foundation

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

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