Metaphysics: Deleuze│Spencer-Brown

Some philosophers, struck with the recent discovery that no one set of metaphysical assumptions is necessary to communication, have fallen into the fallacy of thinking that we can communicate without any metaphysics at all… It is one of the greatest, if not the greatest, of discoveries in 2000 years to have seen that no one metaphysics is necessary; but it is a simple mistake to suppose that none is.

G. Spencer Brown Probability and Scientific Inference, London: Longmans, 1957, pp v-vi

Twelve years after writing those words, George Spencer-Brown published a great metaphysics: Laws of Form. The book can be read as an important and troublesome work of logic and mathematics. It is even more than that.

Among other things, some no doubt still to be discovered, Laws of Form is a touchstone for new theories of autopoiesis – the autonomous construction of life, whether artificial or natural – and cybernetics – the study of control and communication systems. The two are connected because control and communication systems play important roles in the autonomous functioning of autopoiesis.

With his brother, David Spencer Brown, George designed an axle counter for British Rail. These systems are important for safety; for instance, in checking whether a wagon has become detached in a tunnel. The design influenced his work in Laws of Form:

The circuits represented by these equations, the latter being presently in use by British Railways, comprise, as far as we know, a first application of each of the two inventions, notably the first construction of a device which counts entirely by ‘logic’ (i.e. with switches only, and with no artificial time delays such as electrical condensers) and, in addition, the first use in a switching circuit of imaginary Boolean values in the course of the construction of a real answer.

G Spencer Brown, Laws of Form, London: George Allen and Unwin, 1969, p 99

The relatively new study areas of autopoiesis and cybernetics suggest blueprints for the technological, biological and existential revolutions we are undergoing. These interconnected practices are prone to forgetting the contingency of any proposed metaphysics and, hence, of the life and theories growing from those beginnings.

If we follow Spencer-Brown, any fact is contingent on particular observations and metaphysical assumptions. Yet Laws of Form is deeply and perhaps unavoidably conflicted about the chance nature of those assumptions.

Sometimes, he seems to be arguing for the necessity of a particular mathematics as ground for experience. At other times, the contingency of an observation and beginning seems to apply to his philosophy. Perhaps it can be both: a necessary system built on chance, where the beginnings are contingent but all that unfolds thereafter is necessary.

It is not obvious that this solution is available for Spencer-Brown’s metaphysics, since chance and perspectival observation return within the unfolding system. This reappearance is part of its great originality and power, in particular in relation to paradoxes of self-reference, but it raises problems about when contingency is ever absent.

Three philosophical questions guard against the obliteration of the contingency of metaphysics. They are testimony to the everlasting, untimely and uncanny role of philosophical scrutiny (and, therefore, to the hollowness of narrow conceptions of impact around measurable effects and artificial timescales):

  1. What is the implied metaphysics here?
  2. What does it commit us to?
  3. How does it compare to other possible metaphysics?

I’m currently writing a book with the working title ‘Autopoiesis and Existence’. Coming to terms with Spencer-Brown’s dazzling and moving publications is an important part of it.

In this post, I’ll rely on Deleuze’s philosophy to sketch responses to the above questions. My technique will be to contrast two bare metaphysics, one from Spencer-Brown and the other deduced from Deleuze, along with examples to explain both.

Spencer-Brown is a master at stripping down thought. In the opening pages of Laws of Form he gives the following premises, definitions and axioms:

We take as given the idea of distinction and the idea of indication, and that we cannot make an indication without drawing a distinction. We take, therefore, the form of distinction for the form.

Distinction is perfect continence.

The value of a call made again is the value of the call.

The value of a crossing made again is not the value of the crossing.

Laws of Form, pp 1-2

The appearing and disappearing of the hyphen in Spencer-Brown in the passages above is not a mistake. He changed his names many times throughout his life, including writing his novels under a pseudonym. This swapping of names will be one of my examples.

There is a beautiful and – from my observation point – somewhat sad account of these name changes and much else in Graham Ellsbury’s memoir of his friendship with Spencer-Brown (Graham Ellsbury ‘A Brief Personal Memoir’, Cybernetics and Human Knowing, Vol 24 (2017), nos 3-4 pp 103-13). From Ellsbury’s standpoint and, as we’ll see from the theory of naming in Laws of Form, any sadness is extraneous. I recommend this journal volume dedicated to Spencer-Brown and Laws of Form, as well as the forthcoming volume on Spencer-Brown at Esalen (around the conference on Spencer-Brown’s work held in 1973, Esalen, Big Sur – two years before that photograph of Deleuze).

The second metaphysics is deduced from my, perhaps idiosyncratic, reading of Deleuze. In a future blog post on his metaphysics and the transcendental, I’ll explain why, in agreement with Brent Adkins, I cleave him more strongly from Kant than the interpretative mainstream (Adkins, Brent. “To Have Done with the Transcendental: Deleuze, Immanence, Intensity.” The Journal of Speculative Philosophy, vol. 32, no. 3, 2018, pp. 533–543. JSTOR,

‘To cleave’ is a fundamental term in Spencer-Brown’s work. In Deleuze and Deleuze and Guattari’s philosophies a cut is necessary but not fundamental. This is because every cut is on a flow and undone by further flows. Continuity is primary for Deleuze, whereas severance is the sole starting point for Spencer-Brown:

Any division of a space results in otherwise indistinguishable divisions of a state, which are all at the same level, whereas a severance or cleavage shapes distinguishable states, which are at different levels. An idea of the relative strengths of severance and division may be gathered from the fact that the rule of number is sufficient to unify a divided state, but not to void a cloven space.

Laws of Form, p 87

The verb ‘to shape’ is the most important term in this passage. The laws of form are laws of shaping out spaces, of constructing autonomous systems and their universes. For example, to divide a space in two, leads to two parts with connections and shared properties, such as number (‘3 this side and 4 that side’). Whereas to cleave p (<p>) from a space is to fully separate p – a much more fundamental shaping. This is not a divided space with p in part of it, but rather the distinction of p from any other space – hence the definition of distinction as perfect.

It is crucial to note, though, that perfect continence does not mean ‘p without any other space’. In cleaving from a space or other things in that space, these do not disappear; on the contrary, the cleaving depends upon and involves them in special ways covered by the laws of form.

These definitions and their roles for indication are important for later accounts of autopoiesis. Francisco J Varela sees it as giving the structure for any universe, thereby reinforcing the idea that Spencer-Brown’s metaphysics is necessary, prior to the relativity of observations:

I also believe that new possibilities opened, in this and other domains, after the formulation of the calculus of indications by G. Spencer Brown. By succeeding in going deeper than truth, to indication and the laws of its form, he has provided an account of the common ground in which both logic and the structure of any universe are cradled, thus providing a foundation for a genuine theory of general systems.

Francisco J Varela, ‘A Calculus for Self-Reference’, International Journal of General Systems, 2:1, 5-24, 1975, p 6

Perfect continence is also a ground for the importance of boundaries in theories of autopoiesis. In Maturana and Varela, then Evan Thompson, the perfect containment of boundaries provides grounds for a distinction between merely autonomous and fully autopoietic systems. Though a puzzle remains because those boundaries must be crossed, by energy sources for example, but not in a way that alters the autonomy of the system as it maintains and develops itself and its boundary:

An autopoietic system dynamically produces its own material boundary or membrane, but a system can be autonomous without having this sort of material boundary. The members of an insect colony, for example, form an autonomous social network, but the boundary is social and territorial, not material.

Evan Thompson, Mind in life: Biology, Phenomenology, and the Sciences of the Mind, Cambridge MA: Belknap Press of Harvard University Press, 2007, p 44

Despite its apparent simplicity and immediacy, Spencer-Brown’s form is already a point where interpretations bifurcate. Though he influenced a wide range of thinkers on cybernetics, autopoiesis, social systems, logic and mathematics, this is a far from homogenous group.

The next passage, by Niklas Luhmann, while acknowledging Spencer-Brown, introduces ideas that I view as highly controversial. Unlike Varela, Luhmann conflates the form with reference and truth, whereas Spencer-Brown defines the form as prior to truth and indication as prior to reference. Luhmann claims that crossing the boundary takes time. Spencer-Brown makes no such point and is rigorous in not doing so, since that would insert time as prior to the second axiom, or law of crossing, and require a metaphysics of time before the laws of form:

The following reflections arise from the difference-theoretical starting point of our investigations. In other words, they arise from the conception of reference and of truth as form in the sense of Spencer-Brown – as a two-sided form, as difference, as a marking of a boundary whose crossing takes time.

Niklas Luhmann, Theories of Distinction: Redescribing the Descriptions of Modernity, William Rausch (ed.), Trans. Kerstin Behnke, Stanford CA: Stanford University Press, 2002, pp 64-5

In Laws of Form, time is constructed through repetition, memory and imaginary states. Sequences of the same crossings, stored in memory as imaginary, give us a wave form, like the awareness of the ticking of a clock, dependent on imagining ourselves as hearing each tick over a period (a form of paradoxical self-reference): ‘… we have, in time, a succession of square waves of a given frequency.’ (Laws of Form, p 60)

This construction of time ‘can be seen to cover, without inconsistency, all the representative forms hitherto considered.’ (p 62) This in no way means that indication and crossing take time. This would invite the question ‘How much time?’ It means that time can be constructed without inconsistencies with any of the theorems of Laws of Form, thanks to imaginary values.

Spencer-Brown was already well-aware, in 1969, of the immense scope for his work. In the preface to Laws of Form, he highlighted the fundamental importance of severing a space and of assigning boundaries for the sciences and for life itself. It is humbling and somewhat frightening to think of him carrying through his insights with extraordinary conviction and intellectual courage:

The theme of this book is that a universe comes into being when a space is severed or taken apart. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, with an accuracy and coverage that appear almost uncanny, the basic forms underlying, linguistic, mathematical, physical, and biological science, and can begin to see how the familiar laws of our own experience follow inexorably from the original act of severance.

Laws of Form, p v

As much as biography makes me queasy, distracting us from the essence of thought as an exercise on the ragged edge of abstraction, there is a great biography to be written about Spencer-Brown, in addition to the first volume of his autobiography (Autobiography. Volume 1. Infancy and Childhood, Leipzig: Bohmeier Verlag, 2004). His life was dedicated to experiencing those edges and their difficult embodiment.

Perfection in containment and the notion of inexorable necessity in systems are the crux of the differences between two metaphysics. Against perfect isolation, here is my deduced version of Deleuze’s philosophy, conveying continuity as multiplicity and the notion of inexorable disjunction:

We take as given the idea of multiplicity and the idea of drawing, and that we cannot make a drawing without invoking a multiplicity. We take, therefore, the form of multiplicity for the form.

Multiplicity is perfect incontinence.

The value of a repetition repeated again is not the value of the repetition.

The value of a difference made again is the value of all differences.

As stated here, both metaphysics rely on the idea of value. There are risks involved in defining this term, since an overly specific account of value (monetary worth, or moral advantage, or a particular scale or numbering, or even the two values ‘true’ and ‘false’, for instance) would skew each metaphysics and their relations.

I define value as ‘a reason’, in the sense of ‘a reason for a difference’. This is a departure from Spencer-Brown’s definition: ‘… if the content is of value, a motive or an intention or instruction to cross the boundary into the content can be taken to indicate its value.’ (Laws of Form, p 2) So value is the why or wherefore of an intention. However, this motivation requires a difference; the difference is the value of a crossing. My definition is therefore implied, but avoids the philosophically loaded concept of intention.

It is also a departure from Deleuze’s dependence on differences as value, in his transformation of Leibniz’s principle of sufficient reason into a dual function: ‘… sufficient reason has a strange angle to it. On one side, it leans towards what it grounds, towards forms of representation. But on the other side, it diverts and plunges into groundlessness…’ (Gilles Deleuze, Différence et répétition, Paris: PUF, p 352, my translation).

There is a sufficient reason for any represented identity, but this sufficient reason is multiple and beyond representation. Endless differences are the value of any identity – the value of all differences, in the rendering of Deleuze given above. According to my definition of value, these differences are reasons for the value. Again, I view this as implied by Deleuze’s emphasis on the principle of sufficient reason, but avoiding his predilection for baroque folds and complications.

The scope and purity of these bare metaphysics will be distorted by any example, since it will be a restriction on definitions and a narrow interpretation of axioms. Nevertheless, for explanation and practice, I’ll balance two approaches, setting out my interpretations in short glosses, illustrated by examples and followed by critical remarks. Thereby, I hope to skirt the paradox of exemplarity: any metaphysics is for life, but it is skewed when fixed to a life.

Spencer-Brown’s philosophy starts by isolating something from everything else, in order to allow the thing to be indicated or pointed to. His basic intuition is that you cannot indicate without first picking out. Which p? This <p>.

In addition to the ideas of distinction and indication, ‘perfect continence’ means that the thing isolated is completely separated from everything else. This is because distinction and indication don’t work if they are vague or ambiguous. If distinct thing <p> is also partly q, but q is other than p then you have distinguished and indicated p and an overlap with q, not just p.

Due to the process of distinction, as having to take in that which something is distinguished from, there is a temptation to bypass perfection and understand continence as inexact and permeable: many ways of <p> and all that is not contained in <p>. In my view this is an error, since none of Spencer-Brown’s proofs work and neither of his axioms is intuitive, if we begin with this kind of ambiguity.

As an engineer and expert glider pilot, Spencer-Brown is acutely aware of the extreme risks of ambiguity for inputs into systems (Two axles or one? 1000 feet or 970? Unreliable Airspeed?).

There is an important function for ambiguity in his logic and systems, but not at the foundational or input stages; it takes place instead in imaginary values and the capacity to maintain contradictory positions within a system. This is to entertain many inconsistent states prior to resolving them into a simple solution.

Spencer-Brown does not mean entertaining them separately to see which arrives at the best solution, as in a board game taken as a set of possible moves and their results. He was a high-level chess player, familiar with the calculation of outcomes to different moves over long patterns.

It is instead to entertain them together, as interacting within the same system, rather than in distinct lines and tracks, and hence to manage paradoxes, such as those generated by self-reference, or by breaking the law of excluded middle. A game where we assume our opponents are both cowardly and courageous, their cards both strong and weak, with those different emotions and cards having effects on each other:

We may take it that the world undoubtedly is itself (i.e. is indistinct from itself), but, in any attempt to see itself as an object, it must, equally undoubtedly, act so as to make itself distinct from, and therefore false to, itself.

Laws of Form, p 105

Clarifying what he means by act in the above passage, Spencer-Brown traces the Greek roots of ‘act’ to ‘actor and antagonist’ and to the Greek for ‘conflict’ and ‘contest (ἀγώνισμα) . When we calculate in parallel, with distinct and internally consistent lines, we don’t come into conflict with ourselves or a world that is both true and false. However, to get to the world as it is, we must ‘note the identity of action with agony.’ (Laws of Form, p 105) Thereby, Spencer-Brown also reveals one of the secrets of his novels about love and conflict.

For Spencer-Brown, the observer ‘reenters’ the observed system, since the observer not only can, but must be implied in any indicating and distinguishing: ‘Such an expression is thus informed in the sense of having its own form within it, and at the same time informed in the sense of remembering what has happened to it in the past.’ (Laws of Form, p 100)

I’ll return to these remarks on memory and its forms when considering Spencer-Brown along with Deleuze, but it is worth stressing the originality and difficulty of Spencer-Brown’s thought at this point. He is not only making instructions to indicate and distinguish the basis for metaphysics, but imbuing chains of instructions with accurate and yet necessarily self-contradictory memories (my perfect memory of who I was then and there but now and here).

The strangeness of Spencer-Brown’s apparently simple metaphysics comes out here. His laws necessarily imply self-reference, because any calling and crossing is also from what is called or crossed to, since it is from everything. By indicating something, you necessarily implicate yourself, not only as an observer, not only as an actor, but also as acted upon and observed. Furthermore, you can always indicate yourself and, indeed, rename yourself: <James Williams> = <JW>

Dirk Baecker discusses this self-inclusion in a rich paper on the form and Spencer-Brown: ‘Not least the concept of form thus also contains the observer that makes the distinction, because without this observer no distinction would be made.’ (Dirk Baecker, ‘Working the Form: George Spencer-Brown and the Mark of Distinction’ Mousse Magazine. Supplement Settimana Basileia, June 2015, p 7 on PDF)

Baecker interprets Spencer-Brown through the concept of oscillation, such that reintroduced distinctions ‘oscillate imaginarily’ (p 11). Connected to his construction of time, Spencer-Brown shows this by imagining a tunnel that goes back from an indicated form to the outside: ‘… the frequency of its oscillation is determined by the length of the tunnel.’ (Laws of Form, p 59)

Like Luhmann’s reference to time, Baecker’s introduction of oscillation is a limitation on Spencer-Brown’s metaphysics; for instance, by bringing in problems and concepts of periodicity and thereby also many presuppositions about time, space and repetition into the metaphysics. The key questions are again ‘At which point and how does this concept appear? and ‘Is it fundamental?’ Oscillation can be constructed from the form, thanks to imaginary values, but this does not mean the form itself oscillates.

In all these debates, there is a great difference with metaphysics based on atomic facts – ‘p is red’, ‘p is taller than q’ – such as Russell’s logical atomism (Spencer-Brown and Russell knew each other and each other’s work). For Spencer-Brown, <p> is a process, an operation metaphysically and logically very different from simply taking some atomic fact as given. He claims his process is more intuitive than identifying particulars.

Any particular fact has been constructed: ‘Laing suggests that what in empirical science are called data, being in a real sense arbitrarily chosen by the nature of the hypothesis already formed, could more honestly be called capta.’ (Laws of Form, p. xix) The Latin ‘capta‘ means captured, seized or taken, in the sense of shaped and exploited according to an intention.

For Spencer-Brown, we have an intuitive sense of this construction and its strange implications: ‘The act is itself already remembered, even if unconsciously, as our first attempt to distinguish different things in a world where, in the first place, the boundaries can be drawn anywhere we please. At this stage the universe cannot be distinguished from how we act upon it, and the world may seem like shifting sand beneath our feet.’ (Laws of Form, p. v)

An atomic particular such as ‘this green pixel’, or the sense datum corresponding to that pixel, is only arrived at by what Spencer-Brown calls an instruction, such as ‘isolate a green pixel’ or ‘consider the sensed green pixel’.

These instructions presuppose the metaphysics of calling and crossing. You have to cross from a field of pixels to focus on a single one; for instance, when you try to persuade a phone manufacturer to give you a refund by pointing out a flaw in your phone’s screen. You also have to isolate the green pixel from all other things, then call it ‘the pixel stuck on green’.

This would be intuitive, because we have all had to focus someone else on something, then name it. We have all experienced the effort involved in cutting the thing away from everything else. ‘No, I don’t care about the other working pixels, it’s the stuck green one that’s the problem. Look at it.’ The ‘shifting sands’ come from the realisation that we can focus on anything we like and ask others to as well. When we do so, we reappear in what we are observing. The world of the card sharp beckons. A world Spencer-Brown knew well.

Deleuze consistently replaces intuition with radical experimentation. This is because he sees experience as an encounter with the new – even when we are not aware of this novelty. On that view, intuition is deeply unreliable, since it doesn’t even give a good account of its own state, as an approach dependent on ignoring difference, change and novelty.

I say radical experimentation, for Deleuze, because Spencer-Brown emphasises the importance of experimentation in getting at intuitions, but radical experimentation must tease out non-intuitive structures and systems, replacing the test of intuitive feel, built up with a sanitised version of experience, with the test of whether an event triggers interest in things unseen. The former is inward looking (Does the construction accord with what I feel and know?) whereas the latter is a shocking event (Where to, how, and to whom and what does this force us?)

Finally, Spencer-Brown adds two axioms. The first, the law of calling, is that distinguishing something and then distinguishing it again is the same as distinguishing it once. Indicating and naming <water> and then again <water> to someone is the same as indicating <water> to them: < > < > = < >. Nothing changes when the Queen twice says ‘I name this ship Brexit’. The ship is called ‘Brexit’ no matter how many times it is called it.

The second axiom, the law of crossing, is more difficult because it introduces a very puzzling and apparently paradoxical aspect to distinguishing. To distinguish something you cross over a boundary. You go from everything to something: from All and Nothing to <p>. If that’s the case, then there is a second axiom. To cross and cross ‘back’ is the same as doing nothing <<p>p>= . The empty space indicates nothing. When the Queen draws our attention from all other things to a ship, by indicating it and calling it ‘Brexit’, but then says ‘Oh no it’s not’, then no crossing to the ship (or naming) has occurred.

For convenience, I am using the symbol < > for distinguishing. Laws of Form uses the ‘cross’ (others employ a box). There is an amusing discussion of these typographical variations by the mathematician Louis H. Kauffman, a sometime box user and expert on virtual logic of the kind developed after Spencer-Brown’s work on reentry and self-reference. (‘Virtual Logic: Cookie and Parabel Discuss Laws of FormCybernetics and Human Knowing, Vol 24 (2017), nos 3-4, pp 261-68, pp 266-8)

There are subtle differences at stake here, notably in terms of ease of writing long equations, how boundaries are symbolised and problems in topology. Here are the two axioms written in terms of the cross or mark, in Spencer-Brown’s Calculus. The first leads to condensation (two successive marks condense to one) and the second to cancellation (a mark taken within another is cancelled):

Spencer-Brown also talks of condensation, cancellation and proofs as intuitive and based on familiar experiences. There are different kinds of reason for this. First, to justify the extraordinarily wide claims for Laws of Form, its basic laws should be generally shared. Second, his argument consists of proofs that construct complex forms, but then reduce to nothing or to a single mark by appealing to condensation and cancellation in simple operations. He does this with great elegance in his own proofs for theorems.

In the later notes of Laws of Form, this construction of arithmetic and algebra is extended to physics and the universe. These passages of uncommon beauty and quickness might seem mystical, but they shouldn’t be taken as such, if the mystical is associated with mystery and obscurity:

In this way the calculus itself can be realized as a direct recollection. As we left the central state of the form, proceeding outwards and imagewise towards the peripheral condition of existence, we saw how the laws of calling and crossing, which set the stage of our journey through representative space, became fixed stars in the familiar play of time. Our projected hopes and fears of their ultimate atonement, which we called theorems, became their supporting cast. In the end, as we reenter the form, they are all justified and expended. They were needed only as long as they were doubted. When they cannot be doubted, they can be discarded.

Laws of Form, p 102

The reference to recollection suggests Plato’s theory of reminiscence and cycles of birth and death as solution to the problem of knowledge. There is a circular shape to Laws of Form itself. With these final words, Spencer-Brown is closing his work back on itself as ‘justified and expended’ – in a self-consciously different way to the Tractatus. His intention has been to construct from the mark and its two axioms to the universe, but then back again to the mark and to nothing by way of theorems.

These last passages – and other sections of Laws of Form – are responses to Wittgenstein’s Tractatus: ‘What we cannot speak about we must pass over in silence.’ (Ludwig Wittgenstein, Tractatus Logico-Philosophicus, London, Routledge & Keegan Paul, 1961, p 151) The laws of calling and crossing lead to instructions allowing all things to be spoken of, though never completely.

Wittgenstein’s approach is descriptive: ‘2.0201Every statement about complexes can be resolved into a statement about their constituents and into the propositions that describe the complexes completely.’ (Tractatus, p 11) Spencer-Brown’s instructions aren’t descriptions, they don’t show. They are instructions to do something. They tell us to do, to construct, rather than describe: ‘… the primary form of mathematical communication is not description, but injunction.’ (Laws of Form, p 77)

There is a terrific article by Gregory Landini on Wittgenstein and Spencer-Brown that disagrees with everything I have just said about the two of them. Partly I think this is because the emphasis is on logic rather than metaphysics, it is also because the starting point is Wittgenstein and Landini’s reconstruction of his N-operator notation, and it is because Landini equates Wittgenstein’s showing with Spencer-Brown’s instructions:

The Primary Algebra of Spencer Brown’s book Laws of Form has been thought to be an algebra for logic based on the Sheffer stroke. So also had Wittgenstein’s N-operator notation. This paper argues that this is mistaken and that the two systems of calculation are simply notational variants of one another.

Gregory Landini ‘On the Curious Calculi of Wittgenstein and Spencer Brown ‘ Journal for the History of Analytical Philosophy Volume 6, Number 10, pp 2-28, p 3

Landini’s article is invaluable as a logical study, a tracing of the historical documents around Spencer-Brown and Russell, and a discussion of Spencer-Brown and his brother (including taking David’s name in correspondence with Russell). It also makes important points about tautology and Spencer-Brown’s logic and arithmetic. It could well be that all I say here about metaphysics is redundant, if we can limit ourselves and Spencer-Brown to logic.

Turning now to Deleuze’s metaphysics, my version can be understood as a substitution of terms and axioms from Spencer-Brown for ones taken from Deleuze’s work and his Difference and Repetition, in particular. For the closure of distinction, there is the open-ended and manifold definition of multiplicity, an uncountable and continuous multiple.

In the role of indication there is drawing – as in drawing a diagram or picture, or drawing up a description. On this view, even saying ‘This P’ is to draw, using letters or sounds and gestures to draw P over all other things. Spencer-Brown’s mark can be seen as drawing an enclosure, but it does not have a located textured material and virtual support – like a canvas, frame, culture, body, environment and unconscious. Drawing, in the Deleuzian sense, is always on a support that is already textured and marked in multiple ways that are singular to that event of drawing. He called this a diagram, in his work on Francis Bacon.

To support multiplicity as open and many, it is defined as perfect incontinence. This means no multiplicity is closed in any part or place. We saw this incontinence in Deleuze’s version of the principle of sufficient reason, where reason was an endless ungrounding, or open series of differences, resistant to representation, yet underlying any identity.

This lands us with a similar problem to the one set by Spencer-Brown for reliable indication. How do we pick out a multiplicity? The answer is by drawing over it and letting something appear against a background or context, but without cutting away – cleaving – from that context.

Instead of a cross, a symbol for this drawing and its operation is a double opening: >p<. This is to draw p over everything. The ‘over’ is important, since it might be argued that Spencer-Brown’s mark distinguishes something ‘with’ everything and nothing, in the sense of involving them in the distinction. ‘Over’ is much stronger than ‘with’ in terms of the mutual relations and transformations involved. She drew over his face. She distinguished his face from all others.

The first axiom is then not about something being cancelled in repetition, but rather transformed, because each drawing is always a different intervention on the background: >p< >p< = >p'<. For Deleuze, you can never repeat the same, but only repeat through a variation. When the Queen names a ship ‘Brexit’ twice, because the champagne bottle failed to smash the first time, then the good ship Brexit is cursed. Everything has changed between the two namings, despite the name and ship appearing to be the same. Even if superstition about failed christenings is unwarranted, the good ship ‘Brexit’ will carry a faint whiff of difference and perhaps doom with her.

As we found in Spencer-Brown (and setting off many deep philosophical problems) the second axiom is much more difficult. I suspect this is because in both cases the second axiom is not a simple complement to the primary operation, but rather an inventive extension of it allowing for rigorous and powerful constructions based on both axioms.

According to my reading of Deleuze, the second axiom is that a difference cannot be undone, or, in the language of Spencer-Brown, there is no cancellation whereby two crossings cancel one another. So drawing and erasing a drawing is still to have invoked every difference: >>p<< = >p<. There will always be some effective sign – some vestige – of the crossing back or redrawing in the environment.

This leads to two further critical differences between the two philosophies and symbolic systems. For Deleuze, there is no nothing. Moreover, there is also no abstract drawing that isn’t a drawing of something. This means every symbolic operation must involve a variable, so > < and < > and = are meaningless, except as indicators of an account of what nothing might mean, such as vertigo, annihilation, chaos or loss. The existential ramifications of these differences are therefore vast.

>p< >p< = >p'<

>>p<< = >p<

For Spencer-Brown, to name something is to attach a token, a name, to something that has been marked: <Spencer-Brown>. The name is the name of a marked thing: Spencer-Brown = <p>. Accordingly, the name is arbitrary; up to a point, since you cannot use names already in use for something different. All the name needs to do is function as a token. Spencer-Brown’s frequent name changes could then be taken as a way of demonstrating his theory: from George Brown, to George Spencer Brown, to George Spencer-Brown, to David Spencer-Brown, to Maxwell Spencer-Brown.

It is very important to note the care he takes to make the name equal to a mark of indication. If we think of naming, not as calling but as crossing, in the sense of passing from everything to something, then this is in no way arbitrary, since it is to execute the command to change from one state to another. In this case, you cannot continue to use the former name:

… the call may be transmitted in both directions, as with the sign =, so that by calling so-and-so such-and-such we may also call such-and-such so-and-so. Naming may thus be considered to be without direction, or alternatively, pan-directional. By contrast, instruction is directional, in that it demands a crossing from a state or condition, with its own name, to a different state or condition, with another name, such that the name of the former may not be called as a name of the latter.

Laws of Form, p 80

To cross from George to his Brother David and continue to call David ‘George’ is to make a basic error. By indication, David can’t be George. Amusingly, that’s exactly what George did for a while, taking his brother’s name, and leading to deep confusion for the Telegraph obituary writer, who thought the brother had been invented then uninvented by George, until David’s actual existence was pointed out: ‘We have been told by someone who knew George Spencer-Brown that his brother existed and was not ‘invented’ as suggested. We are happy to make this clear.’

For Deleuze’s metaphysics naming is quite different. Instead of being a reliable token, each name must be a drawing that is by the first axiom unrepeatable. This means we never have the same name, each name is singular, because there is no such thing as a free-floating token. Each name comes with baggage and when it is given it acquires a background that inflects it.

In the Deleuzian version, there is no problem with two indicated states having the same token, since by the second axiom two naming events can never be equated. No contradiction can occur. Deleuze took Nietzsche’s claim to be all the names in history seriously, right to the core of his metaphysics, because all names and states are interconnected each time a name is drawn up.

This play of problems and difficulties between the two metaphysics is where the contrast between Spencer-Brown and Deleuze is most fruitful, because the commitments and implications of their philosophies can be teased out.

I’ll begin the contrast with model analogies. Spencer-Brown constructs following the model of a board that can be written on and then wiped clean. There’s a blank space. A mark is drawn on it, then a name in the mark. Then another mark is added around the first mark, and another, and more names, until a system grows from the mark, with the potential to encompass arithmetic, algebra and the universe. Yet after many compensations and cancellations all this can be reduced to the blank board again – ready to start anew, like a computer after a long calculation.

For Deleuze, the model is of a palimpsest. In your prison cell all you have is a knife to draw over all the marks made by former inmates. Your mark changes theirs and theirs foreground yours. No mark disappears fully. It becomes part of an underlying pattern on which every new mark must be made. Your name must join the names of all who came before and be there for all who come after. It never stands alone. There is no blank space on the walls of your cell and there never will be. If you wanted to follow Spencer-Brown’s instructions, you could only do it by abstracting from your situation.

These analogies raise the most significant challenge when considering Spencer-Brown and Deleuze together. A wipeable board can become a palimpsest, but a palimpsest cannot become a blank board. Spencer-Brown’s construction can arrive at complex virtual states that seem very close to Deleuzian complexity, but Deleuze’s model cannot replicate the ground up construction and, hence, autopoietic and autonomous construction offered by Spencer-Brown. More seriously, perhaps, it cannot replicate the testing of logical tautologies.

There have been and no doubt will be more arguments that equate the two thinkers because of the similarities around stages of their philosophies. This has taken place often through work that comes after Spencer-Brown, for instance, in connections made between Luhmann and post-structural theory. For example, Jean-Sébastien Guy argues that Deleuze’s ideas support Luhmann’s analysis of society as functionally differentiated. (J-S Guy, ‘Problems and Differentiation: A Deleuze-Luhmann Encounter’ Cybernetics & Human Knowing, Volume 26, Number 1, 2019, pp. 29-45(17) I hope to show that the notion of support is questionable, because parallels around ideas of difference mask deep divergences in the arguments for those ideas.

Some of these comparisons aren’t claiming simple equivalences. Urs Stäheli brings our attention to the wider scope of Derrida’s deconstruction of meaning. His critical deconstruction would apply to the founding stages of Spencer-Brown’s mark (to his account of intention, for example): ‘If there is a crucial distinction between deconstruction and systems theory, then it is located where the Luhmannian notion of surplus and the Derridean notion of excess meet: surplus connections are all located within the order of meaning, whereas the Derridean excess questions this order.’ Urs Stäheli « The Hegemony of Meaning: Is There an Exit to Meaning in Niklas Luhmann’s Systems Theory? », Revue internationale de philosophie, vol. 259, no. 1, 2012, pp. 105-122, p 32.

Against these trends towards thinking of Spencer-Brown and Deleuze as close, I will turn to a number of areas where the bare versions of their metaphysics are at odds. These can be viewed as traditional existential terms, but it is more productive to think of the possibility of a new, pragmatic and very wide idea of existence, not at all limited to, or based on human lives. The areas are: identity, difference, continuity, nothingness, freedom, time, memory and construction.

Identity is primordial for Spencer-Brown. In the beginning, there must be an actor who identifies something by severing it away from everything else and from nothing. Later, this identity can become lost in imaginary values, but only on condition of the initial operation of identification <p>. For Deleuze, there are no identities, only varying patterns of relative differences. There is no beginning, only a middle. Whenever a claim is made to an identity, it is only against primary multiplicities.

Difference is primordial for Deleuze. Never difference between, or difference from, but rather differences that always elude representation and identity: an uncountable and indivisible many. Both Spencer-Brown’s initial operations, calling and crossing, depend on difference given as distinction, a border must be given first, even if later, through the reentry of forms, differences multiply and become complex.

Continuity is constructed by Spencer-Brown. It is a consequence of reentry into the form, of self-reference, whereby that which was initially distinguished finds itself on both sides of the boundary. This reentry is written into the form, so continuity is necessary but it comes after the instruction to sever: first ‘this’ not ‘the other’, only thereafter, ‘the other’ in ‘this’, and ‘this’ in ‘the other’. There is no ‘other’ for Deleuze, only continuous modulations of divergences. These do not depend on construction, but rather, every construction is drawn upon and transforms continuous multiplicities.

Nothingness has no role to play in Deleuze’s metaphysics, since continuity and multiplicity are always at work. There is no operation, no order and no boundary that could impose nothingness. For sure, identities must disappear, but that’s because they are already gone. Nothingness is essential for Spencer-Brown, not only as the result of cancellation and hence as one of the two conclusions to any calculation, but also as the condition for starting off any process of construction. It will have to wait for later work, but I suspect this means Deleuze and Spencer-Brown will have very different accounts of death, with the former’s two versions of death (death of illusory identities and many ongoing micro-deaths and rebirths) while the latter alternates death as nothingness followed by circular rebirth (But will that involve memory?)

Freedom for Spencer-Brown is in the freedom to observe and distinguish. It is also in the freedom of ambiguous complexity. However, any free construction follows from following instructions. It’s the freedom afforded by the choices outside control systems and on their margins, not within them. For Deleuze, freedom is everywhere, but since there is no nothingness and no identity, it is a freedom always determined in part by an environment and by new events, a freedom to transform, not initiate. For Deleuze and Guattari, order words, or instructions, are designed to hide real freedom.

Time is real for Deleuze. It is a series of multiple dimensions determined by different kinds of repetitions across processes that constitute past, present and future, in relation to each other and themselves. Put more simply, time is made by repetitions in many different ways, but these repetitions are changes, such as the time implied by a new divergence in an historical series, or the time implied by a long series of habitual gestures. For Spencer-Brown time is more simple but also artificial, since it is constructed by retaining successive imaginary crossings in memory. This time has different speeds, but otherwise it is thin on qualities and has nothing of the intensity or drama of real time.

Like time, memory is defined thinly by Spencer-Brown. It is the retention of earlier acts in later ones, earlier callings and crossings in later ones. This simplicity leads to great complexity, since it allows for reentry and self-reference, but it remains thin as a definition of memory when compared to Deleuze, because it is not operative on the past and because memory is active, a decision to retain, with its own symbol bringing crosses together by hooking under them: ‘It now becomes necessary not only to indicate where a re-insertion takes place, but also to designate the part of the expression re-inserted.’ (Laws of Form, p 65)

Against this choice to re-insert and the re-insertion of an earlier state ‘as was’, Deleuze defines memory as dynamic and passive. This means that the past changes dynamically as memory. It also means that the present and future are passive to the work of memory. The past works on them independent of actions and within an action. The past is a record retained through memory for Spencer-Brown. For Deleuze, the past is an actor working on the present, tossing it around and shaping it in new ways.

There should be no unconscious memory in Spencer-Brown, nothing retained unless chosen, nothing retained after cancellation and condensation. Yet, his writing and life are full of examples of this not being the case, of moments when only an unconscious memory can explain retention: ‘… proofs with the purpose of again rendering, to ourselves, irrefutable evidence of what we already knew.’ (Laws of Form, p 106)

Construction is an art for Deleuze. It is located, passionate, complex, innovating, connected to all things, haunted by an unconscious, collective and individual, political and responsible, doomed and destined to renew. For Spencer-Brown, construction is to follow instructions and thereby to reconstruct and validate the universe. This latter has all the power and potential of calculating machines and systems on its side, including living systems and hybrid ones. At its best, the former might have this too, but only by drawing it over real lives and submitting it to their art.