Mathematical logic and multiverses

The concepts of mathematical logic, introduced to explain Godel's theorem, can also be exploited to shed further light on the question of multiverses in mathematical physics.

Recall that any physical theory whose domain extends to the entire universe, (i.e. any cosmological theory), has a multiverse associated with it: namely, the class of all models of that theory. Both complete and incomplete theories are capable of generating such multiverses. The class of models of a complete theory will be mutually non-isomorphic, but they will nevertheless be elementarily equivalent. Two models of a theory are defined to be elementarily equivalent if they share the same truth-values for all the sentences of the language. Whilst isomorphic models must be elementarily equivalent, there is no need for elementarily equivalent models to be isomorphic. Recalling that a complete theory T is one in which any sentence s, or its negation Not(s), belongs to the theory T, it follows that every model of a complete theory must be elementarily equivalent.

Alternatively, if a theory is such that there are sentences which are true in some models but not in others, then that theory must be incomplete. In this case, the models of the theory will be mutually non-isomorphic and elementarily inequivalent.

Hence, mathematical logic suggests that the application of mathematical physics to the universe as a whole can generate two different types of multiverse: classes of non-isomorphic but elementarily equivalent models; and classes of non-isomorphic and elementarily inequivalent models.

The question then arises: are there any conditions under which a theory has only one model, up to isomorphism? In other words, are there conditions under which a theory doesn't generate a multiverse, and the problem of contingency ('Why this universe and not some other?') is eliminated?

A corollary of the Upward Lowenheim-Skolem theorem provides an answer to this. The latter entails that any theory which has a model of any infinite cardinality, will have models of all infinite cardinalities. Models of different cardinality obviously cannot be isomorphic, hence any theory, complete or incomplete, which has at least one model of infinite cardinality, will have a multiverse associated with. (In the case of a complete theory, the models of different cardinality will be elementarily equivalent, even if they are non-isomorphic). Needless to say, general relativity has models which employ the cardinality of the continuum, hence general relativity will possess models of every cardinality.

For a theory of mathematical physics to have only one possible model, it must have only a finite model. A Theory of Everything must have a unique finite model if the problem of contingency, and the potential existence of a multiverse is to be eliminated.

Theories of Everything and Godel's theorem

Does Godel's incompleteness theorem entail that the physicist's dream of a Theory of Everything (ToE) is impossible? It's a question which, curiously, has received scant attention in the philosophy of physics literature.

To understand the question, first we'll need to introduce some concepts from mathematical logic: A theory T is a set of sentences, in some language, which is closed under logical implication. In other words, any sentence which can be derived from a subset of the sentences in a theory, is itself a sentence in the theory. A model M for a theory T is an interpretation of the variables, predicates, relations and operations of the langauge in which that theory is expressed, which renders each sentence in the theory as true. Theories generally have many different models: for example, each different vector space is a model for the theory of vector spaces, and each different group is a model for the theory of groups. Conversely, given any model, there is a theory Th(M) which consists of the sentences which are true in the model M.

Now, a theory T is defined to be complete if for any sentence s, either s or Not(s) belongs to T. A theory T is defined to be decidable if there is an effective procedure of deciding whether any given sentence s belongs to T, (where an 'effective procedure' is generally defined to be a finitely-specifiable sequence of algorithmic steps). A theory is axiomatizable if there is a decidable set of sentences in the theory, whose closure under logical implication equals the entire theory.

It transpires that the theory of arithmetic (technically, Peano arithmetic) is both incomplete and undecidable. Moreover, whilst Peano arithmetic is axiomatizable, there is a particular model of Peano arithmetic, whose theory is typically referred to as Number theory, which Godel demonstrated to be undecidable and non-axiomatizable. Godel obtained sentences s, which are true in the model, but which cannot be proven from the theory of the model. These sentences are of the self-referential form, s = 'I am not provable from A', where A is a subset of sentences in the theory.

Whilst the application of mathematics to the physical world may be fairly untroubled by the difficulties of self-referential statements, undecidable statements which are free from self-reference have been found in various branches of mathematics. For example, it has been established that there is no general means of proving whether or not a pair of 'triangulated' 4-dimensional manifolds are homeomorphic (topologically identical).

Any theory which includes Number theory will be undecidable, hence if a final Theory of Everything includes Number theory, then the final theory will also be undecidable. The use of Number theory is fairly pervasive in mathematical physics, hence, at first sight, this appears to be highly damaging to the prospects for a final Theory of Everything in physics.

However, it is still conceivable that a final Theory of Everything might not include Number theory, and in this case, a final Theory of Everything could be both complete and decidable. In addition, even if a final Theory of Everything is incomplete and undecidable, it is the models M of a theory which purport to represent physical reality, and whilst the theory of a model Th(M) may be undecidable, it is guaranteed to be complete. That is, every sentence in the language of the theory will either belong or not belong to Th(M).

Lee Smolin and the multiverse

Lee Smolin argues in Physics World against the notion that there exists a multiverse of timeless universes. Smolin believes that the need to invoke a multiverse is rooted in the dichotomy between laws and initial conditions in existing theoretical physics, and suggests moving beyond this paradigm.

A choice of initial conditions, however, is merely one of the means by which particular solutions to the laws of physics are identified. More generally, there are boundary conditions, and free parameters in the equations, which have no special relationship to the nature of time. Each theory in physics represents (a part of) the physical universe by a mathematical structure; the laws associated with that theory select a particular sub-class of models with that structure; and the application of a theory to explain or predict a particular empirical phenomenon requires the selection of a particular solution, i.e., a particular model. The choice of initial conditions, or boundary conditions, or the choice of particular values for the free parameters in the equations, is simply a way of picking out a particular model of a mathematical structure. For example, in general relativity, the structure is that of a 4-dimensional Lorentzian manifold, the Einstein field equations select a sub-class of all the possible 4-dimensional Lorentzian manifolds, and the choice of boundary conditions or initial conditions selects a particular 4-dimensional Lorentzian manifold within that sub-class.

As a consequence, any theory whose domain extends to the entire universe, (i.e. any cosmological theory), has a multiverse associated with it: namely, the class of all models of that theory. Irrespective of whether a future theory abolishes the dictotomy between laws and initial conditions, the application of that theory will require a means of identifying particular models of the mathematical structure selected by the theory. If there is only one physical universe, as Smolin claims, then the problem of contingency will remain: why does this particular model exist and not any one of the other possibilities? The invocation of a multiverse solves the problem of contingency by postulating that all the possible models physically exist.

The creation of Brown-man

2:30am.

Gordon Brown stared with dour misery at the digital bedside clock. The red digits burned with laser-like intensity in the dark, third-floor bedroom of Number 10. Gordon noticed, not for the first time, that the digits were actually composed of hexagonal and trapezoidal lozenges, and wondered with fresh irritation why they weren't simply rectangular.

Gordon knew that he wasn't going to get any sleep tonight. He could literally feel his spleen burning in anger, and it was fuelling his brain stem, which wailed like a banshee in his head, a venegeful turbine spinning at full throttle. Ploys, plots and revenge scenarios exploded like fireworks across his mind's eye.

Gordon levered himself onto one elbow, and glanced at Sarah. Her large-boned frame lay motionless across the bed, her breath shallow and regular.

Gordon knew that something had to be done, and it had to be radical, much as he despised that Blairite term.

3:00am

At the wheel of his Jaguar, still dressed in pyjamas, Gordon hurtled down the A12. Chelmsford, Colchester, Ipswich. The road signs zoomed past, but Gordon's focus was fixed in the middle-distance. He knew now what must be done, and would not waver. The speedometer hit 100mph. 105. 110. 115. Gordon's unblinking left eye glinted under the baleful sodium roadlights, hanging like spectres over the central reservation.

5:00am

Sizewell B. Gordon lowered himself with grim determination into the cooling waters of the reactor, and bathed in the rejuvenating and transformative blue glow of the Cerenkov radiation.

5:30am

In a Suffolk forest glade, Gordon knelt, head bowed, on a spongey patch of moss. Gnarled roots and trunks and over-hanging boughs surrounded the Prime Minister. Gordon's head and torso throbbed with an opalescent glow. He twisted and screamed in agony and elation as every cell in his body transformed itself, re-arranging its DNA, creating trillions of super-enzymes and super-proteins. The calcium in Gordon's bones transmogrified into an allotropic titanium-plutonium alloy; his nervous system mutated into a broadband fibre-optic TCP/IP network; his muscles became a carbonfibre-reinforced weave; his organs became high-capacity chemical processing plants.

Gordon raised his head and grinned manically. Standing now upon the mossy platform, his body began to pulse with fetid energy. Blue sparks discharged to the ground, and a growing aura surrounded the Prime Minister. "Behold," he proclaimed with a sonic shock-wave, "I'm a pretty straight sorta guy as well!"

And with that, a blinding flash of blue light exploded from his frame, and a luminous wave of purple coruscation propagated outwards through the forest. In its wake, as the first shafts of dawn sunlight penetrated the glade, the wood was transformed into a jewel-encrusted kaleidoscope of light and colour. The facets of a billion jewels, in a million different shades, shimmered and sparkled and glimmered on trunk and bough; the forest floor became a green velvet carpet; and leafs made of gold and silver shone upon the bejewelled branches above.

The Prime Minister marched out of the clearing. The world needed saving, and Brown-man was here to do it.

Simon Singh and Mr Justice Eady

Mr Justice Eady is, apparently, a "quietly-spoken, shy and precise figure." He is also the most senior libel court judge in the country. Mr Eady presided over Max Mosley's successful libel action against The News of the World last year, and his strong anti-media line is apparently motivated by an intrusion of privacy suffered by the actor Gordon Kaye in 1990. Perhaps he's a big 'Allo 'Allo fan.

Some readers may be aware that the well-known science writer Simon Singh was recently judged to have libelled the Bristish Chiropractic Association (BCA). Although The Times newspaper appears not to have reported the story, readers of other publications will be aware that the purported libel was Singh's claim, in an article for The Guardian, that the BCA "happily promotes bogus treatments". Whilst many people might interpret this claim to mean that the BCA promotes false treatments, the judge presiding over the case chose to provide his own interpretation, in which the "natural and ordinary meaning" of the assertion was that the BCA were being deliberately dishonest.

And the name of the judge in question? Pay attention, for I shall say this only once: Mr Justice Eady.

Peace in our time

It seems that enlightened self-interest has worked its magic, and the Formula 1 teams' association (FOTA) have reached a compromise settlement with the governing body, the FIA, in F1's 'budget-cap' row. Let us recall that FIA President Max Mosley was implacably demanding the imposition of a £40 million budget-cap in 2010. That will now not happen. Rather, a budget cap in the region of that figure will be introduced for 2011, but numerous pieces of expenditure, such as engine, driver, hospitality and marketing costs, will be excluded from the cap. This, then, must constitute a victory of sorts for the teams. Max, however, would no doubt suggest that his initial demands were merely an opening bargaining position, and that he has obtained what he really wanted all along.

So, after three months of remarkable political turbulence in Formula 1, the diffuser issue is sorted, the McLaren issue is sorted for the moment, and the budget-cap issue is also apparently sorted. Which leaves us with a Formula 1 season which is being dominated by a single driver and team. Not good for business, that. As The Times's Edward Gorman comments:

In the past a runaway leader has attracted the attention of the FIA, which has stepped in to find something illegal on a hot car.

It's the type of comment which many people think is true, but which few are willing to state in print. If true, it entails that the Formula 1 World Championship is manipulated by the governing body for financial ends; if false, then it is libellous. Either way, it's the type of comment which could attract the attention of the FIA's lawyers, given the difficulty of proving that the championship is manipulated.

On this occasion, it seems unlikely that the FIA will intervene, given the ongoing need to curtail the power of the manufacturer-teams in FOTA. Moreover, Ferrari are beginning to close on Team Brawn, and whilst the championship may be out of their reach, race victories appear to be imminent.

Is the discussion of free will an illusion?

Biologist Martin Heisenberg writes an article for Nature which purports to address the issue of free will, but ultimately does nothing of the kind.

Heisenberg describes the actual research around which the article is constructed, as follows:

My lab has demonstrated that fruit flies, in situations they have never encountered, can modify their expectations about the consequences of their actions. They can solve problems that no individual fly in the evolutionary history of the species has solved before. Our experiments show that they actively initiate behaviour. Like humans who can paint with their toes, we have found that flies can be made to use several different motor outputs to escape a life-threatening danger or to visually stabilize their orientation in space.

The 'expectations' of fruit flies?

Let us be generous, and accept that this term is used metaphorically. The problem with Heisenberg's article owes far more to the general thrust of the argument, which is merely to claim that animals are capable of adapting their behaviour, that "behavioural output can be independent of sensory input." Yet, as Heisenberg admits himself, "the idea that animals act only in response to external stimuli has long been abandoned, and it is well established that they initiate behaviour on the basis of their internal states, as we do." But given that this fact is well-established, it is difficult to see what Heisenberg thinks has been newly discovered in his lab.

Let us accept that Heisenberg's lab have correctly interpreted their empirical data, and that fruit lies are indeed capable of adapting to their environment. This would constitute a type of learning, but it is difficult to see how this bears upon the issue of free will. Neural networks, for example, are capable of learning, and there is a body of literature which demonstrates that recurrent neural networks can be trained to behave like deterministic finite-state automata (DFA). Fruit-fly learning and subsequent behaviour could be represented by such a neural network, but a neural network that can be trained to behave like a DFA is hardly considered to be the epitome of freely-willed behaviour. Neural networks themselves can be either deterministic or stochastic (i.e., random), but both types of causation are distinct from Heisenberg's notion of freely-willed behaviour as "self-generated," (i.e., neither determined, nor random).

If fruit flies are indeed capable of adapting to their environment, then this would be inconsistent with a behaviouristic interpretation of fluit fly behaviour (i.e., an interpretation which denies that fruit flies possess internal states), but it is perfectly consistent with a deterministic interpretation of their behaviour (as well as being quite irrelevant to the issue of free will). Without internal states, there can be no variation in the output response to input stimuli, but with internal states, the response to a stimulus can vary depending upon the internal state, and the internal state can be the result of prior learning.

So Heisenberg's lab have perhaps found evidence for the existence of internal states in fruit flies, but such a finding is of no relevance to the issue of free will.

Parallel universes and wife-swapping

Today, between fields of luminous rapeseed, I drove up to Oxford to see Peter Byrne deliver a talk on the life of Hugh Everett III, the inventor of the many-worlds interpretation of quantum mechanics. Peter is close to completing a biography of Everett, and has been able to draw upon numerous original manuscripts and notes, retrieved with the assistance of Everett's son, Mark (singer and songwriter of Eels), from the family basement.

One thing which struck most of the audience, I think, was the lengths to which Everett's thesis supervisor, the legendary John Archibald Wheeler, went to re-cast Everett's work, not as a radical alternative to Bohr's prevailing Copenhagen interpretation, but as a generalisation of the existing measurement theory. Wheeler appeared to regard Bohr with great awe, and was most anxious not to upset him in any way.

Everett had been fascinated by game theory and operational research prior to starting his work in quantum theory, and after his PhD he immediately left academia to work for the Pentagon's Weapons Systems Evaluation Group (WSEG). Here, he was involved in tasks such as calculating the number of worldwide casualities from a full-scale nuclear war. The audience were amused to see a certificate that Everett received upon completion of his first course at WSEG, in which the congratulatory text was inscribed upon a schematic drawing of a mushroom cloud.

Everett, however, does not appear to have been a warm, affectionate individual, and I must admit that I recognise in a number of my own colleagues the same semi-autistic symptoms which Everett exhibited. After a time at WSEG, Everett and his wife wrote a letter to their friends, all of whom were fellow WSEG employees, declaring that their marriage was now an open one. Everett then proceeded to sleep with most of his colleagues' wives.

Everett became increasingly unhappy, drinking and smoking copiously until he died of a heart attack in 1982. His daughter committed suicide in 1996, writing in her suicide note that she wished to be placed with the garbage, so that she might inhabit the parallel universe closest to her father's.

Angelons and de-mons

Midnight in a subterranean laboratory straddling the Swiss-French border. Father Vaux Vectra is alone at his desk, transfixed by the image on his display screen. He gazes with growing fear and wonder at the lambent pattern of particle tracks, and struggles to absorb the full implications.

Father Vectra had known for years that his colleagues at CERN were not looking in the right direction, were not searching for the right things in the morass of data generated by the Large Hadron Collider. They might have found the Higgs boson, but Father Vectra knew that there was much more at stake here, that there was something of truly metaphysical significance hidden in the data, waiting to be found.

A knock at the door!

Father Vectra's heart spasms with alarm against his ribcage. No-one else should be here! Vectra worked alone, and had studied the work roster with great care to ensure that he would be totally free from interruption.

"Who's there?"

Silence.

Vectra scowls, levers himself out of his chair, and walks with a mixture of irritation and trepidation to the door. "I say, who is that? Are you security? Work control have cleared me to work the late shift, you know..."

More silence.

Vectra pulls his keys out of his pocket, unlocks the door, and turns the handle. It is the last action that he performs alive.

***********************************************************************************

Robert Wheldon's sense of unease was suddenly transformed into outright alarm. "Surely the Dawkinati cannot still exist?" he muttered aloud. "This is worse than I thought!"

"What does it all mean?" asked Danica.

"It looks like Vectra was murdered by an acolyte of a secret cadre, thought to have dissolved decades ago, devoted to the destruction of the Roman Catholic Church."

"That's terrible!" exclaimed Danica. "But why would they kill a physicist like Vectra?"

"Well, according to Vectra's theories and calculations," said Weldon, holding a sheaf of Vectra's hand-written notes, "the material universe we see around us, with its 4 space-time dimensions, its photons and electrons, its carbon and silicon atoms, its nucleic acids and proteins, its oceans, forests and mountains, its red giants and neutron stars, its galaxies, stellar nurseries, and supermassive black holes: all of that is merely appearance and illusion, a transient, local phenomenon. Beneath it all is a 10-dimensional world, described by a unification of string theory and Manichaeic cosmology, in which the forces of good and evil fight an eternal war. Vectra believed that God has compactified 6 of the dimensions into a Calabi-Yau manifold, forcing the exponential inflationary expansion of the other 4, and the creation of the world we see around us.

"Vectra went looking for evidence of God's signature in the moduli fields which keep the 6 dungeon dimensions compactified, and it looks like he found it just before he was killed. With knowledge of these moduli fields, and their quanta of excitation, the angelons and de-mons, it will be possible for the Dawkinati to open up the 6 hidden dimensions temporarily, allowing the Christ to collide with the AntiChrist!"

"We've got to get to the Vatican before it happens," cried Danica. "I've got a helicopter..."

Jenson's secret

You're Jenson Button, and you have a secret. You have a type of knowledge which the Greeks referred to as a techne, a special skill, which in your case is so finely developed, so sensitive, that you can see and feel things which no other driver in Formula 1 can detect. You've known this all along, and it's provided you with a kernel of inner confidence that has sustained you through the wilderness years. Now, finally, it's time to draw upon that well of secret knowledge.

To understand this, it's necessary to appreciate that each corner is not merely a strip of asphalt with a specific radius of curvature; that might be all that the casual observers are capable of seeing, but in reality each corner, (in combination with the characteristics of the car), provides the conditions which create a sheaf of potential dynamical paths. Each path is defined not just by a geometrical trajectory, but by the speed profile over that trajectory. The telemetry traces retrospectively plot these profiles like paths across a valley, the elevation at each point representing the speed, negative gradients corresponding to deceleration, positive gradients to acceleration. Each path has a different combination of initial braking point, turn-in point, deceleration profile, apex, and acceleration profile, and there's an infinite number of these possible paths through a corner. The points of minimum speed form a basin in the sheaf of dynamical paths, and when the paths are projected down into trajectories through the corner, the set of apices form an extended patch on the road surface. Within the sheaf however, is a single, mathematically optimum path, the one that minimises the time spent in the corner. Your ability as a Grand Prix driver is determined by how closely and how consistently you can approach this optimum path through each corner.

Your mind's eye interacting with your visual field, you can almost see this bundle of possible paths, the optimal one running like a golden thread through a tapestry. You know that you cannot keep your car adhered to the golden thread anything other than briefly, but you can keep it within a small neighbourhood of that path, and if the car is as you like it, you can approach it perhaps closer than anyone else.

With the steering wheel cradled in your hands like you're holding the eggshell of a rare Tibetan albatross, the last of its kind, and your inputs smooth as silk, you avoid the corrections which will send the path of your car zig-zagging across the contours of the dynamical landscape. If you could only see the filigree, gossamer detail of the contours at high resolution within this landscape, you would appreciate the importance of being smooth, of minimising needless deviations. It would almost be vulgar to do otherwise, for this is as much an aesthetic sensibility as a utilitarian method.

It's time to follow that golden thread.

Primeval music

The European Space Agency's Planck satellite is due to launch from French Guiana on May 14th. Pending a successful deployment, Planck will measure the temperature of the cosmic microwave background radiation (CMBR) across the entire celestial sphere, with greater sensitivity and spatial resolution than achieved by its predecessor, NASA's WMAP satellite. The variations in the temperature of the CMBR reflect variations in the density of matter when the universe was 380,000 years old, at the time of so-called 'recombination' when atomic nuclei captured previously free electrons.

New Scientist duly have an article to herald the launch, which claims that "these so-called anisotropies are believed to be due to inflation...During inflation, quantum fluctuations in space-time were extended to cosmological scales: by the time the CMB was released, these fluctuations had led to variations in the distribution of matter across the universe. Denser regions of the universe produced CMB photons slightly colder than average, and vice versa."

In fact, whilst it is claimed by cosmologists that temperature fluctuations more than a few degrees across are the imprint of fluctuations present at the end of the inflationary period, fluctuations smaller than a degree are believed to be the result of acoustic oscillations in the plasma of baryons, electrons and photons present between the end of inflation and the time of recombination. These small-scale fluctuations are therefore the visible remnant of the earliest sound waves in the universe.

For the large angular-scale fluctuations, the denser regions redshifted the light climbing out of those regions, and therefore produce cooler spots in the CMBR; in contrast, for the small angular-scale fluctuations, denser regions were regions where the plasma was hotter, hence these denser regions produce hotter spots in the CMBR.