Changes

==Last Draft==

==Last Draft==

===1.5.  Obstacles to the Project : In the Way of Inquiry===

===3.2.  Obstacles to the Project : In the Way of Inquiry===

In this subdivision I address a set of conceptual and methodological obstacles that stand in the way of this inquiry, that threaten to undermine a reasonable level of confidence in the viability of its proceeding, all of which problems I think can be overcome.

The discussion in this Chapter addresses a set of conceptual and methodological obstacles that stand in the way of the current inquiry, threatening to undermine a reasonable level of confidence in the viability of its proceeding, all of which problems I think can be overcome.

Often the biggest obstacle to learning more is the need to feel that one already knows.  And yet there are some things that a person knows, at least, in comparison to other things, and it makes sense to use what one already knows best in order to learn what one needs to know better.  The question is, how does one know which is which?  What test can tell what is known so well that it can be trusted in learning what is not?

Often the biggest obstacle to learning more is the need to feel that one already knows.  And yet there are some things that a person knows, at least, in comparison to other things, and it makes sense to use what one already knows best in order to learn what one needs to know better.  The question is, how does one know which is which?  What test can tell what is known so well that it can be trusted in learning what is not?

One way to test a supposed knowledge is to try to formulate it in such a way that it can be taught to other people.  A related test, harder in some ways but easier in others, is to try to formalize it so completely that even a computer could go through the motions that are supposed to be definitive of its practice.  Both proposals for testing a supposition of knowledge invoke the critical notion of putting knowledge into a form that is communicable or transportable from one system or one medium of interpretation to another.  If the knowledge is conceived to be residing already in one form or another, then this requirement simply points to a "reformation" or a "transformation" of knowledge, otherwise it demands a more radical metamorphosis, from a wholly disorganized condition to an incipiently communicable facility or an initially portable formulation.

One way to test a supposed knowledge is to try to formulate it in such a way that it can be taught to other people.  A related test, harder in some ways but easier in others, is to try to formalize it so completely that even a computer could go through the motions that are supposed to be definitive of its practice.  Both proposals for testing a supposition of knowledge invoke the critical notion of putting knowledge into a form that is communicable or transportable from one system or one medium of interpretation to another.  If the knowledge is conceived to be residing already in one form or another, then this requirement simply points to a "reformation" or a "transformation" of knowledge, otherwise it demands a more radical metamorphosis, from a wholly disorganized condition to an incipiently communicable facility or an initially portable formulation.

====1.5.1.  The Initial Unpleasantness====

====3.2.1.  The Initial Unpleasantness====

Inquiry begins in doubt, a debit of certainty and a drought of information that is never a pleasant condition to acknowledge, and one of the primary obstacles to inquiry can be reckoned as owing to the onus that everyone feels on owning up to this debt.  Human nature vastly prefers to revel in the positive features of the scientific knowledge it already possesses, and the mind defers as long as possible the revolt it feels arising on facing the uncertainties that still persist, the "nots" and "not yets" that as yet it cannot and ought not deny.

Inquiry begins in doubt, a debit of certainty and a drought of information that is never a pleasant condition to acknowledge, and one of the primary obstacles to inquiry can be reckoned as owing to the onus that everyone feels on owning up to this debt.  Human nature vastly prefers to revel in the positive features of the scientific knowledge it already possesses, and the mind defers as long as possible the revolt it feels arising on facing the uncertainties that still persist, the "nots" and "not yets" that as yet it cannot and ought not deny.

====1.5.2.  The Justification Trap====

====3.2.2.  The Justification Trap====

There is a particular type of "justification trap" that a person can fall into, of trying to prove the scientific method by solely deductive means, that is, of trying to show that the scientific method is a good method by starting from the simplest possible axioms, that everybody would accept, about what is good.

There is a particular type of "justification trap" that a person can fall into, of trying to prove the scientific method by solely deductive means, that is, of trying to show that the scientific method is a good method by starting from the simplest possible axioms, that everybody would accept, about what is good.

Sometimes people express their recognition of this trap, and their appreciation of the factor that it takes to escape it, by saying that there is really no such thing as "the scientific method", that the very term "scientific method" is a misnomer and does not refer to any kind of method at all, in sum, that the development of knowledge cannot be reduced to any fixed method because it involves in an essential way such a large component of non-methodical activities.  If one's idea of what counts as method is fixed on the ideal of a deductive procedure, then it is no wonder that one draws this conclusion.

Sometimes people express their recognition of this trap, and their appreciation of the factor that it takes to escape it, by saying that there is really no such thing as "the scientific method", that the very term "scientific method" is a misnomer and does not refer to any kind of method at all, in sum, that the development of knowledge cannot be reduced to any fixed method because it involves in an essential way such a large component of non-methodical activities.  If one's idea of what counts as method is fixed on the ideal of a deductive procedure, then it is no wonder that one draws this conclusion.

====1.5.3.  A Formal Apology====

====3.2.3.  A Formal Apology====

Using "form" in the sense of "abstract structure", I can state that the focus of my interest in this research is limited to the formal properties of inquiry processes.  Among their chief constituents these include all the thinking and unthinking processes that support the ability to learn and to reason.

Using "form" in the sense of "abstract structure", I can state that the focus of my interest in this research is limited to the formal properties of inquiry processes.  Among their chief constituents these include all the thinking and unthinking processes that support the ability to learn and to reason.

The next several subsections enumerate a few of the ways that I plan to make use of this "formal apologetics".

The next several subsections enumerate a few of the ways that I plan to make use of this "formal apologetics".

=====1.5.3.1.  Category Double-Takes=====

=====3.2.3.1.  Category Double-Takes=====

The first use of the formal apology is to rehabilitate certain classes of associations between concepts that would otherwise go down as category mistakes.  This conversion can be achieved in each detailed case by flipping from one side of the concept's dual aspect to the other as the context demands.  Thus it is possible in selected cases to reform the characters of category mistakes in the manner of categorical "retakes" or "double-takes".

The first use of the formal apology is to rehabilitate certain classes of associations between concepts that would otherwise go down as category mistakes.  This conversion can be achieved in each detailed case by flipping from one side of the concept's dual aspect to the other as the context demands.  Thus it is possible in selected cases to reform the characters of category mistakes in the manner of categorical "retakes" or "double-takes".

=====1.5.3.2.  Conceptual Extensions=====

=====3.2.3.2.  Conceptual Extensions=====

The second use of the formal apology is to permit the tentative extension of concepts to novel areas, giving them experimental trial beyond the cases and domains where their use is already established in the precedents of accustomed habit and successful application.

The second use of the formal apology is to permit the tentative extension of concepts to novel areas, giving them experimental trial beyond the cases and domains where their use is already established in the precedents of accustomed habit and successful application.

This serves to dissipate the essential or "in principle" objection that any category distinction puts a prior constraint on the recognition of similar structure between materially dissimilar domains.  As a result, it leaves this issue as a matter to be settled by a post hoc judgment, one that is directed to the question of what fits best "in practice".

This serves to dissipate the essential or "in principle" objection that any category distinction puts a prior constraint on the recognition of similar structure between materially dissimilar domains.  As a result, it leaves this issue as a matter to be settled by a post hoc judgment, one that is directed to the question of what fits best "in practice".

=====1.5.3.3.  Explosional Recombinations=====

=====3.2.3.3.  Explosional Recombinations=====

Another obstacle to inquiry is posed by the combinatorial explosion of questions that can arise in complex cases.  This embarrassment of riches is deceptively deadly to the ends of inquiry in the very measure that it seems so productive at first.  The formalist strategy provides a way to manage this wealth of material diversity by identifying formal similarities among materially different domains, permitting the same formal answer to unify many contentious questions under a single roof, overall reducing the number of distinct topics that need to be covered.

Another obstacle to inquiry is posed by the combinatorial explosion of questions that can arise in complex cases.  This embarrassment of riches is deceptively deadly to the ends of inquiry in the very measure that it seems so productive at first.  The formalist strategy provides a way to manage this wealth of material diversity by identifying formal similarities among materially different domains, permitting the same formal answer to unify many contentious questions under a single roof, overall reducing the number of distinct topics that need to be covered.

=====1.5.3.4.  Interpretive Frameworks=====

=====3.2.3.4.  Interpretive Frameworks=====

Iterations of this recombinatorial process will generate an alternative hierarchy of categories that helps to control the explosion of parts in the domain under inquiry.  If by some piece of luck this alternative framework is uniquely suited to the natural ontology of the domain in question, then it would be advisable to reorganize the whole inquiry along the lines of its topic headings.  However, a complex domain seldom falls out this neatly.  The new interpretive framework will not preserve all the information in the object domain, but typically capture only another aspect of it.  In order to take the maximal advantage of all the different frameworks that might be devised, it is best to quit depending on any one of them exclusively.  Thus, a rigid reliance on a single hierarchy to define the ontology of a given domain passes over into a flexible application of interpretive frameworks to make contact with particular aspects of one's object domain.

Iterations of this recombinatorial process will generate an alternative hierarchy of categories that helps to control the explosion of parts in the domain under inquiry.  If by some piece of luck this alternative framework is uniquely suited to the natural ontology of the domain in question, then it would be advisable to reorganize the whole inquiry along the lines of its topic headings.  However, a complex domain seldom falls out this neatly.  The new interpretive framework will not preserve all the information in the object domain, but typically capture only another aspect of it.  In order to take the maximal advantage of all the different frameworks that might be devised, it is best to quit depending on any one of them exclusively.  Thus, a rigid reliance on a single hierarchy to define the ontology of a given domain passes over into a flexible application of interpretive frameworks to make contact with particular aspects of one's object domain.

====1.5.4.  A Material Exigency====

====3.2.4.  A Material Exigency====

On the other hand, I have cast this project as an empirical inquiry, proposing to represent experimental hypotheses in the form of computer programs.  At the heart of this empirical attitude is a feeling that all formal theories should arise from and bear on experience.

On the other hand, I have cast this project as an empirical inquiry, proposing to represent experimental hypotheses in the form of computer programs.  At the heart of this empirical attitude is a feeling that all formal theories should arise from and bear on experience.

I describe as "empirical" any method that exposes theoretical descriptions of an object to further experiences with that object.

I describe as "empirical" any method that exposes theoretical descriptions of an object to further experiences with that object.

====1.5.5.  A Reconciliation of Accounts====

====3.2.5.  A Reconciliation of Accounts====

The reader may share with the author a feeling of discontent at this juncture, attempting to reconcile the formal intentions of this inquiry with the cardinal contentions of experience.  Let me try to express this difficulty in the form of a question:  What is the nature of the bond between form and content in experience, between the abstract formal categories and the concrete material contents that exist in experience?

The reader may share with the author a feeling of discontent at this juncture, attempting to reconcile the formal intentions of this inquiry with the cardinal contentions of experience.  Let me try to express this difficulty in the form of a question:  What is the nature of the bond between form and content in experience, between the abstract formal categories and the concrete material contents that exist in experience?

My personal definition of mathematical understanding has long been expressed in the chiasmatic figure of speech:  "the form of experience and the experience of form".  This is not the place to argue for the virtues of this concept, but I thought it would clarify a few points to share it here.

My personal definition of mathematical understanding has long been expressed in the chiasmatic figure of speech:  "the form of experience and the experience of form".  This is not the place to argue for the virtues of this concept, but I thought it would clarify a few points to share it here.

====1.5.6.  Objections to Reflexive Inquiry====

====3.2.6.  Objections to Reflexive Inquiry====

Inquiry begins when an automatic routine or a normal course of activity is interrupted, when agents are thrown into a state of doubt about what is best for them to do next and what is really true of their situation.  If this model applies at level of self-application, then an occasion for inquiry into inquiry arises when an ongoing activity of inquiry into any special area becomes obstructed and agents are obligated to initiate a new order of inquiry in order to obviate the problem.  At such moments, agents must acknowledge the higher order of uncertainty that prevails and accept the interruption of a special inquiry in order to examine their accepted conventions and their antecedent convictions about the appropriate conduct of any inquiry at all.  The new order demands that agents pause and reflect on the assumptions embodied in their previous inquiry, criticizing with a deliberate and reconstructive intent aspects of an activity that formerly proceeded through its paces untroubled by any articulate concern.

Inquiry begins when an automatic routine or a normal course of activity is interrupted, when agents are thrown into a state of doubt about what is best for them to do next and what is really true of their situation.  If this model applies at level of self-application, then an occasion for inquiry into inquiry arises when an ongoing activity of inquiry into any special area becomes obstructed and agents are obligated to initiate a new order of inquiry in order to obviate the problem.  At such moments, agents must acknowledge the higher order of uncertainty that prevails and accept the interruption of a special inquiry in order to examine their accepted conventions and their antecedent convictions about the appropriate conduct of any inquiry at all.  The new order demands that agents pause and reflect on the assumptions embodied in their previous inquiry, criticizing with a deliberate and reconstructive intent aspects of an activity that formerly proceeded through its paces untroubled by any articulate concern.

Second, the interruptive character or escapist interpretation of inquiry is especially significant when contemplating programs of inquiry with recursive definitions, like the motivating case of inquiry into inquiry.  It means that the termination criterion for an inquiry subprocess is whatever allows continuation of the calling process.

Second, the interruptive character or escapist interpretation of inquiry is especially significant when contemplating programs of inquiry with recursive definitions, like the motivating case of inquiry into inquiry.  It means that the termination criterion for an inquiry subprocess is whatever allows continuation of the calling process.

====1.5.7.  Empirical Considerations====

====3.2.7.  Empirical Considerations====

The use of computer programs to represent empirical hypotheses brings with it a number of novel considerations about the nature of hypotheses and the status of theories in relation to phenomena.  It forces a re-examination of several issues whose traditional answers have long been taken for granted.

The use of computer programs to represent empirical hypotheses brings with it a number of novel considerations about the nature of hypotheses and the status of theories in relation to phenomena.  It forces a re-examination of several issues whose traditional answers have long been taken for granted.

====1.5.8.  Computational Considerations====

====3.2.8.  Computational Considerations====

=====1.5.8.1.  A Form of Recursion=====

=====3.2.8.1.  A Form of Recursion=====

=====1.5.8.2.  A Power of Abstraction=====

=====3.2.8.2.  A Power of Abstraction=====

Here's a scenario that often occurs.  Inquiry begins with a question that leads to a number of further questions.  After several iterations of this development a sense of despair sets in that the nominal progress of inquiry is doing more to multiply the tension of uncertainty than to clarify its issues.  The only saving grace that rescues the effort comes from noticing that several groups of materially distinct questions have in fact similar forms.

Here's a scenario that often occurs.  Inquiry begins with a question that leads to a number of further questions.  After several iterations of this development a sense of despair sets in that the nominal progress of inquiry is doing more to multiply the tension of uncertainty than to clarify its issues.  The only saving grace that rescues the effort comes from noticing that several groups of materially distinct questions have in fact similar forms.