## Château Descartes

But if we are to select those dimensions which will be of the greatest assistance to our imagination, we should never attend to more than one or two of them as depicted in our imagination, even though we are well aware that there is an indefinite number involved in the problem at issue. It is part of the method to distinguish as many dimensions as possible, so that, while attending to as few as possible at the same time, we nevertheless proceed to take in all of them one by one. (Descartes, CSM, 63).

The final point we should bear in mind is that among the dimensions of a continuous magnitude none is more distinctly conceived than length and breadth, and if we are to compare two different things with each other, we should not attend at the same time to more than these two dimensions in any given figure. For when we have more than two different things to compare, our method demands that we survey them one by one and concentrate on no more than two of them at once. (Descartes, CSM, 65).

René Descartes, “Regulae ad Directionem Ingenii”, or “Rules for the Direction of the Mind”, pp. 9–78 in John Cottingham, Robert Stoothoff, and Dugald Murdoch (eds., trans., 1985), The Philosophical Writings of Descartes, Volume 1, Cambridge University Press, Cambridge, UK.

## The Difference That Makes A Difference That Peirce Makes : 1

Peircers,

Being one who does not view Peirce’s work as a flickering foreshadowing of analytic philosophy, logical whatevism, or anything else you want to call it, but leans more to thinking of the latter philosophies as fumbling fallbacks losing what ground Peirce had gained for our understanding of logic, mathematics, science, not to mention the life of inquiry in general, I am dropping this thread anchor toward the end of remembering the critical insights Peirce gave us, as they come to mind.

Regards,

Jon

## Ask Meno Questions • Code Meno Code

Adapted from Prospects for Inquiry Driven Systems

1.1.2.3. The Trees, The Forest

A sticking point of the whole discussion has just been reached. In the idyllic setting of a knowledge field the question of systematic inquiry takes on the following form:

What piece of code should be followed in order to discover that code?

It is a classic catch, whose pattern was traced out long ago in the paradox of Plato’s Meno. Discussion of this dialogue and the task it sets for AI, cognitive science, and education, including the design of intelligent tutoring systems, can be found in (H. Gardner, 1985), (Chomsky, 1965, 1972, 1975, 1980, 1986), (Fodor, 1975, 1983), (Piattelli-Palmarini, 1980), and (Collins and Stevens, 1991). Though it appears to mask a legion of diversions, this question will present itself at least twice more in the current engagement, both on the horizon and at the gates of the project to fathom and to build intelligent systems. Therefore, it is worth recalling how this inquiry begins. The interlocutor Meno asks:

Can you tell me, Socrates, whether virtue can be taught, or is acquired by practice, not teaching? Or if neither by practice nor by learning, whether it comes to mankind by nature or in some other way? (Plato, Meno, p. 265).

Whether the word “virtue” (arete) is interpreted to mean virtuosity in some special skill or a more general excellence of conduct, it is evidently easy, in the understandable rush to “knowledge”, to forget or ignore what the primary subject of this dialogue is. Only when the difficulties of the original question, whether virtue is teachable, have been moderated by a tentative analysis does knowledge itself become a topic of the conversation. This hypothetical mediation of the problem takes the following tack:

If virtue is a kind of knowledge, and if every kind of knowledge can be be taught, would it not follow that virtue can be taught?

For the present purpose, it should be recognized that this “trial factorization” of a problem space or phenomenal field is a significant intellectual act in itself, one that deserves attention in the effort to understand the competencies that support intelligent functioning. It is a good question to ask just what sort of reasoning processes might be involved in the ability to find such a middle term, as is served by “knowledge” in the example at hand. Generally speaking, interest will reside in a whole system of middle terms, which might be called a “medium” of the problem domain or field of phenomena. This usage makes plain the circumstance that the very recognition and expression of a problem or phenomenon is already contingent on and complicit with a particular set of hypotheses that will inform the direction of its resolution or explanation.

One of the chief theoretical difficulties that obstructs the unification of logic and dynamics in the study of intelligent systems can be seen in relation to this question of how an intelligent agent might generate tentative but plausible analyses of problems that confront it. As described here, this requires a capacity for identifying middle grounds that ameliorate or mollify a problem. This facile ability does not render any kind of demonstrative argument to be trusted in the end and for all time, but is a temporizing measure, a way of locating test media and of trying cases in the media selected. It is easy to criticize such practices, to say that every argument should be finally cast into a deductively canonized form, harder to figure out how to live in the mean time without using such half-measures of reasoning. There is a line of thinking, extending from this reference point in Plato through a glancing remark by Aristotle to the notice of C.S. Peirce, which holds that the form of reasoning required to accomplish this feat is neither inductive nor deductive and reduces to no combination of the two, but is an independent type.

Aristotle called this form of reasoning apagogy (Prior Analytics, 2.25) and it was variously translated throughout the Middle Ages as reduction or abduction. The sense of reduction here is just that by which one question or problem is said to reduce to another, as in the AI strategy of goal reduction. Abductive reasoning is also involved in the initial creation or apt generation of hypotheses, as in diagnostic reasoning. Thus, it is natural that abductive reasoning has periodically become a topic of interest in AI and cognitive modeling, especially in the effort to build expert systems that simulate and assist diagnosis, whether in human medicine, auto mechanics, or electronic trouble-shooting. Recent explorations in this vein are exemplified by (Peng and Reggia, 1990) and (O’Rorke, 1990).

But there is another reason why the factorization problem presents an especially acute obstacle to progress in the system-theoretic approach to AI. When the states of a system are viewed as a manifold it is usual to imagine that everything factors nicely into a base manifold and a remainder. Smooth surfaces come to mind, a single clear picture of a system that is immanently good for all time. But this is how an outside observer might see it, not how it appears to the inquiring system that is located in a single point and has to discover, starting from there, the most fitting description of its own space. The proper division of a state vector into basic and derivative factors is itself an item of knowledge to be discovered. It constitutes a piece of interpretive knowledge that has a large part in determining exactly how an agent behaves. The tentative hypotheses that an agent spins out with respect to this issue will themselves need to be accommodated in a component of free space that is well under control. Without a stable theater of action for entertaining hypotheses an agent finds it difficult to sustain interest in the kinds of speculative bets that are required to fund a complex inquiry.

States of information with respect to the placement of this fret or fulcrum can vary with time. Indeed, it is a goal of the knowledge directed system to leverage this chordal node toward optimal possibilities, and this normally requires a continuing interplay of experimental variations with attunement to the results. Therefore it seems necessary to develop a view of manifolds in which the location or depth of the primary division that is effective in explaining behavior can vary from moment to moment. The total phenomenal state of a system is its most fundamental reality, but the way in which these states are connected to make a space, with information that metes out distances, portrays curvatures, and binds fibers into bundles — all this is an illusion projected onto the mist of individual states from items of code in the knowledge component of the current state.

The mathematical and computational tools needed to implement such a perspective go beyond the understanding of systems and their spaces that I currently have in my command. It is considered bad form for a workman to blame his tools, but in practical terms there continues to be room for better design. The languages and media that are made available do, indeed, make some things easier to see, to say, and to do than others, whether it is English, Pascal (Wirth, 1976), or Hopi (Whorf, 1956) that is being spoken. A persistent attention to this pragmatic factor in epistemology will be necessary to implement the brands of knowledge-directed systems whose intelligence can function in real time. To provide a computational language that can help to clarify these problems is one of the chief theoretical tasks that I see for myself in the work ahead.

A system moving through a knowledge field would ideally be equipped with a strategy for discovering the structure of that field to the greatest extent possible. That ideal strategy is a piece of knowledge, a segment of code existing in the knowledge space of every point that has this option within its potential. Does discovery mark only a different awareness of something that already exists, a changed attitude toward a piece of knowledge already possessed? Or can it be something more substantial? Are genuine invention and proper extensions of the shared code possible? Can intelligent systems acquire pieces of knowledge that are not already in their possession, or in their potential to know?

If a piece of code is near at hand, within a small neighborhood of a system’s place in a knowledge field, then it is easy to see a relationship between adherence and discovery. It is possible to picture how crumbs of code could be traced back, accumulated, and gradually reassembled into whole slices of the desired program. But what if the required code is more distant? If a system is observed in fact to drift toward increasing states of knowledge, does its disposition toward knowledge as a goal need to be explained by some inherent attraction of knowledge? Do potential fields and propagating influences have to be imagined in order to explain the apparent action at a distance? Do massive bodies of knowledge then naturally form, and eventually come to dominate whole knowledge fields? Are some bodies of knowledge intrinsically more attractive than others? Can inquiries get so serious that they start to radiate gravity?

Questions like these are only ways of probing the range of possible systems that are implied by the definition of a knowledge field. What abstract possibility best describes a given concrete system is a separate, empirical question. With luck, the human situation will be found among the reasonably learnable universes, but before that hope can be evaluated a lot remains to be discovered about what, in fact, may be learnable and reasonable.

## Duality Indicating Unity : 1

A formal duality points to a higher unity — a calculus of forms whose expressions can be read in two different ways by switching the meanings assigned to a pair of primitive terms.

I just ran across an old post of mine on the FOM List where I touched on this theme, so I think I’ll copy that here until I get a chance and the concentration to comment further.

C.S. Peirce explored a variety of De Morgan type dualities in logic that he treated on analogy with the dualities in projective geometry. This gave rise to abstract formal systems where the initial constants — and consequently their geometric or graph-theoretic representations — had no uniquely fixed meanings but could be given dual interpretations in logic.

It was in this context that his systems of logical graphs developed, issuing in dual interpretations of the same formal axioms that Peirce referred to as “entitative graphs” and “existential graphs”. It was only the existential interpretation that he developed very far, since the extension from propositional to relational calculus seemed easier to visualize there, but whether there is some truly logical reason for the symmetry to break at that point is not yet known to me.

When I have explored how Peirce’s way of doing things might be extended to “differential logic” I have run into many themes that are analogous to differential geometry over GF(2). Naturally, there are many surprises.

## Propositions As Types : 1

One of my favorite tricks — it seems almost too tricky to be true — is the Propositions As Types Analogy. And I seem to see hints that the 2-part analogy can be extended to a 3-part analogy, as follows.

$\text{proof hint : proof : proposition ~::~ untyped term : typed term : type}$