Now that we've reviewed the math of particle physics and relativity, we can move on to the math of things we can touch, feel, see, hear and smell.
No, we aren't interested in Newtonian Mechanics at this point. We want to show how chaos theory is a bridge from "simple" systems like atoms to "complex" systems like humans, ecologies, and economies.
Just as patterns in the subatomic world give rise to the mathematics of particle physics, patterns in nature and the cosmos give rise to the mathmatics of complex systems.
A universality has been found in patterns beneath the surface of things like the human heart, weather, internet traffic, the human brain, nature's rhythms, and cosmic expansion.
There is a holism and universality not unlike that found in certain subatomic patterns in the world around us.
Historically, mathematicians and scientists have attemped to deal with the nonlinear patterns of complex systems using standard mathematical structures like transformed differential equations and probability theory. The problem is that complex systems involve patterns that ultimately don't cooperate with such standard approaches.
As was the case in quantum mechanics and relativity theory, a new way of thinking is needed if we are really going to make progress in understanding complex systems. The dramatic increase in computing power in recent years has provided opportunities for such new thinking.
State and phase space, perturbation, fractals, attractors, power laws, cascades, symmetries, topologies, multifractals, and phase transitions are examples of notions which have arisen in the search for patterns in complex systems. If the mathematics underlying those patterns happens to relate to the patterns uncovered in the subatomic world, a metaphysical and scientific revolution of unprecedented power will be unleashed.
A mathematics of reality which begins with the fuzzy patterns in the subatomic world and shows how those patterns, even in principle, can be followed up the chain of reality, could lead us to a picture of human and cosmic reality which is holistic, interconnected and transcendent.
Economic activity, political activity, social activity, intellectual activity and spiritual activity could all be viewed from the perspective of universal patterns emerging into structures of truth beauty and justice. Quite a leap in logic, perhaps, but better than most of the other possibilities.
We have discussed in other blogs, and will discuss in future blogs, how such deep perspective can help us deal with the problems of the day. As a highly complex system, the global economy cries out for holisitc analysis. The tools of complexity science and actuarial science are ready to be combined with those of relativity and quantum mechanics to form an analytical structure for public policy analysis in the twenty first century.
Are we up to the challenge?
Yes we are.
Lee
Showing posts with label computers. Show all posts
Showing posts with label computers. Show all posts
Monday, February 2, 2009
Monday, January 12, 2009
The New Structure of Mathematics
Our blogs will now become somewhat specialized.
Huffington Post will be a general site...Harmony.
This blog, Mathematical Universe, will deal with the underlying mathematics of Harmonious Universe...truth.
Quantum Universe will deal with the underlying science...beauty.
The Art of Hardball will deal with the underlying politics and economics...justice.
The new structure of mathematics will build up from the deep structures of pure and particle mathematics through the cobbled structures of economics, finance and actuarial science.
The unifying structures will be those of complexity science. At it s most basic level, complexity science is the application of the most advanced computer and analytical structures to the most advanced problems of society.
It is the structure which emerges when the best scientists apply their deepest wisdom to the deep problems they attack.
No one will have to go back to school to follow this blog. A major point of the new structure of mathematics is that it be coherent to a general audience.
That doesn't mean sacrificing analytical robustness. It merely means organizing and explaining the tools better.
The new structure itself naturally starts with pure mathmatics. This involves mathematical structures that don't necessarily relate to anything we observe.
It should be noted that much of what is originally considered pure mathematics is ultimately found to apply to real problems. Similarly, much applied math (ie. that being developed in string theory) is often found to be pure.
The first level of mathematics beyond pure math is that being used in particle physics. One doesn't have to be able to manipulate Lagrangians, Hamiltonians. Hermitians, Laguerre's, Legendre's, Fouriers, Path Integrals, Hilbert Spaces, Lie Groups, Topologies, or Tensors to know that Schrodinger Waves, Dirac Seas, Feynman Diagrams, Pauli Exclusions, Planck Lengths, Heisenberg Uncertainties, and Maxwell's Equations help us understand how fundamental particles waves and fields become atoms, molecules and us.
The mathematics of particle physics points us to the mathematics of chemistry which in turn points us to biology. Neurology and psychology are on the same trajectory . Now society itself is in view mathematically.
This is where complexity science really shines. Along with some of the structures of chaos theory (fractals, universality...), complexity science is cobbling together a mathematics of everything.
Skeptics will say that humanity and its institutions are too complicated to approach mathmatically. There is some wisdom in that skepticism if not taken to far.
Remember that if we don't use the best math, we will use an inferior one, like the math behind the models which have missed all the recent global meltdowns.
What we are saying is that complexity science gives us a conceptual framework by which to combine the best math and computer science being used in the labs and universities with that being used in business and government. By considering connections among econometric, actuarial and financial models with those of Quantum Mechanics, String Theory and Relativity we will find better ways of connecting mathematics and our realities.
That will put us in a better position to anticipate problems and resolve them so as to optimize human welfare.
Is that not a worthy goal?
The journey begins.
Lee
Huffington Post will be a general site...Harmony.
This blog, Mathematical Universe, will deal with the underlying mathematics of Harmonious Universe...truth.
Quantum Universe will deal with the underlying science...beauty.
The Art of Hardball will deal with the underlying politics and economics...justice.
The new structure of mathematics will build up from the deep structures of pure and particle mathematics through the cobbled structures of economics, finance and actuarial science.
The unifying structures will be those of complexity science. At it s most basic level, complexity science is the application of the most advanced computer and analytical structures to the most advanced problems of society.
It is the structure which emerges when the best scientists apply their deepest wisdom to the deep problems they attack.
No one will have to go back to school to follow this blog. A major point of the new structure of mathematics is that it be coherent to a general audience.
That doesn't mean sacrificing analytical robustness. It merely means organizing and explaining the tools better.
The new structure itself naturally starts with pure mathmatics. This involves mathematical structures that don't necessarily relate to anything we observe.
It should be noted that much of what is originally considered pure mathematics is ultimately found to apply to real problems. Similarly, much applied math (ie. that being developed in string theory) is often found to be pure.
The first level of mathematics beyond pure math is that being used in particle physics. One doesn't have to be able to manipulate Lagrangians, Hamiltonians. Hermitians, Laguerre's, Legendre's, Fouriers, Path Integrals, Hilbert Spaces, Lie Groups, Topologies, or Tensors to know that Schrodinger Waves, Dirac Seas, Feynman Diagrams, Pauli Exclusions, Planck Lengths, Heisenberg Uncertainties, and Maxwell's Equations help us understand how fundamental particles waves and fields become atoms, molecules and us.
The mathematics of particle physics points us to the mathematics of chemistry which in turn points us to biology. Neurology and psychology are on the same trajectory . Now society itself is in view mathematically.
This is where complexity science really shines. Along with some of the structures of chaos theory (fractals, universality...), complexity science is cobbling together a mathematics of everything.
Skeptics will say that humanity and its institutions are too complicated to approach mathmatically. There is some wisdom in that skepticism if not taken to far.
Remember that if we don't use the best math, we will use an inferior one, like the math behind the models which have missed all the recent global meltdowns.
What we are saying is that complexity science gives us a conceptual framework by which to combine the best math and computer science being used in the labs and universities with that being used in business and government. By considering connections among econometric, actuarial and financial models with those of Quantum Mechanics, String Theory and Relativity we will find better ways of connecting mathematics and our realities.
That will put us in a better position to anticipate problems and resolve them so as to optimize human welfare.
Is that not a worthy goal?
The journey begins.
Lee
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