The following are working speculations
to provoke creativity
Please do not quote or cite:

How Differences in Differences
Generate
Ideas and levels of abstraction

The following thread of thought follows from and assumes knowledge of
Differences in Differences

Yet another way to understand basin sorting is apparent through examination of the four images in Table 3. The essential point is that basins that have identical TAO's patterns at whatever level (TAO-1, TAO-2,...) in Table 3 below will be put in the same category at that level in Table 2 above.

Table 3 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 in Table 2 above and 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). 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.

TAO-1:

Basins 60 and 76 are sorted (see Table2, above) into the same TAO-1category, therefore their corresponding derivative1 columns in Table 3 are identical EXCEPT that one needs to be rotated relative to the other. [For the particular sorting we have done in Table 2 above, all the derivatives were rotated to put them all in phase. Rotating the derivatives leads to a closer correspondence with human perception (which apparently easily shifts phase relations). This rotation is not shown in the images in Table 3.]
Basins 57 and 89 are sorted into different TAO-1 categories in Table 2, above, thus their first derivative columns show different patterns.

TAO-2: Basins 57 and 89 are sorted in Table 2 above into the same TAO-2 category, therefore their second derivative columns here in Table 3 are identical.

TAO-3:
Basins 60 and 76 have identical TAO-3 patterns here in Table 3 and are in the same TAO-3 category in Table 2 above.
Basins 57 and 89 also have identical TAO-3 patterns here and are in the same TAO-3 category in Table 2.
Obviously these two pairs of basins have different TAO-3 patterns here in Table 3 and are therefore put into different TAO-3 categories in Table 2 above.

TAO-4: All differences have disappeared by TAO-4. Therefore all basins have the same, 0, derivative and are joined in a single category consisting of all the basins of the same length.

Something important has been revealed at this point, something that has implications for a general theory of knowledge and is to me quite new and and quite explicit. Before I talk about it, I will quickly give a cursory review of general issues in knowing. [For your convenience you can Pop up a separate, re-sizable page of Table 3.]

What is knowing? One way of thinking about knowing is to say that theories of knowing fall into three broad categories: Deductive, Inductive and Abductive.

Deductive reasoning is the sort that goes "All men are mortal. Socrates is a man. Socrates is mortal." Deduction reasons from general principles to specific cases. Some questions about deductive reasoning are where do the general principles come from originally, from a Kantian type imperative (presumably genetic), from sensory experience? If from an inherent principle of the human being itself, then how did that come about, what are the morphogenetic principles by which the ontology of a single existential being produces such principles? Genetic structures, like the opposable thumb or the universal language generator, are seem oriented toward function. And what are the presumably small set of principles common to all humans from which all others are deduced. How shall we know them? On the other hand, if general principles are leaned from sensory experience, how? Sensory data is not in the form of principles.

Inductive reasoning proceeds from many instances of experiences and by some (not well agreed upon) process arrives at a general principle. That general principle can then be used deductively. Theories of induction, however, are not easily come by. Most theories of learning (association, contiguity) eventually tackle the inductive process but not very successfully. Many end up simply averaging past experience, a procedure that is patently flawed. Modern theories of neural networks starting with Hebb's cell assemblies and continuing through Holland's neural networks do the best job but have important critical limitations.

Abductive reasoning is the application of the the formal relations from one context to another, and is the structural description of metaphor. The formal relations of a mechanical pump can be applied to understanding the heart. The information processing concepts of computing can be applied to human thinking. Again, as with deduction, a fundamental question for abduction is where did the formal principles come from in the first place so that they can be transferred to a new context?

These approaches are not antagonistic and certainly an general approach to knowing will use aspects of all of these approaches.

Eureka!

I will set the stage for the Eureka! which I felt as I thought about the correspondence between the experience of human sorting and the sorting by differences in differences by proposing two manufactured cases of humans learning to abstract an idea. (This set of ideas is exploratory and I'm looking for better presentation in general and for better cases in particular.)

Carburetor. Suppose an adult is working on her car, fixing the carburetor, all the while being watched by a young twelve year old boy. Carburetors are funny things in that most people know the principles of how pumps work, how electrical circuits works, the rain-river-ocean-cloud water cycle works, and so on, but few people know the principles by which a carburetor works. Like many other people the boy has never seen a carburetor before and has no idea what it does; unlike many others, he is very interested. He certainly has a background of general principles learned (somehow) previously in his experience. But Carburetors are new as an idea. The boy watches with great sensory acuity, asking some conceptual questions. When she is done the boy goes away and the next day fixes the carburetor on his mom's car. Being twelve, he doesn't read any reference material in between the two events.

A newly discovered plant. A scientist is working on a difficult problem in the field. She finds a new plant, whose structure is uncatalogued to that time. She thinks about it overnight and in the morning is able to categorize it precisely in the general schema of biology.

A Very General Epistemological Frame. Suppose the world is composed of complex dynamic relations and the those relations are mapped onto the sensory system of, say, humans as differences. These differences are in Bateson's terms "elementary ideas." They are the differences in the world that make a difference to biological being in Grinder's terms.

A more specific epistemological frame. How can we use the representations of E42's logical proofs to make this framework more explicit? It must be the case that transforms of differences, reduced to binary relations, are the unfolding of logical theorems. Which transforms specifically? Which theorems specifically?

Let whatever goes on within the hardware and software of the computer be equivalent to whatever goes on in the world. There is no way to know either the inner workings of a computer or the workings of the world without some representation or other knowledge generating (epistemological) relationship involving we humans as a whole and our sensory neurology in particular. Let the representation we call here the original basin be equivalent to the raw set of differences encoded on a sense organ. (As we noted elsewhere these basins may be encoded auditorily and visually and within each of those representational systems many very different ways of encoding are possible; and so the decision of how specifically to encode dynamic relations makes a profound difference on what flows epistemologically from that encoding. For the moment we will use the encoding described here as Smilie 3.) In short, the Smilie 3 representation of the original basin will be taken to correspond to the "raw neural response" encoded on some sense organ. Note that so early in perceptual processing such a combination of the differences on a sense organ already forms a relational pattern to the degree that the sense organ is successful in encoding a subset of the dynamics available in the universe. (Let us assume that the encoding of a subset of the universe's dynamics are coded into dynamic relations on the sense organ in a way that is functional and useful for the organism, and that this utility is provided by evolutionary processes).

I'm asking you to think of the differences in the original basin (see leftmost column in Table 4, below), as represented by the Smilie 3 output of E42, to be formally equivalent to the differences encoded on a sense organ of a living being. And in both cases (the sense organ and the original basin or E42) the simple sensory representation is already a complex set of relations. The examples here are all visual but I mean the points to be general across sensory systems.

Given that simple set of assumptions, the abstraction of higher order relational clusters, which I'll call "ideas," flows naturally and inevitably. The imperative that I will argue that flows from this simple model is: Use your senses.

Table 4. Three basins Two from the same TAO-3 category, one from a different TAO-3 category. All derivatives from 1 through 8 are shown

These three basins are all different, as are their first and second derivatives, but the top two have the same TAO-3 nd higher TAO's.

Basin 648: Formal Relations in a Ford Truck Carburetor To view the dynamic ongoing process of deriving the TAO levels for basin 648, click here (requires java plugin)

Basin 576: Formal Relations in a GM Carburetor

Basin 328: Formal Relations in a Toyota Carburetor

Basin 440: Formal Relations in a Piper Cub Carburetor

Suppose the original basin structures shown in the leftmost column of Table 4 are abstract representations of some complex area of study--carburators, plant biology, etc. Suppose a person has sensory experience with one single instance, such as a 1975 Ford F100 V8 carburetor or a specific rainforest plant as yet unnamed in European lore. Suppose that a person's sensation of that single instance is followed by many kinds of processes and that of particular important in this discussion the sensation is processed by taking the differences in the differences found in sensation. How specifically does this nervous system process? Specifically via the recursive application of the XOR function: detecting differences in differences in differences in differences in differences, etc.

Given this supposition that one way of biological processing is to transform sensory differences by finding differences in the differences (TAO levels) then the image in Table 4 is an abstract schema of how sensation would be transformed for four different instances (thaw is for four Carburetors, four plants).

A first instance. Within that conceptual frame, if a person encounters and studies one basin, stay basin 648, top, then that person will naturally transform the dynamic relations encoded by the sensory system into the derivatives represented visually in the image. Such a person will for example spontaneously generate the TAO-1 pattern and the TAO-2 pattern, and so on.. In reality, these patterns are dynamic and are shown statically here only as a convenience in communication.

Snapshots. What is shown in Table 4 is snapshots of standing waves (basins) in a flow of process.

Ideas. TAO gives us the abstracted relations among the differences in differences. Any TAO pattern at any level I will call an abstraction, an idea. The richness of differences available at the raw sensory level is reduced to simpler formal relations at higher levels of abstraction. (Recall that this reduction is of the form of the derivation of a logical theorem, that is, it is a formal principle.)

Principles from One Instance. Suppose our person has had contact with a single instance, basin 648, top. Then, through the TAO level processing we have described here, the person will abstract general principles (without experiencing other instances). The ideas our person abstracts are not due to some averaging or other smearing function that combines sensations from multiple instances (induction). Rather they are logical theorems flowing from derivations performed on sensory experience. They may not seem like theorems appearing as they do as music and dance and daydreams, but if the relationship so named "difference" is an elementary idea, then all the complex relations flowing from differences in relation to differences have a boolean, symbolic logic structure that produces derivations of great elegance.

Once you accept differences in differences as your processing primitive, the results of the derivations from such processes are not arbitrary (as is the coincidental association of one neural activity with another). Rather the relational patterns resulting from differences in difference are by the definitions of logic principled abstract forms that must follow from the original differences. Here we have a model of how it might be possible to abstract from raw sensory data higher order principles. To know principles what you really need is high-quality sensation and, and given that elusive prey, to allow differences to be found in differences all the way up.

A second instance from the same TAO-3 category. So our person has come across one instance, we're calling it Basin 648. That person has processed it and produced a set of dynamic relations we call TAO-2, TAO-3 and so on. Now when that person comes across a second instance, say basin 576, and senses and processes it, such experience will quite naturally provoke the same set of abstractions, the same ideas. Basin 576's TAO-1 and TAO-2 will be (slightly in this case) different but its TAO-3 and higher will consist of same structure of formal relations as that of Basin 648. Comparisons of the TAO-3 abstractions given the person that basis to immediately recognize the new instance (576) of as in the same (TAO-3) category as the first instance (648). The abstractions generated by a single instance allow a person to recognize another instance of the same category at a given level of TAO.

A Third Instance from another TAO-3 category but from the same TAO-4 category. A third instance (328, third from top), if encountered would also provoke the same principled ideas (the dynamic patterns we have named TAO-4, TAO-5, TAO-6 and TAO-7). But sensory experience with this instance would derive different principles from TAO-3 down to TAO-1 (compare patterns on the images). A person would derive the same higher level principles but different medium and lower level principles for how the dynamics work.

Instance from a different TAO-7 category. Finally it can happen that sensory experience from a basin (440) within the same dynamic system (as basins 648, 576, and 328) can lead to the derivation of different (to some degree at least) principles of dynamic relations at all levels. Look at the images and notice that ALL levels of TAO for basin 440, bottom, portray different relations than do the top three basins (328, 576, and 648). Suppose two sets of people created their abstractions from instances of the same dynamic system that are sufficiently different (440 versus 648, 576, 328) will create principles that are different all the way up to the highest level. In this framework, its the sensation that you start with that is the important variable.

[For your convenience in the following discussion, you can Pop up a separate, re-sizable page of Table 4.]

Back to the Carburetor.
Let's make this discussion more concrete with one of the cases. Let basin 648 represent a Ford carburetor. Let basin 576 represent a GM carburetor. Let basin 328 represent a Toyota carburetor. The first three are from automobiles. Let basin 440 be a Piper Cub carburetor. This is of course still a carburetor but it is from a different higher level category than are automobile carburetors.

Principles of Auto Carburetors.The boy only needs one first instance (basin 648) of the carburetor to abstract the formal relations that characterize carburetors (TAO's 1 through 7). He has learned on a Ford and so that his TAO-4 abstraction will fit with the higher order principles of all automobile (Fords and Toyotas and Hondas, GM's,etc., i.e., on the image basins 648, 576, and 328 all TAO patterns match at TAO-4 and above). (Notice carefully on the image how the TAO-4 and above formal relations are identical.)

Principles of American versus Japanese Carburetors. The image to the left shows TAO-1 for the two American cars. This is an abstraction of the differences in the differences found in basins 648 (Ford) and 576 (GM). It is a static snapshot of the dynamic relations in a working carburetor and as such qualifies as an abstract "idea" of the structure of that dynamic. The patterns of for the working process of carburetors for the two American cars (basin 648 = Ford and basin 576 = GM) have the identical abstract principles at TAO-1. Learning on a Ford gives you the principles used by GM (although the actual GM carburetor, basin 576) is different so there are still pitfalls.

In contrast, the image on the left shows TAO-1 for the basin 328 (Toyota). But Toyota carburetor (basin 328) has a different set of formal relations in its principles at TAO-1. In fact it has different principles starting at TAO-3 and working down to TAO-2 and TAO-1.

The boy's TAO-1 abstractions from experience with the Ford don't map over precisely to the Toyota. (I've had this experience, having learned about cars made in American cars, primarily Fords.) The engineers at Toyota have their own specific (TAO-3 and below) principles for designing Carburetors (which are consistent with the American cars at the higher (TAO-4 and above) principles of auto arbitration.

• Not Tokens but rather Abstractions. If we accept the metaphor of this model, if we accept the premise of Bateson and Grinder and others that the essence of knowledge is difference and transforms of difference, then these TAO-1 images (right and left) are not mere visual tokens nor signs nor arbitrary names for (metaphorically) carburetors, they are abstractions of the formal relations that make up (metaphorically) carburetors. If we accept the model then these two images reflect a process of refining out low level patterns of difference by taking the differences in those differences to arrive at higher order relations.
• Switching metaphors. Dropping the metaphor of carburetors, recall that, based on broad perspectives on the nature of biology and mental process proposed by Bateson and Kauffman, the sort of N, K Boolean systems we are studying here might well capture some of the abstract aspects of the dynamic complexity of an ecology. Coming to know the principles by which a complex ecology operates is a fundamental epistemological challenge for a sentient being. Recall that N, K Boolean systems in general and E42 in particular produce a deterministic flow of pure relationship and often this flow of relationship produces recursive cycles of the same states (basins). So, the ecology, while dynamic and changing, is demonstrating a dynamic stability, a cyclic pattern of changes whose principles a sentient being might well want to know in order to operate effectively in that ecology.
• A Flow of Formal Relations. How such ecological relations are represented [NOTE 9] is a profound question; and in this discussion they are represented visually by static arrays of black and white squares. Despite the limits of such a representation, it is worthwhile to recall that the black and white arrays are pointing at the structured flow of relationship; they are in effect, a visual mathematico-logical symbol system describing a flow of formal relations. They describe formal relations at one level of abstraction above the formal structure of the basin.
• An elegant conclusion. Taking the simplifying assumption that what the sentient being does is detect differences in an ecology and then find the differences in those differences leads to a simple and elegant (and to me surprising) conclusion: Differences in Differences are a way of abstracting simplified formal structure from the patterns of difference offered by an ecology. To repeat, these images are arbitrary tokens, they represent the abstracted formal structure of the dynamics produce by an N, K Boolean system which is proposed to be a model of mental and biological processes. They represent a possibility of how sentient beings, who only by responding to differences and differences in differences, can extract meaning in the sense of "the formal structure of" a complex and changing ecology.

Back to Carburetors. So when, his mom's Toyota carburetor in hand, the boy applies the Ford TAO-1, TAO-2, TAO-3 level principles (sets of formal relations) to her carburetor, disaster results.

Of course, if he is the quick study we think he is, that is, if does not feed forward the Ford TAO-1, 2, 3 relations so strongly as to block his sensory experience of the Toyota carburetor, then his sensory experience with the Toyota carburetor will generate the Toyota TAO-1, 2, 3 functional relations and he can fix Toyotas thereafter. A single experience (Ford carburetor) gives you all principles all the way up to the highest; AND a single experience (Ford) may not give you what you need when you have another experience (Toyota).

To repeat a crucial point. Sensory experience with a single instance gives him nearly everything. It gives him the abstractions (TAO-1 to 7) of the pattern of that particular carburetor and of all carburetors in the same category (Ford) and of closely related categories (American cars). He can understand arbitration from a single experience. I said "nearly." What a single instance doesn't give him is generalization to the lower level principles specific to other categories (Toyotas) of Carburetors This is a strange thing; he has high order principles and he has some lower level principles, but not all lower level instances. What's easy to get is the higher order principles that will apply to a broad reach of categories. What's hard to get are the particular principles that apply to a diversity of instances. More sensory experience is required for that. Even when you have the idea, you still need lots of experience. This seems to fit many kinds of experience in my past and is a unique imperative of this particular framework to my knowledge.

Principles of Airplane Carburetors. But across which categories of arbitrations are the high-level principles of carburetion the same? Look at the TAO-6's and TAO-7's of the top three basins; they all are identical. In contrast, at these high levels, the principles of airplane carburetion are, to some degree at least, different. The boy's knowledge of principles of automobile carburetors would not agree with the principles of airplane carburetion

In the end, clearing the senses to be open to local experience, even if you've had lots of similar experiences and have abstracted and output great principles from similar context, is the message of wisdom from this frame--any, new specific case may have have specific local principles (TAO-1) which need to be learned for successful work with that case.

Some other related functions of TAO abstraction.

Not Stored. The static images may give the impression that we are talking about "things", such as memories or visual symbols or icons that can be stored somewhere in the brain. The idea here is not that but rather that a person's neurology is a process and these abstract relations are standing waves (basins) within that process. That they are represented statically and visually by the image of the basin or the image of one of the TAO levels is misleading in a fundamental way. This is because the derived principles are dynamic, ongoing processes. These relations, despite being stored here as a static image on this page, are never, in this model, stored in the biological brain in any static form--rather the processing is always ongoing and the TAO abstractions are being generated in context as a natural consequence of processing the context. View the ongoing process of deriving the abstracted relations we are calling ideas or abstractions (TAO levels) by clicking here. (requires java plugin)

Outputting and communicating abstracted principles. Suppose, as seems obviously to be the case, that people have the capability themselves to output external representations of the dynamic abstractions they have generated from their sensory experience. Now our person is capable of outputting abstract representations processed at the level TAO-1, 2, ... Our person can play a melody, speak a sentence, draw a diagram, or dance a dance that maps the structure of the abstracted TAO level relations to a external medium. These can be passed on to other people as the principles that "explain" the existential (and more detailed) instances that those other people could create for themselves if they were to go have the sensory experience. Given the sensory experience and the subsequent availability of TAO level abstractions, a person can talk about, sing about, draw about, program computers about, and generally express the idea abstracted from their sensations. While such communication is useful and often a delight, for the deepest forms of learning, it would be better to arrange that other people experience relevant sensory instances and trust each of these people's processing to generate appropriate abstractions. (However, as we'll mention below due to feed forward loops, and perceptual filters, the same context may not produce the same sensory representation of that context for each person; thus the same ideas and abstracted relational structures may not occur for them.) [NOTE 8, (What am I doing here if sensory experience is better than representations of abstracted higher-order relations?]

Big Picture Summary. The proposed process of abstraction is inductive in the sense that it is triggered by sensory experience. It is different than other inductive approaches in that it does not depend on some abstraction process across multiple instances of a category but rather abstracts the full relational structure from a single instance. This for me is much more like a great deal of my learning--one well structured experiences, one good example can do it. (I'm not against there being cases where some sort of abstraction process could occur across many instances nor against the idea that such a process might be working in parallel with this newly proposed process, but I don't think a theory of abstraction that requires many (possibly hundreds) of instances can explain all, or even much, of learning.)

This framework is simple and is consistent with the experience of quickly grasping higher order relations that allow useful manipulation of many instances (American cars) but yet may not work for many other instances (Japanese cars).

This epistemological frame makes counter intuitive suggestions that are useful. It suggests you can immediately learn the higher order principles the a system from a single instance and that these higher order principles may well apply to a certain sensory contexts that are in the same category of experience. It also suggest that knowing these principles may not be useful in dealing with a similar context that is in a slightly different category. It suggests that the higher up you get in levels of abstraction the more overlap you have in concordance of principles. And it suggests the caveat that the closer you get to sensory experience the more particular the principles of operation are. So, while knowing principles is useful for communicating, in any given existential context it is best to let the sensory data drive the inevitable idea that must occur to you by letting your processing proceed from the sensation deriving the principle anew. If this new principle fits with the principles you have generated before then you have a useful indicator that you are on the right track and you can then utilize previous ideas. And if the new lower order principles (TAO-1, TAO-2) are in fact different from your previous principles you have learned something new and are certified to operate in the current context.

The suggestion from this framework is: Abstraction from sensation first, fitting the new abstraction with old abstractions second. (Unfortunately our civilization, enamored of principles encourages people to the reverse order: Old abstractions first, sensory data second. Such feed forward is about the only thing a person could do wrong.)

A new form. The process of idea abstraction presented here is new in its form. Undoubtedly it could and would work in parallel with other deductive, inductive, and abductive process. Once the abstract structure of formal relations is learned, a person can deduce and abduce from them. Whether we are speaking here of something that should be classified as inductive (because it comes through the senses) or not (because it only requires one instance) I don't know. It could be considered along with other theories of induction and may provide a key step (abstraction without averaging or other smearing techniques) that allows broader theories to work.

FINAL CONCEPUTAL CAVEAT: The the idea of the carburetor as expressed above gives the impression that we are talking about concept learning and abstract ideas. Maybe so. But the actual phenomena we are offering for direct experience (categories of visual stimuli, hierarchies of derivatives) in fact are visual not verbal concepts or abstract concepts. So the carburetor is acting for the moment as a conceptual placeholder for the idea the the visual system may visually abstract visual principles from the retinal image. There are no words for those dynamic visualizations; so the carburetor acts as a verbal metaphor for talking and thinking about visual process. There may indeed be similar processes in verbal and formal realms, but all the examples offered here are visual, and any potential and tentative inferences we may be developing are best kept close to the visual system because that's the kind of phenomena we've offered for examination.

END, for now

Preview of what happens when L is not equal to a power of 2: This is a major puzzle and both extends and qualifies everything we have said up to now.