Difference & Repetition is a wide ranging philosophical exploration, through which Deleuze begins to examine the notion of singularities. The relationship between the early work carried out by Deleuze, and the later writing is not, however, linear or cumulative, but arises from the philosophical focus of Difference & Repetition, which looks to answer two questions;-

1) What is the concept of difference- one which is not reducible to simple conceptual difference, but demands its own Idea, its own singularity at the level of Ideas?

2) What is the essence of repetition- one which is not reducible to difference without concept, and cannot be confused with the apparent character of objects represented by the same concepts, but bears witness to singularity as a power of Ideas?

The arguments presented in Difference & Repetition are a dense maze of interwoven debate. In order to separate the analysis from this maze, and to propose a focused perspective, I have analysed the arguments which clarify Deleuze's manipulation of the singular. The philosophy of Alfred North Whitehead in Process & Reality is a useful tool to unlock the arguments proposed, as the similarities and divergences between the two philosophies are clear. The medieval debate between Nominalists and Realists, and the question of particulars and universals also serve as necessary background to the concept of singularity which is being drawn out through Difference & Repetition. Yet Deleuzean "matter-flow" and materiality, set singularities on a different plane to the Platonic analysis of Whitehead. Deleuze is at pains to resist "idealism", whilst retaining "ideation".

According to Deleuze, the diaphora of Aristotle is a "false transport." Deleuze argues that diaphora never shows difference changing in nature, we never discover a differenciator of difference, which would relate in "respective immediacy", the most universal and the most singular. The differenciator for Deleuze differenciates difference "in-itself", and is a component in his ontological exploration of generalised anti-Hegelianism and the Heideggarian philosophy of ontological Difference. Deleuze reverses the ontological methodology of positing "substance" or "being" as the grounds for asking the questions:- "How does matter change?" (Aristotle), or "How can being gain determinacy?" (Hegel), or "How can being sustain its difference?" (Heidegger); instead, Deleuze asks the question "How can difference sustain its being?" Michael Hardt locates the source of Deleuzean ontology in Bergson, where internal difference is risen to the level of the absolute; in contrast to Mechanism or Platonism, where difference is thought of contingently (per accides). In attempting to think internal difference, Deleuze wishes to ground being in difference, whereupon the internal difference is not conceived as simple determination; but achieves "substantial differences" (per se).

The early Deleuze formulates multiple strategies to untie him from, as he saw it, "Hegelianism and the dialectic";- rejecting prefigured ontology, and the exteriority of difference, which in Hegel's case leads "all the way to contradiction." Leibniz goes deeper than Hegel for Deleuze, through his distribution of distinctive points and the differential elements of a multiplicity, whereas Hegel forces difference into a previously established identity, that of a flat space, where it can be taken where identity wants it to go, into the negative. Deleuze argues that Hegel creates a "false movement", one with words and representation, a movement of abstract universals; difference for Deleuze remains in its own differential reality, always made up of singularities. Representation engenders this false movement for Deleuze, as it is a universal for which there are always unrepresented singularities, who will not recognise the universal precisely because they are singular, the "everyone" of the universal, therefore, does not apply to the singularities.

Hegelians would certainly argue that Deleuze misunderstands Hegel, negation, and interiority as "selfhood". Yet this rarefied "dialectic" debate, does not serve as criticism or analysis of Deleuze, his project is in no way Hegelian; determination, "selfhood" and relations of mediation and identity are not his primary aims. The delineation of strategies to bypass dialectics work throughout Difference and Repetition, in order to prise singularities from their philosophical fields, and to set them into chaotic, creative action, whereby they may be used to positive ends in order to enhance understanding of material fields.

Singularity for Deleuze is beyond particular propositions no less that universality is beyond general propositions. Here the echoes of Whitehead resonate:-Whitehead proposed propositions as hybrids of "pure potentialities and actualities." Singular propositions for Whitehead are the potentiality of an actual world including a definite set of actual entities in complex reactions. Deleuze and Whitehead diverge in that Whitehead wished to extend actual entities to set up relations with eternal objects or "predicates of the proposition." The singular proposition for Whitehead includes in its potentiality the complex predicate finding realisations in the "nexus" of reactions between logical subjects (the definite set of actual entities). The concerns of Deleuze turn to problematic Ideas rather than remaining on the level of singular propositions. Problematic Ideas are not simple entities, but are multiplicities or complexes of relations and corresponding singularities. For Whitehead, the question of the problem is termed in relation to actual entities, each with their own formal existence, entering into objective relations with the actual entity in question. The answer for Whitehead is to posit the "creative action of the universe", always becoming one in a particular unity, and adding to the multiplicity of the universe as many. This for Whitehead is the "concrescence" into unity, which every entity must enter into as a result of its creative action and the establishment of new relation, which Whitehead termed; "innovative becoming." Deleuze characterises the problem of thought by the distribution of the singular and the regular, distinctive and ordinary points; taking place within the description of a multiplicity in relation to the ideal events which constitute the conditions of the problem.

Deleuze and Whitehead are comprehensible as parallel projects, yet I would not wish to posit an identity or assemblage, such as Deleuze-Whitehead, in the manner that Alain Badiou does in his essay concerning, The Fold. The creative difference of Platonic Forms in Whitehead, opposes the actual difference of singularities in Deleuze. Both thinkers are joined more definitely in their appropriation of Bergsonian notions of durée and intuition, and a dynamic relationship to science. In Whitehead's philosophy, temporal endurance (durée) depends on "subjective aim", Whitehead's expression for Bergson's intuition or "conceptual prehension." This temporal endurance selected for any one actuality, determines how the extensive continuum is "atomised" by atomic actualities of a locus in the "unison of becoming." Whitehead's "philosophy of the organism" which presents a coherent cosmology for science in terms of process, then establishes the foundations for mathematical expression of physical science. These complex categoreal conditions, consist generally in satisfying some condition of a maximum, to be obtained by the transmission of inherited types of order.

Deleuze highlights the Bergsonian schema which unites Creative Evolution and Matter and Memory, by beginning with an account of a gigantic memory, a multiplicity formed by the virtual coexistence of all sections of the "cone", each section of the cone being a repetition of all the others and being distinguished from them only by the order of relations and the distribution of singular points. The "one-whole" point of unity, of Whitehead and the Platonists is, for Deleuze, a virtual point, where duration is difference in kind, in itself and for itself. Differences in kind and degrees of difference coexist in a Single Nature, where Bergson spoke of different intensities and degrees in a virtual coexistence, in a single Time or simple Totality. Actualisation of the unity take the form of divergent lines, each of which corresponds to a virtual section and represents the incarnation of the order of relations and distribution of singularities peculiar to the given section in differenciated species and parts.

The direction of the Deleuzean argument leads against multiplicities conceived of as numerical, quantitative multiplicities, of the kind G.B.R. Riemann and Einstein proposed. The Bergsonian proposal, in contrast to quantitative multiplicities, is of qualitative multiplicities which found differences in nature, rather than merely "of degree." However, qualitative and quantitative multiplicities do not act dualistically, but given Bergson's durée, act from unity to multiplicity, virtuality to actuality. This process is, for Deleuze, creative in the Platonic sense, yet without the ordering restraint (final cause) of Platonism, or the restraint of realising the possible (formal cause) in the Cartesian or Scholastic sense. Deleuze does maintain ground for the ideal or transcendental in the virtual, but his process of actualisation is not a degradation, or copy in the real; but it is a creative, immanent, explosive force of life itself.

The analysis of Difference and Repetition also incorporates notions derived from Gilbert Simondon. Deleuze uses Simondon's notion of individuation to effect the passage from the virtual to the actual, in the way that Simondon showed that traditional distinctions between form and matter, individual and milieu, animate and inanimate, specification (of species) and individuation; must be reconceived in terms of information in order to take account of the process of individuation. The prior, metastable, pre-individual state of Simondon sets up "disparateness" of at least two orders of magnitude. These orders possess singularities, which are defined by the existence and distribution of potentials. Therefore, according to Deleuze, an "objective", problematic field appears, determined by the distance between orders. Individuation emerges as the act of solving the problem, the actualisation of the potential, and the establishment of communication between disparateness (Simondon gives the examples of crystallisation and the moulding of clay bricks). Deleuze uses Simondon's individuation as a way of connecting the individual to the pre-individual, without recourse to a formed self, or to a basis in an impersonal pre-individual. The pre-individual is rather associated with singularities, it is still individual, just as the anti-self and the anti-I are. However, in accordance with Deleuze’s strategy to remove any limits, the individual never ceases dividing and changing in nature.

There is an ambiguous nature to the Deleuzean concept of creation, as Todd May indicates. This ambiguity is a tension-limit between concepts, such as the transcendent and the actual, unity and difference, duration and disparateness. Deleuze works to subvert this ambiguity by implanting within it fields of singularities, which subsist below linguistic considerations to mark a coherent perspective, or in Spinozist terms, affirmation. This is not Nietzschean "yea-saying", but it is a characterisation of a transcendental and rigorous empiricism.

The final component in Difference and Repetition which I wish to address is Deleuze's concept of the Idea. Deleuze pays tribute to Leibniz in that his conception of the Idea is an "ensemble of differential relations" and singularities, in which he begins with the inessential and constructs centres of envelopment or folds around these singularities to enact essences. Into this Leibnizian, and Kantian conception of the Idea, Deleuze introduces virtual multiplicities which form through differential relations, "reciprocal determination", and the Idea. The Idea has three dimensions:-

1) Singular points which are embodied in quantities/parts (determinability or the principle of quantifiability).

2) Relations between singular points, which are embodied in qualities/species characteristics (reciprocal determination or the principle of qualitability).

3) Intensities which affect the spatio-temporal actualisation of singular points and their relations (complete determination or the principle of potentiality).

Deleuze qualifies these three aspects in a paraphrase of Kant as being the, "three aspects of multiple reason", which are projected in an ideal temporal dimension; that of progressive determination;- this for Deleuze, is the empiricism of the Idea. The importance of this progressive determination lies in that notions of the regular and the singular, the distinctive and the ordinary, go to make up sense; which depends upon their distribution and distinction. Through the "play" of reciprocal determination of relations, and through the complete determination of singularities, Ideas are formed.

As has been noted, Deleuze pivots on a Kantian axis in his elaboration of the Idea. Sensibility no longer functions as the passive receptor of intuitions, but as the faculty capable of experiencing intensities. It does not present intuitions within the a priori form of space and time, but it encounters a groundless condition of the intensive spatium and the multiple durée. Ideas take pre-eminence as progressive determination, which is a problematic realm of determination inhabited by chaotic fields of singularities.

In the final pages of Difference and Repetition, Deleuze extends his notion of the play of reciprocal determination in terms of chance and nomadic distribution of singularities in the Idea, invoking Heraclitus, Nietzsche, and Mallarmé. However, the extension of the theory outlined above, into the area of art and the multilayered complexes that form around works of art, do not directly concern this argument. The question of singularities in Deleuze does not necessarily entail a "divine game"; but may lead fluidly into a "meticulous material field".

The development of singularities in The Logic of Sense, represents perhaps the clearest exposition of the notion that Deleuze has presented. In the world of sense, with its events and singularities, Deleuze explores the topological landscape of paradoxes, series and a transcendental philosophy, which, following Lewis Carroll, attempts to remain, "on the surface".

Analysing Carroll lead Deleuze to a reading of structure in linguistic terms. The minimal conditions for a structure in general are:- 1) That there are at least two heterogeneous series, one determined as "signifying", the other as "signified": 2) That the series are constituted by terms which exist only through the relations that they maintain with one another: and, 3) That the heterogeneous series converge toward a paradoxical element, their differentiator. The last condition, Deleuze calls "the principle of the emission of singularities." Deleuze’s second condition establishes the consistency by which the concepts of structure and event could be said to be not opposed. Structure includes a register of ideal events, and particular events or singularities corresponding to relations; this combination of events effect an entire internal history (for example, the characters of Lewis Carroll in heterogeneous series connect the singular points of the history and correspond to a linguistic structure). Deleuze essentially derives this notion of structure from his reading of the work of Marcel Mauss and Lévi-Strauss, and his study of the Stoics.

Terms of a similar order to the word Mana of the Melanesian and Polynesia tribes were designated by the structuralists Strauss and Mauss as floating signifiers; which means that they are devoid of meaning, and are "hence able to receive any symbolic meaning whatsoever." This is an instance of incorporeal non-sense. The incorporeal is a Stoic "surface effect", which was divided into four categories by the Stoics;- 1) time or aion, 2) place, 3) the void, 4) lekton or the expressible. Deleuze in The Logic of sense is primarily concerned with aion and lekton, which combine to constitute meaning, or the formation of sense. The reason why Deleuze studied Lewis Carroll was that, for example, the search for the Snark (incorporeal non-sense), sets up a series of "signifying" elements and a series of "signified" elements; which together constitute events through their relations, and converge toward the paradoxical Snark (it does not exist). However, this reading of structure is merely an empirical embodiment of a transcendental structure; the empirical series of the history corresponding to a distribution of singular points, the Snark to an aleatory point.

Classical probability was divided into the epistemic and the aleatory. Aleatory probabilities have to do with the physical state of coins or mortal humans, epistemic probabilities concern knowledge. Aleatory probability thus corresponds to a physical concept of possibility, as indicated by P. M. Boudot (1967), who argued that the aleatory/epistemic distinction derives from de re/de dicto modalities of the seventeenth century. However, as I. Hacking (1975) counters, the de re/de dicto distinction is not identical to either any present distinction, or to any scholastic distinction. Cardano invoked the Aristotelian concept of potentia; the power or ability of an event to happen, which for Cardano explained the aleatory probability of physical events. Jacques Bernoulli, used facility, proclivity and possibility to clarify the same processes of chance. Leibniz introduces the principle of insufficient reason or latterly- indifference. For Leibniz, aleatory and epistemic probability were problems which could be solved via usage of his metaphysical notions; on the one hand we have degrees of makeability in re, which may be taken as tendencies to produce stable frequencies. These are the basis of probabilities in mente. For example, in a particular case if r asserts only that in some chance set-up, the objective tendency is to produce outcome E on repeated trials with relative frequency f; then the probability of the hypothesis that E occurs on the next trial, relative to this data r, is f. This local information had for Leibniz general application, so that in all probabilities which we can distinguish, will have some propensity to be actual. Thus, the calculus of probability, makeable or facile in re, corresponds to that which is probable in mente (the possible worlds metaphysics of Leibniz, where possibilities are conceived by degrees in the divine mind).

The use of an "aleatory point" in Deleuze is a paradoxical appropriation of the Leibnizian hypothesis. Deleuze posits it as the conditions of the question rather than the physical probabilities in re. This point is a mobile element which frames the question, and is developed in problems of singular points corresponding to series. The structures are not stable, but establish nomadic distribution of singular points whereby they may be determined. Traditional transcendental philosophy argued that the relation of empirical consciousness to its objects must be divinely founded, where the epistemic conditions apply. Deleuze argues that the conditions of knowledge must be the same as the real objects of knowledge. Therefore, the divergence of Deleuze is that; while accepting a transcendental foundation, he refuses to model the relation between consciousness and its objects on that which it founds. Deleuzian metaphysics rather treats consciousness and its objects as emissions from a pre-individual world of impersonal singularities, where the aleatory point can traverse "men, plants and animals independently of the matter of their individuation and the forms of their personalities." Deleuze does not curtail the transcendental or divine conditions of knowledge, for him, in the realm of pre-individual singularities, the traditional epistemic distinctions simply do not apply.

The notion of pre-individual singularities owes much to the Leibnizian monad. Through the study The Fold, Deleuze develops monads against the backdrop of the singular. Within each monad exists singularities, which are the "requisites of the individual notion." For Leibniz, at the same time, only the individual exists by virtue of the power of the concept (monad or soul). For Deleuze, the power of the concept does not consist in determining a genre to infinity, but in "condensing and prolonging singularities." The Leibnizian notion that each monad "expresses the world", is for Deleuze the effective extension of constitutive singularities in all directions, up to the singularities of other monads, whereupon the series diverge or converge (the incompossibility or compossibility of singularities). The condensation and prolongation of singularities does not result in generalities for Deleuze, but it results in events, where the individual exists in its own right. The individual is the actualisation of pre-individual singularities, which implies no previous determination. (Deleuze gives the example of predicates in the book of Genesis, which are considered sub ratione generalitatis, tying the predicates to generalities and by degrees to events, by the prolongation and condensation of singularities).

Alain Badiou notes that Deleuze rectifies the traditional objection to monadology, that Leibniz prevents any thought of the relation. In Deleuzian terms, Leibniz thinks nothing but the relation. The relationships he thinks are differential, and link all elements of the expressible world, from the whole to its parts; the predicate is above all relation and event, and not attribute (essentialist and eternitarian judgements of predicate attributes are contrasted to the Leibnizian monadology by Deleuze, in that he characterises monadology through the immanence of the procedure, and in its accomplishment in a flux).

Deleuze holds an elusive relationship to differential calculus and to the mathematical problems involved with series and number. In Expressionism in Philosophy: Spinoza, Deleuze points out that: "Dissociated from any numerical distinction, real distinction is carried into the absolute. It becomes capable of expressing the difference in being and consequently it brings about the restructuring of other distinctions." Real distinction takes Deleuze into his conception of the singular, the singular points determining problems in the series. In differential calculus, distribution of singular points correspond to the values of differential relations. In the Deleuzian rewriting of the monad, the continuum of singularities is entirely distinct from the individuals which envelop it in variable and complementary degrees of clarity. Leibniz thought of the individual as the centre of envelopment, enveloping the singularities inside a world and on its own body. In Deleuzian terms, the individuals are constituted in the vicinity of singularities which they envelop; they express worlds; "as circles of converging series which depend upon these singularities." Differential relations and their contiguous singularities go to make up the expressed world.

The argument pertaining to Leibniz’s theory of infinitesimal calculus, consists of the dispute about real numbers. Cauchy’s real numbers ran through Weierstrassian real numbers and infinitesimals and those numbers which differed from Weierstrassian real numbers by infinitely large numbers and/or infinitesimals: the later Weierstrassian points were finite Leibniz-Cauchy points deprived of their infinitesimal neighbourhoods. Cauchy’s theorem, simply expressed, was that the function sequences must converge everywhere (i.e. on the entire Leibnizian continuum). Bolzano clarified the difference between the rich Leibnizian continuum, and, as he called it, its "measurable" subset, which has become known as the set of Weierstrassian real numbers. For Bolzano, the field of "measurable numbers" constitutes only an Archimedean subset of a continuum, enriched by non-measurable, infinitely small or infinitely large quantities. This distinction has been more latterly expressed by Robinson (1967), as standard and non-standard numbers.

The solution of Cauchy to the problem of infinitesimals in the Leibnizian continuum, was however not tried; because, as Lakatos expresses it, it was untestable in the mathematical sense. Weierstrass’s theory, with easily specifiable non-uniformly convergent sequences, or Robinson’s particular non-standard analysis (which is an elementary extension (in Tarski’s sense) of real analysis), are both testable. According to Lakatos, the criterion of testability is the decisive factor which makes the Leibnizian infinitesimal continuum less heuristic than Weierstrass’s or Robinson’s; however, Deleuze continues in the direction of Cauchy’s theorem, and increases the convergence of series and the distribution of singularities. In a continuum of singularities for Deleuze, continuity has the convergence of series as its ideational criterion. Ideation takes the "real distinction" from arguments about numerical sets, and demarcates fields of singularities as organising factors of other series.

Deleuze also uses the work of Albert Lautman to organise his break from traditional argumentation about differential equations. Lautman distinguished between fields of singularities to which no direction has been attached, and integral curves with the form they take on in the vicinity of the singularities in the field of directions. Lautman designates two distinct mathematical realities as complementary. The nature of the singularities of the field is defined by the form of the curves in their vicinity. For example, a U curve, symmetrically bisected along its vertical axis, is rotated 360º along that axis (hence obtaining a three-dimensional shape). All the points of that curve will move, except for the point at which the axis bisects the curve- the singular point of the curve. One cannot specify the value of that singular point without determining the values of all the other points of the curve. However, one can determine that such a point exists without specifying its value; this for Lautman is the general topology of the problem, the abstract domain of the possible curves that can pass through that singular point. For Deleuze, the problem may be conceived as a virtual structure or relations of singular points; the specific values of the solutions of various equations may be conceived as the actuality of the problem.

At this point it is possible to begin to make connections between the philosophical enquiry of Deleuze, and post-modern science. The development of singular points, has yielded many fascinating results in complexity, chaos, bifurcation and related fields. The philosophy of Deleuze encourages a rigorous empirical investigation, yet retains the grounds by which the research may be organised and distributed. The final question which concerns us from The Logic of Sense, is the question of sense itself.

"Good sense", gives itself a singularity so it may stretch itself over the whole line of ordinary and regular points which depend on it; these points, however, also "avert and dilute it." Deleuze designates the world of sense as "problematic". Like chemical elements, we know the existence and distribution of singular points before we know their nature. This allows Deleuze to give an entirely objective definition of the term "problematic", in respect to the nature of directed singularities and their existence and directionless distribution: all of which depend on objectively distinct instances. The task that Deleuze sets out, is how to "make" the neutral world of sense, with its events and singularities, "the empty square to make pre-individual and nonpersonal singularities." On the level of the event, sense "emanates from nonsense", as if from the always displaced paradoxical instance. This, as we have seen, is the aleatory point, giving a nomadic and non-sedentary distribution, wherein each system of singularities communicates and resonates with the others. For Deleuze, the divergence of the affirmed series forms a "chaosmos", and no longer a world; this as Deleuze terms it sets up the disjunction posed as a synthesis- a "diabolic principle."

Deleuze’s vital engagement with Nietzsche is the ingredient missing in this diagnosis of his philosophy of the singular in The Logic of Sense. Deleuze regarded Nietzsche’s philosophy as an exploration of these pre-individual singularities, the Dionysian free and unbounded energy giving reign to their movement and formation in fields of singular-becoming. However, for this study, the important connection being made via the singular is between the philosophical system which attempts to liberate singularities, and the scientific perspective which may incorporate, digest and apply them.

Deleuze and Guattari distinguish between what they call "nomad" or "minor" science and "State" or "Royal" science. This distinction works on the level of scientific discovery at the bequest of becoming and heterogeneity as opposed to the stable, the eternal, the identical, the constant; which act within the models of science proposed by State science. That becoming is itself a model, and no longer a secondary characteristic, was raised by Plato in the Timaeus, yet dismissed beneath the pressure of Royal science and the need for origination. Plato and Aristotle, supposing a "neutral plastic material", and an original model, instigate the model as eidos and not morphe, and therefore formalise the status of the formal component in opposition to the nomadic formality, which focalises the activity of formation itself. The scientific system of Deleuze and Guattari, beginning with the "nomad", is an attempt to shift science onto the "hydraulic" model. This model "consists in being distributed by turbulence across a smooth space, in producing a movement that holds space and simultaneously affects all of its points, instead of being held by space in a local movement from one specified point to another."

The ideas of Deleuze and Guattari stand in an altogether eccentric relation to, for example the "normal science" of T.S. Kuhn. Although Kuhn's analysis in the Structure of Scientific Revolutions is historical, he does not make the nomad/state science distinction. Science is used by him as a blanket term for research that takes previous theoretical frameworks, finds anomalies, and given sufficient evidence for the anomaly, eventually allows new paradigms to emerge in order to govern the research of the scientific community. The history that Deleuze and Guattari trace is not dependent on a history of "normal science". Nor can we push relativism further and accept Feyerabend's notion of ruleless "anarchist" theory choice as nomad science. The work of Deleuze and Guattari is not a political indictment on choice, their theory is more rule laden than Feyerabend's, rules conforming to mathematical models of seemingly chaotic singularities; these singularities garner a determining role in the shifts of scientific research, rather than the paradigm shifts of Kuhn or the workings of irrational relativism in Feyerabend.

By way of criticism and analysis, I shall compare nomad science with Kuhn and the responses of P.M. Churchland and Hilary Putnam. However, the notion of singularity that is brought out through nomad science cannot be reconciled with a state orientated analysis of the philosophy of science. It must not be forgotten that the analysis of Deleuze and Guattari incorporates the machinations of the so called "war machine", which from the beginning informs the notions and alters the placement of scientific discovery. Nomad science is aligned with the war machine, even though State science appropriates and transforms elements of nomad science. Normal science in the context of Kuhn is the State science of Deleuze and Guattari; as Churchland points out, the paradigm for Kuhn is a prototypical application of some set of mathematical, conceptual, or instrumental resources; an application expected to have distinct but similar instances, which it is the job of normal science to discover or to construct. The groups that Deleuze and Guattari want to select are the minorities, on the fringes of collective bodies, that reconstitute equivalents of the war machine, in sometimes quite unforeseen forms, in other times in specific constructions such as bridges or cathedrals; they even can be seen to be rendering judgements, or making music, or instituting a science.

This criticism and analysis rests on an uneasy alignment which, however, is joined by common concerns for scientific innovation and the desire to comprehend the mechanisms by which innovation takes place. To make this alignment feasible requires scientific theory to be thought of more in the terms that Lakatos spoke, rather than Popper or the logical positivists. Lakatos's interest in the "dynamics" of theories sides more easily with Kuhn and Deleuze and Guattari. The scientific research programs of Lakatos are certainly not nomad science as Deleuze and Guattari would term it, yet State science does seem to conform to the systematic application of positive and negative heuristics which Lakatos investigated. The concern of Deleuze and Guattari to outline their treatise on nomad science, does not ignore State science, or prioritise one over the other. They are two formally different conceptions of science, and ontologically, or in Deleuze and Guattari’s terms "machinically", a single field of interaction, in which Royal science continually appropriates the contents of nomad science, while nomad science continually cuts the contents of Royal science loose. Deleuze and Guattari do see most serious philosophy as concentrating on State science, yet justify their project with names such as Democritus, Menaechus, Archimedes, Vauban, Desargues, Bernoulli, Monge, Carnot, Poncelet, Perronet; all of whom suffered some reprimand at the hands of state authority.

In the language of Churchland, the scientific community of Kuhn is a network, which learns or fails to learn new paradigms. Kuhn's insistence on the typical resistance shown by the scientific community to change the current paradigm, is for Churchland, a process of learning which is characterised by gradients and local and global errors. The most threatening position is one where the error messages are not yet zero, but where every small change in the system produces even larger errors than those currently encountered. Churchland argues that global minima do occur, and that the only way to escape them is to introduce some sort of random noise into the system, in the hope of bouncing the system's configuration point out of a potential dead-end. Even if there is an escape route for the local minimum along one or more dimensions, the system may take a long time to extricate itself from the impasse. Churchland also argues that systems may be inhibited by what he terms "training sets." Diverse examples will not necessarily result in the system progressing away from its earlier configuration, unless the frequency of the examples is high enough to overcome the gravity of the false minimum that captured the initial training set.

Kuhn locates a manipulable "insulation" within the scientific communities, which shields the scientists against problems they will not be able to solve. The insulated scientific communities then "progress" through solving readily achievable problems, rather than taking alternative conceptions seriously. Kuhn suggests that scientific education plays a causal role in this insularity, and exposure to more diverse stimulus, as frequently occurs in the arts, would result in a more rounded and flexible science. Churchland makes much of "conceptual change" in his thesis, which is akin to the paradigm shift of Kuhn, and is beyond the reach of theoretical credibility, which places emphasis on simplicity in syntactic terms, e.g. (Hempel 1965, Scheffler 1963, Salmon 1966), all of whose theories battled with difficulties of rule-governed updating of a system of sentences or propositional attitudes.

Deleuze and Guattari figure the stricture of change on the formalisation of State science. For them, the state does not appropriate the "mobilised projections and inclined planes" of the war machine without submitting them to civil and metric rules that strictly limit, control and localise nomad science. These rules also keep nomad science away from having repercussions in the social field. Deleuze and Guattari give the example of differential calculus, which was for a long time given a merely parascientific status, and was labelled a "Gothic hypothesis". Royal science attempted to improve its status, but only on the condition that continuous variation be eliminated and ordinal rules imposed, (Deleuze and Guattari identify Carnot's ambiguous situation as an example pertaining to this point). Repressing the political or social conceptions of nomadic theories, works for Deleuze and Guattari in favour of the state and logos contra nomos. This nomos is the law creation of the war machine or nomad science, which for Deleuze and Guattari is fundamentally numerical or arithmetic. Contrasting Greek geometrism with Indo-Arab arithmetism, the latter in the terms of Deleuze and Guattari implies a nomos opposing logos, arithmetic and algebra arise in the strongly nomad influenced world.

The distinction between nomos and logos could be rectified in terms of Kuhn's competing paradigms, which prove to be a choice between incompatible modes of community life. Kuhn argued that such incompatibility required the study of socio-historical parameters to elucidate the change in scientific paradigm, as a reflection on scientific communal activity. Criticisms of this approach such as (Fodor 1968), which sought to reduce the importance of perceptual and developmental psychology, and deplored Kuhn's instrumentalism/relativism, could also be applied to Deleuze and Guattari, yet Deleuze and Guattari do not prioritise perceptual and developmental psychology, in that their distinction between nomad and state science does not rest on the perception and application of new theories, but rather on the historical appropriation of theories by various communities or "bodies" as they term them.

Deleuze and Guattari could be cogently termed "fluid realists" in that one scientific model adequately describes objective reality; models themselves, as they indicate, would be switched to the hydraulic-dynamic approach, which is proto-realist and outstrips paradigms and socio-historical perspectives, yet eludes the normative structuration of the state. Nomad science is akin to instrumentalism, in that it is linked to the concerns of the practising engineer or scientist, theories are evaluated in terms of their pragmatics. As Deleuze and Guattari frame the question, it is the problemata of the war machine which is projected into abstract knowledge, eccentrically and without restraining any passages to the limit, vortices, or projections.

J.L. Casti distinguishes between science and pseudo-science. He outlines the characteristics of pseudo-science and in so doing describes the various outlandish attempts at explanation that do not impinge on what he terms the "norms of social activity of science." Nomad science cannot be identified with pseudo-science, as it is quite clearly connected to the "norms of science" via the nomad/state tension limit. Pseudo-science can be better defended using, for example, Feyerabend's "anarchist" theorising, whereas nomad science is more strictly limited and defined by that which escapes the power structure set up by the state. Deleuze and Guattari must define this structure if their nomadic conception is to be clearly understood, and scientific innovation is to be fairly assessed. However, this definition is exactly what reprimands and constrains nomad science: from the perspective of the nomad it is always fluid; in contrast to the state-oriented definition which would have one conception legislated, thereby denying multiple structures the necessary flexibility to encapsulate various socio-historical conceptions and such anomalies as dissent. To make nomad science clearer, Deleuze and Guattari examine more explicitly what it uncovers and the way in which it achieves this "uncovering".

Kuhn emphasised not only the impact of nature and logic, but also the techniques of persuasive argumentation effective within scientific communities. Deleuze and Guattari speak of "journeymen", such as Garin de Troyes, who wrote of operative logic enabling the initiates to draw, then hew "in penetration in space", so that "the cutting line propels the equation." This leads the nomadic science of Deleuze and Guattari to be more immediately "in tune" with the connection between content and expression in themselves; content and expression also encompassing both form and matter. Matter for the nomad science of Deleuze and Guattari is never prepared or homogenised, but is essentially laden with singularities, (which are themselves a form of content). Nomad science follows connections between singularities of matter and traits of expression, and rests on the level of these connections, whether they are natural of forced. This science is not expressed formally, but constitutes a dynamic connection between pertinent traits which it uncovers through its operations.

The content and singularities of Churchland are the structure of the biological brain. Certainly Churchland's science is not nomadic, yet the centrality of dynamism is a pillar of his conceptual framework, binding the tenability of conceptual change to the discovery of complicated networks in the brain. Churchland theorises this dynamism in his PDP (Parallel Distributed Processing) approach, which sets up a position whereby the activation patterns across large populations of neurones portray explanatory understanding and perceptual recognition as being different instances of the general achievement of prototype activation. This prototype activation model consists of what he calls a "novel and unorthodox conception of what cognition consists in." Churchland elaborates his theory in order to detail the way in which prototype activation depends upon input and output vectors, which are perceptually encountered representations in well-trained networks (achieving explanatory understanding). Even though Churchland makes his theory open to conceptual change, he modifies it to be in terms of conceptual redeployment, so that the prototype vectors can be placed as unifying concepts for understanding. This appears to belie Churchland's scientific realism, in contrast to constructive empiricism of the type Van Fraassen expounds (1980). The dynamics of cognitive activity for Churchland depends upon global excellence of theory, to maintain a commitment to truth on levels other than the purely observational. Churchland rejects Van Fraassen's selective scepticism in favour of a broader scepticism, which prioritises the best theories (exhibiting simplicity, coherence, explanatory power), and reconciles unobserved ontologies with observed ontologies. (Churchland argues that Van Fraassen's constructive empiricism collapses into Hume's problem of justifying the inference to unobserved entities). The detailed exploration of the brain structure, is the empirical project that Churchland believed would yield results which are not merely constructive and reconstructive, but have real ontological, epistemological, and perhaps even metaphysical consequences; and will enhance the realist position in favour of a strong scepticism and thoroughgoing understanding of cognitive dynamics.

It would seem that the position of Deleuze and Guattari and nomad science is closer to Van Fraassen's constructive empiricism than to Churchland's sceptical realism; in that no "true", or globally efficient theory is being aimed at. Even if we accept Churchland's claim that his unorthodox realism is more pragmatic that Van Fraassen's, the epistemic endpoint crosses over into Royal science with the legislative logos opposing the nomos of the nomad. Van Fraassen's position is an instrumentalism counter to realism, which, as has been stated, bears an initial resemblance to the nomadic conception of Deleuze and Guattari. Yet Deleuze and Guattari wish to take the conception further, into a field which even cuts away at the "autonomous development" of science. Thus the observational "descriptive excellence" of Van Fraassen is not criteria for epistemic nomad results, but is more appropriate to a state body of knowledge.

Deleuze and Guattari distinguish between scientific procedures which involve "reproduction", and those which engender "following." Reproduction and following, in their terms, are not at all the same, the ideal of reproduction being a part of Royal science along with deduction and induction. Instrumentalism would fold into State science in that its emphasis on observation implies a fixed point of view. This designates an object or fact, which is external to the procedure. Nomad science is the attempt to characterise what is internal to the procedure, and to understand how this internality uncovers the facts without theorising them onto fixed or homogenised planes.

The notion of following links the proposed "hydraulic model" to the flow of matter, corresponding to the Deleuzean reclamation of Pre-Socratic science, and his revaluation of the Pre-Socratic world. With this in place, nomad science situates its field in an "objective zone of fluctuation" that is coextensive with reality itself. Realism then extends to nomad science to the extent that this hydraulic model is realised, and that the singularities in matter may be followed by the nomad scientist. Epistemic considerations appropriate to nomad science are quintessentially problematised, being solved at "ground level" as part of the operation of construction, for example, in the building of Gothic cathedrals. Nomad (or ambulant) science invents problems tied to collective, non-scientific activities, whose scientific solution depends on introducing a theorematic apparatus and organisation of work into the procedure. For Deleuze and Guattari this apparatus and organisation is not the dynamic element within discovery, but is a "laminar movement, that striates space, that goes from one point to another and is weighty." The dynamic space of the nomad is "smooth, tactile, containing manual actions of contact, rather than being visual like Euclid's striated space."

Kuhn states, in a Popperian mode, that normal science does and must continually strive to bring theory and fact in closer agreement, and that activity can easily be seen to be testing, or a search for confirmation of falsification. We have seen how the normal science of Kuhn is assignable to the State science of Deleuze and Guattari. Kuhn's conception of normal science holds until the critical moments, the "paradigm shifts"; whence he is forced to turn to notions of "crisis", "reconstruction", "Gestalt", "extraordinary research"; all of which indicate singular points in Deleuze and Guattari. That Kuhn's socio-historic treatment of the subject of social revolution wells and pivots around these points is testament to the conceptual difficulty which his relativism contests to. In short, normal or State science cannot deal with paradigm shift until after it has been assimilated in to the axiomatic mainstream of the community in which it takes place. Kuhn is forced into vague, unrigorous, multiple conceptual overlaying, to cover up the historical cracks, yet these cracks are exactly the object of his analysis. The alternative of nomad science is not to see them as cracks, but flows, and then not to determine these flows, but to follow them.

It could very well be pointed out that Deleuze and Guattari merely seem to be speaking about art, or the creation of the exceptional craftsman; the question arises:- Why did they lay claim to the term science at all? Yet Deleuze and Guattari disclaim any pretence to be doing science, their analysis does not prioritise either state or nomad science. If the work of the nomad was assigned the nomenclature of art, an implicit prioritisation of state over nomad activity would have occurred. By merely distinguishing between nomad and state, Deleuze and Guattari are examining the nature of science and scientific discovery, without falling into hierarchical socio-economic value systems, and their allied complexes, such as "culture", "utility", or "genius".

By attempting to detail the work of nomad science, Deleuze and Guattari are also exhibiting the following of the hydraulic model, by explaining what they see as the unhinging of conceptual thought from constraint and local minima, they are also attempting to augment this unhinging. It is unfair to suggest that State science does not also unhinge conceptual thought, yet for Deleuze and Guattari the way in which this unhinging is achieved is through the nomad, the state always arrives later on the scene, with its complex legislative apparatus.

Hilary Putnam has attempted to seal up the nonrigorous critical moments in Kuhn's vital thesis in favour of a "sharper formulation." Putnam locates Kuhn's difficulty as being "radical subjectivism", in respect that data is not able to establish the superiority of one paradigm over another, as data itself is perceived through the particular paradigm which one is working through. Putnam suggests that Kuhn's paradigm shifts may be rectified in the direction of Popper's work, and by using the notions of "auxiliary hypothesis" and "conventionalist stratagem." Auxiliary hypothesis spreads the task of falsifiability to encapsulate the "total system under test"; a conventionalist stratagem is needed to save a theory from a contrary experimental result by making an ad hoc change in the auxiliary hypothesis. Popper asserts that it is a fundamental methodological rule of the empirical method to avoid conventionalist stratagems. Putnam, however, advocates ad hoc changes for specific purposes, which are, as he terms it, "highly reasonable". The notion that theories such as the law of Universal Gravitation are fixed systems, strikes Putnam as being absurd, and he therefore repositions them in terms of auxiliary hypotheses (Putnam prefers the term statement to hypothesis, which he deems to be misleading as he wants it to apply, for example, to boundary conditions). Putnam's formulation veers to a highly practical, empirical notion of testing in which scientific theories are vulnerable to data, and in which scientific discovery and change depend upon the close inspection of the world.

Given Putnam's thesis, the hydraulic model of Deleuze and Guattari seems to be ill-conceived on the grounds that it is not open to testing. Yet testing implies a knowledge, or set of principles which the scientist is working on, even if they are merely the principles of tests in general. The hydraulic model, instead of setting up more complex testing units, instigates a problematisation of knowledge, indeed an inventing of knowledge. Yet an ambiguity does remain in that the hydraulic model stands in such a peculiar relationship with the facts. Facts, here loosely termed, are the singularities of which they speak. Nomad science is a treatise which attempts to show how the singularities are uncovered, but not exactly what they are. The "testability" of a Gothic cathedral is ambiguous to a theory about "what a Gothic cathedral is". In order to clarify their position further, we shall look at their example at the end of the Nomadology, that of metallurgy.

Deleuze and Guattari state that it would be useless to say that metallurgy is a science because it discovers constant laws. Metallurgy is inseparable from "several lines of variation"; which can be grouped under two rubrics: 1) singularities, 2) the operations associated with them as processes of deformation or transformation. The singularities are here identified with "traits of expression". The melting of iron at high temperature, successive decarbonisations, in the example of the crafting of an iron sword are its traits of expression, as well as the external piercing rather that hewing and design by inlay. The position of the nomad scientist is one of working through these singularities, in fact "constellations of singularities", and making the operations converge upon several assignable traits of expression. Where the singularities converge, Deleuze and Guattari speak of the "machinic phylum", or technological lineage; which is ideally continuous or a flow of matter-movement conveying singularities and traits of expression. Around this convergence, Deleuze and Guattari also name assemblages, these can be termed "cultures" or even "ages", which can "select, organise, and invent" singularities by bringing them to the surface of the phyla. Assemblages and phyla are inseparable, they form a complex dynamic relationship, yet assemblages organise the direction of the flow across the phyla. The nomad scientist is concerned with the machinic phylum, in using singularities to make the products of their trade, for example, iron swords, following the matter-flow to produce unique artefacts (this is in opposition to, for example, the mass production of cars).

Deleuze and Guattari also speak of the nomad scientist keeping their knowledge of the flow of matter away from state control, which would impose the distinction between matter and form (the hylomorphic model, which Simondon criticised), and, therefore, lose many of the dynamic, topological forms which Deleuze and Guattari term as "variable intensive affects." They argue that this is not initiating an occult or mystical knowledge, but is necessary for the survival of nomad science. Translation to a model is of course conceptually legitimate, but it is not the question they are concerned with; the point at issue is "what gets lost in the translation." The treatise on nomad science is an attempt to reclaim some of the lost effects.

Critical analysis of nomad science rests on comprehension and engagement with the entire project, which is not designed to be artificially separable. Unlike Churchland, Popper, Kuhn or Putnam; Deleuze and Guattari are not ostensibly concerned with constructing a philosophy of science, but render a synthesis which works locally to embrace global features; a Kantian perspective applied to the war machine. They do not construct a globally applicable theory of discovery or change, yet this distance from "the logic of discovery", sets out a tension limit between the uncovering of traits of expression or singularities, and their formalisation into scientific axioms. This tension limit is the working of the socio-economic environment, "the assemblage" which grows around a convergence of singularities and the machinic phylum.

Kuhn's paradigm changes reduce to the problem of historical continuity (a state-oriented difficulty): Popper and Putnam's analysis breaks down into scientific methods of falsifiability and testability: Churchland's philosophy of the mind becomes an empirical quest to integrate findings about the brain into theories of cognition. Deleuze and Guattari do offer an alternative perspective of scientific (nomadic) change and a corroboratory account of State science (though this is not their primary concern). The relationship between the two types of science holds a convincing explanation as to how scientific paradigms change, as it defines a way of looking at science from two universally distinct perspectives. From the fluid perspective of the nomad, science never changes (or is always changing), as they are defining the change through their actions. From the perspective of the state, paradigm shifts offer a legislative problem, which it must at all times appropriate for itself and reconcile to its communal, normative ends.

Hubble’s Law, which states that the universe is expanding symmetrically, posits a single point at the start of the universe, from which it has expanded from. Einstein rejected this view of the universe, as he regarded this barrier to knowledge as being absurd: any calculations before such a singular point would yield nonsensical results. To counter the theoretical logic of equations localising a singular point at the beginning of the universe, Einstein maintained an arbitrary "cosmological constant", which represents a force of unknown nature, that would counteract the gravitational attraction of the mass of the universe.

George Smoot expounds another strategy in order to avoid the singular point; this is that the universe is expanding asymmetrically. The universe could then oscillate, alternately expanding and contracting. The "end-point" at the end of a contracting phase could then be avoided. If one direction in the phase was collapsing more quickly than the others, then the universe would "pan-cake", pass through, and start expanding outward again before the other directions arrived.

The quantum mechanical (Heisenberg) uncertainty principle tells us that both matter-energy and space-time must fluctuate. According to Dennis Sciama, this not only provides the explanation of the origin and isotropy of the cosmic background radiation, but it is also a way of escaping the initial singularity. The fluctuations produce curvature of the space-time continuum, which weakens gravity’s effect. Thus a weakened gravity violates the conditions needed for a singularity theorem to be true.

Inflation theory, of the type Alan Guth calculated, reveals that we can still conserve energy and make everything in the universe starting from practically nothing. For inflation to occur, the energy density in space depends on the fact that the total energy, (i.e.- the energy of space minus the gravitational attraction of the other parts of space), equals nothing. This would seem to concur with the positing of a singularity to produce the mass of the universe. Yet the cosmological statement of the universe as being the product of a singular initial state, shaped by singular laws of nature, leads to speculation as to the possibility that minute variations in the value of series of fundamental properties of the universe, would result in huge fluctuations in the universe as a whole. This is a variant of the Vitalist/Mechanist dispute, which takes speculation about emergence as being either "top-down" or "bottom-up". According to Chris Langton, Mechanists only see the local interaction leading to emergent global structures, while Vitalists only perceive emergent global structures effecting local interaction. Cosmologically, singularities work as a "bottom-up" structure. Theories which contain singular points tend to approach their existence with scepticism, and the inevitable resolution to argue them out of the picture. A point at which laws of nature and mathematical calculation breaks down is an uncomfortable result to achieve if serious scientific determinism is being aimed at.

Roger Penrose (1965), proved the existence of singularities (or black holes), by using the way in which light behaves in general relativity, with the fact that gravity is always attractive. A star collapsing under its own gravity is trapped in a region whose surface eventually shrinks to zero size. Since the surface of the region shrinks to zero, so too must its volume. With all the matter in the star compressed into a region of zero volume, so the density of matter and the curvature of space-time will become infinite. Stephen Hawking points out that any observer remaining outside the singularity would not be affected by the failure of predictability, because neither light nor any other signal could reach him from it. This "obscurity" from any outside view by an event horizon, is known as the "weak cosmic censorship hypothesis". Yet if we maintain a theorem such as Richard Feynman’s idea of;- sum over histories uncertainty principle; which places limitations on the accuracy of all predictive calculation, we may, according to Hawkings, at the same time remove the fundamental unpredictability of the space-time singularity.

Hawkings notes that the singular points, with an infinite gravitational field, indicate that classical gravitational theory does not hold for all the universe. To counter this, Hawkings postulates quantum theory without singularities; quantum gravitational effects alleviating the infinite strain on the space-time continuum of classical gravitational theory. This has important consequences, such as, the universe can only be finite in what he calls "imaginary time", yet it is without boundaries or singularities.

The solution Hawkings offers is a combination of quantum mechanics and general relativity. The possibility that arises is of space and time together forming a finite, four-dimensional space without singularities or boundaries, "like the surface of the earth, but with more dimensions." Using a similar strategy to Sciama, Hawkings looks to avoid the singularities of the general theory of relativity, and its need for the "big bang" or the "big crunch". In part this is an attempt to extend the domain of cosmological science everywhere, and to recover the singularity on behalf of the scientific observer. Even though the observer cannot exist inside a black hole, he or she may then explain its behaviour using four-dimensional space, that, in effect, traverses the singularity.

The strategy of cosmology, whilst recognising singularites, where the laws of science break-down, yet rectifying them by theorising new pictures of the universe; contradicts and compliments the theory proposed in this study. This is due to the theoretical problem of the observer, which the perspective of the singular deals with. The problem of the observer is a problem of knowledge and facts, this is where singularities are the modality of the process. The individual does not exist as an observer, but explores the threshold of pre-individual singularities, which are always converging and diverging in series and explanation. Knowledge is not instituted into a state model, but acts by self-referential difference to follow the traits of expression and phase changes into new scientific areas. New models are set up, yet they are always open-ended, lateral, and not achieving global efficiency of explanation, but local effects of complexity. Cosmology could then be grounded in complexity, and definitively avoid teleology.

The complex mathematics of singularities involves phenomena such as bifurcations, attractors and repellors. These singularities involve points or thresholds in the rate of flow of matter and energy. Scientists have attempted to designate these singularities as mechanism independent, yet structure building, or "mathematical mechanisms", so that they may be incarnated in many different physical mechanisms, for example, at one level in physics, or phase transitions in non-organic matter; and at a higher level in chemistry and biology, or in animal lineage and cultural/technological development.

Poincaré’s formulas, which were an attempt to solve the three-body problem, that have led to the representation of systems via "phase portraits", were, for many, the beginning of singularity mathematics. By creating an abstract space with as many degrees of freedom as the system requires, the system under inspection may be simplified sufficiently for the scientist to observe its general traits. Of special interest are the so called attractor and repellor points, which locate drifting trajectories towards and away from themselves respectively. Simple attractors and repellors are stable, e.g. a thermometer would have its attractor point as 98.4º F. The onset of investigation into this mathematical field of investigation, had yielded many different singular points in addition to the initial simple attractors and repellors. Closed loop attractors define the oscillation of systems between two extremes. "Strange" or "chaotic" attractors represent turbulent behaviour and the boundaries between "basins of attraction" (areas of phase or state space inhabited by simple attractors).

Jacobi invented the term bifurcation for the crossing point between his series of ellipsoids and the Maclaurin series. At this point, one type of attractor is transformed into another. These points are linked with the onset of self-organisation by, for example, Prigogine (1984). In the case of bacteria, at far-from-equilibrium chemistry they show possible "adaptation" of chemical processes to outside conditions. This contrasts strongly with equilibrium situations, in which large perturbations or modifications of the boundary conditions are necessary to determine a shift from one structure to another. Prigogine develops the bifurcation thesis with "cascading bifurcations". For example, the "Feigenbaum sequence" is characterised by a general feature that for a determined range of parameter values, the system’s behaviour is periodic T; beyond this range, the period becomes 2T, and beyond yet another critical threshold, the system needs 4T in order to repeat itself. The system is thus a succession of bifurcations, with successive period doubling, and represents a system going from simple periodic behaviour to the complex aperiodic behaviour occurring when the period had doubled ad infinitum.

The three "levels" of the mathematics of the singular are;- 1) specific trajectories corresponding to objects in the actual world, 2) attractors/repellors corresponding to the long-term tendencies of these objects, 3) bifurcations corresponding to spontaneous mutations of the long-term tendencies of the objects.

Stuart Kauffman has used the science of the singular in his study of evolutionary systems (1993). Kauffman states that in order to study the evolution and behaviour of complex biological systems, a new kind of "statistical mechanics" is necessary. This mechanics defines "ensembles of dynamic systems" and seeks "generic behaviours" in "distinct ensembles". The generic properties then become the macroscopic observables expected from the theory. Kauffman uses random Boolean networks, which are massive parallel-processing systems linking the activities of extremely large numbers of variables. The networks exhibit three realms; 1) ordered, 2) complex, 3) chaotic.

In the chaotic regime, attractors are very large and scale as an exponential function of the number of binary variables in the system. Attractors are small in the ordered regime, containing behaviour into localised regions of state space. The complex regime arises as the border between chaotic and ordered, it is an area of phase transition between, as Kauffman terms it, frozen islands and "unfrozen regions just breaking up." Kauffman argues that throughout complex, ordered and chaotic regions, the systems adapt on correlated fitness landscapes, testable by using large numbers of binary elements. He believes that selective evolution works via attractors, which make the behaviour calculable, however complex, chaotic, or seemingly lacking in order.

The sciences of the singular, conforming to a similar dictate to the theory of Deleuze, do not maintain grounds for their extension in their abstract formulations. These mathematical formulations are themselves at all times "complexive", in that the conditions for positing questions are open to variation. The real conditions examined, become comprehendible not through the legislation of laws, but through the following of singularities in varying degrees of convergence and divergence. The structures formulated are inherently open to traverse physical systems, in a similar manner to the aleatory points of Deleuze. Singularity works in progressing the field of enquiry, but not in a cumulative sense of knowledge or data strorage (which can be nonetheless useful), but rather in the activation of complexive conditions which are "solved" in objective fields in extreme cases, and thereby constituting flows of information or communication. The sciences of the singular are therefore in a state of communication and connective, complex research.

Research using the mathematics of the singular is pushing systems understanding in many directions. The economics of Brian W. Arthur, looks to base complex socio-economic and political relationships on models using dynamic mathematics, and complex adaptive evolution. Christopher G. Langton is using similar strategies to those of Kauffman when dealing with chemical mechanisms and Artificial Life. Other fields using the science of the singular, include, neural networks, genetic algorithms, cellular automata, self-organised criticality, classifier systems and coevolution. The suggestion here is that this explosion of investigation into areas previously considered unreclaimable due to their complexity and chaos, is being galvanised by the singular. Due to mathematical application of complexity principles, systems may be simplified and represented on computers, to reveal their complex or chaotic structures. With such an impetus, the scientist is able to theorise general strategies in order to extend comprehension of the system. The multi-disciplined science of the singular is vital, morphogenic and laterally extending into connected fields.