# Technical Dissociation

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

# Mamikon’s Theorem

Mamikon’s Theorem is the greatest advance in Calculus since the time of Newton. His theorem is the basis of what has become the discipline of Visual Calculus.

# Background and foreground

Distinguishing between background and foreground dispels a lot of confusion. Here, we will focus on the dynamic that the difference between these two realms gives rise to between the two logical connective of ‘and’ and (inclusive) ‘or’. What happens is that the sense of these two connectives flips across the boundary between foreground and background (much like the sense of a mathematical inequality flips when you multiply it through by -1). For example, if you’re the IT person in your company and the boss says to you, “Give me a list of all our salesreps living in Chicago and Boston.”, then you know that as you pass the salesrep file, you change the boss’s intuitive “and” to logical “or”. That is, you output the salesrep’s information if, and only if, the salesrep lives in Chicago OR the salesrep lives in Boston. (The boss was being ‘illogical’, because the boss is always in foreground, and foreground, the terrain of ‘natural language’, is governed not by symbolic logic, but by its own logic of perception and convenience.) So, a foreground ‘and’ becomes a background ‘or’. Also, a foreground ‘or’ becomes a background ‘and’. For example, if you are the hospitality agent for your company and the boss says to you, “We want to offer each guest the choice of coffee or tea.”, then in background (that is, in the company kitchen) you have to prepare a batch of coffee AND a batch of tea. So, a foreground ‘or’ becomes a background ‘and’. Also, problematical situations are alluded to or acknowledged by using this flip-phenomenon. Specifically, tautologies are highly complaisant (indeed, trivial – although the determination that something is a tautology can be highly nontrivial), and the natural habitat of tautologies is, of course, background. Therefore, foregrounding a tautology (by, for example, saying it in the presence of others) alludes to or acknowledges a problem, because the flip-side of complaisance is severity. For example, it is a tautology that boys will be boys, but if you happen to overhear someone actually SAYING so, you can be pretty sure it’s not good news that they are commenting on. A foregrounded tautology is virtually always a vector of bad news. So, when someone says, “Boys will be boys.”, it’s highly unlikely they’re reporting a merit-badge boy-scout activity. To take another example, someone wearing a shirt that says, “It is what it is.” is endemically unhappy about something in the world. Some other thusly-used tautologies are “Kids will be kids.”, “Facts matter.” and “Enough is enough.” (and its over-the-top version, “Too much is too much.”) The purpose and technique of foregrounding some tautology in order to make a negative point in a passive-aggressive manner is understood by everyone at the gut level, but it deepens cognition to treat this matter explicitly. So, if someone says, “We’re going to have some weather next week.”, you know that that implies “We’re going to have some bad weather next week.”, because a foregrounded tautology implies some kind of mischief, and therefore you should ignore those who say that the word ‘bad’ should have been explicitly included.