A Lever for the Mind

Humans are the cleverest animals in the known universe. But our brains, forged over millions of years in the unforgiving furnace of natural selection, are still primitive. They evolved to be good at hunting animals, making tools, recognising facial expressions and maintaining social bonds, but they’re just not very good at working with several difficult ideas simultaneously, and they’re particularly bad at reasoning about the counterintuitive emergent behaviour of complex non-human systems.

That’s a problem, because complex non-human systems now dominate our world.

Abstraction is our solution to this problem. Abstraction works as an adapter between the fearsome complexity of the universe and our simple primate minds; when the real world is too detailed, too confusing or too counterintuitive for us to work with directly, abstraction gives us big friendly levers to pull on instead. This one powerful idea underlies computers, design, culture, language, and everything else we rely on in our daily work.

This is a talk about abstraction: where it comes from, what it’s for, and how we can use it to make our programs better.

Given at RubyConf 2014 (http://lanyrd.com/2014/rubyconf/) and Bath Ruby Conference 2015 (http://2015.bathruby.org/). A video and transcript is available at https://tomstu.art/a-lever-for-the-mind.

Transcript

1. THE A LEVER FOR MIND @tomstuart at RubyConf on 2014-11-19
2. Give me a lever long enough and a fulcrum on which to place it, and I shall move the world.” — Archimedes “
4. ABSTRACTION
5. NUMBERS
17. 4 3 2
21. <
29. 2 3 5 combine add count count count REAL ABSTRACT
30. 2 3 5 add ABSTRACT
31. SUMS
32. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = ?
33. 1 + 2 = 3 3 + 3 = 6 6 + 4 = 10 10 + 5 = 15
34. 15 + 6 = 21 21 + 7 = 28 28 + 8 = 36 36 + 9 = 45 45 + 10 = 55
35. 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
36. 1 + 2 + 3 + 4 + 5 + … … + 98 + 99 + 100 = ?
37. ? 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 =
38. ? 1 + 10 + 2 + 9 + 3 + 8 + 4 + 7 + 5 + 6 =
39. 11 + 11 + 11 + 11 + 11 = ?
40. 11 + 11 + 11 + 11 + 11 = 55
41. 1 + 10 + 2 + 9 + 3 + 8 + 4 + 7 + 5 + 6 = 55
42. 1 + 2 + … + 9 + 10 = (10 ÷ 2) × (10 + 1)
43. 1 + 2 + 3 + 4 + 5 + … … + 98 + 99 + 100 = ?
44. 1 + 100 + 2 + 99 + 49 + 52 + 50 + 51 = ? ⋮
45. 101 + 101 + 101 + 101 = ? ⋮
46. 101 + 101 + 101 + 101 = 5,050 ⋮
47. 1 + 2 + … + 99 + 100 = (100 ÷ 2) × (100 + 1)
48. 1 + 2 + … + n = (n ÷ 2) × (n + 1)
51. 10 11
52. (n + 1) × n ÷ 2 1 + 2 + … + n =
53. 1 + 2 + … + n = (n + 1) × n ÷ 2 ( )
54. 1 + 2 + … + 999 + 1000 = ?
55. 1 + 2 + … + 999 + 1000 = (1000 ÷ 2) × (1000 + 1)
56. 1 + 2 + … + 999 + 1000 = 500 × 1001
57. 1 + 2 + … + 999 + 1000 = 500,500
58. GRAPHS
72. 5 3 3 3
77. DEGREE 2
78. DEGREE 3
79. WALK
80. CONNECTED
81. CONNECTED
82. CONNECTED
83. CONNECTED
84. TRAIL
85. EULERIAN TRAIL
86. EULERIAN TRAIL CONNECTED OF ODD DEGREE AT MOST 2 NODES
88. TRIANGLES
89. 4 3 5
90. 4 4 5.7
91. 4 5 6.4
92. 3 5.8 5
93. a b c a² + b² = c²
96. c²
98. a² b²
99. c² a² b²
100. c² a² b² + =
101. c² a² b²
102. c² a² b²
103. c² a² b²
104. a b c a² + b² = c²
105. MATHEMATICS
106. SPOT PATTERNS, BUILD REUSABLE ABSTRACTIONS
107. BAD MATHEMATICS EDUCATION
110. POKER
111. HIGH CARD ONE PAIR TWO PAIR
112. THREE OF A KIND STRAIGHT FLUSH
113. Card = Struct.new(:rank, :suit) three_of_diamonds = Card.new(3, :diamonds) queen_of_hearts = Card.new(12, :hearts) ace_of_spades = Card.new(14, :spades)
114. Hand = Struct.new(:cards) do include Comparable def <=>(other) # TODO end end
115. Hand = Struct.new(:cards) do include Comparable def <=>(other) highest_card_rank <=> other.highest_card_rank end # … end
116. Hand = Struct.new(:cards) do include Comparable def <=>(other) if highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end # … end
117. Hand = Struct.new(:cards) do include Comparable def <=>(other) if empty? && other.empty? 0 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end # … end
118. Hand = Struct.new(:cards) do include Comparable def <=>(other) if one_pair? if other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end
119. Hand = Struct.new(:cards) do include Comparable def <=>(other) if two_pair? if other.two_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end elsif one_pair? if other.two_pair? -1 elsif other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? || other.two_pair? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end
120. Hand = Struct.new(:cards) do include Comparable def <=>(other) if three_of_a_kind? if other.three_of_a_kind? if highest_triple_rank == other.highest_triple_rank without { |c| c.rank == highest_triple_rank } <=> other.without { |c| c.rank == highest_triple_rank } else highest_triple_rank <=> other.highest_triple_rank end else 1 end elsif two_pair? if other.three_of_a_kind? -1 elsif other.two_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end elsif one_pair? if other.two_pair? || other.three_of_a_kind? -1 elsif other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? || other.two_pair? || other.three_of_a_kind? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end
121. Hand = Struct.new(:cards) do include Comparable def <=>(other) if straight? if other.straight? if highest_straight_rank == other.highest_straight_rank without { |c| c.rank == highest_straight_rank } <=> other.without { |c| c.rank == highest_straight_rank } else highest_straight_rank <=> other.highest_straight_rank end else 1 end elsif three_of_a_kind? if other.straight? -1 elsif other.three_of_a_kind? if highest_triple_rank == other.highest_triple_rank without { |c| c.rank == highest_triple_rank } <=> other.without { |c| c.rank == highest_triple_rank } else highest_triple_rank <=> other.highest_triple_rank end else 1 end elsif two_pair? if other.three_of_a_kind? || other.straight? -1 elsif other.two_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end elsif one_pair? if other.two_pair? || other.three_of_a_kind? || other.straight? -1 elsif other.one_pair? if highest_pair_rank == other.highest_pair_rank without { |c| c.rank == highest_pair_rank } <=> other.without { |c| c.rank == highest_pair_rank } else highest_pair_rank <=> other.highest_pair_rank end else 1 end else if empty? && other.empty? 0 elsif other.one_pair? || other.two_pair? || other.three_of_a_kind? || other.straight? -1 elsif highest_card_rank == other.highest_card_rank without { |c| c.rank == highest_card_rank } <=> other.without { |c| c.rank == highest_card_rank } else highest_card_rank <=> other.highest_card_rank end end end # … end
122. Card = Struct.new(:rank, :suit) class Card::Rank TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } end class Card::Suit CLUBS, HEARTS, SPADES, DIAMONDS = 4.times.map { new } end three_of_diamonds = Card.new(Card::Rank::THREE, Card::Suit::DIAMONDS) queen_of_hearts = Card.new(Card::Rank::QUEEN, Card::Suit::HEARTS) ace_of_spades = Card.new(Card::Rank::ACE, Card::Suit::SPADES)
123. class Card::Rank include Comparable ORDER = ( TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } ) def <=>(other) ORDER.index(self) <=> ORDER.index(other) end end
124. class Hand::Category include Comparable ORDER = ( HIGH_CARD, ONE_PAIR, TWO_PAIR, THREE_OF_A_KIND, STRAIGHT, FLUSH, FULL_HOUSE, FOUR_OF_A_KIND, STRAIGHT_FLUSH = 9.times.map { new } ) def <=>(other) ORDER.index(self) <=> ORDER.index(other) end end
125. class << Hand::Category::HIGH_CARD def matches?(hand) true end def compare(a, b) # … end end
126. class << Hand::Category::ONE_PAIR def matches?(hand) pairs(hand).size == 1 end def compare(a, b) # … end end
127. Hand::Rank = Struct.new(:hand) do include Comparable def category Hand::Category::ORDER. select { |category| category.matches?(hand) }.max end def <=>(other) if category == other.category category.compare(hand, other.hand) else category <=> other.category end end end
128. Hand = Struct.new(:cards) do def rank Hand::Rank.new(self) end end if my_hand.rank > your_hand.rank puts 'I win!' end
129. class Card::Rank ORDER = ( TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } ) def succ ORDER[ORDER.index(self).succ] end end
132. low high
133. class Card::Rank ORDER = ( TWO, THREE, FOUR, FIVE, SIX, SEVEN, EIGHT, NINE, TEN, JACK, QUEEN, KING, ACE = 13.times.map { new } ) def succ ORDER[ORDER.index(self).succ % ORDER.size] end end
134. “DESIGN”?
135. When a thing looks complicated, it’s possible that we’re looking at it wrong and missing some of the pieces of the puzzle.” — Richard Feynman “
136. HOW?
137. FIND THE HONEST ABSTRACTION
138. FIND THE MINIMAL ABSTRACTION
139. FIND THE INTERESTING THING INSIDE THE BORING THING
140. BE EXCITED!
141. KEEP YOUR PERSPECTIVE
142. IF YOU CAN WRITE A PROGRAM, YOU’RE GOOD AT MATHEMATICS.
143. SORRY!* * not sorry
144. THANKS @tomstuart, tom@codon.com, http:/ /codon.com/