Monday, November 14, 2005

What is an abstraction?

This question occurred to me as I was reading a little more about the supposed platonism revival mentioned in the previous post, and alluded to by Steve Esser in a couple of recent posts. Platonism makes what I think to be the "classic" (so to speak) mistake of reifying at least certain kinds of abstractions (mathematical and geometric, principally, but also possibly property-like) -- that is, it makes "things" out of abstractions, and then, having done that, it has the problem of determining the ontological status of the things, including where they exist, etc. Without going any deeper into platonism itself at this point, I'll just say that I don't think abstractions as such, of any kind, are things at all, and so have no ontological status, special realm, etc., to worry about. Apart, that is, from their existence as concepts, or psychological/cultural constructs, in which sense they are part of each individual's cultural imprint and exist as physical states in each individual's brain. Thinking about that, however, made me wonder about how such concepts are formed in the first place, and what that might say about the nature of abstraction itself?

For example, consider what happens when a child is learning to speak (or, for that matter, when an anthropologist is learning a language in another culture) -- someone points to something and utters a word, "dog" say. The pointing is an action that directs or focuses attention, but there's no indication from that alone what the utterance is supposed to "mean" -- within the field of attention, it might refer to a particular object (a proper name), a type of object, a property of an object, or a behavior or a process. In the early stages of language-learning, in fact, these distinctions themselves would be meaningless, and the very notion of an "object" might be unclear (though this might also be hard-wired in some fashion, giving rise to "natural objects", e.g., mama and dada). But after repeated pointing-acts accompanied by the same utterance, it's noticed that there is something in common in the phenomenal field to which attention is directed, and a generalization is made and tested -- the child herself does the pointing action and utters the sound whenever a dog-like object presents itself, and then looks for confirmation. But this "noticing" can't be a simple observation, it has to involve some kind of structuring -- that is, the repeated uttering of a sound in different circumstances generates, in a sense, a commonality to those circumstances. This is because the elicited structure will almost certainly be wrong initially ("wrong" as defined by the ones doing the pointing) -- "dog" will need to be distinguished from "cat", for example, or "squirrel" (or maybe "pillow"), "red" from "orange" or "pink" (or "ball"). So it isn't that the sound simply links to, or labels, a pre-existing structure, it's more like the pointing-and-uttering behavior as a whole has forced experience to take on a structure. And then, in an iterative and essentially social process of generalization and distinction, the structures will be aligned -- the "meaning" of the utterance will be brought into semantic harmony with everyone else (a process that continues all through life, though not with such major adjustments).

After some initial concepts have been formed in this way, more sophisticated abstractions of abstractions can be formed, so that "animal" can be used for both dog and squirrel (but not pillow), or "color" for both red and orange (but not ball), etc. And at some point after that, it starts to become possible to use words themselves to shape experience -- generalizations and distinctions can be made into verbal rules, and both communication and thought can come into being. But initially, and essentially, an abstraction is just a named common feature, element, or aspect of experience -- experience that may include other, already-formed abstractions. This sort of ability to create a semantic structure out of experience is also a linguistic capability just as syntax is, and its potential must be as built-in or hard-wired.


At 11:50 AM, November 15, 2005, Blogger Steve said...

It's fine if you disagree with the platonist arguments, but I think it's important to emphasize that the platonist would find little or nothing to disagree with you about regarding the normal process of concept formation. The point of abstract objects (if they exist) is that we arguably have evidence for their existence independent of our cognitive or psychological processes. When in mathematics (the paradigm example) we discover these objects, they (by hypothesis) outrun any person's individual ability to distill them from the physical environment or other psychological elements the way we typically do in concept formation.

At 1:11 PM, November 15, 2005, Blogger Ellis Seagh said...

Yes, the bulk of this post was concerned with "normal" concept-formation, as you say, and didn't directly address the issue of platonism -- other than to say that I thought its central mistake is a form of reification (and that that then leads to other ontological and epistemological difficulties).

To say just a little more about that, though, I'm not sure quite what you (and others) mean by the claim that mathematical objects "outrun" our ability to "distill" them from our experience (though I like the phrasing)? Clearly, in "discovering" (I would say creating) such objects in the first place, we have the concept of them in our heads. And the deductive procedures that are followed in such discoveries also occur in our heads. True, some of the deduced properties of said objects -- e.g., infinite sets, very large numbers -- would indicate that the objects themselves wouldn't "fit" in a finite space, much less our heads, but the anti-platonist, after all, is arguing that the objects themselves don't exist, and that the concepts of such simply consist of concepts of their properties (formed out of the deductive processes that lead to them), not of the (imaginary) objects themselves. But perhaps I'm missing the point?

At 11:55 PM, November 15, 2005, Blogger Unsane said...

If you look at consciousness as a kind of system with categories which also incorporates an evolutionary "feedback" mechanism to inform us whether we are on target or have made a mistake in our categorisation of any phenomenon, then there is no reason why reified and cultural phenomena should not also be processed in the same way as more concrete and material "objects" are. Where do these categories come from and are they in any way consistent from culture to culture? I suspect there is a large element of local environment adapatation and that they overlap but are not broadly consistent from one culture to another. This implies a materialist (local adaption) base, rather than an Idealist one.

At 6:43 AM, November 16, 2005, Blogger Ellis Seagh said...

Thanks for the comment, Unsane, and I largely agree with it. I think the evolutionary feedback, as you put it, comes from two sources: communication (which tends toward semantic alignment), and experience, or the practical results of using the concepts in thinking and planning.

I had a number of posts earlier on the idea of "culture" (here's a recap if you're interested), and hope to have more soon. Among the questions I want to get to is, how do we know where the boundaries of a culture are -- i.e., how far does one culture extend? My answer brings up that useful idea of a "fractal" again.

At 11:26 PM, November 16, 2005, Blogger Unsane said...

Thanks -- I wrote some comments on that link. I get most of my cues from Antonio Damasio and from my own life's experiences as a migrant trying to adjust.

At 7:52 AM, November 17, 2005, Blogger Steve said...

Here are some questions to try to "pump" intuition about the difference between abstract objects and other concepts.
1. Would mathematical statements be true if there were no humans?
2. Is it math's status as a cultural convention that allows us to refute someone when he or she makes a mathematical error? If it's just a concept, who are we to say someone's concept is in error?
3. You mentioned our logical operators: where does our logical deductive ability come from, anyway? Is it just a product of psychology and cultural convention also?

At 10:21 AM, November 17, 2005, Blogger Ellis Seagh said...

To Jennifer (Unsane): Thanks for the comments -- I wrote some responses.

I know of Damasio but haven't read him (am interested in his apparent opposition to the mind-body split, though). And life's experiences, of any sort, seems like one of the best sources for cues (or clues).

At 11:12 AM, November 17, 2005, Blogger Ellis Seagh said...

To Steve: good questions, thanks. Here are my responses:

1) If there were no mathematical minds (i.e., no humans, aliens, or robots capable of forming mathematical concepts), then who would be making the statements? If the statements themselves are presumed to exist independently of any minds, that kind of begs the question, doesn't it? If there are no statements, then to my mind the question of their truth or falsity really doesn't make sense.

I don't mean this as a quibble -- I think mathematical truth is much more about internal consistency than about an "external" world in the first place, so if there were no minds to form propositions there really isn't an issue of their mutual consistency or "truth". Without minds, in other words, mathematics is neither true nor false -- it simply doesn't exist.

2) Cultural conventions of any sort are social phenomena, not individual (the individual cultural imprint is installed, developed, and maintained through social, communicative processes). So, first, "we" are the authorities on the convention, not any one individual. And second, mathematical conventions are the most abstract and "well-defined" of any conventions, and because of that are best able to cross particular cultural boundaries and stand as general cultural conventions.

3) I do think our deductive abilities are "a product of psychology and cultural convention", yes. I think, in general, that mathematics is just a particular kind of "language-game", played with objects or tokens that we invent, defined as we please, and played according to rules or "axioms" that we make up. Some objects, definitions, and rules are more useful than others. And some of these conventions are very old, and so embedded in our practices that it can seem as though they're written into the nature of things -- e.g., the number system. But this is really just a cultural invention whereby arbitrary names are associated with a process of pointing at similar but distinct objects -- "counting", in other words. It happens to be one of the more useful inventions, but is no less an invention.

And similarly, I think, for the rules of logic, which ultimately are very long-standing, practical rules about relating propositions to one another. From time to time, some of these rules are broken or changed or added to, and this may or may not lead to productive -- i.e., useful -- developments.

Well, sorry to be so long-winded. Let me end, though, by trying to reverse the flow for intuition pump #3: where do you think our deductive ability comes from if it's not, ultimately, a product of psychology and cultural convention?

At 1:30 PM, November 17, 2005, Blogger Steve said...

Well, the main thing I'm arguing for is that the elements of logic and mathematics are necessarily instrinsic to reality but extrinsic to the contingent nature of humans and our culture. In your first answer, you intriguingly use the phrase "no humans, aliens or robots". I think any creature which independently evolved a sufficient mind would discover the same logical and mathematical statements. Therefore there is something independent of us which makes these statements true.

The tokens we use are arbitrary conventions, but the logical and mathematical relations are not arbitrary and would be replicated no matter how we "replay the tape" of the cosmic evolution of the universe.

At 5:01 PM, November 17, 2005, Blogger Ellis Seagh said...

Steve: The tokens we use are arbitrary conventions, but the logical and mathematical relations are not arbitrary and would be replicated no matter how we "replay the tape" of the cosmic evolution of the universe.

I take your point. I think my own view would largely coincide with yours up to a point, but probably for different reasons, and would diverge after that point. That is, I think aliens, robots, etc., would likely also use operations like counting and concepts like sets, but simply because such things are basic and useful, not because the abstractions themselves are "intrinsic" (in any non-trivial sense) to reality. And beyond those basic and likely common abstractions, I, at least, would be very surprised if a "replay of the tape" would replicate even something as simple as i, not to mention some of the real esoterica of contemporary mathematics.

All in all, an interesting debate, but I do think it points to deeper or underlying issues, on both sides -- e.g., the issues surrounding physicalism or naturalism more generally.

At 7:33 AM, November 18, 2005, Blogger Steve said...

Thanks for the opportunity to debate. One add'l comment I can't resist: as far as I can tell i and complex numbers are integral to quantum mechanics. If QM is true of the universe, then we can't do without good old i.

At 9:34 AM, November 18, 2005, Blogger Ellis Seagh said...

Thank you for the debate, Steve -- it's been a pleasure.

And I don't mean to be trying for the last word here, but I also find your last comment irresistable. i certainly has its uses, in QM and elsewhere -- but do you want to say that any abstraction that's useful is an "abstract object", intrinsic to reality? Or only some? Or are some only "sort of" intrinsic and sort of pragmatic? (Is there a difference between saying that a concept is "useful" and saying that it's "intrinsic to reality"?)

Here I think we see some of those larger or underlying issues in back of the platonism/physicalism debate -- I may have to write a post on this at some point.

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