Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory (...) to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are (...) parts of the physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Formal criteria of theoretical equivalence are mathematical mappings between specific sorts of mathematical objects, notably including those objects used in mathematical physics. Proponents of formal criteria claim that results involving these criteria have implications that extend beyond pure mathematics. For instance, they claim that formal criteria bear on the project of using our best mathematical physics as a guide to what the world is like, and also have deflationary implications for various debates in the metaphysics of (...) physics. In this paper, I investigate whether there is a defensible view according to which formal criteria have significant non-mathematical implications, of these sorts or any other, reaching a chiefly negative verdict. Along the way, I discuss various foundational issues concerning how we use mathematical objects to describe the world when doing physics, and how this practice should inform metaphysics. I diagnose the prominence of formal criteria as stemming from contentious views on these foundational issues, and endeavor to motivate some alternative views in their stead. (shrink)

Gödel's Philosophical Legacy Kurt Gödel's contributions to philosophy extend beyond his incompleteness theorems. He engaged deeply with the work of other philosophers, including Immanuel Kant and Edmund Husserl, and explored topics such as the nature of time, the structure of the universe, and the relationship between mathematics and reality. Gödel's philosophical writings, though less well-known than his mathematical work, offer rich insights into his views on the nature of existence, the limits of human knowledge, and the interplay between the (...) finite and the infinite. His work continues to inspire and challenge philosophers, mathematicians, and scientists, inviting them to explore the profound and often enigmatic questions at the heart of human understanding. -/- Kurt Gödel's Broader Contributions to Philosophy Kurt Gödel, while primarily known for his monumental incompleteness theorems, made significant contributions that extended beyond the realm of mathematical logic. His philosophical pursuits deeply engaged with the works of eminent philosophers like Immanuel Kant and Edmund Husserl. Gödel's explorations into the nature of time, the structure of the universe, and the relationship between mathematics and reality reveal a profound and multifaceted intellectual legacy. -/- Engagement with Immanuel Kant Gödel held a deep interest in the philosophy of Immanuel Kant. He admired Kant's critical philosophy, particularly the distinction between the noumenal and phenomenal worlds. Kant posited that human experience is shaped by the mind’s inherent structures, leading to the conclusion that certain aspects of reality (the noumenal world) are fundamentally unknowable. Gödel’s incompleteness theorems echoed this Kantian theme, illustrating the limits of formal systems in capturing the totality of mathematical truth. Gödel believed that mathematical truths exis t independently of human thought, akin to Kant's noumenal realm. This philosophical alignment provided a robust foundation for Gödel's Platonism, which asserted the existence of mathematical objects as real, albeit abstract, entities. -/- Influence of Edmund Husserl Gödel was also profoundly influenced by Edmund Husserl, the founder of phenomenology. Husserl's phenomenology emphasizes the direct investigation and description of phenomena as consciously experienced, without preconceived theories about their causal explanation. Gödel saw Husserl's work as a pathway to bridge the gap between the abstract world of mathematics and concrete human experience. Husserl's ideas about the structures of consciousness and the intentionality of thought resonated with Gödel's views on mathematical intuition. Gödel believed that human minds could access mathematical truths through intuition, a concept that draws on Husserlian phenomenological methods. -/- The Nature of Time and the Universe Gödel's philosophical inquiries extended to the nature of time and the structure of the universe. His collaboration with Albert Einstein at the Institute for Advanced Study led to the development of the "Gödel metric" in 1949. This solution to Einstein's field equations of general relativity described a rotating universe where time travel to the past was theoretically possible. Gödel's model challenged conventional notions of time and causality, suggesting that the universe might have a more intricate structure than previously thought. Gödel's exploration of time was not just a mathematical curiosity but a profound philosophical statement about the nature of reality. He questioned whether time was an objective feature of the universe or a construct of human consciousness. His work hinted at a timeless realm of mathematical truths, aligning with his Platonist view. -/- Mathematics and Reality Gödel's philosophical outlook extended to the broader relationship between mathematics and reality. He believed that mathematics provided a more profound insight into the nature of reality than empirical science. For Gödel, mathematical truths were timeless and unchangeable, existing independently of human cognition. This perspective led Gödel to critique the materialist and mechanistic views that dominated 20th-century science and philosophy. He argued that a purely physicalist interpretation of the universe failed to account for the existence of abstract mathematical objects and the human capacity to understand them. Gödel's philosophy suggested a more integrated view of reality, where both physical and abstract realms coexist and inform each other. -/- Gödel's Exploration of Time Kurt Gödel, one of the most profound logicians of the 20th century, ventured beyond the confines of mathematical logic to explore the nature of time. His inquiries into the concept of time were not merely theoretical musings but were grounded in rigorous mathematical formulations. Gödel's exploration of time challenged conventional views and opened new avenues of thought in both physics and philosophy. -/- Gödel and Einstein Gödel’s interest in the nature of time was significantly influenced by his friendship with Albert Einstein. Both were faculty members at the Institute for Advanced Study in Princeton, where they engaged in deep discussions about the nature of reality, time, and space. Gödel's exploration of time culminated in his solution to Einstein's field equations of general relativity, known as the Gödel metric. -/- The Gödel Metric In 1949, Gödel presented a model of a rotating universe, which became known as the Gödel metric. This solution to the equations of general relativity depicted a universe where time travel to the past was theoretically possible. Gödel’s rotating universe contained closed timelike curves (CTCs), paths in spacetime that loop back on themselves, allowing for the possibility of traveling back in time. The Gödel metric posed a significant philosophical challenge to the conventional understanding of time. If time travel were possible, it would imply that time is not linear and absolute, as commonly perceived, but rather malleable and subject to the geometry of spacetime. This raised profound questions about causality, the nature of temporal succession, and the very structure of reality. -/- Philosophical Implications Gödel’s exploration of time extended beyond the mathematical implications to broader philosophical inquiries: Nature of Time: Gödel questioned whether time was an objective feature of the universe or a construct of human consciousness. His work suggested that our understanding of time as a linear progression from past to present to future might be an illusion, shaped by the limitations of human perception. -/- Causality and Free Will: The existence of closed timelike curves in Gödel’s model raised questions about causality and free will. If one could travel back in time, it would imply that future events could influence the past, potentially leading to paradoxes and challenging the notion of a deterministic universe. -/- Temporal Ontology: Gödel's work contributed to debates in temporal ontology, particularly the debate between presentism (the view that only the present exists) and eternalism (the view that past, present, and future all equally exist). Gödel’s rotating universe model seemed to support eternalism, suggesting a block universe where all points in time are equally real. Philosophy of Science: Gödel’s exploration of time had implications for the philosophy of science, particularly in the context of understanding the limits of scientific theories. His work underscored the importance of considering philosophical questions when developing scientific theories, as they shape our fundamental understanding of concepts like time and space. -/- Legacy Gödel’s exploration of time remains a significant and controversial contribution to both physics and philosophy. His work challenged established notions and encouraged deeper inquiries into the nature of reality. Gödel’s rotating universe model continues to be a topic of interest in theoretical physics and cosmology, inspiring new research into the nature of time and the possibility of time travel. In philosophy, Gödel’s inquiries into time have prompted ongoing debates about the nature of temporal reality, the relationship between mathematics and physical phenomena, and the limits of human understanding. His work exemplifies the intersection of mathematical rigor and philosophical inquiry, demonstrating the profound insights that can emerge from such an interdisciplinary approach. The Temporal Ontology of Kurt Gödel Kurt Gödel's profound contributions to mathematics and logic extend into the realm of temporal ontology—the philosophical study of the nature of time and its properties. Gödel's insights challenge conventional perceptions of time and suggest a more intricate, layered understanding of temporal reality. This essay explores Gödel's contributions to temporal ontology, particularly through his engagement with relativity and his philosophical reflections. -/- Gödel's Rotating Universe One of Gödel’s most notable contributions to temporal ontology comes from his work in cosmology, specifically his solution to Einstein’s field equations of general relativity, known as the Gödel metric. Introduced in 1949, the Gödel metric describes a rotating universe with closed timelike curves (CTCs). These curves imply that, in such a universe, time travel to the past is theoretically possible, presenting a significant challenge to conventional views of linear, unidirectional time. -/- Implica tions for Temporal Ontology Gödel's rotating universe model has profound implications for our understanding of time: Eternalism vs. Presentism: Gödel’s model supports the philosophical stance known as eternalism, which posits that past, present, and future events are equally real. In contrast to presentism, which holds that only the present moment exists, eternalism suggests a "block universe" where time is another dimension like space. Gödel’s rotating universe, with its CTCs, reinforces this view by demonstrating that all points in time could, in principle, be interconnected in a consistent manner. Non-linearity of Time: The possibility of closed timelike curves challenges the idea of time as a linear sequence of events. In Gödel’s universe, time is not merely a straight path from past to future but can loop back on itself, allowing for complex interactions between different temporal moments. This non-linearity has implications for our understanding of causality and the nature of temporal succession. Objective vs. Subjective Time: Gödel’s work invites reflection on the distinction between objective time (the time that exists independently of human perception) and subjective time (the time as experienced by individuals). His model suggests that our subjective experience of a linear flow of time may not correspond to the objective structure of the universe. This raises questions about the relationship between human consciousness and the underlying temporal reality. -/- Gödel and Philosophical Reflections on Time Gödel’s engagement with temporal ontology was not limited to his cosmological work. He also reflected deeply on philosophical questions about the nature of time and reality, drawing on the ideas of other philosophers and integrating them into his own thinking. Kantian Influences: Gödel was influenced by Immanuel Kant’s distinction between the noumenal world (things as they are in themselves) and the phenomenal world (things as they appear to human observers). Gödel’s views on time echoed this distinction, suggesting that our perception of time might be a phenomenon shaped by the limitations of human cognition, while the true nature of time (the noumenal aspect) might be far more complex and non-linear. Husserlian Phenomenology: Gödel’s interest in Edmund Husserl’s phenomenology also informed his views on time. Husserl’s emphasis on the structures of consciousness and the intentionality of thought resonated with Gödel’s belief in the importance of intuition in accessing mathematical truths. Gödel’s reflections on time incorporated a phenomenological perspective, considering how temporal experience is structured by human consciousness. Mathematical Platonism: Gödel’s Platonist views extended to his understanding of time. Just as he believed in the independent existence of mathematical objects, Gödel saw time as an objective entity with a structure that transcends human perception. His work on the Gödel metric can be seen as an attempt to uncover this objective structure, revealing the deeper realities that underlie our experience of time. (shrink)

An old and well-known objection to non-classical logics is that they are too weak; in particular, they cannot prove a number of important mathematical results. A promising strategy to deal with this objection consists in proving so-called recapture results. Roughly, these results show that classical logic can be used in mathematics and other unproblematic contexts. However, the strategy faces some potential problems. First, typical recapture results are formulated in a purely logical language, and do not generalize nicely to languages (...) containing the kind of vocabulary that usually motivates non-classical theories—for example, a language containing a naïve truth predicate. Second, proofs of recapture results typically employ classical principles that are not valid in the targeted non-classical system; hence, non-classical theorists do not seem entitled to those results. In this paper, we analyze these problems and provide solutions on behalf of non-classical theorists. To address the first problem, we provide a novel kind of recapture result, which generalizes nicely to a truth-theoretic language. As for the second problem, we argue that it relies on an ambiguity and that, once the ambiguity is removed, there are no reasons to think that non-classical logicians are not entitled to their recapture results. (shrink)

The paper argues that the formulation of quantum mechanics proposed by Ghirardi, Rimini and Weber (GRW) is a serious candidate for being a fundamental physical theory and explores its ontological commitments from this perspective. In particular, we propose to conceive of spatial superpositions of non-massless microsystems as dispositions or powers, more precisely propensities, to generate spontaneous localizations. We set out five reasons for this view, namely that (1) it provides for a clear sense in which quantum systems in entangled states (...) possess properties even in the absence of definite values; (2) it vindicates objective, single-case probabilities; (3) it yields a clear transition from quantum to classical properties; (4) it enables to draw a clear distinction between purely mathematical and physical structures, and (5) it grounds the arrow of time in the time-irreversible manifestation of the propensities to localize. (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)

In lieu of an abstract, here is a brief excerpt of the content:Kant on the Method of MathematicsEmily Carson1. INTRODUCTIONThis paper will touch on three very general but closely related questions about Kant’s philosophy. First, on the role of mathematics as a paradigm of knowledge in the development of Kant’s Critical philosophy; second, on the nature of Kant’s opposition to his Leibnizean predecessors and its role in the development of the Critical philosophy; and finally, on the specific role of (...) intuition in Kant’s philosophy of mathematics. One of the points that I want to make is that recognizing the importance of these first two issues is essential to giving an adequate account of the third. That is, only by appreciating how Kant uses the example of mathematical knowledge, and in particular the objections he had to previous accounts of such knowledge, can we understand the philosophical role that he ascribes to intuition in his own account of mathematics.I don’t propose to explain the details of Kant’s philosophy of mathematics, but rather to compare his views on mathematics in the pre-Critical Prize Essay of 1764 to those of the Critique of Pure Reason, with the aim of showing how Kant’s doctrine of construction in pure intuition arises out of a combination of two factors: first, his attempt to ground the distinction between the respective methods of mathematics and philosophy, and second, his concern to defend the reality of mathematical knowledge against the views of those he called ‘the metaphysicians.’ The first factor demands an account of the certainty of mathematics, the second requires an account of its content. I shall argue that there is a possible conflict between Kant’s accounts of these features in the Prize Essay, a possibility which is finally ruled out only by means of the Critical doctrine of pure intuition. So I hope to show that there is a significant philosophical role for intuition in Kant’s account of mathematical knowledge (over and above the more logical role attributed to it on readings [End Page 629] such as those of Friedman, Beth, and Hintikka1) in reconciling the content of mathematics and its certainty.2. THE MATHEMATICIANS VS. THE METAPHYSICIANSThroughout his early writings, Kant appealed to the evidence and certainty of mathematics; his attitude towards metaphysics, however, underwent a dramatic shift. In the Physical Monadology of 1756, Kant asked how metaphysics and geometry can be united,[f] or the former peremptorily denies that space is infinitely divisible, while the latter, with its usual certainty, asserts that it is infinitely divisible. Geometry contends that empty space is necessary for free motion, while metaphysics hisses the idea off the stage. Geometry holds universal attraction or gravitation to be hardly explicable by mechanical causes but shows that it derives from the forces which are inherent in bodies at rest and which act at a distance, whereas metaphysics dismisses the notion as an empty delusion of the imagination [1:475–6].2He attempted in this work to reconcile the respective positions of metaphysics and geometry regarding infinite divisibility, thereby demonstrating that “it is neither the case that the geometer is mistaken nor that the opinion to be found among metaphysicians deviates from the truth” [1:480]. But in a series of works from 1762–3, Kant reveals a markedly different attitude towards metaphysics. For example, in “The only possible argument in support of a demonstration of the existence of God,” Kant says:the mania for method and the imitation of the mathematician, who advances with a sure step along a well-surfaced road, have occasioned a large number of such mishaps on the slippery ground of metaphysics. These mishaps are constantly before one’s eyes, but there is little hope that people will be warned by them, or that they will learn to be more circumspect as a result [2:71].In contrast, Kant describes the geometer as uncovering “with the greatest certainty” the most secret properties of that which is extended [2:70]. But the less conciliatory attitude to metaphysics comes out most clearly in Kant’s actual procedure in “The only possible argument.” In the Seventh Reflection, he presents an attempt to explain the origin of... (shrink)

Hopkins begins his study with Plato's written and unwritten theories of eidê and Aristotle's criticism of both. He then traces Husserl's early investigations into the formation of mathematical and logical concepts, charting the critical necessity that leads from descriptive psychology to transcendentally pure phenomenology. An investigation of the movement of Husserl's phenomenology of transcendental consciousness to that of monadological intersubjectivity follows. Hopkins then presents the final stage of the development of Husserl's thought, which situates monadological intersubjectivity within the context (...) of the historical a priori constitutive of all meaning. An exposition of the unwarranted historical presuppositions that guide Heidegger's fundamental ontological and Derrida's deconstructive criticisms of Husserl's transcendental phenomenology concludes the book. By following Husserl's personal trajectory Hopkins is able to show the unity of Husserl's philosophical enterprise, challenging the prevailing view that Husserl's late turn to history is inconsistent with his earlier attempts to establish phenomenology as a pure science. Contents: Introduction Part I Descriptive Psychology 1. Investigation of the Origin of Number 2. Investigation of the Origin of Logical Signification Part II Cartesian Transcendental Phenomenology 3. Investigation of the Origin of Objective Transcendence 4. Investigation of the Origin of Subjective Transcendence Part III Historical Transcendental Phenomenology 5. The Crisis of Meaning in Contemporary European Science 6. Historical Investigation of the Phenomenological Origin Part IV Husserl and his Critics 7. Fundamental Critiques of Transcendental Phenomenology 8. A Husserlian Response to the Critics. (shrink)

In theCritique of Pure Reason,Kant argues for two principles that concern magnitudes. The first is the principle that ‘All intuitions are extensive magnitudes,’ which appears in the Axioms of Intuition (B202); the second is the principle that ‘In all appearances the real, which is an object of sensation, has an intensive magnitude, that is, a degree,’ which appears in the Anticipations of Perception (B207). A circle drawn in geometry and the space occupied by an object such as a book (...) are paradigm examples of extensive magnitudes, while the intensity of a light is a paradigm example of an intensive magnitude. These principles justify and explain the possibility of applying mathematics to objects of experience. The Axioms principle also explains the possibility of any mathematical cognition at all. (shrink)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? Mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, (...) the choice sequence. But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. (shrink)

Daniel Sutherland - Kant on Arithmetic, Algebra, and the Theory of Proportions - Journal of the History of Philosophy 44:4 Journal of the History of Philosophy 44.4 533-558 Muse Search Journals This Journal Contents Kant on Arithmetic, Algebra, and the Theory of Proportions Daniel Sutherland Kant's philosophy of mathematics has both enthralled and exercised philosophers since the appearance of the Critique of Pure Reason. Neither the Critique nor any other work provides a sustained and focused account of his (...) mature views on mathematical cognition, forcing readers to glean what they can from disparate contexts. Despite these hurdles, Kant's views have been of great interest to philosophers of mathematics. They have also been of interest to philosophers wishing to understand Kant's philosophy more generally, since Kant maintains that mathematical judgments are synthetic a priori and that the type of synthesis underlying mathematics is the same as that underlying the perception of objects . Our understanding of Kant's views has been greatly improved during the last four decades by work on Kant's philosophy of mathematics. Despite these advances, however, I think the recent work has largely neglected the importance of Kant's theory of magnitudes and the extent to which that theory is influenced by the Eudoxian theory of proportions presented in Euclid's Elements. As a consequence, an important feature of Kant's.. (shrink)

This chapter considers Kant's relation to Hume as Kant himself understood it when he wrote the Critique of Pure Reason and the Prolegomena. It first seeks to refine the question of Kant's relation to Hume's skepticism, and it then considers the evidence for Kant's attitude toward Hume in three works: the A Critique, Prolegomena, and B Critique. It argues that in the A Critique Kant viewed skepticism positively, as a necessary reaction to dogmatism and a spur toward critique. In (...) his initial statement of the critical philosophy Kant treated Hume as an ally in curbing dogmatism, but one who stopped short of what was really needed: a full critique of reason, to establish the boundaries of metaphysical cognition. Kant found fault with Hume's analyses of cognition and experience, and specifically his failure to see the crucial importance of synthetic a priori cognition in metaphysics. In particular, he held that Hume's empiricist account of cognition could neither explain the synthetic a priori cognition actually found in mathematics and natural science, nor provide a principled account of the limits on what can be known--and what can be thought--through the pure concepts of the understanding. According to Kant, Hume therefore failed in his attempt to determine the limits of metaphysics, whereas he was able to succeed because his transcendental philosophy provided a thorough account of cognition, its structure and limits. In the Prolegomena and the B Critique Kant distinguished his position more sharply from Hume's. He also adopted a more negative attitude toward "skeptical idealism" than before; but he attributed such skepticism to Descartes, not Hume. Prior to the B Critique Kant did not see Hume as attacking natural science or ordinary cognition. In none of the three works was Kant's main aim to "answer the skeptic." His primary aim was to firmly establish the boundary of metaphysics, by discovering the elements of human cognition and fixing its proper domain. His purported discoveries about the limits of metaphysical cognition meant that the traditional objects of metaphysical knowledge, God, the soul, and the world as it is in itself, are unknowable, hence that traditional metaphysics itself is impossible. Besides settling the possibility or impossibility of metaphysics, his findings would also prevent the illegitimate extension of principles of sensibility to God and the noumenal self, an extension that would threaten the metaphysics of morals by incorrectly denying the thinkability of noumenal freedom, and that might otherwise lead to "materialism, fatalism, atheism, and freethinking unbelief" (B xxxiv). (shrink)

In lieu of an abstract, here is a brief excerpt of the content:TOWARD A GENERAL THEORY OF FICTION by James D. Parsons When nelson Goodman writes, "All fiction is literal, literary falsehood," he seems to be disregarding at least one noteworthy tradition.1 The tradition I have in mind includes works by Jeremy Bendiam, Hans Vaihinger, Tobias Dantzig, Wallace Stevens, and a host ofother writers in many fields who have been laboring for more man two centuries to clear the ground for (...) a general theory of fiction. Such a general dieory would, if its exponents are right about its relevemce, be situated at the very heart of die law, literature, religion, science, mathematics — in fact, of humem life itself. Fiction dien would be no neirrow litereiry falsehood, but a kind of pressure, eis Wedlace Stevens said, which die imagination exerts "against die pressure of reality," and which "helps us to live our lives,"2 or, in Hans Vaihingens metaphor, "a scaffolding afterwards to be demolished" 3 — that is, elfter it heis served its constructive purpose. A fiction is a vitally useful creation of die mind corresponding to nothing in the real world. Most people eire aware of some of these fictions, from the perfect geometric figures of Euclid to the black box of contemporary axiomatics. There are many odiers. The notion diat all human beings are equal before die law is a fiction that can serve opposing purposes: it can work as an instrument for securing greater justice in the courts and in society, or it can function as ceimouflage, helping to perpetuate em unjust legal system and em oppressive social order — emd probably in any actual situation it serves both purposes at once. In die science of mechanics the idea of a pure vacuum, in which moving objects meet with no resistance, was for some centuries a useful fiction diat made possible the exact mathematiced analysis of motion in memy forms — another useful fiction. The notion of normality in psychology — mat diere is a universal standard of reference for human behavior — is stdl utilized by clinical and laboratory workers in the field, in spite of the metssive sociological emd ethnographic criticisms that have been leveled against it. In science generally, the idea that 92 James D. Parsons93 diere is a community of experts who by dieir combined influence decide what the trudi is in every field is a fiction diat seems to have come to the end of its usefulness, diough some philosophers have written about it as if it were a reality and not a fiction at edl. The fictions I have cited so far are none of them literary, nor être diey fedse in the ordinetry meaning of the word. They are simply not true; they describe nothing mat exists, except in die mind, emd cem be considered false only if one forgets what they eire emd tries, illegitimately, to apply tests of trudi to them in a direct way. Among the fictions mat might be called literary are the fictions derived from mydis. Now, a myth is a story told in answer to a question or a series of questions that has never been asked, eidier about the real world or (most likely) about the perennial performance of a ritual. The mydi is designed to forestall the questioning, because the questions that might be raised would be too embarrassing or because questions about some aspects of reality ceuinot be formulated without distorting our relation to them. These are the aspects of reedity that can be viewed only out of die corner of the eye, that disappear when we focus our gaze upon diem, and mat can be identified only indirectly.4 The more explicitly fictional devices, too, of poetry, drama, and the novel develop out of the attempt to achieve some sort of cognizeuice of the seune aspects of reality that are related in mydis. I do not meetn to imply that literature, of which mydi is a part, is concerned only with such disconcerting or evanescent aspects of reality. Literature, like all the great realms of culture, is concerned with the whole of reality. It has to be, omerwise its more passionate, its more ediereed concerns would have no lodging place, no environment in which to maintain... (shrink)

Mathematical formalism is the the view that numbers are "signs" and that arithmetic is like a game played with such signs. Frege's colleague Thomae defended formalism using an analogy with chess, and Frege's critique of this analogy has had a major influence on discussions in analytic philosophy about signs, rules, meaning, and mathematics. Here I offer a new interpretation of formalism as defended by Thomae and his predecessors, paying close attention to the mathematical details and historical context. I argue (...) that for Thomae, the formal standpoint is an _algebraic perspective_ on a domain of objects, and a "sign" is not a linguistic expression or mark, but a representation of an object within that perspective. Thomae exploits a shift into this perspective to give a purely algebraic construction of the real numbers from the rational numbers. I suggest that Thomae's chess analogy is intended to provide a model for such shifts in perspective. (shrink)

According to Kant, we gain mathematical knowledge by constructing objects in pure intuition. This is true not only of geometry but arithmetic and algebra as well. Construction has prominent place in scholarly accounts of Kant’s views of mathematics. But did Kant have a clear vision of what construction is? The paper argues that Kant employed two different, even conflicting models of construction, depending on the philosophical issue he was dealing with. In the equivalence model, Kant claims that (...) the object constructed in intuition is equivalent to the properties included in the conceptual rule. In the overstepping model of construction, Kant argues that construction goes beyond the concept which is a “mere definition”. What is more, both models of construction can be found in the Doctrine of Method in the first Critique. The paper examines reasons that have led Kant to adopt the two models of construction, and proposes a reading that alleviates the apparent contradiction between the two models. (shrink)

This course on logic and theory of knowledge fell exactly midway between the publication of the Logical Investigations in 1900-01 and Ideas I in 1913. It constitutes a summation and consolidation of Husserl’s logico-scientific, epistemological, and epistemo-phenomenological investigations of the preceding years and an important step in the journey from the descriptivo-psychological elucidation of pure logic in the Logical Investigations to the transcendental phenomenology of the absolute consciousness of the objective correlates constituting themselves in its acts in Ideas I. (...) In this course Husserl began developing his transcendental phenomenology as the genuine realization of what had only been realized in fragmentary form in the Logical Investigations. Husserl considered that in the courses that he gave at the University of Göttingen he had progressed well beyond the insights of the Logical Investigations. Once he exposed the objective theoretical scaffolding needed to keep philosophers from falling into the quagmires of psychologism and skepticism, he set out on his voyage of discovery of the world of the intentional consciousness and to introduce the phenomenological analyses of knowledge that were to yield the general concepts of knowledge needed to solve the most recalcitrant problems of theory of knowledge understood as the investigation of the thorny problems involving the relationship of the subjectivity of the knower to the objectivity of what is known. This translation appears at a time when philosophers in English-speaking countries have heartily embraced the thoughts of Husserl’s German contemporary Gottlob Frege and his concerns. It is replete with insights into matters that many philosophers have been primed to appreciate out of enthusiasm for Frege’s ideas. Among these are: anti-psychologism, meaning, the foundations of mathematics, logic, science, and knowledge, his questions about sets and classes, intensions, identity, calculating with concepts, perspicuity, and even his idealism. (shrink)

: This paper discusses the origins of two key notions in Foucault's work up to and including The Archaeology of Knowledge. The first of these notions is the notion of "archaeology" itself, a form of historical investigation of knowledge that is distinguished from the mere history of ideas in part by its unearthing what Foucault calls "historical a prioris". Both notions, I argue, are derived from Husserlian phenomenology. But both are modified by Foucault in the light of Jean Cavaillès's critique (...) of Husserl's theory of science. On Husserl's view, we demand that propositions holding of scientific objects be intersubjective and invariant, but this demand conflicts with our immediate experience, which is essentially bound to a subject's perspective. Thus the mathematical and physical sciences must utilise formal languages to fix these truths independently of the thoughts of a particular subject. This necessary procedure leads to the sedimentation of these formal systems: we forget their source in the concrete experiences of individuals, and use them as purely technical means. The technique of reactivating the intentional acts in which sedimented formal systems originated is thus, in Fink's terminology, an archaeological method. Foucault and Cavaillès retain the general outlines of this archaeology of the sciences, but they reject its appeal to conscious acts of meaning, to what Cavaillès calls "the philosophy of consciousness". I conclude by discussing the implicit difficulties in the "linguistic transcendentalism" proposed as an alternative by these French critics of Husserl. (shrink)

Hermann Weyl was one of the early contributors to the mathematics of general relativity. This article argues that in 1929, for the formulation of a general relativistic framework of the Dirac equation, he both abolished and preserved in modified form the conceptual perspective that he had developed earlier in his “analysis of the problem of space.” The ideas of infinitesimal congruence from the early 1920s were aufgehoben in the general relativistic framework for the Dirac equation. He preserved the central (...) idea of gauge as a “purely infinitesimal” aspect of symmetries in a group extension schema. With respect to methodology, however, Weyl gave up his earlier preferences for relatively a-priori arguments and tried to incorporate as much empiricism as he could. This signified a clearly expressed empirical turn for him. Moreover, in this step he emphasized that the mathematical objects used for the representation of matter structures stood at the center of the construction, rather than interaction fields which, in the early 1920s, he had considered as more or less derivable from geometrico-philosophical considerations. (shrink)

A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the (...) ante-rem structuralist fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism. (shrink)

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely geometric (...) and combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

This paper has a two-fold objective: to provide a balanced, multi-faceted account of the origins of logicism; to rehabilitate Richard Dedekind as a main logicist. Logicism should be seen as more deeply rooted in the development of modern mathematics than typically assumed, and this becomes evident by reconsidering Dedekind's writings in relation to Frege's. Especially in its Dedekindian and Fregean versions, logicism constitutes the culmination of the rise of ?pure mathematics? in the nineteenth century; and this rise (...) brought with it an inter-weaving of methodological and epistemological considerations. The latter aspect illustrates how philosophical concerns can grow out of mathematical practice, as opposed to being imposed on it from outside. It also sheds new light on the legacy and the lasting significance of logicism today. (shrink)

Rudolf Carnap’s mature work on the logical reconstruction of scientific theories consists of two components. The first is the elimination of the theoretical vocabulary of a theory in terms of its Ramsification. The second is the reintroduction of the theoretical terms through explicit definitions in a language containing an epsilon operator. This paper investigates Carnap’s epsilon-reconstruction of theories in the context of pure mathematics. The main objective here is twofold: first, to specify the epsilon logic underlying his suggested (...) definition of theoretical terms and a suitable choice semantics for it. Second, to analyze whether Carnap’s approach is compatible with a structuralist conception of mathematics. (shrink)

This paper investigates the role of pictures in mathematics in the particular case of Cayley graphs—the graphic representations of groups. I shall argue that their principal function in that theory—to provide insight into the abstract structure of groups—is performed employing their visual aspect. I suggest that the application of a visual graph theory in the purely non-visual theory of groups resulted in a new effective approach in which pictures have an essential role. Cayley graphs were initially developed as exact (...) mathematical constructions. Therefore, they are legitimate components of the theory (combinatorial and geometric group theory) and the pictures of Cayley graphs are a part of practical mathematical procedures. (shrink)

In his book “Galileo’s Error”, Philip Goff lays out what he calls “foundations for a new science of consciousness”, which are decidedly anti-physicalist (panpsychist), motivated by a critique of Galileo’s distinction into knowable objective and unknowable subjective properties and Arthur Eddington’s argument for the limitation of purely structural (physical) knowledge. Here we outline an alternative theory, premised on the Interface Theory of Perception, that too subscribes to a “post-Galilean” research programme. However, interface theorists disagree along several lines. 1. They note (...) that Galileo’s distinction should be replaced by a truly non-dual account, referring to a difference of degree only. 2. They highly appreciate the role of mathematics, in particular when it comes to actually engage scientifically with consciousness. Some notable features of the interface theory are its skepticism towards our epistemic capacities and its rejection of the existence of a public, mind-independent reality. In addition, some interface theorists further employ a thin concept of “conscious agency” to ground their theory. The interface theory leaves open many of the problems of consciousness science (e.g. what is a “self”?) as questions for further (scientific, mathematical) research. (shrink)

In the Critique of Pure Reason, Kant develops a metametaphysical view concerning the domain and source of a distinctively metaphysical cognition of objects of experience, which is given in terms of an analysis of our representational capacity for thought, namely, the understanding, regarding its sub-capacities and their constitutive abilities and acts. In the Analytic of Concepts, more precisely, in the Metaphysical and Subjective Deductions of the Categories, Kant develops an elaborate account of the content and formation of those (...) metaphysical concepts of an object in general through which there is genuine philosophical cognition of objects of experience, namely, the categories. Relating the Analytic of Concepts to the Doctrine of Method, I give a detailed reconstruction of this account in contrast to both empirical and mathematical concepts. I do so by establishing, for the first time (in English) that and how the representational contents of the categories, as general representations of objects, derive from fundamental abilities and acts of a synthesis of an empirical intuition that the understanding contributes to experiences of objects, which thus yield the original contents of the categories. In particular, I argue that the threefold synthesis of the fundamental act-types of the apprehension, reproduction, and recognition of a manifold of sense-impressions, presented in the A-Deduction, holds the key to understanding the origin of the categories, or concepts of an object in general, investigated in their Metaphysical Deduction. It turns out that the threefold synthesis, as it is exercised regarding a manifold of sense-impressions in an empirical intuition of an object, can indeed account for the representational contents of the categories of Quality (apprehension), Quantity (reproduction), Relation (recognition), and Modality (relating to the object through a manifold of sense-impressions). In this sense, the categories are “concepts of synthesis” (A80/B106) or “concepts of the synthesis of possible sensations” (A723/B751). (shrink)

It is argued that geometrical intuition, as conceived in Kant, is still crucial to the epistemological foundations of mathematics. For this purpose, I have chosen to target one of the most sympathetic interpreters of Kant's philosophy of mathematics – Michael Friedman – because he has formulated the possible historical limitations of Kant's views most sharply. I claim that there are important insights in Kant's theory that have survived the developments of modern mathematics, and thus, that they are (...) not so intrinsically bound up with the logic and mathematics of Kant's time as Friedman will have it. These insights include the idea that mathematical knowledge relies on the manipulation of objects given in quasi-perceptual intuition, as Charles Parsons has argued, and that pure intuition is a source of knowledge of space itself that cannot be replaced by mere propositional knowledge. In particular, it is pointed out that it is the isomorphism between Kantian intuition and a spatial manifold that underlies both the epistemic intimacy of the most fundamental type of geometrical intuition as well as that of perceptual acquaintance. (shrink)

_The Universal_ proposes a radically new philosophical system that moves from ontology to ethics. Drawing on the work of De Beauvoir, Sartre, and Le Doeuff, among others, and addressing a range of topics from the Asian sex trade to late capitalism, quantum gravity, and Merleau-Ponty's views on cinema, Dorothea Olkowski stretches the mathematical, political, epistemological, and aesthetic limits of continental philosophy and introduces a new perspective on political structures. Straddling a course between formalism and conventionalism, Olkowski develops the concept of (...) an ontological unconscious that arises from our "sensible" relation to the world-the information we absorb and emit that affects our encounters with the environment and others. In this "realm of the senses," or the field of vulnerability defined by our experience with pleasure and pain, Olkowski is able to rethink the space-time relations put forth by Irigaray's notion of the "interval," Bergson's "recollection," Merleau-Ponty's idea of the "flesh," and Deleuze's "plane of immanence." This aesthetic sense is shared by all humankind and nonhuman entities in the organic and inorganic world. The sensible universal can be applied to categories of pure and practical reason; experiential binaries of male-female and subject-object; and issues of autonomy, moral laws, and the regulation of perception. (shrink)

In his 1978 paper “Mathematical Explanation”, Mark Steiner attempts to modernize the Aristotelian idea that to explain a mathematical statement is to deduce it from the essence of entities figuring in the statement, by replacing talk of essences with talk of “characterizing properties”. The language Steiner uses is reminiscent of language used for proofs deemed “pure”, such as Selberg and Erdős’ elementary proofs of the prime number theorem avoiding the complex analysis of earlier proofs. Hilbert characterized pure proofs (...) as those that use only “means that are suggested by the content of the theorem”, a characterization we have elsewhere called “topical purity”. In this paper we will examine the connection between Steiner’s account of mathematical explanation and topical purity. Are Steiner-explanatory proofs necessarily topically pure? Are topically pure proofs necessarily Steiner-explanatory? Answers to these questions will shed light on the general question of the relation between purity and explanatory power. (shrink)

In spite of successful tests, the standard cosmological model, the $$\varLambda$$ CDM model, possesses the most problematic concept: the initial singularity, also known as the big bang. In this paper—by adopting the Kantian difference between to think of an object and to cognize an object—it is proposed a degree of scientificity using fuzzy sets. Thus, the notion of initial singularity will not be conceived of as a scientific issue because it does not belong to the fuzzy set of what is (...) known. Indeed, the problematic concept of singularity is some sort of what Kant called the noumenon, but science, on the other hand, is constructed in the phenomenon. By applying the fuzzy degree of scientificity in cosmological models, one concludes that cosmologies with a contraction phase before the current expansion phase are potentially more scientific than the standard model. At the end of this article, it is shown that Kant’s first antinomy of pure reason indicates a limit to our cosmological models. (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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Daniel Isaacson has advanced an epistemic notion of arithmetical truth according to which the latter is the set of truths that we grasp on the basis of our understanding of the structure of natural numbers alone. Isaacson’s thesis is then the claim that Peano Arithmetic (PA) is the theory of finite mathematics, in the sense that it proves all and only arithmetical truths thus understood. In this paper, we raise a challenge for the thesis and show how it can (...) be overcome. We introduce the concept of purity for theories of arithmetic: a theory of arithmetic is pure when it only proves arithmetical truths. Then, we argue that, under Isaacson’s thesis, some PA-provable truths—including transfinite induction claims for infinite ordinals and some consistency statements—are seemingly not arithmetical in Isaacson’s sense, and hence that Isaacson’s thesis might entail the impurity of PA. Nonetheless, we conjecture that the advocate of Isaacson’s thesis can avoid this undesirable consequence: the arithmetical nature, as understood by Isaacson, of all contentious PA-provable statements can be justified. As a case study, we explore how this is done for transfinite induction claims with infinite ordinals below ε0. To this end, we show that the PA-proof of such claims employs exclusively resources from finite mathematics, and that ordinals below ε0 are finitary objects despite being infinite. (shrink)

In 1902 there arrived in Jena a letter from Russell laying out a proof that shattered Frege’s confidence in logicism, which is widely taken to be the doctrine according to which every truth of arithmetic is re-expressible without relevant loss as a provable truth about a purely logical object. Frege was persuaded that Russell had exposed a pathology in logicism, which faced him with the task of examining its symptoms, diagnosing its cause, assessing its seriousness, arriving at a treatment option, (...) and making an estimate of future prospects. The symptom was the contradiction that had crept into naïve set theory in the form of the set that provably is and is not its own member. The diagnosis was that it is caused by Basic Law V of the Grundgesetze (Frege, In: Grundgesetze der Arithmetik: Begrifsschriftlich abgeleitet, Jena: Herman Pohle, 1893/1903. Translated into English as Basic Laws of Arithmetic, also edited by Philip A. Ebert and Marcus Rossberg, with Crispin Wright, Oxford: Oxford University Press, 2013). Triage answers the question, “How bad is it?” The answer was that the contradiction irreparably destroys the logicist project. The treatment option was nil. The disease was untreatable. In due course, the prognosis turned out to be that a scaled-down Fregean logic could have an honourable life as a theory inference for various domains of mathematical discourse, but not for domains containing the logical objects required for logicism. Since there aren’t such objects, there aren’t such domains. On the face of it, Frege’s logicist collapse is astonishing. Why wouldn’t he have repaired the fault in Law V and gone back to the business of bringing logicism to an assured realization? In the course of our reflections, we will have nice occasion to consider the good it might have done Frege to have booked some time with Aristotle had he been able to. By the time we’re finished, we’ll have cause to think that in the end the Russell might well have begged the question against Frege. (shrink)

This volume features essays about and by Paul Benacerraf, whose ideas have circulated in the philosophical community since the early nineteen sixties, shaping key areas in the philosophy of mathematics, the philosophy of language, the philosophy of logic, and epistemology. The book started as a worskhop held in Paris at the Collège de France in May 2012 with the participation of Paul Benacerraf. The introduction addresses the methodological point of the legitimate use of so-called “Princess Margaret Premises” in drawing (...) philosophical conclusions from Gödel’s first incompleteness theorem. The book is then divided into three sections. The first is devoted to an assessment of the improved version of the original dilemma of “Mathematical Truth” due to Hartry Field: the challenge to the platonist is now to explain the reliability of our mathematical beliefs given the very subject matter of mathematics, either pure or applied. The second addresses the issue of the ontological status of numbers: Frege’s logicism, fictionalism, structuralism, and Bourbaki’s theory of structures are called up for an appraisal of Benacerraf’s negative conclusions of “What Numbers Could Not Be.” The third is devoted to supertasks and bears witness to the unique standing of Benacerraf’s first publication: “Tasks, Super-Tasks, and Modern Eleatics” in debates on Zeno’s paradox and associated paradoxes, infinitary mathematics, and constructivism and finitism in the philosophy of mathematics. Two yet unpublished essays by Benacerraf have been included in the volume: an early version of “Mathematical Truth” from 1968 and an essay on “What Numbers Could Not Be” from the mid 1970’s. A complete chronological bibliography of Benacerraf’s work to 2016 is provided.Essays by Jody Azzouni, Paul Benacerraf, Justin Clarke-Doane, Sébastien Gandon, Brice Halimi, Jon Pérez Laraudogoitia, Mary Leng, Antonio Leòn-Sànchez and Ana Leòn-Mejía, Marco Panza, Fabrice Pataut, Philippe de Rouilhan, Andrea Sereni, and Stewart Shapiro. (shrink)

As the word “optics” was understood from antiquity into and beyond the early modern period, it did not mean simply the physics and geometry of light, but meant the “theory of vision” and included what we should now call physiological and psychological aspects. From antiquity, these aspects were subject to geometrical analysis. Accordingly, the geometry of visual experience has long been an object of investigation. This chapter examines accounts of size and distance perception in antiquity (Euclid and Ptolemy) and the (...) Middle Ages (Ibn al-Haytham), before turning to "natural geometry" in Kepler and Descartes. It finds a purely mechanical realization of natural geometry in Descartes (primarily in his L'Homme, 1664), in which he conceives states of the visual system as varying in accordance with distance, much as his geometrical compasses vary their spatial relations in accordance with geometrical proportions. This reading challenges the notion that a two-dimensional sensation (without three-dimensional phenomenality) was universally accepted prior to the nineteenth century by positing a direct psychophysiological account of the experience of distance in Descartes. It thereby challenges the interpretations of Descartes on natural geometry of George Berkeley, Nancy Maull, Margaret Wilson, Erwin Panofsky, and Jonathan Crary. (shrink)

ABSTRACT I suggest how a broadly Kantian critique of classical logic might spring from reflections on constructibility conditions. According to Kant, mathematics is concerned with objects that are given through ‘arbitrary synthesis,’ in the form of ‘constructions of concepts’ in the medium of ‘pure intuition.’ Logic, by contrast, is narrowly constrained – it has no objects of its own and is fixed by the very forms of thought. That is why there is not much room for (...) developments within logic, as compared to the progress in mathematics. Kant’s view of logic remains critical, though – through considerations that are effectively on the scope and limits of classical logic and which play a part in his transcendental idealism. The most important ones are to be found in his critique of the use of reductio ad absurdum proofs in metaphysics and his solutions to the ‘antinomies of pure reason.’ Arguably, these considerations carry over to mathematics as well – by way of ‘analogues’ to the antinomies – in particular in resolving infinity paradoxes. (shrink)

We care not just how things are but how they could have been otherwise – about possibility and necessity as well as actuality. Many philosophers regard such talk as beyond the reach of respectable science, since observation tells us how things are but not how they could have been otherwise. I argue that, on the contrary, such criticisms are ill-founded: possibility and necessity are studied in natural science, for example through phase spaces, abstract mathematical representations of the possible states of (...) a physical system. The possibility is objective, not merely epistemic, though it may be more restricted than pure metaphysical possibility. The elements of a phase space are very similar to Kripke's possible worlds, despite being time slices rather than total histories. The absence of explicit modal operators in the mathematical models used by scientists does not show science to be non-modal; rather, it manifests reliance on a mathematical framework for theorizing about objective possibility similar to the mathematical framework of possible worlds model theory. (shrink)

The fundamental idea of a Neoaristotelian inherence ontology of mathematical entities parallels that of an Aristotelian approach to the ontology of universals. It is proposed that mathematical objects are nominalizations especially of dimensional and related structural properties that inhere as formal species and hence as secondary substances of Aristotelian primary substances in the actual world of existent physical spatiotemporal entities. The approach makes it straightforward to understand the distinction between pure and applied mathematics, and the otherwise enigmatic (...) success of applied mathematics in the natural sciences. It also raises an interesting set of challenges for conventional mathematics, and in particular for the ontic status of infinity, infinite sets and series, infinitesimals, and transfinite cardinalities. The final arbiter of all such questions on an Aristotelian inherentist account of the nature of mathematical entities are the requirements of practicing scientists for infinitary versus strictly finite mathematics in describing, explaining, predicting and retrodicting physical spatiotemporal phenomena. Following Quine, we classify all mathematics that falls outside of this sphere of applied scientific need as belonging to pure, and, with no prejudice or downplaying of its importance, ‘recreational’, mathematics. We consider a number of important problems in the philosophy of mathematics, and indicate how a Neoaristotelian inherence metaphysics of mathematical entities provides a plausible answer to Benacerraf’s metaphilosophical dilemma, pitting the semantics of mathematical truth conditions against the epistemic possibilities for justifying an abstract realist ontology of mathematical entities and truth conditions. (shrink)

This article first refines the question of Kant's relation to Hume's skepticism, and then considers the evidence for Kant's attitude toward Hume in three contexts: the A Critique, the Prolegomena, and the B Critique. My thesis is that in the A Critique Kant viewed skepticism positively, as a necessary reaction to dogmatism and a spur toward critique. In his initial statement of the critical philosophy Kant treated Hume as an ally in curbing dogmatism, but one who stopped short of what (...) was really needed: a full critique of reason, to establish the boundaries of metaphysical cognition. Kant found fault with Hume's analyses of cognition and experience, and specifically his failure to see the crucial importance of synthetic a priori cognition in metaphysics. In particular, he held that Hume's empiricist account of cognition could neither explain the synthetic a priori cognition actually found in mathematics and natural science, nor provide a principled account of the limits on what can be known--and what can be thought--through the pure concepts of the understanding. According to Kant, Hume therefore failed in his attempt to determine the limits of metaphysics, whereas Kant succeeded because his transcendental philosophy provided a thorough account of cognition, its structure and limits. In the Prolegomena and the B Critique Kant distinguished his position more sharply from Hume's. He also adopted a more negative attitude toward "skeptical idealism" than before; but he attributed such skepticism to Descartes, not Hume. Prior to the B Critique Kant did not see Hume as attacking natural science or ordinary cognition. In none of the three works was Kant's main aim to "answer the skeptic." His primary aim was what he said it was: to firmly establish the boundary of metaphysics, by discovering the elements of human cognition and fixing its proper domain. His purported discoveries about the limits of metaphysical cognition meant that the traditional objects of metaphysical knowledge, God, the soul, and the world as it is in itself, are unknowable, hence that traditional metaphysics itself is impossible. Besides settling the possibility or impossibility of metaphysics, his findings would also prevent the illegitimate extension of principles of sensibility to God and the noumenal self, an extension that would threaten the metaphysics of morals by incorrectly denying the thinkability of noumenal freedom, and that might otherwise lead to "materialism, fatalism, atheism, and freethinking unbelief" (B xxxiv). (shrink)

The paper investigates inaccessible set axioms and their consistency strength in constructive set theory. In ZFC inaccessible sets are of the form Vκ where κ is a strongly inaccessible cardinal and Vκ denotes the κth level of the von Neumann hierarchy. Inaccessible sets figure prominently in category theory as Grothendieck universes and are related to universes in type theory. The objective of this paper is to show that the consistency strength of inaccessible set axioms heavily depend on the context in (...) which they are embedded. The context here will be the theory CZF− of constructive Zermelo–Fraenkel set theory but without -Induction . Let INAC be the statement that for every set there is an inaccessible set containing it. CZF−+INAC is a mathematically rich theory in which one can easily formalize Bishop style constructive mathematics and a great deal of category theory. CZF−+INAC also has a realizability interpretation in type theory which gives its theorems a direct computational meaning. The main result presented here is that the proof theoretic ordinal of CZF−+INAC is a small ordinal known as the Feferman–Schütte ordinal Γ0. (shrink)

The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the known examples as (...) well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint, arXiv:1201.5221v1]. (shrink)

Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific (...) issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy. (shrink)

The general framework of this paper is a reformulation of Hilbert’s program using the theory of locales, also known as formal or point-free topology [P.T. Johnstone, Stone Spaces, in: Cambridge Studies in Advanced Mathematics, vol. 3, 1982; Th. Coquand, G. Sambin, J. Smith, S. Valentini, Inductively generated formal topologies, Ann. Pure Appl. Logic 124 71–106; G. Sambin, Intuitionistic formal spaces–a first communication, in: D. Skordev , Mathematical Logic and its Applications, Plenum, New York, 1987, pp. 187–204]. Formal topology (...) presents a topological space, not as a set of points, but as a logical theory which describes the lattice of open sets. The application to Hilbert’s program is then the following. Hilbert’s ideal objects are represented by points of such a formal space. There are general methods to “eliminate” the use of points, close to the notion of forcing and to the “elimination of choice sequences” in intuitionist mathematics, which correspond to Hilbert’s required elimination of ideal objects. This paper illustrates further this general program on the notion of valuations. They were introduced by Dedekind and Weber [R. Dedekind, H. Weber, Theorie des algebraischen Funktionen einer Veränderlichen, J. de Crelle t. XCII 181–290] to give a rigorous presentation of Riemann surfaces. It can be argued that it is one of the first example in mathematics of point-free representation of spaces [N. Bourbaki, Eléments de Mathématique. Algèbre commutative, Hermann, Paris, 1965, Chapitre 7]. It is thus of historical and conceptual interest to be able to represent this notion in formal topology. (shrink)

Gödel’s first incompleteness result from 1931 states that there are true assertions about the natural numbers which do not follow from the Peano axioms. Since 1931 many researchers have been looking for natural examples of such assertions and breakthroughs were obtained in the seventies by Jeff Paris [Some independence results for Peano arithmetic. J. Symbolic Logic 43 725–731] , Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977] and Laurie Kirby [L. Kirby, Jeff Paris, Accessible independence results for Peano Arithmetic, Bull. of (...) the LMS 14 285–293]) and Harvey Friedman [S.G. Simpson, Non-provability of certain combinatorial properties of finite trees, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 87–117; R. Smith, The consistency strength of some finite forms of the Higman and Kruskal theorems, in: Harvey Friedman’s Research on the Foundations of Mathematics, North-Holland, Amsterdam, 1985, pp. 119–136] who produced the first mathematically interesting independence results in Ramsey theory and well-order and well-quasi-order theory .In this article we investigate Friedman-style principles of combinatorial well-foundedness for the ordinals below ε0. These principles state that there is a uniform bound on the length of decreasing sequences of ordinals which satisfy an elementary recursive growth rate condition with respect to their Gödel numbers.For these independence principles we classify their phase transitions, i.e. we classify exactly the bounding conditions which lead from provability to unprovability in the induced combinatorial well-foundedness principles.As Gödel numbering for ordinals we choose the one which is induced naturally from Gödel’s coding of finite sequences from his classical 1931 paper on his incompleteness results.This choice makes the investigation highly non-trivial but rewarding and we succeed in our objectives by using an intricate and surprising interplay between analytic combinatorics and the theory of descent recursive functions. For obtaining the required bounds on count functions for ordinals we use a classical 1961 Tauberian theorem of Parameswaran which apparently is far remote from Gödel’s theorem. (shrink)

François Viète is often regarded as the first modern mathematician on the grounds that he was the first to develop the literal notation, that is, the use of two sorts of letters, one for the unknown and the other for the known parameters of a problem. The fact that he achieved neither a modern conception of quantity nor a modern understanding of curves, both of which are explicit in Descartes’ Geometry, is to be explained on this view “by an incomplete (...) symbolization rather than by any obstacle intrinsic in the system.” Descartes’ Geometry provides only a “clearer expression” of themes already sounded in Viète’s work, one that perfects Viète’s literal calculus and gives it “its modern form”; it merely continues the “‘new’ and ‘pure’ algebra which Viète first established as the ‘general analytic art’.” It can seem, furthermore, that this must be right, that had there been some obstacle intrinsic to Viète’s system that barred the way to a modern conception of quantity and a modern understanding of curves, then Descartes’ Geometry would have had to have taken a very different form than it did. As it was, Descartes had only to improve Viète’s symbolism, free himself of the last vestiges of the ancient view of geometrical and arithmetical objects, and apply the new symbolism to the study of curves in order to achieve what Viète did not but could have. But what, really, is the status of this “could have”; what would it actually have taken for Viète to achieve Descartes’ results in the Geometry? The interest of the question lies in its potential to better our understanding of the nature of modern mathematics. (shrink)

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

What are scientific models? Philosophers of science have been trying to answer this question during the last three decades by putting forward a number of different proposals. Some say that models are best understood as abstract Platonic objects or fictional entities akin to Sherlock Holmes, while others focus on their mathematical nature and see them as set theoretical structures. Although each account has its own strengths in offering various insights on the nature of models, several objections have been raised (...) against these views which still remain unanswered, making the debate on the ontology of models seem unresolvable. The primary aim of this paper is to show that a large part of these difficulties stems from an inappropriate reading of the main question on the ontology of models as a purely metaphysical question. Building on Carnap, it is argued that the question of the ontology of scientific models is either an internal theoretical question within an already accepted linguistic framework or an external practical question regarding the choice of the most appropriate form of language in order to describe and explain the practice of scientific modelling. The main implication of this view is that the question of the ontology of models becomes a means of probing other related questions regarding the overall practice of scientific modelling, such as questions on the capacity of models to provide knowledge and the relation of models with background theories. (shrink)

Neurosemantics is not yet a common term and in current neuroscience and philosophy it is used with two different sorts of objectives. One deals with the meaning of the electrical and the chemical activities going on in neural circuits. This way of using the term regards the project of explaining linguistic meaning in terms of the computations done by the brain. This book explores this second sense of neurosemantics, but in doing so, it will address much of the first as (...) well, for we believe that the capacity of neural circuits to support linguistic meaning, hinges on their peculiar role in coding entities and facts of the world. It is an enterprise at the edge of the available state-of-the-art knowledge in neuroscience and specifically, in the growing understanding of brain computational mechanisms. We conceive neurosemantics, however, as the natural evolution of a long standing project that began in the early days of Boole’s logic, the idea that semantics can be construed and explained in mathematical terms. Classical formal semantics, for a very long time, excluded from the analysis of language any account of mental processes, which on the contrary, became the central focus during the cognitive turn. Cognitive semantics, however, failed to provide a rigorous mathematical framework for semantic processes. Today, it is possible to begin explaining language by way of a new mathematical foundation, one that is empirically grounded in how the brain computes: neurosemantics. The way this book intends to contribute is twofold. One is to present a series of existing examples of neurosemantics in practice: early models addressing aspects of linguistic semantics purely in neurocomputational terms. The other is to try to identify the principles upon which models of this kind can be constructed, and their corresponding neural bases. (shrink)

This paper argues that Weyl took formalism to prevail over intuitionism with respect to supporting scientific objectivity, rather than grounding classical mathematics. This is the respect in which he came to reject pure phenomenology as well.

Show more ▾ There are various dichotomies in Kant’s philosophy: sensibility vs. rationality, nature vs. freedom, cognition vs. morality, noumenon vs. phenomenon, among others. There are also different ways of mediating these dichotomies, which is the systematic undertaking of Kant’s Critique of the Power of Judgment. One of the most important concepts in this work is the sublime, which exemplifies the connections between the different dichotomies; this fact means the concept’s construction is full of tension. On the one hand, as (...) a pure reflection of aesthetic judgment the sublime must be without interest or purpose, but on the other hand it is “based on the concept of reason” (KU AA:292). On the one hand, the sublime “represents merely the subjective play of the powers of the mind (imagination and reason) as harmonious” (KU AA5:258), but on the other hand, reason “exercises dominion over sensibility” and the imagination is “purposively determined in accordance with a law” of reason (KU AA5:268f). Taking into account these problems concerning the essential definition the sublime, this paper will first illustrate how the sublime embodies the a priori principle of aesthetic judgment through contrasting the judgment of the sublime with the judgment of taste in order to establish a basic logical frame for the judgment of the sublime. Second, this paper redefines the boundary between the mathematically and dynamically sublime in order to reveal both the coexistence of contemplation and movement within the sublime and the unrevealed function of reason and imagination. Finally, contrasting the sublime with moral feeling, this paper elaborates the turning-structure (from sensibility to rationality and from object-intuition to idea-exhibition) of the sublime. (shrink)