High abstraction requires lots of words

I’m of the opinion that high-level definitions ought to be as short as possible. A two-word definition is the best possible case. For example,

cranequinier = “mounted crossbowman”

Now, we sometimes don’t know the highly specific vocab, and we often don’t need to because saying “mounted crossbowman” is efficient enough 99% of the time you need to talk about such things.

But such short definitions are much easier to hold in the mind. This becomes important in matters of high abstraction, because we must be able to recall all necessary definitions at a whim.

In abstract algebra, for example, we build up many abstract definitions from relatively simple ones. It is very important to remember all of them in terms of each other.

Set = Common notion
Ordered pair = Ordered set of two elements
Function = Set of ordered pairs

Binary operation = Bijection from the domain to itself
Algebraic structure = Ordered pair of a set and a binary operation
Group = Algebraic structure having closure, associativity, identity and invertibility

Permutation group = Subgroup of a symmetric group

It simply isn’t possible to keep all of this in mind at once, but sometimes we have to trace a definition all the way back to the common notions at the beginning. So maybe I want to know how many subsets are in an algebraic structure. I can walk back through the definitions via simple substitutions:

Algebraic structure = Ordered pair of a set and a binary operation
Algebraic structure = (Ordered set of two elements) of a set and a binary operation
Algebraic structure = (Ordered set of two elements) of a set and a (Bijection from the domain to itself)

Even mediocre minds like mine can recall definitions like this on the spot easily when the definitions are small and tidy. It is basically impossible to understand the inner workings of a high-level definition like this if you start with big, clunky definitions like

“Morality (from the Latin moralitas “manner, character, proper behavior”) is the differentiation of intentions, decisions, and actions between those that are distinguished as proper and those that are improper.”

Too much useless detail. The language of origin is interesting contextual detail, but it is unnecessary for the definition. The descriptive bits are good for grokking the concept image, but they are hindrances to proper understanding at the abstract level. Noise makes a complicated circuit useless much more quickly than a simple circuit. Here’s a good definition

“Morality is about what’s right and wrong.”

Swell. If I don’t remember what “about” is I can look that up, and if I don’t understand one of the words in the definition I can look up that word, and on and on until I have broken down the abstract idea of morality into common notions without recourse to the contortions required to produce new common notions on the spot.

I believe every complicated endeavor should prefer breadth of vocabulary to depth for this reason, and thereby achieve clarity, ease of communication, and brevity. Hence, anyone aspiring to build a new body of knowledge should set themselves the task, first and foremost, of inventing as many simple definitions as possible and assigning each one its own word.


About Aeoli Pera

Maybe do this later?
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2 Responses to High abstraction requires lots of words

  1. Pingback: Daily Linkage – October 4, 2015 | The Dark Enlightenment

  2. Pingback: Studying physics at the three levels of mathematical reasoning | Aeoli Pera

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