was still normal size, although it looked shrunken and atrophic in comparison. The color of the hand was deep blue-gray.

Carefully, Appel picked up the hand, which flopped loosely at the wrist. He checked pulses and found none below the elbow. He touched the fingers of the hand with a pin and asked if Luchesi could feel it; results were confusing, but there appeared to be some loss of sensation. He asked if the patient could move any of his fingers; he could not.

Meanwhile, the orthopedic resident, Dr. Robert Hussey, arrived and examined the hand. He concluded that both bones in the forearm, the radius and ulna, were broken and suggested the hand be elevated; he proceeded to do this.

Outside the door to the room, one of the admitting men stopped Appel. ‘Are you going to take it, or try to keep it?’

‘Hell, we’re going to keep it,’ Appel said. ‘That’s a good hand.’

Once the surgeons had sensed the problem and categorised it as an arm reconstruction, a team of three surgeons, two nurses, and an anaesthetist (all highly trained in their respective fields) then spent more than 6 hours in the operating theatre, repairing bone, tendons, and blood vessels. Certainly not trivial, but a case of professionals doing what they were trained to do.

The Complicated Quadrant

Public Domain image

The Complicated quadrant is the realm of diagnosis. Information is collected – in medicine, that generally means patient history, blood tests, scans, etc. – and is then subjected to analysis. This identifies the nature of the problem (in an ideal world, at least), which in turn indicates the appropriate response.

Diagnosis by physicians typically searches for the cause of an illness, while diagnosis by nurses typically focuses on severity. This reflects differences in the responses that physicians and nurses have been trained to provide (the triage officer in a modern hospital is typically a nurse).

Decades of work have gone into automating the diagnosis process – initially using statistical analysis, later using expert systems, and most recently using machine learning. At present, the tool of choice is still the human brain.

In general, modern medicine excels when it operates in the Obvious and Complicated quadrants.

The Complex Quadrant

The Complex quadrant is the realm of interactions. It is inherently very difficult to deal with, and cause and effect are difficult to disentangle. The paradigm of information collection and analysis fails, because each probe of the system changes it in some way. The best approach is a sequence of experiments, following each probe with a response that seems reasonable, and hoping to find an underlying pattern or a treatment that works. Michael Crichton provides this example:

Until his admission, John O’Connor, a fifty-year-old railroad dispatcher from Charlestown, was in perfect health. He had never been sick a day in his life.

On the morning of his admission, he awoke early, complaining of vague abdominal pain. He vomited once, bringing up clear material, and had some diarrhea. He went to see his family doctor, who said that he had no fever and his white cell count was normal. He told Mr. O’Connor that it was probably gastroenteritis, and advised him to rest and take paregoric to settle his stomach.

In the afternoon, Mr. O’Connor began to feel warm. He then had two shaking chills. His wife suggested he call his doctor once again, but when Mr. O’Connor went to the phone, he collapsed. At 5 p.m. his wife brought him to the MGH emergency ward, where he was noted to have a temperature of 108 °F [42 °C] and a white count of 37,000 (normal count: 5,000–10,000).

The patient was wildly delirious; it required ten people to hold him down as he thrashed about. He spoke only nonsense words and groans, and did not respond to his name. …

One difficulty here was that John O’Connor could not speak, and so could not provide information about where he felt pain. He appeared to suffer from septicaemia (blood poisoning) due to a bacterial infection in his gall bladder, urinary tract, GI tract, pericardium, lungs, or some other organ. Antibiotics were given almost immediately, to save his life. These eliminated the bacteria from his blood, but did not tackle the root infection. They also made it difficult to identify the bacteria involved, or to locate the root infection, thus hampering any kind of targeted response. In the end (after 30 days in hospital!) John O’Connor was cured, but the hospital never did locate the original root infection.

Similar problems occur with infants (Michael Crichton notes that “Classically, the fever of unknown origin is a pediatric problem, and classically it is a problem for the same reasons it was a problem with Mr. O’Connor—the patient cannot tell you how he feels or what hurts”). As Kemperman notes, medical treatment of the elderly often also falls in the Complex domain, with multiple interacting chronic conditions, and multiple interacting drug treatments. Medical treatment of mental illness is also Complex, as the brain adapts to one treatment regimen, and the doctor must experiment to find another that stabilises the patient.

Similarly Complex is the day-to-day maintenance of wellness (see the Food and Wellness section below) which often falls outside of mainstream medicine.

The Chaotic Quadrant

The Chaotic quadrant is even more difficult than the Complex one. Things are changing so rapidly that information collection and experimentation are impossible. The only possible response is a dance of acting and reacting, attempting to stabilise the situation enough that it moves from Chaotic to Complex. Emergency medicine generally falls in this quadrant – immediate responses are necessary to stop the patient dying. In the airline industry, the ultimate (and extremely rare) nightmare of total engine failure shortly after takeoff (as in US Airways Flight 1549) sits here too – each second of delay sees gravity take its toll.

Success in the Chaotic domain requires considerable experience. In cases where the problem is a rare one, this experience must be created synthetically using simulation-based training.

Food and Wellness

Michael Crichton notes that “The hospital is oriented toward curative treatment of established disease at an advanced or critical stage. Increasingly, the hospital population tends to consist of patients with more and more acute illnesses, until even cancer must accept a somewhat secondary position.” There is, however, a need for managing the Complex space of minor variations from wellness, using low-impact forms of treatment, such as variations in diet. Some sections of this field are reasonably well understood, including:

Traditional culture often addresses this space as well. For example, Chinese culture classifies foods as Yin (cooling) or Yang (heaty) – although there is little formal evidence on the validity of this classification.

There remain many unknowns, however, and responses to food are highly individual anyway. There may be a place here for electronic apps that record daily food intake, medicine doses, activities, etc., along with a subjective wellness rating. Time series analysis may be able to find patterns in such data – for example, I might have an increased chance of a migraine two days after eating fish. Once identified, such patterns suggest obvious changes in one’s diet or daily schedule. Other techniques for managing this Complex healthcare space are also urgently needed.

The three men and their sisters

The medieval Propositiones ad Acuendos Juvenes (“Problems to Sharpen the Young”) is attributed to Alcuin of York (735–804), a leading figure in the “Carolingian Renaissance.” He is the middle person in the image above.

Along with the more famous problem of the wolf, the goat, and the cabbage, Propositiones ad Acuendos Juvenes contains the problem of the three men and their sisters. Three men, each accompanied by a sister, wish to cross a river in a boat that holds only two people. To protect each woman’s honour, no woman can be left with another man unless her brother is also present (and if that seems strange, remember that Alcuin was writing more than 1,200 years ago). In Latin, the problem is:

“Tres fratres erant qui singulas sorores habebant, et fluvium transire debebant (erat enim unicuique illorum concupiscientia in sorore proximi sui), qui venientes ad fluvium non invenerunt nisi parvam naviculam, in qua non potuerunt amplius nisi duo ex illis transire. Dicat, qui potest, qualiter fluvium transierunt, ne una quidem earum ex ipsis maculata sit?”

The diagram below (click to zoom) shows the state graph for this problem. The solution is left (per tradition) as an exercise for the reader (but to see Alcuin’s solution, highlight the white text below the diagram).

Miss A and Mr A cross
Mr A returns (leaving Miss A on the far side)
Miss B and Miss C cross
Miss A returns (leaving Misses B and C on the far side)
Mr B and Mr C cross
Mr B and Miss B return (leaving Miss C and Mr C on the far side)
Mr A and Mr B cross
Miss C returns (leaving 3 men on the far side)
Miss A and Miss C cross
Mr B returns (leaving the A’s and C’s on the far side)
Mr B and Miss B cross

Zero in Greek mathematics

I recently read The Nothing That Is: A Natural History of Zero by Robert M. Kaplan. Zero is an important concept in mathematics. But where did it come from?

The Babylonian zero

From around 2000 BC, the Babylonians used a positional number system with base 60. Initially a space was used to represent zero. Vertical wedges mean 1, and chevrons mean 10:

This number (which we can write as 2 ; 0 ; 13) means 2 × 3600 + 0 × 60 + 13 = 7213. Four thousand years later, we still use the same system when dealing with angles or with time: 2 hours, no minutes, and 13 seconds is 7213 seconds.

Later, the Babylonians introduced a variety of explicit symbols for zero. By 400 BC, a pair of angled wedges was used:

The Babylonian zero was never used at the end of a number. The Babylonians were happy to move the decimal point (actually, “sexagesimal point”) forwards and backwards to facilitate calculation. The number ½, for example, was treated the same as 30 (which is half of 60). In much the same way, 20th century users of the slide rule treated 50, 5, and 0.5 as the same number. What is 0.5 ÷ 20? The calculation is done as 5 ÷ 2 = 2.5. Only at the end do you think about where the decimal point should go (0.025).

Greek mathematics in words

Kaplan says about zero that “the Greeks had no word for it.” Is that true?

Much of Greek mathematics was done in words. For example, the famous Proposition 3 in the Measurement of a Circle (Κύκλου μέτρησις) by Archimedes reads:

Παντὸς κύκλου ἡ περίμετρος τῆς διαμέτρου τριπλασίων ἐστί, καὶ ἔτι ὑπερέχει ἐλάσσονι μὲν ἤ ἑβδόμῳ μέρει τῆς διαμέτρου, μείζονι δὲ ἢ δέκα ἑβδομηκοστομόνοις.

Phonetically, that is:

Pantos kuklou hē perimetros tēs diametrou triplasiōn esti, kai eti huperechei elassoni men ē hebdomō merei tēs diametrou, meizoni de ē deka hebdomēkostomonois.

Or, in English:

The perimeter of every circle is triple the diameter plus an amount less than one seventh of the diameter and greater than ten seventy-firsts.

In modern notation, we would express that far more briefly as 10/71 < π − 3 < 1/7 or 3.141 < π < 3.143.

The Greek words for zero were the two words for “nothing” – μηδέν (mēden) and οὐδέν (ouden). Around 100 AD, Nicomachus of Gerasa (Gerasa is now the city of Jerash, Jordan), wrote in his Introduction to Arithmetic (Book 2, VI, 3) that:

οὐδέν οὐδενί συντεθὲν … οὐδέν ποιεῖ (ouden oudeni suntethen … ouden poiei)

That is, zero (nothing) can be added:

nothing and nothing, added together, … make nothing

However, we cannot divide by zero. Aristotle, in Book 4, Lectio 12 of his Physics tells us that:

οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν (oude to mēden pros arithmon)

That is, 1/0, 2/0, and so forth make no sense:

there is no ratio of zero (nothing) to a number

If we view arithmetic primarily as a game of multiplying, dividing, taking ratios, and finding prime factors, then poor old zero really does have to sit on the sidelines (in modern terms, zero is not part of a multiplicative group).

Greek calculation

For business calculations, surveying, numerical tables, and most other mathematical calculations (e.g. the proof of Archimedes’ Proposition 3), the Greeks used a non-positional decimal system, based on 24 letters and 3 obsolete letters. In its later form, this was as follows:

Units Tens Hundreds
α = 1 ι = 10 ρ = 100
β = 2 κ = 20 σ = 200
γ = 3 λ = 30 τ = 300
δ = 4 μ = 40 υ = 400
ε = 5 ν = 50 φ = 500
ϛ (stigma) = 6 ξ = 60 χ = 600
ζ = 7 ο = 70 ψ = 700
η = 8 π = 80 ω = 800
θ = 9 ϙ (koppa) = 90 ϡ (sampi) = 900

For users of R:

to.greek.digits <- function (v) { # v is a vector of numbers
  if (any(v < 1 | v > 999)) stop("Can only do Greek digits for 1..999")
  else {
    s <- intToUtf8(c(0x3b1:0x3b5,0x3db,0x3b6:0x3c0,0x3d9,0x3c1,0x3c3:0x3c9,0x3e1))
    greek <- strsplit(s, "", fixed=TRUE)[[1]]
    d <- function(i, power=1) { if (i == 0) "" else greek[i + (power - 1) * 9] }
    f <- function(x) { paste0(d(x %/% 100, 3), d((x %/% 10) %% 10, 2), d(x %% 10)) }
    sapply(v, f)

For example, the “number of the beast” (666) as written in Byzantine manuscripts of the Bible is χξϛ (older manuscripts spell the number out in words: ἑξακόσιοι ἑξήκοντα ἕξ = hexakosioi hexēkonta hex).

This Greek system of numerals did not include zero – but then again, it was used in situations where zero was not needed.

Greek geometry

Most of Greek mathematics was geometric in nature, rather than based on calculation. For example, the famous Pythagorean Theorem tells us that the areas of two squares add up to give the area of a third.

In geometry, zero was represented as a line of zero length (i.e. a point) or as a rectangle of zero area (i.e. a line). This is implicit in Euclid’s first two definitions (σημεῖόν ἐστιν, οὗ μέρος οὐθέν = a point is that which has no part; γραμμὴ δὲ μῆκος ἀπλατές = a line is breadthless length).

In the Pythagorean Theorem, lines are multiplied by themselves to give areas, and the sum of the two smaller areas gives the third (image: Ntozis)

Graeco-Babylonian mathematics

In astronomy, the Greeks continued to use the Babylonian sexagesimal system (much as we do today, with our “degrees, minutes, and seconds”). Numbers were written using the alphabetic system described above, and at the time of Ptolemy, zero was written like this (appearing in numerous papyri from 100 AD onwards, with occasional variations):

For example, 7213 seconds would be β ō ιγ = 2 0 13 (for another example, see the image below). The circle here may be an abbreviation for οὐδέν = nothing (just as early Christian Easter calculations used N for Nulla to mean zero). The overbar is necessary to distinguish ō from ο = 70 (it also resembles the overbars used in sacred abbreviations).

This use of a circle to mean zero was passed on to the Arabs and to India, which means that our modern symbol 0 is, in fact, Graeco-Babylonian in origin (the contribution of Indian mathematicians such as Brahmagupta was not the introduction of zero, but the theory of negative numbers). I had not realised this before; from now on I will say ouden every time I read “zero.”

Part of a table from a French edition of Ptolemy’s Almagest of c. 150 AD. For the angles x = ½°, 1°, and 1½°, the table shows 120 sin(x/2). The (sexagesimal) values, in the columns headed ΕΥΘΕΙΩΝ, are ō λα κε = 0 31 25 = 0.5236, α β ν = 1 2 50 = 1.0472, and α λδ ιε = 1 34 15 = 1.5708. The columns on the right are an aid to interpolation. Notice that zero occurs six times.