Issue #54 July 2022

Deleuze on Ideas, Dialectics, and Multiplicities

Brian Halsey, "(The Cosmos Suite) Novem I", (1976)

The “Idea” perhaps can boast of being one of philosophy’s first unique concepts. In Plato’s hands, it formed the atoms of the theory of forms, and has circulated ever since. At many stages in philosophy’s history, different thinkers have attempted to transform the concept, forming a dynamic history of the “Idea” that could be seen as the story of philosophy itself.

In chapter 4 of Difference and Repetition, “Ideas and the Synthesis of Difference”, Deleuze outlines a new theory of ‘Ideas’, joining in this long running tradition. So, what is a Deleuzian Idea, and how does it fit into this story?

To summarize quickly, for Deleuze Ideas are complex sets of relations, veritable circuit boards, that condition things in their emergence. However, they are not static blue prints, fixed patterns floating in the world of Forms, but evolving, virtual problems. But to make sense of this, and why these Deleuzian Ideas are interesting or needful, we need to construct them step by step. So let’s do that. To continue the ‘story’ of Ideas into their Deleuzian transformation, we need to establish the threads of the story leading up to Difference and Repetition.



The notion of ‘Ideas’ Deleuze initially borrows from Kant, with all of the connotations Kant gave it, so we should start there. This will be a quick crash course in Kant, and omits many important details, in favour of linking just those points that are of interest to Deleuze’s borrowing of the term.

Kant himself borrows the notion of ‘Ideas’ straight from Plato, favouring using a dusty and outdated term that more-or-less captures what he needs it to, rather than inventing a new term (CPR, A312|B369). Kant, having determined the a priori form of the sensibility, and the categories of the understanding, still requires a something to account for the operations of the understanding in its speculative interest.

In our experience, we have the capacity to be affected by objects – this is the sensibility – and these ‘affectations’ are organized by the sensibility with a spatio-temporal form. This latter point is what answers Kant’s question “How is pure mathematics possible?” (Kant, Pr, [4 : 281]). We can rephrase this question, roughly, as “How is it possible that we, via mathematics (such as geometry), can make interesting, non-trivial (synthetic) discoveries about how things must play out (a priori) in the world according to our experience of it?” The answer is that mathematics (such as geometry) is not really about the world as it is in itself, but rather about the conditions of how any experience could appear to us, given the way we, and our contact with things, is structured.

But we’re not just bombarded by these sensory, spatio-temporal affects, we’re also given to understanding them as coherent objects, as examples and types, as being causally connected, etc. A similar formalization is handled by the structure of the understanding, which makes of our experience a coherent and knowable world amenable to rational investigation. The materials of the understanding, the concepts, have a legitimate and illegitimate employment. The legitimate employment is making sense of the spatio-temporal sensory flux of the sensibility. The illegitimate employment is when these concepts are entertained for their own sake, as perhaps pointing beyond the world of sensation and experience, to something beyond, posited as real and accessible (by the understanding alone). This will always breed phantasms and illusions because our sensory contact with the world always needs to be the touchstone – it’s the only data we have from the world at all.

For example, our way of understanding the world is in terms of ‘objects’. We look out and see a world populated by (and composed of) discrete unities that we can manipulate, count, and investigate. This raw concept of ‘objecthood’ can be the basis of many sciences, and coherent systems of knowledge, just insofar as it is seen for what it is: a function for organizing the spatio-temporal flux of sensation. Accordingly, any hypothesis concerning objects needs to implicate the empirical objects we actually discover this concept being used on: dogs and tables and planets and blood cells. However, the ‘illegitimate’ use would be when we turn inwards, and consider ‘an object, any object, in general’ and then on the basis of these considerations posit in a hypothesis some ‘extra-sensory’ entity that could never, based on its extra-sensory nature, be an object within experience. At best, we can determine ‘transcendental’ conditions from this investigation (what must an object, any object, in general conform to in order to be a part of experience), which is exactly what Kant does. What we can’t do is go further, and ask ‘what is the nature of this object in general, beyond all experiential considerations and modifications?’

Any possible answer to this suspect question is going to be hypothetical in character, but there is no means open to us to assess any hypothesis that necessarily resists determination by the content of experience (our sensory states – our only ‘access’ to the world as it is).

Kant calls these hypothetical assertions ‘assertoric’, which are defined as things we can assert as true. They permit of being true or false. These are contrasted against the ‘apodictic’, that which must be either true or false. Things that can be determined a priori, such as “an object occupies a location”. These two are then contrasted against the ‘problematic’, something of which we cannot ever say if it’s true or false – there can be no resolution on this matter, the best we can do is assume it as possible (Kant, CPR, A75|B100). So, when the understanding entertains the pure concept of ‘an object, in general, beyond experience’ (i.e., the noumenon) it can only do so problematically:

“I call a concept problematic that contains no contradiction but that is also, as a boundary for given concepts, connected with other cognitions, the objective reality of which can in no way be cognized.” (Kant, CPR, A254|B310)

“With all this the concept of a noumenon, if taken only problematically, remains not only admissible but, as a concept to limit the sphere of sensibility, indispensable.” (B311|A256)

What, then, is the point of the ‘problematic’? How can something that cannot be verified in principle be “indispensable” for anything? The problematic doesn’t just mean things that are beyond our abilities to verify, for example “The largest planetary body in the universe appears purple”. This example is perfectly ‘assertoric’ as we can posit it, or deny it, and continue reasoning legitimately about what is more reasonable to believe, or what would need to be the case for it to be true, etc. The ‘problematic’ is of a different nature, yet is an essential category.

It emerges because there are certain postulates that are required by even the legitimate use of concepts on experience, but which nevertheless can never be verified, not even in theory. They enter when we make use of universal conditions (a complete, unbroken series of causes, for example) in order to investigate local conditions (the set of causes of a particular phenomenon, for example).

For example, say we have a general grasp of what factors cause some phenomenon to occur, with a sufficient degree of exactness that we can make predictions about the phenomenon in normal circumstances. However, we then discover random variations in the accuracy of our predictions in certain special circumstances. We could just chalk this up to the phenomenon being wilful, the world being unstable, sometimes things following the rules we have determined, and sometimes just not, for no reason. However, if we think like that, science is over; any phenomenon can be explained by an appeal to the caprice of being. Much more likely is that we infer that in the special circumstances, there are confounding variables present – and these variables are also part of the causal explanation of the phenomenon insofar as it can be modulated by them. We go off in search of them. But on what do we ground this decision? It’s precisely on the Idea that everything is determined by its causes. So, if we discover that the causes we have identified only seem to be determining a portion of the phenomenon, we posit further causes.1placeholder

However, as Hume showed, this assumption, that ‘for everything that is, there is a sufficient reason why it is the way that it is, and not otherwise’ is not something that can be empirically verified. Kant would say we can empirically verify individual things having particular causes up to a degree approaching necessity, because of how ‘things’ are mediated by the understanding, but he would agree with Hume about not being able to verify the ‘everything’ (because we can never experience an ‘everything’). But it’s the ‘everything’ that drives us to continue to search for variables, and rule out the hypothesis that sometimes phenomena are inexplicable. This is the ‘problematic’. It’s something on the one hand that we must presuppose to even begin, yet something we cannot settle once and for all by any method, and something that drives us to continue, that keeps us up at night wondering about how to close any and all explanatory gaps, that problematizes.2placeholder

As Deleuze himself puts it:

“It [reason] must presuppose a systematic unity of Nature; it must pose this unity as a problem or limit, and base all its moves on the idea of this limit at infinity. Reason is therefore the faculty which says: ‘Everything happens as if . . .’ It does not say that the totality and the unity of conditions are given in the object, but only that objects allow us to tend towards this systematic unity as the highest degree of our knowledge.” (Deleuze, KCP, p.18)

In other words, there are certain ‘Ideas’ (the products of Reason) that we have to think, to be able to think about anything, yet whose status is different from an obscure or difficult to verify hypothesis, or from a principle that can be determined with certainty as being necessary, a priori. These are the ‘problematic’, and generally they emerge in order to give a maximum of unity, extension, or scope to the concepts of the understanding. On their own they are ‘undetermined’ (‘everything has a cause’), but, used within the understanding, they are determinable (‘phenomenon A is the cause of phenomenon B if and only if …’), and, when those concepts are then applied in particular cases in the sensibility, they are determined (‘the cause of this particular phenomenon is…’).

“Thus appearances are considered here as given, and reason demands the absolute completeness of the conditions of their possibility, insofar as these conditions constitute a series, hence an absolutely (i.e., in all respects) complete synthesis, through which appearance could be expounded in accordance with laws of the understanding.” (Kant, CPR, A416|B443)

However, as to Ideas  themselves (that on the basis of which reason ‘demands absolute completeness’), “we can have no acquaintance with an object that corresponds to an idea, even though we can have a problematic concept of it.” (Kant, CPR, A339|B397)

An upshot of this progression (from undetermined, to determinable, to determined) is that the ‘problematic’, as being wholly undetermined, becomes backgrounded as we move from Ideas (regulative principles, axioms), conditions (laws of understanding) to the conditioned (actual given objects of experience). The problematic thus has the character of a ‘subjective’ moment, cut off the facts of experience, which themselves are never ‘problematic’, but always fully determined in line with the categories and forms of sensibility. Thus, Kant talks of the ‘illusions’ of reason, but not the objectivity of problems.

One way of seeing Deleuze’s project is to see that he is trying to free up the ‘undetermined’ from its subjective designation, and allow it come into contact with the determined, as its ground (Williams, 2003, p.142). The Kantian way of arranging things leaves all knowledge relative to a subject which forms its ground. Hence, we end up with endlessly reproducing dualisms and clefts between terms. Unknowable noumena and hard problems of consciousness. To undo these clefts, the undetermined needs to be thought in the determined. In figurative language: we need a picture that has things ‘emerging’ from an obscure and open field, not taking fixed ‘things’ for granted as a ground (or appearances as ‘given’), and then placing a mysterious openness and obscurity emanating inexplicably from the side of the subject, of consciousness, etc.3placeholder The picture needs to illustrate how fixed things can emerge out of something unfixed, and subjects emerge from something that is not itself conscious.

Brian Halsey, "(The Cosmos Suite) Suspension IV", (1976)


Deleuze takes Kant’s discussion of the problematic nature of Ideas, these things that go beyond experience, yet push our experience to its maximum of unity, and effects a quick inversion as postulate: “Conversely, problems are Ideas.” (Deleuze, DR, p.214). We can intuit this inversion (from Ideas are problems to problems are Ideas) is justified in that there’s no where else, in the Kantian picture, for problems to be. This is attested to Kant’s insistence on the importance of Ideas for practical reason: the Ideas of, say, utopia and perfect justice and wisdom and virtue etc are, despite not being something we can encounter fully realized in experience, essential for conditioning our action – they make of action a problem in the way that direct empirical experience can’t (because of the is/ought gap). But, with this inversion, of problems being Ideas, not just Ideas being problems, Deleuze opens up connections to the philosophy of mathematics, and the primary place of problems therein, especially the philosophy of Lautman.

Lautman is interested in a much narrower domain than Deleuze – roughly and simply, “what’s math about? How does it work?”4placeholder His answer, equally as roughly and simply, is that math is not about mathematical objects, but theories that then define these objects, creating them from axioms. That is, mathematical objects are always relative to theories that define them. Theories themselves, though, are responses to problems that emerge from considering the relations between the objects that a certain, prior theory defines. The new theory displaces the problem through new axioms, allowing new objects and new relations between them, but it doesn’t solve it directly insofar as it goes beyond it. That is, the problem is reconfigured by its solution, but still persists in the previous situation that gave rise to it.

In his investigations, Lautman posits that these problems are operating at an ‘extra-mathematical’ level. That is, the theories and notions are mathematical, but when we grasp certain problems that emerge from the relating of these mathematical notions we grasp at something beyond just mathematics, that is, metaphysics, or dialectics, that governs and organizes mathematical theories (Lautman, 2011, p.199). For Lautman, problems are dialectical Ideas:

“This dialectic seems to be principally constituted by pairs of opposites and the Ideas of this dialectic present themselves in each case as the problem of connections to be established between opposing notions. The determination of these connections can only be made within the domains in which the dialectic is incarnated, and it is thus that we have been able to follow in a great number of mathematical theories the concrete outline of the edifices whose effective existence is constituted as a response to problems posed by the Ideas of this dialectic.” (Lautman, 2011, p.240)

To make this intuitive, allow me a (no doubt overly) simplified example. If we think the minimum number of assumptions we need to construct a triangle, we probably only need the first couple from Euclid’s Elements. That is, a plane, points, line segments (straightest distances between points), and intersections. With just these, we can construct a triangle, which then produces the notions of area and angle. But if we think about this triangle dynamically, we can quickly see there is some relation between the length of the line segments and the angles of the intersections. Variations in one translate to variations in the other. Length and angle seem like independent notions, emerging from points and intersections respectively, but in a fixed figure, we discover they share a necessary relation: make this line longer, and this line needs to get longer as this angle gets smaller. Or, change this angle and this length here needs to shrink or grow. So, we pose the problem “what is the relation between angles and edge lengths in a triangle, the function?” This problem is one we can discover once we have triangles, based on the notions we set up, but we also discover it dynamically, by thinking through the shape as though it was a rubber band stretched around three pins. Now, whatever the ‘solution’ to this problem is, it will require more resources than the ones we had in our starting assumptions – which is to say the problem can’t be solved at the level it is posed at, but actually engenders a new field, with new notions capable of their own problematic relations. The problem does this while at the same time it conditions this field: the problem has been stated, a new cache of relations and notions can only bear on it insofar as they answer to the problem as it has been formulated. However, because of this correspondence of the new theory to the previous problem through this conditioning, the problem itself is now altered: perhaps with these new resources the relation between line length and angle will be seen to be a mere trivial instance of a more widespread and fundamental symmetry that this problem is now an element of.

For Lautman, problems are dialectical because they always consist of this issue of relation between two given notions, and they are Ideas because they are not just one more notion among the ones defined by axioms, but they are the that which necessitates the hunt for new notions and axioms. Deleuze summarizes Lautman’s view into a tripartite definition of the problem:

“Following Lautman’s general theses, a problem has three aspects:

1) Its difference in kind from solutions;

2) Its transcendence in relation to the solutions that it engenders on the basis of its own determinant conditions;

3) And its immanence in the solutions which cover it, the problem being the better resolved the more it is determined.” (Deleuze, DR, p.226, enumeration added)

The ‘difference in kind’ is that solutions are mathematical theories, but problems are things we grasp when we relate seemingly incommensurate notions together – they are ‘extra-mathematical/theoretical’ for the simple reason that if they were subsumable to extant mathematical theories they wouldn’t be problems.5placeholder The transcendence refers to the conditioning problems provide to the new fields that answer to them, and the immanence refers to how the problem is transformed and reconfigured in and through the fields that develop to answer it.

In addition, Deleuze follows Lautman in the characterization of the dialectic nature of problems:

“Problems are always dialectical”: the dialectic has no other sense, nor do problems have any other sense. (Deleuze, DR, p.227)

However, Deleuze underlines that this is not the dialectic of Hegel:

“By “dialectic” we do not mean any kind of circulation of opposing representations which would make them coincide in the identity of a concept, but the problem element in so far as this may be distinguished from the properly mathematical element of the solutions.” (Deleuze, DR, p.226)

That is, in our simple example above, the relatedness between angle and line that unfurls or points to a new theory or function. The function or theory being established, the problematic relation that we initially grasped is still there, even if now it is better articulated, or easier to express in notions of the new theory or function. Deleuze opposes the relation of A and Not-A, to the relations expressed by the calculus as dx, dx/dy.6placeholderThat is, pure differences that nonetheless can be reciprocally determined.



Deleuze’s innovation with Lautman’s concepts is to give them a full extension. It’s not only mathematics that sustains this transcendent-yet-immanent relationship to problems, but all fields of knowledge. In fact, it is precisely the theoretical spaces that are opened up by the addressing of a problem that determines the differing ‘fields’ of knowledge. After telling us problems are always dialectical, he posits that “[w]hat is mathematical (or physical, biological, psychical or sociological) are the solutions.” (DR, p.227). That is, problems do not belong to particular fields of inquiry, it’s that these different fields of inquiry are so many solution-spaces in which the reciprocal conditioning of a problem can be developed. Why mathematics is privileged is because it most clearly performs the sorts of dynamics in regards to the ‘dialectical Ideas’ (problems) that Lautman described. But, for Deleuze, these same dynamics play out everywhere. So, where Lautman sees in the history of mathematics the development of new theoretical edifices emerging out of problems formed by positing the relation between notions contained in the extant theories, we can generalize out and see entire disciplines being formed in an analogous manner. Freud the neuroscientist developing a psychology that retains the ‘thermodynamics’ of electrical discharge, yet nonetheless goes beyond it. Likewise with the emergence of physics from natural philosophy, etc.

In each of these cases, the ‘dialectic idea’, the problem, is transformed as it is refined down these or those lines, producing novel forms and terms which themselves can enter into new problematic dialectics. Deleuze tells us “an Idea is a complex theme” (DR, p.231), accordingly, problems, as Ideas, sit ‘across’ fields of knowledge and can be thematized by them (and their reserve of notions) variously, and in differing degrees. This is why breakthroughs in science X can have manifold and surprising effects across other areas of knowledge – what unifies them is not so much that they approximate the same ‘object’ (the ‘world’, or ‘electro-magnetic phenomena’, or anything else) from differing directions, but, rather, they answer to, or thematize, the same problems, determined in differing forms and terms.7placeholder

What maintains the separation between disparate fields of knowledge, if it’s not to be their unique objects of study, are various ‘orders’ of problem within the dialectic:

“It is true, however, that on the one hand the nature of the solutions refers to different orders of problem within the dialectic itself; and on the other hand that problems — by virtue of their immanence, which is no less essential than their transcendence — express themselves technically in the domain of solutions to which they give rise by virtue of their dialectical order. … [E]ach dialectical problem is duplicated by a symbolic field in which it is expressed. This is why it must be said that there are mathematical, physical, biological, psychical and sociological problems, even though every problem is dialectical by nature and there are no non-dialectical problems.” (Deleuze, DR, p.227)

Deleuze seems to, later, characterize this ‘order’ within the dialectical Ideas/problems in Leibnizian terms of clarity-obscurity and distinct-confused.8placeholder But for now it is enough for us to pull out the moves so far up to this point.

Deleuze borrows Lautman’s ‘twist of the screw’ regarding ideas: Ideas are not just of the problematic (as Kant argued), but problems themselves are Ideas. This move maintains the ‘undetermined’ nature of Ideas (now problems) insofar as it is of the nature of problems to be undetermined (“what is the relation between x and y?”). The movement from the undetermined problem to a space where it is determinable entails not just the employment of concepts, but the invention of new ones – new theories, new fields, new notions. And, finally, it is via these theoretical innovations that the problem becomes determined in a new set of forms and terms (“the relation works via this”), which are then potential targets for further problematization. Thus, the Deleuzian Idea/problem performs the same function as the Kantian one – as being a horizon or limit that animates the positive propositions of sciences without being reducible to (or articulable within) them. However, the undetermined is no longer a subjective moment – but something that operates at the horizon of our knowledge in a positive sense.

Brian Halsey, "(The Cosmos Suite) Nucleon I", (1976)


Now it begins to get interesting, and the work we’ve done clarifying the concepts begins to pay dividends. Deleuze has so far kept close to Lautman’s characterization of ‘Ideas’ as problems, merely extending it out. But, we’re still within the province of knowledge. If Deleuze wants to find the unconditioned ground of things, it’s not enough to look at the unconditioned horizon of knowledge, and the dialectic whereby problems drive the production of new forms and terms, though it is a good start. To push things further, Deleuze offers further characterization of the idea:

“An idea is an n-dimensional, continuous, defined multiplicity” (DR, p.230)

He immediately unpacks this for us, which is a Deleuzian rarity, so worth quoting at length:

“By dimensions, we mean the variables or co-ordinates upon which a phenomenon depends; by continuity, we mean the set of relations between changes in these variables … by definition, we mean the elements reciprocally determined by these relations, elements which cannot change unless the multiplicity changes its order and its metric.” (DR, p.231)

However, this unpack still needs an unpack.

A multiplicity is a term Deleuze has developed from Bergson, who developed it from Riemann. It’s a designation that is contrasted both to the one, the singular, and the many. But why, we might ask, given the name, does ‘multiplicity’ contrast with the many? Because the ‘many’ can be understood as merely ‘many ones’, so the many and the one are of a kind. What would be opposed to this? Well, a heterogeneous field whereby a ‘one’ is unthinkable, many or otherwise. Deleuze gives us the example of colour: “Colour – or rather, the Idea of colour – is a three dimensional multiplicity.” (DR, p.230). But can’t we just say of colour that there are ‘many’ of them? Only if we limit ourselves to some system of designations (colour words, or paint swatches) that divides the full plenum of colour, given by the three dimensions of hue, saturation, and luminosity, into discrete, countable units. But this is not colour per se, but a system of classification between already defined colours. With this example it’s easy to see how the multiplicity ‘grounds’ the discrete units of, say, a colour swatch library – the three dimensions can be varied to produce countless entities; it’s not the case that the three dimensions are merely a code for the description of the swatches, the swatches instead are ‘halted’ points along the three dimensions.

So, an Idea will have some number of dimensions, but these dimensions need to be related in a structure. Remember it was the apprehending of the tight relation between the line lengths and angles of the triangle that allowed us to apprehend the problem in our simple example above. The dimensions of colour, if you’ve played with HSL graphs in Photoshop or the like, seem quite permissive of any relation between their three values, but this is only a product of how we generally display it in a UI. At the lower limit of luminosity, for example, there is no variation among the other dimensions, just a single black tone. At minimum saturation, the dimension of hue just becomes a singular grey tone along its axis. Thus, the relations between values is more complex than just a smooth, three dimensional graph; there are limits and singularities emerging from the entanglement of the three dimensions. We see this more starkly in the multiplicity of language, at the ‘phonemic’ level, where some vocal features can be combined to create variations, yet others exclude each other (your tongue can only be in one place at one time), or how certain sounds modify other sounds when in conjunction.9placeholder So, the second part of Deleuze’s characterization of Ideas is that a system of relations hold between the dimensions, but we shouldn’t think of this in the simple way of any two dimensions being permitted of relating in any way whatsoever, but rather understanding that the dimensions, being what they are, structure how they relate among each other. Each set of related dimensions plays by its own rules, forms its own ‘figure’, which in turn classify the dimensions (luminosity always is entangled with hue and saturation).

Finally, these relations give rise to ideal elements. We shouldn’t think this reintroduces discrete units back into the multiplicity, because, say, a particular colour is nothing but a point of intersection between the three dimensions; it’s nothing but this intersection, this bundle of relations. Nonetheless, we can speak of ‘points’ emitted by the varying relations, forming a system of elements, much in the way that various functions will give rise to curves, composed of points. The functions are not “really about” those distributed points building the curve, but, nonetheless, they sit there, as elements – a translation of the function. The presence of these elements, and their distribution is what we can use to extract the rules for the relation of dimensions discussed above. Across the colour dimensions, there is a singular entity, black, that occupies the entire matrix when the dimension of luminosity sits at its zero limit. When we unravel and splay out the multiplicity for the ease of a user interface, we may represent it as a black line that runs along the bottom of the spectrum, but in truth it is only a singular point. Within the Idea there are not countless identical blacks, but only one, which is a point formed by a function as a rule: when this dimension reaches its limit of zero, variations in other dimensions (hue, saturation) are empty of elements. But remember how the Idea of colour, with its three dimensions, undergirded the colour swatches as tangible, discrete units? This means the colour swatches themselves, or any other system for extracting discrete units from this multiplicity (the colour terms of a language, for example) are conditioned by how and where ideal entities can form as points of intersection between the dimensions, while they, the ideal entities, themselves define the character and rules for how and where the dimensions intersect – characterizing both curve and space.

Let’s take a look at the characterization again now we’ve unpacked it:

“An idea is an n-dimensional, continuous, defined multiplicity” (DR, p.230)

An Idea has dimensions, those dimensions are related by particular rules, or conditions, and those relations permit of us specifying ‘points’ at their intersections; the presence and absence of those points tracing out limits that define the rules and conditions for the multiplicity.



Now, chances are if you found this article, you were interested in what Deleuze means by ‘Ideas’, and perhaps at this point have gotten something useful. But, maybe you didn’t care one way or the other, and now are wondering “okay, so what?” True, it’s been a long ride, and now we have before us an interesting construction; the Deleuzian ‘Idea’. But the question is now “what’s the point of it?” Let me summarize everything so far in hopes that the answer to that question will emerge, and also to lay some paths of where we go next.

First, we should remember the hand that Deleuze has (strategically chosen to be) dealt, the Kantian picture, and, from there, the plays he needs to make to get where he’s going. Kant has given us a consistent picture of the subject and their relation to the world such that knowledge could be possible. The genius of this picture lies in demonstrating just how much the act of perceiving goes to explain the forms and possibility of knowledge. But we’re not merely, say, a bunch of sensory cells splayed out across the blades of an office fan, sensing and understanding passively – we also take an active and constructive stance towards all this stuff coming in. Hence, Kant introduces the category of both Ideas, and the problematic: horizons, or vanishing points that push the understanding to tend towards an asymptotic limit. But the way this whole structure was arranged gives us a picture of a world already given, whereas the undetermined, the indecisive, the half-formed, is but a subjective moment, percolated by these Ideas, this projection of a horizon or vanishing point, in the functioning of the understanding so it can have a goal of unity. The ‘givenness’ of the world is left as fundamental. But a fully formed world emerges from a half-formed one – it can’t merely be our knowledge of it that is half-formed if we want to explain its emergence.10placeholder

So, Deleuze finds a ‘flip’ switch, where we can reverse the terms here in regards to where the undetermined, the problematic, and the Idea, are located. Why not out there in the world? If our knowledge of mathematics responds to certain objective, extra-mathematical problems, then we have a model for an ‘undetermined objectivity’ in the figure of the ‘problem’. Problems are dialectical Ideas, and Ideas are multiplicities. With this, and on the coat-tails of Lautman, the Deleuzian ‘Idea’ jumps closer to the Platonic ‘Idea’ (which is closer to Lautman’s use of the term). But, now, not as a static realm rendered imperfectly in the world of perception, but rather as complex machines, veritable factories, for the production of phenomena. We need to recall the discussion of Lautman above here: the problem, the Idea, is transcendent in one sense, in how it conditions the theories and notions that answer to it, but is just as immanent in these theories and notions, who determine and modify it. A problem is transformed when a theory is developed in response to it. And, insofar as Deleuze has extended Lautman’s concepts, Deleuzian ‘Ideas’ are not impervious to the fates of the things whose construction they determine. Even for Plato we can now, today, realize this must be the case: if the form (Idea) of “horse” undergirds the galloping creatures we can see, then the forces of evolution act not just on the galloping creatures before us, but the form (Idea) itself: there are no horses but for the adventures and misadventures of galloping proto-horses, from which they, and the multiplicity ‘Horse’ emerges.

What Deleuze has here in outline, with the Idea, is a metaphysical unit with a number of appealing features. Firstly, a multiplicity is a bundle of reciprocally conditioned dimensions, not an object that requires further grounding, so we avoid taking objects as simple (and with them, consciousness and the subject). Secondly, we have no cleft between the Idea and the actual objects it conditions: as problems, Ideas are developed through their ‘solutions’ (actual objects, forms, and terms) just as much as they specify conditions for them. Lastly, in the figure of the Idea/problem we have a category of the ‘objective undetermined’ which prepares the required resources to give an account of the emergence of things not tied to a subject and their epistemic, and even ontological, limits.

However, this description, this characterization and invention, being performed, Deleuze still needs to give an account of how Ideas become incarnated in actual things. This is crucial, as it’s part of the core set of conditions for us being able to say an Idea emerges at all (number 3 below):

“There are three conditions under which together allow us to define the moment at which an Idea emerges:

(1) the elements of the multiplicity must have neither sensible form nor conceptual signification, nor, therefore, any assignable function. …

(2) These elements must in effect be determined, but reciprocally, by reciprocal relations which allow no independence whatsoever to subsist. . . .

(3) A multiple ideal connection, a differential relation, must be actualised in diverse spatio-temporal relationships, at the same time as its elements are actually incarnated in a variety of terms and forms. The Idea is thus defined as a structure … a system of multiple, non-localisable connections between differential elements which is incarnated in real relations and actual terms.” (Deleuze, DR, p.231, enumeration in original, line breaks added)

Even though Ideas distribute elements which have no conceptual signification nor sensible form, i.e., they are prior to the Kantian transcendental subject, they nonetheless must be actualised. This follows from their dual role as transcendent-immanent. The Idea of colour doesn’t sit around and wait for eyes to evolve – it is actualized within them, and undergoes changes as eyes evolve further. Accordingly, Deleuze needs to give an account of this process of actualization from the virtual. He opposes this movement to realization from the possible. That is, Ideas are not merely possibilities, to be realized, for they are already perfectly real as virtual entities. What is needed, then, is an account of how they are actualized in actual, not merely possible, things and processes. However, that will form the subject of a later essay…

John C. Brady is doing his PhD in Philosophy at the University of Sydney, while living in Beijing. He gets most of his reading done in traffic jams. He is also a co-editor of this magazine, by way of full disclosure.

Abbreviations & Works Cited


DR: Deleuze, Difference and Repetition

KCP: Deleuze, Kant’s Critical Philosophy

CPR: Kant, Critique of Pure Reason

Pr: Kant, Prolegomena to Any Future Metaphysics


Works Cited:

Deleuze, G., 2008. Kant’s Critical Philosophy. New York: Continuum.

Deleuze, G., 2011. Difference and Repetition. New York: Continuum.

Kant, I., 1998. Critique of Pure Reason. Cambridge: Cambridge University Press.

Kant, I., 2004. Prolegomena to Any Future Metaphysics That Will be Able to Come Forward as Science. Cambridge: Cambridge University Press.

Lautman, A., 2011. Mathematics, Ideas and The Physical Real. New York: Continuum.

Williams, J., 2003. Gilles Deleuze’s Difference and Repetition: a Critical Introduction and Guide. Edinburgh: Edinburgh University Press.


This also plays out in the search for a grand unifying theory of physics. It’s seen as unacceptable that there is gap between the quantum and Einsteinian levels of description – one that could be filled by a more fundamental theory. Such a theory would reorganize the extant theories in such a way that it wouldn’t be a question of using the extant theories to verify the more fundamental one. Thus, a unifying theory is not simply testable once and for all in a way that would allow us to rule out the counter-hypothesis that there is no more fundamental unity between these ‘levels’. It would be a new opening, a new organization, full of new problems and hypotheses that we would never have been able to entertain if we hadn’t ruled out the possibility of a fundamental lack of cohesion or symmetry in nature by fiat.


Think of a software engineer. Looking at incoming bug and crash reports, they may reason professionally that the number is so low that the software, and the code behind it, effectively ‘work fine’. However, we can imagine this same engineer, outside of their professional capacity, being haunted by the question ‘why?’ in regards to those few bugs and crashes.


For example, taking things for granted, we can, in the Kantian transcendental procedure determine a great deal about the conditions of our conscious experience of things. But in the end we are left with an unknowable residuum (noumenon, bare particular), and a knowledge of objects that is tied to consciousness, which itself cannot be explained as just another object (brain, matter, etc). Since Descartes defined the terms and gave them their more or less modern framing, a bitter dualistic cleft has existed, being shuffled around this way and that, but never cancellable.


Apologies to philosophers of mathematics for this gross simplification.


Lautman also holds a ‘retroactive’ definition: even if no problem is subjectively encountered in the development of mathematical theories and notions, one can be grasped when those theories and notions are understood – as their genetic ground: “It is of the nature of the response to be an answer to a question already posed, and this, even if the idea of the question comes to mind only having seen the response.” (Lautman, 2011, p.204)


In the calculus, dx means ‘a tiny piece of x’, so small (infinitesimal) that it may be disregarded from certain perspectives (as being effectively zero) while at the same time providing a minimum of difference from zero that makes all the difference from other perspectives. Likewise, dy means ‘a tiny piece of y’. By means of dx/dy one can find the slope of a point on a curve, provided one deletes them before concluding the calculation (otherwise, it would not be the slope of a point, but the slope of some minimum of extension). One can freely delete them because they are effectively zero. Yet, one can use them to calculate because they are not quite zero.


This ‘thematization’ needs to be separated from Deleuze’s concept of ‘dramatization’. I use ‘thematize’ here to designate a similar dynamic as dramatization, but playing out here merely within domains of knowledge. ‘Dramatization’ is introduced by Deleuze later in the chapter, as the relation that ‘spatio-dynamisms’ have to the Ideas that they actualize. This ‘actualization’ is performed by these dynamisms and their dramatization of Ideas.


On the surface of it, this seems plausible. Fields can communicate via a common problem, but differing fields allow differing parts of the problem to appear more or less obscure or clear. If the biological is reducible to the chemical which is in turn reducible to the physical, for example, this does not mean that the best way to get at the problem of, say, speciation, is to give a quantum, atomic, account of it. We completely ‘miss’ the phenomenon at this level, despite it also being played out at it.


For example, “hand” and “ham” end with unique sounds, but those sounds turn into the same sound when coming before a ‘b’, as in “handbag” and “ham bag”. This layer of description introduces the entity of the ‘phoneme’ as opposed to the ‘phone’. Phonemics traces the combinatorial rules of individual phones within actual languages, whereas phonetics traces all speech sounds used in any actual languages.


Unless of course, God explains its emergence, and makes it sure it’s ‘complete on arrival’. The given as a ‘gift’. Kant’s connections with Berkeley are clear here. But, unfortunately, our science is ever more interested in how beings ‘shplurp’ out of grounding layers. We can no longer avail ourselves of ‘miraculous’, complete on arrival, solutions.


July 2022


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