The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One (...) particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. (shrink)

The Martin-Steel coarse inner model theory is employed in obtaining new results in descriptive set theory. $\underset{\sim}{\Pi}$ determinacy implies that for every thin Σ 1 2 equivalence relation there is a Δ 1 3 real, N, over which every equivalence class is generic--and hence there is a good Δ 1 2 (N ♯ ) wellordering of the equivalence classes. Analogous results are obtained for Π 1 2 and Δ 1 2 quasilinear orderings and $\underset{\sim}{\Pi}^1_2$ determinacy is (...) shown to imply that every Π 1 2 prewellorder has rank less than $\underset{\sim}{\delta}^1_2$. (shrink)

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely (...) descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL are measurable. John Steel's analysis also settled a number of structural questions regarding HODL, such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals. (shrink)

§0. Preface. There has been an expectation that the endgame of the more tenacious problems raised by the Los Angeles ‘cabal’ school of descriptive set theory in the 1970's should ultimately be played out with the use of inner model theory. Questions phrased in the language of descriptive set theory, where both the conclusions and the assumptions are couched in terms that only mention simply definable sets of reals, and which have proved resistant to purely (...) descriptive set theoretic arguments, may at last find their solution through the connection between determinacy and large cardinals.Perhaps the most striking example was given by [24], where the core model theory was used to analyze the structure of HOD and then show that all regular cardinals below ΘL are measurable. John Steel's analysis also settled a number of structural questions regarding HODL, such as GCH.Another illustration is provided by [21]. There an application of large cardinals and inner model theory is used to generalize the Harrington-Martin theorem that determinacy implies )determinacy.However, it is harder to find examples of theorems regarding the structure of the projective sets whose only known proof from determinacy assumptions uses the link between determinacy and large cardinals. We may equivalently ask whether there are second order statements of number theory that cannot be proved under PD–the axiom of projective determinacy–without appealing to the large cardinal consequences of the PD, such as the existence of certain kinds of inner models that contain given types of large cardinals. (shrink)

One of the standard ways of proving the consistency of additional hypotheses with the basic axioms of an axiom system is by the construction of what may be described as ‘inner models.’ By starting with a domain of individuals assumed to satisfy the basic axioms an inner model is constructed whose domain of individuals is a certain subset of the original individual domain. If such an inner model can be constructed which satisfies not only the (...) basic axioms but also the particular additional hypothesis under consideration, then this affords a proof that if the basic axiom system is consistent then so is the system obtained by adding to this system the new hypothesis. This method has been applied to axiom systems for set theory by many authors, including v. Neumann, Mostowski, and more recently Gödel, who has shown by this method that if the basic axioms of a certain axiomatic system of set theory are consistent then so is the system obtained by adding to these axioms a strong form of the axiom of choice and the generalised continuum hypothesis. Having been shown in this striking way the power of this method it is natural to inquire whether it has any limitations or whether by the construction of a sufficiently ingenious inner model one might hope to decide other outstanding consistency questions, such as the consistency of the negations of the axiom of choice and continuum hypothesis. In this and two following papers we prove some general theorems concerning inner models for a certain axiomatic system of set theory which lead to the result that as far as a fairly large family of inner models are concerned this method of proving consistency has been exhausted, that no essentially new consistency results can be obtained by the use of this kind of model. (shrink)

In this paper we continue the study of inner models of the type studied inInner models for set theory—Part I.The present paper is concerned exclusively with a particular kind of model, the ‘super-complete models’ defined in section 2.4 of I. The condition of 2.4 and the completeness condition 1.42 imply that such a model is uniquely determined when its universal class Vmis given. Writing condition and the completeness conditions 1.41, 1.42 in terms of Vm, we may (...) state the definition in the form:3.1. Dfn.A classVmis said to determine a super-complete model if the model whose basic notions are defined by,satisfies axiomsA, B, C.N. B. This definition is not necessarily metamathematical in nature. If desired, it could be written out quite formally as the definition of a notion ‘SCM’ thus:whereψ is the propositional function expressing in terms ofUthe fact that the model determined byUaccording to 3.1 satisfies the relativization of axioms A, B, C. E.g. corresponding to axiom A1m, i.e.,,ψ contains the equivalent term. All the relativized axioms can be similarly expressed in this way by first writing out the relativized form and then replacing ‘ϕ bywhich is in turn replaced by, and similarly replacing ‘ϕ’ by ‘ϕ’ by ‘), andThusψ is obtained in primitive notation. (shrink)

In this third and last paper on inner models we consider some of the inherent limitations of the method of using inner models of the type defined in 1.2 for the proof of consistency results for the particular system of set theory under consideration. Roughly speaking this limitation may be described by saying that practically no further consistency results can be obtained by the construction of models satisfying the conditions of theorem 1.5, i.e., conditions 1.31, 1.32, 1.33, (...) 1.51, viz.:This applies in particular to the ‘complete models’ defined in 1.4. Before going on to a precise statement of these limitations we shall consider now the theorem on which they depend. This is concerned with a particular type of complete model examples of which we call “proper complete models”; they are those complete models which are essentially interior to the universe, those whose classes are sets of the universe constituting a class thereof, i.e., those for which the following proposition is true:The main theorem of this paper is that the statement that there are no models of this kind can be expressed formally in the same way as the axioms A, B, C and furthermore it can be proved that if the axiom system A, B, C is consistent then so is the system consisting of axioms A, B, C, plus this new hypothesis that there exist no proper complete models. When combined with the axiom ‘V = L’ introduced by Gödel in this new hypothesis yields a system in which any normal complete model which exists has for its universal class V, the universal class of the original system. (shrink)

We discuss some of the relationships between the notion of "mutual stationarity" of Foreman and Magidor and measurability in inner models. The general thrust of these is that very general mutual stationarity properties on small cardinals, such as the ℵns, is a large cardinal property. A number of open problems, theorems, and conjectures are stated.

Assuming $V=L+AD$, using methods from inner model theory, we give a new proof of the strong partition property for ${\sim}{ \delta }^{2}_{1}$. The result was originally proved by Kechris et al.

We consider the possible complexity of the set of reals belonging to an inner model M of set theory. We show that if this set is analytic then either 1M is countable or else all reals are in M. We also show that if an inner model contains a superperfect set of reals as a subset then it contains all reals. On the other hand, it is possible to have an inner model M (...) whose reals are an uncountable Fσ set and which does not have all reals. A similar construction shows that there can be an inner model M which computes correctly 1, contains a perfect set of reals as a subset and yet not all reals are in M. These results were motivated by questions of H. Friedman and K. Prikry. (shrink)

We introduce and consider the inner-model reflection principle, which asserts that whenever a statement \varphi(a) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model W \subset A. A stronger principle, the ground-model reflection principle, asserts that any such \varphi(a) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These (...) principles each express a form of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed \Pi_2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. (shrink)

We extend the theory of “Fine structure and iteration trees” to models having more than one Woodin cardinal.

An inner model operator is a function M such that given a Turing degree d, M is a countable set of reals, d M, and M has certain closure properties. The notion was introduced by Steel. In the context of AD, we study inner model operators M such that for a.e. d, there is a wellorder of M in L). This is related to the study of mice which are below the minimal inner model (...) with ω Woodin cardinals. As a technical tool, we show that the alternative fine structure theory developed by Mitchell and Steel for mice with Woodin cardinals is equivalent to the traditional fine structure theory developed by Jensen for L. (shrink)

In this paper, we sketch the development of two important themes of modern set theory, both of which can be regarded as growing out of work of Kurt Gödel. We begin with a review of some basic concepts and conventions of set theory.§0. The ordinal numbers were Georg Cantor's deepest contribution to mathematics. After the natural numbers 0, 1, …, n, … comes the first infinite ordinal number ω, followed by ω + 1, ω + 2, …, ω (...) + ω, … and so forth. ω is the first limit ordinal as it is neither 0 nor a successor ordinal. We follow the von Neumann convention, according to which each ordinal number α is identified with the set {ν ∣ ν α} of its predecessors. The ∈ relation on ordinals thus coincides with <. We have 0 = ∅ and α + 1 = α ∪ {α}. According to the usual set-theoretic conventions, ω is identified with the first infinite cardinal ℵ0, similarly for the first uncountable ordinal number ω1 and the first uncountable cardinal number ℵ1, etc. We thus arrive at the following picture:The von Neumann hierarchy divides the class V of all sets into a hierarchy of sets Vα indexed by the ordinal numbers. The recursive definition reads: ;Vλ = ∪v<λVv for limit ordinals λ. We can represent this hierarchy by the following picture. (shrink)

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad (...) lines of development. (shrink)

There are two standard ways to establish consistency in set theory. One is to prove consistency using inner models, in the way that Gödel proved the consistency of GCH using the inner model L. The other is to prove consistency using outer models, in the way that Cohen proved the consistency of the negation of CH by enlarging L to a forcing extension L[G].But we can demand more from the outer model method, and we illustrate (...) this by examining Easton's strengthening of Cohen's result:Theorem 1. There is a forcing extensionL[G] of L in which GCH fails at every regular cardinal.Assume that the universe V of all sets is rich in the sense that it contains inner models with large cardinals. Then what is the relationship between Easton's model L[G] and V? In particular, are these models compatible, in the sense that they are inner models of a common third model? If not, then the failure of GCH at every regular cardinal is consistent only in a weak sense, as it can only hold in universes which are incompatible with the universe of all sets. Ideally, we would like L[G] to not only be compatible with V, but to be an inner model of V.We say that a statement is internally consistent iff it holds in some inner model, under the assumption that there are innermodels with large cardinals. (shrink)

We consider the following question of Kunen: Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M $\longrightarrow$ V) imply Con (ZFC + ∃ a measurable cardinal)? We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper (...) class of almost Ramsey cardinals". Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

Let κ be a cardinal, and let H κ be the class of sets of hereditary cardinality less than κ ; let τ (κ) > κ be the height of the smallest transitive admissible set containing every element of {κ}∪H κ . We show that a ZFC-definable notion of long unfoldability, a generalisation of weak compactness, implies in the core model K, that the mouse order restricted to H κ is as long as τ. (It is known that some (...) weak large cardinal property is necessary for the latter to hold.) In other terms we delimit its strength as follows: Theorem Con(ZFC+ω2-Π 1 1-Determinacy) ⇒ ⇒Con(ZFC+V=K+∃ a long unfoldable cardinal ⇒ ⇒Con(ZFC+∀X(X # exists) + ‘‘ $\forall D \subseteq \omega_1 D$ is universally Baire ⇔ ∃r∈R(D∈L(r)))’’, and this is set-generically absolute). We isolate a notion of ω-closed cardinal which is weaker than an ω1-Erd\ os cardinal, and show that this bounds the first long unfoldable: Theorem Let κ be ω -closed. Then there is a long unfoldable ł<κ. (shrink)

We consider the following question of Kunen: Does Con imply Con? We use core model theory to investigate consequences of the existence of such a j : M $\rightarrow$ V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of "there exists a proper class of almost Ramsey cardinals". Conversely, if On is Ramsey, then such a j, M are definable. We construe this as a (...) negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points. (shrink)

According to the math tea argument, there must be real numbers that we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions. And yet, the existence of pointwise-definable models of set theory, in which every individual is definable without parameters, challenges this conclusion. In this article, I introduce a flexible new method for constructing pointwise-definable models of arithmetic and set theory, showing furthermore that every countable model of Zermelo-Fraenkel ZF set (...) theory and of Peano arithmetic PA has a pointwise-definable end extension. In the arithmetic case, I use the universal algorithm and its Σ n generalizations to build a progressively elementary tower making any desired individual a n definable at each stage n, while preserving these definitions through to the limit model, which can thus be arranged to be pointwise definable. A similar method works in set theory, and one can moreover achieve V = L in the extension or indeed any other suitable theory holding in an inner model of the original model, thereby fulfilling the resurrection phenomenon. For example, every countable model of ZF with an inner model with a measurable cardinal has an end extension to a pointwise-definable model of ZFC + V = L[μ]. (shrink)

Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$ , there is a supertransitive inner model of Zermelo containing all ordinals in which for (...) every λ A λ = {α ∣Φ(λ, a)}. (shrink)

Working in Z + KP, we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula $\Phi$ such that for any sequence $\langle A_{\lambda} \mid \lambda \text{a limit ordinal} \rangle$ where for each $\lambda, A_{\lambda} \subseteq ^{\lambda}2$, there is a supertransitive inner model of Zermelo containing all ordinals in which for every $\lambda (...) A_{\lambda} = \{\alpha \mid\Phi\}$. (shrink)

We construct divergent models of $\mathsf {AD}^+$ along with the failure of the Continuum Hypothesis ( $\mathsf {CH}$ ) under various assumptions. Divergent models of $\mathsf {AD}^+$ play an important role in descriptive inner model theory; all known analyses of HOD in $\mathsf {AD}^+$ models (without extra iterability assumptions) are carried out in the region below the existence of divergent models of $\mathsf {AD}^+$. Our results are the first step toward resolving various open questions concerning the length (...) of definable prewellorderings of the reals and principles implying $\neg \mathsf {CH}$, like $\mathsf {MM}$, that divergent models shed light on, see Question 5.1. (shrink)

Aristotle is oftentimes viewed through a strictly philosophical lens as heir to Plato and has having introduced logical rigor where an emphasis on the theory of Forms formerly prevailed. It must be appreciated that Aristotle was the son of a physician, and that his inculcation of the thought of other Greek philosophers addressing health and the natural elements led to an extremely broad set of biologically- and medically-related writings. As this article proposes, Aristotle deepened the fourfold theory of (...) the elements with anatomic and physiologic observations. In books like History of Animals, Parts of Animals, and the lost Anatomai, he actively dissected organisms and recorded his findings. The corpus of medically-related literature Aristotle developed had a direct influence on subsequent Greek thinkers, including Galen, and on Medieval Islamic and modern Western practitioners such as William Harvey. Aristotle's ideas continue to influence modern medical thinkers in Europe and America through both his interpretation by European philosophers gaining the attention of medical humanists, and his writings' enduring impact on medical researchers balancing scientific with more personalistic approaches to medicine. (shrink)

Abstract.Since the gene splicing debates of the 1980s, the public has been exposed to an ongoing sequence of genetic and reproductive technologies. Many issue areas have outcomes that lose track of people's inner values or engender opposing religious viewpoints defying final resolution. This essay relocates the discussion of what is an acceptable application from the individual to the societal level, examining technologies that stand to address large numbers of people and thus call for policy resolution, rather than individual fiat, (...) in their application. A major source of guidance is the “Genetic Frontiers” series of professional dialogues and conferences held by the National Conference for Community and Justice from 2002 to 2004. Genetic testing, human gene therapy, genetic engineering of plants and animals, and stem cell technology are examined. While differences in perspective on the beginning of life persist, a stepwise approach to the examination of genetic testing reveals areas of general agreement. Stewardship of life, human co‐creativity with the divine, and social justice help define the bounds of application of genetic engineering and therapy; compassionate care plays a major role in establishing stem cell policy. Active, sustained dialogue is a useful resource for enabling sharing of religious values and crystallization of policies. (shrink)

Earlier than the Arabic-Latin transfer of Ptolemaic astronomy via the Iberian peninsula, a serious occupation with Arabic astronomy by Latin scholars took place in crusader Antioch in the first half of the twelfth century. One of the translators of Arabic science in the East was Stephen of Pisa, who produced a commented Latin version, entitled Liber Mamonis, of Ibn al-Haytham’s cosmography, On the Configuration of the World. Stephen’s considerations about the physical universe in relation to the doctrines of Ptolemaic astronomy (...) have hitherto received but little attention. The present paper discusses Stephen of Pisa’s treatment of the planetary spheres in regard to Ptolemy’s theory of oscillating deferents. Emphasis is given to geometric arguments in Stephen’s criticism of Ibn al-Haytham’s spherical model of the inner planets and to Stephen’s own attempt at an improved theory based on additional spheres. The paper argues that astronomical studies in Antioch were of an advanced level, involving independent judgement as well as an influence of contemporary trends in Arabic astronomy. (shrink)

The estrangement between genetic scientists and theologians originating in the 1960s is reflected in novel combinations of human thought (subject) and genes (investigational object), paralleling each other through the universal process known in chaos theory as self-similarity. The clash and recombination of genes and knowledge captures what Philip Hefner refers to as irony, one of four voices he suggests transmit the knowledge and arguments of the religion-and-science debate. When viewed along a tangent connecting irony to leadership, journal dissemination, and (...) the activities of the "public intellectual" and the public at large, the sequence of voices is shown to resemble the passage of genetic information from DNA to mRNA, tRNA, and protein, and from cell nucleus to surrounding environment. In this light, Hefner's inquiry into the voices of Zygon is bound up with the very subject matter Zygon covers. (shrink)

We apply molecular code theory to a rule-based model of the human inner kinetochore and study how complex formation in general can give rise to molecular codes. We analyze 105 reaction networks generated from the rule-based inner kinetochore model in two variants: with and without dissociation of complexes. Interestingly, we found codes only when some but not all complexes are allowed to dissociate. We show that this is due to the fact that in the kinetochore (...) model proteins can only bind at kinetochores by attaching to already attached proteins and cannot form complexes in free solution. Using a generalized linear mixed model we study which centromere protein can take which role in a molecular code . By this, associations between CENPs and code roles are found. We observed that CenpA is a major risk factor while CenpQ is a major protection factor . Finally we show, using an abstract model of copolymer formation, that molecular codes can also be realized solely by the formation of stable complexes, which do not dissociate. For example, with particular dimers as context a molecular code mapping from two different monomers to two particular trimers can be realized just by non-selective complex formation. We conclude that the formation of protein complexes can be utilized by the cell to implement molecular codes. Living cells thus facilitate a subsystem allowing for an enormous flexibility in the realization of mappings, which can be used for specific regulatory processes, e.g. via the context of a mapping. (shrink)

This study aimed to explore the role of two models of well-being in the prediction of psychological distress during the COVID-19 pandemic, namely PERMA and mature happiness. According to PERMA, well-being is mainly composed of five elements: positive emotions, engagement, relationships, meaning in life, and achievement. Instead, mature happiness is understood as a positive mental state characterized by inner harmony, calmness, acceptance, contentment, and satisfaction with life. Rooted in existential positive psychology, this harmony-based happiness represents the result of living (...) in balance between positive and negative aspects of one's life. We hypothesized that mature happiness would be a more prominent protective factor during the present pandemic than the PERMA composite. A total of 12,203 participants from 30 countries responded to an online survey including the Depression Anxiety Stress Scale (DASS-21), the PERMA-Profiler, and the Mature Happiness Scale-Revised (MHS-R). Confirmatory factor analyses indicated that PERMA and mature happiness were highly correlated, but nonetheless, they represented two separate factors. After controlling for demographic factors and country-level variables, both PERMA Well-being and MHS-R were negative predictors of psychological distress. Mature happiness was a better predictor of stress, anxiety, and general distress, while PERMA showed a higher prediction of depression. Mature happiness moderated the relation between the perceived noxious effects of the pandemic and all markers of distress (depression, anxiety, stress, and total DASS-21). Instead, PERMA acted as a moderator in the case of depression and stress. These findings indicate that inner harmony, according to the mature happiness theory, is an essential facet of well-being to be taken into consideration. The results of this study can also orient policies aimed to alleviate the negative effects of the pandemic on mental health through the promotion of well-being. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Four Philosophical Models of the Relation Between Theory and PracticeEstelle R. JorgensenSince music education straddles theory and practice, my purpose is to sketch the strengths and weaknesses of four philosophical models of the relationship between theory and practice. I demonstrate that none of them suffices when taken alone; each has something to offer and its own detractions. And I conclude with four suggested ways in which (...) the analysis can be helpful to music teachers and those interested in their work.Clarifying what is meant by theory and practice raises a nest of conceptual problems. Concerning theory, it may be tempting to equate the terms "theory" and "philosophy." Among their shared properties, these terms deal with conceptual and abstract entities that, from the Enlightenment at least, have been held to be distinct from the phenomenal world. They also clarify ideas and distinguish one concept from another. Among their different purposes, the object of philosophy is the asking of questions in a search for wisdom or truth whereas theory is formulated for explanatory purposes, especially within the frame of scientific propositions that can be refuted through empirical observations. The tests of philosophy are, therefore, those of symbolic logic evidenced in principles such as [End Page 21] internal consistency, correspondence with evidence (deductive, inductive, or analogical), and coherence within a unity or whole. Although logical criteria may also be important in formulating and testing theory,1 it is tested and refuted primarily through means of qualitative descriptions and quantitative assessments of observable phenomena.Regarding practice, one is also confronted with an ambiguous construct that may be interpreted descriptively (what is done in the phenomenal world or taken to be the general state of affairs) or normatively (what is desirable or ideal). There is also the distinction between the practice of a seasoned performer and practice in the sense of striving towards the mastery of a competent and expert practice. In this regard, Vernon Howard distinguishes between "practice" (or the skills exemplified by proficient or expert performers) and "practise (or acquiring the level of mastery of proficient or expert performance)."2 For him, the ambiguous and paradoxical nature of artistic practice highlights the complexity of degrees of attaining mastery of skills or exemplary practice and the diverse array of skills ranging from habit to critical thinking involved in the practice of any art. Although each practice is distinguished by its own belief systems, rituals, expectations, and commonsense, boundaries between differing practices are sometimes "soft" or fuzzy, as one seeps into another.3 Consequently, it is often difficult to distinguish one from another at the margins where one practice becomes another.DichotomyThe notion of theory and practice as independent or a dichotomous relationship between the phenomenal world on the one hand and the mind and spirit on the other has its roots in the ancient world. For Plato, the world of "appearances," referring to the phenomenal world understood by imagination (eikasia) and belief (pistis), should be distinguished from the higher, abstract "intelligible world" grasped through thinking (dianoia), knowledge (episteme), and intelligence (noesis).4 René Descartes' later bifurcation of the inner and outer worlds and inversion of Platonic values and understandings resulted in the ascendancy of careful and deliberate observation of the phenomenal world over philosophical reflection.5 His ideas helped to set the stage for the Enlightenment and the Industrial, Scientific, and Information Revolutions. Empiricism, science, and technology gradually displaced philosophy as primary ways of knowing the phenomenal world and understanding the nature of mind, human cognition, feeling, and being. Making hard-boundaried distinctions between within and without one's self, the phenomenal world and the world of spiritual things, and theory and practice were important insights. These distinctions provided for limits on the sorts of ways in which truth could be known. Reasoned deliberation [End Page 22] and imaginative thought have particular usefulness in the world of theory while sensory perception and careful observation are especially useful in the phenomenal world of practice. So while the scholar might speculate about phenomena, observational tools are also useful in coming to know and understand them. Reason and observation each have particular strengths, and it ought not be assumed that philosophy or scientific observation... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Four Philosophical Models of the Relation Between Theory and PracticeEstelle R. JorgensenSince music education straddles theory and practice, my purpose is to sketch the strengths and weaknesses of four philosophical models of the relationship between theory and practice. I demonstrate that none of them suffices when taken alone; each has something to offer and its own detractions. And I conclude with four suggested ways in which (...) the analysis can be helpful to music teachers and those interested in their work.Clarifying what is meant by theory and practice raises a nest of conceptual problems. Concerning theory, it may be tempting to equate the terms "theory" and "philosophy." Among their shared properties, these terms deal with conceptual and abstract entities that, from the Enlightenment at least, have been held to be distinct from the phenomenal world. They also clarify ideas and distinguish one concept from another. Among their different purposes, the object of philosophy is the asking of questions in a search for wisdom or truth whereas theory is formulated for explanatory purposes, especially within the frame of scientific propositions that can be refuted through empirical observations. The tests of philosophy are, therefore, those of symbolic logic evidenced in principles such as [End Page 21] internal consistency, correspondence with evidence (deductive, inductive, or analogical), and coherence within a unity or whole. Although logical criteria may also be important in formulating and testing theory,1 it is tested and refuted primarily through means of qualitative descriptions and quantitative assessments of observable phenomena.Regarding practice, one is also confronted with an ambiguous construct that may be interpreted descriptively (what is done in the phenomenal world or taken to be the general state of affairs) or normatively (what is desirable or ideal). There is also the distinction between the practice of a seasoned performer and practice in the sense of striving towards the mastery of a competent and expert practice. In this regard, Vernon Howard distinguishes between "practice" (or the skills exemplified by proficient or expert performers) and "practise (or acquiring the level of mastery of proficient or expert performance)."2 For him, the ambiguous and paradoxical nature of artistic practice highlights the complexity of degrees of attaining mastery of skills or exemplary practice and the diverse array of skills ranging from habit to critical thinking involved in the practice of any art. Although each practice is distinguished by its own belief systems, rituals, expectations, and commonsense, boundaries between differing practices are sometimes "soft" or fuzzy, as one seeps into another.3 Consequently, it is often difficult to distinguish one from another at the margins where one practice becomes another.DichotomyThe notion of theory and practice as independent or a dichotomous relationship between the phenomenal world on the one hand and the mind and spirit on the other has its roots in the ancient world. For Plato, the world of "appearances," referring to the phenomenal world understood by imagination (eikasia) and belief (pistis), should be distinguished from the higher, abstract "intelligible world" grasped through thinking (dianoia), knowledge (episteme), and intelligence (noesis).4 René Descartes' later bifurcation of the inner and outer worlds and inversion of Platonic values and understandings resulted in the ascendancy of careful and deliberate observation of the phenomenal world over philosophical reflection.5 His ideas helped to set the stage for the Enlightenment and the Industrial, Scientific, and Information Revolutions. Empiricism, science, and technology gradually displaced philosophy as primary ways of knowing the phenomenal world and understanding the nature of mind, human cognition, feeling, and being. Making hard-boundaried distinctions between within and without one's self, the phenomenal world and the world of spiritual things, and theory and practice were important insights. These distinctions provided for limits on the sorts of ways in which truth could be known. Reasoned deliberation [End Page 22] and imaginative thought have particular usefulness in the world of theory while sensory perception and careful observation are especially useful in the phenomenal world of practice. So while the scholar might speculate about phenomena, observational tools are also useful in coming to know and understand them. Reason and observation each have particular strengths, and it ought not be assumed that philosophy or scientific observation... (shrink)

From climate change forecasts and pandemic maps to Lego sets and Ancestry algorithms, models encompass our world and our lives. In her thought-provoking new book, Annabel Wharton begins with a definition drawn from the quantitative sciences and the philosophy of science but holds that history and critical cultural theory are essential to a fuller understanding of modeling. Considering changes in the medical body model and the architectural model, from the Middle Ages to the twenty-first century, Wharton demonstrates (...) the ways in which all models are historical and political.0Examining how cadavers have been described, exhibited, and visually rendered, she highlights the historical dimension of the modified body and its depictions. Analyzing the varied reworkings of the Holy Sepulchre in Jerusalem-including by monumental commanderies of the Knights Templar, Alberti's Rucellai Tomb in Florence, Franciscans' olive wood replicas, and video game renderings-she foregrounds the political force of architectural representations. And considering black boxes-instruments whose inputs we control and whose outputs we interpret, but whose inner workings are beyond our comprehension-she surveys the threats posed by such opaque computational models, warning of the dangers that models pose when humans lose control of the means by which they are generated and understood. Engaging and wide-ranging, 'Models and World Making' conjures new ways of seeing and critically evaluating how we make and remake the world in which we live. (shrink)

Coherence and correspondence are classical contenders as theories of truth. In this paper we examine them instead as interacting factors in the dynamics of belief across epistemic networks. We construct an agent-based model of network contact in which agents are characterized not in terms of single beliefs but in terms of internal belief suites. Individuals update elements of their belief suites on input from other agents in order both to maximize internal belief coherence and to incorporate ‘trickled in’ elements (...) of truth as correspondence. Results, though often intuitive, prove more complex than in simpler models (Hegselmann & Krause 2002, 2006; Grim, Singer, Reade & Fisher 2015). The optimistic finding is that pressures toward internal coherence can exploit and expand on small elements of truth as correspondence is introduced into epistemic networks. Less optimistic results show that pressures for coherence can also work directly against the incorporation of truth, particularly when coherence is established first and new data is introduced later. (shrink)

nature of personality and the structure of personality are distinguished, and the thesis that mainstream personality theories in psychology debate structure but not nature is illustrated with definitions. Mainstream theories assume that person ality is an inner determinant of behavior, but according to views in psychiatry, phenomenology and radical behaviorism the nature of personality is transactional. The theory of reinforcement schedules proposes general mechanisms of transac tions, and phenomenology gives particular transactions meaning. Interactionism, which locates personality in the (...) person and the situation, contributes measurement to transactionalism but its paradigmatic status is ambiguous because it implies an internal storage system for behavior. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability (...) calculus. The previous attempts all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of ℂ replaced by ℤ₂. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability (...) calculus. The previous attempts all required the brackets to take values in ℤ₂. But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T⊆U, the natural definition is <S|T>=|S∩T| (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell's Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over ℂ and QM/Sets over ℤ₂. (shrink)