The solution to McLaren's aerodynamic problems

McLaren finally appear to have solved their crippling aerodynamic problems, the main symptom of which was a diffuser which stalled at high speed. Whilst the package of modifications introduced in Germany included a new double-diffuser and front wing, Mark Hughes reports in The Sunday Times that "the most significant part of the car’s upgrade at the last race is believed to have been the redesigned endplates, which feed the airflow to the modified underfloor in a way that keeps the flow much more consistent through a wider range of speeds."

This solves a long-standing mystery. At the first race of the season, Hughes relayed the following intriguing analysis of McLaren's difficulties:

One rival engineer believes he can see what the McLaren's problem is - but obviously isn't about to spill the beans: "There are a couple of cars, and McLaren is one of them, that in their treatment of one fundamental part of the airflow regime show that their aerodynamicists have come to the fore during the era that's just finished, where gains were made in incremental changes. If they had experience of the previous generation of cars they would have known immediately how to treat one particular area that is probably even more important than the diffuser under these regs." It's something he reckons could be cured very quickly once recognised and is not a fundamental part of a car's design. So don't expect the McLaren to be bad indefinitely. (MPH, Autosport, April 2nd, p23).

The most significant prior change of technical regulations was that introduced for the 1998 season, when the maximum track of the car (the lateral distance between the wheels) was decreased from 220cm to 180cm. One aerodynamic consequence of this was that the front wing airflow now interacted with the turbulent and rotating airflow around the front wheels. The response of the teams then was to re-direct the airflow by curving the front wing endplates towards the inside of the front wheels. The 1998 rule-changes were supplemented in 2001 by the raising of the minimum height of the front wing from 40mm to 100mm, thereby reducing the ground effect of the front wing. This combination of raised front wings, with endplates curving to the inside of the front wheels, defined the previous era of incremental aerodynamic development.

For 2009, however, the front wings were widened from 1,400mm to 1,800mm, spanning the width of the car, and were lowered from a height of 150mm to 75mm. The latter has re-introduced ground effect into front wing aerodynamics, and the wider wing-span has caused designers to use endplates which re-direct the airflow around the outside of the front wheels. One presumes that it is these changes in front wing dimensions which McLaren failed to adapt to.

One of the functions of the endplates is to direct airflow around the front wheels to minimise drag. However, the endplates also reputedly generate four or five separate vortices, which can be used to re-direct turbulent air created by the front wheels, and prevent that turbulent air from feeding the underfloor of the car. If such turbulent air were to feed the underfloor, it could cause the diffuser to stall at high speed.

Is Betelgeuse about to explode?

The red supergiant Betelgeuse is something of a close neighbour in galactic terms. At a distance of only 650 light years, and located prominently on the shoulder of Orion, it's well worth peering through the net curtains to see what it's up to.

In fact, Betelgeuse is sufficently close that it can even be resolved by the Hubble Space Telescope (and interferometers) as an extended disk, with a measurable diameter, rather than a mere point of light.

At 20 times the mass of the Sun, and at an age of 8.5 million years, Betelgeuse is already living on spared time, and it has long been predicted to explode as a supernova sometime within the next thousand years. As a red supergiant approaches the end of its life, it consists of a shrinking core, in which temperatures are sufficiently high for nuclear fusion to take place, surrounded by a much larger envelope. Once the core has been converted to iron, the nuclear fusion ceases, and gravitational attraction overcomes the internal pressure gradient of the core, causing it to collapse. This collapse will only be halted after electrons and protons have reacted in the core to form neutrons, and the quantum mechanical neutron degeneracy pressure then halts the collapse. The star's envelope then falls inward, and rebounds off the rigid core, causing a shock wave to propagate outwards, effectively blowing the star apart. The only residue will be either a neutron star, or if the mass of the core is greater than the Chandrasekhar limit of 1.44 solar masses, a black hole.

Now, a team of astronomers lead by Nobel laureate Charles H. Townes have recently reported that the diameter of Betelgeuse has shrunk by about 15% since 1993, and there is speculation that Betelgeuse may be in the initial stages of collapse to a supernova. John Baez calculates that if Betelgeuse does explode, it will burn in the sky up to 3 times as brightly as the full moon. That might make the news.

It's worth noting that whilst nuclear fusion is capable of synthesizing all the elements up to the iron group, heavier elements require neutron capture reactions (operating in conjunction with beta decay). These reactions go under the name of s-type and r-type processes. The former operate under fluxes of 10⁵-10¹¹ neutrons per cm per second, which is available over thousands of years inside larger stars which have completed their main hydrogen burning phase and have entered a red giant/asymptotic giant branch (AGB) phase. The s-process, however, is only capable of producing isotopes up to bismuth-209. To produce heavier elements such as thorium and uranium, it is necessary for the r-process to operate, which requires fluxes of the order of 10²² neutrons per cm per second. This environment is only available in a core-collapse supernova explosion (or possibly in neutron star collisions).

Hence, if Betelgeuse really is about to explode, not only will it provide a spectacular blaze of light in the sky, but it will also supply a local source of the heavier elements. It might even leave an uncomfortably local black hole in its wake...

The tick

So, I goes for a walk in the fields near where I live, about mid-afternoon, and a lot later in the day, about midnight, I'm feeling very tired, and getting ready for bed, when I feel an itch on the back of me leg. I scratch at it, and notice something black. I take a closer look. Flicking it back and forth, I can see it has little legs, which writhe as I pluck at it. It also probably has a head, although that's currently buried into my flesh, drinking my blood, and probably has been for the past nine hours.

Now, I've heard what to do in circumstances like this, that you shouldn't attempt to pull the tick out because you'll leave part of it imbedded in your skin. "You're supposed to get a match and burn them," I think. "But I haven't got any matches! What am I supposed to do? I'm so tired, I just wanna go to bed, but there's a living thing imbedded in my leg!"

I spend a couple of minutes trying vainly to get my brain into gear, trying to think what I should do. After careful thought, I respond by just yanking out what I can. "Fucking thing!"

Which leaves its head buried in my skin.

"So what now? Have I got something sharp I can use to dig the head out? Ummmmmm...I know: the breadknife!" So I goes into the kitchen and gets the breadknife, sits on the edge of me bath, and I starts digging away with the breadknife. Trouble is, even a breadknife ain't sharp enough when you look at it real careful, and I'm just digging a sort of slightly bloody pit around the black head of the tick.

"What next then? Bed? Yeah, bed." So I wander back into the kitchen, replace the breadknife, and then find that I do have some cooking matches after all! "Maybe I can, er, burn the head or something." So I sits on the edge of me bath, lights a match, and starts dabbing at it with the naked flame.

Dab, "Ow!" Dab, "Ow!" Dab, "Ow!" Dab, "Ow!" Dab, "Ow!" Dab, "Ow!"

That didn't work either.

So I finally goes to bed, and the next morning I goes to work and phones the medical centre. "Can you get foreign bodies out of the skin?" I asks. "What type of foreign bodies?" she replies. Anyway, long story short, I digs it out with a sterile needle, as the watching nurse tells me you can catch lyme disease from ticks. Apparently, the initial symptoms are like the 'flu', so if I gets a fever in the next few weeks, it could be flu, it could be swine flu, or it could be lyme disease. Fucking thing. If one of 'em tries it on again, I'm gonna burn the fucker.

The Sheila Johansson Inadequacy

For a general philosophical outlook, feminist historians have a number of possible options:

(1) History has been largely violent and bad, men are to blame for this, and women have been subjugated victims.

(2) History has been a mix of good and bad, men are responsible for the good and the bad, and women have been subjugated victims.

(3) History has been largely violent and bad, but women have been a civilizing force.

(4) History has been a mix of good and bad, men are responsible for the bad, and women are responsible for the good.

And herein lies the fundamental difficulty for every strain of academic feminism: whilst rigorous scholarship and scientific research requires the open-minded pursual of all possibilities, the persecution complex which underlies feminist thought, automatically introduces a gender-based bias which precludes the possibility of such neutral-minded rigour.

Sheila Johansson, for one, seems to prefer the third option above, writing in her 1976 paper, 'Herstory' as History: A new field or another fad?:

"Even the most chauvinistic male historian might admit that history, if it has been dominated by Man, has also been filled with more heat than light, more injustice, selfishness and violence than enlightened detachment, and certainly more conservatism than progressive creativity. Furthermore...Man, when contemplating Woman, was for the most part a pretty conventional fellow, anything but heroic and objective, quite derivative and often stale in his handling of the woman question.

"The West's current cultural malaise and ecological crisis have led some to reassess the relative merits of 'masculinity' and 'femininity' (as presently defined). Many forces are combining to make a climate more favourable to the appreciation of women and things feminine."

In retrospect, however, this paper has something of an ironic air, given the downward trajectory of Sheila Johansson's own academic career in relation to that of her husband, the economist Paul A. David. After a spell in the UK, she has now moved back to the United States since David became Emeritus Professor of History in the Department of Economics at Stanford University.

But as Sheila wrote back in the Seventies, "in spite of all the discrimination that has hampered women in history, females have a certain amount of built-in power over males, as mothers, wives, lovers and sisters."

Team Brawn, Rossby waves and laser sparkplugs

Despite a dominant start to the season, Team Brawn have struggled somewhat in the last couple of Grands Prix, and, looking at the bigger picture, this actually reveals an inability to cope with Rossby waves.

Rossby waves are kinks or meanders in a jet stream, and if such kinks in the Northern hemisphere's polar jet stream stretch Southward in the Summer, they expose Northern Europe to cold polar air. Such has indeed been the case for the last two Grands Prix, and Team Brawn have struggled to get their tyres up to operating temperature under the cooler conditions.

It is therefore the Rossby waves towards which Rubens Barrichello should be directing his ire, not his own team's strategists.

Meanwhile, The Sunday Telegraph reports that laser ignition will soon be replacing sparkplugs in road cars. Project leader Dr Tom Shenton, of Liverpool University, reports that:

"Lasers can be focused and split into multiple beams to give multiple ignition points, which means it can give a far better chance of ignition.

"This can really improve the performance of the engine when it is cold, as this is the time when around 80 per cent of the exhaust emissions are produced and the engine is at is least efficient.

"The laser also produces more stable combustion so you need to put less fuel into the cylinder."

This is a fabulous idea, and one wonders why Formula 1 no longer has the wherewithal to make this type of breakthrough.

Moreover, given the imminent introduction of laser ignition into motor cars, one might go one step further and replace the cylinders with hohlraums. One could then replace hydrocarbon fuel with pellets of frozen deuterium-tritium, and use the lasers to trigger inertial confinement fusion within the engine. Needless to say, the pistons would have to be made out of a mixture of carbon, beryllium and tungsten to withstand the temperatures produced by a nuclear fusion plasma...

Stuart Kauffman and Reinventing the Sacred

In Reinventing the Sacred, biologist and complexity theorist Stuart Kauffman proposes an alternative to conventional supernatural religions. The basis for Kauffman's proposal seems to be complexity theory and the notion of emergence. In 2008, he wrote in New Scientist that "reinventing the sacred...will require a shift from reductionism...I do not believe that the evolution of the biosphere, economy and human culture are derivable from or reducible to physics. Physicists cannot deduce, simulate or confirm the detailed evolution of the biosphere that gave rise to the organised structure and processes that constitute, for example, your heart. Entities such as hearts, that have causal consequences, are 'real' in their own right."

Kauffman's entire world-view seems to be predicated on this anti-reductionism, yet it betrays a simple misunderstanding: it fails to understand the distinction between ontological reductionism and epistemological reductionism. Andrew Steane reviews Kauffman's book in the July issue of Physics World, and similarly fails to acknowledge the difference.

To digress, then: Ontology pertains to what actually exists and happens; epistemology pertains to what we know, or can know, about what exists and happens. Ontological reductionism merely proposes that the parts of a system, and the way in which those parts are organised and interact, uniquely determine the higher-level states and properties of the system. In particular, ontological reductionism proposes that what ultimately exists are the objects studied in physics, whether they be particles, strings, or whatever. Epistemological reductionism, in contrast, proposes that we can always explain and predict the higher-level states and properties of a system from the parts of the system. In particular, epistemological reductionism proposes that the theories and laws of any branch of science are derivable from those of physics.

Due to the conceptual incompatibilities between different theories, and simple limitations in the tractability of equations when dealing with systems containing billions of particles, epistemological reductionism is false. This is consistent, however, with the truth of ontological reductionism. Kauffman has spotted the falsity of epistemological reductionism (like many authors before him), but seems to think that this invalidates ontological reductionism as well, and bases his new concept of the sacred on this misunderstanding.

Consider, for example, one of Kauffman's central arguments:

"Weinberg rests his reductionism on the claim that the explanatory arrows always point downward, from society to cells to chemistry to physics. But with respect to evolution of hearts by natural selection, do the explanatory arrows actually point downward to string theory or whatever is down there? No. They point upward to the selective historical conditions in the actual evolution of organisms in the specific biosphere that gave rise to hearts...We have now moved beyond reductionism and arrived at emergence, both epistemological and ontological," (p43).

The fact that not all explanatory arrows point downwards, is a fact which mitigates against the possibility of epistemological reductionism, but it remains perfectly consistent with ontological reductionism. Kauffman proclaims that he has established the truth of ontological emergence, but fails to justify this claim. As Steane remarks, "the book implies that a case has been proven when it has not."

Roebuck on Mosley

"My term of office ends in October 2005, and I have a big decision to take fairly soon, to go on or to stop...One should never stay too long, and I am very sensitive to that." (Max Mosley, July 2003).

Separate remarks in the past few days from Autosport's Mark Hughes, and The Times's Edward Gorman, seem to imply that the Mosley camp is in the throes of constructing some sort of legal case against FOTA, on the basis that their efforts to unseat him as President of the FIA constitute a vendetta. It is an approach to governance which would, no doubt, find favour with Robert Mugabe.

One hopes that Mosley is merely attempting to give himself some bargaining chips over the coming months; some leverage, as it were, which will help him to install esrtwhile Ferrari team-principal, Jean Todt, as his successor, come October.

Meanwhile, the August issue of Motorsport magazine contains Nigel Roebuck's epitaph on Mosley, and one feels that Nigel has been waiting not a little time to deliver these words:

So, as Mosley had required the head of Ron Dennis in April, now his own is - at last - on a FOTA platter and it is not impossible that Ron, together with virtually everyone else in F1, has raised a glass or two.

Seventy or so years ago the celebrated diarist James Lees-Milne wrote this: 'It became clear he was a man of overweening egotism. He did not know the meaning of humility. He brooked no argument, would accept no advice. He was overbearing and over-confident. He had in him the stuff of which zealots are made. He was madly in love with his own words. It could be a terrible day, I fancied, when they ran away with him, and took the wrong turning.'

Lees-Milne was writing of Oswald Mosley. On recent evidence it would seem the apple didn't fall far from the tree.

Karen Armstrong and the case for God

[Modern theists] give the name of 'God' to some vague abstraction which they have created for themselves; having done so they can pose before all the world as deists, as believers in God, and they can even boast that they have recognized a higher, purer concept of God, notwithstanding that their God is now nothing more than an insubstantial shadow and no longer the mighty personality of religious doctrines. (Freud, The Future of an Illusion).

In The Case for God: What religion really means, former Catholic nun Karen Armstrong reiterates a now-familiar line of defence against the new wave of atheism. This generally amounts to the complaint that atheists such as Richard Dawkins have a theologiclly-uninformed, and mistakenly literalist interpretation of religious scripture. Thus, amongst those of an educated, literary-ecclesiastical background, religion is defended by advocating a metaphorical interpretation of scripture, and an aesthetic-mytho-poetic concept of God.

Wary of the power of science to overthrow religious worldviews, as demonstrated in the Copernican and Darwninian revolutions, the modern theist ushers God into an ontological safe-zone, where he cannot be subject to refutation by empirical means. Realising, however, that even this stronghold cannot resist the barbs of logic and reason, God is blindfolded, and bundled unceremoniously into a waiting limousine, whence he is taken at breakneck speed to a supra-logical and supra-semantic realm, beyond all human understanding.

"God is, by definition, infinitely beyond human language," writes Christopher Hart in The Sunday Times. "Yet thanks to the misapplication of science to religious faith, we remain literal-minded and spiritually immature, frightened of the silence and solitude in which the Ancient of Days, the Unnameable, might be experienced, though never understood.

"We need to think of God not as a being, but as Being. Armstrong points us towards a vast tradition in all religions in which, in essence, you can ultimately say nothing about God, since God is no thing. In Islam, all speaking or theorising about the nature of Allah is mere zannah, fanciful ­guesswork. Instead, try 'silence, reverence and awe,' she says; or music, ritual, the steady habit of compassion, and a graceful acceptance of mystery and 'unknowing'...God is dead — but, Armstrong suggests, all we have lost is a mistaken and limited notion of God anyway: a big, powerful, invisible man who does stuff."

All of which will come as a surprise to the majority of monotheistic religious believers in the world, who believe that the universe was created by God, that God answers prayers and performs miracles, and provides the means for an afterlife.

Hart's proposition that God is not a being, but Being itself, is the familiar doctrine of pantheism, which is inconsistent with the personal nature of God enshrined in Christianity, Judaism and Islam. The notion espoused by these religions that God is a transcendent, supernatural, personal being, who created the natural universe, is inconsistent with the pantheistic notion that God is an immanent, non-supernatural, non-personal being, equivalent to the natural universe. But, of course, it is precisely the existence of such irritating contradictions which explains the modern theist's desire to push God into a supra-logical realm.

To propose that the notion of God is beyond all human understanding, language and logic, is to acknowledge that there is no coherent, comprehensible content to belief in God. Not only is belief in God belief without reason or evidence, but it is a belief without coherent content. The proponent of the modern educated defence against atheism is, in effect, admitting:

'I have a belief, without reason or evidence, in a meaningless proposition.'

At which point, I rest my case.

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.