This project is about "the integration of conceptual form". In pursuit of this broad ambition, we need a definition of "concept". While respecting existing theories, our methods are stipulative and synthetic. We are forming an algebraic hypothesis based on a suite of definitions grounded in dimensionality. We are striving to be consistent with computer science, semantic ontology, and cognitive science (psychology). We are trying to produce an interpretation that makes sense in all these contexts, in a format that supports cooperation and interaction among academic and technical sectors around a common body of fundamental definitions. We are observing and noting the role of human behavior and motivation, but we are not developing a theory of "how human beings form concepts", because we believe that "human beings form concepts in many ways", no doubt accommodating aspects of the various theories of concepts, depending on factors like context, culture, skill and education, etc. Intuitive definition of concept Alternative theories of concepts ISO definition of concept Integral theory of concepts Is everything a concept? Hypothesis formation Primitive

Intuitive definition of concept Draft | Back The notion of "concept" is controversial and complex. It is debated by scientists and academics from many points of view. In this project, while respecting the issues that are raised in the academic discussion, we generally follow a "common-sense" approach that honors intuition and the popular and intuitive meaning of the term. At an intuitive and introductory level, we generally follow the explanation offered by Wikipedia. In a more technical sense, we follow the ISO definition of concept: “a unit of knowledge created by a unique combination of characteristics” * * * * * From Wikipedia - https://en.wikipedia.org/wiki/Concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They are understood to be the fundamental building blocks of thoughts and beliefs. They play an important role in all aspects of cognition. As such, concepts are studied by several disciplines, such as linguistics, psychology, and philosophy, and these disciplines are interested in the logical and psychological structure of concepts, and how they are put together to form thoughts and sentences. The study of concepts has served as an important flagship of an emerging interdisciplinary approach called cognitive science. In contemporary philosophy, there are at least three prevailing ways to understand what a concept is: Concepts as mental representations, where concepts are entities that exist in the mind (mental objects) Concepts as abilities, where concepts are abilities peculiar to cognitive agents (mental states) Concepts as Fregean senses (see sense and reference), where concepts are abstract objects, as opposed to mental objects and mental states Concepts can be organized into a hierarchy, higher levels of which are termed "superordinate" and lower levels termed "subordinate". Additionally, there is the "basic" or "middle" level at which people will most readily categorize a concept. For example, a basic-level concept would be "chair", with its superordinate, "furniture", and its subordinate, "easy chair". When the mind makes a generalization such as the concept of tree, it extracts similarities from numerous examples; the simplification enables higher-level thinking. A concept is instantiated (reified) by all of its actual or potential instances, whether these are things in the real world or other ideas. Concepts are studied as components of human cognition in the cognitive science disciplines of linguistics, psychology and, philosophy, where an ongoing debate asks whether all cognition must occur through concepts. Concepts are used as formal tools or models in mathematics, computer science, databases and artificial intelligence where they are sometimes called classes, schema or categories. In informal use the word concept often just means any idea. Thu, Apr 29, 2021

Alternative theories of concepts Sketch | Back The notion of "concept" has been developed in various alternative ways, and the subject remains controversial among academics and scientists. Most of these alternatives have emerged for solid reasons, and represent an important aspect of the theory of concepts. Our objective is to respect these concerns and develop a common model that incorporates the main implications of these various theories. To some degree, each theory represents a perspective, a partial explanation, a point of view from some angle. We want to develop a comprehensively inclusive model that meets the common intuitive defintion, while accommodating the specific concerns and perspectives of the various alternative models. Mon, Mar 15, 2021

ISO definition of concept Polished | Back This message is taken from a discussion on the Ontolog email list, with a few edits, posted March 13, 2021. The International Organization for Standardization (ISO) is an international standard-setting body . https://www.iso.org/obp/ui/#iso:std:iso:1087:ed-2:v1:en:sec:3.8.2 * * * * * ISO definition of concept: “a unit of knowledge created by a unique combination of characteristics” Michael: Bruce, that definition works fine for the purposes that ISO has but I think what you are talking about is something different. It seems to me that you intend "concept" to be a term in some theory about computer science and/or psychology or ??? Yes, I think that is right. I am using the word “concept” as a term in a theory about computer science and psychology. But perhaps you can clarify – because for me, the ISO definition -- “unit of knowledge created by a unique combination of characteristics” – makes sense in both a psychological theory and in a mathematical theory. Michael: If that is the case you need a more rigorous definition of what you mean by concept before you can discuss whether concepts are only in the head or are "real" things in the universe. I.e., by "concept" do you mean: 1) Ways that are [our] mind has to make sense of the world (which typically have very loose context dependent definitions) or 2) Terms in scientific theories meant to have rigorous (i.e., mathematical) definitions such as nucleotide, molecule, mass, etc. I want a definition that satisfies both of your criteria, at the same time – a scientifically precise way to describe anything, including the “loose context-dependent definitions” you mention. I argue that "dimensionality" can accomplish both tasks. But the question of whether concepts are “in the head” or “real things in the universe” seems like a philosophical boondoggle – fundamentally unresolvable, and maybe more ideological than scientific. I prefer to go with the simple direct common-sense intuitive definition that is the opening for Categories and Concepts, by Smith and Medin (below). I’m looking for a rigorous definition of “concept”, probably defined in algebra or in “algebraic-like” structures like trees and hierarchies and sets. I want this definition to be “consistent with empirical psychology” -- but my objective is a mathematical idea, like a coordinate frame that interprets and organizes conceptual structure in general terms. Maybe it forms a bridge between major schools of ontology and their intellectual structures and presumptions. For me, the empirical question “How do human beings form concepts?” can probably be answered by saying “In a lot of ways”. These many ways probably include each of the major schools of concept formation – listed by Smith and Medin in 1981 in their influential book Categories and Concepts as The Classical View The Probabilistic View: Featural Approach The Probabilistic View: Dimensional Approach The Probabilistic View: Holistic Approach The Exemplar View I was very influenced by this book, and years ago scanned their PDF into a word.docx, which I put online at http://originresearch.com/docs/SmithAndMedin.docx Their opening words describe my understanding of concepts and “what a concept is”. Without concepts, mental life would be chaotic. If we perceived each entity as unique, we would be overwhelmed by the sheer diversity of what we experience and unable to remember more than a minute fraction of what we encounter. And if each individual entity needed a distinct name, our language would be staggeringly complex and communication virtually impossible. Fortunately, though, we do not perceive, remember, and talk about each object and event as unique, but rather as an instance of a class or concept that we already know something about. When entering a new room, we experience one particular object as a member of the class of chairs, another as an instance of desks, and so on. Concepts thus give our world stability. They capture the notion that many objects or events are alike in some important respects, and hence can be thought about and responded to in ways we have already mastered. Concepts also allow us to go beyond the information given; for once we have assigned an entity to a class on the basis of its perceptible attributes, we can then infer some of its nonperceptible attributes. Having used perceptible properties like color and shape to decide an object is an apple, we can infer the object has a core that is currently invisible but that will make its presence known as soon as we bite into it. In short, concepts are critical for perceiving, remembering, talking and thinking about objects and events in the world. Based on this very clear and direct and simple statement, I would say that “concepts are generalized names for the elements of human experience”, and I am looking for a general method for describing these elements in mathematical terms – which can include an account of “fuzziness” or ambiguity. I want to respond to your further comments, and then I will explain my approach in terms of dimensionality. There is excellent evidence from work in evolutionary psychology that for concept definition 1, not only are they a feature of our brain, but that to some extent they are innate as well as learned. Note: whenever I use the term "innate" I never mean something is "determined by genes" because by that definition, nothing is innate, even the most basic phenotypes such as vision require appropriate environmental stimuli to develop properly. What counts as evidence for innateness is one or more of the following: 1) The concept seems to be universal across all cultures, including hunter gatherer cultures 2) There is evidence that the concept exists in pre-verbal human infants 3) There is evidence that some form of the concept exists in other primates I’ll have to check on this issue of “pre-verbal concepts”, since I am connecting word-definitions to concepts. I should mention that Gregory Murphy’s book The Big Book of Concepts in chapter 11 on “Word Meaning” generally defines “word meaning” and “concept” as the same idea – but I am also looking at Harvard child-psychologist Susan Carey’s book The Origin of Concepts which does discuss pre-verbal infants. Based on this evidence (I'll list a few of the best sources below) there is strong evidence that concepts such as organic vs non-organic, agency, cause and effect, and contact mechanics are to some extent in our genome. Note: this doesn't mean that the concepts are necessarily false. In fact the hypothesis is that these concepts are innate because they facilitated the reproductive success of our ancestors. However, just because a concept was useful for hunter gatherers doesn't necessarily mean it is a good foundation for science. Science often ends up contradicting common sense. It makes sense to argue that there are primary instincts and capabilities built into the organic structure of human beings. But the mapping between words and ideas and neurons is a vexed and complex subject. The symbolic structures of language are not mapped to neural structures in any innate way. Most of our ideas emerge in an evolutionary developmental process as we learn. Of course if you are referring to 2 that is a different question. I think a lot of discussion about theoretical issues in ontology gets confused because people don't make this distinction and ontologies need to refer to both kinds of concepts, both definition 1 and 2. You mean 1) and 2) in your opening – vague common usage versus scientific precision. I've been thinking of this recently in terms of upper models because with upper models such as BFO it isn't clear to me whether concepts such as "continuant" and "occurrent" are meant to be 1 or 2. I would be interested in knowing what others think about this issue and where to look to find the right BFO papers that clarify this. So, to clarify and simplify – I am not really considering the issue of whether “concepts are innate” – not because I don’t think there are innate tendencies and properties within human beings that some people describe as concepts – but because I want to define concepts as crisp mathematical/semantic structures. So, some people might want to call those innate tendencies and properties “concepts” – but generally, I want to preserve that term for semantic and categorically crisp definitions. I’ll try to explain my dimensional model below, that addresses the question of vague reference versus science, and perhaps has implications for the “innateness” of concepts. This is a good list of references. I found several of them online. Of course, you are listing Douglas Medin, co-author of Categories and Concepts. Evidence for innate psychological concepts: Mapping the Mind: Domain specificity in cognition and culture. Edited by Lawrence A. Hirschfeld and Susan A. Gelman. Cambridge University Press. Premack, David (2003). Original Intelligence: Unlocking the Mystery of Who We Are. With Ann Premack. McGraw-Hill. Medin, Douglas L. (1999) Folkbiology. Edited with Scott Atran. The MIT Press. Kurzban, Robert (2011). Why Everyone (Else) Is a Hypocrite: Evolution and the Modular Mind. Princeton University Press. Kindle Edition. Thank you, Michael. These are good sources. * * * * * Now – here’s what’s happening for me with dimensionality. I started out years ago with a little Atari ST hierarchical outline processing program (called “HippoConcept” – which I guess meant “huge concept”) – and I built a glossary of epistemological terms – just a long list of every word and term I came across in epistemology, and I started working on building a consistent interpretation of these terms so that I could “construct” them from basic primitives. I was looking for building blocks. How are these composite abstractions constructed? I went over and over this list, and the entire structure kept coming down to one idea: dimension. All these words and concepts and terms, it seemed, could all be algebraically constructed from this one basic concept: dimension. It was a compound and recursive idea. “Everything (all terms from epistemology and the theory of categories and classes) is made out of dimensions, and dimensions themselves are made out of dimensions.” This gave me a way to define “qualitative dimensionality” in terms of “quantitative dimensionality”. Qualities have internal dimensional structure – BUT – this is only true under stipulative definitions – what Barry Smith calls “fiat” definitions. So, I can take a very abstract qualitative term like “beauty” and give it a stipulative definition – what I mean and intend by the use of that word, in some particular context – and I can probably say exactly what I mean in quantitative definitions. This process creates a “hierarchical decomposition of the abstract term” in a cascade from broad and perhaps vague or ambiguous terms (“beauty means different things to different people”) to some specific measurements and boundaries – not because these definitions are true in all cases, but because in this particular case, I stipulate these boundaries to define my intended meaning with precision. More or less, I would say, people do this all the time in normal conversations when the need for increased clarity and precision emerges. This led me to a concept I called “synthetic dimensionality” – more or less “a dimension where the values are defined in terms of dimensions”. I discuss that idea here: http://originresearch.com/sd/sd2.cfm This is a complex idea with a lot of moving parts – but putting it simply, what I am doing today is exploring the idea that conceptual definition and structure can not only be useful defined in synthetic dimensionality – but that this entire composite structure – capable of defining “absolutely anything, to any desired degree of accuracy (including both scientific and “vague” ideas)” – can be derived from this single underlying and all-containing ontological structure I am now calling the “Closed Loop”. This includes the definition of numbers, arithmetic definitions, sets and classes, Boolean objects, and any kind of abstraction that can be represented symbolically. Supposedly, this entire structure – “containing all concepts” (as per my definition of concepts) can integrate the entire range of analytic thinking – from the foundations of mathematics and continuum and real number line, to every kind of logic including Boolean algebra, unfolding this entire construction from this one “simple” form of the closed loop. Seen in the flat plane, the closed loop is a top-down hierarchy or taxonomy defined across a range of levels from higher (more abstract) to lower (more specific). The “top” is infinite, the “bottom” is infinitesimal and continuous – and “everything is contained within those bounds”. Now, subject this form to “the twist” – which does not change any of its inherent internal structure -- and now suddenly the infinite and the infinitesimal form a single line, a single boundary, “containing everything”. This is totally mind-blowing. This entire structure – with no loss of topological integrity or precision of definition – collapses into a single line – a single boundary -- maybe even a single point. So, the strategy for proof has to be something like Show that you can construct every categorical definition (quantitative and qualitative) from synthetic dimensions without loss of precision Map this system of definitions into the loop I know this is pretty wild and incredibly ambitious. It’s a rather fabulous construction in abstract synthetic objects like matrix rows with internal cellular structures. And yes, the entire idea might explode into total nonsense. But something incremental keeps pushing it. Can we really lock all conceptual form – in any language – into a single algebraic interpretation? Is this what the ancients meant by “Logos”? This entire thing does feel to me a little like the Big Bang – “you can fold absolutely everything up into a straight line of zero width….” And maybe that is a point, not a line…. Mon, Mar 15, 2021

Integral theory of concepts Sketch | Back We are striving to develop an "integral" theory of concepts, which incorporates the major facets of each of the various main approaches to concept definition. Our definition Integral - combining many facets and schools of thinking Holistic - a whole formed from various alternatives where each alternative contributes something important Stipulative - we assert this definition - we construct it intentionally rather than observe it Synthetic - this is a constructed definition, based on various theories and empirical evidence We are intending our Closed Loop model to be an integral and holistic theory - a theory that contains and embodies definitions for all basic terms and concepts in the theory of categories, in one integrated package - indeed, in one single "concept" Mon, May 10, 2021

Is everything a concept? Sketch | Back Should we say, can we say -- that every mathematical idea in the universe is a "concept"? That every page on a mathematical topic in Wikipedia is describing a "concept"? Is that just pushing it too hard? Well, supposedly we are intending to create all essential mathematical definitions in constructivist terms based on dimensionality and the Closed Loop. Point, Line, Set, Element, Number -- all these terms we argue can be usefully defined in terms of digital geometry in ways to flow out of, that flow from, that can be derived from the Closed Loop. Can every mathematical concept described in Wikipedia be described or defined in these terms? For the moment, pressing ahead, the inclination is to say yes. Tue, Mar 16, 2021

Hypothesis formation Note | Back John Anderson's book Cognitive Psychology suggests hypothesis formation as fundamental to the idea of "concepts", and he goes over many of the ideas that are central to this Closed Loop framework. He uses Venn diagrams as basic to his discussion of induction and deduction. This would go to the "origin of concepts" I note that the word "Peirce" does not appear in the name index of the book -- and for concepts, the subject index says "see schemas" -- but he uses the word and there are good example of the dynamic. Where do concepts come from? He has good clear mathematics-based idea on that. Sat, Apr 17, 2021

Primitive Sketch | Back The Closed Loop is a "constructivist" project. We are "building stuff" - from pieces that we put together to form composite systems and "wholes". One thing we are building is "concepts". What are concepts made out of? A "primitive" or a "primitive notion" is supposedly a lowest-level object or element from which abstract ideas can be/are/should be constructed. Mon, May 10, 2021 Reference In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an axiomatic theory, relations between primitive notions are restricted by axioms. Some authors refer to the latter as "defining" primitive notions by one or more axioms, but this can be misleading. Formal theories cannot dispense with primitive notions, under pain of infinite regress (per the regress problem). For example, in contemporary geometry, point, line, and contains are some primitive notions. Instead of attempting to define them, their interplay is ruled (in Hilbert's axiom system) by axioms like "For every two points there exists a line that contains them both". URL https://en.wikipedia.org/wiki/Primitive_notion What is a concept? Semantics Constructivism Equality