Interrogative Imperative Institute

Mapping Mental Spaces: The process of generating, organizing, interpreting, and evaluating:

-- experience, phenomenology, identifying reference, relationships, characterizations, reflexive consciousness, the interrogative imperative, assumptions, implication, entailment, principles, rules, congruence functions, inference, hypotheses, theories, frames of reference, language, emotion, methodology, measurement, and validity.


Both individually and collectively, the set of statements to the right in the main body of 'Mapping Mental Spaces' constitutes something of a mental exercise. Perhaps, what is most important about this exercise -- as with any such exercise -- is that an individual engage the indicated process and critically reflect on not only what is being said by me but, as well, critically reflect on what is going on within you, the reader, as you work your way through the material.

Whether one agrees or disagrees with what is being expressed through the following material is, in many ways, irrelevant. The object of the exercise is to induce a reader to engage, analyze, question, reflect upon, critique, and improve on (where necessary) the process of mapping mental spaces.

There are no definitive answers given here. There are, however, a lot of possibilities which are presented for consideration.

One cannot read this document like a novel. Any given numbered premise or set of premises may require considerable time and effort, so, the engagement process is best pursued through rigor, diligence, and patience.

The format of 'Mapping Mental Spaces' is, in part, homage to, or an acknowledgment of Tractatus Logico-Philosophicus by Ludwig Wittgenstein. However, there is no one-to-one mapping correspondence between the numbered premises in Mapping Mental Spaces and Wittgenstein-s system of numbering premises in his work.

More than thirty years ago, I engaged the Tractatus. Because there were many issues in Wittgenstein-s work that I considered problematic, Mapping Mental Spaces is, in a sense, something of a response in kind to the Tractatus.

Going through Wittgenstein's exercise induced me to begin thinking about a variety of issues which have continued to haunt the corridors of my mind over the more than three decades that have passed since my initial reading of the Tractatus. Perhaps, the present work may help prompt readers to become involved in a process of a similar nature.

Mapping Mental Spaces - An Excerpt

1. The only point(s) of possible contact between understanding and reality is (are) experience(s).

1.01 Initially, we do not know if this possibility is given expression through an asymptote-like relationship (never quite touching although, in some sense, approaching one another as a limit), a tangential link (touching at only one point), multiple-points of contacts, or if understanding and experience constitute the sum total of reality (with nothing independent of such understanding and experience).

1.0101 The term 'manifold' refers to the structural character of such points of contact.

1.0102 Contact constitutes junctures of engagement, interaction, transaction, or contiguity between that aspect of reality which is capable of experience and those facets of what is which makes experience at such junctures possible.

1.0103 Interaction, engagement, transaction and/or contiguity at the junctures of contact between that which is capable of experience and that which makes experience of such structural character possible gives rise to points or clusters of data which are processed by different dimensions of understanding as information of one kind or another concerning the possible nature or structure of such junctures of contact.

1.01031 The term "identifying reference" is a way of alluding to attentional and intentional dimensions of experience. By attending to a dimension or facet of experience and communicating the nature of that attention to another individual, we seek to inform the other person about some aspect of what we are intending in relation to that to which we are attending. The communication which involves conveying the nature of the link between attending and intending gives expression to the process of identifying reference.

1.01032 The process of identifying reference tends to involve pointing toward, or descriptions of, or attempting to draw attention to, the structural character of various kinds of qualities, properties, states of affairs, contexts, experiences, modalities of consciousness, events, objects, phenomena.

1.01033 The idea of "structural character" refers to the nature of the form, logic, framework, format, pattern, figure, latticework, set of relationships, and/or set of degrees of freedom and constraints, through which a given aspect of experience, or that which makes such experience possible, is given expression or is manifested.

1.011 Solipsism is a perspective which maintains that reality is generated as a function of an individual's states of consciousness and all that can be known are such states and, possibly, the nature of the self which gives rise to them.

1.012 The term "relationship" gives expression to the linkage, connection, interface, association, or affiliation of two or more aspects of experience, understanding, or that which makes experience of a certain structural character possible. There are many kinds of relationships which are possible, ranging from: temporal, to: spatial, logical, dialectical, ecological, moral, causal, conceptual, hierarchical, physical, and spiritual.

1.1 Kant might have been wrong, for, it may be possible, after all, to know things in themselves. However, this may be true, if at all, only to extent that we have the capacity to understand the nature, logic, or structural character of such 'things', and only to the extent that these 'things' are expressed through manifestations which can be experienced.

1.11 The phenomenology of the 'manifold' serves as that realm where understanding, experience, and reality are brought into conjunction with one another. Another way of referring to this 'manifold' is by the term: phenomenological field.

1.111 Phenomenology gives expression to a being's capacity to engage experience in a conscious manner.

1.121 Consciousness is a priori ... that is, all experience presupposes its existence. Indeed, consciousness is the ground through which experience is given expression. One cannot deny the existence of consciousness without affirming the very reality which is being denied.

1.122 Consciousness is the awareness of experience.

1.123 Reflexive consciousness is the awareness of such awareness and that such awareness gives expression to different kinds of experience.

1.124 A phenomenological field is a framework whose structural character gives expression to the presence of awareness or consciousness (basic or reflexive) concerning experience at any 'point' (simple or complex) one cares to examine, test, or challenge within the context of that framework. The lines of force which are manifested in such a field are expressions of the dynamics of experience, awareness, understanding, and the impact, if any, of that which lies beyond the horizons of the phenomenological field but which interacts with and affects, in one way or another, the structural character of that field.

1.125 Neither awareness of experience nor reflexive consciousness can guarantee, in and of themselves, that one's understanding of the nature of that of which one is aware, or that which makes possible that of which one is aware, will be correct or accurate.

1.126 Consciousness may, or may not, be shaped by contingencies which lie beyond present or all future modalities of awareness.

1.127 Experience gives expression to the sum total of an individual's interaction with reality.

1.128 Reality is synonymous with whatever is, together with whatever makes being possible, including the being of that which is capable of experience and understanding, on whatever level.

1.1281 Truth refers to an accurate, correct, or non-distorted reflection of one, or another, dimension or facet of reality or what is.

1.1282 Truth may rarely, if ever, be acquired in an ultimate, absolute, definitive, and all-encompassing manner among human beings.

1.1283 For the most part, and at best, human beings tend to acquire truths in tangential, asymptotic, or limited ways. Furthermore, rather than grasping the truth of the entire realm of being, we tend to grasp, within varying degrees, limited aspects of truth involving this or that dimension or this or that facet of experience and/or that which makes experience of such structural character possible.

1.129 Understanding is the process one uses to try to map out the possible relationship between experience(s) and reality.

1.1291 The nature of understanding is to construct mental spaces or possible worlds and compare the logic or structural character of such spaces and worlds with the logic or structural character of experience.

1.13 A possible world gives expression to hermeneutical space.

1.131 Hermeneutical space is a logical form that is generated through understanding.

1.1312 Logic arises through conscious construction, or appears ready made in awareness, or is a combination of conscious construction and ready-made components that arise from beyond the realms of consciousness.

1.13121 Logic concerns: (a) the structural character of a form or process; and/or (b) the relationships of similarity and difference between, or among, structural characters; and/or (c) the causal, temporal, contiguous, dependent, associative (i.e., correlation), and/or theoretical, linkages which are believed to be operative in and/or among different structural forms and processes.

1.131212 Logic is a way of organizing, arranging, relating, valuing, exploring, traveling, and/or generating the structural character of hermeneutical spaces.

1.1312121 Logic gives expression to the degrees of freedom, constraints, operations, functions, rules, principles, relationships, and laws which govern a given hermeneutical space or which are manifested through such a space.

1.312122 Thinking, reflection, inference, interpolation, extrapolation, implication, induction, deduction, abduction, analogy, insight, conceptualization, abstraction, mapping, questioning, believing, assuming, creativity, language, interpretation, hypothesizing, fantasizing, dreaming, feeling, judgment, analysis, evaluation, critical inquiry, and understanding each gives expression to hermeneutical spaces of one kind or another, and logic seeks to chart the structural character (both static and dynamic) of such spaces.

1.322 An idea or concept is a particular kind of hermeneutical space. The structural character of such a space reflects the nature of the idea or concept. Larger hermeneutical spaces are often constructed or generated using various ideas and concepts as 'points', somewhat akin to the manner in which geometric points are said to give expression to, say, a line.

1.3221 The structural character of ideas and concepts tend to be far more complex than the points of geometry -- even the curved points of Riemann geometry -- but are closer in nature to the latter than the former, since the idea of 'curvature' in Riemann's geometry suggests the possibility of an internal structure of varying degrees of complexity that may alter with circumstances and conditions.

1.3222 Reason is the capacity to grasp the structural character of a given hermeneutical space or to follow and/or predict the flow of artificial and/or natural systems of logic as these are given expression through the structural character of such a system being manifested.

1.3223 What cannot be followed through rational means is either irrational (without logical form or unintelligible or trans-rational (that is, beyond the capacity of reason to grasp but not necessarily without logical form, truth, and/or intelligibility).

1.3224 Methodology is a process of evaluation concerning the nature of understanding, experience, and/or that which makes experience of such structural character possible.

1.32241 Evaluation involves the use of reason, hermeneutical spaces, and various systems of logic to establish the value of various aspects of experience or that which makes experience of such structural character possible.

1.32242 The value of an experience or that which makes an experience of such structural character possible is an expression of the way an individual is assisted to understand, adapt, or benefit, in some manner, through such an experience or through that which makes an experience of such structural character possible.

1.32243 The significance of 'value' may be relative to: a given perspective, an individual, a community, or a reflection of the possibilities inherent in a given facet or dimension of the way things are.

1.133 One of the essential questions at the heart of seeking an understanding is to ask: what might give rise to experience(s) of the structural character that are experienced through consciousness.

1.134 One form of mapping gives expression to operations and processes which seek to chart the structural character of one, or more, hermeneutical spaces.

1.1341 Another form of mapping gives expression to those attempts of understanding to establish relationships of congruence, matching, resonance, reflection, and/or similarity between (among) the logical character of possible worlds being constructed and the logical character of experience(s).

1.1342 A third form of mapping gives expression to operations and processes which seek to establish relationships, connections, and links among the structural character of a given hermeneutical space, a given set of experiences, and various aspects of that which makes experiences of such character possible.

2. Facts constitute a logical space which gives expression to and/or represents and/or describes various dimensions of the character of experience.

2.01 Different kinds of experience may, or may not, give rise to different kinds of facts.

2.1 Facts may accurately reflect the structural character of some facet of experience, but this need not entail their accurately reflecting the structural character of that which makes experience of such character possible.

2.2 Facts require context and interpretation in order for their significance to be evaluated.

2.3 The context of facts is the catalog of experiences out of which such facts arise.

2.4 A fact may be a feeling concerning, a belief about, a reflection on, a description of, a reference to, and/or an insight into some aspect of experience.

2.41 Feelings are certain kinds of modality of relating to, and interacting with, various aspects of experience and/or that which makes experiences of such structural character possible. These modalities are non-rational in nature (which does not necessarily mean they are irrational), varying in intensity with circumstances and conditions, and often underwrite, orient, shape, and direct one's commitments and actions.

2.411 Feelings (emotions) must be tasted or experienced in order to grasp something of the structural character of their nature. Just as one can have only very limited understanding concerning the nature of an orange if one has never seen, touched, smelled, or tasted such a fruit, so, too, one can have only very limited understanding concerning the nature of any given emotion, if one has not experienced that emotion from the inside out, as it were.

2.412 Feelings can both help one to better understand the nature of experience, as well as interfere with one's attempt to understand the nature of experience. In the former case, they are complementary to the use of reason and help bring balance to hermeneutical activities. In the latter case, they are antagonistic to and obstacles for, one's attempt to seek understanding.

2.413 When the presence, or expression, of certain kinds of feelings (emotions) dominates or orients hermeneutical activity in a destructive, problematic, or distorting manner, then, one of the biggest challenges to generating hermeneutical spaces which are congruent with, reflect, or mirror the structural character of various dimensions of reality is to find ways of eliminating, containing, or modulating the presence of such feelings in order to limit the extent of bias and error which affects the construction of heuristically valuable hermeneutical spaces.

2.414 A methodology, belief, idea, or activity has heuristic value when it aids the process of discovery with respect to coming to understand the structural character of some aspect or dimension of experience or that which makes experience of such structural character possible.

2.421 Beliefs give expression to hermeneutical spaces which often are not amenable to proofs but, nonetheless, tend to be concerned with the relationship among understanding, experience, and the nature of that which makes experience of such structural character possible. Beliefs are a way of orienting oneself within phenomenological and hermeneutical space.

2.4211 Beliefs are ideas and/or values to which a hermeneutical commitment, of some kind, has been made - the nature of this commitment is to accept or treat the focus of this commitment as if it were true.

2.42112 Beliefs involve commitments which are considered to have some sort of value to the one holding the commitment.

2.42113 Discussions concerning belief frequently involve descriptions of the structural character of the nature of a given belief, or belief system, together with explorations of the assumptions, evidence, arguments, explanations, consistency, coherency, validity, heuristic value, strengths, lacunae, problems, and questions which are, or might be, associated with such a belief or belief system.

2.42114 The use of data, evidence, arguments, demonstrations, and proofs in conjunction with beliefs or belief systems is often, at best, suggestive or leads to inconclusive results as far as verification of the belief or belief system is concerned.

2.42115 In general, showing a belief or belief system to be untenable or problematic tends to be easier to accomplish than showing either of the foregoing possibilities to be plausible, probable, or true.

2.431 Insight is the capacity of intelligence to understand, to varying degrees, the structural character of some aspect, facet or dimension of experience and/or that which makes experience of such structural character possible.

2.5 The possible worlds of hermeneutical space consist of a series of facts, assumptions, interpretations, beliefs, values, and relationships which are arranged into a structure which give expression to both form and process of a given character - namely, the logical character of that hermeneutical space.

2.6 The logical character of a hermeneutical space gives expression to the principles, rules, laws, possibilities, forces, processes, and/or limitations inherent in such a space.

2.7 Objects are forms of a given logical kind which populate a hermeneutical space.

2.8 The logical kind to which an object gives expression is a reflection of the structural character of the role which such an object plays in a given hermeneutical space.

2.81 The role played by an object is an expression of the principles, rules, laws, possibilities, forces, processes and limitations that are operative in a given hermeneutical space.

2.82 The role played by an object is the locus of manifestation through which the logical character of the hermeneutical space is given expression by means of the convergent interaction of the principles, forces, forms, processes, rules, laws, and so that are inherent in that hermeneutical space at a given point in time and at a given location within that space.

2.83 Time and location are a function of the logical character of a given hermeneutical space.

2.9 Language is a species of hermeneutical space.

2.91 Hermeneutical space may not be coextensive with language.

2.92 Emotion, sensation, dreaming, aptitude, interests, motivation, movement, fantasy, creativity, insight, thinking, and spiritual knowledge may, or may not, be expressible, to varying degrees, in terms of language, but the former are not necessarily reducible to the latter.

2.921 Feeling, sensation, dreaming, aptitude, interests, motivation, movement, fantasy, creativity, insight, thinking, and spiritual knowledge may all take place quite independently of language and, in most cases, predate the appearance of language.

2.922 Making experience a function of, and dependent on, language, is to render the process of language completely amorphous and, therefore, oblique to understanding.

2.923 Sometimes language determines what we feel, sense, dream, like, do, create, think or understand, but sometimes the use of language is directed and shaped by what we feel, sense, dream, like, do, create, think, or understand.

2.924. Language is a way of giving public expression to certain dimensions of experience and hermeneutical spaces concerning such experience.

2.925 Language is a tool which can assist in the construction of hermeneutical spaces, and, in turn, hermeneutical spaces can inform the way(s) in which language is used as a tool.

2.926 Language is one mapping medium, among many, through which understanding, experience, and reality may be probed.

2.927 Language without a conscious operator does not have the capacity, on its own, to serve as tool for helping to construct or map hermeneutical spaces.

2.9271 The syntax and semantics of a language are static entities until brought alive through use within a context of consciousness and understanding.

2.9272 Language serves as a catalyst for the constructing and mapping of hermeneutical spaces by conscious beings of some minimal level of understanding and hermeneutical capability.

2.9273 Language serves as a medium of public analysis and comparison for different modalities of hermeneutical space.

2.93 Among those beings who are capable of experience, some degree of understanding concerning such experience, and who have developed a certain proficiency with language to be able to describe both experience and understanding, are some beings who say that the propositions or statements of language constitute a picture of experience and/or understanding and/or those facets of reality which are given expression at the junctures of contact where experience, understanding, reality come together.

2.931 This tends to lead to the questions: What is the nature of a picture, and do the descriptions of language constitute a picture, and, if so, what kind of a picture'

2.932 There are many kinds of pictures - photographs, holographs, mental images, magnetic resonance imaging, art works, positron emission tomography, cartography, X-rays, optical illusions, radio wave imaging, sketches, dreams, hallucinations, stills, movies, television, and so on.

2.933 All pictures involve a methodology (well-conceived or otherwise) for engaging the junctures of contact which bring experience, understanding, and reality together.

2.934 Methodology is an ordered process of understanding whose purpose is to engage experience and that which makes experience of such structural character possible in order to probe, within the capacity of the methodology to do so, the nature, structure, or logic of the relationship, if any, between these two dimensions of being.

2.935 Pictures are generated through a process which affects the quality and character of the images that are produced, as well as imposes a limiting context on the mode of engagement to which the methodology underlying the picture gives expression.

2.936 Pictures are an interpretive mapping of some given juncture, or set of junctures, in which experience, understanding, and reality come together.

2.937 Interpretive mapping gives expression to a methodology's manner of constructing hermeneutical spaces.

2.938 Pictures are hermeneutical spaces, the contents of which are filled up by the data which is generated through the way the methodology of the picture taking engages experience and that which makes experience of such structural character possible.

2.94 Language, to the extent it constitutes a modality of generating pictures, does so according to the methodological properties of the language in question.

2.941 The methodology inherent in any given language is an expression of the rules and principles of syntax and semantics which differentiate one language from another.

2.9411 The rules of a language establish the boundary conditions which cannot be violated without removing one from the way the given language permits one to communicate with others who use the same language. Linguistic rules are like the motor vehicle codes which govern the operation of motor vehicles within a given locality in order for traffic to move smoothly with as few problems as possible.

2.9412 The principles of a language establish the degrees of freedom through which an individual can move creatively and hermeneutically within a given language in order to adapt the rules and principles of syntax and semantics of that language to one's individual desires to communicate about issues which are either meta-linguistic or extra-linguistic. Linguistic principles are like road maps which show you places to which travel is possible but do not specify where one has to go or what routes one must take in order to arrive at one's desired destination.

2.9413 The rules and principles of a given language's syntax and semantics serve as mapping tools which enable an individual to translate, to whatever extent possible, between personal, extra-linguistic hermeneutical spaces and public linguistic hermeneutical spaces.

2.942 Different languages have varying degrees of flexibility concerning the extent to which the syntax and semantics of such languages are able to serve as vehicles of transmission for forms of thought, logic, creativity, understanding, and, methodology which are extra-linguistic.

2.943 Languages and pictures are similar to the extent that each uses mapping methodologies to link together junctures of contact among experiences, understandings, and that which makes experiences and understandings of such structural character possible.

2.944 Languages and pictures are dissimilar to the extent that their respective methodologies give expression to different sets of rules and principles for linking together junctures of contact among experiences, understandings, and that which makes experiences and understandings of such character possible.

2.945 Methodology -- whether linguistic, pictorial, or other -- does not create, construct, or understand, in and of itself, per se. Rather, methodology establishes the limits (or boundary conditions) and degrees of freedom for what can be created, constructed and/or understood using that form of methodology.

2.946 The value of a given form of methodology -- linguistic or otherwise -- is in direct proportion to the capacity of the set of rules and principles inherent in that methodology to enable an individual to probe the relationship between experience and that which makes experience of such character possible. Through this process of hermeneutical probing, one seeks to establish an understanding that accurately reflects the structural character of that which makes experience of a certain nature possible. The greater this degree of accurate reflection, the greater the heuristic value of the methodology.

2.95 Methodology, language, understanding, hermeneutical space, logic, and mapping are different ways of making reference to the process of creating and constructing epistemological mirrors which are capable of reflecting, with varying degrees of accuracy, the nature of the relationship between experience and that which makes experience of such structural character possible.

2.96 The medium of measurement for reflective accuracy is congruency.

2.961 In mathematics, two geometric figures which can be precisely superimposed on one another are said to be congruent.

2.962 In hermeneutics, two spaces which are being compared are said to be congruent to the extent that one can establish mapping relationships which link aspects of respective facets of being in a way that does not generate more problems and questions than the congruency is capable of demonstrating in the way of mapping relationships of a reflective nature.

2.9621 The greater the degree of congruency between spaces being compared, then, the greater will be the degree to which those spaces will be said to merge horizons.

2.9622 A horizon is an expression of the logical nature of some facet of manifested structure. Horizons are boundaries which tend to differentiate what is within a structure from that which is external to such a structure.

2.96221 However, frequently, horizons are not static but shift with perspective, experience, interpretation, and understanding. Facets of experience which, at one time, may have been considered to be separate and independent, may be discovered, at a later time, to have a relationship which requires one to re-work one's understanding of how to differentiate between what is within a structure and what is external to that structure. Like the physical horizon of landscapes, hermeneutical horizons tend to move with us and are shaped and influenced by the nature of that movement.

2.96222 Horizons may be simple or complex. In other words, the boundary conditions which are given expression through the way horizons differentiate between what is within a given structure, and what is external to that structure, may consist of relatively few elements and/or forms of transaction between the 'internal' and the 'external' realms. On the other hand, such boundary conditions may consist of many facets and dimensions -- both with respect to the number and character of elements, as well in relation to the extent of the transactions which transpire across the boundaries marked by the horizons, thereby making it difficult to determine on which side of the boundary a given phenomenon (whether event, object, process, and so on) falls.

2.96223 Most of us have a considerable backlog of experience with, information about, understanding of, and insight into the process of establishing congruency. More specifically, whenever an individual seeks to translate feelings, experiences, thoughts, beliefs, states of consciousness, and other facets of the phenomenological field into public discourse via a language (spoken, written, signed, mathematical, coded), one goes through a process of trying to create logical spaces through the way we utilize and weave together the syntax and semantics of a given language so that the structural character of this space is congruent with, or accurately reflective of, or able to mirror the structural character of whatever aspect of the phenomenological field one to which one is making identifying reference by means of the language.

2.96224 When there is a mismatch between the structural character of the two hermeneutical spaces (one being: that which is meant, intended, understood, or experienced, and the other being: the language used to describe or convey what is meant, intended, and so on), then, the one who is communicating with someone else tends to amend the character of the syntax and semantics being used to better reflect the meaning or sense one wishes to convey to the recipient of the communication.

2.96225 Similarly, when someone receives communication from another individual, and the recipient does not understand the sense of what is meant or intended by the other individual, then, the recipient tends to use the modality of the interrogative imperative to query various facets of what has been communicated. Here, again, there is a mismatch between hermeneutical spaces -- namely, the understanding of the recipient and the structural character of the linguistic spaces generated by the one who is seeking to communicate about some aspect of the latter individual's phenomenological field. 2.96226 Most of us do not tend to think of these processes of translating between phenomenology and language as instances of congruence operations, but, this is what is transpiring irrespective of whether, or not, we use this term. 2.963 The notion of "spaces" need not be restricted to geometric, mathematical, physical, or material modalities. A "space" is anything which has a logical or structural form of whatever kind.

2.964 Since we don't, yet, know where or how creative, interpretive, epistemological, and/or linguistic processes take place, we do not know what the precise nature of the space is through which these phenomena are given expression. However, what we do know is that all of these processes have a logical form or structure to them. 3.01 There are multiplicities of logical systems.

3.011 Some logical systems are invented or created and other logical systems are given expression through the structural character or nature inherent in some dimension of reality being the way that it is.

3.012 Whether created or natural, logic gives expression to the structural character of the forms and/or processes governing a given facet, aspect, dimension, level, or plane of being.

3.0121 All created systems of logic constitute hermeneutical spaces.

3.01212 Created systems of logic involve a hermeneutical process of mapping which is governed by a set of assumptions, principles, rules, and propositions which are ordered in accordance with the constraints and degrees of freedom permitted by the set of assumptions, principles and rules which constitute the given system of logic.

3.0122 Natural systems of logic involve the manner in which some facet, aspect, dimension, or plane of being is manifested or unfolds over time.

3.0123 When the structural character of a created system of logic reflects the structural character of a natural system of logic, then, congruency exists between the two systems of logic to the extent that the reflection of the latter by the former can be shown to be accurate.

3.1 'Characterization' refers to the process of placing an aspect or dimension of experience within hermeneutical space. Assumption, abstraction, categorization, definition, description, belief, faith, and modeling all give expression, in one way or another, to the process of characterization.

3.11 How we emotionally respond to experience forms an important dimension of the characterization process. Liking, attraction, repulsion, hostility, fear, pleasure, pain, trust, avoidance, and so on are all expressions of characterization.

3.112 Characterization is something human beings, along with various other species of life, do in order to help orient oneself within hermeneutical space. Characterization relates us to experience through the construction, creation, and/or generation of modalities of classification concerning such experience.

3.1121 Different systems of created logic employ a variety of mapping techniques -- included among these are: induction; deduction; analogy; abstraction; dialectic; implication; inference; entailment; tautology; validity; consistency; necessity; coherency; assumptions; possibility; plausibility; correlation; probability; causality; conjecture; interpolation; extrapolation; hypotheses; theory; law; formulae; equations; arguments; evidence; demonstration; proof; description; explanation; belief; insight; models; world-making; frames of reference; paradigms, and world-views.

3.1122 Some of these mapping techniques are applied to one, or another, created system of logic as a means of analyzing and/or evaluating such systems. Some of these techniques are applied to the data of experience in order to either map out the structural character of such experience or to generate maps which are intended to account for how experience of such structural character is possible.

3.1123 Induction is a process that uses some set of data as a basis for generating a conclusion concerning the proposed character of similar instances of data not yet encountered. For instance, if all the swans one has seen are white, one might use this base set of data about swans to conclude that all future instances of swan-encounters are likely, as well, to involve white swans.

3.11231 The risk one runs in using induction is that the conclusion one has formed on the basis of what has been observed or encountered may not be correct. For example, black swans do exist, and, therefore, the belief that all future instances of swan-encounters will involve white swans will fall with the first black swan which is encountered.

3.113 Deduction focuses on the kinds of conclusion one can draw about some facet of experience or about a system of logic given certain information concerning both the nature of that facet of being as well as a background of information about a variety of experiences in general. Such conclusions usually are limited to unpacking or delineating the set of constraints and degrees of freedom which are inherent in the available information. Thus, if I know that human beings are capable of carrying on a conversation, and if I am carrying on a conversation, via a telephone, with a voice that is located elsewhere, then, I might deduce that this other voice belongs to a human being.

3.1131 Conclusions reached through the exercise of deduction concerning a given set of data, propositions, experiences, and so on aren't always correct. For instance, if the voice with whom I having a conversation is part of a complex and sophisticated system of software and hardware which constitutes a framework of artificial intelligence, then, the deduction that the other voice belongs to the human being with whom I am having a conversation may not be warranted. Among other things, one might have to determine whether one could extend the category of human beings to include systems of artificial intelligence before making such a deduction. Moreover, whether such a deduction would, then, be correct might depend on whether, or not, the determination concerning the relationship between human beings and any given system of artificial intelligence is warranted.

3.1132 Interpolation is a form of mapping which inserts or computes intermediate values within a given sequence, series, or set of events, operations, or calculations. These values are believed to be related to the rest of the series or sequence in the same way as the present set of events are related to one another. Interpolation may give expression to either inductive and/or deductive processes.

3.1133 Extrapolation is a form of mapping which seeks to determine or estimate the identity of values which extend beyond the horizons or range of some given set of data, and, yet, retain the structural character of the relationship which links the elements within the known set of data. Extrapolation may consist of induction, deduction, or some combination of the two.

3.114 Mapping techniques involving analogy use the structural features and/or relationships within one context to direct attention to possible similarities of structural character and/or relationship within a different context. For example, rivers and arteries constitute different contexts, but they share a variety of similarities. More specifically, they both: involve liquids; the flow of materials within a delimited framework; pressure; currents; a possibility for transport; are part of a larger ecological system; and so on. One might key in on one, or more, of the foregoing features to establish a relationship of analogy between rivers and arteries for purposes of description, explanation, analysis, modeling, and the like.

3.1141 The value of an analogy depends on both the strength of the similarity which is being proposed with respect to the contexts which have been selected for comparison in this manner, as well as the nature of the purpose for which such an analogy is being established and whether, or not, the similarities are capable of sustaining the purpose for which the analogy has been drawn.

3.1142 An analog is a logical system which purports to reflect the structural character, in some way, of some other logical system -- either artificial or natural. Often times, an analog focuses on the manner in which some other system operates or on the kind of relationships that tend to govern the other system, and, usually, the form of an analog keys in on the idea of using the continuous modulation of one, or more, variables as its manner of establishing congruency with the structural character of that system to which the analog makes identifying reference.

3.115 Abstraction is a process of stripping away the details of a given event, object, phenomenon, experience, process, or context, and so on in order to focus on a limited aspect, facet or dimension of such an event, object, phenomenon, experience, process, or context - often times such abstractions are embodied within systems of symbols (e.g., linguistic, mathematical, logical) that are said to represent, or give expression to, the properties or qualities which have been pared down or abstracted in one way or another.

3.1151 Although thinking about objects, phenomena, events, and so on, in the simplified way made possible through abstraction often helps make analysis, evaluation, exploration, experimentation, and/or gaining insight into such objects, phenomena, or events easier to do, the value of such a process tends to depend on the nature of the abstraction, how such abstractions are used, and remembering that simplified systems cannot hope to manifest all of the qualities, properties, and possibilities inherent in the more complex context from which the abstraction has been extracted. As a result, various kinds of error may be introduced into one's mapping program when using: data, ideas, information, and so on, which have been generated through processes of abstraction.

3.1152 Symbols are often used to signify the presence of certain modalities of abstraction. A symbol is not the same as, or synonymous with, that to which it makes identifying reference but, instead, is part of a system of logic which gives expression to a set of abstractions through which hermeneutical spaces are generated that are intended to establish varying degrees of congruency with certain aspects or dimensions of the structural character of experience, or that which makes experience of such structural character possible.

3.11521 Symbols do not necessarily remove one from the context being explored. Rather, they give expression to characterizations of such contexts -- characterizations from which certain details, themes, and so on of the original context have been removed. Symbols permit one to simply the ways in which hermeneutical spaces are described.

3.115211 Some forms of the foregoing sort of simplification have heuristic value while other forms do not.

3.116 A dialectic is a process of hermeneutical mapping which gives expression to a form of argument that links ideas, events, objects, processes, propositions, phenomena, and/or situations in accordance with some rule or principle or set of such rules and principles. One cannot know the nature of the dialectic involved until one understands the character of the rules and principles being used to shape the linkages among ideas, events, objects, and so on, but, usually, the linkages of a given form of dialectic have to do with the manner in which structural relationships are said to direct the flow of unfolding or manifestation of some given set of ideas, events, objects, and so on.

3.1161 The Hegelian dialectic is different from that of Marx's dialectical materialism, and both of these are different from the dialectic of a Socratic dialogue. Each of the foregoing forms of dialectic uses different sets of rules and principles to establish linkages within their respective systems of thought.

Furthermore, the epistemological value of a given instance of dialectics depends on the extent to which the set of rules and principles shaping the flow of hermeneutical linkages within a given kind of dialectic is capable of reflecting the structural character of the way some aspect, facet, dimension, or plane of being actually operates or is manifested and with respect to which the dialectic is being used as a means of explicating the structural character of the aspect or dimension to which the dialectic is giving reference.

3.117 Implication is a process of mapping which points in the direction of other possibilities being connected or related, in some way, to the context out of which the indication of implication arises. The extent and character of such a connection or relationship depends on the nature of the implication and the possibilities to which the implication is being juxtaposed.

3.1171 For example, if one were to enter into a house and find dinnerware and food on the dining room table, then, this information implies there may be a group of people somewhere, nearby, who are preparing to eat. On the other hand, one may have wandered into a nuclear test site in which an atomic bomb is about to be exploded and the table has been set to see what, if any, effects (both short-term and long-term) may result with respect to such a house that contains a dining room with a table set with food and dinnerware.

3.11711 Implications may be strong, weak, or unwarranted. In the latter case, although someone has proposed that a relationship or connection exists between two contexts, events, processes, and so on, in reality, no such relationship or connection exists.

3.118 Inferences are conclusions drawn by an individual concerning some given set of data or body of information or array of propositions. Such conclusions may be causal, relational, hierarchical, or associational in nature. 3.1181 Inferential conclusions are not always correct or warranted.

3.119 Entailment refers to mapping processes which purport to establish that one fact, proposition, event, phenomenon, idea, context, object, or process supports the truth, validity, reality, or existence of some other fact, proposition, event, phenomenon, idea, context, object or process. The nature and strength of such support will depend on the structural character of the entailment relationship which is being proposed.

3.1191 Similar to mappings that involve processes of inference, implication, dialectic, abstraction, analogy, deduction, and induction, so too, entailment proposals may, or may not, be warranted.

3.120 A tautology is a special form of entailment proposal. According to this kind of mapping technique, if one unpacks or delineates the structural character of some given fact, proposition, state of affairs, context, process, event, phenomenon, or object, then, the truth of a given tautology is contained within the structural character being unpacked or delineated. Tautologies are merely re-statements, in altered form, of what is already known about the structural character of some fact, proposition, or issue.

3.1201 Thus, one might say that a tennis ball is yellow, and, then, go on to say that the ball is round and colored. The latter statement is entailed by the first statement - once one understands the nature of tennis balls in general -- because the latter statement is merely re-stating, in altered form, what is known by means of the first statement, and, therefore, is tautological with respect to the first statement.

3.1202 Tautologies are not necessarily about the nature of what makes the structural character of some given experience possible. Tautologies may be part of artificially constructed logical systems (e.g., models, paradigms, frames of reference, world-view, theories, beliefs) which although true in the context of such logical systems have no reference to anything beyond the horizons of those systems.

3.121 Validity is a mapping operation which focuses on the relationship between a given set of data or information and one, or more, deductions, implications, or entailment proposals that are made in conjunction with that set of data or information. The nature of this relationship concerns the degree to which deductions, conclusions, implications, entailments, and/or inferences are warranted as one moves from a given set of data or information to certain deductions, implications, and so on, involving that set of data. Relationships which are warranted, or follow from, or are evidentially supported tend to be referred to as valid.

3.1211 Determining whether, or not, the aforementioned relationships are warranted, or follow from, or are evidentially supported is not always easy or straightforward.

3.1212 Determining validity within artificially constructed systems of logic tends to be an easier problem to solve than trying to determine the validity of statements involving the relationship between ideas or statements about certain dimensions of experience and that which makes experience of such structural character possible.

3.122 Consistency is one test of validity. In order for a series of ideas, propositions, experiences, understandings and so on, to be consistent with one another, there must not be anything within any of the given ideas, propositions, etc., which contradicts -- in part, or in whole -- any aspect, dimension, or facet of any of the other ideas, experiences, or propositions which are in the set or series being considered. In addition, one must be capable of showing there is some degree of relationship among the ideas, propositions, or experiences that ties together, in some fashion, the various items in the series or set.

3.1221 Unrelated ideas, issues, experiences, events, or propositions are neither consistent nor inconsistent. However, there may be varying degrees of consistency -- depending on how weak or strong the relationship is which is said to tie the set or series of ideas, experiences, events, propositions, and so on, together.

3.123 Coherency is an indication of the internal validity of a system of logic. Coherency refers to the manner in which a hermeneutical space hangs together to serve as an account, story, description, or explanation and, as such, appears to possess few, if any, lacunae or gaps in its structural structure -- gaps which would tend to discredit the possible value of the account, story, description, or explanation.

3.123001 The reliability of a methodology, measurement process, or modality of hermeneutical activity points in several directions. On the one hand, reliability concerns the capacity of, say, a given form of methodology to produce results which are relatively consistent with respect to a given phenomenon under similar conditions of engagement. On the other hand, reliability raises the issue of whether, or not, a given methodology or form of measurement has the capacity to accurately reflect, mirror, or establish congruency with some aspect or dimension of the structural character of some given experience, or that which makes experience of such structural character possible.

3.123002 Replication, confirmation, and verification are all different ways of referring to the issue of reliability in both its inward pointing sense (the first aspect noted above), as well as its outward pointing sense (the second aspect outlined in the foregoing.)

3.124 Necessity gives expression to the way logical systems manifest themselves such that the manifesting could not have been other than what it is. The necessity of artificial and natural systems of logic both are functions of the structural character of such systems.

3.1241 The necessity of artificial systems of logic may not extend beyond the horizons of that system.

3.1241 Necessary conditions refer to those facets of a logical system - whether artificial or natural - which, if not present, will impede something within that system from taking place or being manifested or continuing or proceeding, but, if present, may help provide for the possibility of something transpiring without necessarily guaranteeing such an outcome. Thus, with respect to the lighting of a match - oxygen, a match head with the right composition and quality of sulfur and phosphorus, a minimal degree of dryness, a striking surface of the appropriate properties, and the presence of someone or something to strike the match against such a surface. All of the foregoing conditions are considered necessary since if any of them are absent, the lighting of the match may be impeded, and, yet, if they are all present, there is no guarantee that the match will light since the person or device used to strike the match may not be active, or even if active, the match may not strike the surface in the way which is required for the match to light.

3.125 Assumptions are mapping operations which serve as starting points for exploration, analysis, evaluation, measurement, methodology, and, in general, constructing or creating hermeneutical spaces. Initially, assumptions tend to be not provable but provide one with conceptual direction with respect to subsequent hermeneutical activity and one proceeds 'as if' the assumption were true in order to see where -- conceptually or hermeneutically speaking -- one might journey from such a starting point.

3.1251 Assumptions may, or may not, accurately reflect -- partly or wholly -- the structural character of some aspect, facet, or dimension of experience or that which makes experience of such structural character possible. However, assumptions -- even if not true -- may be utilized for their heuristic value in suggesting possible avenues of hermeneutical consideration that, eventually, may lead to results which do bear on some dimension, facet, or aspect of being in an accurately reflective manner. Thus, the idea of a geometric point which is without dimension does not necessarily have any counterpart in reality, but it serves as a starting point of considerable heuristic value in relation to constructing artificial systems of geometric logic.

3.126 Possibility refers to mapping operations which entertain various facets of a logical system and treat these facets as if they might be true because nothing that is known to be true contradicts such a consideration.

3.1261 Just as experience, belief, understanding, and knowledge change, so too the character of what one will entertain as being possible may also change. However, what one considers possible may, or may not, accurately reflect what, in reality, is actually possible.

3.1262 Plausibility is a mapping operation or process that renders a judgment concerning not only the validity, consistency and coherency of a given hermeneutical space, but, as well, maps out a degree of confidence one might have with respect to whether, or not, such a space may serve as a candidate which has congruency with some given aspect of experience and/or that which makes experience of such structural character possible.

3.1263 The foregoing sort of judgment assigns a value that is greater than mere possibility but less than certainty. Consequently, depending on circumstances, there are many values of confidence which might be assigned to such a judgment, and while all such judgments have some degree of reflective capacity or sense to them, not all such judgments are equally plausible.

3.127 Correlation involves mapping operations which seek to establish the degree to which, say, two objects, events, phenomena, processes, or contexts are manifested, occur, or appear together -- either simultaneously, or contiguously, or sequentially.

3.1271 Correlation says nothing about the structural character of the relationship between such objects, events, phenomena, and so on. Rather, it is a measure of the likelihood that if one encounters one of these objects, events, etc, one also will encounter the other object, event, etc -- whether simultaneously, contiguously, or sequentially. Thus, although night and day have a high degree of correlation, night does not cause day, nor does day cause night, but, instead, both are related to a further set of phenomena concerning, among other things, the rotation of the Earth, the movement of the Sun, the propagation of photons across a vacuum, the dispersion of such photons by the atmosphere of the Earth, and the existence of beings capable of discriminating between light and darkness.

3.128 The idea of randomness is an assumption which alludes to the presence of a principle within reality which says there are no dimensions of hidden variables governing a given system and that the structure of such a system is entirely the result of events and processes that, although caused, are not ordered in accordance with any preexisting pattern which is imposed on those events and processes -- other than the fact that such events and processes having the character which they do.

3.1281 An algorithm is a determinate array of operations which are performed on a body or set of data. Although the array of operations is determinate, the outcome may not be predictable (as in non-linear and chaotic systems) because of the synergy -- both negative and positive -- with which the operations feedback into themselves and the data on which they operate.

3.1282 Randomness is an assumption which can never be proved since there is always the possibility that the series or array or set of events which are being called random is a function of an algorithm whose presence and nature has not, yet, been detected.

3.129 Probability encompasses a variety of artificial systems of logic which seek to assign degrees of likelihood to expectations concerning the way a given system or hermeneutical space will be manifested over time. The manner in which these degrees of likelihood are determined and assigned depends on the structural character of the methodology governing a given framework of probability. Irrespective of the method used, the assumption of randomness is often used to establish base lines against which expectations and outcomes may be compared for purposes of analysis.

3.1291 Probability is a way of modeling certain dimensions of a system -- for example, the likelihood that various kinds of event or process will be given expression at different junctures as the system is manifested during its operations or functioning.

3.1292 As is the case with all models, the value of a given probability framework depends on the tenability of mapping processes such as assumptions, abstractions, deductions, analogs, and so on, which are being used to create the structural character of the hermeneutical space that constitutes a probability model.

3.1293 Statistics is a form of mapping which seeks to quantitatively describe, analyze, organize, and interpret a given body of data and/or information, especially in relation to issues of average, frequency, distribution, distance from some standard feature, correlation, trends, and reliability of such quantitative treatments. Statistics is often used as basis for informing, shaping, and directing various kinds of inductive, deductive, and modeling processes, as well as serving as a possible approach to the interpretation and evaluation of experimental data.

3.1294 Although related, in various ways, to probability frameworks, statistics is a different kind of quantitative description than the latter. However, statistics shares many of the same strengths and weaknesses as do mapping operations involving probability.

3.130 Information refers to the ways in which the structural character of experience is characterized, analyzed, interpreted, and organized. Information does not exist in that which is being characterized, rather the structural nature of the logical form of that which is being explored and delineated through the process of characterization serves as the focus of engagement for various processes, operations, functions, and methods which are artificially generated. Each of the foregoing has its own modality for creating the data that become the points -- simple or complex -- from which the hermeneutical space of some system of logic is constructed.

3.131 Information may, or may not, be accurately reflective -- in part or in whole -- of that to which the information makes identifying reference.

3.132 Objectivity is a process which seeks to eliminate as many sources of bias, prejudice, distortion, undue influence, obfuscation, corruption, misunderstanding, and error from the construction, creation, or generation of hermeneutical spaces in conjunction with both experience, as well as that which makes experience of such structural character possible.

3.1321 Hermeneutical filters are used to process experience, data, information, and so on in a way that emphasizes, or brings out, some features of that experience, etc., while eliminating other facets of such experience. Photographers use various kinds of lenses to filter out certain wavelengths or conditions of lighting. In chemistry, one uses filters to eliminate certain ingredients whose size is larger than the holes of the filter. Audio technicians filter out noise to enhance the quality of sound.

3.13211 All filters have a bias to them which is inclined to some forms, or aspects, of experience, to the exclusion of others, according to the structural character of a filter.

3.13212 Sometimes such biases serve a useful function in conjunction with the quest for objectivity, and sometimes they do not. In either case, one needs to make note of the filters in use and how they shape, color, and orient experience.

3.132121 Calibration is a process that is intended to enable some form of methodology, instrumentation, or hermeneutical activity to function in an optimal way. Being 'optimal' is a function of the capabilities inherent in the given methodology, instrumentation, or hermeneutical activity, together with the skill and artistry of the individuals who are using such methodology, etc.

3.132122 Part of the process of calibration involves establishing, under specified conditions, base lines of performance and outcomes against which subsequent performance and outcomes generated through such methodology, instrumentation and hermeneutical activity can be compared and assigned meaning and significance.

3.132123 A given base line is not necessarily a reflection of the structural character of some aspect or dimension of experience, or that which makes experience possible, which is independent of the base line. Rather, base lines are established in order to give one a place of known properties and conditions from which to operate and through which one can explore, probe, and experiment with various facets of experience.

3.132124 Base lines and calibration are part of a filtering process.

3.132125 Measurement is a process which seeks to quantify the extent to which some aspect or dimension of experience, or that which makes experience of such structural character possible, gives expression to some quality, property, state, activity, value, or feature in which one is interested. Generally speaking, measurement depends on the existence of some kind of standard unit that either remains consistent over time and across conditions, or fluctuates in known, regular ways according to circumstances.

3.132126 Measurement is another kind of filtering process. The properties of this filter will vary with: (a) the modality of measurement; (b) the nature of, and the problems surrounding, the 'standard unit used by a given form of measurement; (c) the extent to which such a modality interferes with the way in which that which is being measured is manifested; (d) the capacity of the modality of measurement to generate relevant data which serve as hermeneutical entry points through which one might gain insight into the structural character of that which is being measured; (e) the degree of resistance inherent in the structural character of that which is to be measured to the modality of measurement being employed (i.e., some modes of measurement are more compatible with certain dimensions of experience, or that which makes experience possible, than are other modes of measurement.

3.132127 Unobtrusive measures are those forms of measurement which do not interfere with, or influence, the way some given phenomenon, event, process, object, condition, state, or the like, is manifested during the time in which the modality of measurement engages such a phenomenon, event, etc..

3.132128 At least since the work of Heisenberg, there has been an awareness that the very act of observing a system, phenomenon, and so on, can alter the way in which the system, phenomenon, etc., is given expression during the process of observation. The nature of such alterations may mask, to varying degrees, the actual character of certain dimensions or facets of the system being observed, and, as a result, affect the quality and accuracy of the hermeneutical spaces generated with the assistance of such processes of observation.

3.132129 Quantifying a given property has at least two aspects. The first aspect is to establish a modality of measurement which is capable of reflecting relevant data concerning such a property. The second aspect involves the mathematical treatment of that data.

3.13212901 Methodology, measurement, quantification, and mathematics do not guarantee that the experience or data which is processed through such means will be understood. As Richard Feynman is reported to have once told a student who was anguishing over the nature, meaning and significance of quantum mechanics - 'Look, no one understands it, just do the calculations."

3.1321291 Relevancy is not a matter of what is of value to a given form of methodology, measurement, or hermeneutical activity. Relevancy is determined by the actual nature, logic, or structural character of that which is being explored.

3.13212911 The ultimate baseline for all methodology and measurement is reality itself.

3.132130 Not all facets or dimensions of experience, and/or that which makes experience of such structural character possible, are amenable to processes of measurement and/or mathematically tractable.

3.133 The interrogative imperative refers to a dimension of human existence that is, on the one hand, rooted in curiosity and the desire to know the truth concerning the nature of experience and/or that which makes experience of such structural character possible. On the other hand, the interrogative imperative is rooted in the awareness that there are many ways in which objectivity can be compromised during the process of engaging, exploring, characterizing, analyzing, interpreting, evaluating, modeling, understanding, and applying experience - such awareness contains the desire to eliminate as many of these kinds of problems as possible.

3.1331 Much of the focus of the interrogative imperative is to determine the extent, if any, to which a claimed insight is possible, plausible, probable, or accurately reflective with respect to that to which the alleged insight makes identifying reference.

3.134 Ockham's razor stipulates that one should not multiply terms, concepts, and assumptions beyond what is necessary to explain or account for a given phenomena. An alternative way of alluding to the same sort of principle is that when comparing two explanations, ideas, assumptions, etc., then, all other things being equal, the simpler of the two is to be preferred.

3.1341 Some of the problems with the foregoing are as follows: what is necessary is often at issue; moreover, 'all other things' often are not equal and how such inequalities affect the process of identifying what is necessary or simpler is not always easily, if at all, capable of being sorted out; in addition, finding reliable measures of simplicity which are independent of the eye of the beholder (i.e., some artificially constructed system of logic) is a complex and difficult process.

3.135 Evidence refers to the set of assumptions, data, information, facts, beliefs, values, judgments, interpretations, understandings, methodologies, mappings, questions, and so on, which have been woven into a framework of reference through which certain kinds of experiences are considered to have some degree of congruency with either an aspect of experience or an aspect of that which makes experience of such structural character possible.

3.136 The manner or modality of weaving together such evidence is often given expression in the form of a mathematical, logical, or rigorous argument, demonstration, proof, or explanation, of some kind. These 'forms' are ways of ordering, structuring, arranging, and/or relating the elements of evidence so that the structural character of such a form may be seen, or understood, to have a certain degree of congruency with the structural character of that to which the form of evidence makes identifying reference.

3.137 Forms of tenable argument, demonstration, proof, or explanation are ones which is capable of standing up under the scrutiny of the interrogative imperative over time.

3.1371 Allegedly tenable arguments, and the like, are not necessarily true, for the value and strength of a given judgment of tenability is dependent on the strength and value of the questions which are asked. If the right questions are not asked, then, a given argument or explanation is only as good as the quality and rigor of the questions which have been raised concerning it ... which may, or may not, be all that good depending on circumstances.

3.1372 Proof can be a relative thing which depends on an individual's acceptance of the assumptions, evidence, arguments, propositions, mapping operations, and conclusions contained in the proof.

3.13721 The fact someone accepts a proof as valid, adequate, consistent, coherent, and so on does not, in and of itself, confirm the proof as true, logical, substantiated, and/or legitimate.

3.137211 Before Riemann and Lobachevski, people generally accepted Euclid's geometric proofs and made the latter the cornerstone of a great deal of subsequent work in both mathematics and science. After the work of the two aforementioned mathematicians, people approached the idea of geometric proof differently.

3.137212 Prior to the time when G�del's notions of incompleteness and inconsistency arrived on the scene, many people regarded the proofs of mathematics as certain and reliable. After G�del, people looked at the idea of proof very differently.

3.13722 The fact most people believe something to have been proven does not, in and of itself, mean the proof is beyond warranted criticism. Similarly, the fact few people believe in a given proof, does not, in and of itself, negate the value of such a proof.

3.137221 Some proofs are entirely about the internal properties of a given system of artificial logic, and have little, if anything, to do with reality beyond the horizons of such a system.

3.137222 Some proofs focus on seeking to determine the structural character of various facets, aspects, or dimensions of experience.

3.137223 Some proofs are concerned with the relationship among understanding, experience, and the nature of that which makes experience of such structural character possible.

3.138 Falsification is an idea introduced by Karl Popper which, in simplified terms, stipulates that while only one contraindication with respect to some given conjecture, hypothesis, principle, or the like, is enough to falsify claims concerning the correctness or truth of such a conjecture or hypothesis, no amount of positive evidence is sufficient to prove the truth of a given conjecture or hypothesis because there is always the possibility that some form of contraindication with respect to such a conjecture or hypothesis might arise in the future.

3.139 Human beings seek out certainty, but, in general, are immersed in uncertainty, unanswered questions, inconclusive evidence, and problematic proofs.

3.140 Hermeneutical spaces can be divided up into linear and nonlinear systems. Linear systems are those that tend to be tractable to mathematical treatment because of the regularity or repetitive nature of the patterns and features to which such a system gives expression The task, then, becomes one of trying to establish some degree of congruency between the structural character of some form of mathematical system of logic and the structural character of the facets of hermeneutical space and/or phenomenology of experience which one seeks to understand. One uses such congruency as the manifold of commonality through which one generates abstractions, models, logical frameworks, and so on, as a basis for mirroring the properties, structure, and logical nature of a given linear system.

3.141 Non-linear systems refer to contexts in which the forms, patterns, and structures to which such systems give expression tend to be irregular in character and oftentimes exhibit anomalous behavior of one kind or another. The properties manifested by such systems over time are said to be self-similar rather than self-same (as in the case of linear systems), and, consequently, such systems are not easily, if at all, tractable through most mathematical systems.

3.142 Non-linear systems are determinate in nature. This means that such systems are governed by a set of principles of identifiable nature, but the systems in question tend to be unpredictable because of the manner in which the various dimensions of the system are extremely sensitive to fluctuations taking place within that system (as well as around the system). Therefore, such systems exhibit complex forms of feed-back loops that are not readily amenable to mathematical treatment, and even when such treatments are available, the latter tend to be limited to very specific contexts and subject to a considerable amount of constant manual adjustments in the formulae and equations of such treatments in order to keep up, somewhat, with the changes being manifested in nonlinear systems.

3.1421 Most of life consists of non-linear phenomena.

3.15 Mathematical formulae and equations are expressions of different facets and dimensions of the structural character of the artificial systems of logic to which they give expression.

3.151 The value of a formula, equation, or set of formulae and equations, lies in the degree of congruency which can be established or exists between the structural character of a formula or equation (or set of them) and the structural character of the aspect of experience to which such mathematical forms make identifying reference in a given context.

3.1511 Mathematical and non-mathematical languages, alike, seek to establish congruency among understanding, experience, and that which makes such experience possible.

3.1512 In some cases mathematical language accomplishes the task of establishing congruency far more precisely and rigorously than non-mathematical languages do. In other instances, the reverse may be true (e.g., in the realms of, say, creativity, love, emotion, morality, spirituality, poetry, identity, justice, faith, art, community, belief, purpose, parenting, psychological therapy, and so on).

3.16 All methodologies are subject to the limitations of incompleteness. In other words, no methodology is self-contained and self-sufficient, but, instead, one must journey beyond the horizons of any given methodology in order to discover the value of that methodology.

3.161 Methodology tends to stand in need of, and presupposes, experience and/or that which makes experience of such structural character possible.

3.162 Although methodology arises out of experience, not all experience is necessarily reducible to such a methodology or capable of being grasped through such a methodology.

3.163 Methodology, like language, and systems of logic in general, does not move itself. They require the presence of consciousness (basic as well as reflexive) and intelligence to invent, generate, create, construct, apply, understand, and critique them.

3.17 Frames of reference, belief systems, hypotheses, theories, models, paradigms, and world-views are the hermeneutical spaces created or constructed by intelligence as it engages experience through the phenomenological field -- which is the point of conjunction of understanding, experience, and that which makes experience of such structural character possible.

3.171 A hypothesis is a conjecture concerning the way in which certain facets of experience, or that which makes experiences of such structural character possible, are related.

3.1712 Oftentimes, the nature of this relationship is expressed in terms of independent and dependent variables.

3.17121 Something is considered an independent variable when: (a) it can change in value under different circumstances, and (b) the value is not affected by changes to the dependent variable with which it is associated by means of the hypothesis.

3.171211 Among various possibilities one might cite, global economics, chaotic systems, and mysticism as tending to suggest that few things in the universe may actually be fully independent of changes elsewhere in a given context or system. As such, there are degrees of relative independence and relative dependence.

3.171212 Causation refers to the idea that the relationship between two events, objects, contexts, states, and so on is governed by the manner in which one pole of the relationship is prior to (both logically and physically), as well as, directs, shapes, orients, alters, transforms, changes, and/or helps give rise to the other pole of the relationship.

3.171213 The interdependent nature of many facets and dimensions of experience and/or that which makes experience of such structural character possible - as is suggested by, among other things: life, Bell's theorem, quantum physics, the stock market, politics, gravitation, education, peace, cybernetics, ecology, jurisprudence, consciousness, intelligence, understanding, illness, and happiness - indicates that isolating something as 'the', or even 'a' cause, may not be a straightforward matter, and may be, in many instances, quite arbitrary.

3.1713 A theory is a belief or set of beliefs concerning the structural character of some facet of experience and/or that which makes experience of such structural character possible.

3.17131 Some theories are more rigorous than others in the sense that the former: (a) tend to be supported by more well-considered evidence than the latter; (b) may be more coherent and consistent; (c) may have been subjected to closer and more exacting scrutiny through the interrogative imperative than have weaker theories; (d) are more likely to be accepted as heuristically valuable guides to subsequent exploration by the prevailing community of experts who deal with such matters; (e) tend to have a more precise, and less problematic, ability to describe and/or account for certain phenomena than do weaker theories.

3.17132 However, rigorously developed, a theory is still a belief system which embodies a certain amount of knowledge and has, within limits, a capacity to accurately reflect various facets of experience and/or that which makes experience of such structural character possible.

3.17133 Hypotheses are used to help confirm or refute various dimensions of a theory by stating issues in a narrow fashion which is both capable of becoming actively operational in the form of testable proposition (or set of them), and, as well, is likely to lead to results that provide data which can serve as evidence to help confirm or refute some aspect of a given theory.

3.17134 Theories rarely stand or fall due to the outcome of a single experiment which is devised to test a given hypothesis. Oftentimes, if experimental results are inconsistent with a particular theory, the theory may be revised or re interpreted in order to accommodate the new data.

3.17135 Theories, however, may come into disfavor as the result of a series of contraindications which arise from experimental data. A certain theory also may come into disfavor because there some other theory, seeking to account for similar phenomena and/or data, which is considered, rightly or wrongly, to be more heuristically valuable, in some sense, than is the previously accepted theory. One theory may gain in general acceptance over a competing theory because of the influence of certain centers of learning in setting hermeneutical trends that tend to propagate such perspectives to the next generation of researchers. The popularity of one theory may increase at the expense of a competing theory due to the politics of hiring and publishing. Finally, one theory may gain in ascendency relative to a competing theory because the proponents of one theory die off, leaving the field relatively clear for another theory to establish itself and begin to flourish through the activity of its still living proponents.

3.17136 A paradigm is a theoretical framework which serves as a work in progress that shapes the methodology, experimentation, interpretation, understanding, politics, and education of those who come under its influence. A paradigm is the hermeneutical filter through which certain facets of experience -- and/or that which makes experience of such structural character possible -- are engaged, processed, and understood.

3.172 Some people argue that one cannot derive 'ought' from 'is'. In other words, just because some dimension of experience, and/or that which makes experience of such structural character possible, has a certain nature does not, in and of itself, necessarily warrant the inference that one ought to behave in certain ways which are said to follow, or are derivable, from experience or things being the way they are.

3.1721 Whether, or not, the foregoing contention is correct really depends on the extent to which some form of 'ought' is inherent in the logical character of that which makes experience possible.

3.17211 If there is a dimension of 'ought' to what is, then, there is a directional potential which is built into being and existence.

3.17212 In one sense there is such a directional component inherent in being -- namely, reality is what it is. If one wishes to have any hope of understanding various facets and dimensions of that reality, then, one ought to seek generating hermeneutical spaces which have a structural character that has congruency with the structural character of the aspect of experience to which identifying reference is being made through the hermeneutical space and/or the structural character of that which makes such experience possible.

3.17213 If there are one, or more, dimensions of ought to being, then, this, in and of itself, does not necessitate what one will choose to do with respect to such an 'ought'. Ought is a suggestion with a certain degree of moral direction and force (or warrant) with which one complies or ignores at one's own risk - just as truth, knowledge, and understanding (of whatever kind, and on whatever level) are hermeneutical vectors with a certain degree of moral direction and force (warrant) with which one complies or ignores at one's own risk - the risk one runs in the latter case is ignorance, misunderstanding, error, bias, or the like.

3.18 The primary task of education is to provide a means for individuals to explore, gain facility with, learn how to critique, and generate (or adopt) useful applications as a result of the capacity, and inclination, of human beings to generate hermeneutical spaces. The essence of this generation process is a function of the interplay of the following processes: identifying reference; characterization; the interrogative imperative; mapping operations; and establishing congruencies.

3.19 As such, facts, per se, are less important than understanding the processes which gave rise to, shaped, colored, and oriented those facts. Information, per se, is less important than grasping the structural character of the processes that generated data of such structural character. Facts and information, together with their perceived value or reliability, often change over time, but the general features of the structural character of generating and evaluating the nature of hermeneutical spaces do not change with time.

3.21 Logic is an expression of the manner in which the different, aforementioned components involved in generating hermeneutical spaces are employed by a given intelligence within the context of engaging the phenomenology of the experiential field in the attempt to understand that which makes experience of such structural character possible.

3.211 There are many kinds of logic and one of the challenges with which all human beings are confronted -- and with which education ought to be concerned -- is to try to discover which system(s) of logic is (are) most congruent with, or reflective of, the structural character of various realms of experience, together with the nature of that which makes experience of such structural character possible.

3.3 Education is a medium for learning about the possibilities, problems, and methods that are associated with trying to understand the logical nature or structural character of hermeneutical spaces which arise in conjunction with various kinds of experience, together with that which makes experiences of such structural character possible.

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