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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Planetary Intelligences

In a book review of Out of the Silent Planet, I mentioned last year that C. S. Lewis had pioneered the science fiction sub-genre of a planetary intelligence or sentient planet which resists outsiders. A planetary intelligence provides a way of exploring colonisation and other issues, while still having a positive ending to the story.

The chart above (click to zoom) shows a timeline of the concept. Although there are many other stories based on the idea, these six seemed particularly noteworthy (star ratings out of 5 are from GoodReads and RottenTomatoes):

• ★★★★ Out of the Silent Planet by C. S. Lewis (1938).
• ★★★★ The Lonely Planet by Murray Leinster (1949).
• ★★★★ Solaris by Stanislaw Lem (1961).
• ★★★★ “Vaster than Empires and More Slow” by Ursula Le Guin (1971).
• ★★★★ Powers that Be by Anne McCaffrey and Elizabeth Ann Scarborough (1993) and its sequels.
• ★★★★ The film Avatar by James Cameron (2009) and its upcoming sequels.

Solaris was filmed in 1968, 1972 (★★★★☆), and 2002 (★★★☆). Here are trailers for the last two films:

Readers, how do you feel the various books and films compare?

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Fictional Scientists and Mathematicians

I have been reflecting on fictional mathematicians and scientists. The image above shows four:

• Victor Frankenstein from Frankenstein by Mary Shelley.
• Hari Seldon from Foundation by Isaac Asimov.
• Ian Malcolm from Jurassic Park by Michael Crichton.
• Max Cohen from the film Pi by Darren Aronofsky.

None of these are terribly good role models, it seems to me. Literature and cinema have some better examples, but on the whole, mathematicians and scientists are not treated well by fiction authors.

Milton Millhauser, in “Dr. Newton and Mr. Hyde: Scientists in Fiction from Swift to Stevenson” (Nineteenth-Century Fiction, 1973, Vol. 28, No. 3, pp. 287–304) notes that, in literature, “broadly speaking the scientist is either disregarded or held up to contempt and ridicule.” But perhaps that is because some writers (Michael Crichton is a notable example) see dangers and dilemmas that demand exploration.

However, female scientists and mathematicians seem to be portrayed somewhat more positively. For example:

• Eleanor Arroway from Contact by Carl Sagan.
• Sarah Harding from The Lost World by Michael Crichton.
• Catherine Llewellyn from the play (and later film) Proof by David Auburn.
• Grace Augustine from the film Avatar by James Cameron.

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Race the Sun: where are they now?

The 1996 movie Race the Sun is almost sacred in the solar car racing community. It fictionalises the true story of a Hawaiian high school team racing in the World Solar Challenge in Australia. But where is the cast now?

Teachers

• Halle Berry (Sandra Beecher): became a major star, playing Storm in X-Men, Ginger Knowles in Swordfish, and many other roles.
• Jim Belushi (Frank Machi): already well known in 1996, he has acted in multiple later films, such as The Ghost Writer.

Left: Halle Berry at the 2017 San Diego Comic-Con (cropped from a photo by Gage Skidmore); Centre: Pritzker School of Medicine in Chicago, where Sara Tanaka did her initial medical training; Right: Casey Affleck in 2016 (cropped from a photo by Bex Walton)

Solar Car Race Team

• Casey Affleck (Daniel Webster): has acted in multiple films. For his role in Manchester by the Sea, he won several awards.
• Eliza Dushku (Cindy Johnson): continued on to Buffy the Vampire Slayer and various film and TV work. In later years, she became politically active.
• Anthony Ruivivar (Eduardo Braz): has acted in various films and has frequently played policemen on television.
• Sara Tanaka (Uni Kakamura): graduated in Medicine from the University of Chicago in 2008, and then specialised in cardiology. She now practices, I believe, in New York.
• Dion Basco (Marco Quito): had a few other acting roles.
• J. Moki Cho (Gilbert Tutu): became a musician, and is on Instagram and on YouTube.
• Nadja Pionilla (Oni Nagano): had a few other acting roles, and is on Twitter.
• Adriane Napualani Uganiza (Luana Kanahele): I don’t know what happened to her.

Other Characters

• Steve Zahn (Hans Kooiman): various film and TV work, including a role as an ape in War for the Planet of the Apes (2017).
• Joel Edgerton (Steve Fryman): acted in multiple films, including playing the young Owen Lars in several Star Wars films.
• Kevin Tighe (Jack Fryman): acted in film and television, as well as on stage.
• Bill Hunter (Commissioner Hawkes): after acting in numerous films, he died in 2011.
• Jeff Truman (Ed Webster): after a career of acting and writing, he died in 2014.

With the obvious exceptions, I would like to see one of these people at the start (or finish) of a major solar car race.

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Looking back: 2001

The 1968 film 2001: A Space Odyssey suggested that we would have extensive space flight in 2001. That turned out not to be the case. What we did get was the September 11 attacks on the USA and the military conflicts which followed. Nevertheless, NASA commemorated the film with the 2001 Mars Odyssey orbiter.

Films of 2000 included the superb The Lord of the Rings: The Fellowship of the Ring, several good animated films (including Monsters, Inc., Shrek, and Hayao Miyazaki’s Spirited Away), the wonderful French film Amélie, some war movies (Enemy at the Gates was good, but Black Hawk Down distorted the book too much for my taste), the first Harry Potter movie, and an award-winning biographical film about the mathematician John Nash.

In books, Connie Willis published Passage, one of my favourite science fiction novels, while Ian Stewart explained some sophisticated mathematics simply in Flatterland.

Saul Kripke (belatedly) received the Rolf Schock Prize in Logic and Philosophy for his work on Kripke semantics, while Ole-Johan Dahl and Kristen Nygaard (also belatedly) received the Turing Award for their work on object-oriented programming languages (both these pioneers of computing died the following year).

The year 2001 also saw the completion of the Cathedral of Saint Gregory the Illuminator in Armenia, which I have sadly never visited.

In this series: 1978, 1980, 1982, 1984, 1987, 1989, 1991, 1994, 2000, 2001, 2004, 2006, 2009.

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Houston, we have a problem

Some years ago, I posted the chart above, inspired by a classic XKCD cartoon. The infographic above shows the year of publication and of setting for several novels, plays, and films.

They fall into four groups. The top (white) section is literature set in our future. The upper grey section contains obsolete predictions – literature (like the book 1984) set in the future when it was written, but now set in our past. The centre grey section contains what XKCD calls “former period pieces” – literature (like Shakespeare’s Richard III) set in the past, but written closer to the setting than to our day. He points out that modern audiences may not realise “which parts were supposed to sound old.” The lower grey section contains literature (like Ivanhoe) set in the more distant past.

The movie Apollo 13 has now joined the “former period piece” category. Released in 1995, it described an event of 1970, 25 years in the past. But the ill-fated Apollo 13 mission of 11–17 April 1970 is now 51 years in the past; the movie is closer to the event than it is to us (although the phrase “Houston, we have a problem” – in real life, “Houston, we’ve had a problem” – has become part of the English language).

The image shows the real-life Apollo 13 Service Module, crippled by an explosion (left), together with a poster for the 1995 movie (right). Maybe it’s time to watch it again?

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History, geography, and the Western genre

Once Upon a Time in the West, Rio Grande, High Noon. We know the films – and the many books.

“The bray of a lazy burro broke the afternoon quiet, and it was comfortingly suggestive of the drowsy farmyard, and the open corrals, and the green alfalfa fields. Her clear sight intensified the purple sage-slope as it rolled before her. Low swells of prairie-like ground sloped up to the west. Dark, lonely cedar-trees, few and far between, stood out strikingly, and at long distances ruins of red rocks. Farther on, up the gradual slope, rose a broken wall, a huge monument, looming dark purple and stretching its solitary, mystic way, a wavering line that faded in the north. Here to the westward was the light and color and beauty. Northward the slope descended to a dim line of canyons from which rose an up-flinging of the earth, not mountainous, but a vast heave of purple uplands, with ribbed and fan-shaped walls, castle-crowned cliffs, and gray escarpments. Over it all crept the lengthening, waning afternoon shadows.” – Zane Grey, Riders of the Purple Sage (1912)

But why are the films and books all set in the United States? Didn’t the very similar continent of Australia have similar stories? Well, up to a point.

Click map to zoom

If we want to know why things are the way they are, the answers often lie in history and geography (Jared Diamond makes an especially strong case for geography in his Guns, Germs, and Steel). European settlement in the US began several centuries ago. The Appalachian Mountains (rising to 2,037 m or 6,684 ft) formed a barrier to westward expansion, but hardly in insurmountable one. The eastern US is also blessed with many navigable rivers, especially the Mississippi and tributaries such as the Ohio, Missouri, Platte, and Arkansas. The eastern US is also blessed with good rainfall.

Click map to zoom

Western expansion in the US constantly outran organised government. This created a degree of chaos that lasted for a surprisingly long time. The Oklahoma Panhandle, for example, was “No Man’s Land” from 1850 until 1890 – not part of any state or territory. The western part of the Minnesota Territory had the same status between 1858 and 1861. In addition, some of the organised territories in the contiguous US (Arizona and New Mexico) did not become states until 1912.

One tool for dealing with this situation was the resurrection of a thousand-year old English law enforcement strategy: posse comitatus or “power of the county.” Law enforcement was provided by a sheriff, who was authorised to call on armed citizens as needed. Part of the drama of Western stories lies in the sheriff deciding when this was actually needed.

Click map to zoom

In contrast to the US, Australia is significantly drier. The Great Dividing Range in the east is somewhat loftier than the Appalachians, with the highest point being Mount Kosciuszko (2,228 m or 7,310 ft). A significant part of the water falling on the west of the range winds up underground in the Great Artesian Basin, a vast bed of porous sandstone holding up to 64,900 cubic kilometres (15,600 cubic miles) of water, capped by an impermeable layer of rock. In places, the basin is 3 km (2 miles) deep. The basin was discovered in 1878, and only after that date did cattle stations or sheep stations in certain parts of the country become feasible, thanks to water from deep bores.

Click map to zoom

Politically, the Australian situation was quite different from the US as well. The entire continent east of 135°E was initially part of the British colony of New South Wales, and by 1829 all of the continent had been claimed. Colonial boundaries shifted several times before Federation in 1901 (and the Northern Territory was transferred to federal control in 1911), but the US situation of unorganised territory was nonexistent.

Law enforcement in Australia was initially military, and early police forces were composed of military personnel. In 1853, Victoria was the first colony to merge law enforcement into one colonial police force. However, law enforcement was never decentralised, as it was in the US.

Mounted police escorting gold, 1852 (British Museum)

The vast size and relatively small population of Australia meant that there was plenty for law enforcement to do, of course. Stage coaches and gold miners were robbed, and what Americans call “rustling” also took place. In 1870, a daring theft of around 800 head of cattle took place at Bowen Downs Station in Queensland. Harry Redford and four accomplices overlanded the stock to outback South Australia, where the brands would not be recognised (a distance of about 1,300 km or 800 miles). Employees of Bowen Downs successfully tracked the herd, but Redford was acquitted by a working-class jury who didn’t much mind rich graziers being robbed.

Tom Roberts, Bailed Up (1895)

The Western genre tells stories of human drama and resourcefulness on the frontier, and in that it resembles the science fiction genre. But to a large extent the Western genre is also a celebration of the land. To quote one of my favourite contemporary short stories (a Christmas story, actually), from novelist Elisabeth Grace Foley:

“A million diamonds glinted in the smooth, untouched white curve of snow in the basin, struck out by the sun that pierced the bright silver-white sky. The bitter wind whisked across it, kicking up little powdery swirls. Cal Rayburn turned up the collar of his sourdough coat with one hand, hunching his shoulders a little so the collar half covered his ears. He squinted at the blinding-bright landscape, and one side of his cold-numbed lips twisted back a little in a half-smile.” – Elisabeth Grace Foley, “The Bird of Dawning”

Australians may have lost contact with the land to a greater extent than Americans have, so that the genre of Australian colonial stories has largely faded away. Australia was formed as a collection of colonies with coastal capitals (and with the national capital only 100 km or 60 miles inland). That, together with the dryness of the interior, facilitated a drift to the cities, so that 70% of the population now lives in the 8 capitals.

In contrast, the US has many landlocked states which seem to retain a greater connection to the land. The state flag of Kansas, to pick just one state, seems to tell an entire story, including Indians hunting bison on the Great Plains, a steamboat on one of the navigable rivers, a settler ploughing his field, and a wagon train heading west. There is scope for all kinds of literature and cinema right there (as Laura Ingalls Wilder, Louis L’Amour, Howard Hawks, John Sturges, Clint Eastwood, and many others have shown). Let us hope that people will keep telling those stories.

Flag of Kansas

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Looking back: 1987

In 1987, my PhD work at the University of Tasmania was beginning to take shape, and I produced a technical report with some preliminary results. I also started a side-project on functional programming language implementation which was to result in the design of a novel computer (a computer, sadly, that was never actually built, although many people joined in on the hardware aspects).

Also in that year, Supernova 1987A became visible within the Large Magellanic Cloud (picture above taken by the Kuiper Airborne Observatory). The programming language Perl also appeared on the scene, and Per Bak, Chao Tang, and Kurt Wiesenfeld coined the term “self-organized criticality.” Prompted by a discovery in 1986, physicists held a conference session on high-temperature superconductivity, billed as the “Woodstock of physics.” The immediate benefits were somewhat over-hyped, however.

The usual list of new species described in 1987 includes Fleay’s barred frog from northern New South Wales and south-eastern Queensland (picture below taken by “Froggydarb”).

In the world of books, James Gleick popularised chaos theory with his Chaos: Making a New Science, Allan Bloom wrote The Closing of the American Mind (which Camille Paglia called “the first shot in the culture wars”), and Donald Trump co-wrote Trump: The Art of the Deal (nobody imagined that he would be President one day).

Horror writer Stephen King had a good year, with The Tommyknockers and several other novels being published. The term “steampunk” was coined in 1987, and Orson Scott Card’s Speaker for the Dead, the sequel to Ender’s Game, won the Hugo Award for best science fiction or fantasy novel (it also won the Nebula Award in 1986, the year it was published).

In music, The Alan Parsons Project released their album Gaudi (which included the single below), U2 released The Joshua Tree, and Linda Ronstadt, Emmylou Harris, and Dolly Parton released Trio. The Billboard top song for 1987 was the rather silly 1986 single “Walk Like an Egyptian.”

Films of 1987 included 84 Charing Cross Road (based on the wonderful 1970 book by Helene Hanff), Bernardo Bertolucci’s The Last Emperor, Japanese hit A Taxing Woman (マルサの女), sci-fi action film Predator, Australian film The Year My Voice Broke and, of course, the cult classic The Princess Bride (based on the 1973 novel by William Goldman).

In this series: 1978, 1980, 1982, 1984, 1987, 1989, 1991, 1994, 2000, 2004, 2006, 2009.

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Looking back: 2009

Washington, DC in June 2009

In 2009, I had the privilege of visiting the United States twice (in June and November).

This was the year that saw the launch of the Lunar Reconnaissance Orbiter (which imaged, among other things, the Apollo 11 landing site), the Kepler space telescope (designed to look for exoplanets), the Herschel space observatory (an infrared telescope studying star formation), the Planck spaceprobe (which studied the cosmic microwave background), and the Wide-field Infrared Survey Explorer (an infrared telescope looking for minor planets and star clusters).

Apollo 11 landing site, imaged by the LRO (with photographs from 1969 inset)

More metaphorically, Bitcoin and the programming language Go were also launched. US Airways Flight 1549, on the other hand, was skillfully landed in a river. In archaeology, hoards were discovered in Staffordshire (gold and silver metalwork) and Shrewsbury (Roman coins). Australian Hospital Ship Centaur, torpedoed in 1943, was discovered off the Queensland coast.

Books of 2009 included Wolf Hall by Hilary Mantel (set in 1500–1535; a TV series of 2015), The Windup Girl by Paolo Bacigalupi (dystopian science fiction; Nebula Award winner), and The Maze Runner by James Dashner (young adult dystopian sci-fi; a film of 2014). Books that I later reviewed include The Lassa Ward by Ross Donaldson and God’s Philosophers by James Hannam.

Movies of 2009 included Avatar (rather disappointing), 2012 (a little silly), Angels & Demons (a travesty), Up (Pixar/Disney), Coraline (designed to give children nightmares), District 9 (designed to give adults nightmares), Julie & Julia (a film about cooking), The Imaginarium of Doctor Parnassus (a film about mirrors), and Sherlock Holmes (a lot of fun). On the whole, a good year for films.

In this series: 1978, 1980, 1982, 1984, 1989, 1991, 1994, 2000, 2004, 2006, 2009.

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Looking back: 2004

xkcd comments on the Spirit rover (click to zoom)

In 2004, I was privileged to visit Middle Earth (aka New Zealand) with a colleague and to present the paper “Network Robustness and Graph Topology.” A major event of that year was the landing of the Mars rovers Spirit and Opportunity. Intended to operate for 90 Martian days (92 Earth days), Spirit kept going until 2010 (as xkcd remarked on in the comic above) and Opportunity set a record by operating until 2018. Also in 2004, the Stardust spaceprobe collected some comet dust.

On a more sombre note, 2004 saw the Boxing Day Tsunami. In the field of technology, Facebook and Gmail both launched in 2004, and Vint Cerf and Bob Kahn shared the Turing Award (for having invented the Internet).

This was an excellent year for cinema. Examples from different genres include National Treasure, Troy, Van Helsing, Man on Fire, Hotel Rwanda, The Village, Howl’s Moving Castle, and The Passion of the Christ. I certainly have memories that I treasure.

In this series: 1978, 1980, 1982, 1984, 1989, 1991, 1994, 2000, 2004, 2006.

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