(tbc)

contents

topology: shape, manifold, dimension

analysis: infinity, continuum, maps

algebra: abstraction, structure, inference

foundation

modeling: model, automata, science

Theory proved in the book

  • there are infinite number of shapes (infinite family argument: systemic process of churning out descendant)
  • continuous >> infinity (diagonal indexing technique)
  • every complete-information game without luck is “solvable.”
  • 1+1 = 2

Quotes

  • Topology
  • two shapes are the same if you can turn one into the other by stretching and squeezing, without any ripping or gluing
  • The circle (aka S one) and the infinite line (named R one) are the only manifolds in the first dimension.
  • shape is called a manifold if it has no special points: no end-points, no crossing-points, no edge-points, no branching-points (corresponding open set from manifold to euclidean space)

  • Analysis
  • analysis deals with infinity and continuum the way jouranlist deal with vowels and consonants
  • abstraction (reduction – the same are the same): get the general concept of flow without committing to any particular flowing substance
  • flowing substance inside a rigid container has a fixed point
  • maps are used to analyze projection, transformation, dynamic changes, geometric curves, physical system states

    Algebraic structure – abstraction of pattern, regularity, relation
  • every symbol is a general placeholder for an infinite cast of possible replacements.
  • form? What makes this structured world, with the partner-things, qualitatively different from a collection of unrelated objects?
  • abstract algebra’s big idea is that math is complicated version of basic partner world
  • isomorphic: two structure with the same structure
  • categories of structure—fields, rings, groups, loops, graphs, lattices, orderings, semigroups, groupoids, monoids, magmas, modules, and then a whole bunch that we just lazily call algebras.
  • structure examples: set(simplest structure), graph(set with additional structure), game/predator-prey tree,
  • symmetry types: flip, rotational, transitional, dilational
  • reduction; all arithmetic relation could be reduced to axiom system with five statements: 1) zero is a number 2) if x is a number, the successor of x is a number, 3) zero is not the successor of the number, 4) two numbers with the same successor are the same number 5) if a set S contains zero and S contains the successor of every number in S, then S contains every number

* Godel proved “every possible model of arithmetic is imcomplete”, “no formal system of proof can prove all mathematical truths”

  • math philosophy types:
    • platonism – mathematical objects really exist in some “platonic realm”
    • intuitionism -math is an extension of human intuition and reasoning
    • logicism – math is an extension of logic, which is objective and universal
    • empiricism – math is just like science: it must be tested to be believed
    • formalism – math is a game of symbolic manipulation with no deeper meaning
    • conventionalism – math is the set of agreed-upon truths within the math community
  • We’re not inventing math to fit our world—we’re discovering what math is out there, and then later realizing that our world happens to look exactly like it.