Spencer-Brown is wrong: The distinction is not ‘perfect continence’!

I’m troubled by a recent realization of a fundamental error that my until-now near-idolized inspiration, George Spencer-Brown, has made ab initio and which has unravelled my deepest convictions about the soundness of mathematics…

He begins with the premise:

We take as given the idea of distinction and the idea of indication, and that one cannot make an indication without drawing a distinction.

Ok, so far, I totally agree. He goes on:

We take, therefore, the form of distinction for the form.

Again, I can accept that.

The problem is in his definition of ‘the distinction’: ‘The Distinction is perfect continence.‘ (continence: archaic sense ‘that which contains’ – not ‘sexual abstinence’)

Yes, while a distinction is said to ‘cleave’ the boundless space in two, thereby ‘distinguishing’ two spaces – one could argue that the distinction ‘perfectly encloses’… but the problems arise immediately in the rest of his text:

He presumes that one space is ‘inside’ another! The first distinction cannot permit such a claim. At best, at this point, all that can be said is that both spaces are ‘beside’ one another:

One boundary separates two spaces
One boundary separates two adjacent spaces

Whichever of them you ‘choose’ to ‘indicate’ is completely arbitrary. A is ‘Not-B’, and B is ‘Not-A’. That’s all we ‘know’ at this point. Ok, so admit we can’t ‘say’ that A is ‘inside’ B or vice-versa at this point… what if we were to have a second distinction? Now can we say one is ‘inside’ the other? No! Look:

Two boundaries separate three spaces
Three spaces are still nevertheless only ‘adjacent’

I’ve drawn two variants because at first I thought the error was to ‘require’ that no boundaries intersect (it’s a curious convention among GSB fans for some reason – like an ‘unspoken rule’) – but one quickly realizes that it’s meaningless: When we’re considering ‘spaces’, then each space, being ‘distinct’ is _defined_ as being ‘not’ either of the other spaces (to lean on Leibniz’s PII: if they weren’t different then they wouldn’t exist – they’d merely be one of the other two spaces). This means that they are necessarily adjacent because they must all be simultaneously distinct from one another. So – no matter how many ‘boundaries’ you care to draw – at the ‘basest’ most fundamental level, any configuration of boundaries will always be a single, massively-interconnected boundary: the “not”. This means that every single distinct existent (used as a noun here – a ‘thing’) is and will only ever be ‘adjacent’ to every single other existent.

So now, I’ve found myself incapable of building any kind of orientable geometry from this.

Yes, I’ve gone up to six distinct spaces, such that each is adjacent to the 5 others. This in attempting to draw (pen-and-paper) it, leads to a tetrahedral ‘shape’, where each space shares its boundary with all others:

Six 'adjacent' spaces
Six ‘adjacent’ spaces

Here is where GSB’s ‘a space becomes a boundary’ concept (law of calling) seems to arise where to cross from ‘space’ E to F, one has to go ‘through’ one of A-D… but given the arbitrary nature of these spaces, ‘E’ and ‘F’ could just as well be completely turned ‘inside-out’! Don’t be fooled by the seeming ‘rigidity’ of the straight lines here: this is still a ‘flat’ 1D arrangement of ‘spaces’ separated by the 0D ‘reflection’ transformation – the ‘not’ – or ‘boundary’…

And here’s where it begins to get deeply troubling for me: if by epistemological ‘bare necessity’, any one ‘thing’ must at least be ‘defined’ (bounded by) as being ‘not’ everything else, then:

a) I can’t figure out how we can claim the existence of or describe ‘one thing’, ‘twice’. Multiplication seems a human conceit…

b) I can’t see at what point anything can ever be considered ‘inside’ another…

c) From b above, I can’t build any kind of ‘geometry’… I can’t even get to ‘translations’ as a transformation – since I’m stuck at ‘reflections’, which are dimensionless by nature…

This can be iterated over as many ‘distinctions’ as you please – I still can’t ‘get to’ translations as the ‘next level’ of transformation…

Now: GSB’s work was intended as a system of reasoning – so my metaphysics is possibly a bit ‘overkill’, but still, we are talking absolutes and fundamentals here… so I don’t see why I shouldn’t attempt to understand the ‘cleaving of space’ out there in the real, metaphysical ‘nature’ of reality…

Help! I’m stuck in a dimensionless (ok, 1D because ‘adjacent’) space!

This does all hell to my sense of ‘the number line’ – since all numbers are ‘not’ every other, one might be able to develop a ‘line’ of ‘points’ (infinitesimally-small spaces) where each is ‘not’ its neighbour, and you might be tempted to use that as a metric to begin to build a kind of geometry – but you’d be ignoring the fact that the point ‘one-over’ from the point next to you is also your neighbour! So the number-line is shot. So ‘coordinate’ geometry is shot. So Euclidean and non-Euclidean geometries are shot.

So what the hell am I describing then? Because each space is by definition ‘distinct’, at first I thought I was building a description of ‘gauge symmetries’ or whatever physics uses when they talk of ‘degrees of freedom’… but I had nearly forgotten that this is a universal truth – these are absolutes! So every ‘space’ here is also every instant of past existence too. If you ‘merge’ two spaces (by removing the boundary that separates them) then you merely go back to a state of one-fewer-spaces… So I don’t know at what point ‘inside-ness’ arises from this if it ever does..

If this boundary were a recursive involutory function (which ‘not’ is – c.f. the Liar’s Paradox), continually inverting itself (where each ‘iteration’ merely wraps a new boundary all previous spaces) this could be a fine definition of the successor function… Interestingly, this applies to all sorts of things – except that it seems so ‘useless’ too because I can’t get anywhere with it! I’m just getting a more and more complex object composed of mutually-extant (distinct) spaces.

6 thoughts on “Spencer-Brown is wrong: The distinction is not ‘perfect continence’!

  1. I like the way try to model reality. The thing with me is that people confuse the model with actual reality when it’s a symbolic model. We try to make sense of a virtual reality using only the rules of that virtual reality, and leave out the bigger picture of consciousness. Consciousness created this “physical world” in order to evolve itself by splitting itself into individuated pieces of consciousness in a dense physical world. Time is a tool used by humans to make sense of the changes occurring, but it’s a necessary tool because without change their could be no evolving. But math and science is poorly equipped to understand the reality that lies beyond our physical matter reality. We are just not quite able to know what seems to be unknowable. But even the Buddha 5,000 years ago said that life is an illusion (something Einstein was quoted as saying also). Science refuses to look at the measurement problem or the hard problem. Dr Rupert Sheldrake explains this in his 10 dogmas of science. Keep seeking knowledge Tom W. There’s nothing more important than knowing the truth. Respect! (think of geometry as fractals and it can model anything)!

  2. You wrote about building a geometry or building a physics.. I was wondering if you could explain to me as if I was quite simple what it is you’re trying to do? I gather that it has something to do with Taoism, and using the idea of transformation as part of a theory of fundamental aspects of the world, but not much more than that. I’d like to understand your position. (I am a philosophy undergraduate who flunked high school math, by the way. I studied Propositional Logic but still I hardly know any math.)

    1. Hi again Eilish! (Azalea? confused) And thanks for writing 🙂

      (I’m not a philosophy anything, and I flunked first year engineering school… I’ll do my best 😉 )
      You’re pretty-much spot-on… but the ‘taoism’ aspect is really quite minimal. It’s just that the more I thought about these things, the more I keep happening across ideas that were already expressed in the tao te ching (from what little of it I could grasp). I also really like the symbolic quality of the tai chi symbol, and its aesthetic.

      So at the risk of merely repeating what you already said:

      My work really all starts from the idea “The only constant is change”. It just struck me as soooo true and I ran with it. Then in my first year of engineering school (where we were all re-learning linear algebra and analytical geometry, calculus, etc.) I heard two ideas that blew my mind: 1) vectors exist independently of the basis in which they’re described, and 2) the way my physics teacher explained the boundary between glass and air in optical geometry (the word in French is “Dioptre” and is misleading because it is also used in the English sense in optics). The second one really opened my eyes to an ‘interpretation’ of reality as ‘it’s all just boundaries between things’ – so when I began writing about these ideas, and posting to philosophy forums (in particular this one), a few people who read my ideas pointed me to George Spencer-Brown’s “Laws of Form” (among many – there was also Henri Bergson, Gilles Deleuze, Hegel, and others whom I still haven’t read – I know, ‘for shame!’). GSB seemed to really ‘get it’ – in that ‘it’s all just boundaries’ sense, but it seemed to me that he hadn’t yet gotten the ‘it’s all just transformations’ sense… so my work is far from done. I’m trying my best, despite my ignorance and self-taught amateurism, to be as rigorous and methodical as possible – all while not letting myself be trapped in ‘old thinking’ which has led to so much confusion in QM and other physics (‘Dark’ Energy/Matter? Puh-lease…). All this in the hopes of maybe getting to a way of looking at things which clears our heads and sheds new light on what we call reality… Humble, but ambitious!

  3. On Louis Kauffman’s own blog, I commented concerning the above post. After a little back-and-forth, here’s his reply:
    Dear Tom,
    I have looked at your site.
    You are engaged in producing some very nice structures that arise when one considers the boundaries of distinctions. This reminds me of the algebraic topology where we consider (for example) simpliicial complexes where each basic n-simplex has a boundary that is a union of (n-1) simplices. Then there is a hierarchy of distinction making that occurs in both the adjacenies in a given dimension and the ascent and descent among dimensions. The key formula d^2 = 0 (the boundary of a boundary is zero (empty)) governs the structure and has a role similar but not the same as GSB’s Law of Crossing. I could go on here and talk about many structures that involve adjacency and boundary. Once one includes the boundary even in a simple distinction such as a circle in the plane, the distinction becomes a multiple entity – there is the inside, the boundary and the outside. Thus one is not, in articulating the boundary in this way, talking about the simplest possible idea of a distinction. The simplest idea of a distinction is a two-valued one. Marked or unmarked. Even or odd. Inside or outside. I am sure that we can both agree that (at least in the context of somewhat more complex situations) simple distinctions do occur and that we idealize them in mathematics. Thus when I let P be the set of all prime natural numbers (with the convention that 1 is not prime), then every natural number is either prime or it is not prime. This is a simple mathematical distinction.

    Even with a simple mathematical distinction such as the primes, one may argue and discuss boundary issues. For example, I can ask if there are infinitely many Fibonacci numbers that are prime numbers. We do not know the answer to this question and so the relationship of P and the set of Fibonacci numbers is problematical. Nevertheless, it is a finite computation to determine if a given number N is prime and so (theoretically) we regard the matter of primality as a simple distinction.

    If I want to regard disks in the plane as giving simple distinctions then I can either ignore the boundary of a given disk, or decide to include it on the inside of the distinction. This is what we do when we think of the disk as a closed set in planar topology. The the complement of the disk is an open set and it does not contain its boundary. For any given closed disk in the plane, every point in the plane is either inside the disk (this includes the boundary) or outside of the disk.

    Spencer-Brown takes the idea of a distinction to be a zero-one notion. One stands either inside or outside the distinction at the beginning of this idea. The idea is not something that is right or wrong. It is an idea, and one can follow it along to see what happens with it. Thus in set theoretic mathematics we start with exactly this idea and we say that a given set S either does or does not have another set T as a member. Aha! You say. But this is known to lead to paradox such as the Russell paradox of the set R of all sets that are not members of themselves. Is R a member of R? The apparent perfect continence of R has been controverted by R itself!!
    How do we take this situation? These days, people are rather sanguine about it.
    They make rules to that can keep culprits like the Russell Set out of their theories so that it would seem that all the sets in standard ZF set theory have “perfect continence”. But it takes some study to learn how to talk the set theoretic language that keeps you from paradox.

    Spencer-Brown suggests in his Chapter 11 another way out of the Russell paradox.
    He says that the solution is in a temporal dimension. If R is all the sets we know right now (time t) that are not members of themselves, then we find that once R is created, then R is not a member of itself, and so at time t+1 there must be a new R’
    that has the old R as a member. But now the new R’ does not belong to itself and so we must add it in and this process goes on forever, time after time. Language can be performative. An act of language can create an on-going process and our perfect continences will appear to leak.

    Let me repeat. We start with the ideal of perfect continence and then we find out how far we can go with it. We always find that it starts to fail and then we construct new mathematics to keep it present. It is not wrong to ask for perfect continence.
    But you have to understand that mathematics is the deliberate creation of ideal worlds, conceptual worlds, where such simple distinctions can live Mathematics is conceptual fiction aiming to create logical worlds. Such worlds are like the tangent lines to complex curves. They always shoot off from ‘reality’ and the way they do tells us much about the ideal worlds and about the ‘realities’ from which they have sprung.

    This is all interesting stuff and it warrants a new post, which I’m working on. I will link the relevant discussion points there

  4. IANAE (I am not an expert) by any means. I feel GSB’s notes for Chapter 14 clarifies things pretty well. In your first diagram you assume A is contained in B by the boundary. But if GSB’s complete treatise is followed, I believe it is impossible to say whether A contains B or B contains A. The only thing that can be said about them is that the circle distinguishes one from the other.

    1. Hi Tom,

      First, thanks for taking the time to comment!
      I would like to correct you however – in my first diagram, and to be clear, this is my whole point – I do not assume any ‘continence’ whatsoever. The phrase before the diagram states: “The first distinction cannot permit such a claim. At best, at this point, all that can be said is that both spaces are ‘beside’ one another”. So in effect we both agree: like you said – “It is impossible to say whether A contains B or B contains A”.
      But the impact of the title of this post – where I feign the hubris of being able to say that GSB is ‘wrong’ – is that in the very first pages he ‘defines’ the distinction as being ‘perfect continence’, and that I disagree with for the very reason we’ve agreed upon.
      I have advanced in my thinking since I first wrote this post – however I have remained steadfast in my conviction that ‘continence’ is not the right word – and that saying which is ‘inside’ and which isn’t – is GSB’s arbitrary contrivance.
      I encourage you wholeheartedly to continue to work-out what problems that poses us with however, because we are really stuck… 🙂
      Best regards,
      Tom W.

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