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What are Laws of Nature?


 
Regularities/structures in a highly complex universe

 
    Allow for predictions
  • Dependent on only a small set of conditions (i.e., independent of very many conditions which could possibly have an effect)

  ...but why are there laws of nature and how can these laws be discovered and understood by the human mind?

No One Knows!

  • G.W. von Leibniz in 1714 (Principes de la nature et de la grāce):
    • Why is there something rather than nothing? For nothingness is simpler and easier than anything
  • E. Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", 1960:
    • [...] the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and [...] there is no rational explanation for it
    • [...] it is not at all natural that "laws of nature" exist, much less that man is able to discover them
    • [...] the two miracles of the existence of laws of nature and of the human mind's capacity to divine them
    • [...] fundamentally, we do not know why our theories work so well

In a Nutshell

  • We happen to live in a structured, self-organizing, and fine-tuned universe that allows the emergence of sentient beings (anthropic principle)
  • The human mind is capable of devising formal thought systems (mathematics)
  • Mathematical models are able to capture and represent the workings of the universe
See also this post: in a nutshell.

The Fundamental Level of Reality: Physics


 
Mathematical models of reality are independent of their formal representation: invariance and symmetry

 
  • Classical mechanics: invariance of the equations under transformations (e.g., time => conservation of energy)
  • Gravitation (general relativity): geometry and the independence of the coordinate system (covariance)
  • The other three forces of nature (unified in quantum field theory): mathematics of symmetry and special kind of invariance
See also these posts: funadamental, invariant thinking.

Towards Complexity

  • Physics was extremely successful in describing the inanimate world the in the last 300 years or so
  • But what about complex systems comprised of many interacting entities, e.g., the life and social sciences?
  • The rest is chemistry; C. D. Anderson in 1932; echoing the success of a reductionist approach to understanding the workings of nature after having discovered the positron
  • At each stage [of complexity] entirely new laws, concepts, and generalizations are necessary [...]. Psychology is not applied biology, nor is biology applied chemistry; P. W. Anderson in 1972; pointing out that the knowledge about the constituents of a system doesn't reveal any insights into how the system will behave as a whole; so it is not at all clear how you get from quarks and leptons via DNA to a human brain...

Complex Systems: Simplicity

The Limits of Physics
  • Closed-form solutions to analytical expressions are mostly only attainable if non-linear effects (e.g., friction) are ignored
  • Not too many interacting entities can be considered (e.g., three body problem)
The Complexity of Simple Rules
  • S. Wolfram's cellular automaton rule 110: neither completely random nor completely repetitive
  • [The] results [simple rules give rise to complex behavior] where were so surprising and dramatic that as I gradually came to understand them, they forced me to change my whole view of science [...]; S. Wolfram reminiscing on his early work on cellular automaton in the 80s ("New Kind of Science", pg. 19)

Complex Systems: The Paradigm Shift

  • The interaction of entities (agents) in a system according to simple rules gives rise to complex behavior
  • The shift from mathematical (analytical) models to algorithmic computations and simulations performed in computers (only this bottom-up approach to simulating complex systems has been fruitful, all top-down efforts have failed: try programming swarming behavior, ant foraging, pedestrian/traffic dynamics,... not using simple local interaction rules but with a centralized, hierarchical setup!)
  • Understanding the complex system as a network of interactions (graph theory), where the complexity (or structure) of the individual nodes can be ignored
  • Challenge: how does the macro behavior emerge from the interaction of the system elements on the micro level?
See also these posts: complex, swarm theory, complex networks.

Laws of Nature Revisited


 
So are there laws of nature to be found in the life and social sciences?

 
  • Yes: scaling (or power) laws
  • Complex, collective phenomena give rise to power laws [...] independent of the microscopic details of the phenomenon. These power laws emerge from collective action and transcend individual specificities. As such, they are unforgeable signatures of a collective mechanism; J.P. Bouchaud in "Power-laws in Economy and Finance: Some Ideas from Physics", 2001

Scaling Laws

Scaling-law relations characterize an immense number of natural patterns (from physics, biology, earth and planetary sciences, economics and finance, computer science and demography to the social sciences) prominently in the form of
  • scaling-law distributions
  • scale-free networks
  • cumulative relations of stochastic processes
A scaling law, or power law, is a simple polynomial functional relationship f(x) = a x^k     <=>   Y = (X/C)^E

Scaling laws
  • lack a preferred scale, reflecting their (self-similar) fractal nature
  • are usually valid across an enormous dynamic range (sometimes many orders of magnitude)
See also these posts: scaling laws, benford's law.

Scaling Laws In FX

  • Event counts related to price thresholds
  • Price moves related to time thresholds
  • Price moves related to price thresholds
  • Waiting times related to price thresholds
FX scaling law
(Figs. by jbg under the Creative Commons Attribution-NonCommercial2.5 License)

Scaling Laws In Biology

So-called allometric laws describe the relationship between two attributes of living organisms as scaling laws:
  • The metabolic rate B of a species is proportional to its mass M: B ~ M^(3/4)
  • Heartbeat (or breathing) rate T of a species is proportional to its mass: T ~ M^(-1/4)
  • Lifespan L of a species is proportional to its mass: L ~ M^(1/4)
  • Invariants: all species have the same number of heart beats in their lifespan (roughly one billion)
allometric law (Fig. G. West)
G. West (et. al) proposes an explanation of the 1/4 scaling exponent, which follow from underlying principles embedded in the dynamical and geometrical structure of space-filling, fractal-like, hierarchical branching networks, presumed optimized by natural selection: organisms effectively function in four spatial dimensions even though they physically exist in three.

Conclusion

  • The natural world possesses structure-forming and self-organizing mechanisms leading to consciousness capable of devising formal thought systems which mirror the workings of the natural world
  • There are two regimes in the natural world: basic fundamental processes and complex systems comprised of interacting agents
  • There are two paradigms: analytical vs. algorithmic (computational)
  • There are 'miracles' at work:
    • the existence of a universe following laws leading to stable emergent features
    • the capability of the human mind to devise formal thought systems
    • the overlap of mathematics and the workings of nature
    • the fact that complexity emerges from simple rules
  • There are basic laws of nature to be found in complex systems, e.g., scaling laws



This was originally posted on Wednesday, July 1, 2008 in my tech blog...

 

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