This article focuses on how young children acquire concepts for exact, cardinal numbers. I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey. In this framework, the (...) counting list and the counting routine form a placeholder structure. Over time, the placeholder structure is gradually filled in with meaning to become a conceptual structure that allows the child to represent exact numbers A number system is a socially shared, structured set of symbols that pose a learning challenge for children. But once children have acquired a number system, it allows them to represent information that they had no way of representing before. (shrink)

We present a new logic-based approach to the reasoning about knowledge which is independent of possible worlds semantics. \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document} is a non-Fregean logic whose models consist of propositional universes with subsets for true, false and known propositions. Knowledge is, in general, not closed under rules of inference; the only valid epistemic principles are the knowledge axiom Kiφ → φ and some minimal conditions concerning common knowledge in a group. Knowledge is explicit (...) and all forms of the logical omniscience problem are avoided. Various stronger epistemic properties such as positive and/or negative introspection, the K-axiom, closure under logical connectives, etc. can be restored by imposing additional semantic constraints. This yields corresponding sublogics for which we present sound and complete axiomatizations. As a useful tool for general model constructions we study abstract versions of some 3-valued logics in which we interpret truth as knowledge. We establish a connection between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document} and the well-known syntactic approach to explicit knowledge proving a result concerning equi-expressiveness. Furthermore, we discuss some self-referential epistemic statements, such as the knower paradox, as relaxations of variants of the liar paradox and show how these epistemic “paradoxes” can be solved in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document}. Every specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\in_K}$$\end{document}-logic is defined as a certain extension of some underlying classical abstract logic. (shrink)

This paper sets out to accomplish two goals. First, drawing on the Irish perspective, it reconceptualises one of the enduring legacy-based harms of epistemic colonisation, in this case, ‘linguicism’, in terms of ‘hermeneutical injustice’. Second, it argues that otherwise well-meaning attempts to combat epistemic colonisation through the inclusion of marginalised testimony can, in certain circumstances, lead to cases of ‘epistemic exploitation’, which, in turn, can result in ‘ontic burnout’. Both linguicism and epistemic exploitation, this paper theorizes, have the potential to (...) bring about ontic burnout. Here is the plan: to set the scene, section I briefly surveys some key arguments supporting the call for higher education institutions (HEIs) to engage in epistemic decolonising, focusing on the importance of striking a balance between two extremes: (i) abandoning all forms of Eurocentric/colonial knowledge through the elimination of inherited thinking and conceptual schemes which result from colonization, in particular, those which remain in our thinking “owing to inertia rather than to our own reflective choices” (Wiredu 2002, p.56); (ii) the other—pursuing a more positive project where we seek to, “exploit as much as is judicious the resources of indigenous conceptual schemes” (Wiredu 1996: p.136). From here, it moves to consider harms associated with a subset of ‘epistemicide’, that is, ‘linguicism’—a phenomenon where the language and culture of the colonizer become part of the ‘civilising project’ and, in more extreme cases of linguicide, forcibly replaces indigenous languages. Section II then explores the subtleties of harmful by-products deriving from ‘inclusion’, namely that which scholars refer to as ‘epistemic exploitation’. Situations where marginalised scholars are finally ‘at the table but still on the menu’ arise when privileged persons, compel marginalised knowers to educate them [and others] about the nature of their oppression (Berenstain, 2016). Habitual requests to turn themselves inside out for emancipatory principles give rise to a phenomenon known as ontic burnout (Dunne and Kotsonis, 2022, p.9)—“a form of dissociative explanatory fatigue culminating in oppressed persons engaging in a conscious or unconscious decoupling from their socially constructed identity, from the burdens and injustices associated with being a member of a certain social kind”. This harm is, I argue, rarely discussed in the literature and worthy of further analysis. (shrink)

The natural epistemology of W. V. Quine has not been well understood. Critics argue that Quine's scientific approach to epistemology is circular and fails to be normative, yet these criticisms tend to be based on the very presuppositions concerning language, theory, and epistemology that Quine is at pains to reject or alter. ;Quine's views on the meaningfulness of language use imply a breakdown in the dichotomy between language as a theoretically neutral instrument and theory as the commitment to some (...) class='Hi'>subset of the sentences allowed by a language. This breakdown, and the view of language which drives it, result in Quine's doctrine that speaking a language involves substantial theoretical commitments, and that inquiry must begin from within some ongoing theory of the world. That is, our epistemic practices, including epistemology itself, must be globally circular. The traditionally fundamental norm of linear propositional support, which drives the circularity objection, is thus seen as based in a misconception of language and theory. This not only explains Quine's lack of concern with the circularity of his view, but also illuminates the fundamental motivation for his general naturalism. ;Natural epistemology cannot be normative in the traditional sense of providing an independent ground for science. However, having accepted the "terminal parameter" of prediction as the test of theory, normativity becomes a matter of describing which procedures best promote predictively successful theory. Thus, epistemic norms emerge from science as hypotheses concerning the effectiveness of various methods in promoting predictively successful theory. Not only can natural epistemology prescribe and proscribe various epistemic practices, it offers the promise of an explanatory understanding of the strengths and weaknesses of those practices. Based on this understanding, natural epistemology further involves the improvement of current practices and the development of new ones. Crucial to this reconception of epistemology is the reconception of the knower as an adaptive organism employing the language/theory complex as an adaptive tool for interacting with the environment. ;The naturalism/anti-naturalism debate is advanced by acknowledging the substantial theoretical commitments implicit in the traditional view, and by examining Quine's reconception of language, theory, and the human knowing subject. (shrink)

This paper propounds a systematic examination of the link between the Knower Paradox and provability interpretations of modal logic. The aim of the paper is threefold: to give a streamlined presentation of the Knower Paradox and related results; to clarify the notion of a syntactical treatment of modalities; finally, to discuss the kind of solution that modal provability logic provides to the Paradox. I discuss the respective strength of different versions of the Knower Paradox, both in the (...) framework of first-order arithmetic and in that of modal logic with fixed point operators. It is shown that the notion of a syntactical treatment of modalities is ambiguous between a self-referential treatment and a metalinguistic treatment of modalities, and that these two notions are independent. I survey and compare the provability interpretations of modality respectively given by Skyrms, B. (1978, The Journal of Philosophy 75: 368–387) Anderson, C.A. (1983, The Journal of Philosophy 80: 338–355) and Solovay, R. (1976, Israel Journal of Mathematics 25: 287–304). I examine how these interpretations enable us to bypass the limitations imposed by the Knower Paradox while preserving the laws of classical logic, each time by appeal to a distinct form of hierarchy. (shrink)

According to the subset account of realization, a property, F, is realized by another property, G, whenever F is individuated by a non-empty proper subset of the causal powers by which G is individuated (and F is not a conjunctive property of which G is a conjunct). This account is especially attractive because it seems both to explain the way in which realized properties are nothing over and above their realizers, and to provide for the causal efficacy of (...) realized properties. It therefore seems to provide a way around the causal exclusion problem. There is reason to doubt, however, that the subset account can achieve both tasks. The problem arises when we look closely at the relation between properties and causal powers, specifically, at the idea that properties confer powers on the things that have them. If realizers are to be ontically prior to what they realize, then we must regard the conferral of powers by properties as a substantive relation of determination. This relation of conferral is at the heart of a kind of exclusion problem, analogous to the familiar causal exclusion problem. I argue that the subset account cannot adequately answer this new exclusion problem, and is for that reason ill-suited to be the backbone of a non-reductive physicalism. (shrink)

Research on the status and experience of women in academia in the last 30 years has challenged conventional explanations of persistent gender inequality, bringing into sharp focus the cumulative impact of small scale, often unintentional differences in recognition and response: the patterns of 'post-civil rights era' dis­crimination made famous by the 1999 report on the status of women in the MIT School of Science. I argue that feminist standpoint theory is a useful resource for understanding how this sea change in (...) understanding gender inequity was realized. At the same time, close attention to activist research on workplace environment issues suggests ways in which our understanding of standpoint theory can fruitfully be refined. I focus on the implications of two sets of distinctions: between types of epistemic injustice (and correlative advantage) that may affect marginalized knowers; and between the resources of situated knowledge and those of a critical standpoint on knowledge production. (shrink)

Defenders of the subset view of realization have claimed that we can resolve well-known worries about mental-physical causal overdetermination by holding that mental properties are subset realized by physical properties, that instances of subset realized properties are parts of physical realizers, and that part-whole overdetermination is unproblematic. I challenge the claim that the overdetermination generated by the subset view can be legitimated by appealing to more mundane part-whole overdetermination. I conclude that the subset view does (...) not provide a unique solution to overdetermination worries. (shrink)

Recent discussions in critical social epistemology have raised the idea that the concept 'knower' is not only an epistemological concept, but an ethical concept as well. Though this idea plays a central role in these discussions, the theoretical underpinnings of the claim have not received extended scrutiny. This paper explores the idea that 'knower' is an irreducibly ethical concept in an effort to defend its use as a critical concept. In Section 1, I begin with the claim that (...) 'knower' is an irreducibly normative and social concept, drawing from some ideas in Wilfrid Sellars. In Section 2, I argue that one's being a knower involves demands for various sorts of ethically-laden recognition. I develop this thought by arguing that Axel Honneth's threefold typology of recognition—love, respect, and esteem—finds clear expression within the context of socio-epistemic practice. I conclude in Section 3 by arguing that Miranda Fricker's proposed " analogy " between epistemic and moral perception should be modified to indicate a closer relationship than mere analogy. (shrink)

In this book, authors, for the first time, introduce the new notion of special subset linguistic topological spaces using linguistic square matrices. This book is organized into three chapters. Chapter One supplies the reader with the concept of ling set, ling variable, ling continuum, etc. Specific basic linguistic algebraic structures, like linguistic semigroup linguistic monoid, are introduced. Also, algebraic structures to linguistic square matrices are defined and described with examples. For the first time, non-commutative linguistic topological spaces are introduced. (...) The notion of a linguistic special subset of doubly non-commutative topological spaces of linguistic topological spaces is introduced in this book. (shrink)

The subset realization view proposes to solve the causal exclusion problem of non‐reductive mental instances by taking the mental instance as a part of its physical realizer. Many philosophers have argued that such a part‐whole relation will undermine physicalist realization because parts are ontologically prior to their wholes and the subset view is thus flawed. I argue that the relation that the subset view should propose is different from the ordinary part‐whole relation. What they should propose is (...) another kind of part‐whole relation under which parts are posterior to wholes. So, the subset view is not flawed in the aforementioned aspect. But discussion of the new part‐whole relation reveals that the subset view is in lack of a crucial argument against the exclusion reasoning, thus cannot solve the exclusion problem. Adopting the principle of causal proportionality (a route subset realizationists sometimes take) will not help the subset view either. So, my conclusion is, the subset view is yet to find a way to solve the exclusion problem. (shrink)

The Knower Paradox has had a brief but eventful history, and principles of epistemic closure (which say that a subject automatically knows any proposition she knows to be materially implied, or logically entailed, by a proposition she already knows) have been the subject of tremendous debate in epistemic logic and epistemology more generally, especially because the fate of standard arguments for and against skepticism seems to turn on the fate of closure. As far as I can tell, however, no (...) one working in either area has emphasized the result I emphasize in this paper: the Knower Paradox just falsifies even the most widely accepted general principles of epistemic closure. After establishing that result, I discuss five of its more important consequences. (shrink)

We reformulate a key definition given by Wáng and Ågotnes to provide semantics for public announcements in subset spaces. More precisely, we interpret the precondition for a public announcement of ???? to be the “local truth” of ????, semantically rendered via an interior operator. This is closely related to the notion of ???? being “knowable”. We argue that these revised semantics improve on the original and offer several motivating examples to this effect. A key insight that emerges is the (...) crucial role of topological structure in this setting. Finally, we provide a simple axiomatization of the resulting logic and prove completeness. (shrink)

Brian McLaughlin has objected to Sydney Shoemaker’s subset account of realization, posing what I call the problem of property entailment. Recently, Shoemaker has revised his subset account in response to McLaughlin’s objection. In this paper I argue that Shoemaker’s revised view fails to solve the problem of property entailment, and in fact makes the problem worse. I then put forward my own solution to the problem.

According to the subset view of realization, a property realizes another if the causal powers of the latter are a subset of those of the former. Against this view, some authors (in particular, Kevin Morris and Paul Audi) have argued that it has an untenable consequence that realizing properties are less fundamental than the properties they realize, because the subset view characterizes realized properties as parts (subsets) of their realizers whereas it is generally true that a part (...) is prior to its whole. This paper defends the subset view from this “part-whole” objection, by arguing that if we compare individual powers of realizer and realizee with particular attention to their manifestation conditions, it turns out that each power of a realizee is grounded in some power of its realizer. This grounding relation between powers, I shall argue, allows subset theorists to explain why a realizer is more fundamental than its realizee, even while having the latter as a part. (shrink)

In Bias: A Philosophical Study, Thomas Kelly offers a response to epistemological skepticism grounded in his account of bias. According to Kelly, the classic argument for skepticism is best understood as an attempt to show that our commonsense beliefs are biased against the skeptic. Kelly grants that this is true but argues that biased beliefs can still be knowledge. I offer two objections. First, if we are applying Kelly’s theory of bias to skepticism, it is best to think of the (...) skeptic’s challenge to be that our anti-skeptical beliefs are based on what we know to be biased reasons. Kelly has not shown that this sort of bias is compatible with knowledge. Second, Kelly’s approach to the problem of skepticism is an example of what I have called “unambitious epistemology.” And, for that reason, it is not a satisfactory answer to skepticism. (shrink)

This paper is a presentation and discussion of Spinoza's view on the knower, or the mind, as an agent. The knower is on his view to be regarded as an active or generative complex cognitive experience. Imagination, reason and intuition are the cognitive principles. On account of their intrinsic relation to “the first law of nature”, that of selfpreservation, together with the thesis of the mind as constituted by ideas or knowledge, these principles function at the same time (...) as moral principles. Consequently, it makes sense to speak of an individual's moral attitude toward everything be knows. A discussion of imagination, reason and intuition mainly as cognitive principles is followed by some concluding remarks on the cognitive and moral relation between human beings: To know another human being is to know his knowledge. The moral attitude of individuals to each other is a function of their mutual knowledge of knowledge. (shrink)

This essay addresses the connections between Emersonian and Wordsworthian concepts and Norman Maclean’s novella A River Runs through It, specifically those ideas of the Knower, the Sayer, and the Doer from Emerson’s “The Poet,” Emerson’s concept of what constitutes poetry and “The Poet,” as well as Wordsworth’s notions of poetic creativity. As discussed in the essay, Emerson’s concepts of the Knower, the Sayer, and the Doer line up with the three central characters of the novella—The Reverend Maclean, Norman (...) Maclean (both the author and the narrator), and Paul Maclean respectively. It is the unique blending of Romantic poetic leanings and religion that make all three the characters they are, which the author represents through the use of fishing as metaphor. The difficulties faced by the Sayer, as he tries to relate the story of past events and “spots in time,” are central to my argument, as they present the central conflict of the novel and offer the readers a contradiction as well, as the character who’s known as the Sayer should have no trouble expressing himself. However, it is with Wordsworth’s notion of ideas reflected on in tranquility that allow the Sayer to tell the story of his brother, the great poet/artist in a Romantic sense. (shrink)

According to a prominent line of thought, we can be physicalists, but not reductive physicalists, by holding that mental and other ‘higher-level’ or ‘nonbasic’ properties — properties that are not obviously physical properties — are all physically realized. Spelling this out requires an account of realization, an account of what it is for one property to realize another. And while several accounts of realization have been advanced in recent years,1 my interest here is in the ‘subset view,’ which has (...) often been invoked explicitly in defense of nonreductive physicalist positions.2 The subset view holds that property realization consists in the powers of one property having as a subset the powers of another; .. (shrink)

The Subset View of realization, though it has some attractive advantages, also has several problems. In particular, there are five main problems that have emerged in the literature: Double-Counting, The Part/Whole Problem, The “No Addition of Being” Problem, The Problem of Projectibility, and the Problem of Spurious Kinds. Each is reviewed here, along with solutions (or partial solutions) to them. Taking these problems seriously constrains the form that a Subset view can take, and thus limits the kinds of (...) relations that can fulfill the realization relation on this view. (shrink)

The knower paradox states that the statement ‘We know that this statement is false’ leads to inconsistency. This article presents a fresh look at this paradox and some well-known solutions from the literature. Paul Égré discusses three possible solutions that modal provability logic provides for the paradox by surveying and comparing three different provability interpretations of modality, originally described by Skyrms, Anderson, and Solovay. In this article, some background is explained to clarify Égré’s solutions, all three of which hinge (...) on intricacies of provability logic and its arithmetical interpretations. To check whether Égré’s solutions are satisfactory, we use the criteria for solutions to paradoxes defined by Susan Haack and we propose some refinements of them. This article aims to describe to what extent the knower paradox can be solved using provability logic and to what extent the solutions proposed in the literature satisfy Haack’s criteria. Finally, the article offers some reflections on the relation between knowledge, proof, and provability, as inspired by the knower paradox and its solutions. (shrink)

Fricker‘s Epistemic Injustice discusses the idea of testimonial injustice, specifically, being harmed in one‘s capacity as a knower. Fricker‘s own theory of testimonial injustice emphasizes the role of prejudice. She argues that prejudice is necessary for testimonial injustice and that when hearers use a prejudice to give a deficit to the credibility of speakers hearers intrinsically harm speakers in their capacity as a knower. This paper rethinks the connections between prejudice and testimonial injustice. I argue that many cases (...) of prejudicial credibility deficits do not intrinsically harm speakers. Further, I suggest that prejudice is not necessary for harming speakers. I provide my own proposal on which testimonial injustice occurs when speaker‘s capacity as a giver of knowledge is interfered with in important ways. My proposal does not give prejudice any essential role. (shrink)

Hamkins and Kikuchi (2016, 2017) show that in both set theory and class theory the definable subset ordering of the universe interprets a complete and decidable theory. This paper identifies the minimum subsystem of,, that ensures that the definable subset ordering of the universe interprets a complete theory, and classifies the structures that can be realised as the subset relation in a model of this set theory. Extending and refining Hamkins and Kikuchi's result for class theory, a (...) complete extension,, of the theory of infinite atomic boolean algebras and a minimum subsystem,, of are identified with the property that if is a model of, then is a model of, where M is the underlying set of, is the unary predicate that distinguishes sets from classes and is the definable subset relation. These results are used to show that that decides every 2‐stratified sentence of set theory and decides every 2‐stratified sentence of class theory. (shrink)

The article states that the Knower paradox exposes the circularity of the conceptualization of knowledge. The article provides an explanation of the Knower paradox for the purpose of demonstrating how the paradox arises. A modification to the concept of knowledge is presented to solve the paradox and structure knowledge.

Adherents of the epistemological position called internalism typically believe that the view they oppose, called externalism, is such a new and radical departure from the established way of seeing knowledge that its implications are uninteresting. Perhaps itis relatively novel, but the approach to knowledge with the greatest antiquity is the one that equates it withcertainty, and while this conception is amenable to the demands of the internalist, it is also a non-starter in the opinion of almost all contemporary epistemologists since (...) obviously it directly implies that we know nothing about the world. Perhaps skepticism is correct, but there are conceptions of knowledge at least as plausible as the certainty equation that do not obviously land us there. It is its promise along these lines that makes the so-called traditional conception of knowledge initially interesting. But contrary to popular belief, the traditional conception cannot be claimed by internalists if it is to have any chance at all in avoiding skepticism; to avoid skepticism, I shall argue, it has to have an externalist element.Moreover, each of the departures from the traditional view that appears in the Gettier literature is externalist as well, or at least all of the ones of which I am aware. The only genuine forms of internalism are those held by philosophers who draw a fairly sharp line between knowledge and justified belief, ignore the former, then offer an internalist account of the latter. This approach is very common and very plausible. But it is not as useful as is often thought; in particular, I shall suggest, it must succumb to a form of skepticism. (shrink)

In this essay I present a new version of the Paradox of the Knower and show that this new paradox vitiates a certain argument against epistemic closure. I then prove a theorem that relates the new paradox to epistemological scepticism. I conclude by assessing the use of the Knower in arguments against syntactical treatments of knowledge.

_ Source: _Page Count 11 Epistemic Authority: A Theory of Trust, Authority, and Autonomy in Belief. By Linda Trinkaus Zagzebski. New York: Oxford University Press, 2012. Pp. xiii +279. isbn 978–0–19–993647–2.

Might there be extended cognition and thereby extended minds? Rightly, that possibility is being investigated at present by philosophers of mind. Should epistemologists share that spirit, by inquiring into the possibility of extended knowing and thereby of extended knowers? Indeed so, I argue. The key to this shift of emphasis will be an epistemologically improved understanding of the implications of epistemic externalism.

We characterize the (κ, Λ, < μ)-distributive law in Boolean algebras in terms of cut and choose games $\scr{G}_{<\mu}^{\kappa}(\lambda)$ , when μ ≤ κ ≤ Λ and κ<κ = κ. This builds on previous work to yield game-theoretic characterizations of distributive laws for almost all triples of cardinals κ, Λ, μ with μ ≤ Λ, under GCH. In the case when μ ≤ κ ≤ Λ and κ<κ = κ, we show that it is necessary to consider whether the κ-stationarity (...) of Pκ+Λ in the ground model is preserved by B. In this vein, we develop the theory of κ-club and κ-stationary subsets of Pκ+Λ. We also construct Boolean algebras in which Player I wins $\scr{G}_{\kappa}^{\kappa}(\kappa ^{+})$ but the (κ, ∞, κ)-d.1, holds, and, assuming GCH, construct Boolean algebras in which many games are undetermined. (shrink)

Assuming the existence of a supercompact cardinal, we construct a model of ZFC + ). Answering a question of Uri Abraham [A], [A-S], we prove that adding a real to the world always makes P ℵ 1 - V stationary.

This essay corrects an error in the presentation of the Paradox of the Knowledge-Plus Knower, which is the variant of Kaplan and Montague’s Knower Paradox presented in C. Cross 2001: ‘The Paradox of the Knower without Epistemic Closure,’ MIND, 110, pp. 319–33. The correction adds a universally quantified transitivity principle for derivability as an additional assumption leading to paradox. This correction does not affect the status of the Knowledge-Plus paradox as a rebuttal to an argument against epistemic (...) closure, since the quantified transitivity principle is true in the standard model of arithmetic and therefore innocuous. (shrink)

Let us sum up. The paradox of the Knower poses a direct and formal challenge to the coherence of common notions of knowledge and truth. We've considered a number of ways one might try to meet that challenge: propositional views of truth and knowledge, redundancy or operator views, and appeal to hierarchy of various sorts. Mere appeal to propositions or operators, however, seems to be inadequate to the task of the Knower, at least if unsupplemented by an auxiliary (...) recourse to hierarchy. But the cost of hierarchy appears to be an abandonment of any notion of all truth or of omniscience. What the contradictions of the Knower seem to demand, then, is at least an abandonment of these. As noted in the introduction, the argument is complicated enough that one must be wary of dogmatic and precipitate conclusions. One may legitimately wonder whether some new response, or some variation on an old one, will yet offer a way out.Far too often, however, it is asked what has gone wrong with paradox rather than what paradox may have to teach us. What the Knower may have to teach us, I think, is that there really can be no coherent notion of all truth and really can be no coherent notion of omniscience. In its own way that conclusion is perhaps as humbling as is any traditional notion of God. (shrink)