Part 4. Abstraction and Compression

A. Abstractions

abstraction (n) Something that exists only as an idea. Not concrete.
 "to abstract" (v) To extract or remove (something).
 "an abstract" (n) A summary of the contents of a book or research article.

"one entity ...represented symbolically by another"
 -"Constructor Theory of Information", on abstraction
   Deutsch/Marletto (2014)

i. Abstract vs Concrete

The idea of “two”, is an example of an abstraction:

"The abstract must be made concrete by examples."
 -Definition search for "abstract".
  Google (2019)

You can have “two” of anything – two rocks, two fish, two people, two cars. Across the set of all possible things, the idea of “two” would apply to a situation when you first have one thing, and second have another thing that arbitrarily-resembles the first thing.


Above: Wikipedia image for “generalization process”. Visually describing how the one might [1] observe several trees in reality, and [2] generalize these experiences so as to construct an abstract “tree” with abstract attributes (eg: green leaves above, trunk below; trunk is linear and perpendicular to ground; leaves expand omni-directionally and occlude trunk; etc).

The opposite of the abstract is the concrete – specific things that you actually experience. You will never meet an “abstract person”. (When you meet a person in real life, it is always a specific, “concrete” person.)

Of course, all words are abstractions. (They aren’t limited to concepts like “three”, or “people”.)

ii. Essentialism

Imagine if you asked me to bring you “two apples”.

This request implies “essentialism” – the not-strictly-true idea that each and every thing has an “essence” that makes it either be “apple” or “not apple”.

Essentialism cannot be taken literally, because it is not true. If it were true, it would be possible to answer the following questions:

  • When, exactly, does a fruit bud “become” a real apple? And at which point has an apple become ripe (or so rotten), that it stops becoming an “apple”?
  • If an apple is very small, does it still count as “one apple”? (At what size would it not?) If an apple is very very large (or a kind of “Siamese twin apple”), is that “one” apple or “two”?

But these questions cannot be answered – not in any “a priori” sense. Because the concept of “apple” is a human-constructed abstraction.

These categories are simplifications – where many individual particular features have been intentionally stripped away [from our set of individual concrete experiences]. This is a very helpful thing – it prevents us from having to describe each apple’s features in 100% exhaustible detail. It lets us compress the information, by removing irrelevant details. It lets us notice that some things are similar, and that they will share properties with each other.

Without essentialism, we couldn’t even talk about “electrons” in the plural, because each particle would be a fresh, new experience. We certainly couldn’t talk about “the sun”, “birth”, or “temperature” as what they ‘really are’. In fact, we couldn’t use words to talk about anything!

B. Compression

Science may be described as the art of systematic
over-simplification — the art of discerning what we
may with advantage omit.
-Karl Popper, The Open Universe (1992), p.44

When comparing the time scales of genetic and cultural evolution, it
is useful to bear in mind that we today — every one of us — can easily
understand many ideas that were simply unthinkable by the geniuses in
our grandparents' generation!
-Daniel Dennett

i. “Good Enough” Summaries

The compression/abstraction process is enormously helpful. Information-processing is impossible without it.

But compression necessarily introduces error – the information that we deleted [during the compression process] might suddenly become relevant again, at any time.


Above: The word “with” was compressed, and an error resulted. From jakelikesonions.

A good compression will be more helpful, than the error that it introduces. It will be like a leaky water pump – some fluid leaks out, and runs downwards; but even more fluid makes it into the pump and is instead sent upwards.

ii. Accuracy vs User-Friendliness

"Everything simple is false. Everything complex is useless."
-Bonini's Paradox

The Map Metaphor has a well known saying: “the map is not the territory”.

But it has a slightly-less-well-known rejoinder: “… but you can’t fold up the territory and put it in your glove compartment.” Ie, “The only 100% accurate map of California, is California.”

We use “compressed” explanations of reality, both out of necessity but also out of efficiency. The trick is to delete the most redundant information first.

Check it out:


Above: Two kinds of SVD-based image compression, from this paper. Notice the rank-30 image (using 30 units of information) is very effectively reproduced using 20 and 14 units of information. But as we move from rank 8 to rank 2, we start to delete information that is crucial (ie, “non-redundant”).

See Also: Good Approximations (?)


Above: Scatterplot of weight-height-gender relationship. Instead of explaining everything in full detail, we explain it “to a first approximation” or to a “first couple of approximations”. For example when we say “men are usually taller and heaver than women”.

In some sense, Italy is “shaped like a boot”. Simultaneously, however, it is shaped nothing like a boot. This contradiction is resolved by examining shapes at different levels: a “geo-political” level (in which countries are pastel shapes, on a zoomed-out globe, outlined in with dark bold lines), and other levels (such as photographs of a few square-km of the actual country of Italy).

iii. Regress

So-called “infinite regress” is total non-explanation. It is answering the question “What is 2+2?” by saying “It is equal to 2+2” (or by saying “it is the answer to the question which you just asked). It is the useless Library of Babel, where everything might mean everything else (and therefore, what it really means is nothing).

However, in a “finite regress”, the explanation is not completely circular. It is like the leaky water pump mentioned above. As each question is answered, some of the explanatory “buck” is “passed” (to a new, deeper question); but some of the curiosity has been genuinely alleviated, as well.

See also:

Part 5, the final part, is about omniscience and communication.

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