Duality of the Universe

Inspired by some ideas from Shahn Majid and Michael Heller, I've just written a new paper, 'The Duality of the Universe'. Using the concepts of mathematical category theory, and the mathematical concept of a dual, it proposes that the physical universe is an instance of a mathematical structure which possesses a dual structure, and that this dual structure is the collection of all possible knowledge of the physical universe. In turn, the physical universe is then the dual space of the latter. Contrary to the proposals of Majid and Heller, it is postulated that the dual space of the physical universe possesses a distinct structure to that of the physical universe. In particular:

Epistemology is the dual of metaphysics.

It is proposed that knowledge consists of various types of mathematical representation, and this proposition incorporates the representational theory of the mind (RTM), an approach to the mind-brain relationship which falls under the umbrella of functionalism. The latter holds that the mind possesses a structure which cannot be identified with the structure of the brain. In other words, functionalism holds that the structure of the mind is non-isomorphic to the structure of the brain.

Supervenience holds that any change in the higher level properties, states and processes of a composite system, must correspond to a change in the lower level properties, states and processes. All functionalist accounts accept that the mind supervenes on the brain, but they reject the idea that the higher-level structure can be defined in terms of the lower-level structure. One can accept that the mind is essentially subjective, but one can still hold that it possesses a structure, a distinct structure from the structure of the brain. The structure may well be a non-spatial structure, which cannot be reduced to any of the structures which characterise the brain, but it is, nevertheless, the structure of subjective experience. Intentional mental states are not observable by means of the sense organs, are directly accessible only to their owners, and do not occupy space, yet they nevertheless possess a structure which functionalism and the RTM sets out to capture. The proposal made in this paper entails that this structure is part of the dual structure of the physical universe.

A self-explaining universe?

Next week I'm attending a symposium at the Royal Society on the origin of time and the universe, in honour of Michael Heller's 2008 Templeton Prize. There'll be talks by Michael himself, John Barrow, Andreas Doring, Paul Tod, and Shahn Majid. Majid proposed some years ago that the mathematical concept of self-duality can be used to provide an ultimate explanation of the universe in terms of the universe itself; a type of self-explanation. Michael Heller wrote a lovely paper on Majid's ideas, (Algebraic self-daulity as the 'ultimate explanation', Foundations of Science, Vol. 9, pp369-385) in 2004, and I've been reviewing this ahead of next week's symposium.

Majid illustrates his idea with the notion of a self-dual bicrossproduct Hopf algebra, (although he doesn't believe that this specific mathematical structure is a candidate for a theory of everything). To understand what this is, we first need to understand what a Hopf algebra is.

An algebra is a structure which possesses both the addition operation '+' of a vector space over a number field K, and a product operation '⋅'. An algebra is unital if there is a mapping η:K → H. A bialgebra possesses both the structure of an associative unital algebra (H,+,⋅,η,K), and the structure of a coalgebra (H,+,Δ,ε,K), where Δ:H → H × H is the coproduct, and ε:H → K is the counit. Given an algebra or coalgebra A, the dual A* is defined to be set of linear functionals on φ: A → K. The dual of an algebra is a natural colagebra, and the dual of a coalgebra is a natural algebra, hence a bialgebra possesses within itself the dual of both its algebra and coalgebra. A Hopf algebra H is a bialgebra equipped with an special map S:H → H called the antipode map.

Now, suppose that H and A are both Hopf algebras, and suppose that there is a left-action of H on A, and a right-coaction on A on H. This means that there is a mapping α: H × A → A, and a mapping β: H → H × A, respectively. One can then form something called the cross-product H ♦ A, which will also be a Hopf algebra. This cross-product has a dual cross-product, and the combination of the two cross-products provides the structure of a bicrossproduct Hopf algebra.

The dual of this bicrossproduct gives another bicrossproduct, but with H exchanged for the dual A*, and with A exchanged for H*. Thus, if H is isomorphic to A* (and A isomorphic to H*), then one has a self-dual bicrossproduct Hopf algebra.

Majid's proposal is that such a structure is self-explaining, in the sense, I think, that the A-part explains the H-part, the H-part explains the A-part, the whole thing H ♦ A is explained by either H or A, and either H or A is explained by H ♦ A. It's a type of Yin-Yang idea.

Now, this is fascinating stuff, but I think the overall objective of seeking a self-explaining universe, is doomed to failure. Suppose, for the sake of illustration, that our universe were a self-dual bicrossproduct Hopf algebra. This would not eliminate the question of contingency. One could still ask: why does this particular self-dual bicrossproduct Hopf algebra physically exist, and not one of the many other self-dual bicrossproduct Hopf algebras? The question is only eliminated if one proposes that our universe is an object in a mathematical category containing only one object. Even then, however, the problem of contingency is merely shifted, and one can pose the question: why is our universe an object in this mathematical category rather than an object in one of the many other mathematical categories? 'Why is the universe not a topos, or a highly structured pseudo-Riemannian manifold?' one could ask.

Ultimately, I think, one has to adopt Max Tegmark's proposal that all mathematical structures physically exist. There is, one can propose, only one type of existence: absence of contradiction. What we deem the physical universe is simply one particular mathematical structure, which exists by virtue of being free from contradiction. All other mathematical structures exist, but only those containing cognitive self-aware substructures possess observers capable of asking what the mathematical structure of their universe is.

Motorway aesthetics

The best place to go this Spring is the A34 around Tot Hill Services.

No, really. At this time of year, this otherwise moribund dual carriageway cuts a path between spectacular efflorescent banks of buttercup-yellow gorse.

And this, I propose, should be taken as a precedent. Far too much of our motorways are lined by the chlorophyllically-dominated greens of various passe shrubs and trees. As a first step towards a more colourful motorway experience, the M4 around Swindon should be lined by swathes of pyrophoric tulips; the M1 around Birmingham should be lined by roses; and the M25 in Kent should be surrounded by purple rhododendrons.

Don't Speak

If you turned onto Virgin Radio for more than half-an-hour or so in 1997 or 1998, the chances are that you would've heard Don't Speak, sung by Gwen Stefani when she was in the band No Doubt. One reason for this is that Virgin Radio never had more than 5 songs on their playlist. The other is that it's a genius pop-song.

Despite being about the break-up of a relationship, it's got what I believe music journalists call a brilliant 'hook', and Stefani's voice melds perfectly with the melody. It's simply one of the best pop songs you could hope to hear, yet it's often neglected in Top-100 compilations. Perhaps now that it's over that crucial 10-year horizon, it will acquire some nostalgic value, and begin to be recognised as a pop classic.

Michio Kaku

The indefatigable Bryan Appleyard rightly criticises Michio Kaku's latest book, Physics of the Impossible, for its naive technophilic populism.

Appleyard is interested in putting such work into an overall cultural context. Science, however, can also put human culture into context, and those seeking the most sophisticated field of view will be rewarded by flipping back and forth between the two perspectives.

Kaku is one of those individuals who has forged an international media career by making wildly exaggerated, and often inaccuarate claims, about the content of modern physics. For a more reliable guide to the science, John Barrow's eclectic 1998 book, Impossibility: The Limits of Science and the Science of Limits, should be consulted. It can be freely downloaded here.

The Orphanage

Last night I travelled via about 400 sets of traffic lights along the A4 to the Oracle Centre, in Reading, to see The Orphanage.

Almost every set of traffic lights was at red when I reached it, and there was almost no transverse traffic at any of those traffic-lighted junctions. The clear purpose of the traffic lights and their Saturday-evening timing is to break up the traffic-flow along the A4 into bunches. Before many of the traffic lights, I'd be travelling along serenely, with nothing close in front, or behind, but after waiting for an interminable length of time at a number of the junctions, there would then be an impatient little pack of cars, which would jostle and jockey for position as two-lanes funelled into one on the other side of the lights.

Upon reaching the Reading Vue Cinema, we tried to take our seats, only to find that a smartly dressed man, in late middle-age, and with a mysterious foreign accent, was sitting in one of the seats, and refusing to budge.

"'Scuse me, what's your seat number?"

"No"

"I think this is our seat!"

"I don't care."

Eventually he shifted, but to an adjacent seat which also wasn't his. Presently, the occupant of that seat arrived, but this time the mysterious aristocrat totally refused to move, even when the usher came to assist.

"I'm not moving when the film's started!"

"The film hasn't started Sir."

Eventually he moved, and this time to the seat specified on his ticket.

Half-way through the film, a group of lads at the front appeared to throw a drink over two girls in front of them. The girls got up and left, the guys laughing raucously. A security man arrived, and asked them to leave. They refused. The security man said he'd call the police. The lads stayed where they were. Eventually, two luminescently-bibbed policemen arrived, and rather than immeditaely removing the lads from the cinema, appeared to argue with them for 5 to 10 minutes. Eventually, the lads were compelled to get up and leave, one of them with a hood cloaking his features in a manner which, according to David Cameron, is supposed to evoke our sympathy more than anything else.

And the film? Brilliant, without question. It's everything you've heard about it, both scary and moving. By the end, your nerves are absolutely on edge. In more ways than one...

Snow!

Holy crap! Snow in April! That hasn't happened since the last time it happened.

En route to oblivion

There are many people I've met and known in my life, who I expect I will never meet again. Placed in its full context, this is an odd thought.

Those people I once knew are now just specks in the rear-view mirror of my mind's eye, as they, and I, hurtle by different paths to personal oblivion. We met once, but we are now separate, and in a finite time, they, and I, will cease to exist, and perhaps an infinite time will then pass during which our meeting will never be repeated. Viewed in that context, those transient meetings and acquaintances seem remarkably poignant.

Jody Scheckter's Buffalo Burgers

South African Jody Scheckter is perhaps best-known as the 1979 Formula 1 World Champion, and the winner of World Superstars in 1981. He is currently, however, the owner of Laverstoke Park in Hampshire, a flagship organic farm. "Modern farming is chasing profit, and the animals are bred to grow bigger faster," says Scheckter in this month's Motorsport magazine. "We're going in the opposite direction. We go for smaller and slower, because it tastes better. There are only 45 pure uncrossed Angus cattle left in the world, and we've got 13 of them here. There are probably 500 pure-bred Herefords left in the world, and we've got 80. We have 1000 head of buffalo, which is most of the UK population. Our sheep, our chickens, all are feeding totally naturally. I employ a full-time Doctor of Microbiology working in our own laboratory here...so we can get the soils and the grasses, herbs and clovers back to how they used to be. We hired the world's top animal psychologist to help us design our abattoir, so there is no stress on the animal when we kill it. We purify our own water - no chlorine, no fluoride - and within two years we want to be running all the farm machinery on non-fossil fuels, like rape seed oil. We want to be totally self-sustaining."

Imagine, the top animal psychologist in the world! Must be the only branch of psychology where 'talking about your problems' isn't considered to be a therapeutic solution. However, if this man can stop animals from fretting about the existence of the afterlife as they are led to the guillotine, then I too would like to hire him, mainly to help my colleagues get to work on a Monday without the normal levels of rancour and stress.

Anyway, I was most impressed with Mr Scheckter's organic farm, and when I read that Waitrose are now stocking his buffalo burgers, I decided, on this balmly Spring day, to take a trip to my local store.

On arriving at the store's entrance canal, I paused momentarily to pick up my cellophane-cradled copy of The Sunday Times, before forging inward to something called the Fresh Meat section. Here, I marvelled at the cuts of naked meat, stripped of their normal breadcrumb coating. Sadly, I could find no buffalo burgers. Instead, however, my eyes alighted upon a pack of 4 quarterpounder Aberdeen Angus (99%) burgers. These I cooked with the grill in my oven on returning home. I placed the burgers on a type of grid in a pan, which was inserted into a slot under the grill. At this stage, it all began to get quite complex. There was a half-grill option, a full-grill option, and a 'thermal grill' option where the grill alternates with the fan. There were different recommended temperatures and cooking times for each option, so I just had to make my best guesstimate, and see what happenend. After about 10 minutes, a type of greasy smoke began to issue forth from slots in the hob of the cooker, which I instinctively took to be a sign that I should turn the temperature down, and open the window. Having done so, the burgers sizzled happily to a cooked state. I was even able to clean the grid and pan afterwards without using any Acetonitrile.

And the burgers? Well, they tasted rather nice actually. But I do want those buffalo burgers...

Terminal 5 and Systems Engineering

Those seeking to understand the ultimate root cause of the debacle at Heathrow's Terminal 5 this week, might perhaps wish to consult British Airways' 'Baggage Handling Project Management Manual'. The key sentence can be found on p5:

The [project] process has a similar structure to the systems engineering principles used throughout industry.

Systems engineering is a top-down approach to large, complex engineering projects, which expends great time and effort in codifying and formalising the blindingly obvious, usually in the form of UML (Unified Modelling Language) diagrams.

The fundamental fallacy of Systems Engineering is the tacit belief that complex engineering systems are best managed by defining, often in very abstract terms, top-level requirements, capabilities and stakeholders, and by then breaking those top-level entities down into sub-requirements, sub-capabilities and sub-stakeholders, all considered in abstraction from specific technologies. Systems Engineering often requires systems houses to expend large amounts of time and money before crucial procurement decisions are made, and therefore constitutes a form of institutionalised procrastination. And, crucially, the top-down dogma has a tendency to produce systems lacking in practical operational effectiveness. True, practical, rapid and effective engineering uses both top-down and bottom-up approaches, in combination, and without any of the wasteful and vacuous formalising used by systems engineers.

Intellectually, Systems Engineering has never progressed beyond the facile observation that complex engineering systems are really 'systems of systems'. On Friday's edition of Newsnight, Allen Fairbairn, Systems Engineering Manager for the Channel Tunnel Project, offered the useful, and entirely non-specific diagnosis that large engineering systems often fail at the point where the various subsystems interact. This is the self-sustaining aspect to Systems Engineering: when it is responsible for the failure of a project, it provides generalised diagnoses of why the project failed, and prescribes more Systems Engineering as the solution.

Digital radio

I'm fundamentally opposed to digital radio. Let me explain:

The idea of analogue radio is that the frequency and amplitude of sound waves at the source can be transmitted over great distances by mapping them to the frequency and amplitude of electromagnetic waves, and thence re-created as sound waves again at their destination. As such, analogue radio exploits isomorphisms between sound waves and radio waves. Digital radio, in contrast, turns continuous sound waves into binary digits, encodes those binary digits into electromagnetic waves, and then decodes and re-creates the sound waves at their destination. This improves the quality of the sound but destroys the feel of radio.

The essence of radio lies in the hiss. I want to listen to the radio late at night, with the lights off, my head on the pillow, and feel that there's something magic in the way my little radio catches the hidden, delicate, fleeting signal from the omnipresent background of electromagnetic noise. I want to feel the distance over which the signal has travelled, the variations in the volume and quality of the signal as the meteorological conditions fluctuate. I want to manually tune my radio, not have the radio tune itself. Digital radio, like radio over the internet, just feels like a voice in my own head. I want to sense how many different voices and channels there are out there, competing for space in the electromagnetic spectrum. And I want to hear the background hiss, the hiss that was present before humanity evolved, and the hiss that will prevail after humanity has passed away. I want to hear the voices climbing out of the stochastic abyss.