As we move upwards socially, things come together. This is a process of integration, famously described by Abraham Maslow’s hierarchy of needs. The shape is like a pyramid, an inverted ‘V’. However, as we move upwards technically, things come apart. This is a process of dissociation, of a parting of ways of things that were previously conflated, or synonymous. The shape is like a normal ‘V’. (Superimposing the inverted ‘V’ and the normal ‘V’ gives a simultaneous picture of what is happening.)
A few examples of technical dissociation:
* 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)
* apparent astronomical movement versus actual astronomical movement
* in foraging theory, time minimization versus energy maximization
(pp. 8-9 of the book ‘Foraging Theory’ by Stephens and Krebs)
* 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.
* In foraging, path depletion not equal to negative acceleration of the energy gain function.
* complete information versus perfect information
* a function being analytic versus being infinitely differentiable
* two types of paraboloid (elliptic and hyperbolic)
* ‘heavy-tailed’ distribution has several meanings
* MAD (median absolute deviation) has more than one meaning
* non-unique generalization of the single-variable derivative
* The domain of a partial function is ambiguous, depending on the discipline
(Logic or Mathematics).
* multiple, and only partially satisfactory, definitions of tortuosity
* general life situation (gls) versus momentary life situation (mls) (terminology of Kurt Lewin)
* singularities of solutions not necessarily occurring only at singularities of the equation
* inequality of the types of cardinality for surface area and volume (e.g.: Gabriel’s horn)
* Homeomorphism type is not necessarily determined by homotopy type.
* Coverage probability splits into ‘actual’ and ‘nominal’.
* 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)
* having to choose between a statistical estimator that is unbiased or which has better
mean squared error
* There are two types of Hermite polynomials: the “probabilists’ Hermite polynomials”
and the “physicists’ Hermite polynomials”.
* canonical form versus normal form (see the Wikipedia article on computer algebra)
* A subgroup of a finitely generated group need not be finitely generated.
* exploiting prey versus exploiting patches
* the zero-one law in foraging theory versus Kolmogorov’s zero-one law – the former
being prescriptive, and the latter being descriptive
* how the product topology is defined for finitely many spaces versus how it is defined
for infinitely many spaces
* a series converging versus getting arbitrarily many digits correct
* 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)
* dice equivalence versus dice winning against each other with equal probability
* whether energy is present versus whether it is available
* sidereal time versus solar time
* a removable versus a non-removable discontinuity
* continuity versus differentiability
* perception controlling behavior versus behavior controlling perception
* radiant energy versus heat
* topological convergence versus convergence in measure
* sub-sonic versus super-sonic explosions
* cycloid versus circular arc
* longitudinal versus transversal waves
* traditional versus public-key cryptography
* Nash equilibrium for a game repeated finitely many times versus infinitely many times
* conceptual simplicity versus computational simplicity (e.g., n! versus Stirling’s formula)
* Spheroidal coordinates are of two types: oblate and prolate.
* how symmetric groups behave on finite versus on infinite sets
* optimal behavior in the Prisoners’ Dilemma in the short run (betrayal)
versus in the long run (cooperation)
* 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.
* topological definition of an object versus geometrical definition
* defining fields by polynomials giving different results in the finite and infinite cases
* temperature versus conductivity
* intensive versus extensive properties
* amortized update time of an algorithm versus worst-case update time
* impossibility versus probability of 0 (Things of probability 0 happen all the time.)
* conservation as wilderness preservation versus as resource management
* duality in terms of polar reciprocation versus topological duality
* powdered chocolate mix for a cold drink versus for a hot drink
* non-uniqueness of tetration
Here is my big list of such:
Technical Dissociation Big List