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