Syndiffeonesis, Again.
Stream of thought this week, apologies about the scatter brained-ness. All references in here are from MP 6: An Introduction To Mathematical Metaphysics unless otherwise stated.
Syndiffeonisis is weird. On the surface, I get it. Any relation between two things requires they exist in the same ontological medium. Differing is a relation, hence differences imply sameness - at least in a way. Dig a little deeper though and it starts to get pretty trippy.
So what do we have? Syndiffeonesis has two “parts” or “phases” or… what would you even call the attributes of syndiffeonesis? Anyway. These are the synetic and diffeonic phases, with the synetic distributing over the diffeonic phase. The synetic phase is the syntax that distributes over the diffeonic phase and is stratified according to the order of the relationship or attribute. Cool. The diffeonic phase being a “differentiative function of arity or cardinality“. Let’s look at that for a moment.
The diffeonic phase is a “differentiative function of arity or cardinality“. Also: “the diffeonic level consisting of some number of discernable instances thereof (this being the arity of the relationship)”. Good to know. Let’s remember that and come back when we can confuse ourselves more thoroughly.
Langan gives a simple example for how the generic syndiffeonic relationship can be written":
s(d1, d2, d3…)
He writes it this way intentionally referencing notation for certain distributive operators:
n(a1 + a2 + a3 + …) = na1 + na2 + n*a3 + …
This makes me think that s = the relationship = the synetic phase, which is equivalent to shared syntax (I have questions but we’ll get there), and that the number of d’s between the parentheses is the arity of the syndiffeonic relationship.
But wait. What if we’re talking about the ‘>’ relationship, just for kicks and giggles? 3 > 2. Great! We could also write this >(3, 2) if we were computer scientists, or Polish. Fun fun. Looking at >(3, 2) I don’t exactly see how ‘>’ distributes over both ‘3’ and ‘2’. Certainly some properties of both 3 and 2 are implied by knowing they can be relands of >, but that’s not the same thing as saying > distributes over them, is it? Especially since position matters. following our s(d1, d2) example >(3, 2) should unfold into >3, >2. But this feels way off. If, however, we’re taking ‘>’ to be the synetic component and {(x, y) : >(x, y)} to be the diffeonic component then things make a bit more sense - obviously > distributes over that set. (The issue with needing every relationship, including >, to distribute over its relands comes from the folowing: “Other ways of saying this are that every intelligible relationship is a syndiffeonic relationship,…”. Hence the need to make > distribute over 3 and 2)
But then why did he say “the diffeonic level consisting of some number of discernable instances thereof (this being the arity of the relationship)”? That would seem to imply the diffeonic level should only have 2 elements since > has arity 2, while our set has countably infinite members (granted, each a tuple of 2 elements).
I’m going to assume that what he means is the set of all of them since that accounts for the most explanation with the least information. Then why did he say “… (this being the arity of the relationship)“? I’m going to chalk this up to a typo or a some explanatory error on his part. Typos and oddities do exist in the MP, they don’t have the benefit of peer revew. Look for example at the second page of The Resolution of Newcombs Paradox where he writes “…$1,001 million or $0.001 million…“; that ‘,’ should be a ‘.’. check the paper if you don’t believe me. (I realize I have made many, many more typos, lol. But I am not writing a technical paper right now). Another example from the very paper we’re talking about today is:
“…whenever, in the course of perception and/or cognition, one simultaneously recognizes any number of things as different from each other (discernible), one is necessarily bringing to bear a cognitive-perceptual syntax in order to accept both of them as cognitive or perceptual input,…“
I think the “both” in there should be an “all”, since it was “any number of things”. Things like this don’t help his case, but I’m inclined to overlook them since again, he’s not a professional academic and has real life to attend to. On the other hand, when he’s so far ahead of his readership it matters a lot: for all I know that’s exactly what he means and I’m just an idiot. Errors, or slight carelessness, or a lack of pedagogy on his part makes it harder for me to discern when I’m on the right path, and when I’m a dumbass. (it’s usually the dumbass one, but not always).
Anyway, going with the set interpretation of diffeonic component. This also kind of does allow us to look at things like:
>(3, 2)
In this case > is a “lower” or “more specific” version of > that only applies to the specific elements 3 and 2. In that sense our “lower” > does distribute over 3 and 2, it just wouldn’t distribute over 5 and 4, you’d need a different lower >. This brings us to stratification. The synetic phase is stratified according to the order of the relationship or attribute. In the case of our having several “lower” >, the regular > could be seen as an inclusion or identification relation on all valid lower >. We could then say perhaps there is an inclusion or identification relation IsMath, which distributes over >, <, =. as well as N, and in particular: N**2 - of which our diffeonic set for the > relation is a subset. Higher order logics talk about this kind of thing. Second order logic adds predicate variables to the first order logic and allows one to talk about expressions of the first order. Third order logic allows us to talk about relationships of the 2nd order, and so on until one gets the theory of types.
This also suggests that what is synetic at one level may be diffeonic at another.
But wait, there’s more. More? You ask. More. I say. Syndiffeonesis is self-dual. Langan give three senses in which it is self-dual:
The synetic ordinal dimension is dual to the diffeonic arity dimension.
It has dual active and passive interpretations.
It relates cognition and perception.
Since the formal aspect right now seems most important to our understanding let’s focus on that. The synetic and diffeonic axes are dual to each other. Cool. What does that mean? Exactly? Also what is duality? Did he make this up? No. Duality can be found in the CTMU itself in section 4.6.8. One example he gives there is “points are functions of lines” <-> “lines are functions of points”. A good example external to the writings of Langan comes from S. C. Kleene in his “Mathematical Logic” (section 6), here is an excerpt:
“Suppose a visitor from Mars is confused by what he observes upon his arrival on Earch, and mistakes our true “t” for false “F” , and our false “f” for true “T”; i.e. let F = t and T = f. Then our table for AND would read for him as our toble of OR for us, and vice versa….”
Trust me, it gets way more technical and in depth than the cute little setup but that excerpt illustrates the point. My own summary is this: Duality is when two theoretical systems can be transformed into one another with a simple, invertible replacement, like replacing true with false and vice versa, or points with lines.
Neat! Duality is a thing! So wait: The synetic and diffeonic axes are dual to one another? How would this even work? What invertible swap could you find? Well, my simplistic first attempt went like this:
Set up ordinary syndiffeonic relation. >(a, b) for example.
Switch positions: (a, b) now distributes over > and IsMath (and anything in between but whatever), giving us (a, b)(>, IsMath).
But this doesn’t account for order. Hmmmm…
>(a, b) -> b(a(>)). This gives us two levels and accounts for the “cardinality”? of the dimension. a is the first order synetic, b is the second order synetic, > is the singular item in the arity axis.
But wait: b doesn’t really distribute over a unless you already have > in mind. I’m sure you could force it somehow but it feels unnatural. There might be some duality there, but I am nowhere near the mathematician required to find it - if it exists.
So back to (a, b)(>, IsMath). This properly swaps the “ordinal direction” or “synesis chain” or whatever into the diffeonic chain but it seems to miss something in swapping the diffeonic axis into the synetic chain…
Also note: we should probably be writing this: {(a,b): >}(>, IsMath) to acknowledge we are using the full set, rather than just an individual instance. But we weren’t doing that when we were going b(a(>)). Hrmmm… Let’s keep the set for now and just write {a, b} as an abbreviation for that singular object.
Back to 6, to get a better swap we would go {a, b}(>) for the basic >(a, b) relation, and perhaps {a, b}(>(IsMath)) when we want to go all the way up. Hey, that actually does kinda work out. Or at least appears to.
But then… IsMath(>, <, =, N, N**2) [0] … -> IsMath(>, <, =, N, N**2, >(a, b)) [1] -> IsMath(>(a,b), <, =, N, N**2) [2] ??. I am inclined to say that [1] is an error, since it doubly includes >, and that [2] is a more slightly explicit version of [0]. This chain of nonsense makes the term “syntactic covering” more sensible. IsMath is a syntactic covering of everything in the parentheses, > is a syntactic covering for that set {(a,b)} to which it applies.
This also lets us swap the synetic and diffeonic axes under certain restrictions… But not, while IsMath covers everything, {(a, b)} for > and some {(c,d)} for < will not cover eachother if the axes are dualized at their locals, at least not how I’m doing it, so…?
But wait, there’s more! More?! We already had more! You’re telling me there’s more more?! Yes!! There’s more more!! Here’s a neat little line:
“Syndiffeonesis breaks down into sameness or synesis, which is stratified by order of relation or attribute,…”
Did you see it? Did you spot it? Here, let’s zoom in:
“which is stratified by order of relation or attribute,”
Let’s zoom in a bit farther:
“relation or attribute,”
Oh… Oh no… If the rest of this wasn’t tricky enough we get to this, relationship or attribute. As I’ve read, in logic and model theory you have predicates on the language side, and relations on the model or universe side. This adds another layer of duality. If before we had a “vertical” duality, this appears to be a “horizontal” duality. This notion should not surprise us, since the CTMU is all about the relationship between theory and universe, but it does add an extra layer of “we don’t understand this thing we thought was simple” to the mix. On top of this, he seems to equate the horizontal and vertical splits here in the next section when talking about intension and extension couplings. That’s for the next section.
So, after all that: What is syndiffeonesis?
Yeah, I don’t know either. Best I can do right now is this: Syndiffeonesis is a meta-syntactic property of relationships or a meta-relationship that applies to all relationships, including itself. Oh shoot! How would we even do that? Well, let’s try it:
Let U be a unary relationship for “Is a syndiffeonic relationship”
Clearly U(>(a, b)) (where (a,b) just represents the domain of >)
Also clearly U((U))
Oh, but we need to include the domain
So: U(U(all relationships, somehow including this one))
Hmm… This feels a bit like that self-reference girl uncle Bertrand warned me about.
But wait a second, that might not be fatal in this case. Reason being that there’s a sort of break; U(all relationships) is a unitary object and the “all relationships” isn’t actually included in U(U(all relationships)) the same way the “set of all sets” would need to include itself. U(whatever) kinda just means true if whatever is a valid relationship and false otherwise. I’m sure someone objects, but again, it would take a better mathematician than I to sort it out formally.
So then we have U distributing over U and U being a valid argument for U. Again, I’m having trouble keeping R(domain) and R(specific argument) straight. Let’s remove the domains. (really remove the diffeonic relands of the diffeonic reland of our U, keeping things to just one level at a time).
U(>). Cool.
U(U) Also looks pretty dope.
Then U is both synetic and diffeonic as well…
That’s enough of an attempt for now. We’ll come back to it later. Hopefully you see how deep this goes and that syndiffeonesis is definitively not some shallow blabberings devised by a nut.
