The natural objects that I propose to consider are broadly those physical objects which are studied and referred to by science and by a common sense view, informed by science, of the world. Natural objects are, philosophically speaking, individuals; they are involved as units in dynamic, causal processes. I shall draw a distinction between natural objects and the abstract objects of mathematics, in particular set theory.
Natural objects encompass atoms and molecules; cells and organisms, including you and me; the objects of everyday life such as chairs and automobiles; nations, continents, ecosystems, mountain ranges, geological faults; planets, stars and galaxies. Each natural object, when regarded internally, is a dynamic system with various interacting parts and components (some of which may be natural objects in their own right); when regarded externally, a natural object acts as a unit with respect to a larger system or systems (which may again be natural objects) of which the given object forms a part or component.
It is sometimes argued that objects such as atoms or galaxies are theoretical constructs, as much so as mathematical objects (or even, according to some, more so). It is true that any reference to an object rests on epistemological assumptions. The approach here will be not to belittle these important epistemological questions but to leave them aside, and accept as a working assumption the practical viewpoint of people who are dealing with the world: that natural objects exist, act, and are acted upon, independently of the observer -- although any description of them or of their actions is dependent on the describer.
Scientific theories refer to natural objects as well as to abstractions of them; they also refer to and quantify over other entities, in particular, the real numbers, points in space, time, and other mathematical spaces such as vector spaces, Hilbert space, etc. -- these are abstract objects which are not necessarily abstractions of any natural objects.
Thus the ontology of natural objects may be said to be more restrictive than, say, a Quinesque one. My point, however, is again not to dispute the validity of this or that ontology, but rather to describe a level of existence. No judgment will be presumed about the question of what ontological commitment should be made to abstract objects: rather, our focus of attention will be elsewhere.
In the examples above and in what follows, I concentrate on physical objects. I do not mean to rule out psychic or spiritual or intellectual objects. You and I, as objects, have these aspects; French society, Indian classical music, the theory of natural selection, King Lear, King Lear, the collective unconscious may all be regarded as natural objects. Their ties to physical existence (in the sense of matter and energy) or to extensional (spatial) existence are various, and there is scope for varied theories about their ontological status; I tend to follow Nicolai Hartmann (1949) in assigning to spiritual and psychic strata "categorial dependence" on the strata of physical existence in spaces: thus these objects are tied, however indirectly, to the physical level of existence. What I have to say will apply to such objects, in so far as they are acknowledged to exist; and even to purely spiritual or psychic objects not tied to the physical level of existence at all, if they are admitted to exist. But I shall keep largely to physical and biological examples as being less problematic. This will also insulate the discussion from problems of consciousness and free will which inevitably come up but are not my concern here.
Thus the natural objects to be considered here may correspond more or less to the "natural bodies" of Francis Bacon: "Toward the effecting of works, all that man can do is to put together or put asunder natural bodies." (Bacon 1620, 39)
An interesting example of a natural object is a hurricane. It is an open system, exchanging matter, energy and information with its environment, and acting as a whole. It is certainly accorded the status of an individual, in everyday language and in meteorology, even to the extent of being given a proper name (such as Floyd, Mitch, Zeb or Babs). It cannot be precisely identified with any set of atoms or molecules; and any boundary in space or time that you try to draw for the hurricane will be arbitrary. Meteorologists will draw such a boundary -- say, the point in time when a tropical storm is upgraded to a hurricane -- according to conventions about energy content and wind speeds, but the passing of a boundary is merely the employment of a convenient yardstick and is not supposed to have absolute significance.
Natural objects may usefully be described as systems. Von Bertalanffy (1968) defines a system as "a complex of interacting elements" (although von Bertalanffy's elements are not to be equated with elements in the set-theoretic sense, and to avoid confusion I use "components"). He goes on to study systems defined in terms of partial differential equations. Mesarovic and Takahara (1975) develop a more general, set-based concept of system. They are able to define such notions as cascading, feedback, and open systems.
Natural objects can be named, or
they can be defined by ostension. Often, naming and pointing will be essentially
the only ways to refer unambiguously to a particular object.
2. Natural and abstract objects
The words "a logic" of the title are intended to suggest the aim of depicting how we reason or infer about natural objects. Set theory, as a foundation for mathematics, is a way of doing this for abstract objects. Before examining natural objects more closely, I wish to compare them with abstract objects -- which I shall identify, for concision, with sets -- in order to bring out some properties relevant to both.
Cantor (1895) introduced the concept of set as "any comprehension into a whole M of definite and separate objects m of our intuition or our thought", and the modern development of set theory has retained the spirit of Cantor's conception, in particular through the fundamental role of the Comprehension Axiom. Even in the limited form necessary to avoid the paradoxes, Comprehension formalizes the grouping together of objects with like properties into a single entity. The objects -- sets, classes -- of set theory are abstractions, the product of our mental processes ("our intuition or our thought") which impose an organization on the world we perceive. Except under a naive Platonist interpretation, sets are not required to have an autonomous existence outside the formal system that creates them: that is to say, outside the language of the observer. (The same goes for all other mathematical objects, including categories.)
Thus set theory may be said to be a part of epistemology, the study of our knowledge of the world. Following a hint of Graham C. D. Griffiths (1974, 87) I wish to contrast this with ontology, the study of what things exist, independently of any observer. From a scientific perspective, objects such as atoms, molecules, organisms (including persons), stars, galaxies may be said to have an objective individual existence -- although there are borderline cases where the object's existence as a separate individual may be questionable, and the individuality of, say, a quark is of a very different nature from yours or mine. On the other hand, sets of objects, such as the set of all stars, or an arbitrary collection of, say, 100 people chosen at random, can exist as individuals only in an abstract sense (though the former example may be viewed as a natural kind). In addition, the "definite and separate" character of the elements of a set is not always found in nature where boundaries are generally ill-defined and separation may be relative and time-dependent. Natural objects interact, are born of others, and die.
To bring out how natural objects in general differ from sets, let me pause to consider the nature of one particular object, myself, whose autonomy and continuity I feel so clearly (but proceeding in a different direction from Descartes). Can I precisely be identified with a set? What about the set of cells that contain my unique DNA? This set provides a good approximation to my physical extent, but there are many problems with identifying "me" with this set:
I am not trying to attack set theory as such. I simply wish to note that the common mode of talking about the world leaves philosophical doubts aside and names natural objects; that the language of science is full of them (as well as of abstract objects); and that set-theoretic language does not capture the nature of natural objects, as of course it was never intended to. It therefore seems worthwhile to try and develop a theory of natural objects.
As Jody Azzouni puts it (1994, 4), mathematical objects are metaphysically inert. Natural objects, by contrast, interact intensively with the rest of the world, and this interaction in fact cannot be separated from their nature. But great care will be needed because once a mathematical framework for referring to these objects has been established, the referents themselves will be mathematical objects. Rigorous separation of levels of discourse will be necessary -- but this is nothing new in logic and metamathematics.
Sets are abstracted (from the Latin: drawn away) from direct physical experience. A common example frequently seen in expositions of elementary (naive) set theory illustrates the abstract nature of the set concept by explaining how a collection of disparate objects can be grouped together into a set merely by enumerating them or by specifying a common property -- no matter how incongruous and varied their natures or how widely separated they are in space and time. This kind of construction is counter-intuitive to a student who is not accustomed to thinking in terms of abstractions; the natural objects one is used to dealing with in less abstract contexts do not have these properties. Even the Pair Set Axiom does not hold, in general, for natural objects. A pair of objects is not, usually, a single object (although in special cases one will consider certain pairs as a unit; for example a married couple is an object in legal considerations). The same goes for other elementary set-theoretic constructions such as unions and intersections.
The distinction between individuals and classes is relevant here. (Some philosophers talk of natural kinds; others oppose individuals to "sets".) This distinction, which goes back to Plato and Aristotle, was blurred by Cantor: his sets are both individuals and classes. Indeed, the principle of Comprehension is a formalization of the notion that a class of objects is defined, as an individual, by a common property that they share. The subsequent development of set theory reinstated the individual/class distinction in a different form with sets and proper classes; this happened as early as Cantor (1899). Different versions of modern set theory (Zermelo-Fraenkel, von Neumann-Bernays-Gödel) vary in the ontological status accorded to proper classes, as do different interpretations of the formalisms. However, the basic point here is that sets are essentially collections ("multiplicities" -- Cantor) treated as individuals. A natural object is never merely a collection; the interaction of its components is an essential part of its individuality.
It is perhaps of interest to note that Ernst Zermelo, in his fundamental paper (1908) on axiomatizing set theory, seemed to be open-minded about whether all objects are sets: "Set theory is concerned with a domain B of individuals, which we shall call simply objects and among which are the sets." (Emphases in original.) For Zermelo, the membership relation is fundamental; sets are defined in terms of this relation, as objects which have elements, so that a special case is needed for the empty set. So sets are logically posterior to, not prior to, the fundamental binary relation. But unlike later authors Zermelo does not rule out the possibility that some objects are nonsets. Of course the nonsets which he had principally in mind were atoms or urelements; these have continued to play a role in set theory, but only in connection with the membership relation. Other possible relations or properties of urelements are not considered. Natural objects may be urelements -- the raw material for forming sets -- for example, the set of all hurricanes in 1999 may be used, and abstracted, in a statistical analysis. What interests me here is not the properties of natural objects qua urelements, but what other properties they may have, orthogonal as it were to their set-theoretic ones, because not expressed in terms of the membership relation.
As a first approximation, let us suppose that an appropriate language for natural objects has a binary relation , analogous to the fundamental binary relation ("x is an element of y") of set theory, but stating rather "x is a component of y". This relation will have very different properties from . In fact partakes of some of the properties of natural objects: it changes through time and space. Its properties in the subatomic and the macroscopic realms diverge. But we can certainly see whether set-theoretic ideas apply to , exploring the correspondence, more than an analogy, between and .
As intimated above, basic set-theoretic axioms including Comprehension and the Pair Set Axiom do not hold for natural objects under this -interpretation. There is something of a tradition of alternate set theories in which some of the standard axioms are negated. We may gain clues about reasoning with natural objects from what has been learned about reasoning with weak sets. One important line of work in this direction negates the Axiom of Foundation. This has been most successfully done by Peter Aczel (1988), building on work of Boffa (1969), Forti and Honsell (1983), Finsler, Scott, and others. Aczel's work in turn has had important applications in situation theory, theory of communicating systems, and elsewhere (e.g. Barwise 1989, Barwise and Moss 1998). Aczel has built a universe of set theory, extending and not supplanting the standard universe of set theory, that provides a useful formalism for certain ways of looking at the world (e. g. situation theory).
The Axiom of Foundation was historically
one of the last of the standard axioms of set theory to be added to the
canon. The Axiom of Choice was introduced earlier (Choice: Zermelo 1904;
well-foundedness and Foundation: Mirimanoff 1917, Skolem 1922), but its
separate status was recognized from the start, and has, of course, been
a central theme in 20th century set theory. Other authors have more recently
examined the independence of some of the core axioms of set theory from
weak base systems, thus bringing into question even these axioms that were,
unlike Choice, readily accepted. For example, Boffa (1972) proved an independence
result concerning the Pair Set Axiom; González (1992) showed the
independence of the Union axiom in Zermelo set theory, using a permutation
method; Zarach (1998) used forcing to show that Collection is not implied
3. Species and the indescribability
In this and the next section I turn from a comparison with set theory to motivating influences from the sciences. Problems of behaviour of natural objects, and of the constitution of an individual, have arisen in concrete situations, and I shall attempt to draw some working principles from these lessons.
An important case study in the investigation of natural objects is the work of philosophers of biology on the nature of individuals. This has arisen in large part because a set-theoretic framework has been found by many to be inadequate for the ontological status of taxonomic groups. In particular, the traditional conception, which goes back to Aristotle, of each species as a class has been questioned, first by Michael Ghiselin (1966; see also 1974, 1981, 1987), and then by many other authors (e. g. Holsinger 1984, Hull 1978, Sober 1984, Sober 1993, Griffiths 1974). They point out that to consider a species as a class, or as defined by a set of properties, is to impose on it a character that it does not have, and to miss the actual nature of species. Since Darwin we know:
This has the added advantage of unifying the biological hierarchy of entities: at each level -- cell, organism, population, species, etc. -- there occur individuals that are defined, not by Comprehension in terms of their elements (this is impossible), but as dynamical, changing systems whose behaviour can be characterized internally by interactions among their components, or externally by their interactions with their environment. Indeed, one can extend this hierarchy to the physical level(s) as well (molecules, atoms, . . . stars, galaxies, . . .) with the same remarks still holding (Griffiths 1974). Ghiselin (1987, 128) goes so far as to say that treating species as individuals opens "the prospect that we can develop a single body of knowledge for the entire universe."
Thus the practical pursuit of systematics in biology broadens into a picture of the universe as consisting of individuals, each individual being bounded in space and time but not (necessarily) contiguous or connected, characterized by a dynamical interplay of its components and its environment, and definable by ostension or by naming and not by Comprehension:
When anyone tries to find the "defining properties" of an individual, he is wasting his time. This is equally true for Homo sapiens, tellurian life, human language, French, and Noam Chomsky. (Ghiselin 1981, 283)Another biological science, ecology, similarly encourages us to view biological systems as wholes, with components that are dynamically interdependent (e. g. Levins and Lewontin 1980).
The notion of supervenience (Kim 1978) is a good candidate for formalization in any attempt to describe the articulation of the different levels of the hierarchy of natural objects.
An important principle suggested by all the above discussions is the inadequacy of any (finite) language to describe any natural object fully, other than by naming or ostension. Everyday life, and the example of "me", tend to confirm this principle; and of course it is fundamental to quantum mechanics as expressed in the Uncertainty Principle and complementarity, construed (as in the Copenhagen interpretation) as precluding a complete objective description of phenomena: I shall say more on this in the next section.
The principle of the indescribability
of objects provides a contrast between natural objects and abstract objects:
in set theory, the exact specification and construction of objects using
Comprehension and the other constructive axioms is essential.
A second principle is, in contrast, shared by natural and abstract objects. In fact it is suggested both by set theory and by recent work in cosmology. Cosmologists are facing more and more consciously the problem of describing everything. Set theory, among other branches of mathematical logic, has a similar ambition in that an entire universe is to be constructed -- not in this case the physical universe but the "universe of discourse" of mathematics, the aggregate of all the abstract objects studied by mathematicians. Incidentally, the habitual use of the term "universe" for the class of all sets is relatively recent. Zermelo (1908) merely spoke, as we have seen, of a "domain" of individuals; Russell and Whitehead in Principia Mathematica (1910) referred to the "universal class" V, and even Gödel in his work on V=L, which set the tone for all subsequent set theory, refers to the "universal class" and not the "universe" (Gödel 1940, 40).
The principle I am proposing here is the impossibility of describing everything. In set theory, it was necessitated by the early paradoxes and is articulated in a fundamental series of theorems, including those of Löwenheim-Skolem, of Gödel, and the related result of Tarski on the undefinability of truth (these are not confined to set theory). Even before these results, it was realized that sets have a way of spilling over any circumscription that one may attempt to put on them:
. . . quel que soit l'ensemble qu'on envisage (pourvu qu'il existe), des individus nouveaux surgissent, et un ensemble plus vaste apparaît nécessairement; on est bien en présence d'une extension indéfinie qui ne comporte pas d'arrêt ni borne. (Mirimanoff 1917, 48)Since Gödel's work, there has been what Akihiro Kanamori (1996, 46) calls a "cornucopia of models of set theory" (i. e. models of everything) and in fact these are a basic research tool.
In cosmology, the attention of some researchers has been focussed more recently on the problem of everything. It arises acutely when the universe (defined, perhaps, as what emerged from the Big Bang) turns out not to be everything: when in order to account for the observed properties of our universe, it is found that the best explanation is a theory incorporating the existence of many universes, of which ours is just one. Several modern cosmological theories call for multiple universes, including Lee Smolin's theory of the creation and natural selection of universes (Smolin 1997). Andrei Linde's "chaotic inflation" and "eternally self-reproducing universe" (e. g. Linde 1990) also call for many, perhaps infinitely many, universes -- if by universe we mean everything that we can in principle communicate with.
Smolin (1997, 14) has addressed the issues that emerge when attempting to describe everything:
The problem of how to make a theory of the whole universe is thus the problem of how to construct a theory without making any reference to anything that exists, or anything that we might have imagined happened, outside of the system we are describing.A related lesson is taught by quantum theory; thus Heinz R. Pagels (1982, 103):
Bohr's principle of complementarity implies that knowing everything at one time about the world -- a requirement of determinism -- is impossible because the conditions for knowing one thing necessarily exclude knowledge of others.The separation of observer and observed, in the Copenhagen interpretation, carries the same implication.
The "universe" of natural objects ought to be described in such a manner that there is no possibility to interpret it in terms of an observer outside the world. As in quantum mechanics, the observer is inextricably tied to the description; but unlike classical quantum mechanics (and as in cosmology), the observer is not outside the system being studied.
Interestingly, in both cosmology and set theory the intent is to construct or describe something called the universe, i. e., everything, but in both fields the practitioners have been impelled to construct many universes with different properties. Thus, belying the original meaning of "universe," one ends up with lots of them, and no particular universe is privileged. In the case of cosmology, this is the continuation of a long historical process, beginning with Copernicus, whereby the place of the human species in the cosmic scheme of things has been progressively found to be smaller, less central, and less important than previously imaginable.
One way that the impossibility of describing everything has been dealt with in mathematical logic has been to restrict the language (or the logic or the axioms) used to describe and specify mathematical structures; the price has been, among other things, the lack of uniqueness of set-theoretic "model universes" which are referred to from outside rather than from within. I believe that another way to address the problem is to undermine the distinction between language and reality.
By this I mean that, traditionally, there has been a sharp distinction within mathematics between the structures that mathematics describes (this is what I am referring to as "reality") and the "language" (in which I include logic and syntax) in which it describes them. But if the reality that one wishes to describe is to include everything, how can it be described in terms of a language specified in advance -- that is, "outside" the reality being described?
There is an another analogy with physics here: Newtonian physics was expressed in terms of a space and time that were absolute and prior to the physical universe. Newton himself realized this and had recourse to a metaphysical notion of space and time. Leibniz also realized it and made it a central part of his criticisms of the Newtonian cosmos, proposing instead a relational view of space and time. It was part of the achievement of Einstein to remove space and time from their privileged position as the "stage" on which the events of the universe are played out, and replace them by a concept of space and time as part of the fabric of the universe, acting and acted upon by matter and energy in a dynamical system.
Analogously, in "classical" logic one first specifies the language (syntax and rules of inference) and then constructs "universes" with this language as the stage-setting. A way to avoid the paradoxes and limitations to which this gives rise is a description in which the language is no longer separate from and prior to the object being described. (Barwise and Etchemendy (1987) use just such a strategy to deal with the Liar paradox, although the limitations of logic to abstract objects are not challenged. I comment further on situation semantics below.) Such descriptions would be impredicative -- physicists might call them non-local -- in that the specification of a single object would involve the whole universe or a substantial part of it:
That each singular substance expresses the whole universe in its own way, and that in its concept are included all of the experiences belonging to it together with all of their circumstances and the entire sequence of exterior events. (Leibniz 1686, 308)Mach's Principle may be regarded as an expression of this in dynamics.
In quantum theory, Bohr frequently noted the following corollary to, or instance of, complementarity:
On the lines of objective description, it is indeed more appropriate to use the word phenomenon to refer only to observations obtained under circumstances whose description includes an account of the whole experimental arrangement. (Bohr 1961, 73)This means, in fact, since it is not clear where to draw the line that separates the "whole experimental arrangement" (and the laws governing it) from the rest of the world, that the description of a phenomenon implicitly includes the whole universe.
Such impredicativity has the consequence that one cannot construct from scratch models of a universe consisting of "singular substances". This would be considered by logicians a drawback of impredicativity; but in the context of the problem of Smolin, it may be an advantage that not only are there no obvious models of the system which are describable from outside, but perhaps such models are in principle not possible.
Situation semantics (Barwise and Perry 1983; Barwise 1989) is a different kind of attempt to address the problem of the impossibility of describing everything in the context of the analysis of natural language. The approach of viewing utterances and inferences as situated activities differs from the present one in that it is epistemological rather than ontological. It "shifts attention from truth preservation to information extraction and information processing" (Barwise 1989, xiv; emphases in original). Furthermore, its syntax presupposes the existence and discreteness -- the abstractness -- of objects. Ian Hacking (1972, 148), discussing Leibniz's notion of individual substance, makes an important point:
Which bundles [of qualities] are substances? Only those bundles that are active, in the sense of having laws of their own. Laws provide the active principle of unity. There is a tendency in much analytic philosophy to conceive things as given, and then to speculate on what laws they enter into. On the contrary, things are in the first instance recognized by regularities. (Emphasis in original)Thus a logic of natural objects will put priority on addressing the question: what makes an individual an individual? As Hacking points out, Leibniz shares with Berkeley the view that substances are bundles of qualities; the important question, as in the quote above, is the converse.
With the principle that each object (or singular substance, or thing, or individual, or phenomenon) irreducibly reflects its entire surroundings, we have come to a fusion of the first two principles mooted: the indescribability of the individual and of the universe. In combination with either of these, it implies the other.
In Leibniz's case, this third principle is a consequence of the principle of reason (or of "predicate-in-notion") -- as is another principle, that no two individuals are exactly alike. The identity of indiscernibles has not received attention here but is indeed logically linked to the principles adduced above. The principle of reason does not stand up to scrutiny in the light of contemporary logic, but some of its consequences may be recast for our use.
In the Novum Organum Bacon (1620, 41) criticizes Aristotelian logic thus:
The syllogism is not applied to the first principles of sciences, and is applied in vain to intermediate axioms, being no match for the subtlety of nature. It commands assent therefore to the proposition, but does not take hold of the thing.A modern logic of natural objects would aim to take hold of the thing, aiming to secure sound principles for reasoning with individuals without need of Leibniz's metaphysical starting point.
1. I am grateful to
John Wahlert for elucidating this example for me.
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