This volume contains an English translation of Edmund Husserl’s first major work, the Philosophie der Arithmetik, (Husserl 1891). As a translation of Husserliana XII (Husserl 1970), it also includes the first chapter of Husserl’s Habilitationsschrift (Über den Begriff der Zahl) (Husserl 1887) and various supplementary texts written between 1887 and 1901. This translation is the crowning achievement of Dallas Willard’s monumental research into Husserl’s early philosophy (Husserl 1984) and should be seen as a companion to volume V of the Husserliana: (...) Collected Works series (Husserl 1994b), which already contained selected translations from Hua XII. As Willard re- marks on the inner cover of the volume, it is “a window on a period of rich and illuminating philosophical activity”, to which I wholeheartedly agree. Willard’s two volumes of translations open this window on the beginnings of Husserl’s philosophy for the English-speaking world. This earliest period of Husserl’s philosophy has often been unjustly ignored and Willard provides the English reader with an excellent starting point. Husserl’s first steps into phenomenology and the philosophy of logic and mathematics contain many promising seeds that will flower later on, in the Logische Untersuchungen and beyond. (shrink)

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality (...) of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy [1991]; Shin [1994]; and Giaquinto [1994]) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.

Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped (...) with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent. (shrink)

Since Peter van Inwagen in his book Material Beings raised The Special Composition Question, many lines have been written about the conditions for two or more material objects to compose a further object. This paper proposes a study of the relationship between composition and the notion of boundary. The question about composition entails drawing a boundary to determine when it occurs and when it does not. There are three traditional answers: restrictivism, universalism, and nihilism. While according to restrictivism, composition occurs (...) occasionally, according to universalism and nihilism, it occurs either always or never respectively. Nonetheless, despite the opposition of these two answers, both have a common target: to avoid the existence of ontologically vague boundaries for material composition. Thus, how a boundary is drawn will determinewhat kind of things we accept as components of the world. (shrink)

PLEASE NOTE: This is the corrected 2nd eBook edition, 2021. ●●●●● _Critique of Impure Reason_ has now also been published in a printed edition. To reduce the otherwise high price of this scholarly, technical book of nearly 900 pages and make it more widely available beyond university libraries to individual readers, the non-profit publisher and the author have agreed to issue the printed edition at cost. ●●●●● The printed edition was released on September 1, 2021 and is now available through (...) all booksellers, including Barnes & Noble, Amazon, and brick-and-mortar bookstores under ISBN 978-0-578-88646-6. ●●●●● In light of the length of this book, readers who would like to have a more detailed description of the book's objectives and method may find it helpful to read the detailed and clearly written Wikipedia entry about this work: From the Wikipedia search page, use the search phrase "Critique of Impure Reason". At least at the time of this writing (11/29/2021), the Wikipedia entry is well-researched and accurate. ●●●●● In addition, a "Primer on Bartlett's CRITIQUE OF IMPURE REASON" has been made available by the author. It is available under its title through PhilPapers and other philosophy online archives. ●●●●● COMMENDATIONS OF THIS WORK, from the back cover of the published edition: ●●●●● “I admire its range of philosophical vision.” – Nicholas Rescher, Distinguished University Professor of Philosophy, University of Pittsburgh, author of more than 100 books. ●●●●● “Bartlett’s _Critique of Impure Reason_ is an impressive, bold, and ambitious work. Careful scholarship is balanced by original analyses that lead the reader to recognize the limits of meaning, knowledge, and conceptual possibility. The work addresses a host of traditional philosophical problems, among them the nature of space, time, causality, consciousness, the self, other minds, ontology, free will and determinism, and others. The book culminates in a fascinating and profound new understanding of relativity physics and quantum theory.” – Gerhard Preyer, Professor of Philosophy, Goethe-University, Frankfurt am Main, Germany, author of many books including _Concepts of Meaning_, _Beyond Semantics and Pragmatics_, _Intention and Practical Thought_, and _Contextualism in Philosophy_. ●●●●● “[This work’s] goal is of a unique and difficult species: Dr. Bartlett seeks to develop a formal logical calculus on the basis of transcendental philosophical arguments; in fact, he hopes that this calculus will be the formal expression of the transcendental foundation of knowledge.... I consider Dr. Bartlett’s work soundly conceived and executed with great skill.” – C. F. von Weizsäcker, philosopher and physicist, former Director, Max-Planck-Institute, Starnberg, Germany. ●●●●● “Bartlett has written an American “Prolegomena to All Future Metaphysics.” He aims rigorously to eliminate meaningless assertions, reach bedrock, and place philosophy on a firm foundation that will enable it, like science and mathematics, to produce lasting results that generations to come can build on. This is a great book, the fruit of a lifetime of research and reflection, and it deserves serious attention.” — Martin X. Moleski, former Professor, Canisius College, Buffalo, NY, studies of scientific method, the presuppositions of thought, and the self-referential nature of epistemology. ●●●●● “Bartlett has written a book on what might be called the underpinnings of philosophy. It has fascinating depth and breadth, and is all the more striking due to its unifying perspective based on the concepts of reference and self-reference.” – Don Perlis, Professor of Computer Science, University of Maryland, author of numerous publications on self-adjusting autonomous systems and philosophical issues concerning self-reference, mind, and consciousness. ●●●●● ●●●●● The _Critique of Impure Reason: Horizons of Possibility and Meaning_ comprises a major and important contribution to philosophy. Thanks to the generosity of its publisher, this massive 885-page volume has been published as a free open access eBook (3.75MB) as well as an open access printed edition. It inaugurates a revolutionary paradigm shift in philosophical thought by providing compelling and long-sought-for solutions to a wide range of philosophical problems. In the process, the work fundamentally transforms the way in which the concepts of reference, meaning, and possibility are understood. The book includes a Foreword by the celebrated German philosopher and physicist Carl Friedrich von Weizsäcker. ●●●●● In Kant’s _Critique of Pure Reason_ we find an analysis of the preconditions of experience and of knowledge. In contrast, but yet in parallel, the new _Critique_ focuses upon the ways—unfortunately very widespread and often unselfconsciously habitual—in which many of the concepts that we employ _conflict_ with the very preconditions of meaning and of knowledge. ●●●●● This is a book about the boundaries of frameworks and about the unrecognized conceptual confusions in which we become entangled when we attempt to transgress beyond the limits of the possible and meaningful. We tend either not to recognize or not to accept that we all-too-often attempt to trespass beyond the boundaries of the frameworks that make knowledge possible and the world meaningful. ●●●●● The _Critique of Impure Reason_ proposes a bold, ground-breaking, and startling thesis: that a great many of the major philosophical problems of the past can be solved through the recognition of a viciously deceptive form of thinking to which philosophers as well as non-philosophers commonly fall victim. For the first time, the book advances and justifies the criticism that a substantial number of the questions that have occupied philosophers fall into the category of “impure reason,” violating the very conditions of their possible meaningfulness. ●●●●● The purpose of the study is twofold: first, to enable us to recognize the boundaries of what is referentially forbidden—the limits beyond which reference becomes meaningless—and second, to avoid falling victims to a certain broad class of conceptual confusions that lie at the heart of many major philosophical problems. As a consequence, the boundaries of _possible meaning_ are determined. ●●●●● Bartlett, the author or editor of more than 20 books, is responsible for identifying this widespread and delusion-inducing variety of error, _metalogical projection_. It is a previously unrecognized and insidious form of erroneous thinking that undermines its own possibility of meaning. It comes about as a result of the pervasive human compulsion to seek to transcend the limits of possible reference and meaning. ●●●●● Based on original research and rigorous analysis combined with extensive scholarship, the _Critique of Impure Reason_ develops a self-validating method that makes it possible to recognize, correct, and eliminate this major and pervasive form of fallacious thinking. In so doing, the book provides at last provable and constructive solutions to a wide range of major philosophical problems. ●●●●● CONTENTS AT A GLANCE ▪▪▪▪▪ Preface ▪▪▪▪▪ Foreword by Carl Friedrich von Weizsäcker ▪▪▪▪▪ Acknowledgments ▪▪▪▪▪ Avant-propos: A philosopher’s rallying call ▪▪▪▪▪ Introduction ▪▪▪▪▪ A note to the reader ▪▪▪▪▪ A note on conventions ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART I ▪▪▪▪▪ ▪▪▪▪▪ WHY PHILOSOPHY HAS MADE NO PROGRESS AND HOW IT CAN ▪▪▪▪▪ 1 Philosophical-psychological prelude ▪▪▪▪▪ 2 Putting belief in its place: Its psychology and a needed polemic ▪▪▪▪▪ 3 Turning away from the linguistic turn: From theory of reference to metalogic of reference ▪▪▪▪▪ 4 The stepladder to maximum theoretical generality ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART II ▪▪▪▪▪ THE METALOGIC OF REFERENCE ▪▪▪▪▪ A New Approach to Deductive, Transcendental Philosophy ▪▪▪▪▪ 5 Reference, identity, and identification ▪▪▪▪▪ 6 Self-referential argument and the metalogic of reference ▪▪▪▪▪ 7 Possibility theory ▪▪▪▪▪ 8 Presupposition logic, reference, and identification ▪▪▪▪▪ 9 Transcendental argumentation and the metalogic of reference ▪▪▪▪▪ 10 Framework relativity ▪▪▪▪▪ 11 The metalogic of meaning ▪▪▪▪▪ 12 The problem of putative meaning and the logic of meaninglessness ▪▪▪▪▪ 13 Projection ▪▪▪▪▪ 14 Horizons ▪▪▪▪▪ 15 De-projection ▪▪▪▪▪ 16 Self-validation ▪▪▪▪▪ 17 Rationality: Rules of admissibility ▪▪▪▪▪ ▪▪▪▪▪ ▪▪▪▪▪ PART III ▪▪▪▪▪ PHILOSOPHICAL APPLICATIONS OF THE METALOGIC OF REFERENCE ▪▪▪▪▪ Major Problems and Questions of Philosophy and the Philosophy of Science ▪▪▪▪▪ 18 Ontology and the metalogic of reference ▪▪▪▪▪ 19 Discovery or invention in general problem-solving, mathematics, and physics ▪▪▪▪▪ 20 The conceptually unreachable: “The far side” ▪▪▪▪▪ 21 The projections of the external world, things-in-themselves, other minds, realism, and idealism ▪▪▪▪▪ 22 The projections of time, space, and space-time ▪▪▪▪▪ 23 The projections of causality, determinism, and free will ▪▪▪▪▪ 24 Projections of the self and of solipsism ▪▪▪▪▪ 25 Non-relational, agentless reference and referential fields ▪▪▪▪▪ 26 Relativity physics as seen through the lens of the metalogic of reference ▪▪▪▪▪ 27 Quantum theory as seen through the lens of the metalogic of reference ▪▪▪▪▪ 28 Epistemological lessons learned from and applicable to relativity physics and quantum theory ▪▪▪▪▪ ▪▪▪▪▪ PART IV ▪▪▪▪▪ HORIZONS ▪▪▪▪▪ 29 Beyond belief ▪▪▪▪▪ 30 _Critique of Impure Reason_: Its results in retrospect ▪▪▪▪▪ ▪▪▪▪▪ SUPPLEMENT ▪▪▪▪▪ The Formal Structure of the Metalogic of Reference ▪▪▪▪▪ APPENDIX I ▪▪▪▪▪ The Concept of Horizon in the Work of Other Philosophers ▪▪▪▪▪ APPENDIX II ▪▪▪▪▪ Epistemological Intelligence ▪▪▪▪▪ References ▪▪▪▪▪ Index ▪▪▪▪▪ About the author. 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The purpose of this paper is to suggest that we are in the midst of a Cantorian bubble, just as, for example, there was a dot com bubble in the late 1990’s.

Are there different sizes of infinity? That is, are there infinite sets of different sizes? This is one of the most natural questions that one can ask about the infinite. But it is of course generally taken to be settled by mathematical results, such as Cantor’s theorem, to the effect that there are infinite sets without bijections between them. These results settle the question, given an almost universally accepted principle relating size to the existence of functions. The principle is: for (...) any sets A and B, if A is the same size as B, then there is a bijection from A to B. The aim of the paper, however, is to argue that this question is in fact wide open: to argue that we are not in a position to know the answer, because we are not in one to know the principle. The aim, that is, is to argue that for all we know there is only one size of infinity. (shrink)

The topic of our analysis is the argument for the existence of substances given by Bernard Bolzano in Athanasia, where he essentially employs two ontological categories: substance and adherence. Bolzano considers the real and conditioned Inbegriff of all adherences, which are wirklich and nicht selbst bestehen. He claims that the formed collection is dependent on something external and non-adherential, which therefore is a substance. Bolzano’s argumentation turns out to be structurally similar to his argument for the existence of God from (...) Lehrbuch der Religionswissenschaft, but in each of these reasonings, we find different plausible interpretations of the key concept “Inbegriff”. The latter argumentation refers to the mereological totality of existentially conditioned objects. We propose the explication of the Bolzanian Inbegriff of all adherences using two types of predication: we consider its extension as composed of certain intensional counterparts of adherences. In our approach, we use a fragment of the theory of abstract objects formulated by E. Zalta, describing two different relations between individuals and properties: extensional exemplification and intensional encoding. We put our reconstruction in a wider context of Bolzano’s ontology, formulating the needed axioms with two primitive predicates of second order... is an adherence,... is conditioned by something real as well as the conditionally introduced first order predicate constant \ for Inbegriff of all adherential ideas. Finally, we sketch a model for our theory. (shrink)

open journal of Philosophical Investigations (PI) is an international journal dedicated to the latest advancements in philosophy. The goal of this journal is to provide a platform for academicians all over the world to promote, share, and discuss various new issues and developments in different areas of philosophy. -/- All manuscripts to be prepared in English or Persian and are subject to a rigorous and fair peer-review process. Generally, accepted papers will appear online. The journal publishes papers including the following (...) fields: -/- · Analytic Philosophy · Ancient Greek and Roman Philosophy · Art Aesthetics · Comparative Philosophy West and Chinese · Eastern Philosophy . Islamic philosophy · Epistemology · Ethics · Hermeneutics · History of Religious Thought · History of Western Philosophy · Language Logic . (shrink)

This paper presents an overview of several years of my research into individuals’ reasoning, argumentation, and bias when addressing problems, scenarios, and symbols related to mathematical infinity. There is a long history of debate around what constitutes “objective truth” in the realm of mathematical infinity, dating back to ancient Greece. Modes of argumentation, hindrances, and intuitions have been largely consistent over the years and across levels of expertise. This presentation examines the interrelated complexities of notions of objectivity, bias, and argumentation (...) as manifested in different presentations and normative interpretations or resolutions of well-known paradoxes of infinity. Paradoxes have been described as occasioning major epistemological reconstructions, and I highlight such occasions as they emerged for both novices and experts with connection to current conceptualisations of objectivity. Of interest is the perception that one single objective truth about “actual” mathematical infinity exists – indeed, this is brought to question at an axiomatic level with both theoretical and empirical research implications. (shrink)

Cantor showed that there are infinite sets that do not have one-to-one correspondences between them. The standard understanding of this result is that it shows that there are different sizes of infinity. This paper challenges this standard understanding, and argues, more generally, that we do not have any reason to think that there are different sizes of infinity. Two arguments are given against the claim that Cantor established that there are different such sizes: one involves an analogy between Cantor’s result (...) and Russell’s paradox, another is more direct. (shrink)

Our research concerns a formal representation of Bolzano’s original concepts of Substanz and Adhärenz. The formalized intensional theory enables to articulate a question about the consistency of a part of Bolzano’s metaphysics and to suggest an answer to it in terms of contemporary model theory. The formalism is built as an extension of Zalta’s theory of abstract objects, describing two types of predication, viz. attribution and representation. Bolzano was aware about this distinction. We focus on the consistency of this formalism (...) and the description of its semantics. Firstly, we explore the possibility to reconstruct a Russellian antinomy based on the concept of the Bolzano’s Inbegriff of all adherences. Our aim is to show limitations of his theory that prevent a contradiction when the Inbegriff consists of non-self-referential adherences. Next, we discuss two competing semantics for the proposed theory: Scott’s and Aczel’s semantics. The first one yields a problematic result, that there are no models for the considered theory, containing a non-empty collection of all adherences. This is due to the fact that Scott’s structures verify the formula on reloading abstracts in extensional contexts. We show that Aczel’s semantics does not contain this difficulty. There are described Aczel’s models with a non-empty set of all adherences. The self-referentiality of such a collection becomes irrelevant here. Finally, we show that there are Aczel’s structures verifying the formula on reloading abstracts and we exclude them from the class of models intended for our theory. (shrink)

This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...) those that are more readily available for mathematico-scientific use. (shrink)

This paper investigates the ontological presuppositions of quantifier logic. It is seen that the actual infinite, although present in the usual completeness proofs, is not needed for a proper semantic foundation. Additionally, quantifier logic can be given an adequate formulation in which neither the notion of individual nor that of a predicate appears.

In this paper, I present a proposal for a Formal Natural Theology. The approach employed for this task is through a first-order theory, in which fundamental concepts such as divine, necessary, and supreme beings, are formally introduced, which allow obtaining the theorems of existence and uniqueness of a divine being, according to the perspective of classical theism.

Concepts of infinity have been subjects of dispute since antiquity. The main problems of this paper are: is the mind able to acquire a concept of infinity? and: how are concepts of infinity acquired? The aim of this paper is neither to say what the meanings of the word “infinity” are nor what infinity is and whether it exists. However, those questions will be mentioned, but only in necessary extent.