The definition of "abstract"

[blowfly]

I posted this at CARM before the great meltdown. The definition of "abstract" really interests me, because it seems misconceptions about it lie at the heart of many bad philosophies - including platonism, dualism, transcendental arguments, reification of numbers etc. Consider this definition:

Abstract: able to be instantiated in a wide variety of spatial structures.

IOW, if X can be instantiated in a wide variety of different spatial structures, then it is highly abstract (eg. a wide variety of spatial structures fulfill our definition "raincloud" (think about all the possible spatial structures (ie. shapes) a raincloud could take), so "raincloud" is reasonably abstract). Or if X can only be instantiated in a narrow set of spatial structures, then it is highly concrete (eg. only a very small set of structures qualify as "The Eiffel Tower", so "The Eiffel Tower" is highly concrete).

Exactly what quantifies a "wide" vs. "narrow" variety should be defined more precisely (perhaps even mathematically or computationally, I see a few possibilities here), but the intuitive sense can suffice for now. Importantly, this distinction is a scale, not a dichotomy ("How narrow?" instead of "Wide or narrow?", allowing us to talk in terms of the degree of abstraction ("How abstract?"), rather than some abstract/concrete dichotomy ("Abstract or concrete?").

Given the number of different of ways to instantiate "The number 3" - neurologically, electrically, abacus, etc etc, we can consider it extremely abstract.

What do you think?

Cheers,
-blowfly

[ydoaPs]

It seems to me that your definition of abstract only really applies to objects which vary in size/shape/texture/colour/etc. What about ideas such as Freedom, Love, etc?

[blowfly]

Dec 3, 2009 2:20:18 GMT ydoaPs said:

It seems to me that your definition of abstract only really applies to objects which vary in size/shape/texture/colour/etc. What about ideas such as Freedom, Love, etc?

Perfect examples, I think. How many different structures or scenarios would you call "Freedom"? There's a massive variety. Therefore according to this definition, Freedom is highly abstract. Which we intuitively thought anyway.

Thinking about abstract in terms of variety of spatial structures seems a bit odd at first. But I'm increasingly convinced that it's correct - correct, as in accurately summing up our intuitive sense of the concept.

[vazscep]

Dec 3, 2009 2:07:03 GMT blowfly said:

I posted this at CARM before the great meltdown. The definition of "abstract" really interests me, because it seems misconceptions about it lie at the heart of many bad philosophies - including platonism, dualism, transcendental arguments, reification of numbers etc. Consider this definition:

Abstract: able to be instantiated in a wide variety of spatial structures.

IOW, if X can be instantiated in a wide variety of different spatial structures, then it is highly abstract (eg. a wide variety of spatial structures fulfill our definition "raincloud" (think about all the possible spatial structures (ie. shapes) a raincloud could take), so "raincloud" is reasonably abstract). Or if X can only be instantiated in a narrow set of spatial structures, then it is highly concrete (eg. only a very small set of structures qualify as "The Eiffel Tower", so "The Eiffel Tower" is highly concrete).

Exactly what quantifies a "wide" vs. "narrow" variety should be defined more precisely (perhaps even mathematically or computationally, I see a few possibilities here), but the intuitive sense can suffice for now. Importantly, this distinction is a scale, not a dichotomy ("How narrow?" instead of "Wide or narrow?", allowing us to talk in terms of the degree of abstraction ("How abstract?"), rather than some abstract/concrete dichotomy ("Abstract or concrete?").

Given the number of different of ways to instantiate "The number 3" - neurologically, electrically, abacus, etc etc, we can consider it extremely abstract.

What do you think?

Cheers,
-blowfly

Hey blowfly.

What would you say about infinite sets? It seems to me that these are highly abstract, and yet I don't know of a single instantiation of them.

[blowfly]

Dec 3, 2009 10:18:17 GMT vazscep said:

Dec 3, 2009 2:07:03 GMT blowfly said:

I posted this at CARM before the great meltdown. The definition of "abstract" really interests me, because it seems misconceptions about it lie at the heart of many bad philosophies - including platonism, dualism, transcendental arguments, reification of numbers etc. Consider this definition:

Abstract: able to be instantiated in a wide variety of spatial structures.

IOW, if X can be instantiated in a wide variety of different spatial structures, then it is highly abstract (eg. a wide variety of spatial structures fulfill our definition "raincloud" (think about all the possible spatial structures (ie. shapes) a raincloud could take), so "raincloud" is reasonably abstract). Or if X can only be instantiated in a narrow set of spatial structures, then it is highly concrete (eg. only a very small set of structures qualify as "The Eiffel Tower", so "The Eiffel Tower" is highly concrete).

Exactly what quantifies a "wide" vs. "narrow" variety should be defined more precisely (perhaps even mathematically or computationally, I see a few possibilities here), but the intuitive sense can suffice for now. Importantly, this distinction is a scale, not a dichotomy ("How narrow?" instead of "Wide or narrow?", allowing us to talk in terms of the degree of abstraction ("How abstract?"), rather than some abstract/concrete dichotomy ("Abstract or concrete?").

Given the number of different of ways to instantiate "The number 3" - neurologically, electrically, abacus, etc etc, we can consider it extremely abstract.

What do you think?

Cheers,
-blowfly

Hey blowfly.

What would you say about infinite sets? It seems to me that these are highly abstract, and yet I don't know of a single instantiation of them.

Let's separate a few things here - with "cat". There are actually 3 separate things at play:

"HRG's cat" - highly concrete. Very few spatial structures could qualify as "HRG's cat". I mean there's a little flexibility - he could lose a limb, ear, or lose some fur, and it would still be "HRG's cat", but change too much and it ceases to qualify.

"cat" - less concrete, somewhat abstract. Many different structures qualify as "cat" - all kinds of shapes, sizes, colors, locations, genetic makeup etc.

"the concept of cat" - highly abstract. Literally the neurological or electronic representation of what "cat" means. A wide variety of neurological structures could qualify as "the concept of cat", essentially any set of symbols defining cat-ness, instantiated in any medium, could qualify. *)

(On the last point - if we take the sentence "The Eiffel Tower is a 19th century iron lattice tower located on the Champ de Mars in Paris", the number of different physical, spatial structures which can constitute the storage of this sentence is huge - we're literally talking about which sectors it spreads across on the RAM/hard drive, the conventions of the language/file system, or which neurological structures in which part of the brain make it up. So the sentence itself is hugely abstract.)

Now WRT to infinities - can we work entirely with "the concept of infinity", without any reference to an actual instantiation of it? "Infinity", rather than being some divine, transcendent, and massive, is simply a definition of a concept, and how it relates to other mathematical concepts.

Of course arguably, this is exactly what we already do for everything in mathematics, not just infinities. An instantiation of "the concept of 3" is found in my brain, whether it's meaningful to talk about an instantiation of the number 3 itself (eg. in 3 apples sitting on my desk) is dubious, probably completely misguided.

Cheers,
-blowfly

*) Of course to discuss it, we're actually employing "the concept of (the concept of cat)", and I've now employed "the concept of (the concept of (the concept of cat))", ad infinitum or until we get bored.

[blowfly]

Hm, if I can ramble for a moment...

"Abstractness" is a property of concepts, not instantiations of the concepts. * (If "abstractness" is defined as spatial/structural variety among all possible instantiations then of course it has to be a property of the concept, not any individual instantiation.)

In which case I think I see your point - if "infinity" has no (known) instantiations, yet the definition of abstractness is wide spatial/structural variety of instantiations, then infinity isn't abstract.

Yet, "the concept of infinity" (literally the neurological/electrical structures making it up) can still be considered abstract, which I think is the important thing.

Not that mathematicians should be burdened with prefixing infinity with "the concept of" all the time, but philosophically it seems that there's nothing more to infinity than our conceptual representations.

Mildly confusing, I have to admit...

Cheers,
-blowfly

*) Thinking back to your terrific explanation of FOL and SOL: if we formalized all this in predicate logic, it seems we'd need SOL - is this right?

The relationship between a concept and an instantiation is basically a predication relationship. So if A = somewhat abstract, C = cat, and B= Bagheera, "A(C) & C(B)" would only be possible in SOL.

[vazscep]

Your thesis raises a few more questions for me. Firstly, you talk of the concept of cat as any set of symbol defining catness. I'd like to know more about such sets of symbols. Presumably, the symbol "cat" on its own does not define catness. So which symbols do and how many of them do I need?

Secondly, how do we get from a concept to one of its instantiations? What is the relationship between an actual cat and the set of symbols defining a cat?

I only ask in case we're just replacing one mystery (abstracta) with another, in this case symbols (another example of abstracta) and their instantiations.

PS: If we identify concepts with predicates, then you are right that talk of concepts could be done in second-order logic. We could also use first-order set theory and identify a concept with its set of instantiations.

[dahduh]

Is it possible to pin down the idea of 'abstract' more precisely? For example, let F represent an abstraction such that for any instantiation x, F(x) = 1 if x is a member of that abstraction and zero otherwise. Then, you will have a set S = { x : F(x) = 1 }. S will have a cardinality, and you could use that as a measure of 'abstractness'.

But this can't be the full story. The function F: x -> 1 for all x will then be the most abstract of all abstractions, but that's not useful. What seems more interesting to me is the usefulness of abstractions. We as humans invent abstractions because they serve to summarize some essential feature of what we are manipulating, while discarding all the irrelevant features. So when we speak of 'cloud' we mean a nebulous substance in the upper atmosphere that blocks sunlight; it's shape, color or even composition is not important (thus we can say 'cloud' when we see smoke).

So blowfly, when you speak of 'wide' and 'narrow' are you actually trying to get measure of the generality and usefulness of a powerful abstraction (like numbers), as opposed to an abstraction that is maybe not so powerful or general?

[blowfly]

Sorry for the absence, very busy recently.

Your thesis raises a few more questions for me. Firstly, you talk of the concept of cat as any set of symbol defining catness. I'd like to know more about such sets of symbols. Presumably, the symbol "cat" on its own does not define catness. So which symbols do and how many of them do I need?

Let's assume the string of letters "cat-concept" (the symbol) represents the set of properties "small furry animal, pointy ears, long whiskers, chases mice, looks like image X" (the semantics). It's easy to find exceptions (if you cut off your cat's whiskers, it's no longer a cat!?), but this doesn't stop this set of properties being roughly the intuitive cat-concept we learn as we grow up. *)

If necessary, we could be even more precise and say "a living organism whose DNA matches 99.99999% of the following string..."

Or ultimately, we should talk in terms of "how cat-like" instead of "cat or not-cat" - eg. "the percentage of DNA matching this string determines how cat-like the organism is".

Secondly, how do we get from a concept to one of its instantiations? What is the relationship between an actual cat and the set of symbols defining a cat?

In my view, the relationship is descriptive, ie. the relationship between "cat-concept" and "that thing over there" is that you and I are using "cat-concept" to describe/label "that thing over there". So the relationship is only possible within the context of cognition, and it's the natural result of memory and empirical data.

I only ask in case we're just replacing one mystery (abstracta) with another, in this case symbols (another example of abstracta) and their instantiations.

There's a certain fuzziness about the exact semantics of a symbol (must cats really have whiskers?), and there's also an element of circularity (symbols defined in terms of other symbols). Also, the relationship between a concept and an instantiation is probabilistic ("I'm pretty sure that's a cat", or "it's 99.99% cat-like"). I can't see a mystery here, just an acknowledgment that our internal library of concepts is typically a bit of a mess.

*) The goal of all this isn't to provide a rigid foundation for clearly defining concepts. I think of concepts literally as bundles of neurons or bits, and there's nothing particularly neat about them.

Cheers,
-blowfly

[blowfly]

Dec 4, 2009 19:40:01 GMT dahduh said:

Is it possible to pin down the idea of 'abstract' more precisely? For example, let F represent an abstraction such that for any instantiation x, F(x) = 1 if x is a member of that abstraction and zero otherwise. Then, you will have a set S = { x : F(x) = 1 }. S will have a cardinality, and you could use that as a measure of 'abstractness'.

Assuming "x" represents a physically distinct instantiation, then yes, the cardinality of S would be a measure of the abstractness of F.

(Remember, I suggest abstractness relates to the variety of possible physical instantiations, rather than number)

But this can't be the full story. The function F: x -> 1 for all x will then be the most abstract of all abstractions, but that's not useful.

(Useful? What does useful have to do with any of this? )

The actual concept of "F: x -> 1 for all x" would be highly abstract, in that it can be instantiated in wide variety of physically distinct structures (neurology, circuitry etc), though no more abstract than anything else in mathematics.

What seems more interesting to me is the usefulness of abstractions. We as humans invent abstractions because they serve to summarize some essential feature of what we are manipulating, while discarding all the irrelevant features. So when we speak of 'cloud' we mean a nebulous substance in the upper atmosphere that blocks sunlight; it's shape, color or even composition is not important (thus we can say 'cloud' when we see smoke).

Sure. The only use my suggested definition of "abstract" has is to clear up a few other things in philosophy. Abstractions themselves are useful to us for exactly the reasons you mentioned.

So blowfly, when you speak of 'wide' and 'narrow' are you actually trying to get measure of the generality and usefulness of a powerful abstraction (like numbers), as opposed to an abstraction that is maybe not so powerful or general?

Certainly not. "Wide" and "narrow" varieties of possible physical instantiations indicate purely a measure of abstractness, not usefulness, or power, or generality.

(Actually maybe "generality", depending on what we mean by this.)

Cheers,
-blowfly