micro reality |

macro reality |

local internal symmetry |

local external symmetry |

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quantum mechanics |

general relativity |

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reality

mathematical
reality

physical
theory

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gauge invariance |

covariance |

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group theory |

differential geometry |

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Mathematical Models of Reality

- Science as the quest to capture the processes of nature in a formal mathematical representation ("maths as the blueprint of reality")

- Physics as a subset of this general class of problems (physics: maths = reality)

- Idea of natural systems (ontological and epistemological issues)

- Natural system as subset of reality which is encoded (through observation and measurement) into a formal system which can make predictions about the natural system (decoding) via logical rules of inference

- Rule: the mathematical model should
not depend on its choice of representation of reality =>
*invariance*=>*symmetry*

- Notions of
*observables*,*states and equations of state*arise naturally (in quantum mechanics these ideas are applied in a specific way)

- E.g.:

- Ref.: J.L. Casti (Institute of Operations Research and System Theory, Vienna), "Mathematical Models of Nature and Man", Wiley, 1989

Systems Theory

- Scientific approach to complexity and real natural phenomenon

- A gravitational 3-body problem is not analytically solvable

- A system is taken as a whole, not the parts which constitute it

- Again: Notion of a natural system (mathematical model)

- Levels of reality (complexity):
- suborganic: reality, space and time, matter...
- organic: life, evolution...
- metaorganic: consciousness, group dynamical behavior, financial markets...

- Emergence of new structures and phenomenon ("the whole is more than the sum of parts")

- Valid from natural to social sciences and finance theory

- Algorithmic instead of analytical behavior (cellular automata)

- E.g.: Nobody knows how to get from: matter -> atoms -> molecules -> DNA -> cells -> organisms -> mind, but systems theory allows one to choose every level as system and formalize its patterns and structure

Category Theory

- Result of "unification of mathematics" by Eilenberg and McLane in 1940s (systems with algebraic structure are combined with systems of topological structure)

- Categories most basic structure in mathematics

- A category is a set of objects and a set of morphisms (maps)

- A
*functor*is a structure-preserving map between categories

- Physical observables are functors: independent of a chosen representation or reference frame, i.e. gauge-invariant and covariant

- 2nd quantization of quantum theory is a functor

- E.g.: "Category of tangles = Chern-Simons theory (topological quantum field theory)"

- 1963 Lawvere uses category theory
as foundation for mathematics:
*topos theory*

- Topos: self-contained mathematical "universe", model of first order logic, rich structure capable of modeling any situation which can be discussed in mathematical terms

- Ref.: McLarty, "Elementary Categories, Elementary Toposes", Oxford, 1992