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
   

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2. Give me a lever long enough
   and a fulcrum on which to place it,
   and I shall move the world.”
   — Archimedes
   “
   

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4. ABSTRACTION
   

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5. NUMBERS
   

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21. <
    

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29. 2
    3 5
    combine
    add
    count count count
    REAL
    ABSTRACT
    

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30. 2
    3 5
    add
    ABSTRACT
    

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31. SUMS
    

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32. 1 + 2 + 3 + 4 + 5 +
    6 + 7 + 8 + 9 + 10 = ?
    

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33. 1 + 2 = 3
    3 + 3 = 6
    6 + 4 = 10
    10 + 5 = 15
    

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34. 15 + 6 = 21
    21 + 7 = 28
    28 + 8 = 36
    36 + 9 = 45
    45 + 10 = 55
    

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35. 1 + 2 + 3 + 4 + 5 +
    6 + 7 + 8 + 9 + 10 = 55
    

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36. 1 + 2 + 3 + 4 + 5 + …
    … + 98 + 99 + 100 = ?
    

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37. ?
    1 + 2 +
    3 + 4 +
    5 + 6 +
    7 + 8 +
    9 + 10 =
    

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38. ?
    1 + 10 +
    2 + 9 +
    3 + 8 +
    4 + 7 +
    5 + 6 =
    

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39. 11 +
    11 +
    11 +
    11 +
    11 = ?
    

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40. 11 +
    11 +
    11 +
    11 +
    11 = 55
    

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41. 1 + 10 +
    2 + 9 +
    3 + 8 +
    4 + 7 +
    5 + 6 = 55
    

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42. 1 + 2 + … + 9 + 10 =
    (10 ÷ 2) × (10 + 1)
    

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43. 1 + 2 + 3 + 4 + 5 + …
    … + 98 + 99 + 100 = ?
    

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44. 1 + 100 +
    2 + 99 +
    49 + 52 +
    50 + 51 = ?
    ⋮
    

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45. 101 +
    101 +
    101 +
    101 = ?
    ⋮
    

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46. 101 +
    101 +
    101 +
    101 = 5,050
    ⋮
    

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47. 1 + 2 + … + 99 + 100 =
    (100 ÷ 2) × (100 + 1)
    

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48. 1 + 2 + … + n =
    (n ÷ 2) × (n + 1)
    

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52. (n + 1) × n ÷ 2
    1 + 2 + … + n =
    

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53. 1 + 2 + … + n =
    (n + 1)
    ×
    n ÷ 2
    ( )
    

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54. 1 + 2 + … + 999 + 1000 =
    ?
    

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55. 1 + 2 + … + 999 + 1000 =
    (1000 ÷ 2) × (1000 + 1)
    

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56. 1 + 2 + … + 999 + 1000 =
    500 × 1001
    

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57. 1 + 2 + … + 999 + 1000 =
    500,500
    

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58. GRAPHS
    

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77. DEGREE
    2
    

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78. DEGREE
    3
    

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79. WALK
    

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80. CONNECTED
    

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81. CONNECTED
    

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82. CONNECTED
    

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83. CONNECTED
    

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84. TRAIL
    

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85. EULERIAN TRAIL
    

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86. EULERIAN TRAIL
    CONNECTED
    OF ODD DEGREE
    AT MOST 2 NODES
    

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88. TRIANGLES
    

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90. 4
    4
    5.7
    

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91. 4
    5
    6.4
    

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92. 3
    5.8
    5
    

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93. a
    b
    c a² + b² = c²
    

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96. c²
    

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98. a²
    b²
    

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99. c²
    a²
    b²
    

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100. c²
     a² b²
     + =
     

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101. c²
     a²
     b²
     

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102. c²
     a²
     b²
     

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103. c²
     a²
     b²
     

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104. a
     b
     c a² + b² = c²
     

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105. MATHEMATICS
     

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106. SPOT PATTERNS,
     BUILD REUSABLE
     ABSTRACTIONS
     

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107. BAD
     MATHEMATICS
     EDUCATION
     

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110. POKER
     

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111. HIGH CARD
     ONE PAIR
     TWO PAIR
     

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112. THREE OF
     A KIND
     STRAIGHT
     FLUSH
     

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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)
     

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114. Hand = Struct.new(:cards) do
     include Comparable
     def <=>(other)
     # TODO
     end
     end
     

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115. Hand = Struct.new(:cards) do
     include Comparable
     def <=>(other)
     highest_card_rank <=> other.highest_card_rank
     end
     # …
     end
     

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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
     

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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
     

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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
     

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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
     

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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
     

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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
     

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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)
     

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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
     

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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
     

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125. class << Hand::Category::HIGH_CARD
     def matches?(hand)
     true
     end
     def compare(a, b)
     # …
     end
     end
     

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126. class << Hand::Category::ONE_PAIR
     def matches?(hand)
     pairs(hand).size == 1
     end
     def compare(a, b)
     # …
     end
     end
     

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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
     

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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
     

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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
     

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132. low
     high
     

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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
     

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134. “DESIGN”?
     

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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
     “
     

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136. HOW?
     

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137. FIND THE
     HONEST
     ABSTRACTION
     

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138. FIND THE
     MINIMAL
     ABSTRACTION
     

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139. FIND THE
     INTERESTING THING
     INSIDE THE
     BORING THING
     

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140. BE EXCITED!
     

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141. KEEP YOUR
     PERSPECTIVE
     

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142. IF YOU CAN
     WRITE A PROGRAM,
     YOU’RE GOOD
     AT MATHEMATICS.
     

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143. SORRY!*
     * not sorry
     

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144. THANKS
     @tomstuart, tom@codon.com, http:/
     /codon.com/
     

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