It’s very easy for discussions to get hung up on definitions, so a serious approach to debating a subject begins with synchronizing everyone’s vocabulary watches. Accurate and nuanced communication requires mutually understood ideas and terminology for expressing those ideas.
Yet some concepts seem almost impossible to define clearly. The idea of “consciousness” is notorious for being a definition challenge, but “morality” or “justice” or “love” are also very difficult to pin down. At the same time, we seem to share mutual basic intuitions of these things.
So the question today is: why are some concepts so hard to define?
I think there at least two problems with defining some concepts. The first involves reduction; the second involves configuration space.
Reduction analyzes something in terms of its comprising parts. A key premise is the parts combine to fully explain the whole. A second premise is the parts reduce to sub-parts which themselves reduce and so on. Reduction is recursive.
Recursion requires a condition that halts the process. Otherwise, it’s “turtles all the way down” — infinite recursion that never ends. In computational recursion the halt condition depends on a computational condition.
Definitional recursion ends where the sub-parts are atomic — indivisible, per the original Greek meaning of the word. It’s not possible (or sensible) to further sub-divide them.
(Ironically actual atoms are not atomic in that original sense — atoms reduce to electrons and nuclei; atomic nuclei reduce to protons and neutrons; and those reduce to quarks. We believe electrons and quarks are the end of the line; no more turtles.)
A more familiar example: A book reduces into chapters, which reduce into paragraphs, which reduce into sentences, then into words, and finally into individual characters.
Recursion ends there because characters are atomic, they don’t divide into anything (not in printed books, anyway — one could argue characters in electronic books are comprised of bits).
We could talk about the ink (or bits), or the atoms involved, or even take it to the quantum level of electrons and quarks, but there is no “bookness” below the level of a sequence of characters. All books look the very much the same at the quantum level.
(To be precise, the quantum state of different books would be different, but picking out the book text would be just about impossible. The text information is vastly swamped by the information of the constituent particles.)
The key point is this: Definitions consist of (reduce to) other, presumably simpler, definitions.
This has two consequences:
Firstly, that a definition must not circle back to what it’s defining. That creates a loop of definition with no grounding.
Secondly, that necessarily (to avoid loops), at some point there must be atomic definitions that cannot be defined by simpler concepts. There must be atomic definitions that ground everything else.
The question is how we construct those atomic definitions.
They are just Cartesian spaces with axes — as many as necessary — that represent traits applying to the subject.
If the subject is cars, one obvious axis is number of doors. Others include (but aren’t limited to): number of wheels, number of cup holders, number of engine cylinders, engine displacement, fuel tank size, wheel size, fuel economy, height, width, weight, model, age, and color.
Note that some axes are smooth (e.g. age, fuel economy) whereas others are lumpy (e.g. numbers of things). It just means that certain points in the space jump. Cars, for instance, jump from three wheels to four wheels — there are no cars with 3.137 wheels. Fuel economy, on the other hand, can be any reasonable value.
Given the right set of axes, a given car is a point in the configuration space. We can’t visualize such a space, but we can intuit the general idea from 2D and 3D examples.
What’s important about a configuration space is that similar objects (for instance, cars) are “close together” in the space. Cars with the same model and year form tiny clusters of points separated only by the small variations among them.
In the 3D Neapolitan ice cream configuration space, people who really like vanilla and chocolate, but not strawberry, create a cloud of close points in one corner of the cubical space. (The high-vanilla, high-chocolate, low-strawberry, corner.)
Mathematically, the distance between two points is the square root of the sum of the square of their distances along each axis — in other words, the Pythagorean distance. (Configuration spaces use Cartesian coordinates in Euclidean spaces, so naturally Pythagoras applies to measuring distance.)
The key is that a collection of similar objects forms a fuzzy cloud of close points — a fuzzy region in configuration space.
The first two are precisely defined. Seconds and meters are also simply defined — their definitions are easily stated. More involved definitions can still be precise. The definition of an electron, for example, or of a 1968 Volkswagen Beetle.
Baseball bats are precisely defined, but in a way that allows variation. In contrast, the definition of the baseball is narrower — it allows almost no variation. (There is also room for variation among ’68 Beetles.)
Frying pans aren’t as precisely defined. The definition describes a wide flat pan with a single long handle. There is a typical size range, but there are large and small variants.
(Measurements aren’t enough. We took camping with us “The Frying Pan From Hell” — a cast iron monstrosity 30″ in diameter with a bolt-on handle just as long. It cooks 8-10 pancakes at once. Or a pound of bacon.)
What’s relevant to the definition is the ratio between width and height. Frying pans are characteristically wide and flat. They have a characteristic shape.
More general definitions tend to involve characteristics rather than specific properties. (Characteristics are properties, of course, but they imply less requirement and more tolerance.)
Seconds, meters, and electrons, have specific definitions that make them points in their respective configuration spaces. Objects strictly are, or are not, seconds, meters, or electrons — they have to hit the bullseye.
Baseballs and ’68 Beetles have definitions with little variation, so they form rather small volumes in their spaces. The definition of baseball bat allows more variation, so the volume of baseball bats is larger.
The more general definition of frying pans creates a very large and fuzzy space volume. Note, too, that the definition begins to shift from properties to (characteristic) functionality — a frying pan is a pan for frying things.
Something as simple as a chair has a definition that is almost entirely characteristic and functional. A chair is a flat-ish surface a human can sit on. (And yet, is a rock or log a chair?)
The chair definition has a huge volume in configuration space — think of all the things that are legit chairs, from bean bag to folding to thrones.
Here’s a key point: The more specific a definition is, the smaller the region in configuration space (including possibly just a point). The more general something is, the larger its region is.
But general definitions involve multiple properties that interact. Pans have width and height. Is a pan flat enough to be a frying pan (or is it just a low-rider sauce pan)?
Such concepts tend to make the region boundaries fuzzy and vague. Determining if an object fits in borderline cases becomes something of a judgement call.
As was notoriously said regarding pornography, ‘We know it when we see it.’
There is truth to that. It’s another general concept with lots of interacting axes and a very large, vague, configuration space.
So how do we ‘know it when we see it’?
We do the same thing as Artificial Neural Nets (ANNs): We train our minds with all the examples of movies, books, videos, various forms of erotica, opinions of others, plus our life experiences. As a result, we make judgement calls on whether something is in the region or not.
Given fuzzy boundaries, some things are hard to judge. When an ANN considers unknown input, it provides a confidence percentage: “I’m 87.4% certain this is a picture of a cat.”
Likewise, when we consider something, our minds are testing for a match. Is this porn? Is this justice? Is this art? Or even: is this a frying pan?
Obviously the accuracy of judgements depends on the quality of the training. It also correlates with quantity given equal quality input — more (quality) experience is better than less.
So to put the pieces together, some concepts are irreducible yet general. This may seem contradictory, but many of our most basic ideas are abstract notions about something.
A basic characteristic of irreducible notions is, because definition requires reduction, they can’t be defined, only described. This involves labeling example points in the configuration space in terms of quality. Providing both good and bad examples defines the boundaries of the concept.
The more examples provided, the more the definition converges on a clear gestalt. The region of configuration space remains fuzzy because of interacting axes, but discrimination — whether a thing is or isn’t — can still become quite acute.
This all only scratches the surface, but it introduces the basic parts. I’ll pick up the threads another time.
Stay safe, my friends! Wear your masks — COVID-19 is airborne!