That sounds like math Math and Science Mathematics as the universe It’s a Mathematical World Cristian Rios University of Calgary PIMS Lunchbox Lecture Series April 16 2015 Cristian Rios Mathematics 1 / 38
That sounds like math Math and Science Mathematics as the universe Cristian Rios Mathematics 2 / 38
That sounds like math Math and Science Mathematics as the universe Mathematics everywhere Cristian Rios Mathematics 3 / 38
That sounds like math Math and Science Mathematics as the universe Celestial mechanics Cristian Rios Mathematics 4 / 38
That sounds like math Math and Science Mathematics as the universe General relativity Cristian Rios Mathematics 5 / 38
That sounds like math Math and Science Mathematics as the universe Particle scatter Cristian Rios Mathematics 6 / 38
That sounds like math Math and Science Mathematics as the universe Wave interference Cristian Rios Mathematics 7 / 38
That sounds like math Math and Science Mathematics as the universe Quantum interference Cristian Rios Mathematics 8 / 38
That sounds like math Math and Science Mathematics as the universe Quantum scattering Cristian Rios Mathematics 9 / 38
That sounds like math Math and Science Mathematics as the universe Crossing a street Cristian Rios Mathematics 10 / 38
That sounds like math Math and Science Mathematics as the universe The streets of Buenos Aires Cristian Rios Mathematics 11 / 38
That sounds like math Math and Science Mathematics as the universe Social Mathematics Every decision taken, or action performed, as a result of the use of our frontal cortex has a mathematical base. Basic Principles: Minimize risk. 1 Accident prevention or avoidance. 1 Minimize losses. 2 Self preservation. 3 Maximize profit. 2 Economical advancement. 1 Social advancement. 2 Professional advancement. 3 Betterment of society. 4 Cristian Rios Mathematics 12 / 38
That sounds like math Math and Science Mathematics as the universe Social Mathematics Every decision taken, or action performed, as a result of the use of our frontal cortex has a mathematical base. Basic Principles: Minimize risk. 1 Accident prevention or avoidance. 1 Minimize losses. 2 Self preservation. 3 Maximize profit. 2 Economical advancement. 1 Social advancement. 2 Professional advancement. 3 Betterment of society. 4 Deterministic: Optimization problems. Probabilistic: Computing the odds, choosing the path with best expected outcome. Cristian Rios Mathematics 12 / 38
That sounds like math Math and Science Mathematics as the universe Social mathematics Example of a social mathematics problem : What is the optimal time to call for an election? Cristian Rios Mathematics 13 / 38
That sounds like math Math and Science Mathematics as the universe Mathematics of sports Cristian Rios Mathematics 14 / 38
That sounds like math Math and Science Mathematics as the universe Money ball math " Sabermetrics is the empirical analysis of baseball, especially baseball statistics that measure in-game activity. The term is derived from the acronym SABR, which stands for the Society for American Baseball Research. It was coined by Bill James, who is one of its pioneers and is often considered its most prominent advocate and public face." (Lewis, Michael M. (2003). Moneyball: "The Art of Winning an Unfair Game") Cristian Rios Mathematics 15 / 38
That sounds like math Math and Science Mathematics as the universe That sounds like math Cristian Rios Mathematics 16 / 38
That sounds like math Math and Science Mathematics as the universe Cristian Rios Mathematics 17 / 38
That sounds like math Math and Science Mathematics as the universe The miracle of feeding the multitude Cristian Rios Mathematics 18 / 38
That sounds like math Math and Science Mathematics as the universe The Banach-Tarski Paradox - Paradoxical sets Cristian Rios Mathematics 19 / 38
That sounds like math Math and Science Mathematics as the universe The Banach-Tarski Paradox (paradoxical sets) Cristian Rios Mathematics 20 / 38
That sounds like math Math and Science Mathematics as the universe Foundations of mathematics Principia Mathematica (1910-1913), by Alfred North Whitehead and Bertrand Russell . Intended to describe a set of axioms and inference rules in symbolic logic from which all mathematical truths could be proven. Cristian Rios Mathematics 21 / 38
That sounds like math Math and Science Mathematics as the universe THE ROOTS (DNA) - Zermelo-Frankel axioms - Axiomatic set theory. Axiom of extensionality ∀ A ∀ B ( ∀ X ( X ∈ A ⇐ ⇒ X ∈ B )) = ⇒ A = B Axiom of empty set ∃ X ∀ Y ¬ ( Y ∈ X ) Axiom of pairing ∀ A ∀ B ∃ C ∀ D [ D ∈ C ⇐ ⇒ ( D = A ∨ D = B )] Axiom of union ∀ A ∃ B ∀ c ( c ∈ B ⇐ ⇒ ∃ D ( c ∈ D ∧ D ∈ A )) ∃ I ( ∅ ∈ I ∧ ∀ x ∈ I (( x � { x } ) ∈ I )) Axiom of infinity Axiom of replacement "the image of a set is a set" Axiom of power set ∀ A ∃ P ∀ B [ B ∈ P ⇐ ⇒ B ⊂ A ] ⇒ ∃ y ∈ x ( y � x = ∅ )) Axiom of regularity ∀ x ( x � = ∅ = Axiom schema of specification "a subclass of a set is a set" � � ⇒ ∃ f : X �→ � X ∅ / / ∈ X = Axiom of choice ∀ X ∀ A ∈ X ( f ( A ) ∈ A ) Cristian Rios Mathematics 22 / 38
That sounds like math Math and Science Mathematics as the universe One odd family Equivalence classes Z = { integers } = { 0, 1, − 1, 2, − 2, 3, − 3, . . . } . 2 Z = { even integers } = { 0, 2, − 2, 4, − 4, 6, − 6, . . . } . Define the equivalence p ≈ q ( p is "related" to q ) if p − q ∈ 2 Z . 2 equivalence classes (families): even integers, and odd integers. Representatives of the classes M = { 0, 1 } . Cristian Rios Mathematics 23 / 38
That sounds like math Math and Science Mathematics as the universe One odd family Equivalence classes Z = { integers } = { 0, 1, − 1, 2, − 2, 3, − 3, . . . } . 2 Z = { even integers } = { 0, 2, − 2, 4, − 4, 6, − 6, . . . } . Define the equivalence p ≈ q ( p is "related" to q ) if p − q ∈ 2 Z . 2 equivalence classes (families): even integers, and odd integers. Representatives of the classes M = { 0, 1 } . Each "family" is affinely equivalent to the whole set Z . q ∈ 2 Z (even family) define T 0 : q �→ q /2. Then T 0 ( 2 Z ) = Z (biyective) p ∈ 2 Z + 1 (odd family) define T 1 : p �→ ( p − 1 ) /2. Then T 1 ( 2 Z + 1 ) = Z (biyective) Cristian Rios Mathematics 23 / 38
That sounds like math Math and Science Mathematics as the universe 5 Families Z = { integers } = { 0, 1, − 1, 2, − 2, 3, − 3, . . . } . 5 Z = { multiples of five } = { 0, 5, − 5, 10, − 10, 15, − 15, . . . } . Define the equivalence p ≈ q ( p is "related" to q ) if p − q ∈ 5 Z . 5 equivalence classes (families): 5 Z , 5 Z + 1, 5 Z + 2, 5 Z + 3, 5 Z + 4. Representatives of the classes M = { 0, 1, 2, 3, 4 } . Cristian Rios Mathematics 24 / 38
That sounds like math Math and Science Mathematics as the universe Group of rotations in the circle � p � Q = { rationals } = q , p , q ∈ Z , q � = 0 T = { group of rotations } = [ 0, 2 π ) . R = { group of rational rotations } = 2 π Q mod 2 π . Define the equivalence . p , q ∈ T , p ≈ q if p − q ∈ R . ∞ equivalence classes (families). Axiom of choice : We can build a set M of representatives of the classes. Given a rotation p , its family is p + R . · · � � T = disjoint union of all families = p ∈ M { p + R } = p ∈ R pM . Cristian Rios Mathematics 25 / 38
That sounds like math Math and Science Mathematics as the universe Rationals are countable � p � Q = { rationals } = q , p , q ∈ Z , q � = 0 = { p 1 , p 2 , p 3 , p 4,... } Cristian Rios Mathematics 26 / 38
That sounds like math Math and Science Mathematics as the universe A paradoxical decomposition We enumerate the rationals Q = { p 1 , p 2 , p 3 , p 4 , . . . } Then, we let M i = p i M ( i th rotation of the representatives) � · � � · � � ∞ · · � � � T = (circle) = i = 1 M i = i even M i i odd M i . Cristian Rios Mathematics 27 / 38
That sounds like math Math and Science Mathematics as the universe A paradoxical decomposition We enumerate the rationals Q = { p 1 , p 2 , p 3 , p 4 , . . . } Then, we let M i = p i M ( i th rotation of the representatives) � · � � · � � ∞ · · � � � T = (circle) = i = 1 M i = i even M i i odd M i . The paradox · · � � 2 p − 1 2 p − 1 T = i even p i M i and T = i odd p i + 1 M i . i i Cristian Rios Mathematics 27 / 38
That sounds like math Math and Science Mathematics as the universe Cristian Rios Mathematics 28 / 38
That sounds like math Math and Science Mathematics as the universe Banach-Tarski Paradox for the unit ball in 3-space Cristian Rios Mathematics 29 / 38
That sounds like math Math and Science Mathematics as the universe Mathematics evolution Axioms are the "atoms" of mathematics. Basic accepted truths. These atoms "interact" and combine, according to the rules of logic. These interactions lead to new "molecules" (theorems). Molecules combine and group into "proteins", (theories, areas). Cristian Rios Mathematics 30 / 38
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