I posted the following in the Yahoo Adult Development forum and am cross-posting here. I'll keep you apprised of some key responses, provided I get any: 

Building on the post below* regarding Lakoff's embodied reason, he seems to call into question the type of abstract reasoning usually found at the formal operational level. This appears to be false reasoning based on the idea that reason is abstract, literal, conscious, can fit the world directly and works by logic (also see for example this article ). If formal reasoning is false wouldn't this call into question some of the assumptions of the MHC? That perhaps this "stage" is a dysfunction instead of a step toward post-formal reasoning? 

Now Lakoff has his own hierarchy of how embodied reason develops: image-schematic, propositional, metaphoric, metonymic, symbolic. (See for example "Metaphor, cognitive models and language" by Steve Howell.) So I'm wondering how the MHC takes into account Lakoff's work here and how it answers his charge of false reason? Terri Robinett noted in his Ph.D. dissertation (at the Dare Association site) that "work has already begun by Commons and Robinett (2006) on a hierarchically designed instrument to measure Lakoff’s (2002) theory of political worldview." So perhaps you can shed some light on this? 

* This is the referenced post: 

Since Michael brought up Lakoff as perhaps being "at right angles to the stage dimension" I read this by Lakoff this evening: "Why 'rational reason' doesn't work in contemporary politics." He distinguishes between real and false reason, the former being bodily based and the latter existing is some sort of objective, abstract realm. Very interesting indeed. Here are a few excerpts: 

"Real reason is embodied in two ways. It is physical, in our brain circuitry. And it is based on our bodies as the function in the everyday world, using thought that arises from embodied metaphors. And it is mostly unconscious. False reason sees reason as fully conscious, as  literal, disembodied, yet somehow fitting the world directly, and working not via frame-based, metaphorical, narrative and emotional logic, but via the logic of logicians alone."
 
"Real reason is inexplicably tied up with emotion; you cannot be rational without being emotional. False reason thinks that emotion is the enemy of reason, that it is unscrupulous to call on emotion. Yet people with brain damage who cannot feel emotion cannot make rational  decisions because they do not know what to want, since like and not like mean nothing. 'Rational' decisions are based on a long history of emotional responses by oneself and others. Real reason requires emotion."

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Some further comments:

Me: Some of the references I've provided show that cognitive functioning, and natural phenomenon generally, operate via multifractal cascades, not linear, repeated, monofractal similarities. The latter are an imposition created by formal mathematics under the guise of ideal Platonic forms and/or ideal Aristotelian rules and categories. As much is admitted in this MHC paper (pp. 113-15). Even advanced maths operate via such cascades and not formal necessary and sufficient conditions that fit into tidy, reiterated sets. It seems to me that any model of complexity should be based on how dynamic systems actually operate rather than trying to fit them into a formal ideal.

I'm looking over Sara's paper on fractals, which purports to address my inquiry as it discusses the non-linear fractal nature of dynamic systems (DS). So please help me understand Sara how a monofractal recurring of self-similar patterns handles the multifractal nature of DS in the message 35 article? It would be much appreciated.

Cory: It didn’t handle multifractal nature of DS of behavior, but it does allude that it can be done.

Sara: Indeed, wrapping our heads around the fact of, and implications of, cascading recursions throughout the entire system - and the information processing & coordination the entire system is doing thereby!!! - is truly key to understanding system complexity in general, and developmental complexity even more so.

I’m underwater with deadlines and can’t join discussion but these might help.  The first paper listed is addressing whole human system dynamics, and if you read the section on Major Theoretical Orientations, I think you’ll see how the recognized dynamics of cascading recursions, the structure of the system with “decider subsystem” echelons, and the fact that information processing/coordination is done via these means, the multifractal reality is well recognized.

My work discussion coarse-grained and fine-grained analysis & scoring is another illustration of recognized levels of fractal existence and dynamics. The MHC theory needs the addition/revision of the fractal transitions, which I began calling the Model of Fractal Action Complexity (MFAC).  It demonstrates the limit of using it to score anything is one of the user’s analytical devotion to incredible detail, not a limit of the theory or method. One has to have the expertise for analysis at different levels of analysis, e.g., biochemistry and molecular and cellular biology at micro levels, all the way up to super complex capacities at the high-order ranges. Cory and I worked for a while on explicating this, till we didn’t. I believe he’s continued in some way, and notably, uses his Spectrum of Human Imagination  while presumes recursive dynamics as core to increases in complexity.

 These are the recognitions and methods that exist.  It all sounds abstract – because it is – and the best, most accessible way in my view to internalize this stuff to know it from the inside out is my long-time mantra of “use the 24/7 laboratories of our own lived experience.”  I.e., analyzing the heck out of one’s long term processes of making a wrenching decision, figuring out how something works the way it does and why, watching the fractal processes we are always using to coordinate information throughout such processes. For someone who’s got hierarchical complexity level-know-how, this becomes easy (and yeah, incredibly detailed so writing everything down and scoring it is needed if you want to get fine-grained).

Finally, as a reminder of setting scale and scope in any analysis, I include a source on domains of action that below too.

The bottom line, in my opinion, is that we have the laboratories and methods we need to ascertain the existence of the multifractal reality that comprises our existence as living systems and the fractal nature of each and every micro to macro increase – and in degenerations, decrease.

 

Sara: I don’t have time to get into this but just based on what you wrote below, calculating the fractal dimension (fd) of (usually large-sized) sample data over time to test for a power law is “one thing” (let’s say apples) i.e., making a math calc on fixed historical data  (which is by its nature got homogeneity characteristic by virtue of how data were aggregated to begin with!) vs analyzing the real-time "interactions across multiple scales and among fluctuations of multiple sizes" that account for the system behaviors over time (let’s say, apple seeds).  Different units of analysis on data that “became” at different points in time.  Digest that difference. It might be the core of your Q.

There was a lot of hullaballoo in the early days of nonlinear sci when fd got ID’d and everyone spent ad nauseam amounts of effort cal’c’g it on everything under the sun.  never meant anything, really. Have to get into the apple seeds to understand what accounts for the power laws. Two different units of analysis, for starters.

However, I too cal’cd the inherent fd of the orders of hierarchical complex _orders being the general theory part, not the actions of actors_ and yup, it’s got an inherent power law too. So when you have a method like HC that enables scoring using same method from micro to macro with unlimited levels of nesting and most importantly disjunctures in time – NOT enslaved to sequential time – (see ‘fractal transitions / fractals of themselves” artice) then we’ve got analytical power – without even bothering to calc a fd. The fractal patterns are obvious on the face of the data.  We can however calc the fd of any HC-scored data. Sometimes there’s a scientific argument for certain audiences where that’d be valuable.

Cory: Compare with https://en.wikipedia.org/wiki/Scale_invariance

Me: I did. While fractals might have self-similar scaling, multifractals have "multiple scaling factors at play at once." So when you have several fractals interacting with each other, each with their own self-similar scaling, aren't they going to affect each? If so how?

E.g., the section on fractals also has a link to multifractals, the first line of which reads: "A multifractal system is a generalization of a fractal system in which a single exponent (the fractal dimension) is not enough to describe its dynamics; instead, a continuous spectrum of exponents (the so-called singularity spectrum) is needed." That article also addresses the difference between mono- and multifractals.

Also the article on scale invariance says this about fractals, which perhaps you can explain to me:

"It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. A fractal is equal to itself typically for only a discrete set of values λ, and even then a translation and rotation may have to be applied to match the fractal up to itself."

Cory:The function f may perform operations that alter x in a self-similar way such that its output incremental differences alter the overall output as being different from the function that produced it on some value such as rotation. Rotating a degree for example. So if you scale one level up or down, you have to rotate a degree in addition to scaling factor. All λ means is measurement scaling. 

Me: As I have no math background this is difficult for me to understand when that language is used. So I've been looking around for some basic introductions to the topic to get the gist. I found this series to be helpful. In the link it relates fractal scaling with levels of complexity. It talks about self-similar monofractals like the familiar Koch fractal. (It calls them monofractals at the bottom.) This seems to be the sort of fractal used in the MHC analysis, at least what I can gather from reading two of Sara's papers so far.

Also at the bottom is a link to the section on multifractals. It appears that monofractals use one scaling rule, whereas multifractals use multiple scaling rules (per this definition). Then the math discussion goes over my head. It concludes: "To put this all into perspective and make the point that it helps relate the 'multi' in multifractal to the 'mono' in monofractal—multifractals have multiple dimensions in the D(Q) vs Q spectra but monofractals are rather flat in that area." I can't decipher exactly what that means but it seems multifractals are more complex?

On this page it talks about lacunarity, i.e., rotational and translational invariance as Cory explained above. It seems that monofractal scaling can be accounted for by a rotation, i.e., they look the same with rotation. But that doesn't seem to be the case with multifractals. It seems that when multifractals are rotated the lacunae present a different, variable image. Their images are certainly far different than the monofractal Koch fractal. Is this due to using multiple scaling rules?

So again,as a novice in this domain I can only get general intuitions at this point, hence my questions for the experts. I'm just wondering how a multifractal analysis might change the fractal analyses already done for the MHC. Isn't that what you're working on Cory?

The discussion of smash sub-tasks in Sara's 2010 paper is the closest thing to multifractal analysis I can see with my limited understanding. It seems between the abstract and the metasystematic stages a task might require coordination of sub-tasks within the smash steps from each of those stages to achieve the task performance. So its fractals within fractals nature appears multifractal given the inclusion of multiple scaling rules. And this cross-level task occurs in a nonlinear manner, with ups and downs and sideways hits and misses.

So it seems even within the different fractal step sub-tasks they too use the same scaling rule, i.e. a monofractal self-similar scale like the stages. Granted it seems to be a different scaling, but both the stage and steps scales seem monofractal. So does a multifractal analysis simply mean the coordination of multiple monofractals using the same self-similar scaling? It seems multifractals don't exhibit self-similar scaling?
Cory: Okay here is a super basic example of a multifractal. Imagine a line and you have 10 increments, and between each increment you have 10 increments. So that's a fractal dimension. Now put a point on 2 and a point on 5. For point two, between 2 and 3, count 2 smaller increments and put a point. For point five, between 5 and 6 count 5 increments and put a point. The zoom on the 2 and the zoom on the 5 have different fractal measurements and don't scale with each other but are on the same fractal dimension. 
 
If it was just 2 or just 5, it is monofractal because only one scaling with one pattern. If the 2 and 5 both happen at both 2 and 5, it can be considered monofractal and multifractal since the overall dimension has different scaling factors but has same pattern in both cases.
 
There is no explicit MHC math in the literature to account for fractal transitions or stages beyond what you've now seen. You can learn fractal math and come up with your own ideas for how to advance or wait for one of us to publish. Until then, this is more than less the constraint on what is available on the fractal properties of human development.
 

The crux of the issue is that monofractals are symmetrically self-similar at each iteration. Multifractals are not. Multifractals weave 2 or more fractals which each iterate differently creating asymmetry. Which is indeed the case with natural phenomenon; there is no ideal hill that expresses a perfect, mono- or multifractal parabola.

The link for the last post is here. The following is from this link:

"Hierarchical organization is a corner stone of complexity and multifractality constitutes its central quantifying concept. For model uniform cascades the corresponding singularity spectrum are symmetrical while those extracted from empirical data are often asymmetric."

This recent thread is relevant to this one. E.g., from the 2nd comment:

"There has been a profound, even revolutionary, shift in our theory of developmental psychology. The revolution began with challenges to Piaget's theory of cognitive development, particularly his views of infancy. As everybody who has attended scientific conferences, read technical journals, or monitored the popular media knows, modern research has discovered that young children know more at earlier ages than had been predicted by classical theory. These new findings led to the gradual weakening, and finally the collapse of, classical Piagetian theory.

"There is now a furious search for a new framework. An analogy can be drawn to the early part of this century when classical Newtonian mechanics was overthrown and physicists were searching for a new model. In our field, we know that the classical framework of developmental psychology, which has reigned for almost 50 years, does not work; we have crucial experiments that have uncovered surprising facts; and we have great excitement in both the laboratories and in society at large, as competing views of early human development are being thrashed out."

This post and following are also relevant to this thread. Fischer's "constructive web" and inhibitory-control theory's "overlapping waves within a dynamic system" remind me quite a bit of image schema and Edwards' various lenses, how they relate and interact. It seems the strict ladder of stages metaphor is not apt to handle such interactivity. The reality is that we are a hot mesh.

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