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Thu, 17 Dec 2009
In my past writings I have had a lot to say about the reality of abstract objects. I have argued that these entities are real, though not in the same way that concrete things (like tables and chairs) are real. This is one of the standard philosophical positions about abstract objects . One of the main lines of argument against this position is that accepting abstract objects adds unnecessary new things to the world. (In other words, accepting abstract objects supposedly violates Occam's Razor.) I do not believe that abstract objects pose any threat of this sort. The claim that abstract objects exist tells us little beyond what we already know when we say that objects have properties or relationships. See my earlier writings (here, here and here) for more about these ideas.
In this post I'd like to explore another, far more daring question about abstract objects: Might everything be abstract?
The speculation that everything might be abstract has precedents in philosophy. One precursor is the idea that the world is basically mathematical. This goes back to Greek philosophy (especially the Pythagoreans). The idea that the world might actually be a mathematical object occurs in modern times . At first glance, the possibility that everything might be abstract seems implausible. How could everything be abstract when there's this solid, obviously concrete world around us? How could physical objects be abstract when abstract objects seem to be placeless, timeless, and devoid of the ability to cause events? (These are negative features that philosophers often attribute to abstract objects .) Also, how could everything be abstract when abstract objects are mainly just features of concrete objects? Wouldn't there have to be some concrete objects to begin with?
These worries become less pressing if we can begin to overcome the habit of picturing all abstract objects as intangible, ethereal, or not quite real. The worries might lose force even more when we explore the relationship between ordinary physical objects and their properties.
Before starting this exploration I'll make a few preliminary remarks about abstract objects.
If we accept the reality of abstract objects, then our picture of reality has room for many different kinds of existence. Tables and chairs are real, but so are properties like shapes and colors. Stars and galaxies are real, but so are relationships like being more massive than and being hotter than. Atoms and molecules are real, but so are mathematical items, like the set of all atoms in a DNA molecule (not the same as the molecule itself!) and the number of atoms or nuclei in a hydrogen molecule.
Mathematical logicians often think of abstract objects as forming systems of "logical types," or domains of abstract objects of different levels. For example, we might take physical objects to have logical type 0. Then a property of physical objects (like rectangularity) is of logical type 1. A property of a property of physical objects (like the property of being a shape property) is of logical type 2. And so on through type 3, 4,.... There also can be other types not in this series, such as types of relations. Logicians sometimes visualize an endless tower of logical types starting with the world of concrete things. Set theorists use a more flexible idea of levels, but the core idea is the same.
Often it's convenient to think in terms of towers of types, but we shouldn't get stuck thinking that every abstract object has to belong to one of these types. We need to keep an open mind and consider other possibilities - like sets that are members of themselves, or properties that are properties of themselves, or perhaps even two sets that are members of each other. Some logicians and mathematicians study things like these, mostly under the banner of "non-well-founded sets."
With these preliminaries in mind, I'd like to ask a key question: Where is the dividing line between concrete things and abstract objects?
I tried to answer this question in a talk called "Abstract Objects and Physical Reality," which you can find in a book of mine called The Unfinishable Book. (As of the date of this post, the book is downloadable for free - where "free" means "free except for the usual internet charges.") The gist of my answer is that there is no uniquely determined boundary between concrete objects and abstract objects.
This idea is not really new. Both Carnap's Aufbau  and Quine's thesis of ontological relativity  recognized that the choice of a domain of "concrete" entities might not be unique. However, what I'm proposing here is different from these earlier ideas. I am not embracing Quinian ontological relativity, and I am not proposing to use logical constructs, Aufbau-style, as substitutes or ersatzes for anything. I am only proposing that there is no unique domain of objects which alone are objectively "concrete."
Here is a brief introduction to the argument.
First, a bit more background on abstract objects. Concrete physical objects seem to be very different from abstract objects. However, when we begin to analyze physical objects, we find out that a physical object is an item that unites, or joins together, several properties and relations. A physical object has properties and relations which determine what the object is like. A physical object without properties and relations would be essentially "nothing." The most useful thing we could say about such an indefinite object is that it is able to hold several properties and relations together.
If we mentally distinguish a concrete object from its properties and relations, what is left of the object? Almost nothing! There would be only the factor that unites the properties and relationships.
Some philosophers have held that this uniting factor is a "bare particular" - a sort of simplified concrete object that does little more than hold together its properties and relations. Other philosophers (the "bundle theorists") deny bare particulars and view the uniting factor as something abstract, like a class.
Now here is the key insight behind my answer: It's hard to believe that a "bare particular" really is a concrete object. A bare particular looks more like a special property, shared in common by the properties and relations that it unites. I would argue that there is no significant contrast between a bare particular and a special property of this kind. Thus, if there is a bare particular, a concrete object is, at bottom, an abstract object. If there is no bare particular (and bundle theory is right), then a concrete object is, at bottom, abstract too. Either way, a concrete physical object can be analyzed into a combination of abstract objects of some kind.
If this view of existence is true, then there is no strong dividing line between the concrete and the abstract. The difference depends on where you begin your analysis. If you take concrete physical objects as the starting point, and don't try to analyze them into entities of other kinds, then you get the usual picture of abstract objects. You find that there are concrete objects, and then there are the various properties, relations, properties of properties, etc. of those objects. However, if you start your analysis with entities usually called abstract, you can portray concrete objects as properties or classes of them.
An obvious (but weak) objection to this view is that it can lead to circles of attribution, in which a physical object is a feature or class of abstract objects, while abstract objects are in turn features of physical objects. I don't think we should worry about these circles. They aren't vicious. They are no more illogical than the non-well-founded sets that I mentioned earlier. We can always postpone the circle in practice, by taking some fixed domain of objects as "concrete" and building up from there. The fact that everything is analyzable into other items doesn't make anything less real.
Another obvious objection is that physical objects have features that abstract objects don't have. For example, physical objects have spatial and temporal locations and causal powers - features that philosophers often deny to abstract objects. I don't think this objection is fatal either. Bare particular and bundle theories have to face this objection too. (Where is a bare particular located, when its spatial location has been stripped away along with its other properties? The "bundle" in bundle theory is an abstract object such as a class; how can it "be" a locatable, causality-ridden physical thing?) If this objection is not fatal to those theories, then it is not fatal to what I am proposing here.
For further details of the above argument, read Talk #9, called "Abstract Objects and Physical Reality" in The Unfinishable Book.
What, then, is the answer to our initial question, "Might everything be abstract?" The answer is "yes - in a way." Any given thing might be analyzable in a way that shows it to be an abstract object. However, that wouldn't prevent us from labeling some objects as concrete and using them as a "ground floor" for defining further abstractions. Normally we use physical objects as the concrete objects, but do we have to? Physical objects are the objects that seem most concrete to us - but perhaps that's just because we can detect many of them through our sense organs, which also can be analyzed as abstract objects if we have the nerve! (Pun intended.)
Concluding Cryptic Remark:
And now, for something really strange to think about.
Normally we equate the "world" to the world of concrete physical objects. We tend to regard other entities as mere features of that world. To borrow a term from mathematics, this amounts to taking a kind of "section" through the universe - picking out a preferred set of objects and treating them as basic. Usually the entities we treat as basic are the concrete physical entities. We regard them as the basic "world," with all other things mere features of that world.
What if we broke away from this unnecessary practice? What if we took a different "section" by choosing some different family of abstract objects as basic? What if we regarded all other entities, including our familiar physical objects, as mere features of those other objects? What would the resulting view of reality be like? Can we even do this without inconsistency?
This proposal sounds a bit like certain relativism-soaked ideas in philosophy - I'm thinking especially of Quine's ontological relativity thesis. However, it's not the same, because my proposal begins with a domain of real entities. All the abstract objects really exist, each in its own way. What is "relative" is the classification into abstract and concrete.
A conjecture: I suspect that if we began with data structures in the human brain as basic objects, we might arrive at a picture of reality in which the "basic" entities are experiences or their contents, and the physical world is a system of features of experiences or of their contents . This would be a form of metaphysical idealism. However, this idealism would be fully compatible with naturalism, because both these viewpoints are just alternative analyses of the same world. See Chapter 13 in my book From Brain to Cosmos for my early thoughts on the possibility of a naturalistic idealism.
End of Cryptic Remark.
 The position that abstract objects are real is called ontological realism or platonism. My version of it is a modest version.
 Rudy Rucker wrote about the idea that everything is a set. See pp. 200-201 in Rucker's book Infinity and the Mind: the Science and Philosophy of the Infinite (Boston, Basel and Stuttgart: Birkhäuser, 1982).
 See the following article: Rosen, Gideon, "Abstract Objects", The Stanford Encyclopedia of Philosophy (Fall 2009 Edition), Edward N. Zalta (ed.), URL = [http://plato.stanford.edu/archives/fall2009/entries/abstract-objects/].
 Carnap's Aufbau exists in several editions, including the following: Carnap, Rudolf (1928). The Logical Structure of the World. In The Logical Structure of the World; Pseudoproblems in Philosophy, trans. R. A. George (Berkeley: Univ. of California Press, 1969).
 Quine, W.V.O. Ontological Relativity and Other Essays (New York: Columbia University Press, 1969).
 This seems reminiscent of what Carnap tried in the Aufbau, but actually it's different from the ground up (pun intended again). We aren't substituting logical constructions for anything. There is just a domain of objects, and there are different ways of classifying them.
posted at: 23:58 | path: /ontology | persistent link to this entry
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