MATHEMATICS AND DESCRIPTIVE PSYCHOLOGY

by F Richard Singer III         edition date 11/07/07

website: www.conceptualstudy.org      email: richardsinger3@sbcglobal.net

Purpose: The purpose of this paper is to indicate my perspective as a mathematical logician and a descriptive psychologist on the similarities between mathematics and descriptive psychology. My main point is that both are best characterized as ways of thinking. Much of this paper is a condensed revision of parts of my paper The Potential Impact of Descriptive Psychology.

Ways of Thinking: In his preface to Place, Tony Putnam indicates that Ossorio’s work presented special challenges to most readers, who are accustomed to getting a basic understanding of books the first time thru. However a few individuals who had extensive background in extremely rigorous and complex logical systems found his work straightforward and understandable. They read it with the care and attention one gives to the presentation of new material in mathematics, making sure they had a deep grasp of each idea before going on, but also assuming that their understanding of each concept would unfold and deepen over time. Below two are excerpts from this preface. Later I will expand on them.

It is not hard to see why these readers chose the approach they did: there is a mathematical "feel" to Ossorio’s writings that is both pronounced and hard to pin down.

Ossorio’s writings are exactly like mathematics in this way: they richly reward, indeed virtually require, deep study before the sense they make becomes evident. The unusual structure of Place can be recognized as alerting us to the need for a deep and careful approach to understanding it. What kind of understanding will this deep study yield? The parallel with mathematics is instructive here: the test of whether one truly understands mathematical concepts is whether one can use them to do mathematics. That is, one can understand derivations (proofs, problem-solving) using these concepts, and can use them (within the limits of one’s own ability, of course) to create new derivations.

PNDP & PNCM: By a net, I mean a network of related concepts and conceptual distinctions that can be used to think and communicate about some realm of interest. Contemporary Mathematics and Descriptive Psychology are both public nets, i.e. they are used in similar ways by a significant number of people and are designed for use by a wide public. To focus on this I refer to them as PNCM (Public Net for Contemporary Mathematics) and PNDP (Public Net for Descriptive Psychology).

Altho PNDP and PNCM are both public nets, there are many other public nets, so this is only a small part of the story. It is the way they were designed, and how this effects the way that they are public, which allows a similar perspective on both. Unlike most other extensive public nets, PNDP was designed using mature conceptual study. Thus PNDP does not constitute a psychological theory or a theory of any type. It is net, and thus theory neutral. In this it resembles the only other extensive theory neutral public net, PNCM. Being theory neutral makes a net public in a special way. Persons having very different beliefs about the world can use the conceptual tools of such a net. About all they need are some very ordinary beliefs shared by almost everyone. For instance to use PNCM, P needs to believe that some aspect of the world lend themselves to mathematical considerations. Likewise to use PNDP, P needs to believe that persons engage in courses of action and that this can sometimes be described in ways that are useful for some purposes.

To further examine how PNCM and PNDP were developed using mature conceptual study, and why this sets them apart, I introduce some epistemic concepts. I first indicate why I became interested in PNDP.

Like many mathematical logicians, I am interested both in philosophy and mathematics, and hence in the philosophy of mathematics. The evolution of my philosophy of mathematics led me to what I call conceptual philosophy, and my deepest interest in PNDP relates to the connection I see between it and conceptual philosophy. Conceptual Philosophy involves a type of activity during which the want parameter is to enhance a conceptual understanding of our most ubiquitous concepts. Unlike most work

in philosophy, conceptual philosophy focuses on concepts rather than on philosophical theories. Instead of asking about the nature of reality, it studies the most ubiquitous concepts we can use for thinking about what happens in the world. Thus, encountering the reality concepts from PNDP, I considered them as a core part of conceptual philosophy. I also regarded other portions of PNDP as part of a net that I would like to see designed for conceptual philosophy. This net would use the epistemic concepts that I will now sketch in preparation for a further comparison of PNDP and PNCM.

Conceptual vs Paraceptual: That first cousins share a pair of grandparents is a proposition from within our public net for ordinary family relationships. It is conceptual, since it is independent of any state of affairs in the realm of families. A statement such as ‘Bill and Jane are first cousins’ uses this net to tell about a state of affairs that this net is intended to help us think about. Such information is paraceptual in relation to this net. The concept of paraceptual includes what is often called empirical, but I will use the term empirical in a narrower manner that relates to verification. Specifically empirical information is paraceptual information that has been verified by careful or systematic observation or study of some non-conceptual realm.

Conceptual Study: In CS (Conceptual Study), the state of affairs being studied is in some realm that is a net. The want parameter IS guided by a desire to understand some portion of the net, and often by a desire to modify the net. CS also involves a special condition on the performance parameter. In doing CS, the only significant claims are conceptual, i.e. they are made from within the net being studied. Other types of study involve significant paraceptual claims, i.e. claims that apply some portion of a net to some state other than the net. Many study episodes are a mixture of paraceptual study and conceptual study, with the emphasis being on the paraceptual components. Contemporary mathematics is the most prevalent exception in the academic world, with most of the study being conceptual. Of course, CS presupposes some paraceptual knowledge, but to qualify as CS this must only be ordinary reliable knowledge. Without some such knowledge, P would not even know that P was studying. CS also uses the concepts being studied to makes paraceptual remarks, as long as it is understood that this is done merely to add perspective on these concepts.

Evolution of Conceptual Study for a Realm: The division into the following stages from C0 to C3 is a rough attempt to give some linear order to developments in doing CS in relation to a realm R.

C0: CS episodes are brief and not recognized as such. There is little or no attempt to focus directly on concepts apart from paraceptual applications.

C1: Concepts are examined in their own right, with an attempt to clarify them by definition. Conceptual considerations are subordinated to paraceptual ones, with concepts recognized as legitimate only if they represent something considered as existing in R. Axioms may be formulated for R, but are regarded as true of R, rather than as a means of concept clarification.

C2: There is an attempt to formulate all concepts for R, using axioms and definitions or other conceptual methods. Altho paraceptual restrictions are not placed on the concepts in the net, the net is judged by paraceptual considerations outside the net. This is done using various abstract concepts that are intertwined with certain paraceptual beliefs.

C3: At this level, there are extended episodes of CS. Nets are formulated with or without regard to any particular realm of application. Deliberate efforts are made to not favor paraceptual beliefs about any intended realm of application, except perhaps a belief that the net might be useful for thinking about this realm. This level will be referred to as mature CS.

Types of Conceptual Study: Conceptual study IS pre-paraceptual if a noteworthy part of the significance parameter involves becoming better prepared to apply the concepts being studied to paraceptual concerns. If that concern also involves empirical concerns pre-paraceptual study is pre-empirical. Conceptual study is pure if the significance parameter relates in a noteworthy manner mainly

to conceptual concerns. Since significance is can be complex, an episode of conceptual study can be both pure and pre-paraceptual. The study of the metric system in American elementary schools is seldom pre-paraceptual, altho similar study would be in many countries. Altho pre-paraceptual and pure conceptual study differ in the significance parameter, when successful their achievement parameters share a very important feature. Achievement involves shaping tools that are theory neutral.

Mature Conceptual Study PNCM is the most widely known net being developed by mature conceptual study. Most of this study is not pre-paraceptual, since even when it provides concepts for paraceptual use, this is not part of the significance parameter for this type of study. Work in PNCM may be suggested by any human activity but it takes a life of its own that is independent of paraceptual theories or any attempt to mirror our other forms of experience. The tongue-in-cheek toast which says "Here is to pure mathematics, may it never be of use to anyone." and the quip "There can be no dispute between pure and applied mathematicians, they cannot talk to each other" are indicative of the purely conceptual attitude prevalent among pure mathematicians.

Even in mathematics, CS has not always been mature. Pre-Greeks; mathematics was at stage C0. The Greek perspective considered geometry as the study of an ideal space that was more basic than physical space. It was mixed study, with CS as a major secondary unrecognized component at about level C1. The realm was imagined as platonistic rather than personal or natural. Understanding this provides a perspective on what they were attempting, but does not mean that their platonistic realm of ideal space is essential. Hilbert’s work in euclidean geometry was mature CS, and he even describes it as such. He used definitions and axioms without any commitments to this being a theory of either an ideal or a physical space. Mathematical study from then on is mature CS. In the late 19th century, we were mostly at level C2. I base this estimate on some of the prevalent attitudes.

The conceptual part of Men and Women: Partners and Lovers by Mary Kathleen Roberts (Advances in Descriptive Psychology Volume 2) is an excellent example of pre-empirical study. This paper is systematic and well organized. It uses concepts from PNDP to develop other specialized concepts. The conceptual part is a prelude to the empirical part and is clearly separated from the empirical predictions. Similar remarks could be made about other papers from Advances, many of which have conceptual parts that are pre-empirical.

Not only has study within PNDP been mature conceptual study, I think it has primarily been largely pre-empirical. This is a matter of the significance parameter for those doing Descriptive Psychology, and much of the work has been intended as a basis for paraceptual investigation in the behavioral sciences. Given the dominant paradigms in the behavioral science, this is not surprising. Lacking a net like PNDP, there is little recognition of the restrictive nature of the underlying epistemological presuppositions of most of the work being done. Nor is there an understanding of the role that an understanding of ordinary reliable knowledge could play in clarifying or broadening these paradigms.

If there was a public net for conceptual philosophy, and if that net had been developed using mature conceptual study, then this net would share the same features as shared by PNCM and PNDP. My work in conceptual philosophy has been a tentative step in proposing such a net. This has mostly been pure conceptual study, with some pre-paraceptual study, but no pre-empirical study. This is not to say that study in conceptual philosophy cannot be pre-empirical. If philosophy of science was developed using mature conceptual study then this area of epistemics could be explored either as pure or as pre-empirical. If this were done in order to use these concepts in some area of empirical science then it would be the latter.

Axioms and Maxims: In a net some concepts may be so basic that they provide a substrate supporting each other and all the other concepts in the net, altho often only implicitly. Since these concepts lie submerged and since they implicitly support a significant part of the meaning of other concepts used, I refer to them as subconcepts. Clearly a subconcept cannot be explicated via analytic definitions. In PNCM, axioms provide specifications for working with subconcepts. Maxims play a similar role in PNDP, altho excepts from the preface to place indicate more of a difference between axioms and maxims than fits with my perspective on axioms.

Ossorio does not deal in definitions or axioms or postulates, nor does he prove theorems – indeed he vigorously disputes the need for or desirability of using any of these in a fundamental approach to persons and behavior. But equally certainly his writings have the clarity, precision and careful articulation of an interconnected set of concepts that we associate with mathematics. The similarity is even more pronounced in Place, where the form of the work almost invites misunderstanding as a set of axioms or postulates about persons and behavior – a Principia Persona, as it were. Nothing could be further from what Ossorio intends; to see why requires a careful understanding of the difference between axioms and maxims.

Axioms have their place in the realm of "pure reason", maxims in the realm of "practical reason" . Both axioms and maxims "bound" their respective domains; in this structural way they are similar, and any discussion of maxims might well have that familiar mathematical feel as a result. But what they bound, and how, are quite different. Pure reason fundamentally is concerned with establishing "truth" via logical proof; axioms state what is taken to be absolutely true within this domain of reason, thereby establishing the logical "structure" of the domain. Maxims serve a similar function in the domain of practical reason, which is concerned with establishing what is to done in a given situation. Maxims codify our understanding of persons and behavior; as such, they establish a "structure" for what qualifies as an adequate description of behavior in any particular instance.

The Maxims in Place "reflect our competence in regard to the concept of a person, which encompasses the concepts of (a) individual persons, (b) human behavior, (c) the real world, and (d) language." (Ossorio, p. XX) Understanding these Maxims increases our competence in regards to these, which on the face of it is no small matter.

For a formalist, axioms are not true. Rather they are specifications for what can be inserted into a proof, and thus are prescription for how one is proceed in a particular formal system. What a proof establishes is not the truth of a theorem T, but the truth that T is a theorem. For a formalist, truth is in what Curry calls the metatheory and mathematical truth has nothing to do with the content of any mathematical theorems. What is true about an axiom is that it is a theorem and that it has a one line proof.

To look for truth in the theorems of mathematics, at least in the last 100 years, one could be a platonist. For a platonist mathematical statements are about some transcendent universe of sets, unobservable by the senses, but observable by mathematical intuition. Furthermore the only axioms that can be regarded as true are the axioms of set theory. So called axioms-outside-of-set-theory are true only if they can be modeled as theorems in set theory. Why I find platonism unappealing and why I consider the truth of axioms as irrelevant to mathematics is developed in My Net for Understanding Mathematics. I adopt a conceptualist perspective, in which axioms are true in the mundane sense that they state what is true of a math net we are using. In essence, formalists and platonists and others do the same mathematics, and they use the axioms to specify what counts as doing mathematics.

What is different is the scope of PNCM and PNDP and the way that they are developed. The scope of PNDP is potentially everything, while the scope of PNCM is limited to what can be developed axiomatically. Altho the mature conceptual thinking characterizes both, most mathematicians prefer to work with a more limited type of net. In answer to a request by FOM (An Internet group concerned with Foundations Of Mathematics) for information about significant foundational work outside of mathematics, I pointed to Ossorio’s work. This was ignored. All of their discussion centered on work that could be axiomatized.

Comprehensive Paradigms: The most important link that I see between PNCM and PNDP may seem farfetched to most. I see the evolution of CS in mathematics and its expanded use into the more extensive realm of interest for PNDP as an important step towards a potential comprehensive paradigm shift. It is the epistemic and ontic components of PNDP, along with the person related concepts that are central to the new type of net that I hope may emerge. I act from the conjecture that the paradigm shift in mathematics and the paradigm shift suggested by the creation of PNDP could be a prelude to one of the most significant comprehensive paradigm shifts in human history. This would be a shift away from the traditional types of comprehensive paradigms towards a comprehensive net. To explain this I will sketch the concept of the comprehensive paradigm that I have developed in Comprehensive Paradigms.

Altho a communities C’s ordinary net contains all of the concepts C needs for thinking about ordinary knowledge, it may not contain concepts used to think about all of the beliefs in C’s realm of certainty. Thus C will usually have a broader net that is used in relation to cosmic concerns. The beliefs and concepts are likely to be so intertwined that they will normally be used unreflectively. Separating them will not even be considered. This conglomerate, along with some other kinds of commitments, IS a comprehensive paradigm for C. The realm of interest for such a paradigm potentially includes everything. This means that altho it may not explicitly focus attention on various matters, it influences everything that those in C do and think about, and there is no act or thought that cannot be judged by the paradigm. Its concerns are broader than those in a limited paradigm, as are its prescriptive attitudes towards behavior. In the paradigm case indicated below, C is a culture.

(1) The paradigm has sufficient influence to dominate some culture for generations, primarily by being internalized by almost everyone in the culture. It thus provides internal cohesiveness and unity and stability and a sense of security and wellbeing.

(2) The paradigm will be strongly defended if seriously challenged from without, and any adherent who seriously challenges its core components will be severely sanctioned.

(3) The paradigm has a set of primary core beliefs about how things are, and these are considered essential. One of its core components is a preeminent cosmic version, including beliefs about the origin of the universe, the nature of reality, what can and what must exist, humanities place in the universe. Another of its core components is a set of epistemological beliefs that govern acceptable practices for obtaining and verifying what is known and what can be known.

(4) An essential component of its core provides a foundation for the values. This component includes ways for thinking about human activity and criteria for making value judgments about such activity. This has strong prescriptive implications for personal behavior norms, especially those involving principles for behavior. This component is the one most essential to cultural stability. Many social practices and institutions are organized around it, and it provides a rationale and support for them, including means for making judgements about any social practice or institution.

Allowable Transformations:

Change culture in (1) to a subculture or community or even a single person, and even allow different communities to have versions of the same paradigm. Weaken (1) by omitting cohesiveness or unity or stability, or allow its influence to last for a shorter time.

Weaken (2) by changing strongly defended to something like defended, or by changing seriously sanctioned to something like ignored or ridiculed. Even omit (2).

Change beliefs in (3) to considerations, and omit preeminent in respect to a cosmic version or perhaps change version to image or images and epistemological beliefs to epistemic considerations.

Omit (4) or weaken it in any way.

Conceptual Paradigms: A conceptual paradigm is a limiting case of a comprehensive paradigm, obtained by using most of the allowable transformations. It provides a conceptual net and some attitudes for thinking about some realm of interest. It does not contain any essential beliefs about this realm. To be comprehensive, the realm of interest for a conceptual paradigm must include everything. This does not mean that its net contains all the concepts needed to think about everything. It merely means anything considered could involve and be mediated by concepts from its net. Peter Ossorio developed a net now used by The Society for Descriptive Psychology. With some augmentations, I would regard this net as comprehensive.

Traditional comprehensive paradigms emerged in slowly changing cultures, where they functioned as a support for fixed values and as a rationale for rigid social practices and institutions. This may not be a useful function in a more rapidly evolving pluralistic world culture. Values can be acquired and maintained without the support of a comprehensive paradigm. Perhaps all the values needed for social stability would better evolve if they were seen not to be dependent on any particular paradigm. If so traditional types of comprehensive paradigms are not as important in supporting such values as it might seem from examining the role they have previously played. A useful comprehensive paradigm may differ radically from the paradigm case. It would emphasize tolerance, a value that does not depend on any particular paradigm and could be a sufficient stabilizing value if widely adopted. Not being committed to paraceptual beliefs, other than vital knowledge, a comprehensive net could play a central role in a stable pluralistic world culture.