##### vol **6:** Essays

### How much freedom should we trade for our security?

Essay submitted to the Shell and The Economist writing competition, August 2002.

#### Contents

1. Summary

2. Network

3. The physical world is a network . . .

4. And the spirtual world also

5. The human condition

6. The principle of least action

#### 1. Summary

Do we need to sacrifice freedom to gain security? The word 'trade' in the topic seems to suggest that we do: pay freedom to buy security. To the contrary is evidence flowing from the diversity and durability of the world's states. Hitler minimised freedom, and his thousand year Reich lasted little more than a decade. Free states, on the other hand, generally enjoy longer periods of stability.

To gain insight into the relationship of freedom and security, I assume a network model of the world and identify freedom with entropy and security with the ability of the network to survive damage and noise induced error. The mathematical theorems of communication tell us that entropy may be exploited to overcome error, so this model raises the possibility that increasing freedom may increase security.

Can we apply this result in the real world? My hunch is yes. I hope this rather abstract and impressionistic essay will move you in the same direction. The security of a community depends upon its ability to solve the problems that face it. If we increase community entropy by giving everybody an active voice in the search for solutions, we increase the experience and brainpower available for survival, increasing security. back

#### 2. Network

The dotcoms and telcos may be a little tarnished, but the mathematical foundations of their technology remain forever brilliant, and can tell us much about freedom and security. Let us assume that all the world can be seen as a network. This is not implausible, as we will see below. In the abstract mathematical world, a network is a graph, a set of points (or nodes) connected by a set of lines (or edges) obeying certain axioms. When we apply the mathematics to real networks, the elements may be anything that obeys the axioms, atoms, people, nations . . . , railways, phone lines, treaties . . . .

There is a rich literature on networks and the internet is a handy practical model. The nodes are computers (and behind them, users), and the edges are communication channels. The most important feature of networks is their ability to work perfectly despite partial failure. The first computer networks were designed to preserve the defences of the United States even if many cities were destroyed by nuclear weapons. An insecure network may break down, a state that manifests as war, disaster, accident, loneliness or any of the other difficulties that beset us.

The security of a network is bought with redundancy, which we might think of as spare entropy, freedom or headroom. First, in a well designed network the nodes are connected to one another by multiple edges, so that if we lose a few, there is still a good chance that the remaining nodes can still communicate. Second, the mathematical theory of communication shows us how to exploit the redundancy in messages to guarantee error free transmission over noisy channels. Finally, security against unauthorised listeners can be achieved using special encodings. back

#### 3. The physical world is a network . . .

We study of history with an eye to predicting and controlling the future. Here we seek a cosmological foundation, placing human history in the context of the whole Universe Cosmology is founded on three disciplines: quantum mechanics, relativity and quantum field theory. These theories are now so trusted that they, like the theories of networks and communications, lie at the theoretical heart of national defence systems.

Quantum mechanics postulates an infinite world of states animated by a boundless zoo of operators. To see the quantum world as a network, we may make the operators nodes and the states edges. Quantum networks are subject to an important constraint: the operators transforming an isolated system must be unitary.

Unitarity corresponds to the common observation that physical resources are locally limited. The quantum formalism recognises that while the variety of possible events is without limit, the probability of something happening at any point in spacetime is normalised to one. The physical world is full. This has the consequence, recognised by Malthus, that every potential being must compete for the physical resources to realise itself. This constraint, plus the infinite combinatorial creativity of the world, has given us the tree of life in which we nest.

Newton assumed that information propagates through space instantaneously. In fact it takes time: space and time are intermingled. The geometrical structure of spacetime is described by relativity. Relativity is consistent with the network view, since networks take time to encode, transmit and decode messages. The space/time relationships in a network create a metric space which enables us to define and measure the geometry of networks (and physical spacetime, as Einstein did so brilliantly).

Quantum field theory lies at the intersection quantum mechanics and relativity. These two theories, plus natural selection, seem sufficient to explain the evolution of our Universe. Quantum field theory holds that all communications involve the exchange of physical particles, a fact widely verified by experience. Many feel, nevertheless, that there exists a spiritual world in which communications (eg revelations from a god outside the Universe) move by non-physical means. The needs of spirituality, however, might also be met in the network model. back

##### 4. And the spiritual world also . . .

For historical reasons, the term 'wave function' is often applied to the mathematical mechanism of quantum mechanics. This arises because a central construct of mechanics is the 'harmonic oscillator' which explains such things as musical instruments and light. Quantum mechanics works in Hilbert space, which is rather like ordinary space, but may have any number of dimensions. The quantum harmonic oscillator (revealed by Planck in 1900) has as many states as there are integers, 1, 2, 3, . . . , a 'countably infinite' number. To construct the wave function of the Universe, we use the fact that as networks can be joined to form bigger networks, quantum systems can be joined to form bigger systems working in bigger spaces.

We construct and measure such spaces using set theory. We call the
cardinal number of the set of natural numbers 'aleph zero'
(ℵ_{0}), the first transfinite number. The natural numbers have a natural order, 1, 2, 3, . . . , but we can rearrange them in many
different orders, like 1, 3, 2; 2, 3, 1; . . . called permutations. The
cardinal number of the set of all permutations of the natural numbers
is a transfinite number (called ℵ_{1}) strictly greater than
ℵ_{0}. By considering all permutations of ℵ_{1} numbers, we arrive at ℵ_{2}, and so on without end. This construction is often called the Cantor universe.

When we unite quantum systems, we find that the corresponding
Hilbert spaces grow like the transfinite numbers. The product of two
ℵ_{0}-dimensional spaces is ℵ_{1}-dimensional, the product of two ℵ_{1}-dimensional spaces an ℵ_{2}-dimensional space, and so
on. For quantum mechanics, the physical Universe looks very like the
Cantor universe.

Is the real Universe this big? It looks to be everywhere finite, and the unitarity constraint seems to explain why. A century of quantum theory, however, has forced us to accept that there are transfinite processes going on behind the scenes. Physicists have constructed a quantum model of the mind of the Universe just as we, reading someone's words, try to picture their invisible mind. Like human activities, quantum events are beset by uncertainties because the huge space of mental possibilities can never be fully represented by observables.

Quantum information theory makes the picture intuitively clearer because it interprets wave functions in terms of information rather than physical motion. Vast amounts of information may be stored and manipulated with quantum systems. Many hope to exploit this quantum resource to construct very powerful computers. Quantum information theory, like its classical antecedent, relies on redundancy to overcome error. back

#### 5. The human condition

How do we link such physics (and metaphysics) to human freedom and security? Our connection between Hilbert spaces and the Cantor Universe is consistent with the observation that the world is a layered network, fundamental particles being assembled into atoms, atoms into molecules, molecules into cells, organisms, communities and so on through solar systems to galaxies and the Universe as a whole. Each layer is populated by entities of the same power, sometimes called peers.

Somewhere in all this we find networks whose nodes are human and whose edges are human communication. Nodes come and go as people enter or leave communities. We notice, in most human networks, that all nodes are not equal. We are not all peers. Some, like newborn children, are still learning their network protocols and have limited functionality. Others, at the far end of their life cycle, are fading out.

Beside these natural processes, there are vast differences between people in their economic and political power, and so in their ability to access the goods of the world. Further, many weak spots in the human network are caused by geographical, religious and political barriers. To maximise the security of the human network, we must eliminate these weak spots. Forging human networks it not easy however, since it is hard to reach agreement about communication protocols. Babel complicates connectivity. This difficulty is compounded because some (despots, warlords) seek freedom at the expense of others (slaves, cannon fodder). There is, however, a further feature of the world which may simplify the task of making us all peers. back

#### 6. The principle of least action

The self-similarity of networks tells us that what we learn about one we may apply to others. This suggests that apart from questions of scale, the intractably complex human world echoes the simpler world of physics. The mathematical wormhole which joins classical to quantum physics is Hamiltonian dynamics, at whose heart we find Hamilton's principle of least action. Action entered physics with Newton's second axiom of change: to every action there is an equal and opposite reaction. Classical action carries over to quantum physics where it is quantised (Planck's quantum of action) and conserved. Hamilton's principle tells us that physical systems that obey the second law evolve so as to minimise action.

Capitalism creates wealth at an unprecedented rate, but stability
requires an effective distribution system as well. Newton's second
axiom holds in the ideal of free trade between equals, but
disparities in power lead to its frequent violation in other cases.
*A *may bomb *B* into the stone age without *B* being
able to react, or even make its plight public.

The beauty of the second axiom is that it is local. We do not need to take on the whole world. We can generate freedom in the world by generating freedom in our local environment. This move from local to global is reminiscent of Einstein's extrapolation of his local special theory of relativity to the global general theory, which constrains the overall structure of the physical Universe Globally we can strengthen the human network by using wealth to heal the weak spots.

We want a free and secure world, a dream. To get there we must implement the human equivalent of Newton's second axiom at every point: a combination of an eye for an eye with universal love. We live in a system stabilised by mutual assured destruction. Here I see a path to a world stabilised by mutually assured construction, the physical expression of the spirit of love.