The Journal of Philosophy Vol

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 fight. 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 idem 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 middie-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 monadie 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 groun& 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 ex* press 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") /s 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 lan~ guage 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, asymmetic 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: