Saturday, 26 January 2013

64. Russell's Paradox

Constituents of a complex system interact with one another, and with the surroundings. This interaction can be viewed as communication of information.

I introduced the basics of information theory in Part 21 and algorithmic information theory in Part 23. Shannon’s formulation of information theory was originally meant for designing better communication channels. An interesting aspect of modern communication theory is that the merit of a particular communication-channel design lies not just in how well the actual message is sent, but also in how well the channel could have sent all the other messages it might have been asked to convey.

It is instructive to adopt a historical approach and trace the development of ideas regarding the nature of information and information-processing (or what is now better known as computation).

Let us begin by talking about 'sets'. [A set is a unification of well-defined objects of thought into a whole. Here are some examples: a set of chairs; a set of red chairs; a set of integers; a set of odd integers; the set of all odd integers.]

At the beginning of the 20th century the mathematician David Hilbert asked a question in the theory of infinite sets: Starting from 1, 2, 3, . . ., what is the largest integral number one can think of? Whatever that integer is, let ω be the first number after all the finite numbers we can think of. So we get 1, 2, 3, … , ω.

We need not stop there, and can go on adding the next higher integers to the set: 1, 2, 3, … , ω, ω+1, ω+2, … . Let 2ω be the next integer after all the integers in this set: 1, 2, 3, … , ω, ω+1, ω+2, … 2ω. We can go on and on:

1, 2, 3, .., ω, ω+1, ω+2, .. 2ω, . . 3ω, . . 4ω, .. ω2, ω3, .. ωω, ...

Even this can go on. We shall reach ω to the power ω to the power ω  . . .; and so on, forever. The set is infinite.

Georg Cantor proved a theorem which said that for any infinite set there is a larger infinite set which is the set of all its subsets. This is sometimes referred to as Cantor’s diagonal argument. [A subset is a portion of the set.]

Now suppose we apply this theorem to the infinite 'universal' set; by definition it is the set comprising of all the sets. Application of Cantor’s theorem here leads to a paradox because the theorem says that there should be a set still larger than the universal infinite set.

This paradox was noticed by Bertrand Russell. The seminal work Principia Mathematica by Whitehead and Russell (1925-1927), the first edition of which was published during 1910-1913, included a new formulation of set theory by Russell; he was led to it during his sustained efforts to solve the following problem posed by him about sets:

Consider a set A, defined as a set containing all sets that are not members of themselves. Does A contain itself?

Unless you are a politician, the two possible answers are Yes or No (a politician may say 'yes and no'). If the answer is Yes, there is a contradiction because set A is defined as a set containing sets which are not members of themselves.

If the answer is No, again there is a contradiction. Since A is defined as a set comprising of all sets which do not belong to themselves, it should contain itself. But according to the second answer, A does not contain itself.

This means that we can have incompatible propositions that imply one another.

This famous paradoxical situation goes by the name of Russell's paradox. Apparently, Yes implies No, and No implies Yes.

Here is another example of the Russell paradox, called the Barber Paradox: There is a barber in a small town where every man is clean-shaven. The barber shaves all men who do not shave themselves and only those men who do not shave themselves. The question is: Who shaves the barber? There is a paradox here, no matter what answer you give:

If the barber does shave himself, then the barber (himself) must not shave himself; and if the barber does not shave himself, then he (the barber) must shave himself.

Russell's work for resolving this paradox that now goes by his name led to a reformulation of mathematics in terms of his new theory of sets, mentioned above. After much mental agony and loss of sleep, his final resolution of the problem was as follows:

He first imagined what we now call a theoretical computer. It was essentially a sequential logic machine. This imaginary machine carries out one logical operation at a time, i.e. it operates in discrete time. The answers about a set are dealt with sequentially, and one at a time. Thus, at any given point of time the answer may be, say, Yes. But the theoretical computer keeps running, and a few time steps later the answer becomes No. The program runs in an infinite loop, alternating between Yes and No. Thus there is said to be no paradox because the answer is never Yes and No at the same time! Are you convinced by this explanation?

Hilbert applied an approach different from that of Russell for tackling the problems posed by Cantor in set theory. He tried to use, and improve upon, Euclid’s axiomatic method. The idea was to take the apparatus of symbolic logic to its extreme. He argued that one reason we get into contradictions in set theory is that words often have a vague meaning. So why not come up with a finite set of formal axioms and an artificial language for doing mathematics? This artificial language must have strictly precise grammatical and other rules.

Did Hilbert succeed? I shall answer that next time.

Saturday, 19 January 2013

63. A Possible 'Heliocultural' Energy Regime

The currently dominant carbocultural energy regime is turning against itself by overshooting the carrying capacity of the habitat. This cannot go on, and a new energy regime must emerge. I discussed the possibility of a nucleocultural regime in Part 62. But if humans do not make that choice fully, what else is possible?

Niele (2005) has argued in favour of drawing inspiration from what was done by the blue-greens two billion years ago. They were the rulers of the phototrophic regime (cf. Part 57), just as we are the rulers of the current carbocultural regime. They went for a partnership, or symbiosis: The self-induced crisis of oxygen emission, which was poison for the blue-greens, was overcome by the evolution of a new type of cell: the eukaryotic cell, which had organelles limited by membranes. In the new (aerobic) regime, respiration provided the main fuel-burning mechanism: The atmospheric oxygen was conducive to the aerobes, but poison for the blue-greens. The evolution of a symbiotic ‘pact’ between oxygenic photosynthesis and aerobic respiration was at the heart of the oxo-energy revolution, resulting in the emergence of the aerobic regime. The eukaryotic cell design embodied sunlight-harvesting photosynthesis, as well as protection against oxygen toxicity.

A similar symbiosis can happen again, this time for saving man from the consequences of the loud and clear macroscopical signal of unsustainability. We may be heading for the emergence of the Symbian Man, who will effect a symbiosis of the various energy options. It will be a symbiosis of many things, born out of our perception that system Earth is one big complex superorganism.

It would be a symbiosis between:
  • Imperial man and Arcadian man.
  • Scientific reductionism and scientific holism; or simplicity and complexity.
  • Knowledge of natural disciplines and knowledge of cultural disciplines.
  • Anthropocentrism and ecocentrism.
  • 'Nature mastery' and 'back to Nature'.
  • Techno-scientific virtues and socio-ethical virtues.
Although energy from the Sun will form the backbone of this regime, there are other renewable-energy options also. To quote Smil: 'Beyond the fossil fuels the world can tap several enormous renewable flows: direct solar radiation and wind energy in the accessible layer of the troposphere are both several orders of magnitude larger than the current global total primary energy supply and they can be supplemented by hydroenergy and geothermal flows'. The watchword will be: sustainable energy.

The Symbian Man will consciously bring about the 'Heliocultural Energy Revolution' (Niele 2005). The aim will be to develop 'closed-loop' technologies enabling the solar-driven recycling of matter. Even wind energy is of solar origin. There will be partnerships or symbiotic relationships of all kinds: from local to regional to global, and emphasis will be on integrated and cascaded flows of renewable energy and recyclable matter.

The energy carriers will be green electricity, solar hydrogen, and green biofuels. 'The beauty of green biofuels is that Nature looks after carbon recycling through photosynthesis, with energy storage for free. And albeit efficiencies are relatively low, residues of food and wood production nevertheless grow. In the Sun Valley, socio-metabolisms could differ regionally with ecological circumstances and heliocultures. They show optimised configurations of large-, small- and also medium scale solutions . . ' (Niele 2005).

The Symbian approach favours distributed small-scale and decentralised medium-scale socio-metabolic sites and corresponding infrastructures.

To suppress the discharge of the greenhouse gas carbon dioxide into the environment, the Symbian Man will seek to exploit geological and chemical sequestration.

An interesting aside regarding carbon dioxide is the large amount of this and other greenhouse gases released by cattle: They emit from both ends! Their population should be reduced. In fact, there is a strong case for reduction in the use of food products of animal origin. Their production is very energy-intensive, with a very large carbon footprint. Humans should move towards a larger use of foods of plant origin. Not to mention the fact that a voluntary and phased reduction of the total human population will also be an important step in the right direction.

Centralized/remote production or processing of electricity, drinking water, sewage, food, and fuels results in huge transportation and loss problems. Self-sufficient local communes with closed-loop economy are the answer. This will also help in the use of direct local use of heat produced in industrial processes, instead of first converting heat to electricity, transporting the electricity over long distances, only to convert it back to heat. All this would call for Symbian partnerships between governments, NGOs, universities, and R&D companies.

What we have at present are three possible approaches: the Green Valley, the Nuclear Valley, and the Sun Valley. There may be an evolutionary battle between the Arcadian Man, the Imperial Man, and the Symbian Man. Better still, a symbiosis may emerge wherein the best features of the Green Valley approach and the Nuclear Valley approach are adopted and subsumed in the Heliocultural Energy Regime.

The Masdar City in Abu Dhabi is an indicator of the shape of things to come.

Efforts are already afoot for spreading awareness regarding a variety of conservation measures. For example, a case is being made for 100% utilisation of 'wet waste' from kitchens for composting etc.


NB. From the next post onwards I shall take a break from the description of biological evolution of complexity, and move towards a discussion of artificial evolution. A major landmark in the biological-evolution story was the emergence of humans, with brains so advanced that they ultimately became aware of this evolution, courtesy Charles Darwin. Our brains have also made us develop computers with ever-increasing capabilities in terms of speed and computing power. And we have been investigating evolution using computer algorithms. We build computational models, and study evolution occurring inside a computer. This is artificial evolution because humans rather than natural processes are causing it. And we are evolving computer-based robots of ever-increasing sophistication. It is only a matter of time before the robots (our 'mind children') become superior to us in every respect. The possibilities are truly exciting, even grave. The least the public at large can do is to be aware of how it is being done, and what is at stake.