## Relationalism

Under Construction

In this section I’m going to deal with something that I call relationalism. It originates from a preoccupation I’ve had for years over how much weight the ontological category of “relations” can be made to carry – for example, properties could be regarded as relational, internally to an object or in terms of the relations between the object and others; indeed, the concept of object could be regarded as not a fundamental category.

My idea of relation, here, admits of no further differentiation. You might say “Ah, but it’s no good talking about relations as such – you have to talk about what type of relation.” This is a reasonable point, so I’d best come clean and say that what I’m talking about is the idea of a fundamental relation, having one type, with its opposite as lack of relation.

Of particular interest to me are these two ideas –

1) that graphs can represent any structure, of any complexity, and

2) that all graphs are equivalent to graphs where each node has three and only three connections. (Nodes and Connections are often called Vertices and Edges in graph theory).

The interest of the second point is that it shows a way of making nodes immediately locally identical, i.e. without taking into account their larger-scale context; they can be differentiated only with respect to their positioning within the larger scale networks. The diagram above is my attempt to get this point across – all nodes are locally identical, but differentiated by how they sit within the whole picture. I think that each node is, within the whole picture, different; in other words, I think that the graph is completely non-symmetrical (though I don’t have the mathematical expertise to be absolutely sure.) This, to me, indicates a way of getting rid of intrinsic properties altogether.

The importance of this is that, in principle, it indicates a way of modelling the world which has the minimal number of categories (basic ontological kinds); there is a reduction to two – “vertex” and “edge”. All other categories (such as properties) can be regarded as emergent from complex graphs – as patterns.

Though I arrived at these concerns fairly independently, I eventually found some back-up in the ideas of Randall Dipert, an American philosopher based at West Point, whose approach has been termed by Nicholas Shakel “metaphysical graphical structuralism”, and the scientist Stephen Wolfram.

Links to their presentations of their ideas are in –

Appendix 1 (for Dipert) and Appendix 2 (for Wolfram) below.

For their own sites, see –

Text in bold in the appendices is my emphasis, for ideas I regard as important. Unfortunately, the diagrams referred to have not carried across, and there are other formatting issues.

APPENDIX 1

The Journal of Philosophy Vol. XCIV, No. 7, July 1997, pp. 329-358

THE MATHEMATICAL STRUCTURE OF THE WORLD:

THE WORLD AS GRAPH*

By Randall R. Dipert

United States Military Academy, West Point

Many theories of logic and metaphysics since Plato and Aristotle (in fact, since Heraclitus) have hinted at claims concerning relations as well as the “connectedness of all things”. Nevertheless, few theories have forthrightly, and with rigor and clarity, addressed the jointly metaphysical and logical issues of relations and this supposed connectedness–in large part because developed theories of relations were not available until the nineteenth century, after work by Augustus De Morgan, C. S. Peirce, Ernst Schröder, and less self-consciously, Gottlob Frege. Although many philosophical theories have mixed logical observations with metaphysical ones, especially in this century, few if any metaphysical theories have made their treatment of relations a touchstone.

I. THE SHORTCOMINGS OF LOGIC AND “LOGICAL” METAPHYSICS

The view of metaphysics I propose is relational and holistic: the concrete world is a single, large structure induced by a single, two-place, symmetric relation, and thus best analyzed as a certain sort of graph. Every concrete entity “in” the world is a part of this structure and is a structure (subgraph) in its own right. Such entities are individuated (and hence contemplated) solely by their graph-theoretic structural features. A motivation for this admittedly strange proposal begins, first, by claiming that both reality and thoughts are structures of certain sorts and, then, by arguing that the correct or most perspicuous portrayal of this structure is purely relational and, in fact, best portrayed by graphs.

There are components of this view in the relationalism of Peirce, in the metaphysical holism of Baruch Spinoza, and in what we could call the “structuralism’ of metaphysically inclined set theorists and mereologists. Karl Popper advocated at least a methodological relationalism, and David Mertz has taken logico-metaphysical relationalism seriously.[1] T. L. S. Sprigge [2] has recently proposed a doctrine of “holistic relations” in his examination of F. H. Bradley’s theory of relations. Among recent authors, D. M. Armstrong [3] has most clearly considered some of the issues I shall investigate. His conclusion is that every fundamental property could ultimately be a “relationally structural property” (ibid., p. 8). A whole school of researchers in artificial intelligence and cognitive science have developed positions in “knowledge representation” that utilize diagrams and networks of conceptual relations. The proposal with which I am most familiar–and which partly inspired my own approach is the semantic network processing system (SNePS). [4]

Finally, a number of researchers have suggested that traditional logical representations are inadequate, or misleading, in portraying conceptual and especially “perceptual” information. Jon Barwise and others [5] have suggested this view and urged the importance of diagrams in teaching, learning, and thinking about logical structures. Connectionist theories of mind, and neurological or neural-network analyses of mental processes and states, have still more strongly urged a “network” approach to these phenomena.

My own proposal differs from these diverse suggestions in at least three ways. First, mine is a metaphysical proposal: it is a theory about the world and its structure, not just about how the mind or brain is organized. Second, I do not take diagrams and two-dimensional structures as merely notationally or pedagogically convenient tools for portraying traditionally “logical” structures: I propose diagrams, graphical structures, as a serious alternative to logic as it has been traditionally conceived. Finally, I seek to return metaphysical structure to something that can be clearly and indeed mathematically thought about and discussed, and thereby avoid both a merely suggestive holism and the pseudo-mathematical formalisms of logic or networks, while retaining their respective assets.

We may, crudely to be sure, divide an ontology into two parts: the kind of entities proposed to exist, an ontological inventory if you will, and a structural system or theory of structure by which these entities are constituted and related. In most materialist systems, for example, the ontological inventory consists of “material objects,” while the structural system consists in their organization by spatiotemporal relations.

Much ingenuity has gone into devising subtle, alternative proposals for materialist and nonmaterialist ontological inventories. Alas, remarkably little energy has gone into developing articulated, alternative systems of ontological structure. Other than materialist spariotemporal models, the primary program for ontological structure in the twentieth-century analytical tradition centers on logical structure. This latter approach is the strongest current in analytical metaphysics, is epitomized by the extreme position of logical atomism, and is crystallized in Rudolf Carnap’s famous fide, Der logische Aufbau der Welt (1928) (The Logical Structure of the World). By logics of the sort on which philosophical analysis has been based, I mean the following. First, they are formal, and usually symbolic theories, built from symbolic (categorematic) constituents intended to have metaphysical correlates, combinations of which form the most basic, complete ingredients necessary for the description of the concrete world: sentences, propositions, states of affairs, or situations.[6] Second, they have uniformly presumed that there exist basic entities—individuals–that “have” properties, which are reflected in sharply varying accounts of individual constants (proper names, rigid designators, Urelementen, and so on) and of the predicates that apply to them. Third, “logical” structure is portrayed by strictly linear strings of symbols that both capture the content of, and have the same superficial form as, natural language. Second-order, “deviant,” modal, and intensional logics, and even set and mereological theories can all be seen as variations of this basic approach. With respect to the crucial second feature, involving a contrast between individuals and properties, even Aristotle’s metaphysics counts as a logical theory.

There is a widespread presumption that these logical theories are now perfectly well understood in the sense that a mathematical theory is, such as number theory, Euclidean geometry, or the theory of groups. The “term” logics of Aristotle, G. W. Leibniz, and G. Boole articulate the straightforward structure of a Boolean algebra, but the more recent quantified predicate logics–and especially set theory-have, by comparison with theories originating in pure mathematics, such as group theory, a very messy and ill-understood mathematical structure. It is thus perhaps understandable that many mathematicians have been suspicious of a “foundational” approach to their discipline that roots crisp, well-understood mathematical structures in poorly understood theories with comparatively large numbers of complexly interacting axioms, such as set theory. I would maintain that the very possibility of a clear understanding of the world requires the possibility that it/s a simple mathematical structure, and that creating complex, ad hoc, or hybrid structures for this task constitutes negative progress.

Logics, as well as their extensions in set and mereological theories, have been especially dominant in the twentieth century as proposals for metaphysical and conceptual structure. The various alternative set theories seem to have enormous expressive capacity, and have attracted serious proponents of such views as that there ex/st sets (of simple things, often material ones), or even (W. V. Quine) that there exist only sets. This latter view has the attractive feature that every existent’s only features are simultaneously its structural features as described by “the” axioms of set theory. Unfortunately, set theory is, in terms of its own mathematical structure, a notorious and ill-understood behemoth, about which there is much quibbling, artfully hidden from and by many philosophers, over “the” nature of sets as described by various proposed axioms of choice, infinity, foundation, the continuum hypothesis, and so on. [7]

One should, of course, not underestimate the enormous progress that has been made in the last two centuries within this logico-metaphysical tradition. One goal of set theory has been to “unify” the language and deductive techniques of the sciences, or at least mathematics. [8] Because of its now advanced technical sophistication, the sheer investment put into it by logicians and philosophers to date, a certain pragmatic measure (loosely speaking) of “success” in dealing with problems, and a certain accrued historical legitimacy–none of which a “start up” proposal like mine is likely initially to have—philosophers have not exactly driven themselves to attack logic’s premiere status as a structural system. There is also a certain sense—which I shall not contest–that, if any articulated, formal theory of a metaphysical structural system is possible, then a logic is in certain ways “adequate” to express it.

We can, however, observe the following weaknesses in logic as a structural system for metaphysics (and cognition).9 First, much of our thought seems to be directed toward, and to manipulate, aural and visual structures, or similarly textured entities. These do not seem to be in a logical form, even if they can be wrestled into it. A logical coding of this content–such as digitalization–may indeed be possible, but is often unnatural and unperspicuous in exhibiting structure. Likewise, we can express an English sentence as a numerical structure, such as through the trick of identifying positive whole numbers with sentential structure (for example, Gödel numbers or Boolos’s simpler scheme), but this does not entail that arithmetical structures display clearly semantic, syntactic, or pragmatic linguistic structure. This objection focuses on logic’s portrayal of logical form by linear sequences (strings) of predicates, individual constants, and logical symbols. Such an observation seems to be at the heart of gestures toward two-dimensional logical diagrams that we see, for example, in Peirce and even Frege, and more recently in semantic-network representations of logical structure. The world might have structure–and surely does have structure if it is graspable–and yet this might not be a logical structure: it has patterns that are “mathematical” but not perspicuously rendered into traditional logical notation. Perhaps the most important such features of experience–for example, music or visual beauty–have a definite and easily cognized structure which is not helpfully described as “logical” structure—or in which their value is not preserved in logical transcription.

Second, logical structure is historically associated with highly conceptualized thought, and in fact with thoughts that are easily expressed in natural languages. The theory of logical form has thus imitated the structure of this linguistic surface. Such a methodology begins with Aristotle, in which his conception both of logic and of metaphysics is intimately connected to Greek grammar. Even the rare concern that has been expressed about this feature, such as Bertrand Russell’s worries about Western subject-predicate forms, or efforts to develop an adverbial account, often merely appeal to (other) features of (other) natural languages. Such linguistic-representational interests may have twisted logic away from its core concern–which I, with Aristotle, Frege, and Peirce at their best, take to be the analysis of demonstration and, more narrowly, deduction. Following other such interests may ultimately make logic a good theory of the linguistically expressible, a good philosophy of language, but a poor theory of the metaphysical Structure of the world, or of nonlinguistic thought and mental content. It would be very surprising indeed if one single formal theory, such as first-order predicate logic, would simultaneously be the ideal theory for the theory of demonstration, a foundation for all of mathematics, ideal for a theory of both linguistic and all mental content, and for an account of metaphysical structure of the world.

Third, current logical notations, as linear sequences of symbols or “strings,” seem to be irredeemably awkward, or even inadequate, at representing certain quantificational phenomena. One such feature is indicated by the phenomenon of branched quantifiers.[11] Quine [12] has remarked upon the awkward nature of all commonly-used notations for quantifiers and other variable-binding operators. Such inelegances, while initially appearing merely to be aesthetic shortcomings in isolated areas, may indicate deeper difficulties[13]

Fourth, the notion of a logical individual, the notion of a distinct and identifiable thing that “has” properties or enters into relations in either its metaphysical or cognitive aspect that is, what it is that individual constants or proper names denote–has been enormously problematic. Logical theories, and especially our now dogmatic introductions to logic, often casually equate them with middle-sized, common-sense objects, or even with persons. We might read that New York City or George Washington are “individuals” for logical purposes. Yet these “ordinary” entities–grotesque and naive reifications of a supposedly common-sense metaphysics–are widely known to have enormous internal structure, and are most likely to be best analyzed as composites made up of other “individuals” in more sophisticated thinking.[14] Taking logical individuals to be the smallest subatomic particles that we have identified is only slightly less naive. The most serious, open, and sustained debate over this issue took place within the logical atomism of Russell and the early Ludwig Wittgenstein. But their efforts were eventually regarded, even by themselves, as having been grossly inadequate: Russell was left conjecturing that there existed logically perfect individual names, even if he could not otherwise describe them, while Wittgenstein came to regard this difficulty as crucial and insurmountable, and thus to cast the whole approach aside. [15] This difficulty with the nature of individuals has come to be regarded strictly as a problem for logical atomism. In more recent logico-philosophical accounts, this fundamental problem has simply been ignored, not solved. I wish to suggest, however, that it is a very basic and serious problem with (almost any) logic as a philosophical scaffolding for a theory of world or thought: What exactly are the individuals that have properties? What are they and how are they individuated? The ultimate members of sets, not themselves composite sets (if there are any), pose a similar and largely ignored puzzle.

Fifth, logic provides only the framework of a general series of proposals for the structure of thought or reality. Predicate logic does not answer, or even frame the question, for example, of which one-place properties are basic, and which reducible, or which (if any) two-place relations are basic, and so on. Furthermore, modem predicate logic has been remarkably blasé about questions of reducibility among the arities of various properties: Are basic one-place properties sufficient (exclusive monadism)? Are basic two-place relations required or perhaps even sufficient (exclusive dyadic relationalism)? Are basic three-place relations reducible to two-place ones? And so on. We might call this feature of predicate logic polyadism. Set theory and mereology offer some guidance in the polyadic wilderness and hint that a two-place, nonsymmetric relation is required, and perhaps even sufficient. In set theory, nonmembership relations of various arities, including beyond two-places ones, can be expressed–or at least extensionally described–by means of various Wiener-Kurotowski devices [16] Peirce proposed that three-place relations are required, but that higher-order relations are reducible to relations of less than four places;[17] he seems to have had no concrete proposals for which one-, two- and three-place relations are basic. In short, modem logic has encouraged, through its arity-neutrality, logic-driven metaphysics to avoid facing certain basic structural questions about properties and relations [18].

Finally, logic gives nonperspicuous accounts of large and important structural features in the world–the organization of the planetary system or of a Ludwig Beethoven symphony. Most such large phenomena would be treated in logic as giant conjunctions (of “facts” about “individuals”), while reality and cognition of them seem to involve features that incorporate holistic and hierarchical notions of pure relational structure which do not always attend to the exact nature of the ultimate constituents (for example, particles in the planets or the pitches in a symphony). Our thinking should be more, and more rigorously, directed toward the larger and more significant structures of the world, not at the bolts and welds of isolated connections, hoping that depth and insight will somehow emerge if we inspect enough of them.

II. CONSIDERATIONS IN FAVOR OF RELATIONALISM

If exclusive monadism is the position that reality requires for a description of its structure only one-place basic properties, I shall call relationalism the view that it requires at least one two- or higher place basic relation. Exclusive relationalism would be the view that two- or higher place relations are basic and thus required for a structural account of the world, and that no one-place properties are required. Exclusive monadism is a common view in the history of philosophy, endorsed by Plato, Leibniz, and many others [19].

In this section, I shall present a number of considerations that suggest that relationalism is true and that relations are necessary for a structural account of the world. It also contains suggestions that monadic properties are derivative phenomena, that is, arguments for exclusive relationalism.

Numbers. It is now quite common to think of the natural numbers, together with “logical” tools for describing patterns among them (for example, functions or sets[20]), as the basis of all mathematical, or at least all quantitative, thought. It is also natural to think of the natural numbers as “entities’ in their own right witness our names for them and the appeals of naive set theory and even of mathematical (better: arithmetical) Platonism.

With the advent of the Peano postulates, and even more clearly in the formalization by Kurt Gödel, we see that a single given number, even the ostensibly unique zero, quite remarkably lacks, within the most sophisticated of our formal theories, identifying (monadie, one-place) propert/es. A number’s nature is instead exclusively specified by relationships with other similar entities, notably by the successor relation. In a sense, we cannot say what numbers are–at least not without reference to other numbers. This difficulty can be exposed most notoriously through the fact that 3, 5, 7… equally well constitute a “model” of the entities described in the postulates as do the intended 0, 1, 2, 3,…; namely, in the former model, 3 plays the role of the entity that is no number’s successor within this set, a structural “zero,” and “next odd number” plays the role of the successor.

This cluster of observations leads us to the hypothesis that all formalizable accounts of “the” numbers, such as the Peano postulates, identify only the relations among numbers—the numerical structure, so to speak—not unique properties “true of” numbers but not of other things. There are no distinctive “things” that are numbers, anymore than there is a distinct class of entities that are, by their very nature, “larger than” other things. There are numerical relationships;, there are various collections of entities–many of them–that exhibit this structure. The system of things with these relationships is itself a structure (a structure-abstract), and this is what the number system is. We might even reify the structure-abstracts themselves, but must keep in mind that they have no distinctive monadic properties other than derivative ones in virtue of their structural nature.

To put the matter another way, numbers are (just) nodes in certain kinds of patterns [21]. They are like the internally indistinguishable Urelementen of Zermelo-Frankel set theory. “Numbers” are properly identified not by what they are but by their relationships to other things. Their identity consists entirely in their “external” relationships. Our reification of numbers–talking of their “existence,” “properties,” and so on–is a convenient shorthand, of the “place” they occupy in a system of prototypical “things” ordered in certain ways.

Spatial location and other “physical” properties. Leibniz’s relational view of space has had a special place in modern science. It has motivated whole research programs in physics, including Ernst Mach’s and Albert Einstein’s. It is the view that the “location” of things—and their velocities, accelerations, and so on–are not “absolute”: they are not objective, intrinsic properties of things.[22]. Instead, locations, sizes, velocities, accelerations, and all other spatiotemporal qualities of things are properly to be understood as shorthand for much longer relational descriptions: ‘North America’ means to the “north” of South America, “west” of Europe, and the locations of these entities are in turn relationally interdefined. Our usual nomenclature and “common-sense” view of such things retain an absolutist view that spatiotemporal properties are one-place, objective, and intrinsic. No one doubts that it is often quite useful to continue to think and speak in this manner, although many theorists now claim that the “real” nature of things underlying our quaint speech and thought about space and time has a quite different structure: it is purely relational [23]. Now, since virtually all of our common-sense thought about the physical world involves implicit or explicit reference to spatiotemporal properties, and since our best available scientific accounts rely heavily upon them, it follows that a large part of common-sense and scientific pictures of the world is purely relational (if space-time is): things are “where” they are in virtue of their (spatial) relations to other things.

We might still believe that some aspects of concrete, experienced things, and our thoughts about these things, are the way they are due to one-place properties, and not just to relational properties. Pondering the common sensibles of Aristotelian metaphysics (the primary qualities of John Locke) at a common-sense level, we might think of the “shape” of a thing as intrinsic and one-place. But shape is, again, a set of spatiotemporal relations, albeit “internal” to the object. Still worse, even to mark out an everyday thing–an apple, for example–from the rest of the world, from its “background,” is something we usually do effortlessly and unconsciously through perception. But to perceive an object, to consider a thing as an object, is to notice or attribute contrasts of various perceivable “properties” against a background. An apple is considered by us an “object” because we can see its outline, and because it is easy to “separate” it from its surroundings by moving it. These contrasts and separability that appear so phenomenologically immediate and monadic (and are the bane of computer optical recognition) are however really relational phenomena: the apple is red compared with the brown ground on which it lies,[24] and it is easily and separately movable from the grass and ground. Thus, even when an aspect of an object does not seem overtly spatiotemporal, and seems monadic (being a distinct object, having a shape), our perception of material objects seems to depend crucially on—or even be entirely constituted of–relational features. Both its monadic properties, and its very objecthood, seem rooted in relations.

At quite a different level and kind of understanding of material objects, our best scientific theories attribute various “properties” of electrical charge, mass, spin, and so on to the finest microspatial (microphysical) constituents of reality. But how we come to attribute these “properties” to these “things” is only through what we might call their interactivity. The notion of a “particle” in physics has difficulties precisely analogous to those of an “individual” in logic. Traditional explanatory systems have an understandable penchant for such footholds in analysis and explanation. It is not difficult to see, however, that our regarding a particle to have the “property’ of a certain mass is our explanation of why it interacts in certain ways with other similarly interactive entities. We should perhaps instead express ourselves in terms of the root phenomenon, rather than its convenient monadistic shorthand, and say that certain entities interact with other entities in certain ways: this relational interactivity (and a “disposition” for this interactivity over “time”) is the underlying phenomenon. Objects supposedly having masses, charges, spins, and so on are much like objects having “locations”: they are our ways of handily referring to deeply relational phenomena using conveniently monadic expressions.

Words, languages, and meaning. It is a long-standing custom to speak of a word or phrase as having meaning; this everyday approach, of treating words as having semantic and syntactic (monadic) properties, has been aided and abetted by theories of language since the Middle Ages. As is often briefly noted but rarely developed, however, this is quite misleading: no series of marks, or of sounds, really–intrinsically–has a meaning. A word has a meaning only in the bosom of a language. And the concept of a natural language itself refers to a vast and conceptually ill-understood “system” of teachings, understandings, practices, and even past or present explicit agreements, beliefs, and intentions across a certain social group. In practice, the “meaning” of a word also typically involves its relationship to other words, with their meanings. (This position is broader than what Michael Devitt [25] calls “semantic holism.” It is difficult to imagine that a word such as ‘gift’ has (roughly) the meaning in some language that it does for us without there being a word or expression in that language with the meaning ‘possession’ has for us; similarly for ‘swim’ and ‘water’ (or some fluid). Furthermore–and this is still less dwelt upon–what counts as a “word” or morpheme (spoken, written, or other) is itself a highly, if not exclusively, relational notion: it is to have a visual or audible contrast with a “background,” to be identifiable as a certain “type”–itself a very difficult notion rooted in “similarity” to other tokens and differentiation from (tokens of) other types. This loose talk about “words” somehow involves shapes of physical object or images, or involves pauses between sounds, thus utilizing a physical notion whose relationality we have already discussed, or in phonological contrasts and relations of pitch, voicing, and so on. Here (“contrastive” phenomena in linguistics), as in music, we note the importance of sound relations more than “properties” of sounds. In music, it is pitch relations, not pitch properties, that determine the salient features of identity and similarity of theme, melody, and harmony in pieces, heard in a context of a scale and style system. Our conclusion may well then be that the features of words (indeed, the very identity and individuation of “words’) are extensively, even thoroughly, relational: a word exists (has identifiable character as a distinct word), and means, signifies…what it does because of its relation to other words, activities, and things. [26]

Summary: considerations. In our discussion of these three classes of disparate phenomena, the numerical, the spatiotemporal and physical, and the linguistic, we notice two features. First, there are powerful reasons for believing that many significant features of these phenomena are extensively, or even completely, relational. This claim is strongly suggested by a deeper phenomenological investigation as well as by our most sophisticated theories of number, of space and time, of physical things, and of words, languages, and their features. Second, there is nevertheless a tendency to continue to speak, even among those who know these theories well, of “properties” of numbers, physical objects, and linguistic entities. I would even admit to an explanatory and computational (or psychological) bias in favor of such modes or presentation: Aristotelian and monadic modes of inference and conceptualization may be our default mode of our reasoning and naming. The world might also be so structured that such simplified–monadized–accounts cause comparatively little damage for everyday accuracy or usefulness; and computation itself might be such that dealing with relatedness in all of its glory (that is, in the full quantified, relational predicate calculus) might involve insuperable constraints on our ability to reach answers and effectively assess such structures–for example, Church’s Theorem and the nondeterministic polynomial time (NP) completeness of many properties of relational structures.[27] Our minds and traditions may have opted for “reasonable” success in thought and speech, over mind-boggling and frequently ineffective efforts at accurate representation and manipulation of relational phenomena. This is not to be construed as admitting that we cannot ever conceive of, or talk clearly about, relatedness, but only that we often do not–and sometimes need not. We can, after all, think and talk about some mathematics, even if we often need the help of diagrams, and we easily use indexicals in natural language.

III. STRUCTURE, ASYMMETRIC GRAPHS, AND ARISTOTLE REFUTED

A graph is often defined set theoretically as a set of points, typically called “vertices” or nodes, and a set of “edges” connecting these points. The graph G can also be identified (misleadingly, as I shall soon argue) using set theory as an ordered pair consisting of the vertex set V and the edge set E, that is, G = < V, E >. If the graph is a “simple” graph, an edge is an unordered pair, a doubleton set consisting of two vertices. Such an entity may be considered to be the portrayal of the structure induced by a single two-place, symmetric relation. If the graph is “directed,” or a digraph, then an edge is an ordered pair. A directed graph is a structure formed by a single two-place, asymmetric relation. We may also allow an edge to connect a vertex with itself, in which case a graph is reflexive. (Standard graphs are irreflexive: no edge joins the selfsame vertex.) Further extensions are possible. We might, for example, formulate a notion of edge that simultaneously connects three or more vertices: a hypergraph.

General graph theory is the theory of structures induced by a single relation (two- or more place, symmetric or not, and so on) and integrally involves what we might think of as “relational combinatorics”: the ways in which entities (here, vertices) may be connected by a relation. Observe, however, that it is the theory of how entities may be connected by just one relation, not many relations of diverse arities. As we shall see, even with this restriction, graphs embrace a high degree of complexity, diversity, and hence information. Even with small graphs–say, graphs with just twenty vertices—the diversity of structures that are in some sense distinct is dazzling and, indeed, largely uncontemplated.

Defining graphs in this set-theoretic way retains, however, some of the distinguishing flaws of normal logical and set-theoretical notation, namely, set-theoretic entities are distinguished even when we can see they have the same graph-theoretical structure. Two structurally identical graphs will be considered distinct if their individual vertices are different (or appear to be labeled differently). For example, consider two graphs, both with three vertices, A, B, C: graph 1 contains only the edges {A, B} and {B, C}, while graph 2 contains only the edges {A, C} and {B, C}. Since their respective edge sets {{A, B}, {B, C}} and {{A, C}, {B, C}} are different, then–if we define graphs as set-theoretic structures–graphs I and 2 are different. In a sense, however, the two have the same graph structure: they are just looked at, or labeled, in different ways.

Two graphs are structurally identical just when their labels can be rearranged so that they are set-theoretically identical. This relationship between two graphs is normally described as an isomorphism: two “distinct” graphs are structurally isomorphic just when there exists a 1-1 function, f, between the nodes of the first graph, and those of the second, that is edge preserving. Conversely, two graphs are structurally distinct if they are not isomorphic: if no function exists that maps vertices onto vertices in an edge-preserving way. We are primarily interested in structurally distinct graphs rather than in artifacts of the labeling of their vertices, or the way we have diagrammed them or happen to be viewing them. The task of determining, visually or computationally, whether two representations of graphs are isomorphic (structurally identical) is nevertheless not a trivial one.

What we would ideally wish for the theory of pure graph structure is an account of graphs such that every difference between two graphs’ descriptions entails a difference in structure, and vice versa. Existing set-theoretical and diagrammatic notations do not do so because differences in vertex labels result in distinct (that is, non-set-theoretically equivalent) expressions, or in diagrams that look different but are not structurally distinct [28]. This obscurity arises because of the presumed “individual’ vertex labels.

If we include quasi-distinctness—that is, distinct only in virtue of having differently-named vertices–the number of simple graphs is given by the formula 2(p(p-1))/2 where p is the order, or number of vertices (or rather, vertex names!). Even this combinatorial explosion is limited by the assumption that we are choosing from a canonically ordered set of vertex labels. For p — 2 vertices, this is 2; for p = 3, this is 8; for p = 4, 64; for p = 5, 1024; p = 6, 32,768; and so on. The number of nonisomorphic graphs initially rises much less steeply, demonstrating the nuisance that mere vertex naming causes. It is given by a difficult formula due to George Pólya, [29] and for increasing values of p starting with 2, generates the sequence:

 Order(Number of Vertices) Number of non-isomorphic graphs 2 2 3 4 4 11 5 34 6 156 7 1,044 8 12,346 9 308,708 [30]

This sequence asymptotically approaches 2P/2/p!

Even the theory of “unlabeled” or non-isomorphic graphs has undesirable features for a theory of pure structure, I now want to argue. Here, I go beyond conventional graph theory and develop a way of looking at graphs for specific philosophical purposes that is not part of the usual mathematical theory of graphs (polluted as it has become by set-theoretic notational devices). The crucial concept for our purposes here is structural distinctness, free from all artifacts of labeling and perspective. The first phase of this investigation, discussed above, observed that mere aspects of vertex labels (or ways we draw or perceive diagrams) does not coincide with this interest. We home in on non-isomorphic graphs as paradigmatic of structural distinctness.

The second phase of this investigation attends further to the issue of vertex labels. Consider now only graph 1. Examine carefully the features of vertices A, B, and C, trying to ignore information misleadingly imported through vertex labeling. B is different-structurally different from both vertex A and vertex C, namely, vertex B has two edges emanating from it: it is of degree 2. Both vertex A and vertex C have only one edge. In fact, on close examination, once we ignore their different labels, vertices A and C begin to fade into one another. They have the same structural features: both have only one edge emanating from them (they are of degree 1), and both are directly connected to a single vertex that has degree 2. Each is indirectly connected (by a two-step path through the unique vertex of degree 2) to the other. We might call the property that A and C have graph-structural symmetry: they occupy places in a graph that cannot be described, using only structural features, in any way that distinguish them. In the completely connected graph of order 3:

Here in Graph 3, none of the vertices is structurally distinct: each is connected by two edges, and each is connected to every other vertex.

Observe that what I have called the two phases of our investigation of structural distinctness involve two different notions of distinctness. The first applies to graphs, and may be called internal distinctness:, two entities (graphs) are distinct if and only if they contain features that structurally distinguish them. The second applies to vertices (and subgraphs) and may be called external (or contextual) distinctness:, vertices have no internal structure, but are distinct if they are structurally related to other entities in a unique way. We do not rely upon our labelings to assert or assume they are distinct.

The exact criteria for distinctness and structure of such representations are of singular importance if we identify the concrete world, and its constituents, as graphs–or as any formal structure. We should not import assumed individuation of entities, whether they be logical individuals, graph labels, or proper names, when a key component of metaphysics is precisely the individuation of entities. A key metaphysical and methodological principle is that we shall not assert, or assume, metaphysical distinctness unless we can show (internal or external) structural distinctness. Traditional logical metaphysics has used the deus ex machina strategem for individuation (and thence for structure), while I propose lifting ourselves up by the bootstraps, using nothing but notions of mathematized structure. The method I am proposing for formal (not “logical”) metaphysics has clear affinities with constructivism in the philosophy of mathematics.

A graph-structural symmetry between two vertices, X and Y, occurs when there is a 1-1 function, such that all nodes are mapped to themselves, except that X is mapped to Y, and Y to X, and that applying this function to the edge set of the original graph results in a graph that is isomorphic to the original. Other, more systematic symmetries may occur, even where there are no two nodes that are symmetric. The test for these broader (subgraph) symmetries will be whether or not there is a 1-1, nonidentity function, f, whose domain and range is the vertex-set, and which results (when applied to, or “translating,” the edge-set) in a graph structurally identical to the original. If there is such a function, we shall say there is an auto-isomorphism, and more exactly that the graph is auto-isomorphic under f. Where there exists one or more auto-isomorphism, we shall say that the graph itself is symmetric, or contains a symmetry. This symmetry may consist either in containing one or more vertex symmetry, or in containing one or more subgraph symmetry, or both. A graph for which there is no such fwill be called non-auto-isomorphic or, simply (following P. Erdös [31]), asymmetric the graph contains no symmetries.

```It is a surprising fact that there are no asymmetric graphs of orders 2, 3, 4, or 5; that is, all graphs of these orders contain some symmetries. We might then even suggest that these small graphs lack (nonexternally imposed) structural distinctness and thus are not well-defined structures. The first asymmetric graphs are of order 6, one of which is:

In an asymmetric graph, it is possible to give a unique, purely structural description for each vertex. For this asymmetric graph, one such description is:
A: Unique vertex of degree 1

B: Vertex [of degree 4] adjacent to A

C: Vertex of degree 2 adjacent to two degree4 vertices (B,D)

D: Vertex of degree 4 not adjacent to A

E: Unique vertex of degree 3

F: Vertex of degree 2 not adjacent to two degree4 vertices
The literature on asymmetric graphs is extremely sparse. The number of structurally distinct, asymmetric graphs of order 6 is 7 (out of 156 non-isomorphic
graphs). There are 152 structurally distinct, asymmetric graphs of order 7 (out of 1,044).[32] I do not have a guess concerning an enumeration theorem for asymmetric graphs, and there is no proof or speculation in the literature (to my knowledge): that is, how one calculates the number of structurally distinct asymmetric graphs for order p.
The theory of "pure" formal structure is for me the theory of non-isomorphic, asymmetric graphs, and their subgraphs. Graph theory, as it is usually investigated and described, contains too many artifacts of vertex naming. The study of graph theory through unlabeled diagrams is thus purer than that using conventional set-theoretical representations. Consequently, when I speak of what I propose is the conceivable structure of the world the world graph for short--it is surely an asymmetric graph (if it is a graph at all). It is only asymmetric graphs that have structural distinctness that is not imposed from without, such as by naming.```

Aristotle’s principle refuted. The locus classicus for a discussion of relations in undoubtedly section 7 of Aristotle’s Categories.[33] Aristotle there states a number of conjectures and arguments that have had a deep impact on the limited clear thinking about relations in Western philosophy. For instance, he assumes that no “primary substance” is relational, but broaches the possibility that some secondary substances are, such as “head [of] ~ or “hand [of]” (8a14f). At 8a37f, he begins an argument that no substance, primary or secondary, is relational. Apparently, then, Aristotle regarded relations just as ways of speaking that are ultimately reducible to monadic aspects of substance.

His argument is as follows. Anytime we speak of an entity being related to another, we presume the distinctness of the relata–those things which are related. This distinctness cannot be further established by relations, since they would require, for their definition, the distinctness of their relata. Hence, relations require monadic properties in order to define their relata. His argument seems to be that irreducible relations are conceptually incoherent, or result in an infinite regress. Although Aristotle himself speaks rather tentatively of the implications of his arguments, most who have thoroughly contemplated the issue of the foundational nature of relational versus monadic properties have probably followed a similar pattern of reasoning. And it is this reasoning which has given the primary impetus toward what I have called monadism.

The main steps of this reasoning seem to be:

`1. The definition, and an understanding, of relations requires the distinctness of relata [34].`

2. This distinctness of relata cannot be established except by monadic properties, on pain of incoherence or an infinite regress, [35].

3. So, monadic properties have a more fundamental metaphysical importance than relations, since relations depend on monadic properties, but not vice versa, [36].

4. Consequently, there are irreducible monadic properties, and all relational properties are reducible to monadic properties.

Thesis 1 seems to be the heart of Aristotle’s anti-Platonism, here extended to relations: that relations are known through their extensions, and not “directly.” We might call this extensionalism, or particularism, or something of the sort. I need not dispute it to make my point.

`I assert that the existence of asymmetric graphs shows conclusively, for the first time in the history of philosophy, that thesis 2 is incorrect: namely, distinct relata--vertices in asymmetric graphs, for example---can be distinct (Aristotle: known definitely), and that this distinctness can arise through relations alone. With this evidence, I believe that most of the impetus toward monadism immediately cob lapses. (Another alleged argument against relationalism, a puzzle over asymmetric relations, is addressed below.) Indeed, I shall soon show that exclusive relationism is superior to exclusive monadism on at least one metaphilosophical criterion, a variant of Ockham's Razor. Furthermore, ! believe that in section I, I have shown that Aristotle's assumption--that many or all "primary substances" are monadic is also probably mistaken. Perhaps none is.`

#### IV. THE WORLD AS ASYMMETRIC GRAPH

`The view I have proposed is a mathematical, formal, and structural metaphysics, but it is anti-atomistic and, in a precise sense, non-logical. I would admit that I have not yet given a single, clear argument that the world is, or can only be conceived clearly as, an asymmetric graph induced by a single, two-place, symmetric relation. I have presented, first, what I believe are powerful considerations that most phenomena in the world and mind are ultimately relational. Second, I have presented a rarefied account of relational structure that avoids difficulties that I had pinpointed in logical metaphysics, notably, individuation by fiat. This account shows exclusive relationism, pace Aristotle and many others, to be conceptually coherent. In this last section, I would like to add considerations in favor of relationalism and exclusive relationalism, and to the view that one, symmetric, dyadic relation is necessary and sufficient for describing the structure of the world. I would also like to connect this extraordinarily abstract conception of the world, so distant from the way we speak and the way we believe we think, to more common-sense and scientific conceptions of the material and phenomenological world.`

Relations not only can define basic entities, but they do so more efficiently and elegantly than do monadic properties. Using only monadic properties, it is necessary that there be, for example, four “primary” properties in order to define sixteen basic entities in the world. In general, we require a minimum of log2N monadic primary properties in order to distinguish N “basic entities”—such as ultimate particles (quarks) at a space-time point. The number of basic entities (N) in the universe is very large (we may suppose), so the number of primary properties necessary to distinguish them is also large. On the other hand, a single relation can be used to distinguish an arbitrarily large number of basic entities?

This is, I believe, quite a powerful point. If one is an anti-relationalist, that is, a monadist, then one must believe—if one believes there are many distinct entities in the world that they are ultimately distinguished by the different combination of monadic properties each has. Far from being a minimal, Ockhamite assumption, this entails a commitment to a very large number of irreducible, exemplified, monadic properties. Reduction of arity (from 2 or higher to 1) of basic predicates in the world is thus possible only if one tolerates an explosion of the number of basic monadic predicates. This is artfully hidden from us by the casual assumption–in materialist metaphysics, for exampie–that entities are ultimately and simply distinguished by their locations in space, time, or space-time. For this account not to turn out to be disguised relationalism, however, we would require a commitment to a view of space (and time) as being constituted by unique one-place spatial properties: absolute locations–and a sufficient number of them to distinguish all basic material entities [38]. While it is possible that some such view is palatable, it seems wrongheaded to have such a view unwittingly forced on us by a commitment to a monadistic metaphysics without conscious regard to this commitment’s far-reaching implications for other kinds of metaphysical inflation.

The theory of graph-theoretic structure is sufficient to account for all structure in thought or world. Minimally, it has the information-theoretic content to describe the complexity of the apparent world, it mirrors the “computational” difficulty we have in grasping this world, and it has the combinatoric texture to give a theoretically satisfying account of the nature of the world. That is, the world is of daunting size and complexity, parts of it are difficult precisely to isolate and conceive, but it is fundamentally made up of parts arranged in simple, graspable arrangements.

This is an extremely speculative assertion that a graph–large graphs anyway–have the same compositional “feel” as the world; and that the “facts” or sentences of first-order predicate logic of logico-metaphysical analysis do not. Could the universe be a graph-a particular asymmetric graph? This breaks into two questions. First, is it possible that the universe is just a formal structure, and that we can only think about the world as such a structure? Second, is it conceivable that the structure that is the world is just a graph-theoretical structure [39]?

The graphical description of a world is an asymmetric graph. An asymmetric graph is a possible world, structurally described. An entity in such a world is any subgraph of a world graph. Such a subgraph is a subset of the vertices of the world graph, together with all of the edges in the world graph’s edge set such that both vertices of each edge are in this subset. The entities closest to a common-sense notion of an entity will be connected subgraphs: subgraphs in which there exists a path through its edges from every vertex in the subgraph to every other vertex in the subgraph. A great many of these entities in the world are not themselves asymmetric graphs. Indeed, many of these subgraphs are isomorphic to other but ultimately distinguishable subgraphs.

I would like to make a claim that might seem preposterous: that even moderate-sized asymmetric graphs, on the order of forty to one hundred vertices, have compositional attributes like the perceivable world. Structures of similar sizes composed from monadic properties emphatically do not have these features. My guess would be that the world is a larger structure than one formed by one hundred connected vertices, but the point I want to make can be made about relational structures of even this “small” size. How many distinguishable entities would a world graph of this order contain? How many different asymmetric graphs–possible worlds of this order are there? A graph of order forty contains 240 subgraphs. (We shall consider the matter of their individuation and identification in a moment.) This is a very large number. Even if our own world contains a still greater number of basic particles in our best available account of microphysical structure, it is clear that even relatively small such graphs approximate the number of basic components in our universe (physically interpreted). One peculiarity of the graph-theoretic notion of an entity is that while there are 24° distinguishable subgraphs, a great many of these subgraphs “overlap”: they contain the same vertices and indeed may overlap in terms of graph structure, mutually containing one or more of the same further subgraphs. Far from being an undesirable oddity of graph-theoretic relational structure, I believe this phenomenon in fact mirrors difficulties in individuation versus apparent individuation in the actual world. The “two” electrons in the double-slit experiment used to explore quantum mechanics may be entities that share a great deal of metaphysical structure in precisely this sense, even if they are in other ways distinct. (This overlap of metaphysical structure plays the role of speculations about their wave-like nature, dropping their strictly particle-like, individual nature.)

The information content of such a possible graph world can be roughly equated with the number of nonidentical entities of that size and kind. This amounts to the number of nonisomorphic graphs of order forty. F. Harary and E. M. Palmer (op. ciL, p. 242) give the number of order twenty-four graphs as about 1.95 · 1057 and for order forty, the number is a great deal larger than that. In other words, there is a great deal of information content in a graph of that order [40] These two points support my contention that a large graph is somewhat like the apparent gross morphology of the world itself in terms of number of distinguishable entities and in basic complexity or information content what together we might call the world’s metastructural texture. Indeed, these features are duplicable in strictly mathematical structures of a remarkably small absolute size, such as graphs of merely forty vertices.

It is true that a great many of the “distinct” entities in the world graph, namely, many of its subgraphs, are in fact isomorphic. How then can such entities be said to be distinct? Subgraphs can be individuated in one of two ways. First, two subgraphs may have different internal structures: they are nonisomorphic. We might term this internal or absolute (structural) individuation. Second, they may occupy different “places” in the asymmetric world graph. We have built into the very nature of a world graph the fact that different locations, down to the level of vertices themselves, are distinguishable. Some subgraphs, while internally indiscernible, are individuated by their differing locations in the world graph. We may call this external or contextual individuation. Every subgraph in the world graph is individuated, either by its internal structure, by its “external” structure–or both.

A “property” of the entity identified with a subgraph is a pattern, either within that subgraph (an “internal” property) or in a larger subgraph (or subgraph skeleton) of which the given subgraph is a proper part (an “external” property) [41]. A relation between two subgraphs (entities) is the pattern of a subgraph that connects the two given subgraphs or, less interestingly, is a formal relation that holds between descriptions of the two subgraphs considered in isolation. (The former is a type of a posteriori relation, that requires for its confirmation an examination of the world graph, not just the relata; the latter is an a priori or formal relation between the two relata.) A pattern is a subgraph skeleton: a subgraph in which all connected subgraphs of a certain specification are replaced with a single vertex. It is this feature of patterns/properties which will make the world seem to be organized hierarchically. There will be an extraordinarily large number of such possible patterns even in relatively small, connected graphs. This mirrors the plurality of distinct-seeming properties and relations in the world.

One respect in which the real world does not seem much like a graph is in this. In the real world, many and, in fact, the most interesting, relations are asymmetric, while in the world graph, the only basic relation is symmetric. How can so many asymmetric relations arise out of a symmetric relation, and why is the graph-theoretic account not more true to the phenomenal world? If entities are described as subgraphs, and relations are (extensionally) described as subgraph paths that connect these entities (at certain points), then a relation is asymmetric just when the subgraph path is asymmetric at the vertices where it joins its relata. (This is what we might call internal-structural relational asymmetry; there is a trivial sense in which all subgraph paths are externally asymmetric, since the points at which they join their relata are distinguishable in the asymmetric world graph.) The simplest such asymmetric subgraph path connecting entities A, B, C, and D will look like this:

`where X can either be a single node or a subgraph used to mark asymmetries? A similar technique can be used to describe other features of relations, such as transitivity [42]. The second question, of why we should not try harder to be more true to the apparent world, is far more difficult: it is in part the question of why one should use undirected rather than directed graphs--or of why one should not abandon graph-theoretic monorelationalism altogether. My best answer has to do with the desirability of seeking ultimate theories of metaphysical structure which are tractable and perfectly explain, and hence are mathematical accounts of discrete, probably finite, structures. Graph theory is far simpler, better understood, and more theoretically translucent than its structural competitors, including nominally the theory of directed graphs. (Either approach is far ahead of the polyadism of predicate logic.) The second reason is likely to be far less convincing: it is that symmetries have usually been regarded as the more convincing explanatory devices, and that asymmetries are symptoms of a missing variable or an ill-understood phenomenon. This may be a hint of a sophisticated methodological principle, or "merely" an aesthetic desideratum, but a related methodological principle has successfully driven modern theoretical physics. The suspicion of relationalism since Plato has arisen partly because of a mystery posed by asymmetric relations, for example: Which of two mutually entailing, converse relational facts, "S is shorter than T" or "T is taller than S,' is more metaphysically basic? Neither seems metaphysically privileged, and yet it seems odd to admit both as basic facts when they are mutually entailing. One traditional resolution of this puzzle has been to use it to reject relationalism altogether and embrace monadism for example, that each has a monadic property of height that grounds both relations. A less desperate response is to take this puzzle as evidence that all basic relational facts are symmetric, which do not allow the puzzle over relational converses to arise, and from which asymmetric relations may be derived.`

I need to sketch how structuralism could plausibly analyze physical (and phenomenal) objects, their structure (what we might call physical structure), including redescribing space, time, fields and forces, and quanta. In short, how would my “theory of everything’ (TOE)link up with theories of everything (TOEs) in contemporary physics? My speculations are these. Physical objects, even the finest subatomic particles, certainly do not correspond to vertices. Instead, they themselves are composite entities, subgraphs of the world graph. Physical microstructure is graph-theoretic macrostructure.” Space and time are frameworks or grids that may conveniently, if sometimes imperfectly, be laid on the world graph [45]. If there is something that is an aspect of things or experience called “time,’ and if it is asymmetric, this is ultimately because there are asymmetric paths connecting certain subgraphs that are the entities and their states [46]. Forces, fields, and causal chains are also such paths among the subgraphs we identify as physical objects/states, and each is distinguished (if at all) by its structure. Some of the stranger physical phenomena are, I believe, accommodated without too much difficulty: the two electrons in the double slit experiment are not, for example, the fully distinct entities that we imagine them to be (or location and momentum are not independent properties). They graph theoretically overlap. Calling them wave-like is, I think, a less perspicuous metaphor than observing that metaphysical, graph-theoretic proximity or distinctness may not always perfectly correspond

to spatial distance or distinctness, even for ostensibly physical entities [47]. Although physical structure is ultimately graph-theoretic structure if it is thinkable at all, this is not to say that we should always strive to analyze physical phenomena in terms of their ultimate graph-theoretic structure. All sciences and their goals of theoretical explanation aim for a certain level of useful description, and we might be perfectly satisfied with physicists’ focus on levels of description that are no finer than useful for a theoretic account of “physical’ objects [48].

No theory of the ontological structure of the world could be considered complete unless it also connects to, and hopefully sheds light on, difficult issues in epistemology and the philosophy of mind. Here, I can only gesture toward possible conclusions, since my focus in this essay was of necessity on metaphysics rather than on the collateral topics of mind and thought. Thoughts are also subgraphs of certain sorts. Thoughts are about a phenomenon–refer to it–if they occupy a certain place in the world graph (in a ~mind~) and share certain structural features with the thought-about object: a thought, per impossibile; is perfectly about an object just when its internal graphical structure is the same as the object’s, and when the object occupies a location in the system of all objects that is like the thought’s location in the system of (that mind’s) thought. This is an iconic view of semanticity which includes both internal and external iconicity [49]. Such a conception has two advantages over current theories of reference and semanticity: first, it attempts to give a serious account of reference without appealing to the black box of denotation or to the camera obscura of causation, or scientistic “naturalized” or evolutionary epistemology. Second, it gives us a plausible handle on the many ways our concepts and thoughts can fail to be precisely about items in the world. That is, it gives an account of the types of defective thoughts that have intrigued but eluded modern philosophy from René Descartes to the latest theories of vagueness. Concepts are generally clearer the more we have of them: this is due to the necessity of external iconicity in disambiguating them. Even with respect to their internal structure, they may specify patterns (subgraph skeletons) that overshoot or undershoot their marks in various ways, that is, there are many, computationally distinguishable, ways in which thought subgraphs can be like–or fail to be like—their object subgraphs.

There might at first seem to be no place in these cold graphs for minds, consciousness, and other mental phenomena–unless, that is, everything is mental. Although within the dialectic of this essay it is wild and possibly irresponsible speculation, we should perhaps consider seriously the possibility that something like the pan-psychism of Spinoza, Leibniz, or Peirce is true, and that vertices are pure feelings (Peircean “firstnesses’), constituting a distinct thought or object only when connected to other such entities.

RANDALL R. DIPERT, United States Military Academy/West Point

NOTES

* An earlier version of this paper was presented at the September 1994 Marvin Farber Conference on “The Metaphysics and Epistemology of Relations” at SUNY/Buffalo, organized by Kenneth Barber, at the SIGMA XI Society at SUNY/Fredonia, and in 1997 to the Department of Logic, Charles University (Prague). I thank Michael Grady, H. Joseph Straight, Morton Schagrin, Kenneth Lucey, WJ. Rapaport, Peter Hare, Fabio Morales, Barry Smith, Don Roberts, D.M. Armstrong, James A. Davis, and Darry Johnson for comments. I especially thank Wallace Matson and Tibor Machan for comments and encouragement.

[1] Popper, Popper Selections (Princeton: University Press, 1983), pp. 165-66; Mertz, Moderate Realism and Its Logic (New Haven: Yale, 1996).

[2] James and Bradley (La Salle, IL: Open Court, 1994).

[3] A Theory of Universals (New York: Cambridge, 1978).

[4] John Sowa, ed., Principles of Semantic Networks (San Matco, CA: Morgan Kauf-

mann, 1991), Conceptual Graphs for Knowledge Representation: First International Conference on Conceptual Structures, ICCS’93 (New York.’ Springer, 1993); S.C. Shapiro and WJ. Rapaport, “SNePS Considered as a Fully Intensional Propositional Semantic Network,” in N. Cercone and G. McCalla, eds., The Knowledge Frontier. Essays in the Representation of Knowledge (New York: Spring, 1987), pp. 262-315; and W.J. Rapaport, “Logical Foundations for Belief Representation, ‘ Cognitive Science x (1986): 371-422.

[5] Barwise and Gerard Allwein, eds., Logical Reasoning with Diagrams (New York: Oxford, 1996);J. Glasgow, N.H. Narayanan, and B. Chandrasekaran, Diagrammatic Reasoning: Cognitive and Computational Perspectives (Cambridge: MIT, 1995).

[6] Another feature of most logics is that they are “discontinuous”: the grammar of syntactic items and metaphysical status of their correlates involves a break between terms, on the one hand, and formulae made up of terms, on the other, or between individuals and predicates, and sentences formed from them. The view I am proposing is, perhaps like the Lockean use of “idea,” continuous: there is no metaphysical demarcation between the nature (or concept) of a thing and of a state of affairs or sentence.

[7] I allude here to critical histories of set theory, such as M. Hallett, Cantorian Sets and Limitation of Size (New York: Oxford, 1986); S. Lavine, Understanding the Infinite (Cambridge: Harvard, 1994); and my “Peirce’s Philosophical Conception of Sets,” in N. Hauser and J. van Evra, eds., Studies in the Logic of C.S. Peirce (Bloomington: Indiana UP, 1997).

[8] This unification of science and mathematics seems to be the chief desideratum of Quine’s program. How much set theory has actually helped us to understand branches of science and mathematics as “one” is questionable, but there is little doubt that set theory has made possible the uniform description of various mathematical structures. I describe below one of the undesirable consequences of such descriptions.

[9] Compare Mark Wilson, “Can We Trust Logical Form?” this JOURNAL, XCI, 10 (October 1994): 51944.

[10] For Peirce, especially in his Beta and Gamma existential graphs, this diagrammatic structure is genuinely metaphysical (or at least conceptual in being “semiotic”) structure. Other logical diagrams, such as those of Venn, Euler, Lambert, Leibniz, and Sturm, seem to be merely pedagogical. See my “History of Modern Logic,” in the article “Logic” in the Macropedia of Encyclopedia Britannica (1904), and “Logic Machines and Diagrams,” in the Routledge Encyclopedia of Philosophy (forthcoming).

[11] Cf. Quine, Philosophy of Logic (Cambridge: Harvard, 1986, 2nd ed.); and Jaakko Hintikka, Philosophy of Mathematics (New York: Cambridge, 1996) for a system Hintikka calls independence friendly (“IF”) logic that allows the specification of nonlinear dependence of quantifiers and even connectives.

[12] “Peirce’s Logic,” in K.L. Ketner, ed., Peirce and Contemporary Thought (New York: Fordham, 1995).

[13] For a discussion of how such ambivalences in choosing logical reductions may be symptoms of deeper problems in a logical theory–and specifically with regard to the status of relations in set theory–see my “Set-Theoretical Representations and Their Adequacy for the Logic of Relations,” Canadian Journal of Philosophy, XI (1982): 353-74.

[14] A similar view is voiced in Sprigge.

[15] See R. Monk, Ludwig Wittgenstein: The Duty of Genius (New York: Penguin,

1990). The exact nature of individuals was viewed as an acute problem for late Boolean logic in the works of Peirce, Schr6der, and O.H. Mitchell.

[16] See my “Set-Theoretical Representations and Their Adequacy for the Logic of Relations.”

[17] Robert W. Burch, Peirce’s Reduction Thesis: The Foundations of Topological Logic (Lubbock: Texas Tech UP, 1991). Further literature and other issues are discussed in an unpublished manuscript, “Reduction to a Dyadic Predicate Revisited” by A.P. Hazen, provided to me by the author.

[18] See, for example, the polyadic metaphysics in “The Nature of Relations,” in R. Chisholm, A Realistic Theory of Categories: An Essay on Ontology (New York: Cambridge, 1996), pp. 51-54.

[19] For Plato’s views, see H.-N. Castaneda, “Plato’s Phaedo Theory of Relations,” Journal of Philosophical Logic, x (1972): 467-80; and a contrary view in David Gallop, “Relations in the Phaedo,” Canadian Journal of Philosophy Supplement, n (1976): 149-63. For Leibniz’s views, see Castaneda, “Leibniz’s 1686 Views on Individual Substances, Existence, and Relations,” this JOURNAL, LXXn, 19 (November 6, 1975): 687-90; J.A. Cover, “Relations and Reduction in Leibniz,” Pacific Philosophical Quarterly, LXX (1989): 185-211. For a glimpse at medieval views, see M.G. Henninger, Relations: Medieval Theories 1250-1325 (New York: Oxford, 1989). Bertrand Russell’s apparent relationalism (“we cannot prove that there are such entities as qualities,…whereas we can prove that there must be relations…”) is discussed in The Problems of Philosophy (New York: Oxford, 1912), p. 95, and elsewhere. On reductions of relations, see Castafieda, “Relations and the Identity of Propositions,” Philosophical Studies, xxwIl (October 1975): 237-44; I.L. Humberstone, “Comparatives and the Reducibility of Relations,‘ Pacific Philosophical Quarterly, LXXVI (June 1995): 117-41; and Milton Fisk, Nature and Necessity (Bloomington: Indiana UP, 1974).

[20] The philosophical implications of proposals in category theory are less well established. See Colin McLarty, “Axiomatizing a Category of Categories,’ Journal of Symbolic Logic, LVl, 4 (December 1991): 1243-60, and “Category Theory in Real Time,’ Philosophia Mathematica, II, 1 (1994): 36-44.

[21] Cf. M. Resnik, “Mathematics as a Science of Patterns: Ontology and Reference,” Nous, xv (1981): 529-50; “Mathematics as a Science of Patterns: Epistemology,” Nous, xw (1982): 95-105; as well as remarks in his Frege and the Philosophy of Mathematics (Ithaca: Cornell, 1980). See also Charles Parsons, “Structuralism and the Concept of Set,” in W. Sinnott-Armstrong, ed., Modality, Morality, and Belief. Essays in Honor of Ruth Barcan Marcus (New York: Cambridge, 1995). Resnik talks suggestively of a similar general philosophical account of numbers, and informally uses notions of “structures” and “patterns.’ He seems to lack a precise definition of these notions which I take myself to be supplying in the account of graphs and structure below.

[22] L. Sklar distinguishes location, velocity, and acceleration absolutism, rejecting location absolutism but proposing velocity or acceleration absolutism–Space, Time, and Space-Time (Berkeley: California UP, 1976).

[23] For a survey of the literature on Leibniz’s views and relationalist accounts of space see J. Earman, World Enough and Space-Time (Cambridge: MIT, 1989) and Andrew Newman, “A Metaphysical Introduction to a Relational Theory of Space,” Philosophical Quarterly, xxxix (1989): 200-20.

[24] See also E.W. Averill, “The Relational Nature of Color,” Philosophical Review, cx (1992): 551-88; as well as Russell, pp. 95-102.

[25] “At its most extreme, semantic, or meaning, holism is the doctrine that all of the inferential properties of an expression constitute its meaning’-Devitt, “A Critique of the Case for Semantic Holism,’ in J. Tomberlin, ed., Philosophical Perspectives, vi: Language and Logic (Atascadero, CA: Ridgeview, 1993).

[26] Similar relational notions are historically suggested in the slogan “meaning is use” that we see in views of J. L. Austin, the later Wittgenstein, and others. Relational views of beliefs are developed in works by Quine and J. Ullian in The Web of Belief (New York: Random House, 1970). Quine and Ullian do not seem to have a serious proposal for formalizing their “web,” and they give a structure only for beliefs, not their composing concepts, according to certain entailment and theory-dependence relationships.

[27] More exactly, it is a combinatoric explosion that we see, for example, in the NP-completeness of the identification of graph isomorphism and of determining subgraph relations. See Michael R. Garey and David Johnson, Computers and Intractability: A Guide to the Theory of NP-Corapleteness (New York: Freeman, 1979).

[28] Since the graph isomorphism problem is NP-complete, there exists no canonical way of representing graphs so that we can always see when two graphs are isomorphic (or putting them into this canonical form is itself NP complete).

[29] See N.L. Biggs, E.K. Lloyd, and R.J. Wilson, Graph Theory, 1736-1963 (New York: Oxford, 1976).

[30] F. Harary and E.M. Palmer, Graphical Enumeration (New York: Academic, 1973).

[31] P. Erdös and A. Rènyi, “Asymmetric Graphs,” Acta Mathematica Academicae Scientiae Hungaicae, XlV (1963): 293-315. See also M. Capabianco and J. Molluzzo, Examples and Counterexamples in Graph Theory (New York: North-Holland, 1978), pp. 89-101.

[32] I have explored these graphs and their properties in a computer-assisted way. I have written a large number of programs in Prolog that first identify nonisomorphic graphs and then manipulate symmetries, asymmetries, and subgraph properties of these graph~

[33] See a very fine article by Fabin Morales, “Relational Attributes in Aristotle, Phronesis, XXXlX, 3 (1994): 255-74; Frank A. Lewis, Substance and Predication in Aristotle (New York: Cambridge, 1991); and J.P. Ackrill’s Aristotle’s Categories and De Interpretatione (New York: Oxford, 1963). Quotations are from J.D. Barnes, ed., The Complete Works of Aristotle (Princeton: University Press, 1984), whose translation of the Categories is by Ackrill.

[34] “It is clear from this that if someone knows any relative definitely he will also know definitely that in relation to which it is spoken of” (8a37-38).

[35] This claim is explicitly attributed to Aristotle in Morales. “But as for a head or a hand or any such substance, it is possible to know it-what it itself is–definitely, without necessarily knowing definitely that in relation to which it is spoken of…. And if they [these relata] are not relatives, it would be true to say that no substance is a relative” (8b16-91).

37] Namely, through the method of unique position in an asymmetric graph. This claim relies on the easily proven theorem that there exists at least one asymmetric graph for every order greater than 5.

[38] Even if one tolerates one basic space-time relation, little-discussed constraints must be imposed on it so that it ensures the individuation of all material entities. If the material structure is infinite–either spatially or temporally–this is extremely problematic, since there may be repetitions that defeat individuation. Even if it is finite, uniqueness is not guaranteed, except by some feature such as the asymmetries of graphs; but few have suggested that material entities can only arrange themselves in such asymmetric arrangements.

[39] True, graph theory is at best the theory of discrete structure based on one symmetric, dyadic relation. Basing apparently continuous phenomena, such as spacetime, in discrete structures such as the lattices of quark chromodynamics is becoming more common, although an argument for a full-fledged reduction is rarely made. Compare D.W. Kueker and M.C. Laskowski, “On Generic Structures,” Notre Dame Journal of Formal Logic, xxiii, 2 (Spring 1992): 175-83.

`[40] Strictly speaking, we should be interested in the number of connected asymmetric graphs of order forty. I lack an enumeration theorem for asymmetric graphs, but would conjecture that they, too, exponentially increase and, in fact, asymptotically approach the number of non-isomorphic graphs for large orders. I owe this conjecture to conversation with graph theorist and my former colleague, H. Joseph Straight.`

[41] In the physical or common-sense case, these would correspond to examples such as “having a flu virus” (internal) and “being in New Jersey” (external).

[42] A. P. Hazen later pointed out to me that D. Scott and M.O. Rabin had discovered a similar technique, described in Rabin, “A Simple Method for Undecidability Proofs and Some Applications,” in Y. Bar-Hillel, ed., Logic, Methodology and Philosophy of Science: Proceedings of the 1964 International Conference (New York: North-Holland, 1965), pp. 58-68, although not the general theory for the description of asymmetric relations in graphs.

` [43] The views we see in a spate of recent works such as F. David Peat, Superstrings and the Search for a Theory of Everything (Chicago: Contemporary, 1988) and P.C.W. Davies and J. Brown, eds., Superstrings: A Theory of Everything (New York: Cambridge, 1988).  Compare Armstrong: "It may be that at some deep level of analysis we come to particulars which are not spatio-temporal and spatio-temporally related," p. 90.`

[45] Just as I have given what I think is a compelling argument why the world, if it is a graph, is an asymmetric graph, I also believe there are further such a priori restrictions on what the world, or the thinkable world, may be. I am not committed to precise Leibnizian claims about the “best” of all possible worlds, or about the plenitude (maximization) of “being” in the world, but I believe there are further such a priori, graph-theoretic reasons why the actual world graph is the single one that is. I believe (although as a regulative or “pragmatically justified” principle) that everything thinkable does have an ideal explanation and that all ideal explanation is ultimately a priori, indeed mathematical. There is one and only one asymmetric graph that has these features and constitutes the world, and thus only one alethically “possible” world or graph. Our sense of possibility and freedom is due to ignorance and computational intractability. For reasons that I do not understand, but suspect exist and are provable, there are in this unique graph (at least fairly) regular connecting paths among a certain subset of objects (subgraphs) that we conceive as mid-sized, phenomenal “physical objects,” and that these paths have the structure we identity as space and time. Perhaps in very large, well-connected, asymmetric graph-theoretic structures, it is always possible to identify paths with certain loose specifications (transitivity, irreflexivity, and so on) that we associate with the somewhat vague, macrostructural notions of space and time.

[46] See Robin Le Poidevin, “Relationalism and Temporal Topology: Physics or Metaphysics?” Philosophical Quarterly, XL (October 1990): 419-32.

[47] What some metaphysics txeat as “individual” discrete physical objects are really quasi-distinct, in appearing at some level (for example, to casual perception)–that is, with respect to some of their internal or external structure—to be metaphysically distinct. But they are not so, just as I share some DNA-patterns with my parents, my siblings, and beyond. More notoriously, the paradoxes of quantum mechanics crucially rely upon what I believe is the false assumption of complete metaphysical, physical distinctness of particles (in, for example, the double-slit experiment). I interpret quantum mechanics in fact, as a reductio of certain assumptions of metaphysical distinctness, or of the assumption of no action at a distance, or both. Similar views have led David Bohm, in inter alia, Wholeness and the Implicate Order (New York: Routledge, 1980), away from formalistic thinking altogether and toward a suggestive holistic Buddhism. (See Paul Teller, ‘Relational Holism and Quantum Mechanics,~ British Journal for the Philosophy of Scien~ xxxvn (1986): 71-81.)

[48] On the concept of nature, see “Natural and Artificial: The Fateful and Obscure Contrast,” in my Artifacts, Art Works, and Agency (Philadelphia: Temple, 1993), chapter 12, and compare my holistic conception of nature with the suggestive views of Arne Naess (and some Eastern conceptions) in David Rothenburg, “No World but in Things: The Poetry of Naess’s Concrete Contents,” Inquiry, xxxix, 2 (June 1996): 255-72.

[49] This unpopular view, and its place in the theory of meaning and reference, is described more extensively in my “Reflections on Iconicity, Representation, and Resemblance: Peirce’s Theory of Signs, Goodman on Resemblance, and Modem Philosophies of Language and Mind,” Synthese, cvi, 3 (March 1996): 373-97.

APPENDIX 2

from Stephen Wolfram – A New Kind Of Science – Chapter 9 Fundamental Physics – Section 7 Space as a Network – p.475

Space as a Network

In the last section I argued that if the ultimate model of physics is to be as simple as possible, then one should expect that all the features of our universe must at some level emerge purely from properties of space. But what should space be like if this is going to be the case?

The discussion in the section before last suggests that for the richest properties to emerge there should in a sense be as little rigid underlying structure built in as possible. And with this in mind I believe that what is by far the most likely is that at the lowest level space is in effect a giant network of nodes.

In an array of cells like in a cellular automaton each cell is always assigned some definite position. But in a network of nodes, the nodes are not intrinsically assigned any position. And indeed, the only thing that is defined about each node is what other nodes it is connected to.

Yet despite this rather abstract setup, we will see that with a sufficiently large number of nodes it is possible for the familiar properties of space to emerge–together with other phenomena seen in physics.

I already introduced in Chapter 5 a particular type of network in which each node has exactly two outgoing connections to other nodes, together with any number of incoming connections. The reason I chose this kind of network in Chapter 5 is that there happens to be a fairly easy way to set up evolution rules for such networks. But in trying to find an ultimate model of space, it seems best to start by considering networks that are somehow as simple as possible in basic structure–and it turns out that the networks of Chapter 5 are somewhat more complicated than is necessary.

For one thing, there is no need to distinguish between incoming and outgoing connections, or indeed to associate any direction with each connection. And in addition, nothing fundamental is lost by requiring that all the nodes in a network have exactly the same total number of connections to other nodes.

With two connections, only very trivial networks can ever be made. But if one uses three connections, a vast range of networks immediately become possible. One might think that one could get a fundamentally larger range if one allowed, say, four or five connections rather than just three. But in fact one cannot, since any node with more than three connections can in effect always be broken into a collection of nodes with exactly three connections, as in the pictures on the left.

So what this means is that it is in a sense always sufficient to consider networks with exactly three connections at each node. And it is therefore these networks that I will use here in discussing fundamental models of space.