Differences in Differences
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Web Address: http://www.psych.utah.edu/stat/dynamic_systems/Content/examples/TAO/Differences_in_Differences.html

Part 1: Introduction: Differences in Differences and TAO Tool

Part 2: Recursive Application of TAO Tool--Generating Differences in Differences

Part 3: Finding Basins with Identical TAO patterns

Part 4 : Principle of Dynamic Constancy--Pattern Sorting by TAO levels

Part 5 : A Hierarchy of Categories Generated by Differences in Difference

Part 6: The Basin Length Hypothesis

The Next Conceptual Step

Notes

Interactive Java Applet for Experiential Learning:
Dynamics of recursively taking differences of differences (requires java plugin)

Part 1

Introduction:
Differences in Differences
and TAO Tool Top

Knowledge and Difference. Bateson, Grinder and his associates and Malloy have proposed that difference is a relationship with fundamental and special epistemological status. Since the late 1880's perceptual researchers have proposed the most basic unit of perception as the Just Noticeable Difference (JND). Bateson proposes that transforms of differences are elementary ideas. Bateson and Grinder both stress that knowledge starts with those differences in world that make a difference in human sensory systems. Once differences have occurred in sense organs then further neurological transforms of these differences occur as the differences wend their way through the complex network of human neurology. Eventually these differences in differences will make a difference in humans' first moment of access to information about the world (Bostic St. Clair & Grinder, 2001). [NOTE 5] This web page focuses on an explicit model of differences in differences

Bostic St. Clair and Grinder (2001), p. 10ff, p. 57ff, provide a detailed discussion both of the epistemological issues involved in the various transforms from sense organ to the point of first (conscious?) access and the epistemological issues involved with subsequent transforms of difference. This page will explore the nature of the relationship named "difference" and how models of the flows and the transforms of difference do or do not correspond to human experience. The emphasis on this web page is on direct human experience.

Hierarchies of difference: Logical? Emergent? In as much as emergent hierarchies aren't yet and certainly weren't then well defined in 1970 when Bateson proposed the psychological experience which begins this page, he chose to define his hierarchies of difference in terms of Russellian logical types (levels). In the context of current ideas thirty years later, I propose that the kind [NOTE 7] of levels to which his quote refers be considered an open question. And I propose that we consider the possibility of categorizing levels of difference in terms of emergent hierarchies. To do so implies a definition of what emergence is (particularly in relation to difference as an epistemological primitive). And, given a definition of emergence, a proposal for what would be an emergent hierarchy would be a second important consideration. This page will address neither of those questions. As preparation for a basis upon which to answer those questions, this page will focus on a model of the structure of transforms of difference and on how these transforms of difference correspondence (or do not correspond) to direct experience.

Interactive Java Applet for Experiential Learning:
Dynamics of recursively taking differences of differences (requires java plugin)

The TAO tool is described in detail elsewhere (TAO Tool). For the moment, it is sufficient to say that TAO uses the XOR logical operator to detect changes in the flow of difference. The discrete dynamic systems studied here have two states: 0 or 1, ON or OFF, BLACK or WHITE. The XOR operator compares two states; it returns a 0 if they are the same and returns a 1 if they are different. So TAO examines the differences present in the states of all the nodes of a system at time T and determines whether each of those differences are different at T+1. This is equivalent to taking the first derivative.

TAO can be applied recursively to its own output TAO-1, TAO-2, TAO-3, ... This is equivalent to taking the first, second, third... derivatives. The details of this recursive application of difference to difference is given on the TAO Tool page which ha a link at the top to that discussion. Top

Part 2

Recursive application of the TAO tool to the output of the TAO tool
Differences taken on Differences

For Basin Lengths that are powers of 2 (L = 2, 4, 8, 16, 32, ...) Top

 

Static Representations of TAO recursions (for Basin 24)

Basin Length = 4, Static Image of first four derivatives. The mage to the left shows one basin, arbitrarily named Basin 24, from a small dynamic system. This system has 36 nodes; the vertical axis represents the 36 nodes. Notice that the horizontal axis is organized into five sets of four columns, each set of four separated by a gray bar. The first set of four columns (left) is labeled "Or" which refers to the original basin, the second cluster of four columns is labeled "1" which refers to the first derivative (TAO-1), and so on. In the figure a black square indicates a given node (vertical coordinate) is "on" for a particular iteration (horizontal coordinate).

The basins length for Basin 24 is 4 iterations. As mentioned above, these four iterations are shown along the horizontal axis. For example, within the first set of columns (original basin), the horizontal axis shows the four iterations that define Basin 24. You can see each node iterating four times across the four Original basin columns; after that fourth iteration Basin 24 would repeat the pattern in the first column and so on endlessly. [Image taken from 03-06-05_N36-K3-SR100-In50_Powers-of-2_95-Basins-L4-8-16-32-64.tao.]

To be very specific, look at the very top row--just below the solid grey line in the "Or" set of columns. The top row represents the successive states of Node 1 across the four iterations of Basin 24. Node 1 is always OFF, and so is represented by four successive white squares. The next node down (Node 2) is always ON and is represented by four black squares. The third row down (Node 3) is, once again, always OFF, and so shows up as four white squares. The fourth row down (Node 4) is OFF, ON, ON, OFF; this is represented as WHITE, BLACK, BLACK, WHITE. And so on, down the vertical axis to Node 36 on the bottom.

In the next set of four columns, labeled "1," you can see the discrete analogue of the first derivative. That is you can see the differences in the differences which exist in the original Basin 24. This format allows a quick visual search of the meta-patterns in the differences in differences over time.
(Fuller explanation, after you navigate to linked page, click on Recursive TAO.)

Taking visual derivatives in your mind. Notice the second node (vertically) from the top in the original basin is ON (black) across all four iterations (thus forming a black bar). Consequently, in the TAO-1 column that black bar in the original basin 24 is transformed into a white bar (all off) by the XOR function. That is, there are no detectable differences in the state of that node across the four iterations in the original basin.

The fourth node (vertically) from the top is OFF-ON-0N-OFF or visually WHITE-BLACK-BLACK-WHITE. Therefore the first derivative compares the first and second iteration (WHITE-BLACK) and returns a 1 because there is a difference. Visually this 1 is expressed as BLACK square in the first of the TAO-1 cluster of columns. Then the XOR function compares the original basin states of the second and the third iteration (BLACK-BLACK) and returns a 0 = WHITE in the second of the TAO-1 cluster of columns. And so on. (Note, the fourth iteration, WHITE, is compared to the first iteration, WHITE, and so returns WHITE for lack of difference in the fourth TAO-1 column.) Therefore visual representation for TAO-1 for the fourth node is BLACK-WHITE-BLACK-WHITE. What happens when we apply the XOR function to that sequence to get TAO-2?

The column labeled "2" shows TAO-2. The fourth node we just examined will have a solid black bar for TAO-2. That is because the BLACK-WHITE-BLACK-WHITE sequence of TAO-1 has constant changes of state so TAO-2 will return a BLACK-BLACK-BLACK-BLACK to indicate constant change.

The column labeled "3" shows TAO=3, the application of XOR recursively three times to the original basin. Notice that only two nodes have a sufficiently complex pattern of change in the original basin so that the differences (TAO-3) in the differences (TAO-2) in the differences (TAO-1) of those differences persist to TAO-3.

The column labeled "4" shows TAO-4. All nodes have a TAO-4 = 0. Top

Part 3

Finding Basins with Identical TAO patterns Top

Table 1. Derivatives 1 through 4 for four basins
Visual comparisons of patterns in various TAO columns makes evident how categories of basins are generated.
NOTE: The categories used to sort basins throughout this running example are based on rotating all derivatives so that they begin arbitrarily with the iteration that has the lowest Boolean value (starting the Boolean expression from the first node, top, down to the last node bottom). [Note there is a minor error in the graphics: TAO-2 for basin 57 has not been rotated.] In the visual representation above white equals 0 and black equals 1. For example, scan down the first column of Basin 60's TAO-1 column and notice that when you get to the little checker board area TAO would rotate first derivative so that first row of the checker board starts with a white square, i.e., a Boolean 0. That would start basin 60's derivative with its lowest possible Boolean value and would adjust its phase with the TAO-1 pattern for basin 76. [Image taken from 03-06-05_N36-K3-SR100-In50_Powers-of-2_95-Basins-L4-8-16-32-64.tao.]

TAO-3. Notice that the two basins on the left (Basins 60 and 76) have identical patterns in the TAO-3 column. Similarly, the two basins on the the right (Basins 57 and 89) also have identical patterns in the TOA-3 column.

But the TAO-3 columns for Basins 60 and 76 are DIFFERENT THAN the TAO-3 columns for Basins 57 and 89.

In the model being explored on this web page, basins that have identical patterns for any TAO level can be placed in the same category. Conversely, Basins with the different patterns for a given category are placed in different categories.

Therefore Basins 60 and 76 are placed in the same TAO-3 category and Basins 57 and 89 are placed in a different TAO-3 category. Later we will examine the correspondence of such categories to human perceptual experience.

The above example is incomplete and is meant only to give the general idea. The general idea is that basins that have identical TAO-1 patterns are put in the same TAO-1 category (and sorted from other basins that have different TAO-1 categories). Similarly, basins that have identical TAO-2 patterns are put in the same TAO-2 category (and sorted from other basins that have different TAO-2 categories). The above introductory example only examines this categorization process for the perceptually simple case of TAO-3.

The discussion below will based on the general idea we've started to exemplify here. Top

Part 4

The Principle of Dynamic Constancy:

Pattern Sorting by TAO Levels
When Basin Length is a Power of 2 Top

Very interesting correspondences happen between human experience in sorting visual representations of basin patterns and how TAO sorts basin patterns. This will lead to the proposal of a whole new principle of perceptual organization akin the Gestalt principles of grouping. Let's begin with an example.

Nine basins of length L=4. Table 1 below shows results from a TAO analysis of a system that has many basins. How many? The number of basins the TAO tool finds depends on how carefully the user asks it to search. A moderate but somewhat cursory search found 95 basins in this system of which 9 were of length L=4. [The reason for a cursory rather than a more thorough search is that for expository reasons, it easier to make the perfectly general point using a small set of basins whose names and whose visual representations can be conveniently presented on this web page.]

Basin Names. Of the 95 found basins (numbered from 1 to 95), the number (name) TAO arbitrarily assigned to the nine basins of length L=4 are: 38, 89, 76, 67, 74, 60, 95, 57, and 78. These basin numbers are just categorical names and have no properties of numbers so their order does not matter. They are assigned arbitrarily by the TAO tool in the order that it found them in its pseudo-random search.

Sorting visual patterns using TAO-1. Along its top row Table 1 shows static visual representations of all nine L=4 basins. Along the bottom row these nine static patterns have been sorted by their first derivative (TAO-1) into three categories. [TECHNICAL NOTE 1] Notice for yourself whether the patterns that are chunked together by TAO-1 on the bottom row are cogent to you? After the fact do they make sense as a sort? Look at the top row of all nine basins, would you sort those patterns differently? If so, how specifically?

We have found an interesting correspondence between our own sorting of basin patterns and the kind of sorting done based on systemic dynamics that is performed by TAO. One of the original motivations for the series of choices that led to the specific E42 model was the epistemological framework laid out by Bateson (Mind and Nature) and Grinder and his associates (Turtles All the Way Down, Whispering in the Wind), as well as others, that proposes that knowledge can be usefully grounded in differences transforming as they circulate through networks. If that framework is indeed useful, it is less surprising that a model of differences circulating in networks would have correspondences with human experience. This correspondence is grounded in a larger epistemological context.

Table 2. All nine L=4 basins (top) sorted by the first derivative into six categories (bottom) Examine the TAO categories in the bottom row. Notice the consistency within categories and the differences between categories. Study the whole set of patterns on the top row. Would you sort them differently than TAO-1 does if you were looking to group like things together and different things separately?
TAO-1 Category 1		TAO-1 Category 2		TAO-1 Category 3		TAO-1 Category 4		TAO-1 Category 5		TAO-1 Category 6

The Principle of Dynamic constancy:
A hypothesis about a new principle of human pattern perception.

While there are important boundary conditions mentioned below and described in detail elsewhere, we have found a useful correspondence between how, based on differences in differences, the E42 model sorts basins and how humans perceive and categorize patterns. This correspondence hints at what may be a principle of human perception, much like the various Gestalt grouping rules (e.g., proximity, closure, similarity of size, similarity of orientation, symmetry, parallelism, etc.). It is hypothesized that a new and important principle of human perceptual processing is based on invariances in the dynamics of systems which transform differences. As will be discussed below, this principle applies given that other considerations are held equal (ceteris paribus).

The Principle of Dynamic constancy. Other influences being equal, things and events that share the same pattern of change over time are liable to be perceived by humans as similar and will, if it is useful, be sorted into the same categories.

Ecological Connections. We'll at least tentatively join the tendency of people with models to fit their model post hoc in very general ways to ecological considerations. In a natural ecology one important way to separate one "thing" or event from another must surely be to notice differences in differences over time. Conversely, lack of differences in differences over time (the seasons, the daily routines of prey, the rhythmic beat for a dancer) are the basis of grouping "things" or events together. We live in a world of change and noticing when the pattern of change over time itself is constant in two contexts is a key to discovering functionally equivalent contexts. The principle of dynamic constancy says that humans will categorize, as similar, things or events whose changes over time are the same (given that other ways of categorizing are held equal). Dynamic constancies are important cues for sorting the world.

Generalization. The patterns generated by E42 are representations of the dynamics of specific small discrete systems. We propose to generalize the correspondence between human sorting and sorting on the basis of the logic of dynamics with such well-defined and well-specified representations to the perception by humans of their representations of the dynamics of the world in general.

Common Fate. Of the classic principles of perceptual organization, the principle of dynamic constancy is most related to the principle of common fate, which says that objects that move together are grouped together. The principle of dynamic constancy extends such cursory dynamic considerations to more general cases with more fully specified processes.

Limitations and Boundary Conditions for the Principle of Dynamic constancy. There is something intriguing perceptually about how these static visual snapshots of system dynamics in Table 1 are sorted based on the identity of derivatives. Yet, while this TAO sorting corresponds in certain cases and in certain ways to human sorting, the correspondence does not hold in other cases and other ways. What is this about? First of all, we propose that the Principle of Dynamic constancy, like most principles of perceptual organization, is a Ceteris Paribus (all other things being equal) Principle. That is, the principle holds when the influences of other perceptual principles do not over-ride it. This is a typical condition of perceptual theories. For example, the principle of proximity can be over-ridden by the principle of closed figures. In short, the principle of dynamic constancy is not the only, in fact not the primary, perceptual process conditioning human perceptual sorting.

Click here for a discussion of the Boundary Conditions and Limitations on the Principle of Dynamic constancy.

We now turn to a consideration of the differences in the way that TAO-1 sorts basin patterns from how TAO-2, TAO-3, TAO-4 etc. sort basin patterns. We will find is hierarchy of perceptual categories.

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Part 5

A Hierarchy of Categories
Generated by Recursively Applying
The Difference Operator (XOR)
to Differences
(When the Basin Length is a Power of 2) Top

We can take the derivative of the first derivative, thereby getting the second derivative. And we can take the derivative of the second derivative, thereby getting the third derivative. Once that conceptual tool is present, then we can sort the patterns that have identical second derivatives into the same category, we can sort patterns that have identical third derivative into the same category, and so on. This creates a hierarchy of categories. An example of such a hierarchy is shown in Table 2, below.

[An ASIDE: While I will present a hierarchy of categories immediately below, later, farther below, I will argue that such hierarchies do not involve a shift of the level of analysis, whether such a shift of level be construed as a logical level, a part/whole level, an emergent level or other type level. In fact, in my opinion it is a rare case when someone presents "levels" that meet what I think are deep criteria for shifting a level. Bateson sets a high standard when he presents his mind experiment in the quote at the start of this page. Later on this page I will present material that I think meets that standard.]

Organization of Table 3. As we have noted, sometimes different basins have the same derivatives. TAO can sort basins by any level of derivative, putting those with the same derivatives in the same category. Table 2 below shows the 9 original L=4 basin names (numbers) highlighted in yellow in its bottom row. Above the bottom row those basins names have been sorted into categories based on their having the same derivative as other basins. For each level of TAO, the categories are separated by gray bars. The TAO-1 row is the same sort was was detailed in Table 1 and the discussion above. To make this sorting cognitively manageable, Table 2 shows these categories by basin name only. But you can click on the basin names in any category and see the actual patterns that have been put together in the same category.

Differences in Differences applied recursively generates a Hierarchy of Categories. TAO sorts basins by invariances in their dynamics. That is, basins are put with other basins that have the same pattern of change over time. Conversely, basins are separated from other basins that have different patterns of differences over time. Or, put yet another way, using the language in the Bateson quote at the beginning of this web page, TAO can look at the differences in the differences among basins. When basins do not have any differences in their differences over time those basins are candidates for humans to perceive them as having "something in common."

Click here to see a pop-up image of all the images for all the categories in Table 2

See Dynamic applets showing all the TAOs (TAO-1 through TAO-4) for each basin in the above category scheme.

Basins 38, 60, 76, 57, 89, 74, 95, 67, 78

It requires a fair amount of cognitive load, but you can use any two or three (or more) applets above to examine the TAO-level patterns whose invariances are the basis of the categories. Each applet must be opened in a new instance of your browser. Opening too many applets at once can eat a fair amount of memory.

See Dynamics of TAO levels when L NOT equal to a power of 2 (requires java plugin)

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Part 6

For Basins Lengths that are a power of 2
Differences in Differences will Diminish to Null
At or Before the Length of the Basin Top

More Examples

Let's at examine a couple more examples with basin lengths of 8 and 16.

Basin Length = 8, Static Image of first eight derivatives. This shows the same pattern as above but with the next power of 2 as the basin length. The image on the left shows the recursive application of the XOR operator to the dynamics of a system that has fallen into a basin of length 8. Notice that all differences disappear before 8 recursive applications of the XOR.

Basin length L=16. Below is an example of a system that has fallen into a basin of length L=16. Notice that by the tenth recursive application of the TAO (XOR) function all differences in differences have disappeared. (You most likely will have to scroll the window in your browser to see out to TAO-11) Note, this L=16 image is taken from a different dynamic system than the images above for L=4 and L=8, which were from the same system.

This image shows interesting and quite general patterns of how differences in differences transform themselves, arriving eventually at no difference. The last vestige of difference is always a black bar, prior to a black bar is a set of alternating black and white squares. Prior to that is a set of alternating two-black-squares-together and two-white-squares-together. Prior to that are sets of three-black-squares-together alternating with a single white square. And so on.

The pattern that is apparent in this image is that the differences in the changes over time that define the dynamics of the original basin eventually disappear with successive application of the XOR operator. This conclusion is quite general in our experience for all basin lengths that are powers of 2. This leads to a hypothesis:

Hypothesis: Let TAO be the application of the XOR operator to the dynamic changes in a system that is in a basin. Let TAO-M be the recursive application of the XOR operator basin dynamics M times. When Basin Length, L, is a power of 2 (2, 4, 8, ...) the recursive transformation by the XOR operator of the differences that exist in a system that has fallen into that basin will diminish to no differences at or before TAO-L. That is, all difference will diminish to nothing at or before M = L.

Across many simulations, we have found no counter example to the above hypothesis.

In search of a theorem. We assume that a theorem in boolean or symbolic logic either exists or could be proven that confirms our experimentally observed hypothesis. Alternatively, there may exist or yet be proven a theorem that would disconfirm our hypothesis. We would be more than happy to collaborate with anyone in this regard.

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The Next Conceptual Step: Abstracting Principles and Ideas from Experience: The ideas on this web page imply a model of the processes by which humans abstract principles which define the categories that they use. The basic Batesonian process of finding differences in differences leads to a hypothesis about how formal principles may be abstracted from a single experience without the necessity of multiple examples from which to abstract those formal principles.

NOTES Top

[NOTE 1: Hierarchy of differences. Difference (c) is hierarchically imbedded under difference (b); that is, there is a difference (b) between pattern and quantity and among the differences in quantity, there is a difference (c) between ratios of quantities and the subtractive difference between quantities. But see Angels Fear, pp. 62, 63, and the balance beam because there the distinction between subtraction and ratio seems more complexly developed than what I said the the prior sentence.]

[NOTE 2: Bateson's choice of differences to compare: Differences in sensory representation. Differences quantity (size) versus differences in quality (pattern), for example in thinking about bilateral symmetry and crab claws (which differ in size on the right and left side) size is a quantity while bilateral symmetry refers to the mirroring of pattern and to find pattern we look to the set formal relations among the parts of each claw and not to their size. Differences in how science measures differences in quantities (subtraction, ratios).]

[NOTE 3: Quality versus Quantity: Bateson comes down on the side of the usefulness of distinguishing how we think about pattern (e.g., shape, e.g., the formal relations among parts that define bilateral symmetry) from how we think about quantity. In Mind and Nature he argues that "quantity does not determine pattern," (p. 58). He argues, also (p. 111), that difference is qualitative not quantitative and so belongs to the world of pattern not to the world of quantity. (Note, Bateson distinguishes between integers (counting numbers, e.g., 3 tomatoes) which are part of the world of pattern and quantities (e.g., 3.0000001 gallons). Integers are "the product of counting." Quantities are "the product of measurement," (p. 53f). He points out that you can have exactly three tomatoes while you can never have exactly three gallons. So boolean numbers and the expressions of boolean and symbolic logic which underlay E42 are part of pattern not quantity.)]

[NOTE 4: What might be Bateson's reasons for choosing those particular differences? I think Bateson chose those three types of difference (differences in sensory representation, the difference between pattern and quantity, and, among quantities the difference between ratios and subtractions) because he has found them to be basic to his epistemology. In Angels Fear, p. 63, he states that "Our entire epistemology will take a different shape as we look for subtractive or ratio differences." In Mind and Nature, p. 58f, as mentioned above, he notes that quantities do not determine pattern, the consequence being that the quantification of patterns through measurement scales is likely to be a futile enterprise. Finally, of course, the differences in sensory modality are fundamental to how we gain experience of the world and therefore to our epistemological inferences about the world. Thus these three types of differences are not an arbitrary choice in creating an example to think about. They are quite possibly very basic types of differences in the nature of knowing, and that is one reason why thinking about the differences in these differences is so difficult.]

[NOTE 5. These issues of transformation from sense organ to knowledge are nontrivial. A vision science approach to our "seeing" an object such as a drinking glass is typically broken down into four stages: The retinal image-based stage of transforms, the surface-base stage of transforms, the object-based stage of transforms and the category-based stage of transforms. The deep issues of how each stage happens have not yet been fully understood. The discussions on this page in Whispering in the Wind provide meta-frames for seeking useful solutions to the issues of moving from sense organ to knowledge.]

[NOTE 6. The oddness of the question, how shall we know differences that make a difference, strikes to the heart of the nature of the epistemological stance that a person wants to take in thinking about knowledge. I for one do not think that knowing can be usefully separated from the person knowing. Knowing is a relationship between the knower and and the known. There is always a loop; and knowing is part of the relating that happens in the knower-known loop. Bateson (ref?) provides a brilliant analysis of the epistemological pitfalls of assigning characteristics of a relationship to to one player in the relationship. He argues that dependency is not a characteristic of a person but rather a characteristic of how two or more people are relating. In this regard I propose that a similar stance be taken with knowing. Warren McCulloch, who was among the pioneers in computing of thinking about knowledge as differences in differences in "neural networks" addressed the issue of epistemological stance in his essay "What is a Number that a Man May Know It, and a Man that he May Know a Number? The epistemological stance is that the nature of numbers and the nature of humans mutually inform each other and know what either is requires knowing the relationship between the two. In that sense I propose the discussion on this web page which focuses on the question "How shall we know differences in differences?" is always to be understood in terms of both sides of the relationship]

[NOTE 7. Bostic St. Clair and Grinder discuss the issues of deciding differences of kind (p. ?ff) and also different kinds of levels, p. 30ff, p. 230ff. The current version of this web page simply lays out the logic by which transforms of difference will produce different kinds of hierarchies, without taking the next step of relating these to established kinds of levels.]

[NOTE 8. I focus as much as possible on sensory based visual and auditory representations of dynamics. What are all these images of black and white dots about, anyway? My instance on grounding all discussion to patterns of dots is that they are nearly free of content (we have to accept BLACK-WHITE as our sole content) and they (I assume) can be encoded sensorily in (reasonably) the same way by everyone. They are prime candidates for patterned stimuli that produce very similar sensory encodings across a broad range of people. My abstraction are open to discussion and argument; the Bateson-Grinder "difference that make a difference" frame may be open to discussion, how E42 simulates that (Bateson-Grinder) frame is open to discussion, but my hope is that the basic data (black and white squares) is as likely to be encoded in highly similar ways across people as any form of data found in any scientific discussion.]

[NOTE 9]. Actual processing occurring in a living being's neural representations (e.g., retinal responses and subsequent transforms) is a profoundly different representation than these static visual images expressed in the modality of neural activity. As a minimum a sentient being's processes would be dynamic. Given such profound differences in representation, the proposal is that the living being's representation will have a similar task of capturing the kind of abstract structure we see in these static images.

TECHNICAL NOTES

TECH NOTE 1. See Note in Table 3 above for a concrete example. The vectors for each derivative were rotated in their Node by Iteration Matrix, so that the first row (vector) of the matrix was shifted to the vector with the lowest Boolean value. This in no way changes the basin or its derivative; it is merely as if the derivative were taken starting on the a different iteration. (A basin repeats state vectors in an endless cycle that can be thought of as a circle. So starting an analysis at one point on that circle is equivalent to starting it at any other point.) But... such adjustments have an impact on the correspondence between human perceptual sorting and formal logical (TAO) sorting because many basins are simply out of phase with each other. If the rotation adjustment is not done, TAO places highly similar (in the perception of humans) basins that are out of phase into separate categories; but if they are rotated, they are placed in the same category.

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References

 

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