The total disorder in the universe, as measured by the quantity that physicists call entropy, increases steadily as we go from past to future. On the other hand, the total order in the universe, as measured by the complexity and permanence of organized structures, also increases steadily as we go from past to future. Freeman Dyson
Beneath the surface of the world are the rules of science. But beneath them there is a far deeper set of rules, a matrix of pure mathematics which explains the nature of the rules of science and how it is we can understand them in the first place. Dangerous Knowledge, BBC 2007
Georg Cantor: how big is infinity? ibid.
A whole hierarchy of different infinities. ibid.
After his breakdown everything about Cantor is transformed. ibid.
Like Cantor, Boltzmann’s ideas were out of step with his times. Cantor had undermined the ideal of a timeless and perfect logic in maths; Boltzmann’s formula and his destiny was to undermine the ideal of a timeless order in physics. Together their ideas were part of a general undermining of certainty in the wider world outside maths and physics. ibid.
Boltzmann in essence captured mortality in an equation. ibid.
Kurt Godel and the work that he did here brought that dream of finding the perfect system of reasoning and logic crashing down. Dangerous Knowledge II
Godel’s Incompleteness Theorem: ‘the obvious logic of Godel’s work is that logic is a failure.’ ibid.
In 1934 he had his first breakdown. ibid.
He [Turing] is also the man who made Godel’s already devastating incompletely theorem even worse. ibid.
Turing recast incompleteness in terms of computers … There would always be some problems they would never solve. ibid.
This is a story about the rise of machines. And our belief in the balance of nature. How the idea of the ecosystem was invented. How it inspired us. And how it wasn’t even true. Adam Curtis, All Watched Over by Machines of Loving Grace II: The Use and Abuse of Vegetational Networks, BBC 2011
In the mass democracies of the west a new ideology has risen up. We have come to believe that the old hierarchies of power can be replaced by self-organising networks. ibid.
This is the story of the rise of the dream of the self-organising system. And the strange machine fantasy of nature that underpins it. ibid.
It was part of what [Arthur] Tansley called, ‘the great universal law of equilibrium’. All these systems, he wrote, are constantly tending towards positions of balance or equilibrium … There was an underlying mechanism that regulated nature as if it were a machine. But it was only an hypothesis. ibid.
Cybernetics saw human beings not as individuals in charge of their own destiny but as components in systems. At its heart, Cybernetics was a computer’s idea of the world. And from that perspective there was no difference between human beings and machines. They were just nodes in networks acting and reacting to flows in information. ibid.
Cybernetics transformed the idea of the eco-system because it seemed to explain how the system stabilises. ibid.
‘I will make my life an experiment to search for the principles that govern the universe.’ ibid. Buckminster Fuller
What began to rise up in the 1970s was the idea that we and everything else on the planet are connected together in complex webs and networks. Out of that were going to come epic visions of connectivity. ibid.
Eco-systems did not tend towards stability but the very opposite was true. That nature far from seeking equilibrium was always in a state of dynamic and unpredictable change. ibid.
Complexity Theory: Murray Gell-Mann: he believed that there were underlying patterns at every level of the universe, not just in the particles but in the way people think, and the structure in human societies … ‘And it’s fascinating to try and figure out what these laws are.’ Adam Curtis, Can’t Get You Out of My Head VI Are We a Pigeon? Or Are We Dancer? ***** BBC 2021
All those woodland walks and conversations came to a brilliant climax in 1927. Early that year Werner Heisenberg gave a new characterisation of the electron: yes, it is a particle, he said, but a particle which yields only limited information. That is, you can specify where it is at this instant, but then you cannot impose on it a specific speed and direction of setting off. Or conversely, if you insist that you’re going to fire it at a certain speed and a certain direction then you cannot specify exactly what its starting point is, or its end point. Jacob Bronowski, The Ascent of Man 11/13: Knowledge or Certainty ***** BBC 1973
Heisenberg called this the Principle of Uncertainty. ibid.
The theory of probability is the only mathematical tool available to help map the unknown and the uncontrollable. It is fortunate that this tool, while tricky, is extraordinarily powerful and convenient. Benoit Mandelbrot, The Fractal Geometry of Nature
The theory of probabilities is at bottom nothing but common sense reduced to calculus; it enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which of times they are unable to account. Pierre-Simon Laplace, Introduction to Théorie Analytique des Probabilitiés
Nature prefers the more probable states to the less probable because in nature processes take place in the direction of greater probability. Heat goes from a body at higher temperature to a body at lower temperature because the state of equal temperature distribution is more probable than a state of unequal temperature distribution. Max Planck
The mind is not designed to grasp the laws of probability, even though the laws rule the universe. Steven Pinker, How the Mind Works
We have a closed circle of consistency here: the laws of physics produce complex systems, and these complex systems lead to consciousness, which then produces mathematics, which can then encode in a succinct and inspiring way the very underlying laws of physics that gave rise to it. Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe