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Link To #Describe for more about these sections.
mailto: richardsinger3@sbcglobal.net
Materials for conceptual study in mathematics are currently available in the six categories above. For a discussion of the perspective involved, see Manifest Math. The first three mathematical categories are intended primarily as a resource for parents interested in home schooling. The last four sections above involve conceptual study outside of mathematics. Descriptions of the types of materials in these categories are given below.
Most of are materials in these sections are in a continual stage of development and we welcome input from readers. We also welcome materials for conceptual study from anyone sharing our perspective. We can either post them on this site or provide links to sites containing them.
Altho the materials available on
this site are copyrighted,
they are free and may be modified and shared with anyone.
The only restriction on their use is that it must be non-commercial.
The Activity of Conceptual Study Conceptual study is a type of study suggested by a common thread running thru contemporary mathematics and descriptive psychology. Conceptual Study focuses on presenting-clarifying-refining concepts. A conceptual net is a network of concepts and conceptual relationships. Conceptual Study is guided by a one main purpose, namely to understand some portion of a conceptual net. This involves exploring relations between concepts within some net, either to clarify or refine the net or to create a new or alternative one. The goal is to obtain better competence in regards to the net being considered. The concepts and conceptual relations in a net can be used to think and communicate about some realm of interest. The concept of a realm is extremely broad, including both impersonal and personal phenomena.
That first cousins share a pair of grandparents is information about the relationship between concepts used in our public net for ordinary family relationships. Such information is conceptual, since it is independent of any state of affairs in the realm of families. Information such as ‘Bill and Jane are first cousins’ uses this net, but tells about a state this net is intended to help us think about. Such information is paraceptual in relation to this net.
Conceptual information is about concepts and
relationships between concepts within some net.
Paraceptual information presupposes some
net, but is about some particular state of affairs that the net is intended to
help access.
In conceptual study only conceptual claims are made within the net being studied. For instance, the claim that at least one of the parents of anyone who has a first cousin has at least one sibling is a purely conceptual claim within the net mentioned above. Other types of study focus on paraceptual claims, i.e. claims that apply some state of affairs that is not purely conceptual. CS does not yield paraceptual knowledge about any intended realm of application, altho it may form a foundation for organizing and obtaining such knowledge.
Conceptual study also makes use of paraceptual statements about matters that illustrate how concepts can be used, but these are not claims within the net. Normally, these paraceptual statements will seem plausible, but as long as they enhance understanding, whether or not they are correct is irrelevant to the purpose of conceptual study.
Pure CS involves conceptual study of some realm that is a net. Contemporary mathematics uses the most mature form of pure CS. Pure CS is also a central focus of Descriptive Psychology. Descriptive Psychology does not propose any paraceptual theory for psychology. Instead, it provides a net to talk about persons, behavior, states. Furthermore Peter Ossorio, the founder of Descriptive Psychology, deliberately adopts a purely conceptual perspective. More information about descriptive psychology can be found in "My Net for Philosophy".
The perspective that underlies conceptual study is not widespread. Most mathematicians implicitly work as if mathematics is pure conceptual study. However many of them still adopt a platonistic perspective that is irrelevant to the results of their work. In explaining his perspective on Descriptive, Ossorio say that sometimes it is better to make a fresh start, and that to try to explain what he is doing in traditional terms is to encounter what he calls The Tar Baby Problem. Once you touch, you never become unstuck. So the one thing I recommend when reading conceptual papers is do not expect the perspective to be what you would expect elsewhere. As you read, remember no significant paraceptual claims are being made and none are implicit. If there seem to be then this is because communication has failed.
Descriptions of Materials: All materials in these categories can be downloaded either as HTML or as MS Word files.
Conceptual Philosophy: A concept is crucial for a person P if the network of conceptual distinctions and conceptual relationships that P routinely uses would lack coherence without it. A conceptual philosophy for P is an organized conceptual net that centers on those concepts that P finds crucial along with those concepts that these crucial concepts most directly support. Since a conceptual philosophy is a net, it has no theories. With few exceptions, it involves only conceptual claims. For instance, it does involve the non-conceptual claim of being a person with at least some conceptual competence, because doing conceptual philosophy involves presenting-clarifying-refining concepts and relations between concepts. In a personal approach to conceptual philosophy, P would normally illustrate the role concepts might play in P’s thinking and actions. This is one reason that a personal approach to conceptual philosophy will include considerable text that may not seem directly conceptual. Context should allow the reader to distinguish between text focusing directly on concepts and text that add perspective. The core of my work in conceptual philosophy is given in my book entitled A Personal Approach to Conceptual Philosophy. It can be accessed as an htm file or downloaded as a word document. Many of the paper listed in both the conceptual philosophy and in the conceptual papers section are versions or augmentations of some of the material in this book. The conceptual philosophy section of the website also provides access to some other work in conceptual philosophy.
A Personal Approach to Conceptual Philosophy.htm
A Personal Approach to Conceptual Philosophy.doc
Conceptual Papers: This section includes a miscellaneous collection of papers that focus on the study of a variety of concepts and conceptual relationships.
Descriptive Psychology: This section includes a sketch of the concepts from Descriptive Psychology that I have found most useful in my conceptual studies outside of mathematics. In spite of its name, the concepts developed go far beyond those usually associated with psychology, as indicated in something Ossorio said about Descriptive Psychology.
As a pre-empirical conceptual system, Descriptive Psychology provides us with the resources to bring together science and art, religion and the behavioral sciences, history and law, fairy tales and everyday living. It does so in a way that preserves the uniqueness of every domain and individual yet does not leave them isolated from one another. Because it is reflexive and recursive, it is unlimited in its scope and precision. Because it is content-free, it is not culture-bound in the usual sense, and is non-committal with respect to anything empirical: to repeat, it does not "preempt the answers to any questions that could be settled empirically". It is a resource designed to increase our behavior potential, not a way to limit it by imposing a set of theories or a sequence of behaviors, like answers at the back of the book.
Understanding Fractions: Most people learn how to calculate with fractions primarily through drill and practice. Because of this, most adults do not have the kind of understanding of fraction concepts that allows for easy application to a variety of situations. These materials and ideas relate fraction concepts to a simple understanding of counting and ordinary skills in visualization. They are intended primarily for adults who want a better understanding of fraction concept. One purpose might be so they can to help children obtain a foundation for work that they will encounter in calculating with fraction.
Understanding Ordinary Algebra: This section contains a related collection of units on the concepts central to understanding ordinary algebra. Materials are written primarily for adults, with more emphasis on ideas than is usually taught in middle or secondary school. These materials should be especially useful for adults who are interested in helping children understand ordinary algebra.
Enrichment Ideas for Math: This section contains topics that can be used for enriching the study of secondary level mathematics. Some of the units and papers involve ideas that may be subtle, but the mathematical background needed to understanding it involves nothing more than a good understand of ordinary algebra. This section includes are some modifications of some units from the areas of logic and boolean algebra and abstract with commentary on there use for enriching the understanding of ordinary algebra
Boolean Algebra: This section includes a
collection of units on boolean algebra, including applications to digital devices
and to solving logic puzzles. Some units designed for enrichment ides for
students as they study ordinary algebra.
Other units are suitable for math majors or math graduate unit
Mathematical Logic: This section includes a system of units on mathematical logic. It provides a variety of materials for relating the fundamental concepts of contemporary mathematical logic to concepts that are more manifest
Abstract Algebra: This section includes a collection of abstract algebra units that focus on groups and rings. Some units focus on the relationship between abstract algebra and number theory. All units are developed as manifest resources for mathematicians and students of mathematics, i.e. examples are explored prior to general results from which they follow as special cases.
mailto: richardsinger3@sbcglobal.net
