**contents** - [[#**Topology**|**Topology**]]: shape, manifold, dimension - [[#**Analysis**|**Analysis**]]: infinity, continuum, maps - [[#**Algebraic structure** - abstraction of pattern, regularity, relation|**Algebraic structure** - abstraction of pattern, regularity, relation]]: abstraction, structure, inference - modeling: model, automata, science --- Theory proved are as follows - 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