The unsubstitutable, the unique, the distinct, the inexchangable; that which has no equal or equivalent. It is what cannot be subsumbed under a regime of generality because it cannot be understood as one particular among an equivalence of particulars (although one can treat singular things as though they are particulars). The singular belongs to the order of novelty: to the degree that it can be thought of in general terms, it brings with it it's own index of generality. It's like what Nietzsche once wrote concerning 'great human beings': "Every great human being exerts a retroactive force: for his sake all of history is put on the scale again".

One of the paradoxes of singularity - and what accounts for it's fragility as a category - is that precisely because it's bears in itself the index of it's own recognition, it can only be recognized as a singularity only retroactively, by means of a repetition. Zizek - whose reference here is Hegel rather than Deleuze - gives the example of both Julius Caesar and Margaret Thatcher, the novelty of whom had to each be affirmed by a repetition: that of Augustus in Caesar's case - who assumed the title of the first 'caesar' - and that of Tony Blair in Thatcher's case, who was said to have institutionalized Thatcherism as a philosophy of government. In both cases it's only in the light of the 'repetition' that the novelty of the 'original' can be recognized. Without Augustus one could not speak of Caesarism, without Blair one could not speak of Thatcherism.

Again the reason for this is that the singular brings with it it's own index of generalization, the novelty of which is ungraspable without a repetition which brings it to light: "the murder of Caesar - the historical individual - ended up resulting in the establishment of Caesarism; Caesar-the-person is repeated as Caesar-the-title. The crucial point here is the changed symbolic status of [the new]: when it erupts for the first time it is experienced as a contingent trauma, as an intrusion of a certain non­symbolized Real; only through repetition is this event recognized in its symbolic novelty" (Zizek, The Sublime Object of Ideology). Deleuze himself, for exactly these reasons, will link singularity indissolubly with the concept of repetition: "To repeat is to behave in a certain manner, but in relation to something unique or singular which has no equal or equivalent... If repetition exists, it expresses at once a singularity opposed to the general... In its essence, repetition refers to a singular power which differs in kind from generality." (Difference and Repetition)

I mention this because it's easy to see, because of this, how singularity can be so easily mistaken or miscrecognized for particularity. Because repetition retroactively constitutes the singular status of the novel event, it becomes easy to treat both things as particulars, and from there, extract - by means of an imaginary extrapolation - an order of generality to which both might be said to belong. But it's clear that any operation of this kind is simply a kind of epistemological back-formation, an attempt to expel any consideration of novelty by 'levelling' the field of events in order to embed them into an artificial coordinates through which they can be compared, and subsequently particularized. This is why Deleuze warns that "generality, as generality of the particular, thus stands opposed to repetition as universality of the singular."

Eh, your random opinion doesn't really matter tho. It's true that the notions discussed here are pretty abstract, but they are so of necessity, given how widely they can be cashed out. I already gave the example of an economy, here's a few more, courtesy of John Protevi from his Deleuze and the Sciences, regarding other fields:

Neural functioning: "We can see the embodied and embedded nervous system as a preindividual virtual field: (1) a set of differential elements (reciprocally determined functions —in other words, neural function is networked: there is no such thing as the function of “a” neuron; some argue the same for higher-level cognitive processes, i.e., that they emerge from global brain activity and hence cannot be understood in isolation) (2) with differential relations (linked rates of change of firing patterns) (3) marked by singularities (critical points determining turning points between firing patterns). The dynamics of the system as it unrolls in time are intensive processes or impersonal individuations, as attractor layouts coalesce and disappear as singular thresholds are passed (Varela 1995)"

A football game: "Let's take the Idea of football games. ... What are the [virtuals] that conditioned the genesis of American football? Well, it would be a multiplicity of differential elements, differential relations, and singularities. The differential elements would be the players, the field, and the ball. They are differential elements because they are defined only in relation to each other. A prolate spherical of pigskin leather is only a football in relation to the players, who are only players when the entertain a certain relation to each other and to the ball, and of course, to the field, which in turn. The differential relations are what the players are able to do with the ball and with each other. They are differential in that they are relations of change in the elements: how they are able to move, to advance and retreat. And these relations are strewn with singularities, or sensitive points: when the ball moves between players across a certain threshold of the field, a touchdown or field goal is scored."

Ecology: "An example here would be a Deleuzian understanding of niche construction: the activities of organisms change the selection pressures for future generations. The ecological web of relations that we describe as “selection pressures” is not ghostly, it is perfectly real, but for Deleuze, it does not have the same ontological status as a single individuated act (e.g., a predator devouring a prey animal). Rather, the web is virtual, that is, composed of the relations of dynamically interactive processes. The virtual field is not composed of the processes themselves but by the differential elements, relations, and singularities of the processes. These elements, relations, and singularities are progressively determined by intensive individuation processes so that at critical points in the relation of predator and prey activity — at a singular point in the linkage of the rates of change of those processes — we can find the triggering of an event such as a population explosion or, in the opposite direction, an extinction."

But 0/0 is the limit. So the point never exists - except as an idea, a goal, a virtual object (in the way that singularities, event horizons, virtual particles, renormalised fields, etc, are all virtual objects in physics).

So all we can do is imagine the point as the virtual locus - a bare property-less location - to which we can then start artificially gluing dimensionality (the general or global quality that is constraint!) back on to.

Yes - it's a limit precisely from the perspective of the already-individuated, which in this case would be the primitive function. But the whole point is to reverse the order of priority in order to think the construction of the primitive out of the virtual, which, although not actual, is in every regard real: "The virtual is fully real in so far as it is virtual [it is] real without being actual, ideal without being abstract [...] The reality of the virtual consists of the differential elements and relations along with the singular points which correspond to them". Indeed the reason you can only think in terms of mutually-constraining limits is precisely because you are unable to countenance exactly this reality of the virtual.

Which is just another way of saying that you are unable to properly consider the process of individuation because you can only ever look at it from the perspective of the already-individuated. And from that POV, all you will ever see is limits and a process of othering. As Deleuze puts it, "Negation is difference, but difference seen from its underside, seen from below. Seen the right way up, from top to bottom, difference is affirmation." In other words, if we reverse the picture and look upon individuation from the perspective of individuation, what you see instead are differential relations - coupled rates of change - and distributions of singularities which define thresholds of mutation.

One can think of an economic system this way: flows of labour and capital, rates of birth and death, employment and wage (all of which reciprocally determine each other as coupled rates of change), together with thresholds of mutation (environmental carrying capacity, minimum survival income, etc): these are the parameters out of which 'economic individuals' are crystallized from - companies, trade agreements, tax rates, etc. The 'virtuals' here are not 'possibilities' which are then culled by a process of mutual limitation to give rise to actualities: the virtualities are fully real and they engender creativity at the level of the actual. Given these rates of change, given these singularities which define thresholds of tolerance, in what way should 'economic individuals' go about achieving whatever it is they do - in what manner do they become the individuals that they are ?

They is why Evens - inspired entirely by Deleuze from whom these terms are borrowed - speaks of individuation as a matter of 'problem solving'. Not 'symmetry-breaking' but problem-solving is the model for the process of individuation: the differential qua genetic element of quality defines an open-ended problem (like the coupled rates of change in an economy) that can be 'solved' in multiple ways (two different companies might attempt to 'solve' an inequality of supply and demand in two different ways, even though the 'problem' itself is entirely determined); a distribution of singularities can give rise to varied curves, so long as the points of that curve behave in roughly the right ways around those singularities themselves.

Evens: "The function thus takes shape gradually, progressively, as the singular points shift and glide relative to each other, tense and relax to alter their configuration. A problem forms like a soap bubble stretched across the wire outline of an abstract geometric figure. How to connect the vertices most efficiently, how to find the correct degree of curvature, how to distribute density so as to bend without breaking? And when a weakness is stretched beyond its breaking point, the bubble snaps into a new shape, determining new criteria, new boundary conditions, posing and solving a new problem in a flash, so that one would never suspect the whole network of differential calculations that take place in this instant. Problems determine themselves incrementally and always in relation..."

Individuation as symmetry-breaking in comparison is an incredibly basic and rather naive approach to the whole issue. Indeed, the entire model is a back-formation, a retroactive posit that takes the actual and, through a fantasy or a daydream that operates by means of an hallucinatory extrapolation, imagines that there must be some vague, indeterminate potentiality out of which individual, 'crisp' things are coalesced. But just as the 'limits' which apparently drive this process are fictional, so too is the entire edifice a just-so story that weaves itself in order to justify itself. As a good King Lear might have said to you: only fantasy comes of fantasy - speak again. This is why what you bring to the table is not philosophy but taxonomy: you have a sterile descriptive framework under which various entities and processes can be slotted into as so many indifferent elements, but which itself does nothing to account for their ontogenesis except by means of an ineffectual handwave. The Apeiron, like the Anaximandian myth that spawned it, is exactly that: a fable told in the place of any concretely engaged analysis.

I'm a little lost here, but the claim that you can generate a polynomial function from its differential is wrong.

For example, f(x) = 3x + 1 and f(x) = 3x + 2 are different functions, but their derivative is the same: f(x) = 3. You cannot go 'backwards' from 3 to either of these lines, not even around a single point (they're parallel and share no points in common). The lines can be specified without reference to the differential, as I just did.

Also, the claim that the differential is not a number is confusing: if by 'differential' you mean the result of performing the differential operation on some function, then of course it's not a number, it's another function. The result of differentiating is of the same sort as the thing differentiated, it's just of a lower power.

If by differential you mean the infinitesimal, I don't know what people think about it generally, but certainly you don't need to treat it as a number. You seem to be saying you don't want to treat it as an ideal limit, either, but then, I'm not sure what you're proposing instead. — The Great Whatever

Okay, I wanna backtrack here a little because a) we've both misread the passage on generation, because of my out-of-context quote, and b) I wanna deal more precisely with the Deleuzian treatment of the differential, which I wasn't clear enough about. First, the passage on the creation of the polynomial refers not to (re)creating the entire function, but the function around the point in question: "You are given one point on a function and a sequence of numbers representing the values of the derivatives of the function at that point, and from these numbers, you can reconstruct an approximation of the whole function not just at that one point, but also in an area around that point." So the reconstruction referred to is pretty local rather than global, but the thrust and stake of the passage is that this locality is nonetheless not correlative to a single, particular point: the successive derivatives express the overall behaviour of points around any one particular point.*

We can bring out the importance of this seemingly trivial point however if we turn again to Deleuze's reading of the calculus. I said originally that "the differential must differ in kind from the numbers that make up the primitive curve" - this was ambiguous and you were right to call me out on this. It's indeed far more precise to say that the derivative of f(x) yields another function f'(x): what I wanted to convey is that on Deleuze's reading, the difference between these two functions is not simply quantitative but rather qualitative. What does this mean? Negatively, that the differential cannot be a magnitude or a quantity: at the point at which dy/dx = 0/0, the value of the derivative is itself neither zero nor an infinitesimal. As Sean Bowden puts it, "dx represents only the cancellation of quantity in general"; instead, Deleuze's argument is that while it cannot be determined in the form of quantity, it can (only) instead be determined in "qualitative form".

And what does it mean that the differential can be determined only in qualitative form? Simply that, as we've said, the derivative is never simply a value that correlates to a single, particular point on a primitive function, but instead defines the qualitative character of the function around a particular point. In Simon Duffy's words, "the differential relation characterises or qualifies not only the distinctive points which it determines, but also the nature of the regular points in the immediate neighbourhood of these points" (Duffy, "The Mathematics of Deleuze's Differential Logic and Metaphysics"). This is the import of the Aden quote above. Now, the point of this giant mathematical detour is that insofar as the differential is understood as this element of pure quality ('the cancellation of quantity in general'), it serves as the model for Deleuze's notion of pure relationality. Again in Bowden's words: "even though dx is totally undetermined with respect to x, as is dy to y [[dy/dx can only be determined in relation to each other, without each each value is nothing], since the relation subsists, they are in principle determinable with respect to each other" (my emphasis).

Why is this reciprocal determinability of the differential important to Deleuze? For two reasons: first, not only does it provide a model for pure relationality, but second and even more importantly, this model itself has a distinctive trait that allows Deleuze to set himself against a position that his entire oeuvre pitches itself against: the idea that what exists prior to individuation is an indeterminate generality which is then progressively differentiated though limitation or negation (which itself calls for a correlative abandonment of any hylomorphic model of individuation). In other words, Apo's entire metaphysical picture. What's at stake here? It's this: while Deleuze agrees that one must begin any approach to individuation from the perspective of the undifferentiated (at the point at which there are not yet 'crisply defined individuals', to use Apo's parlance), it is nonetheless a complete mistake to think that this undifferentiated realm is indeterminate. On the contrary, he will argue that this pre-individual, undifferentiated sphere of being is entirely determined - and determined precisely in the qualitative form as outlined above: this is it's 'distinctive trait' that I mentioned.

To bring it all together then, the determination of the pre-individual realm means that it is characterized buy the distribution of singular and ordinary points. And what does this mean? Again, back to the differential: if we accept that the differential characterizes the qualitative behaviour of a primitive function, then one can argue for the 'existence' of two kinds of behaviours: singular and ordinary. Points with 'singular behaviours' are, as we've said before, things like inflexion points and stationary points (where the value of a gradient changes or equals to zero or infinity); points with 'ordinary behaviours' are those that remain relatively continuous to their neighbouring points. So with the calculus as his model, Deleuze will refer individuation to the manner in which singular and ordinary points are distributed among a series, and from which, taken together, a primitive function can be generated. Hence Deleuze's affirmation that "the reality of the virtual [the pre-individual] consists of the differential elements and relations along with the singular points which correspond to them […] Far from being undetermined, the virtual is completely determined.”*

--

It's ultimately over the question of the determination of the pre-individual that the debate between me and Apo turns. Apo is unable to recognize - perhaps because he's never encountered it before - the idea of a determinate but undifferentiated realm of the pre-individual. The terminological disputes over the general and the particular, the singular and the universal, and pretty much the rest of it, all turn upon this difference. The idea that the pre-individual is a vague generality is the 'null hypothesis' which Deleuzian metaphysics tests itself against, even as it responds to a similar motivation - which accounts for the closeness and the distance between our respective position. Anyway, sorry for the long reply, but I'm working on two fronts at once - if you haven't noticed already not all of this post is for you TGW - so I don't have to post multiple times.

*See also Gil Morejon's paper, "Differentiation and Distinction: On the Problem of Individuation from Scotus to Deleuze" (on academia.edu), which helped me clarify alot of these issues to myself (sneak peak: "What we will suggest is that the positive notion of common nature or virtuality, as something both completely determined and undifferentiated, far exceeds in its explanatory capacities a negative notion of possibility as purely indifferent... This is an important insight because it forces a re-evaluation of the idea of possibility, which was classically understood as the negation of actuality. Hegel’s famous critique, that Schelling’s philosophy of Indifferenz amounted to ‘the night where all cows are black’, exposes the paucity of such a conception, through which we ultimately are left incapable of accounting for the reality of actual individuals or distinguishing between empirical instances"

(^ i.e. accounting for the singular - this explains why so many of Apo's posts end up being a simplistic 'taxonomy' of being: things and processes only have 'value' in his system to the degree that they correspond to one or another of his pre-established categories - in themselves, they are meaningless, devoid of significance. I wasn't just being snide when I said earlier that the whole edifice is self-referential - it really is).

You must have a dim view of 'everyone' if you think 'everyone' is as rigidly dogmatic in their approach to conversation as you. But as we've established, you can't think singularity, so even your view of 'everyone' is tainted by that self-same monotony. Thankfully, almost no one I know approaches discussion in the way you tend to do, so you're wrong about that too.

You have a hard life ahead if you can't tell the difference between a challenge to your arguments and an attack on your person. — apokrisis

A challenge? Please, don't flatter yourself. Your modus operendi consists of waltzing into a thread, declaring a position wrong from the point of view of your already-established orientation, then proceeding to pontificate on how that orientation works itself out. After which you conclude, thanks to this circluar hop on the spot, that the original position is wrong. You wouldn't know how to effectively engage with another position on it's own terms if your life depended on it. So no, your don't offer challenges, you self-aggrandize by using other people's posts as platforms to preach your system-mormonism from.

Conversely I have to spend most of my time explaining how your two-bit categories of thought are generally entirely inadequate to the discussion at hand, and then have to deal with you mounting rearguard actions to fit things into your misshapen boxes. The blunt hammer of your 'systemetizing' treats everything it encounters as a nail, and you lack the very ability to imagine that not everything amenable to it's bluntness. This is nicely dramatized in your request for a 'generality that does not come trailing the other for context': as I've told you time and time again, this isn't what it's meant by a singular, but because you literally lack the capacity to think outside of your pre-fab categories, you take your own failure of imagination for a failing on my part.

So it's cute that you think you're trying to help me, but until you develop some sense of just how inadequate your categories of thought are, I'm afraid that all you're doing is consistently charging me with not living up to an extraneous position, which, as far as I'm concerned, gets everything entirely bottom-up to begin with. Your 'help' is of the same kind offered by Mormons at the door - mostly irrelevant and preaching for the sake of conversion.

You can actually use a Taylor series to reconstruct a primitive curve (locally, around a singularity) with a single derivative. I couldn't tell you the details, but that's what the paper is referring to.

As for the derivative, you have to remember that we're speaking about dy/dx under the double condition that dy/dx = 0/0, AND dx > 0. That's what at stake here, not just the derivative of any particular function.

I don't recall my calculus well, but so far as I know, the derivative still uses the function-argument schema, and I'm not sure what you were trying to say about it: the dx/dy are essentially notations for explicitly binding variables. The rate of change with one value w.r.t. another still just means, if the argument were to change 'infinitesimally,' how the value would change, and then extrapolating from that. — The Great Whatever

The issue is that much like the quantum formalism, there are differing interpretations of the differential, and the general attitude in math is mostly 'shut up and calculate'. While Newton and Leibniz - who 'discovered calculus' - appealed to infinitesimals to explain the efficacy of the calculus, modern interpretations like the epsilon-delta method get rid of any such notion and appeal to the idea of limits instead. Basically the e-d method gets rid of any reference to geometry. There is also non-standard analysis, which formalizes the notion of the infinitesimal, but there's lots of people who're are pretty sus about it.

Deleuze sits somewhere in the middle here. He refuses the turn to either infinitesimals and limits to explain the differential, and instead argues that the differential must differ in kind from the numbers that make up the primitive curve; negatively, the differential is not another number. Positively and specifically, the differential must be productive and generative of the primitive curve, and indeed, of number more generally. So while infinitesimals posit the existence of 'really little numbers', and limits do away with any reference to such numbers, Deleuze - whose reading is indebted to Jean-Baptise Bordas-Demoulin and Hoene-Wroński - instead reads the differential as determining the behaviour of the curve around a singularity. Singularities are points at which the curve changes it's overall behaviour: generally a change in the value of the slope of the curve (from positive to negative, or from either one to zero and back again, for example)

What's important about this is that the differential no longer simply corresponds in a one-to-one manner with a value on the primitive curve. Rather, what's important is that the differential determines the overall character or quality of the curve. Here's Evens: "This is the sense in which the differential is a universal: the differential packs into each point the nature of the entire function, for the differential relation generates not the value of the function, but its behavior, its character, what the function is doing at each point. It’s not that the differential relation represents the slope of the function at each point; it’s that by representing the slope of the function at each point, the differential relation presents or characterizes the whole function in each of its points... The differential relation captures... how many times [the primitive function] changes direction, how many bumps it has and how regularly they occur, how often it becomes infinite, and how often it crosses the x-axis." (my emphasis).

The point - after this long digression - is this: the differential - interpreted thusly - is not a matter of binding variables. On this reading, it is explicitly the opposite of that. Evens again: "In other words, the differential relation is not a formula that relates x to y over some range of values for x, though this is how we are taught to interpret it: in school, the differential relation, or derivative, is just another formula, another function akin to the primitive function. Rather, the differential relation relates x to y not in breadth, over a range of values, but in depth; it operates in each point on the function, condensing the quality, the character of the entire function into every point... In fact, if you know the values of all the derivatives of a function at a given point, you can construct a polynomial, another function, that is equivalent to the primitive function near that point."

This universality of the differential - the fact that it determines behaviour over singular points rather than explicit values in a one-to-one manner - is why Deleuze will explicitly set this understanding against any which would take the differential to be a matter of a general formula generating particular values: "The relation dy/dx is not like a fraction which is established between particular [values], but neither is it a general relation between variable algebraic magnitudes or quantities. Each term exists absolutely only in its relation to the other: it is no longer necessary, or even possible, to indicate an independent variable. ... The zeros involved in dx and dy express the annihilation ... of the general as well as the particular, in favour of the universal..." Anyway, I hope this hasn't gotten too technical, and I'm kind of writing on the edges of my math knowledge as well, but I hope it constitutes something of an answer to your concern here.

Explicit in my description of limits is that they don't "actually exist". Limits are what actuality can approach - with asymptotic closeness. But by the same token, actuality can never arrive at the limit. The limit is where existence ceases to be an intelligible possiblity.

Thus a limit is virtual - in having this kind of negative reality. The reality of a general constraint on actualisation or individuation. — apokrisis

This must be why they say ignorance is bliss. It saves you from this kind of embarrassment. As usual, the terms in play aren't so easily coopted into your pre-fab categories: the virtual - which refers here to the register of coupled rates of change - is precisely opposed to the possible, and in fact is more or less defined directly in distinction to it: “The only danger in all this is that the virtual could be confused with the possible. The possible is opposed to the real; the process undergone by the possible is therefore a 'realisation'. By contrast, the virtual is not opposed to the real; it possesses a full reality by itself…. Any hesitation between the virtual and the possible, the order of the Idea and the order of the concept, is disastrous, since it abolishes the reality of the virtual.” Your conception of limits - as having ‘negative reality’ that constrains a general ‘vagueness’ could not be better described as exactly what Deleuze considers to be the entirely wrong approach to things.

Elsewhere again: "The notion of 'generality' here suffers the disadvantage of suggesting a confusion between the virtual, in so far as it is actualised by a process of creation, and the possible, in so far as it is realised by limitation.” And I’ve already quoted it, but since you seem to have a selective reading problem, here it is again: on your view, "difference can no longer be anything but the negative determined by the concept: either the limitation imposed by possibles upon each other in order to be realised, or the opposition of the possible to the reality of the real.... It is contradictory to speak of 'potential' ... and to define differenciation by the simple limitation of a global power, as though this potential were indistinguishable from a logical possibility.”

So no, the only idiocy here is yours, thanks to your ever-reliable inability to think beyond the six or seven words you have at your disposal to talk about anything whatsoever. Thanks too for affirming beyond doubt that you take the entirely wrong view of how to understand the problem of relationally here. As for the Evens paper, the irony of complaining that I have a comprehension problem is kinda hilarious considering that the whole paper is geared towards treating the differential not as a question of limits, but as a question of generative production that is everywhere opposed to understanding the differential in terms of limits. But please, don’t let that stop you from trying to continually jam your misshapen pegs in to spaces where they don’t fit.

As for me occasionally adopting the parlance of Bateson and the cyberneticists - yeah, I do actually recognize the usefulness and importance of such concepts, when taken in certain contexts. However I'm not so fool as to pretend that they constitute anything close to a reasonable metaphysics, and I'll be the first to tell you that they are only useful within a very limited and circumscribed domain of application. And besides, if we're simply 'talking about the same thing with different jargon', then one wonders about the pathetic arrogance of your introduction to thread by declaring that the OP is 'a dangerous pipe dream' which is 'particularly wrong headed'. Perhaps the vacillation is function of your literal inability to understand most anything of what's going on here coupled with the need to preach your gospel despite it's utter intellectual poverty.

And speaking of preaching - dude, if it were up to me I wouldn't engage with you ever, except you can't help but spew your babble in every thread I post in. Trust me, I have never once initiated a conversation with you except when you barge in telling me how I got it all wrong from the perspective of your ready-made monotone pseudo-system. The only one who incessantly rocks up time and time again to spread the gospel of symmetry-breaking and general-particular bullshit here is you. So if you feel hard done by feel free to fuck off any time - you won't exactly be missed.

Doesn't matter: The very fact that you're speaking of limits at all is to go awry.

Except I have no problems with reciprocal relations on the condition that what is reciprocally related are themselves relations. What's at stake here are not reciprocal relations between terms - and especially oppositional or dichotomous terms - but a reciprocal relation operating already at the level of relation. Every time you cash out reciprocity at the level of generalities, you go wrong. Which is basically all the time.

Deleuze warns exactly against this conflation - of which you engage in every time - of what he calls the virtual with the actual, wherein the terms of the reciprocal relation are taken to be themselves terms rather than relations. On such a compromised view, "difference can no longer be anything but the negative determined by the concept: either the limitation imposed by possibles upon each other in order to be realised, or the opposition of the possible to the reality of the real.... It is contradictory to speak of 'potential' ... and to define differenciation by the simple limitation of a global power, as though this potential were indistinguishable from a logical possibility."

Perhaps - and this just struck me - the best way to get a handle on this is to speak in terms of coupled rates of change. A rate of change, we can recall, is already a 'derivative': it is a change of a second order, a change that measures a change. Now, rates of change are interesting because they are not simply measurements of on-going processes so much as they define those processes themselves. For example, a population (of cells, of animals in an ecosystem, of a nation-state) can be defined not simply in terms of it's numbers - in fact a rather poor definition - but in terms of it's rates of change with respect to parameters like births, deaths, migration flow and resource availability.

Even more importantly, these rates of change can be said to be coupled, which is just to say that rates of change very in relation to each other. The higher the birth rate, the higher the change in resource consumption, for example. What we're dealing with here, in other words, is relations between relations; moreover, it's these relations which define the very 'objects' of which they are said to be relations 'of'. In embryogenesis for example, it's the differing rates of change between cell birth and death, along with the synthesis and degradation of so called 'adhesion molecules' (which bring cells together), that define what kind of cell (brain cell, skin cell, etc) will be formed. These coupled relations - which are productive and not merely derivative of their relata - have the 'form' of the relation prescribed in the differential calculus: dy/dx (the ratio between the change in one series and the change in another).

Note that one can of course, artificially reverse this whole enterprise so that rates of change are mere 'properties' of self-identical substances. But everything that is in any way important is thereby missed: the entire process of individuation whereby a thing 'takes on' an identity is missed. Any attempt to treat these relations as properties - which is entirely possible - simply misses the becoming of the entity or process at hand. Recall too that in the OP, I marked a distinction between becoming and change. In a process such as embryogenesis, becoming is taking place 'all the time': change, however, only occurs when coupled rates of change cross certain thresholds, when enough cells accumulate in a certain point at a certain speed in order to trigger certain reactions which in turn engender cell mutation, etc. Predicate logic basically operates entirely at the level of 'change': it literally cannot see, by design, the intensive becomings which operate at the level of relations.

I should also note that none of this is particularly 'out of the box': the sciences have been operating in this domain for decades, and continental philosophy has never taken predicate logic seriously. I would suggest instead that the whole institution of formal logic has on the contrary 'boxed itself in', playing formal-logical games without actually attending to the world about it.

This is possible, but I'm not sure what it buys you. For example, one can 'Montague-lift' an individual, to turn it into what's called a 'generalized quantifier -' that is, the set of properties true of that individual (which includes its relations to other things - these being properties once you saturate the first term). In fact, the originator of the device, Richard Montague, proposed that the meaning of say a proper name is not the individual which it denotes, but rather the set of properties that individual bears.

You could also create a logic in which properties are primary and individuals are secondary, reversing the role of function and object we've had since Frege. But I think ultimately this is a terminological quibble and it's unclear to me how it genuinely rephrases the problem. The point is that properties and individuals interact in a certain functional way: whether one takes individuals or properties/relations as fundamental probably won't change that. — The Great Whatever

The point of much of this is to see how one would approach concepts from the point of view of genesis: that is, if we don't take for granted the individuality of any-one-thing and instead try and approach from the point of view of things-coming-into-being. From such a perceptive, basically the entire edifice of formal logic is more or less inadequate to the task, precisely because it can only 'think' in terms of a subject-predicate coupling, and consequently, in terms of the already-individuated. The very form of thought that it engages in is compromised. As such, it's not enough to 'swap' the priority from individual to property, which simply keeps the form in place while reversing out the contents, as it were. Every time a relation is treated as a property, one gives up on thinking relation.

From the point of view of genesis and individuation, to say something like: "The point is that properties and individuals interact in a certain functional way...", is basically anathema. There's no point in beginning with your set of properties, and your set of individuals, and then combining and breaking them apart, lego-like. Doing this takes for granted individuation, and no amount of combinatoric cleverness will ever attain the point of view of genesis.

Peter being taller than Paul isn't, I would say, a property of the dyad <Peter, Paul>. Perhaps I would say that the dyad takes part in the relation 'taller than,'... — The Great Whatever

Interestingly, this was almost exactly Plato's solution: to posit the (supersensible) Idea of the Small and the Idea of the Large which things could 'participate' in - as in, to say Peter is taller than Paul is to say Peter participates in the Idea of the Large in relation to Paul and that Paul participates in Idea of the Small in relation to Peter: but of course this just kicks the problem down a level because this 'taking part' or 'participating in' is itself a relation - and it's no good to account for a relation in terms of a relation.

This is why I think relations are troublesome: either one erases their specificity by treating them as a property, or one ends up recoursing to some Platonic notion of Participation which just makes the whole thing mysterious to begin with. The upshot of treating relations as external to their terms, on the other hand, is to grant relations a kind of autonomy with respect to their terms, or rather, it reverses the relation: rather than the relation being defined by it's terms, terms themselves become defined by their relations. This is the link between relations and becoming: if understood on this model, a change in a relation would imply a change in the relata (rather than the other way around): "If relations are external to their terms, and do not depend on them, then the relations cannot change without one (or both) of the terms changing. A resembles B, Peter resembles Paul: [if] this relation is external to its terms, it is contained neither in the concept of Peter nor in the concept of Paul. If A ceases to resemble B, the relation has changed, but this means that the concept of A (or B) has changed as well. If properties belong to something solid, relations are far more fragile, and are inseparable from a perpetual becoming" (Dan Smith, The New).

One way to cash this out a little more solidly is - as Deleuze does - is to turn to the differential (dy/dx) in differential calculus as a model for a "pure relation" without terms that is at the same time generative of the curve or solution-series which it is the (supposed) 'derivative of'. I won't go too far into this as it's perhaps a bit more math-y than is necessary for a general discussion, but if you have access to Academia.edu, check out Aden Evens's paper on this: https://www.academia.edu/1084825/Math_Anxiety ; long and short of it is that one can look to calculus as model for what it would mean to have a relation without relata, and which also plays the function of generating relata (quick quote for preview's sake: "the differential relation, dy/dx precedes the “primitive” function whose slope it is said to represent. In calculus class we are presented with a function and told to differentiate it, to take the derivative or produce the differential relation. In Deleuze’s rereading of the calculus, the primitive function does not precede the differential relation, but is only the ultimate result or byproduct of the progressive determination of that relation.The differential is a problem, and its solution leads to the primitive function".)

Weren't you talking, literally, about any thing at all? And wouldn't that include complex negentropic objects? & The problem with my discussion of singular objects is it that's not general enough [for what]? — csalisbury

General enough to fit into the artificial coordinates of his 'system' of course. The whole thing is a kind of watered-down Hegelianism: if the singularities don't fit the system, simply throw away the singularities. There is simply no 'place' in Apo's 'system' to accommodate the singular: it - and consequently he - cannot think in terms of the specificties of a given, concrete situation because the whole thing is designed to 'level' singularities and subject them to a (general) order of equivalence in order to render them into particulars. That's why you never actually learn anything from Apo's posts except how the system itself works - it's a self-referential mess that basically ends up talking about itself more than the phenomena it supposedly accounts for. Hence also their mind-numbing monotony.

It's what Deleuze speaks of in the opening pages of D&R, where he specifically points out how such conceptions render themselves blind to both the singular and the universal:

"There is no reason to question the application of mathematics to physics: physics is already mathematical, since the closed environments or chosen factors also constitute systems of geometrical co-ordinates. In these conditions, phenomena necessarily appear as equal to a certain quantitative relation between the chosen factors. Experimentation is thus a matter of substituting one order of generality for another: an order of equality for an order of resemblance. Resemblances are unpacked in order to discover an equality which allows the identification of a phenomenon under the particular conditions of the experiment. Repetition appears here only in the passage from one order of generality to another, emerging with the help of - or on the occasion of - this passage.

... However, If repetition exists, it expresses at once a singularity opposed to the general, a universality opposed to the particular... It puts law into question, it denounces its nominal or general ... In its essence, repetition refers to a singular power which differs in kind from generality, even when, in order to appear, it takes advantage of the artificial passage from one order of generality to another."

There's nothing intrinsically 'taller than' about Peter, but there is something intrinsically 'taller than' about the dyad <Peter, Paul>. Increasing the number of substances by one doesn't seem to change anything. — The Great Whatever

This is a valid move I think, but I also think that it comes with a trade off, which is precisely to give up thinking about relations. That is, it's possible to translate: "Peter is taller than Paul" to: "Peter being taller than Paul is a property of the dyad <Peter, Paul>", but at this point, you've lost the specificity of relationality. I mean, if you expand this 'translation strategy' to include the whole universe (that is, if you take any relation that <Pr,Pl> might enter into and then make that a property of a larger dyad and so on ad infinitum), you'd end up with something like a set U with elements (x,y,z) where each element is a relation-turned-into-a-property like (<P,P>(P>p)). Not unlike - or pretty much exactly like - a Leibnizian monad.

But the whole point is to think relation outside or beyond the subject-predicate model such that - to use the Deleuzian phrasing - 'relations are external to their terms'. I think that terms can always 'co-opt' relations in precisely the way you've proposed, but in order to secure the autonomy of relation, one ought to resist that kind of move. Of course at this point I'm not trying to adjudicate between the 'two paths', as it were, but just exploring where this particular one might take me.

Sorry it's taken a while to reply, yours was a great reply which I had to think about a bit and I've been a tad busy recently.

Did they program that one for you too?

Or at least that's the long-held consensus of everyone in the Deleuze Studies department ;)

Just kidding, sort of, I really do like Deleuze, but do you know what I mean? — csalisbury

Yeah but who cares unless you're invested in that little cottage industry to begin with? Honestly, it's these self-referential loops that get us stuck in these situations in the first place. Really, if your thinking isn't being forced by the exigency of the situation, if it's not imposing itself upon you in order to reorient your sedimented categories of thought, then what's the point? I mean honestly, Apo literally does not think as far as I'm concerned. He's like one of those conversation-bots you used to come across a few years ago that kind of just spat out pre-fab lines depending on the keywords it came across. I mean he is literally incapable of understanding what it means for something to be singular and not - because this is the only word his fifteen word vocabulary allows him - particular.

This just again confirms Deleuze to be a donkey. There couldn't be a more precise movement than a reciprocal or inverse relation. — Apo

Yes, precise because analytically so, and thus completely incapable of engaging at a singular level, and thus philosophically impotent. And who says donkey, lol.

Again, if you could present a valid example of a singular conception - one that somehow exists alone without being reciprocal to a context - then you might have something to get started with here. But you don't.

And who says singularity is something that 'exists alone without being reciprocal to a context'? Again, It's cute how you like to jam things into the three of four categories of thought you are capable of, but it makes for very boring conversation.

The only thinking is thinking again, thinking otherwise : ) Everything else is doxa.

So you don't recognise this as a distinction between syntax and semantics? — apokrisis

It isn't though. Maybe one day it'll hit you that your pre-fab Apospeak isn't applicable here. Maybe one day you'll even respond to the singularity of the discussion, but it's no surprise that one committed to modelling reality after the image of thought is incapable of novelty in thought. Deleuze understood the dangers and inadequacies of exactly your approach better than anyone, and it's unfortunate that his warnings are less heeded than they should be:

"Of what use is a dialectic that believes itself to be reunited with the real when it compensates for the inadequacy of a concept that is too broad or too general by invoking the opposite concept, which is no less broad and general? The concrete will never be attained by combining the inadequacy of one concept with the inadequacy of its opposite. The singular will never be attained by correcting a generality with another generality.... the dialectic [is] a false movement, that is, a move­ment of the abstract concept, which goes from one opposite to the other only by means of imprecision."

How better to capture the poverty of your entire thought process? And as for the conjunction of 'singular and reductionist thinkers...' - well, that's just embarrassing.

Medium <> Message. Again the Kantian conflation.

Not at all curious - it isn't science's job to think the singular - it is methodologically bound to ignore it! - and no one with a taste for philosophy would expect it to. All the more reason not to confuse the two.

But what justifies that when any one term can only have cogent definiteness or counterfactuality in terms of its "other"? — apokrisis

But this is just a warmed-over Kantianism that gets everything back to front. As if the world cares about the definteness of terms. Nah mate, its you who's wearing your knickers on your head. The whole edifice - generality, symmetry-breaking, vagueness, dichotomies and dialectics - are so many backward projections that compensate for an inability to think the singular.

I am reluctant to separate it along the lines of natural and unnatural, as I consider humans part of nature. — Jeremiah

I wouldn't use the term unnatural though, or rather, I wouldn't set them in opposition to each other; I would say instead that the laboratory setting - in which the category of 'possibility' is an apposite concept - is a kind of 'subset' of nature, embedded in, but not coincident with, the wider world (Not A∨B, but A⊂B.). It's to the degree that philosophy deals with precisely this 'wider' subject matter that I call it a poor - or maybe rather limited - philosophical tool.

Sure, if it makes you sleep better dude.

Lol, jargon jumping by me when you're the one who can't read and impute terms to me that I never used in the first place. Like three times. Please. Your illiteracy is not my problem. 'Muddying the waters'. Maybe I'll sculpt you an irony prize with the thick slabs of it you're serving up.

Cheers. One of the advantages in thinking of randomness in terms of equipotential is that is allows us to bypass many of the tricky debates about causality in a rather clear and unambiguous manner. It also has nice physical applications, as in information theory where one can leverage the random/probable distinction in order to think in terms of noise (randomness/static) and signal (information).

One thing to note however, is that insofar as probability is premised on 'all things being equal' (ceteris paribus) across the repeated exercise of a certain event (as you said), probability is concept far better suited to the laboratory than to nature: by design, it can only operate in the context of a stable, unchanging, and already individuated state if affairs, explicitly relegating any emergence of the new. It can only ever yield generalities rather than singularities. It's a wonderful scientific tool, but a poor philosophical one.

Again, the term spontaneity is yours. I did not use it in my original post(s), and I would prefer to avoid the word altogether if I could. Novelty - which I mistakenly understood you to be asking after - is my preferred term, and in any case, if I were to make a point about randomness here, it would simply be a negative one: that randomness has nothing to do with novelty. And insofar as there is novelty in nature, randomness and it's corollary, possibility, are not very good ways about thinking of nature. They are always behind the curve, one step too late, as it were.

But you are asking a bad question. The question itself is wrong. It's like asking: is the rock falling off the cliff ethical, or not ethical? The right answer is: neither, your question is bad. Substitute ethical for random, and you get the same thing - a category error, a misuse of grammar, a fallacy of misplaced concreteness (three ways of saying the same thing). I'm not sure there are any more ways to convey it.

So when one observes the world, how can one tell the difference between a random event and a spontaneous event? — apokrisis

But this is a misformed question: the point is that randomness (qua equiprobability) is indexed to motivations and expectations of an inquirer (not 'mind', btw, a word with much too much metaphysical baggage), and that it's category error to speak of nature as being either random or not random. Nature is a-random, if you like. Hence the malformed question. Perhaps the reason you are so perplexed is that your whole framework of thought is tethered to the notion of possibility, but that's not my problem.

As an aside, the very idea of the 'will' is also one of the worst posed notions in all of philosophy, so despite your attempt - as per your usual manner - to pin things on me that I don't hold, well... nope, no will here.

One can accept Bergson's critique of possibility without subscribing to his metaphysics wholesale. Nuance isn't hard. As for your question, I never said anything about predictability or unpredictability - this is a term you seem to have read in to my comments. Feel free however, to clarify what you think you mean by the distinction between ontic and epistemic unpredictability. Given that novelty just is the interruption of any regime of predictability, I don't really have much use for that concept. But then, singularity does seem a concept alien to your whole framework...

Because as per Bergson, possibility as a modal category is always a back-formation: it takes what exists and then retrojects it's possibility as an explanation for it's being. 'Spontinaiety' or rather novelty isn't the realization of abstract possibilities - it's the very creation of possibilities to begin with. Thinking in terms of possibility always forecloses any thought of novelty. It can be useful, but it isn't good metaphysics.

It's the word Bateson uses - and yes, I know you like 'constraints'. And I do believe that spontaneity is a part of nature, but probability or possibility isn't the best way to go about thinking it.

Is random a situation with various possible outcomes? — Jeremiah

Something like this is probably the most useful way to grasp the concept of randomness: as 'equiprobability', or the equality of probable outcomes. Conversely, 'non-randomness', or order, would be the culling of probable outcomes so that some are more likely than others. This 'culling' can be thought of in terms of putting 'restraints' on the range of possible outcomes. Gregory Bateson uses the lovely old image of the monkeys on a typewriter to get the point across:

"If we find a monkey striking a typewriter apparently at random but in fact writing meaningful prose, we shall look for restraints, either inside the monkey or inside the typewriter. Perhaps the monkey could not strike inappropriate letters; perhaps the type bars could not move if improperly struck; perhaps incorrect letters could not survive on the paper. Somewhere there must have been a circuit which could identify error and eliminate it." (Steps To an Ecology of Mind, "Cybernetic Explanation").

If, however, one thinks of randomness in terms of equiprobability, the properly interesting question is what the 'ontological status' of 'probability' is. Are there probabilities in nature, or is probability an epistemic concept that has to do with the motivations of an inquirer? I lean towards the latter answer, but there you go.

But surely such 'active pessimism' would simply no longer be pessimism any more? Wouldn't 'active pessimism' simply be.... optimism? One of my favorite little tracts I read this year was Eugene Thacker's Cosmic Pessimism. What I very much like about it is the way it conveys pessimism as consistent precisely to the degree that it fails to be so:

"Had it more self-assurance and better social skills, pessimism would turn its disenchantment into a religion, possibly calling itself The Great Refusal. But there is a negation in pessimism that refuses even such a Refusal, an awareness that, from the start, it has already failed, and that the culmination of all that is, is that all is for naught. Pessimism tries very hard to present itself in the low, sustained tones of a Requiem Mass, or the tectonic rumbling of Tibetan chant. But it frequently lets loose dissonant notes at once plaintive and pathetic. Often, its voice cracks, its weighty words abruptly reduced to mere shards of guttural sound".

..."The very term “pessimism” suggests a school of thought a movement, even a community. But pessimism always has a membership of one — maybe two. Ideally, of course, it would have a membership of none, with only a scribbled, illegible note left behind by someone long forgotten. But this seems unrealistic, though one can always hope"

"And all of this shadowed by an impasse, a primordial insignificance, the impossibility of ever adequately accounting for one’s relationship to thought — all that remains of pessimism is the desiderata of affects — agonistic, impassive, defiant, reclusive, filled with sorrow and flailing at that architectonic chess match called philosophy, a flailing that pessimism tries to raise to the level of an art form (though what usually results is slapstick)."

I'm just saying - doesn't an 'active pessimism' betray... pessimism?

That's the price we pay for civility around here.

You can call our vulgarity without yourself being vulgar. Watch this: Augstino, I find your rhetoric vulgar. Ta da.

Using the word "fuck" is not against the rules. Using it as much as you have been recently though to refer to the sexual act along with "sluts"/"pussy"/"dong" etc in what is supposed to be a serious discussion is low class and does detract from the quality of your posts. — Baden

This. For all your 'conservatism' your rhetoric tends to be shot through and through with sheer vulgarity ("You'll marry some slut who fucks left and right, just like you", etc), especially when characterizing the positions you do not like. It is unproductive, off-putting, and makes the forum a worse place for discussion. That, and not your politics is the reason you are 'on notice'.

Haha, fair enough. Sometimes I get carried away and forget to parse things down to make 'em more digestible. Its hard in this case too because the ideas are so interesting and abstract!

Now think of a cone (a 3D shape). It could be used to represent the the sentence ''I'm good'' instead of individual sounds and then we could point it to the left to convey anger, to the right to convey joy and so on. Simpler and more compact isn't it? — TheMadFool

Like this?: ▲ ►◀

Again, almost everything that can be represented in a higher-dimensional language can be presented in a lower-dimensional one in a simpler way. Indeed, the three triangles here are in fact simpler than cones precisely because they discard the additional 3rd dimension of depth. If you want your example to work, you need to utilise the unique dimension of depth as a vector of information, not orientation: the differences between the three sentences will need to be encoded in the volume of your cone, not in it's orientation, because this is the one thing your lower-dimensional language, by definition, doesn't have.

Indeed, one can imagine something like a 'topological language', in which every slight transformation of a three dimensional shape might correspond to a different 'word' or meaning. This, for example:



...might be an entire essay. The problem of course is that this kind of language, if you could call it one, would be unimaginably hard to learn or understand. If the syntax of this language are the transformation rules between one shape and another (when the torus begins to thicken on the left, say), and the semantics the distinct shapes the model takes on as it transforms, one would have to learn and know (and have the recognition capacity for!) an almost infinite amount of minute, impossible-to-reasonably-track changes in volume. It's possible that some higher being might be able to communicate with changing shapes, but we certainly wouldn't be able to. Or maybe just a super-advanced computer.

This is why even Chinese, which has a very tight morpheme-grapheme ratio (i.e. every character tends to designate a single word) still needs to have multiple characters. Think of it as a trade off along two axes: one can either have a limited alphabet base and employ combinatorics to craft meaning (as with English with it's limited letters and varied manners of combining them), or one can have a very large character base which employs combinatorics in a limited manner. With a very high-dimensional language like the topological one above, what you're asking for is a something like a language with ONE character and no combinatorics - only topological transformation (assuming one can take a shape as a character) - it would be (almost?) entirely analog language (only degrees of transformation form one shape to another, rather than discreet units).

The problem, of course, is that we are simply not cognitively equipped to deal with any kind of language of that sort. Most fluent Chinese speakers know roughly 8000 or so characters, beyond which the words begin to get pretty obscure (Chinese having 20000 characters give or take). What you're after though, would be to multiply this to infinity, as it were: there would no doubt be a certain simplicity to it (especially one you scale your dimensions up beyond 4 or 5) in which case the 'changes' would be practically imperceptible. But it would also be unlearnable, and more or less unrecognizabe as a language at all.

This is why, as humans, who can only do so much with language, we have an entire 'extra-linguistic' scaffolding, that allows us to makes sense and multiply the meanings that language can generate. This is what I was getting at when I brought in the notion of play and satire, which has to do, once again, with human behaviour. We employ our environments as much as our words to communicate, because we know that words can only do so much given our particular cognitive capacities. We 'compensate' for our inability to deploy higher-diemsional languages by employing the entire world around us instead, as it were, as a means of communication, to bring it 'into the fold' of communication as it were, rather than 'sticking to the written word'. This is equally why we are so much more than symbol-manipulating machines. Anyway, super interesting stuff to think about.