The Rafa rhyme

The Rafa-lution is wearing thin,
Sir Alex Ferguson is sporting a large grin,
The fans are tiring of Dirk Kuyt,
The journalists fear your sanity is slight.

You think that football is a game of chess,
But your rotation policy just creates an awful mess,
Your zonal marking cannot work,
Your substitutions are beserk.

You want the freedom to buy and sell at will,
But your team selections make our chances nil,
You've never understood the English game,
Your post-match excuses sound really lame.

After 5 years, I think the jury is in,
And I'm sorry to say, Rafa, the Premiership, you're never going to win.

Rafa Benitez

Grand Designs and the Pantheon

The people featured on Grand Designs are awfully bourgeois and middle-brow, don't you think? Their property aspirations, I find, are generally rather banal. The term 'budget' (a typical bourgeois concern), often seems to enter the fray.

This is all very infra dig. If I designed a house, it would have a voluminous and labyrinthine library; a wine cellar, with a tunnel leading to a secluded balcony overlooking the local river; a retractable domed emplacement on the roof, housing a powerful telescopic observatory; a particle accelerator in the basement; multiple spiral staircases; glass elevators; conveyor belts; a grand piano in the foyer; and lots of secret doors.

The entire building would also function as a sundial, much like the Pantheon. On last October's trip to Rome, we were fortunate enough to stay in a hotel opposite the Pantheon. We did not know it then, but it transpires that the geometry of the interior is so designed that, every equinox, and only on the equinox, if the day be not cloudy, then the sunlight will shine through the ocular at the top of the domed ceiling (and I thought an 'ocular' was the thing darts players stand on), onto a small grille placed at the junction between the roof and the wall. The sunlight will thence pass through the grille, and shine onto the floor of the colonnaded courtyard outside, the only time during the year that the sun will fall thusly.

Now that's proper architecture.

Grand Designs Pantheon

Cognitive horizons

Whilst the death of an individual constitutes a mental singularity, it's also worth noting that prior to reaching such an endpoint, different minds not only have different contents, but also possess differently-sized cognitive horizons.

Each cognitive individual has a total mind-map, a fluctuating space of ideas and experiences in memory, linked into a network by the relationships of association, causation and entailment. This space of ideas can be equipped with a metric, which defines the mental distance between any pair of mental items. The distance between two mental items can be defined to be the minimum number of links between those items, taken over all the paths joining those elements in the mind-map. The cognitive horizon of an individual can then be defined to be the maximum distance between any two items in that individual's mind-map. With this concept, we have a precise definition of small-mindedness (and large-mindedness, for that matter).

Following from this, an important question for the philosophy of mind and for neuroscience, is to ask:

Is there an isomorphism at some level between the state of an individual's neuronal network and the state of their mind-map?

If there is such an isomorphism, then an exact brain scan would enable one to read off the state of an individual's mind-map. Conversely, if there is no such isomorphism, then what are the implications of this for the relationship between the mind and the brain?

Cognitive horizon Mind-map

How to live forever

I am sitting alone in the small seminar room on the tenth floor. This is known as 'Harry's room'. I am at the head of a long oak table, working at a laptop computer. The door is at my back and the single window at the other end of the room sheds a thin, early evening light. There are glass-fronted oak cabinets along the walls, left and right. On the shelves are rows of display jars containing specimens of human brain, each suspended in a liquid the colour of watery piss. This is Harry's collection. The specimens are arranged according to pathology: tumours, cerebrovascular disease, degenerative disorders, and so on. There are whole brains, half brains, and parts of brain, sliced and segmented. Close to my right shoulder, there swims a cerebellum.

The room is ineffably still.

'I could be bounded in a nutshell and count myself a king of infinite space,' said Hamlet, 'were it not that I have bad dreams.' The infinite space was within the shell of his head. And so, inescapably, were the dreams. But looking around now at these dead still, grey-beige objects it is hard to see them as erstwhile progenitors of infinite space. They each represent the opposite: a singularity. A point at which the universe has collapsed.

(Paul Broks, Into the Silent Land).

From our perspective, it takes a finite time for other people to die. Each death represents the end-point of a mental world-line in space-time, a type of mental singularity, as Paul Broks puts it. Perhaps, however, there is a sense in which we can use a well-known psychological phenomenon, and the mathematics of conformal transformations, to rescue the dead from these singularities.

It's a familiar truism that time passes more quickly when you're concentrating on something, and more slowly when you're bored. The more you think, the less subjective time appears to elapse, and the less you think, the more subjective time appears to elapse. A greater amount of thought corresponds to greater brain activity, a higher rate of information processing, whilst a smaller amount of thought corresponds to less brain activity, and a lower rate of information processing.

Taking the limit in which the amount of brain activity, and therefore the rate of information processing, tends to zero, as one approaches death, the amount of subjective time which elapses becomes infinite. Thus, subjectively, we never die.

We can retain the general relativistic representation of physical space-time as a 4-dimensional manifold, equipped with a metric tensor of Lorentzian signature, but for each worldline γ representing a physical system with a mind supervening upon it, we can introduce a subjective time tensor. Let τ denote the proper time along the worldline; this is determined by the metric of the physical space-time, and a finite proper time will elapse along the worldline of any mortal. However, if we let dI/dτ denote the rate of information processing along the worldline of a mental system, then the reciprocal of this (dI/dτ)^-1 measures the lapse of subjective time, and we can treat this as a conformal factor Ω which enables us to define the subjective time tensor as follows:

-Ω(τ) dτ ⊗ dτ ,

where

Ω(τ) =(dI/dτ)^-1(τ) .

The conformal factor Ω blows up towards the end of the worldline of a mental system, providing such a system with an infinite lapse of subjective time, even though, to an external observer, a finite time lapses before the death of that system.

Possible objections? Well, for a start, one needs to explain why the years seem to pass more rapidly as we get older. And presumably, there is a lower threshold on the rate of information processing which a brain is capable of, below which the continuum approximation tacitly assumed by this use of calculus breaks down. Below this lower threshold, the brain has ceased to operate. Moreover, an infinite lapse of subjective time, in which a vanishing amount of experience is possible, seems to constitute something of a living purgatory...

How to live forever Paul Broks Conformal Death

Coconut ice cream and Wagamama

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The tree of life?

The tree of life, it seems, is no more than a first approximation, and must be supplanted by the web of life.

This week's New Scientist contains a fascinating article which explains that, due to horizontal gene transfer, the evolution of life does not necessarily possess a tree-like, branching structure, but can also contain horizontal links, when different species exchange genetic material, and loops when two different species coalesce. The structure of living species, then, is not so much dendritic as reticulate.

Horizontal gene transfer (HGT) applies primarily to microbial species, rather than multicellular eukaryotic species such as plants and animals. Whilst microbes undergo self-reproduction rather than sexual reproduction, they are capable of exchanging genetic material during their individual lifetimes, which can be incorporated into their genomes, and transferred to their descendents through self-reproduction. It's been known for a while that microbial evolution is strongly influenced by HGT, but it was thought that this web-like structure was confined to the base of the tree of life. In multicellular eukaryotes, HGT appears to be difficult because the germ cells are separate and protected; even if a virus infects a subsystem of the body, and inserts its genetic material into the cells there, that genetic material will only affect the function of the subsystem, and will not be inherited by the descendants of the multicellular organism. The New Scientist article, however, suggests that there is evidence of prolific HGT amongst plants and animals.

This is interesting in itself, but immediately made me wonder if HGT has a cosmological analogue. If we combine braneworld cosmology and cosmological natural selection, then we obtain a scenario in which self-reproducing universes not only evolve by natural selection, but in which they collide and interact. Perhaps such interactions would permit physical information to be exchanged between different universes, altering the parameters of physics in the colliding universes. Lee Smolin's proposal for cosmological natural selection suggests only that the parameters of physics undergo random mutations when a new universe is born inside a black hole. Whilst Smolin's scenario produces a tree-like structure to the population of evolving universes, what we might dub horizontal cosmogenic transfer (HCT) produces more of a web-like structure.

Horizontal Cosmogenic Transfer Horizontal Gene Transfer Tree of Life Web of Life

Fundamentals

Psychological time is a notoriously malleable quantity. I hardly seem to spend any time waiting in supermarket checkout queues these days, but that's mainly because I tend to slip into a type of absent-minded reverie. Sometimes I imagine winning the Monaco Grand Prix from 17th place on the grid; on other occasions, I come from 4-down with 5 holes to go to beat Tiger Woods and win the Ryder Cup for Europe. On very rare occasions, I ponder more difficult tasks, such as answering Martin Heidegger's question: What is the thingness of a thing? In other words, What is it for a thing to exist?

Someone in front seems to have something with a barcode missing, so I've got the opportunity to recall that there's a number of concepts which can be identified as fundamental in the human conceptual system:

(i) Objects.
(ii) Properties and relationships.
(iii) The possession of properties and relationships by objects.
(iv) Existence.

All other concepts appear to be built from these, but are these concepts independent, or can we define them in terms of each other, and reduce the number of fundamental concepts?

First of all, properties and relationships can themselves possess properties and relationships, hence properties and relationships can be treated as special types of objects. One could define zeroth-order objects to be objects which are not predicated of other objects; first-order objects (properties) to be those which are predicated of other objects; second-order objects (binary relationships) to be those which are predicated of pairs of other objects, and so on.

Conversely, however, one can define objects to be merely bundles of properties. Moreover, if existence is merely absence from contradiction, then an object exists by virtue of the fact that the bundle of properties which define it is free from contradiction. This would re-define existence as a logical property of bundles of properties. Whilst existence might not be a property of zeroth-order objects, it could be a property of the bundle of properties which defines a zeroth-order object.

(I'm vaguely irritated by the fact that some silly person in front wants cashback). Even the possession of properties by objects can be re-cast as a relationship itself, but a relationship between objects and properties rather than a relationship between (zeroth-order) objects.

There is no need for one of these approaches to be right, and the others to be wrong; it may be that the human conceptual system is cyclic, in the sense that each one of the fundamental concepts above can be defined in terms of the other concepts. But just as this idea solidifies, it's time to ask for some plastic bags, and to admit once more that, no, I don't have a Sainsburys Nectar card thanks.

The Primordial Existence Question

Modern philosophical legend, Adolf Grunbaum, considers the question, Why is there something rather than nothing? to be a pseudo-problem, and the theistic response to the question to be a pseudo-explanation. I would agree with Grunbaum on the latter, but I think that what Grunbaum dubs as Leibniz's Primordial Existence Question (PEQ) does pose a legitimate philosophical problem.

If we can equate nothing with the empty set and something with a non-empty set, (not a trivial assumption, given that mathematics extends beyond set theory), then it is significant to note that whilst every set contains the empty set as a subset, the converse certainly isn't true. This asymmetry can be used, I suggest, to construct an answer to the PEQ.

Because every set contains the empty set, the existence of something is consistent with the proposition that everything logically possible does actually exist. In contrast, the existence of nothing is clearly inconsistent with the proposition that everything logically possible actually exists; the existence of nothing excludes all other possibilities.

So the PEQ, Why is there something rather than nothing? receives the answer Because all logical possibilities actually exist. Needless to say, however, this then invokes the further question, Why do all logical possibilities actually exist? This can, in turn, be answered by equating actual existence with possibility, i.e., with absence from contradiction.

In particular, mathematical objects and structures exist because they are free from contradiction, and the physical universe, if identified with its mathematical structure, exists simply by virtue of the fact that that mathematical structure is free from contradiction.

Adolf Grunbaum Leibniz Nothing Something

The flying car - an update

A couple of years ago, McCabism featured the Terrafugia Flying Car, and the promise that it would be available for purchase in 2009.

Sadly, it still appears to be 18 months away from 'appearing in the showrooms', but nevertheless warranted two separate articles in The Sunday Times, here and here.

Disappointingly, there's no mention in either of these articles of Terrafugia's 2007 VP of Sales and Marketing, Alex B.Min. Maybe he's not 'B min' like a cheshire cat after all.

Bryan Appleyard and creationism

Bryan Appleyard has an article on Darwin and natural selection in The Sunday Times newspaper, which quotes, without reproach, Dr David Menton of Answers in Genesis, and David Rosevear, chairman of the British Creation Science Movement. Appleyard also quotes the anti-evolutionary opinions of James Le Fanu, (a medical doctor and journalist, no less):

[Le Fanu] insists that new biological discoveries have overthrown Darwin. The old man is "screwed", he says gruffly.

Perhaps most startling is the discovery from the deciphering of the human genome that we have only between 20,000 and 25,000 genes. We were previously thought to have 100,000. A mere 25,000 doesn’t seem to be enough to sustain our vast complexity and yet genes are supposed to be the heavy lifters of the Darwinian enterprise.

"I wouldn’t get out of bed for 25,000 genes," says Le Fanu, "and we don’t find form in the genome. We share most of our DNA with chimpanzees, but nowhere in the genome have we found what it is that makes us so different from chimps."

It's an interesting point. But here's a good response:

It is unarguably true that the differences between a monkey and a human are huge...The point is - as every geneticist...knows perfectly well - that a small number of nucleotides can make a very big difference...The fact that, at the molecular level, the difference appears small is irrelevant because, at the molecular level, everything appears small. And, besides, the whole of modern science from quantum theory to chaos theory has successfully persuaded us of the fact that small things make big differences.

And the writer of these wise words? Bryan Appleyard, in fact, on p102 of his 1998 work Brave New Worlds. It's strange that Bryan didn't choose to raise this point in his Sunday Times article.

Moreover, Bryan uses today's article to wheel out the most famous evolutionary canard of the creationist movement, namely the claim that natural selection cannot explain the evolution of the eye:

It’s all very well to talk of small mutations changing an organism, but how do such changes make, for example, an eye? Without all its bits and pieces, an eye does not work. It is, in the terms used by the biochemist Michael Behe, author of Darwin’s Black Box, "irreducibly complex", beyond the reach of blind, random mutation.

On the contrary, as Richard Dawkins has pointed out, "intermediates are not only easy to imagine: they are abundant all around the animal kingdom. A flatworm has an eye that, by any sensible measure, is less than half a human eye. Nautilus...has an eye that is intermediate in quality between flatworm and human. Unlike the flatworm eye, which can detect light and shade but see no image, the Nautilus 'pinhole camera' eye makes a real image; but it is a blurred and dim image compared to ours." (The God Delusion, p124).

The creationist/intelligent design attack upon evolutionary theory is two-pronged: not only does it seek to imply that evolution is incomplete or flawed as a theory, but it hedges its bets by also arguing that the consequences of evolutionary theory have been detrimental to society. Appleyard duly follows this rhetorical template, and quotes David Rosevear's assertion "If [we are not the children of God], then there is no right or wrong – we can do what we like."

This argument, another popular canard of the creationist/intelligent design movement, presupposes that the existence of God is necessary for the truth of ethical principles. As philosopher Adolf Grunbaum pointed out in The Poverty of Theistic Morality, this fallacy was exposed by Socrates in Plato's Euthyphro:

Is the conduct approved by the gods right ("pious"), because of properties of its own, or merely because it pleases the gods to value or command it? In the former case, divine omnibenevolence and revelation are at best ethically superfluous, and in the latter, the absolute divine commands fail to provide any reason at all for imposing particular kinds of conduct.

For if God values and enjoins us to do what is desirable in its own right, then ethical rules do not depend for their validity on divine command, and they can then be independently adopted. But, on the other hand, if conduct is good merely because God decrees it, then nowadays we also have the morally insoluble problem of deciding, in a multi-religious world, which one of the conflicting purported divine revelations of ethical commands we are to accept.

Appleyard tells us that Darwinian evolution "was and is, for many, a grim vision," but also points out that "Darwinism remains only a small part of the popular imagination." So the claim, then, is that we are depressed by something we are largely unaware of!

This contradiction is important because it points to the fundamental fallacy of Appleyard's worldview, the general thrust of which is to characterise science as providing a threat to human well-being, with the ulterior motive of promoting, as the remedy, an aesthetic-religious worldview. The general theme permeating the Appleyard oeuvre, is to suggest that humans have a need for 'the sacred', and for 'spiritual depth', and to suggest that science is a danger because it threatens those needs. Appleyard wishes us to believe that science is depriving us of something essential.

On the contrary, the needs Appleyard speaks of are not general truths about the human condition, or what it feels like to be human, but truths about what it feels like to have inherited a certain religious worldview. Darwinian evolution entails neither a 'grim vision', nor, as Darwin himself suggested, a type of grandeur; these are extra-theoretical valuations, which can be tacked-on, depending upon the subjective personality of the individual.

Ultimately, on the subject of evolution and religion, creationists and their journalistic apologists, would do well to heed the verdict of the Reverend Michael Heller ('Where physics meets metaphysics', p272-273, On Space and Time, Cambridge University Press, 2008):

Correct theology is obliged to take into account what science has to say to us...And the verdict of science...is clear. The Universe we live in is an evolutionary process, and the thread leading from the plasma of primordial stuff, through chemical elements, galaxies, stars and planets, to more and more complex systems, intelligent life included, is but a fibre in this overwhelming process. And theology that would choose to ignore this magnificent process is a blind way to nowhere.

Ron Dennis and integrity

I love listening to Ron Dennis. Whilst Ronspeak comes with a certain level of corporate cliche, and its use of prosaic circumlocution can be irritating to some, the scale of Ron's achievement in the most competitive business in the world, bestows upon his utterances a sense of weight and solemnity, perhaps in the same way that the Higgs field purportedly bestows mass upon electrons and quarks.

Ron answered questions from the audience this week at the Autosport International show. Asked about his future plans, he intriguingly hinted at a semi-philanthropic role, helping to provide opportunities in life to young people. And in a year when others involved in the administration of Formula 1 continue to find themselves guided by dishonesty, expedience, vanity and self-interest, Ron remains a beacon of integrity:

Our company suffered a phenomenal fiscal penalty that would normally bring any organisation of our size to its knees, but in the end the one thing that came out – and we are not judging ourselves but we are judged by those people who really immersed themselves in the situation and really analysed everything about it, I believe the company emerged with its integrity, and I emerged with my integrity, and to me that is a priceless thing. Priceless.

Ron Dennis

Cosmogenic drift and stochastic processes

I recently proposed an extension to Lee Smolin's theory of cosmological natural selection, in which I attempt to explain the evolution of a quantum relativistic universe, a necessary precondition for a population of universes evolving by natural selection. I propose that the parameters of physics not only change when a new universe is born, but undergo random diffusion even in universes which cannot initially reproduce. I refer to this process as cosmogenic drift, by analogy with the process of genetic drift in evolutionary biology.

Random diffusion is a type of stochastic process, so if the theory of cosmogenic drift is to be developed, and if observable predictions are to be derived from it, then it will be necessary to employ the mathematics of stochastic processes. Random diffusion comes in a number of different varieties, so the first question one might pose is to ask which particular type is being suggested here. The simplest type of diffusion is Brownian motion, (also termed a Wiener process), which is a simple random walk in which the increments between random variables S_t have a normal distribution with a mean value of zero. Geometric Brownian motion, in contrast, is such that each random variable S_t has a lognormal distribution. Moreover, Brownian motion and geometric Brownian motion can possess a drift, which ensures that the mean values of the random variables S_t evolve as if under the action of an external force. For example, the Black-Scholes equation, used to calculate the price of financial options, assumes that the value of underlying stocks will evolve according to geometric Brownian motion with drift. This stochastic process is typically denoted as

dS_t = μ S_tdt + σ S_t dW_t ,

where μ is the percentage drift due to the expected risk-free rate-of-return on the underlying stock, μ S_t is the drift-rate, σ is the volatility of the stock, σ S_t is the diffusion rate, and dW_t is a standard Wiener process.

In terms of the probability distribution ρ on the range of the random variables S_t, in the case of simple Brownian motion it evolves according to the diffusion equation,

∂ ρ/∂ t = D ∇² ρ ,

whilst in the case of Brownian motion with drift tending towards a terminal drift-rate ν, it will evolve according to the diffusion equation with drift:

∂ ρ/∂ t = D ∇² ρ - ν ∇ ρ .

D here is the so-called diffusion coefficient, ∇² is the Laplacian, and ∇ is the gradient.

To explain the prior evolution of a quantum relativistic universe, diffusion seems to work equally well as diffusion-with-drift. Simple random diffusion will eventually evolve a universe into a quantum relativistic universe, at which point a population of reproducing universes can be created, and evolution by natural selection can kick-in. Diffusion-with-drift towards the relevant part of the cosmic fitness landscape will produce a quantum relativistic universe in shorter time, but given eternity, this seems unnecessary. Similar considerations suggest that the form of the stochastic process distribution, whether it is normal, lognormal, or otherwise, is superfluous. All of which seems to mitigate against the possibility of deriving potentially observable predictions from the theory. It might still be possible to observe the variations in the parameters of physics within our own universe, and to infer the nature of the stochastic process, but that information would then have to be fed back into the theory, rather than derived from it.

Cosmogenic drift Cosmological natural selection Lee Smolin Stochastic processes

Stochastic processes

Mathematicians define a stochastic process as a time-ordered family of random variables X_t upon a probability space Ω. In itself, this is not particularly illuminating. However, the basic idea is that Ω is the path-space for the system under consideration. In other words, each point in this probability space, ω ∈ Ω, represents a possible history of the system. (In the case of a stochastic process, these histories will typically be non-differentiable).

By definition, a random variable X is a function on a probability space Ω which possesses a probability distribution over its range of possible values by virtue of the probability measure on the subsets of the probability space Ω. In the case of a stochastic process, X_t is a function on the path-space of the system, which represents the position of the system at time t. ('Position' here can be taken to be spatial position, or any sort of state-defining value, such as the price of a financial stock). Thus X_t(ω), the value of the random variable X_t at the point ω ∈ Ω, is the position of the system at time t in the history ω. X_t takes different values at different points because the different points in Ω correspond to different histories of the system. The probability measure on Ω, the space of histories, determines the probability distribution over the range of each random variable X_t, and thereby determines a probability distribution over position at each time t. Different positions at time t have different probabilities because different histories have different probabilities.

A stochastic process can also be defined by a function G(x,x'; t) which specifies the probability of a transition from x to x' over a time interval t. Given an initial probability distribution ρ(x,0), this determines the probability distribution ρ(x',t) at a future time t.

ρ(x',t) = ∫ G(x,x'; t)ρ(x,0) dx

In fact, given the transition probabilities and an initial probability distribution, a probability measure is determined on the path-space, and the time evolution of the probability distribution over position x is determined. In the special case of a discrete stochastic process, with the transition probability of going from x to y in one time-step denoted as T(x,y), the probability p(γ) of a path γ defined by the sequence of positions (x₀,...,x_n) is defined to be

p(γ) = ρ(x₀,0)T(x₀,x₁)...T(x_n-1,x_n)

It is these concepts which tacitly lie behind the mathematicians' definition of a stochastic process.

Stochastic Processes

Morning

I squeeze some GlaxoSmithKline Aquafresh Fresh & Minty onto my Reach toothbrush. With its blue and red gel stripes, the Aquafresh provides triple protection for clean strong teeth, healthy gums and fresh breath. After brushing, I rinse my mouth with water. Then, I fill my armitage shanks enamel wash-basin with hot water, and wash my face with water-activated Safeway honey-almond soap. I rinse my face to remove the soap. I then apply Nivea Creme moisturiser. Containing Eucerit, a special moisturising ingredient, Nivea Creme is free from preservatives, and skin compatibility is dermatologically approved. Truly multi-functional, it’s perfect for all the family, all year round. I then wash my mouth with Pfizer’s Freshburst Listerine antibacterial mouthwash. Containing Eucalyptol and menthol, it reduces plaque by up to 56% more than brushing alone. I rinse my mouth again with water. Then I apply Unilever Lynx Africa deodorant bodyspray, containing butane, isobutane, and polyaminopropyl Biguanide Stearate, holding the can 15cm from the body, and spraying in a ventilated place. I avoid prolonged spraying. Walking back to the bedroom via the living room, I pass the Sony Bravia 32” LCD HD-Ready digital TV, sitting in an arched recess beside the fireplace. Back in the bedroom, I don a pair of black Marks and Spencer briefs, 100% cotton, made in Sri Lanka; a white British Home Stores Menswear shirt, 65% polyester, 35% cotton, made in Indonesia; and black UrbanSpirit trousers, 65% polyester, 35% viscose, made in China. My shoes are scuffed, black leather Hush Puppies, the soles of which are beginning to perish. I drive to work.

(With apologies to Bret Easton Ellis)