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:
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:
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:
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.