Directory talk:Jon Awbrey/Papers/Inquiry Driven Systems : Part 1

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1.3.

1.3.5. Discussion of Formalization : Specific Objects

| "Knowledge" is a referring back:  in its essence a regressus in infinitum.
| That which comes to a standstill (at a supposed causa prima, at something
| unconditioned, etc.) is laziness, weariness --
|
| (Nietzsche, 'The Will to Power', S 575, 309).

With this preamble, I return to develop my own account of formalization,
with special attention to the kind of step that leads from the inchoate
chaos of casual discourse to a well-founded discussion of formal models.
A formalization step, of the incipient kind being considered here, has
the peculiar property that one can say with some definiteness where it
ends, since it leads precisely to a well-defined formal model, but not
with any definiteness where it begins.  Any attempt to trace the steps
of formalization backward toward their ultimate beginnings can lead to
an interminable multiplicity of open-ended explorations.  In view of
these circumstances, I will limit my attention to the frame of the
present inquiry and try to sum up what brings me to this point.

It begins like this:  I ask whether it is possible to reason about inquiry
in a way that leads to a productive end.  I pose my question as an inquiry
into inquiry, and I use the formula "y_0 = y y" to express the relationship
between the present inquiry, y_0, and a generic inquiry, y.  Then I propose
a couple of components of inquiry, discussion and formalization, that appear
to be worth investigating, expressing this proposal in the form "y >= {d, f}".
Applying these components to each other, as must be done in the present inquiry,
I am led to the current discussion of formalization, y_0 = y y >= f d.

There is already much to question here.  At least,
so many repetitions of the same mysterious formula
are bound to lead the reader to question its meaning.
Some of the more obvious issues that arise are these:

The term "generic inquiry" is ambiguous.  Its meaning in practice
depends on whether the description of an inquiry as being generic
is interpreted literally or merely as a figure of speech.  In the
literal case, the name "y" denotes a particular inquiry, y in Y,
one that is assumed to be plenipotential or prototypical in yet
to be specified ways.  In the figurative case, the name "y" is
simply a variable that ranges over a collection Y of nominally
conceivable inquiries.

First encountered, the recipe "y_0 = y y" seems to specify that
the present inquiry is constituted by taking everything that is
denoted by the most general concept of inquiry that the present
inquirer can imagine and inquiring into it by means of the most
general capacity for inquiry that this same inquirer can muster.

Contemplating the formula "y_0 = y y" in the context of the subordination
y >= {d, f} and the successive containments F c M c D, the y that inquires
into y is not restricted to examining y's immediate subordinates, d and f,
but it can investigate any feature of y's overall context, whether objective,
syntactic, interpretive, and whether definitive or incidental, and finally it
can question any supporting claim of the discussion.  Moreover, the question y
is not limited to the particular claims that are being made here, but applies to
the abstract relations and the general concepts that are invoked in making them.
Among the many additional kinds of inquiry that suggest themselves at this point,
I see at least the following possibilities:

   1.  Inquiry into propositions about application and equality.
       Just by way of a first example, one might well begin by
       considering the forms of application and equality that
       are invoked in the formula "y_0 = y y" itself.

   2.  Inquiry into application, for example, the way that
       the term "y y" indicates the application of y to y
       in the formula "y_0 = y y".  

   3.  Inquiry into equality, for example,
       the meaning of "=" in "y_0 = y y".

   4.  Inquiry into indices, for example,
       the significance of "0" in "y_0".

   5.  Inquiry into terms, specifically, constants and variables.
       What are the functions of "y" and "y_0" in this respect?

   6.  Inquiry into decomposition or subordination, for example,
       as invoked by the sign ">=" in the formula "y >= {d, f}".

   7.  Inquiry into containment or inclusion.  In particular, examine the
       claim "F c M c D" that conditions the chances that a formalization
       has an object, the degree to which a formalization can be carried
       out by means of a discussion, and the extent to which an object
       of formalization can be conveyed by a form of discussion.

If inquiry begins in doubt, then inquiry into inquiry begins in
doubt about doubt.  All things considered, the formula "y_0 = y y"
has to be taken as the first attempt at a description of the problem,
a hypothesis about the nature of inquiry, or an image that is tossed out
by way of getting an initial fix on the object in question.  Everything in
this account so far, and everything else that I am likely to add, can only
be reckoned as hypothesis, whose accuracy, pertinence, and usefulness can
be tested, judged, and redeemed only after the fact of proposing it and
after the facts to which it refers have themselves been gathered up.

A number of problems present themselves due to the context in which
the present inquiry is aimed to present itself.  The hypothesis that
suggests itself to one person, as worth exploring at a particular time,
does not always present itself to another person as worth exploring at
the same time, or even necessarily to the same person at another time.
In a community of inquiry that extends beyond an isolated person and
in a process of inquiry that extends beyond a singular moment in time,
it is therefore necessary to consider the nature of the communication
process that the discussion of inquiry in general and the discussion of
formalization in particular need to invoke for their ultimate utility.

Solitude and solipsism are no solution to the problems of community and
communication, since even an isolated individual, if ever there was, is,
or comes to be such a thing, has to maintain the lines of communication
that are required to integrate past, present, and prospective selves --
in other words, translating everything into present terms, the parts of
one's actually present self that involve actual experiences and present
observations, do present expectations as reflective of actual memories,
and do present intentions as reflective of actual hopes.  Consequently,
the dialogue that one holds with oneself is every bit as problematic
as the dialogue that one enters with others.  Others only surprise
one in other ways than one ordinarily surprises oneself.

I recognize inquiry as beginning with a "surprising phenomenon" or
a "problematic situation", more briefly described as a "surprise"
or a "problem", respectively.  These are the types of moments that
try our souls, the instances of events that instigate inquiry as
an effort to achieve their own resolution.  Surprises and problems
are experienced as afflicted with an irritating uncertainty or a
compelling difficulty, one that calls for a response on the part
of the agent in question:

   1.  A "surprise" calls for an explanation to resolve the
       uncertainty that is present in it.  This uncertainty
       is associated with a difference between observations
       and expectations.

   2.  A "problem" calls for a plan of action to resolve the
       difficulty that is present in it.  This difficulty is
       associated with a difference between observations and
       intentions.

To express this diversity in a unified formula:  Both types of inquiry
begin with a "delta", a compact term that admits of expansion as a debt,
a difference, a difficulty, a discrepancy, a dispersion, a distribution,
a doubt, a duplicity, or a duty.

Expressed another way, inquiry begins with a doubt about one's object,
whether this means what is true of a case, an object, or a world, what
to do about reaching a goal, or whether the hoped-for goal is really
good for oneself -- with all that these questions lead to in essence,
in deed, or in fact.

Perhaps there is an inexhaustible reality that issues in these
apparent mysteries and recurrent crises, but, by the time I say
this much, I am already indulging in a finite image, a hypothesis
about what is going on.  If nothing else, then, one finds again the
familiar pattern, where the formative relation between the informal
and the formal merely serves to remind one anew of the relationship
between the infinite and the finite.

1.3.5.1. The Will to Form

| The power of form, the will to give form to oneself.  "Happiness"
| admitted as a goal.  Much strength and energy behind the emphasis
| on forms.  The delight in looking at a life that seems so easy. --
| To the French, the Greeks looked like children.
|
| (Nietzsche, 'The Will to Power', S 94, 58).

Let me see if I can summarize as quickly as possible the problem that I see before me.
On each occasion that I try to express my experience, to lend it a form that others
can recognize, to put it in a shape that I myself can later recall, or to store it
in a state that allows me the chance of its re-experience, I generate an image of
the way things are, or at least a description of how things seem to me.  I call
this process "reflection", since it fabricates an image in a medium of signs
that reflects an aspect of experience.  Very often this experience is said
to be "of" -- what? -- something that exists or persists at least partly
outside the immediate experience, some action, event, or object that is
imagined to inform the present experience, or perhaps some conduct of
one's own doing that obtrudes for a moment into the world of others
and meets with a reaction there.  In all of these cases, where the
experience is everted to refer to an object and thus becomes the
attribute of something with an external aspect, something that
is thus supposed to be a prior cause of the experience, the
reflection on experience doubles as a reflection on that
conduct, performance, or transaction that the experience
is an experience "of".  In short, if the experience has
an eversion that makes it an experience of an object,
then its reflection is again a reflection that is
also of this object.

Just at the point where one threatens to become lost in the morass of
words for describing experience and the nuances of their interpretation,
one can adopt a formal perspective, and realize that the relation among
objects, experiences, and reflective images is formally analogous to the
relation among objects, signs, and interpretant signs that is covered by
the pragmatic theory of signs.  One still has the problem:  How are the
expressions of experience everted to form the exterior faces of extended
objects and exploited to embed them in their external circumstances, and
no matter whether this object with an outer face is oneself or another?
Here, one needs to understand that expressions of experience include
the original experiences themselves, at least, to the extent that
they permit themselves to be recognized and reflected in ongoing
experience.  But now, from the formal point of view, "how" means
only:  To describe the formal conditions of a formal possibility.

1.3.5.2. The Forms of Reasoning

| The most valuable insights are arrived at last;
| but the most valuable insights are methods.
|
| (Nietzsche, 'The Will to Power', S 469, 261).

A certain arbitrariness has to be faced in the terms that one uses
to talk about reasoning, to split it up into different parts and
to sort it out into different types.  It is like the arbitrary
choice that one makes in assigning the midpoint of an interval
to the subintervals on its sides.  In setting out the forms of
a nomenclature, in fitting the schemes of my terminology to the
territory that it disturbs in the process of mapping, I cannot
avoid making arbitrary choices, but I can aim for a strategy
that is flexible enough to recognize its own alternatives and
to accommodate the other options that lie within their scope.

If I make the mark of deduction the fact that it reduces the
number of terms, as it moves from the grounds to the end of
an argument, then I am due to devise a name for the process
that augments the number of terms, and thus prepares the
grounds for any account of experience.

What name hints at the many ways that signs arise in regard to things?
What name covers the manifest ways that a map takes over its territory?
What name fits this naming of names, these proceedings that inaugurate
a sign in the first place, that duly install it on the office of a term?
What name suits all these actions of addition, annexation, incursion, and
invention that instigate the initial bearing of signs on an object domain?

In the interests of a "maximal analytic precision" (MAP), it is fitting
that I should try to sharpen this notion to the point where it applies
purely to a simple act, that of entering a new term on the lists, in
effect, of enlisting a new term to the ongoing account of experience.
Thus, let me style this process as "adduction" or "production", in
spite of the fact that the aim of precision is partially blunted
by the circumstance that these words have well-worn uses in other
contexts.  In this way, I can isolate to some degree the singular
step of adding a term, leaving it to a later point to distinguish
the role that it plays in an argument.

As it stands, the words "adduction" and "production" could apply to the
arbitrary addition of terms to a discussion, whether or not these terms
participate in valid forms of argument or contribute to their mediation.
Although there are a number of auxiliary terms, like "factorization",
"mediation", or "resolution", that can help to pin down these meanings,
it is also useful to have a word that can convey the exact sense meant.
Therefore, I coin the term "obduction" to suggest the type of reasoning
process that is opposite or converse to deduction and that introduces
a middle term "in the way" as it passes from a subject to a predicate.

Consider the adjunction to one's vocabulary that is comprised of these three words:
"adduction", "production", "obduction".  In particular, how do they appear in the
light of their mutual applications to each other and especially with respect to
their own reflexivities?  Notice that the terms "adduction" and "production"
apply to the ways that all three terms enter this general discussion, but
that "obduction" applies only to their introduction only in specific
contexts of argument.

Another dimension of variation that needs to be noted among these different types
of processes is their status with regard to determimism.  Given the ordinary case
of a well-formed syllogism, deduction is a fully deterministic process, since the
middle term to be eliminated is clearly marked by its appearance in a couple of
premisses.  But if one is given nothing but the fact that forms this conclusion,
or starts with a fact that is barely suspected to be the conclusion of a possible
deduction, then there are many other middle terms and many other premisses that
might be construed to result in this fact.  Therefore, adduction and production,
for all of their uncontrolled generality, but even obduction, in spite of its
specificity, cannot be treated as deterministic processes.  Only in degenerate
cases, where the number of terms in a discussion is extremely limited, or where
the availability of middle terms is otherwise restricted, can it happen that
these processes become deterministic.

1.3.5.3. A Fork in the Road

| On "logical semblance" -- The concepts "individual" and "species"
| equally false and merely apparent.  "Species" expresses only the
| fact that an abundance of similar creatures appear at the same
| time and that the tempo of their further growth and change is
| for a long time slowed down, so actual small continuations
| and increases are not very much noticed (-- a phase of
| evolution in which the evolution is not visible, so
| an equilibrium seems to have been attained, making
| possible the false notion that a goal has been
| attained -- and that evolution has a goal --).
|
| (Nietzsche, 'The Will to Power', S 521, 282).

It is worth trying to discover, as I currently am, how many properties of inquiry
can be derived from the simple fact that it needs to be able to apply to itself.
I find three main ways to approach the problem of inquiry's self-application,
or the question of inquiry's reflexivity:

   1.  One way attempts to continue the derivation in the manner of a
       necessary deduction, perhaps by reasoning in the following vein:
       If self-application is a property of inquiry, then it is sensible
       to inquire into the concept of application that could make this
       conceivable, and not just conceivable, but potentially fruitful.

   2.  Another way breaks off the attempt at a deductive development and puts forth
       a full-scale model of inquiry, one that has enough plausibility to be probated
       in the court of experience and enough specificity to be tested in the context
       of self-application.

   3.  The last way is a bit ambivalent in its indications, seeking as it does
       both the original unity and the ultimate synthesis at one and the same
       time.  Perhaps it goes toward reversing the steps that lead up to this
       juncture, marking it down as an impasse, chalking it up as a learning
       experience, or admitting the failure of the imagined distinction to
       make a difference in reality.  Whether this form of egress is read
       as a backtracking correction or as a leaping forward to the next
       level of integration, it serves to erase the distinction between
       demonstration and exploration.

Without a clear sense of how many properties of inquiry are necessary
consequences of its self-application and how many are merely accessory
to it, or even whether some contradiction still lies lurking within the
notion of reflexivity, I have no choice but to follow all three lines of
inquiry wherever they lead, keeping an eye out for the synchronicities,
the constructive collusions and the destructive collisions that may
happen to occur among them.

The fictions that one devises to shore up a shaky account of experience
can often be discharged at a later stage of development, gradually coming
to be replaced with primitive elements of less and less dubious characters.
Hypostases and hypotheses, the creative terms and the inventive propositions
that one coins to account for otherwise ineffable experiences, are tokens that
are subject to a later account.  Under recurring examination, many such tokens
are found to be ciphers, marks that no one will miss if they are cancelled out
altogether.  The symbolic currencies that tend to survive lend themselves to
being exchanged for stronger and more settled constructions, in other words,
for concrete definitions and explicit demonstrations, gradually leading to
primitive elements of more and more durable utilities.

1.3.5.4. A Forged Bond

| The form counts as something enduring and therefore more valuable;  
| but the form has merely been invented by us;  and however often
| "the same form is attained", it does not mean that it is the
| same form -- what appears is always something new, and it
| is only we, who are always comparing, who include the new,
| to the extent that it is similar to the old, in the unity of
| the "form".  As if a type should be attained and, as it were,
| was intended by and inherent in the process of formation.
|
| (Nietzsche, 'The Will to Power', S 521, 282).

A unity can be forged among the methods by noticing the following
connections among them.  All the while that one proceeds deductively,
the primitive elements, the definitions and the axioms, must still be
introduced hypothetically, notwithstanding the support they get from
common sense and widespread assent.  And the whole symbolic system
that is constructed through hypothesis and deduction must still be
tested in experience to see if it serves any purpose to maintain it.

1.3.5.5. A Formal Account

| Form, species, law, idea, purpose -- in all these cases the same error
| is made of giving a false reality to a fiction, as if events were in
| some way obedient to something -- an artificial distinction is made
| in respect of events between that which acts and that toward which
| the act is directed (but this "which" and this "toward" are only
| posited in obedience to our metaphysical-logical dogmatism:
| they are not "facts").
|
| (Nietzsche, 'The Will to Power', S 521, 282).

In this Section (1.3.5), I am considering the step of formalization that
takes discussion from a large scale informal inquiry to a well-defined
formal inquiry, establishing a relation between the implicit context
and the explicit text.

In this project as a whole, formalization is used to produce formal models
that represent relevant features of a phenomenon or process of interest.
Thus, the formal model is what constitutes the image of formalization.

The role of formalization splits into two different cases depending on
the intended use of the formal model.  When the phenomenon of interest
is external to the agent that is carrying out the formalization, then
the model of that phenomenon can be developed without doing any great
amount of significant reflection on the formalization process itself.
This is usually a more straightforward operation, since it can avail
itself of automatic competencies that are not themselves in question.
But when the phenomenon of interest is entangled with the conduct of
the agent in question, then the formal modeling of that conduct will
generally involve a more or less difficult component of reflection.

In a recursive context, a principal benefit of the formalization
step is to find constituents of inquiry with reduced complexities,
drawing attention from the context of informal inquiry, whose stock
of questions may not be grasped well enough to ever be fruitful and
the scope of whose questions may not be focused well enough to ever
see an answer, and concentrating effort in an arena of formalized
inquiry, where the questions are posed well enough to have some
hope of bearing productive answers in a finite time.

1.3.5.6. Analogs, Icons, Models, Surrogates

| One should not understand this compulsion to construct concepts, species,
| forms, purposes, laws ("a world of identical cases") as if they enabled us
| to fix the real world;  but as a compulsion to arrange a world for ourselves
| in which our existence is made possible: -- we thereby create a world which is
| calculable, simplified, comprehensible, etc., for us.
|
| (Nietzsche, 'The Will to Power', S 521, 282).

This project makes pivotal use of certain formal models to represent the
conceived structure in a "phenomenon of interest" (POI).  For my purposes,
the phenomenon of interest is typically a process of interpretation or a
process of inquiry, two nominal species of process that will turn out to
evolve from different points of view on the very same form of conduct.

Commonly, a process of interest presents itself as the trajectory
that an agent describes through an extended space of configurations.
The work of conceptualization and formalization is to represent this
process as a conceptual object in terms of a formal model.  Depending
on the point of view that is taken from moment to moment in this work,
the "model of interest" (MOI) may be cast as a model of interpretation
or as a model of inquiry.  As might be anticipated, it will turn out
that both descriptions refer essentially to the same subject, but
this will take some development to become clear.

In this work, the basic structure of each MOI is adopted from the
pragmatic theory of signs and the general account of its operation
is derived from the pragmatic theory of inquiry.  The indispensable
usefulness of these models hinges on the circumstance that each MOI,
whether playing its part in interpretation or in inquiry, is always
a "model" in two important senses of the word.  First, it is a model
in the logical sense that its structure satisfies a formal theory or
an abstract specification.  Second, it is a model in the analogical
sense that it represents an aspect of the structure that is present
in another object or domain.

1.3.5.7. Steps and Tests of Formalization

| This same compulsion exists in the sense activities that support reason --
| by simplification, coarsening, emphasizing, and elaborating, upon which
| all "recognition", all ability to make oneself intelligible rests.  Our
| needs have made our senses so precise that the "same apparent world"
| always reappears and has thus acquired the semblance of reality.
|
| (Nietzsche, 'The Will to Power', S 521, 282).

A step of formalization moves the active focus of discussion from
the "presentational object" or the source domain that constitutes
the phenomenon of interest to the "representational object" or the
target domain that makes up the relevant model of interest.  If the
structure in the source context is already formalized then the step
of formalization can itself be formalized in an especially elegant
and satisfying way as a structure-preserving map, a homomorphism,
or an "arrow" in the sense of mathematical category theory.

The test of a formalization being complete is that a computer program could
in principle carry out the steps of the process being formalized exactly as
represented in the formal model or image.  It needs to be appreciated that
this test is a criterion of sufficiency to formal understanding and not of
necessity directed toward a material re-creation or a concrete simulation
of the formalized process.  The ordinary agents of informal discussion
who address the task of formalization do not disappear in the process
of completing it, since it is precisely for their understanding that
the step is undertaken.  Only if the phenomenon or process at issue
were by its very nature solely a matter of form could its formal
analogue constitute an authentic reproduction.  However, this
potential consideration is far from the ordinary case that
I need to discuss at present.

In ordinary discussion, agents of inquiry and interpretation depend on
the likely interpretations of others to give their common notions and
their shared notations a meaning in practice.  This means that a high
level of implicit understanding is relied on to ground each informal
inquiry in practice.  The entire framework of logical assumptions and
interpretive activities that is needed to shore up this platform will
itself resist analysis, since it is precisely to save the effort of
repeating routine analyses that the whole infrastructure is built.

1.3.5.8. A Puckish Ref

| Our subjective compulsion to believe in logic only reveals that,
| long before logic itself entered our consciousness, we did nothing
| but introduce its postulates into events:  now we discover them in
| events -- we can no longer do otherwise -- and imagine that this
| compulsion guarantees something connected with "truth".
|
| (Nietzsche, 'The Will to Power', S 521, 282-283).

In a formal inquiry of the sort projected here, the less the discussants
need to depend on the compliance of understanding interpreters the more
they will necessarily understand at the end of the formalization step.

It might then be thought that the ultimate zero of understanding expected
on the part of the interpreter would correspond to the ultimate height of
understanding demanded on the part of the formalizer, but this assumption
neglects the negative potential of misunderstanding, the sheer perversity
of interpretation that our human creativity can bring to bear on any text.

But computers are initially just as incapable of misunderstanding as they
are of understanding.  Therefore, it actually forms a moderate compromise
to address the task of interpretation to a computational system, a thing
that is known to begin from a moderately neutral intitial condition.

1.3.5.9. Partial Formalizations

| It is we who created the "thing", the "identical thing",
| subject, attribute, activity, object, substance, form,
| after we had long pursued the process of making identical,
| coarse, and simple.  The world seems logical to us because
| we have made it logical.
|
| (Nietzsche, 'The Will to Power', S 521, 283).

In many discussions the source context remains unformalized in itself,
taking form only according to the image it receives in one or another
individual MOI.  In cases like these, the step of formalization does
not amount to a total function but is limited to a partial mapping
from the source to the target.  Such a partial representation is
analogous to a sampling operation.  It is not defined on every
point of the source domain but assigns values only to a proper
selection of source elements.  Thus, a partial formalization
can be regarded as achieving its form of simplification in
a loose way, ignoring elements of the source domain and
collapsing material distinctions in irregular fashions.

1.3.5.10. A Formal Utility

| Ultimate solution. -- We believe in reason:
| this, however, is the philosophy of gray concepts.
| Language depends on the most naive prejudices.
|
| (Nietzsche, 'The Will to Power', S 522, 283).

The usefulness of the MOI as the upshot of the formalization arrow is
that it provides discussion with a compact image of the source domain.
In formalization one strives to extract a simpler image of the larger
inquiry, a context of participatory action that one is too embroiled
in carrying out step by step to see as a whole.  Seen in this light,
the purpose of formalization is to identify a simpler version of the
problematic phenomenon or to fashion a simpler image of the difficult
inquiry, one that is well-defined enough and simple enough to assure
its termination in a finite interval of space and time.  As a result,
one of the main benefits of adopting the objective of formalization
is that it equips discussion with a pre-set termination criterion,
or a "stopping rule".

In the context of the recursive inquiry that I have outlined,
the step of formalization is intended to bring discussion
appreciably closer to a solid base for the operational
definition of inquiry.

1.3.5.11. A Formal Aesthetic

| Now we read disharmonies and problems into things
| because we think only in the form of language --
| and thus believe in the "eternal truth" of
| "reason" (e.g., subject, attribute, etc.)
|
| (Nietzsche, 'The Will to Power', S 522, 283).

Recognizing that the Latin word "forma" means not just "form"
but also "beauty" supplies a clue that not all formal models
are equally valuable for a purpose of interest.  There is
a certain quality of formal elegance, or select character,
that is essential to the practical utility of the model.

The virtue of a good formal model is to provide discussion with
a fitting image of the whole phenomenon of interest.  The aim of
formalization is to extract from an informal discussion or locate
within a broader inquiry a clearer and simpler image of the whole
activity.  If the formalized image or precis is unusually apt then
it might be prized as a gnomon or a recapitulation and be said to
capture the essence, the gist, of the nub of the whole affair.

A pragmatic qualification of this virtue requires that the image be
formed quickly enough to take decisive action on.  So the quality of
being a result often takes precedence over the quality of the result.
A definite result, however partial, is frequently reckoned as better
than having to wait for a definitive picture that may never develop.

But an overly narrow or premature formalization, where the nature of
the phenomenon of interest is too much denatured in the formal image,
may result in destroying all interest in the result that does result.

1.3.5.12. A Formal Apology

| We cease to think when we refuse to do so under the constraint of language;
| we barely reach the doubt that sees this limitation as a limitation.
|
| (Nietzsche, 'The Will to Power', S 522, 283).

Seizing the advantage of this formal flexibility makes it possible
to take abstract leaps over a multitude of material obstacles,
to reason about many properties of objects and processes
from a knowledge of their form alone, without having
to know everything about their material content
down to the depths that matter can go.

1.3.5.13. A Formal Suspicion

| Rational thought is interpretation according to a scheme that we cannot throw off.
|
| (Nietzsche, 'The Will to Power', S 522, 283).

I hope that the reader has arrived by now at an independent suspicion that the
process of formalization is a microcosm nearly as complex as the whole subject
of inquiry itself.  Indeed, the initial formulation of a problem is tantamount
to a mode of "representational inquiry".  In many ways this very first effort,
that stirs from the torpor of ineffable unease to seek out any sort of unity
in the manifold of fragmented impressions, is the most difficult, subtle,
and crucial kind of inquiry.  It begins in doubt about even so much as
a fair way to represent the problematic situation, but its result can
predestine whether subsequent inquiry has any hope of success.  There
is very little in this brand of formal engagement and participatory
representation that resembles the simple and disinterested act of
holding a mirror, flat and featureless, up to nature.

If formalization really is a form of inquiry in itself, then
its formulations have deductive consequences that can be tested.
In other words, formal models have logical effects that reflect on
their fitness to qualify as representations, and these effects can
cause them to be rejected merely on the grounds of being a defective
picture or a misleading conception of the source phenomenon.  Therefore,
it should be appreciated that software tailored to this task will probably
need to spend more time in the alterations of backtracking than it will have
occasion to trot out parades of ready-to-wear models.

Impelled by the mass of assembled clues from restarts and refits to the
gathering form of a coherent direction, the inkling may have gradually
accumulated in the reader that something of the same description has
been treated in the pragmatic theory of inquiry under the heading
of "abductive reasoning".  This is distinguished from inductive
reasoning, that goes from the particular to the general, in
that abductive reasoning must work from a mixed collection
of generals and particulars toward a middle term, a formal
intermediary that is more specific than the vague allusions
gathered about its subject and more generic than the elusive
instances fashioned to illustrate its prospective predicates.

In a recursive context, the function of formalization is to relate a
difficult problem to a simpler problem, breaking the original inquiry
into two parts, the step of formalization and the rest of the inquiry,
both of which branches it is hoped will be nearer to solid ground and
easier to grasp than the original question.

1.3.5.14. The Double Aspect of Concepts

| Nothing is more erroneous than to make of
| psychical and physical phenomena the two faces,
| the two revelations of one and the same substance.
| Nothing is explained thereby:  the concept "substance"
| is perfectly useless as an explanation.  Consciousness in
| a subsidiary role, almost indifferent, superfluous, perhaps
| destined to vanish and give way to a perfect automatism --
|
| (Nietzsche, 'The Will to Power', S 523, 283).

This project is a particular inquiry into the nature of inquiry in general.
As a consequence, every concept that appears in it takes on a double aspect.

To illustrate, let us take the concept of a "sign relation" as an example
of a construct that appears in this work and let me use it to speak about
my own agency in this inquiry.  All I need to say about a sign relation
at this point is that it is a three-place relation, and therefore can
be represented as a relational data-base with three columns, in this
case naming the "object", the "sign", and the "interpretant" of the
relation at each moment in time of the corresponding "sign process".

At any given moment of this inquiry I will be participating in a certain
sign relation that constitutes the informal context of my activity, the
full nature of which I can barely hope to conceptualize in explicitly
formal terms.  At times, the object of this informal sign relation
will itself be a sign relation, typically one that is already
formalized or one that I have a better hope of formalizing,
but it could conceivably be the original sign relation
with which I began.

In such cases, when the object of a sign relation
is also a sign relation, the general concept of
a sign relation takes on a double duty:

   1.  The less formalized sign relation is used to mediate the
       present inquiry.  As a conceptual construct, it is not yet
       fully conceived or not yet fully constructed at the moments
       of inquiry being considered.  Perhaps it is better to regard
       it as a "concept under construction".  Employed as a contextual
       apparatus, this sign relation serves an instrumental role in the
       construal and the study of its designated objective sign relation.

   2.  The more formalized sign relation is mentioned as a substantive object
       to be contemplated and manipulated by the proceedings of this inquiry.
       As a conceptual construct, it exemplifies its intended role best if it
       is already as completely formalized as possible.  It is being engaged
       as a substantive object of inquiry.

I have given this inquiry a reflective or recursive cast, portraying it
as an inquiry into inquiry, and one of the consequences of this picture
is that every concept employed in the work will take on a divided role,
double aspect, or dual purpose.  At any moment, the object inquiry of
the moment is aimed to take on a formal definition, while the active
inquiry need not acknowledge any image that it does not recognize
as reflecting itself, nor is it bound by any horizon that does
not capture its spirit.