The diagrams of the two cultures noted by C.P. Snow are the inverses of one another. On the humanities side, things come together as we progress, resulting in a pyramid (e.g., Maslow’s hierarchy of needs). On the technical side, things come apart as we progress (former synonyms no longer holding), resulting in an inverted pyramid. I call these instances of things coming apart ‘technical dissociations’. Here is a big list of technical dissociations (no doubt grossly incomplete):

1. number versus numeral (a distinction everyone learns in the third grade – except for those who played hooky that day)

2. loss of unique-factorization as you move from the reals to the complex numbers – e.g., 26 = (2)(3) and 26 = (1 – 5i)(1 + 5i)

3. For two distinct numbers, in the set of positive numbers “is to the left of” and “is closer to 0” are synonymous, but are not synonymous for the set of arbitrary real numbers.

4. For two distinct lines, in the plane “are everywhere equidistant from each other” and “do not intersect” are synonymous, but are not synonymous for 3D space.

5. rotation versus revolution

6. apparent astronomical movement versus actual astronomical movement (e.g., ‘sunrise’ and ‘sunset’)

7. dependence of final result on initial conditions, versus, independence

8. rational versus irrational numbers

9. a polynomial versus the polynomial function associated with it

10. In foraging theory, time minimization versus energy maximization (pp. 8-9 of the book ‘Foraging Theory’ by Stephens and Krebs)

11. explanation versus prediction

12. tracking versus detection

13. detection versus recognition

14. ordinal numbers versus cardinal numbers

15. In foraging, path depletion not equal to negative acceleration of the energy gain function.

16. complete information versus perfect information

17. a function being analytic versus being infinitely differentiable

18. two types of paraboloid (elliptic and hyperbolic)

19. wave/particle duality of light

20. MAD (median absolute deviation) has more than one meaning.

21. inertial mass versus rest mass

22. Domain of a partial function is ambiguous, depending on the discipline (logic or mathematics).

23. multiple, and only partially satisfactory, definitions of tortuosity

24. general life situation versus general life situation (terminology of Kurt Lewin)

25. singularities of solutions not necessarily occurring only at singularities of the equation

26. inequality of the types of cardinality for surface area and volume (e.g., Gabriel’s horn)

27. Sometimes homeomorphism type is not determined by homotopy type.

28. “There are several definitions of *R*^{2} that are only sometimes equivalent.”

(Wikipedia’s article on coefficient of determination)

29. Coverage probability splits into ‘actual’ and ‘nominal’.

30. utility versus exactness – e.g., Agresti and Coull’s 1998 paper “Approximate is Better than ‘Exact’ for Interval Estimation of Binomial Proportions.” (cited in the Wikipedia article on binomial proportion confidence intervals)

31. having to choose between a statistical estimator that is unbiased or which has better mean squared error

32. There are two types of Hermite polynomials: the ‘probabilists’ Hermite polynomials and the ‘physicists’ Hermite polynomials.

33. good for exploratory data analysis versus good for classification applications – e. g., Sammon mapping

34. canonical form vs normal form (see the Wikipedia article on computer algebra)

35. A subgroup of a finitely generated group need not be finitely generated.

36. exploiting prey versus exploiting patches

37. the zero-one law in foraging theory versus Kolmogorov’s zero-one law – the former being prescriptive, and the latter being descriptive

38. double-entry bookkeeping versus single-entry bookkeeping

39. agent-designer’s goals versus agent’s goals

40. how the product topology is defined for finitely many spaces versus how it is defined for infinitely many spaces

41. ‘heavy-tailed’ distribution has several meanings.

42. non-unique generalization of the single-variable derivative

43. a series converging versus getting arbitrarily many digits correct

44. convexity of a set versus convexity of the region bounded by the set (e.g., a convex Jordan curve)

45. dice equivalence versus dice winning against each other with equal probability

46. connectedness versus i-connectedness

47. connectedness versus path-connectedness

48. leaves versus structures that look like leaves (such as that possessed by mosses and leafy liverworts)

49. the polyphyletic nature of algae versus the situation of living in water and performing photosynthesis

50. whether energy is present versus whether it is available

51. sidereal time versus solar time

52. rolling friction versus static friction

53. planet versus star

54. elastic versus inelastic collision

55. heat versus temperature

56. a removable versus a non-removable discontinuity

57. blood versus type-distinguished blood

58. vapor versus gas

59. air versus oxygen

60. whale versus fish

61. gold versus fool’s gold

62. rocket propulsion versus friction-based propulsion

63. physical change versus chemical change

64. chemical combustion versus stellar dynamics

65. warm-blooded versus cold-blooded creatures

66. robustness versus anti-fragility

67. linear response versus nonlinear response

68. chaotic versus non-chaotic phenomena

69. continuity versus differentiability

70. speed of sound in air versus speed of sound in water

71. how others hear us versus how we hear ourselves

72. chemical compound versus chemical element

73. Bronze-Age creation myth versus Evolution

74. jealousy versus envy

75. perception controlling behavior versus behavior controlling perception

76. conscious versus unconscious mind

77. momentum versus energy

78. potential energy versus kinetic energy

79. radiant energy versus heat

80. 24 hour period versus calendar day (as in ‘Around the World in 80 Days’)

81. mass versus weight

82. currency versus money

83. sub-sonic versus super-sonic explosions

84. cycloid versus circular arc

85. coma versus death

86. medical intervention versus palliative care

87. data versus information

88. macro versus micro economics

89. weather versus climate

90. strategy versus tactics

91. longitudinal versus transversal waves

92. traditional versus public-key cryptography

93. the definition of uniform integrability in measure theory versus probability theory

94. Nash equilibrium for a game repeated finitely many times versus infinitely many times

95. looking only at truth values versus looking at content (material implication)

96. synonymity of cardinality and measure for finite sets

97. how symmetric groups behave on finite versus on infinite sets

98. optimal behavior in the Prisoners’ Dilemma in the short run (betrayal)

versus in the long run (cooperation)

99. If W is a generalized complex subspace of a generalized complex vector space V,

then V/W is not necessarily a generalized complex quotient of V.

100. topological definition of an object versus geometrical definition (e.g., is the circle a 1-sphere or a 2-sphere?)

101. stable, versus merely long-lived

102. defining fields by polynomials giving different results in the finite and infinite cases

103. temperature versus conductivity (why water feels colder than air at the same temperature)

104. intensive versus extensive properties

105. addition of heat versus a rise in temperature (i.e., the phenomenon of ‘latent heat’)

106. Subsystems can be connected in series or parallel.

107. distinct types of rock (igneous, sedimentary, metamorphic)

108. musical time versus metronome time

109. amortized update time of an algorithm versus worst-case update time

110. absence of evidence versus evidence of absence

111. simple versus small (e.g., uncompressed data versus compressed; cf: Pascal’s remark

about not having time to make his letter shorter)

112. ‘no students’ (but staff present) versus ‘no school’ (nobody present)

113. a commonly accepted proxy for a hard-to-pin-down concept (e.g., variance as a

proxy for risk); also: measure of an entity versus a (more convenient) proxy

measure for the entity (e.g., weight as a proxy measure for mass)

114. ‘good’ cholesterol versus ‘bad’ cholesterol

115. cis versus trans configurations of unsaturated fatty acids

116. direct current versus alternating current

117. animal classification by its diet versus by the nature of its digestive system (e.g., the panda)

118. frequency of a burden versus its intensity

119. specification versus implementation

120. testing versus debugging

121. impossibility versus probability of 0 (things of probability 0 happening all the time, actually)

122. de jure versus de facto

123. topological convergence versus convergence in measure

124. frontier versus wilderness

125. revenue versus profit

126. voltage versus current

127. dictionary versus encyclopedia

128. conservation as wilderness preservation versus as resource management

129. background versus foreground

130. short term versus long term

131. duality in terms of polar reciprocation versus topological duality

132. a candid photo versus a posed photo

133. mixture versus solution

134. billable time versus elapsed time

135. independent generalizations (such as how Newton’s work stands in

relation to Kepler’s work) versus dependent generalizations (such as

how the Law of Cosines stands in relation to the Pythagorean Theorem,

or how Taylor’s Theorem stands in relation to Maclaurin’s Theorem)

136. ability as a conversationalist versus ability as a lecturer

137. archival storage versus working storage

138. complex versus complicated

139. powdered chocolate mix for a cold drink versus for a hot drink

140. form determined by gravity versus by capillarity

141. vector as an arrow (‘something with magnitude and direction’) versus as an element

of a vector space

142. an analytic set versus a Borel set (Suslin’s theorem)

143. catastrophe versus tragedy (“If he were to fall in the Thames it would be a catastrophe. If someone were to pull him out again, it would be a tragedy.”)

144. The subgroup identity is equal to the group identity, but when we move up to rings, we find that a subring can have an identity different from the ring.

145. for organizations, normative control and a regime of collective interest versus rational

control and a regime of self-interest (as noted in ‘Metaphor and the Embodied Mind’ –

Boland and Tenkasi)

146. dice equivalence versus dice winning against each other with equal probability

147. conceptual simplicity versus computational simplicity (e.g., n! versus Stirling’s formula)

148. non-uniqueness of tetration (i.e., repeated, or iterated, exponentiation)

149. inequality between the Hausdorff dimension of a set and its topological dimension

150. multiplication as repeated addition, versus multiplication as an independent operation with a life of its own

151. Spheroidal coordinates are of two types: oblate and prolate.

152. divergence of direction between percentage retention (goes to zero) and amount of retention (goes to infinity). (e.g., progressive income tax)

153. ‘one’ now also means not only ‘exactly one’, but also ‘at least one’, as in the requirement that your password contain ‘one of the following special characters’.

154. danishes dissociate into donuts and cores

155. harmonic mean: distinction between the natural form of the formula and the pattern form of the formula

156. As a system becomes more sophisticated, its nodes undergo bifurcation. This may be why we have two forms of possessive in English: analytic (expressed by ‘of’) and synthetic (expressed by an apostrophe + ‘s’.

157. simple interest versus compound interest

158. mind versus body

see also the website ‘DifferenceBetween.net’: