(2008) As simple as one, two, three, four … mathematics and murders in which, theoretically, a butterfly's wings imperceptibly flapping might cause a hurricane on the other side of the world.

An overseas student, Martin (Elijah Wood), an American from Arizona, arrives in Oxford in 1993 with the intention of completing his doctorate thesis under the supervision of Prof Arthur Seldom (John Hurt). He takes a room in the home of the elderly Mrs Julia Eagleton (Anna Massey), whose deceased husband had been among the code breakers during World War II (a miniature replica of the Enigma machine captures Martin's interest along with a photograph of Alan Turing) and daughter Beth (Julie Cox) is caring for her.

A cellist in an amateur orchestra, Beth, feeling constricted by an overly protective parent, soon becomes infatuated with the young boarder for his cheerful, go-with-the-flow attitude: "You should try it," he recommends. On a squash court, Martin with his racquet makes acquaintance with another available female with an impressive rack, Lorna (Leonor Watling).

His office mate, Yuri Podorov (Burn Gorman), expresses a grudge against Dr Seldom, blaming the professor for having discouraging his efforts at proving Bormat's last theorem (a reference to Fermat's Last Theorem), the proof of which Henry Wilkins demonstrates at Cambridge to everlasting fame and Podorov's shame.

In the audience during Dr Seldom's lecture, in which the professor begins with an anecdote from the life of Ludwig Wittgenstein, philosopher and author of Tractatus Logico-Philosophicus - "Can we know the truth?" - Martin speaks up, taking issue with Dr Seldom's pronouncement: "There is no such truth outside mathematics. There is no way of finding a single absolute truth, an irrefutable argument to help answer the questions of mankind. Philosophy therefore is dead. Because whereof we cannot speak, thereof we must be silent."

Declaring his loyalty to logic, "The essence of nature is mathematical" because there is "hidden meaning beneath reality," Martin receives a harsh rebuttal from his idol, dismissively - "As if numbers were pre-existing ideas in reality" - followed by a rhetorical retort that chance reigns supreme, for reality is unpredictable, as with hurricanes and cancer, thus fear rather than truth is the outcome.

After the lecture, both Dr Seldom, who has just received a note with a circle, and Martin arrive ("By pure coincidence?" Inspector Petersen later questions) at Mrs Eagleton's residence to find her murdered. Anyone with any involvement shares some responsibility in this psychological thriller and brain teaser, directed by Alex de la Iglesia, screenplay co-written with Jorge Guerricaechevarria, based on Argentine author Guillermo Martinez's novel, which includes references to Gödel's Incompleteness Theorem (axioms must be assumed and obtaining all the facts is impossible) and Heisenberg's Uncertainty Principle.

Upon learning of her mother's death, Beth, feeling guilty for her evil thoughts, confesses to Martin of having waited five years for the old woman to drop dead, freeing her to having a life of her own; she angrily reacts to his resistance to her sudden sexual openness: "Treat me like a woman!"

Having published a book in which he compared a logical series with serial murders, Dr Seldom assumes that the murderer as a purely intellectual exercise wants to embarrass him by proving him wrong. Martin, however, points out that Mrs Eagleton's death, considering her frail health, is "an imperceptible murder" that might not have been realized had they not found her. Lacking material evidence, the police suspect Beth.

Visiting Lorna, a nurse at the hospital where Dr Seldom frequently spends time with his one-time brilliant friend, Kalman, who obsessively investigated mathematical series (ravaged by bone cancer, become a head on a torso with one arm who performed self-lobotomy with a nail gun), Martin becomes briefly acquainted with a man who expresses the view that Jesus was a terrorist who came back from the dead to get revenge on those who crucified him. Lorna relates to Martin that the man he's just met, a bus driver for severely handicapped children, is the father of a daughter who desperately needs an organ transplant.

"Another imperceptible crime" occurs at the terminal ward preceded by a note Martin discovers with the Christian symbol of a fish (two intersecting half circles). Expecting a third victim, Dr Seldom challenges Martin to predict the next symbol in the series, observing that we can always find a rule to justify whatever it might be: unlike the world of semiotics, the real world has irreversible consequences.

During an interrogation with the police, Beth implicates Martin; a psychological profile of the killer suggests a homosexual trying to impress Dr Seldom. With everyone becoming a suspect, Dr Seldom appears to be everywhere that Martin makes his presence, especially with Lorna (who likes mysteries and has a copy of the professor's book). The perfect crime, Dr Seldom describes to Martin with the story of Howard Green's scheming to do away with his wife, involved the crime being solved with the wrong culprit.

A third death, involving the symbol of a triangle, takes place during the Guy Fawkes celebration, where Beth performs with the orchestra in her last concert, Dr Seldom dons the costume of the famous 16th-century leader of the Gunpowder Plot, and another costumed figure draws Martin's attention, along with Inspector Petersen's, away from the central stage and fireworks. Afterward a fourth symbol needs to be found in anticipation of yet more victims.

A little while back I reviewed this mystery/thriller movie, but here I want to revisit it after reading Rebecca Goldstein's Incompleteness. In September 1993 at Oxford, Professor Arthur Seldom (John Hurt) is lecturing about Ludwig Wittgenstein's philosophy, which "set the limits on our thoughts," as elucidated in his Tractatus Theologico-Politicus. "Can we know the truth?" Seldom asks the question, which the great minds have sought to answer with "a single certainty."

Employing the immutable language of mathematical logic, free from the passions of mankind, Wittgenstein came to a "terrifying conclusion," expounds the professor: "There is no such truth outside of mathematics. There is no way of finding a single absolute truth, an irrefutable argument which might help answer the questions of mankind. Philosophy, therefore, is dead. Because whereof we cannot speak, thereof we must be silent."

In the audience Martin (Elijah Wood), an American graduate student from Arizona, who has come to England expecting Seldom will be his advisor for his PhD thesis only to be disappointed, raises his hand, which elicits a sardonic remark from the professor as to whether the young man in breaking the silence has discerned a contradiction in Wittgenstein's logic or perhaps an absolute truth. (Wittgenstein was certain "that there can be no genuine surprises in mathematics," observes Goldstein: "When, therefore, a surprise on the order of Gödel's result arrived, the thing had to be argued away.")

Martin proclaims as a Platonist: "I believe in the number pi…. the golden section, the Fibonacci series. The essence of nature is mathematical. There is a hidden meaning beneath reality. Things are organized following a model, a scheme, a logical series. Even a tiny snowflake includes a numerical basis in its structure. Therefore, if we manage to discover the secret meaning of numbers, we will know the secret meaning of reality."

(To a large extent the German logician Kurt Gödel would be in agreement with Martin, for Goldstein writes of his having "believed his first incompleteness theorem supported Platonism's insistence on the existence of a suprasensible domain of eternal verities.")

Seldom translates the American's speech for the British audience: "We find ourselves faced with a fresh, rousing defense of mathematics as if numbers were pre-existing ideas in reality. Anyway, this is nothing new." He further mocks Martin: "Since man is incapable of reconciling mind and matter, he tends to confer some sort of entity on ideas because he cannot bear the notion that the purely abstract exists only in our brain."

Such people have suggested that a butterfly flapping its wings on the other side of the planet can effect the development of a hurricane, yet no one has ever accurately predicted well in advance the occurrence of a hurricane. "Where is beauty and harmony in cancer?" (Seldom has in mind his deceased friend Harry Eagleton's wife.) When logic comes up against mere chance, the overwhelming randomness of nature, he lectures Martin, we find ourselves in the hands of destiny, facing not truth but fear.

In her book Goldstein writes: "Wittgenstein was later to reject many of the assertions of his Tractatus. In fact, the discontinuity in his thinking was judged so radical that he was bifurcated into 'early' and 'later' Wittgenstein…. In the early Wittgenstein the interesting nonsense (so to speak), characteristic of philosophy, derives from the violation of the rules that govern the bounds of all meaningfulness; in the later Wittgenstein, the interesting nonsense is a result of confusing the rules of one language-game with those of another."

This movie thrills as a result of confusing the rules of one game with those of another while serendipity plays its coincidental part. As a true believer like Gödel in the pure reasonableness of the universe, Martin isn't convinced as he says to Mrs Eagleton's daughter Beth (Julie Cox): "Nothing happens by chance."

Beth's father and mother had been involved, along with Seldom and logician Alan Turing, in deciphering the code of the Nazis' Enigma machine; Harry Eagleton had been the discoverer of "fractional dimensions" (which may refer to "fractal dimensions," the mathematical basis of snowflakes and other objects having irregular or fragmented shapes). Turing, by the way, notes Goldstein, "ignored Wittgenstein and went on to produce another extraordinary proof which shares many attributes with Gödel's; so many, in fact, that it yields an alternate proof for the incompleteness of formal systems rich enough to express arithmetic."

Following the murder of Mrs Eagleton, Arthur and Martin collaborate with each other on solving the case, and in preventing another anticipated victim's death, while Inspector Peterson (Jim Carter) focuses on Beth as the prime suspect. Following two more "imperceptible murders," other persons - Lorna (Leonor Watling), a nurse and Martin's lover; Yuri Podorov (Burn Gorman), Martin's office mate and student of mathematics with a grudge against Seldom; and a bus driver with a daughter in need of an organ transplant - also have motives, as Seldom and Martin appear to have as well from others' perspectives.

A message has preceded each death with a symbol; Martin proposes using logic to determine the next symbol in the series along with linking the clues together, but Seldom reminds him that one can never be certain of having all the facts necessary to connect the dots.

"What if I say," says Seldom, paraphrasing Epimenides's famous paradox, "'All Britons are liars'? True, false, or impossible?" Martin recognizes this self-referential statement as an "indeterminable proposition." Referring to Gödel's incompleteness theorem, Seldom (who, by the way, mispronounces Gödel) adds: "There are things that cannot be proven."

(While superficially Gödel and early Wittgenstein were in accord, their views differed in that Gödel demonstrated that some mathematical truths are expressible but unprovable while Wittgenstein insisted that the most important truths are completely inexpressible.)

When Arthur brings up Heisenberg's uncertainty principle, Martin correctly points out that Beth and other persons involved are not the same as electrons. Seldom remarks that symbols and puzzles aren't the same as real life with its consequences either.

Employing the idiot series (mirror images of the first four natural numbers), Seldom elucidates the fact that with any series of numbers or ciphers "we can always find a rule that justifies" the next character or numeral. Further, Arthur demonstrates how, as with the case of Howard Green, the perfect crime is a crime solved erroneously.

As the symbols appear one after the other, they seem to reveal themselves to be three figures leading up to the Pythagorean tetraktys or quarternary (a temple which has one spire, two pillars, three steps, four gates), provoking Martin to exclaim of Arthur: "As if the damn Pythagoreans were on your side!"

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