This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open (...) versus closed intervals is discussed. Finally, it is suggested that one reason there is a common structure between Aristotle's account of the continuum and that found in Cantor's definition of the real number continuum is that our intuitions about the continuum have their source in the experience of the real spatiotemporal world. A plea is made to consider Aristotle's abstractionist philosophy of mathematics anew. (shrink)
In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s (...) theory of rest and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises. (shrink)
We often speak of 'Eastern' and 'Western' philosophy, yet it is not always easy to distinguish the key factors that justify this distinction. This essay explores the very different conceptions of the continuum that underlie these two traditions of thought and knowledge. The views of Hermann Weyl are given and it is proposed that they are correct. Attention is drawn to the mutually-exclusive visions of the continuum that separate the philosophies of East and West, and that give us (...) a way of pinning down a definition of these vague geographical terms so as to give them, in at least one respect, a clear philosophical and scientific meaning. (shrink)
In a recent article, Christopher Ormell argues against the traditional mathematical view that the real numbers form an uncountably inﬁnite set. He rejects the conclusion of Cantor’s diagonal argument for the higher, non-denumerable inﬁnity of the real numbers. He does so on the basis that the classical conception of a real number is mys- terious, ineffable, and epistemically suspect. Instead, he urges that mathematics should admit only ‘well-deﬁned’ real numbers as proper objects of study. In practice, this means excluding as (...) inadmissible all those real numbers whose decimal expansions cannot be calculated in as much detail as one would like by some rule. We argue against Ormell that the classical realist account of the continuum has explanatory power in mathematics and should be accepted, much in the same way that "dark matter" is posited by physicists to explain observations in cosmology. In effect, the indefinable real numbers are like the "dark matter" of real analysis. (shrink)
In this paper, I propose a logical-cognitive approach to argumentation and advocate an idea that argumentation presupposes that intelligent agents engaged in it are cognitively diverse. My approach to argumentation allows drawing distinctions between justification, conviction and persuasion as its different kinds. In justification agents seek to verify weak or strong coherency of an agent’s position in a dialogue. In conviction they argue to modify their partner’s position by means of demonstrating weak or strong cogency of their positions before a (...) ‘rational judge’. My approach to argumentation employs a ‘light’ version of Dung’s abstract argumentative frameworks. It is based on Stich’s idea of agents’ cognitive diversity the epistemic aspect of which is argued to be close to Pavilionis’s conception of meaning continuum. To illustrate my contributions I use an example based on the Kitchen Debate (1959) between Khrushchev and Nixon. (shrink)
We characterize those identities and independencies which hold for all probability functions on a unary language satisfying the Principle of Atom Exchangeability. We then show that if this is strengthen to the requirement that Johnson's Sufficientness Principle holds, thus giving Carnap's Continuum of inductive methods for languages with at least two predicates, then new and somewhat inexplicable identities and independencies emerge, the latter even in the case of Carnap's Continuum for the language with just a single predicate.
This paper discusses an argument for the reality of the classical mathematical continuum. An inference to the best explanation type of argument is used to defend the idea that real numbers exist even when they cannot be constructively specified as with the "indefinable numbers".
This book is about experiential content: what it is; what kind of account can be given of it. I am concerned with identifying and attacking one main view - I call it the inferentialist proposal. This account is central to the philosophy of mind, epistemology and philosophy of science and perception. I claim, however, that it needs to be recast into something far more subtle and enriched, and I attempt to provide a better alternative in these pages. The inferentialist proposal (...) holds that experiential content is necessarily under¬pinned by sophisticated cognitive influences. My alternative, the continuum theory, holds that these influences are relevant to experience only at certain levels of organisation and that at other levels there are contents which such features do not capture at all. Central to my account is that there are degrees to which cognitive influences affect experiential content; indeed, for the most part, experience is an amalgam of both inferential and non-inferential features. I claim that the inferentialist proposal is fundamentally flawed and deserves replacement, and I argue that my alternative fills the hollow that remains. The book is divided into four sections. In Part I, Chapter 1, I introduce two traditionally rival views of experiential content. In Chapter 2, I develop my continuum alternative. Chapter 3 assesses the relationship between experience and language, while Chapter 4 explores the relationship between beliefs and experience. The overall argument is that it has been a mistake to understand experience simply in inferential or non-inferential terms. In Part II, I examine the structure of mental content. Chapter 5 is concerned with the kinds of experiences which escape the inferentialist analysis. Chapter 6 considers Kant’s metaphysic of experience counterpointed to Lorenz’s reading of his work in the light of evolutionary biology. Chapter 7 treats animal experience in relation to the continuum view I am developing, while Chapter 8 reviews Fodor’s contribution to perceptual psychology. It is argued that the view of experiential content being developed is both consistent with empirical data on informationally local perceptual sub-systems, but also accords well with evolutionary theory and a naturalist interpretation of Kant’s taxonomy. Part III deals with inferentialism in the philosophy of science. In Chapter 9, I assess the theory dependence of observation thesis as it is advanced by Paul Feyerabend. I bring out of his account a subtle confusion concerning the importance of inference in the context of scientific inquiry. Part IV deals with the issue of experience in the philosophy of mind. In Chapter 10, I look at Wilfred Sellars’s attack on sense data theories. Chapter 11 confronts Paul Churchland’s treatment of ‘folk psychology’ while Chapter 12 isolates the issue of experiential qualia and the position of property dualism. I offer a critical review of Thomas Nagel’s work in this chapter and claim that his position can be read in a way which is consistent with the continuum account I am developing. I conclude the book in the usual fashion with a summary of the central claims. (shrink)
We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)
What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock (...) of higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)
In this paper a new model of mind is proposed, to do so, at first it was assumed that our physical world a new structure and the mind defined in this context. In this model, the planets are massive curvature of time-space continuum that has made a trapping physical reality that we are located within. Then the mind is defined as an hourglass structure with half bulb within the physical reality and half out of it. This model with attention (...) to figures perhaps might be able to explain some mental states and disorders. (shrink)
Thaler and Sunstein advocate 'libertarian paternalism'. A libertarian paternalist changes the conditions under which people act so that their cognitive biases lead them to choose what is best for themselves. Although libertarian paternalism manipulates people, Thaler and Sunstein say that it respects their autonomy by preserving the possibility of choice. Conly argues that libertarian paternalism does not go far enough, since there is no compelling reason why we should allow people the opportunity to choose to bring disaster upon themselves if (...) sometimes they will make the wrong decision. She defends 'coercive paternalism'. The present paper argues that errors in reasoning are not due only to cognitive biases. People also make errors because they have an insufficient level of general intelligence. Intelligence is distributed on a continuum. Those who fall on higher levels of the continuum have greater abilities, in certain contexts, to reason about both their own and others' interests. Coercive paternalism may sometimes be appropriate to prevent less intelligent people from engaging in self-destructive behavior due to errors of reasoning. (shrink)
This dynamical interpretation of the continuum is based on a threefold perspective. First, detailed differentiation of all standard realms of Leibnizian Weltanschauung – (R real), (P phenomenal), (I ideal). Second, analysis of the scope of the Law of Continuity famously formulated by Leibniz and mapping it onto this (RPI) structure. Third, finding the precise place of dynamics and force in this (RPI) continuum.
When so much is being written on conscious experience, it is past time to face the question whether experience happens that is not conscious of itself. The recognition that we and most other living things experience non-consciously has recently been firmly supported by experimental science, clinical studies, and theoretic investigations; the related if not identical philosophic notion of experience without a subject has a rich pedigree. Leaving aside the question of how experience could become conscious of itself, I aim here (...) to demonstrate that the terms experience and consciousness are not interchangeable. Experience is a notoriously difficult concept to pin down, but I see non-conscious experience as based mainly in momentary sensations, relational between bodies or systems, and probably common throughout the natural world. If this continuum of experience — from non-conscious, to conscious, to self-transcending awareness — can be understood and accepted, radical constructivism (the “outside” world as a construct of experience) will gain a firmer foundation, panexperientialism (a living universe) may gain credibility, and psi will find its medium. (shrink)
Albert Lautman. Mathematics, Ideas and the Physical Real. Simon B. Duffy, trans. London and New York: Continuum, 2011. 978-1-4411-2344-2 (pbk); 978-1-44114656-4 (hbk); 978-1-44114433-1 (pdf e-bk); 978-1-44114654-0 (epub e-bk). Pp. xlii + 310.
This thesis is about experiential content: what it is; what kind of account can be given of it. I am concerned with identifying and attacking one main view - I call it the inferentialist proposal. This account is central to the philosophy of mind, epistemology and philosophy of science and perception. I claim, however, that it needs to be recast into something far more subtle and enriched, and I attempt to provide a better alternative in these pages. The inferentialist proposal (...) holds that experiential content is necessarily under¬pinned by sophisticated cognitive influences. My alternative, the continuum theory, holds that these influences are relevant to experience only at certain levels of organisation and that at other levels there are contents which such features do not capture at all. Central to my account is that there are degrees to which cognitive influences affect experiential content; indeed, for the most part, experience is an amalgam of both inferential and non-inferential features. I claim that the inferentialist proposal is fundamentally flawed and deserves replacement, and I argue that my alternative fills the hollow that remains. The thesis is divided into four sections. In Part I, Chapter 1, I introduce two traditionally rival views of experiential content. In Chapter 2, I develop my continuum alternative. Chapter 3 assesses the relationship between experience and language, while Chapter 4 explores the relationship between beliefs and experience. The overall argument is that it has been a mistake to understand experience simply in inferential or non-inferential terms. In Part II, I examine the structure of mental content. Chapter 5 is concerned with the kinds of experiences which escape the inferentialist analysis. Chapter 6 considers Kant’s metaphysic of experience counterpointed to Lorenz’s reading of his work in the light of evolutionary biology. Chapter 7 treats animal experience in relation to the continuum view I am developing, while Chapter 8 reviews Fodor’s contribution to perceptual psychology. It is argued that the view of experiential content being developed is both consistent with empirical data on informationally local perceptual sub-systems, but also accords well with evolutionary theory and a naturalist interpretation of Kant’s taxonomy. Part III deals with inferentialism in the philosophy of science. In Chapter 9, I assess the theory dependence of observation thesis as it is advanced by Paul Feyerabend. I bring out of his account a subtle confusion concerning the importance of inference in the context of scientific inquiry. Part IV deals with the issue of experience in the philosophy of mind. In Chapter 10, I look at Wilfred Sellars’s attack on sense data theories. Chapter 11 confronts Paul Churchland’s treatment of ‘folk psychology’ while Chapter 12 isolates the issue of experiential qualia and the position of property dualism. I offer a critical review of Thomas Nagel’s work in this chapter and claim that his position can be read in a way which is consistent with the continuum account I am developing. I conclude the thesis in the usual fashion with a summary of the central claims. (shrink)
According to Grice's “Modified Occam's Razor”, in case of uncertainty between the implicature account and the polysemy account of word uses it is parsimonious to opt for the former. However, it is widely agreed that uses can be partially conventionalised by repetition. This fact, I argue, raises a serious problem for MOR as a methodological principle, but also for the substantial notion of implicature in lexical pragmatics. In order to overcome these problems, I propose to reinterpret implicatures in terms of (...) implicature-like effects delivered by non-inferential processes. (shrink)
Since the 1950`s in Britain, and perhaps in the rest of the world, the term pluralism is almost invariably associated with the name of Isaiah Berlin and his formulation of ‘value pluralism’. The core idea is that values (but also, on some interpretations, ends, duties and obligations) are irreducibly plural and heterogeneous, and nevertheless objective.
A non-relativistic quantum mechanical theory is proposed that describes the universe as a continuum of worlds whose mutual interference gives rise to quantum phenomena. A logical framework is introduced to properly deal with propositions about objects in a multiplicity of worlds. In this logical framework, the continuum of worlds is treated in analogy to the continuum of time points; both “time” and “world” are considered as mutually independent modes of existence. The theory combines elements of Bohmian mechanics (...) and of Everett’s many-worlds interpretation; it has a clear ontology and a set of precisely defined postulates from where the predictions of standard quantum mechanics can be derived. Probability as given by the Born rule emerges as a consequence of insufficient knowledge of observers about which world it is that they live in. The theory describes a continuum of worlds rather than a single world or a discrete set of worlds, so it is similar in spirit to many-worlds interpretations based on Everett’s approach, without being actually reducible to these. In particular, there is no splitting of worlds, which is a typical feature of Everett-type theories. Altogether, the theory explains (1) the subjective occurrence of probabilities, (2) their quantitative value as given by the Born rule, and (3) the apparently random “collapse of the wavefunction” caused by the measurement, while still being an objectively deterministic theory. (shrink)
ABSTRACT In previous work I explored how Integral Theory can be applied as a metatheoretical and transdisciplinary framework, in an attempt to arrive at an integrally informed metatheory of addiction. There was an overemphasis on Integral Methodological Pluralism in that thread of research, without clarifying the ontological pluralism of addiction as a multiple object enacted by various methodologies. To arrive at a comprehensive integral metatheory and integral ontology of addiction, I believe it is necessary to include the conception of Integral (...) Pluralism and Integral Enactment Theory as posited by Sean Esbjörn-Hargens. Integral Enactment Theory highlights the phenomenon of addiction as a multiple and dynamic object arising along a continuum of ontological complexity; it adeptly points out how etiological models “co-arise” in relation to methodology (methodological pluralism) to enact a particular reality of addiction (ontological pluralism) while being mediated by the worldview of the subject (epistemological pluralism) applying the method. This article briefly explores the significance of including Integral Pluralism and Integral Enactment Theory in the quest of an integral metatheory and integral ontology of addiction. (shrink)
La conciencia falogocéntrica se funda en el a priori incondicional de carácter ético-metafísico que sostiene la criminalización universal del ser como efecto de una culpa o caída original. Nacer supone para esta conciencia un pecado y la realidad material representa para ella un lugar de exilio, extrañeza y alienación. Sin embargo, cuando uno retrocede sobre la protohistoria de la conciencia humana, sorprende la determinación de integridad y sacralidad que el pensamiento primitivo le atribuye a lo real. Para la conciencia primitiva, (...) la realidad nace en el parto de un seno infinito que la alimenta, cobija y bendice sus propios frutos. En lo que sigue, intentaremos remontarnos a la protohistoria de esta conciencia a fin de reconsiderar las condiciones de posibilidad de la inocencia metafísica y ética, o bien, el modo de ser de la inocencia original. (shrink)
It is often alleged that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a (...) mathematical hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)
This paper suggests that time could have a much richer mathematical structure than that of the real numbers. Clark & Read (1984) argue that a hypertask (uncountably many tasks done in a finite length of time) cannot be performed. Assuming that time takes values in the real numbers, we give a trivial proof of this. If we instead take the surreal numbers as a model of time, then not only are hypertasks possible but so is an ultratask (a sequence which (...) includes one task done for each ordinal number—thus a proper class of them). We argue that the surreal numbers are in some respects a better model of the temporal continuum than the real numbers as defined in mainstream mathematics, and that surreal time and hypertasks are mathematically possible. (shrink)
This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.
It is traditionally thought that metaphorical utterances constitute a special— nonliteral—kind of departure from lexical constraints on meaning. Dan Sperber and Deirdre Wilson have been forcefully arguing against this: according to them, relevance theory’s comprehension/interpretation procedure for metaphorical utterances does not require details specifi c to metaphor (or nonliteral discourse); instead, the same type of comprehension procedure as that in place for literal utterances covers metaphors as well. One of Sperber and Wilson’s central reasons for holding this is that metaphorical (...) utterances occupy one end of a continuum that includes literal, loose and hyperbolic utterances with no sharp boundaries in between them. Call this the continuum argument about interpreting metaphors. My aim is to show that this continuum argument doesn’t work. For if it were to work, it would have an unwanted consequence: it could be converted into a continuum argument about interpreting linguistic errors, including slips of the tongue, of which malaprops are a special case. In particular, based on the premise that the literal–loose–metaphorical continuum extends to malaprops also, we could conclude that the relevance-theoretic comprehension procedure for malaprops does not require details specifi c to linguistic errors, that is, details beyond those already in place for interpreting literal utterances. Given that we have good reason to reject this conclusion, we also have good reason to rethink the conclusion of the continuum argument about interpreting metaphors and consider what additional (metaphor-specifi c) details—about the role of constraints due to what is lexically encoded by the words used—might be added to relevance-theoretic comprehension procedures. (shrink)
It is argued that colour name strategy, object name strategy, and chunking strategy in memory are all aspects of the same general phenomena, called stereotyping, and this in turn is an example of a know-how representation. Such representations are argued to have their origin in a principle called the minimum duplication of resources. For most the subsequent discussions existence of colour name strategy suffices. It is pointed out that the BerlinA- KayA universal partial ordering of colours and the frequency of (...) traffic accidents classified by colour are surprisingly similar; a detailed analysis is not carried out as the specific colours recorded are not identical. Some consequences of the existence of a name strategy for the philosophy of language and mathematics are discussed: specifically it is argued that in accounts of truth and meaning it is necessary throughout to use real numbers as opposed to bi-valent quantities; and also that the concomitant label associated with sentences should not be of unconditional truth, but rather several real-valued quantities associated with visual communication. The implication of real-valued truth quantities is that the Continuum Hypothesis of pure mathematics is side-stepped, because real valued quantities occur ab initio. The existence of name strategy shows that thought/sememes and talk/phonemes can be separate, and this vindicates the assumption of thought occurring before talk used in psycho-linguistic speech production models. (shrink)
Leeuwenhoekin kokeilut mikroskoopilla 1600-luvun lopulla olivat G. W. Leibnizille suuri innoituksen lähde. Monadologia-teoksessaan Leibniz hehkutti keksinnön merkitystä ja antoi ymmärtää, että sillä löydetyt pikkuruiset eliöt todistivat hänen metafyysisen pluralisminsa oikeaksi. Hänen mukaansa "huomataan, että pienimmässäkin osasessa ainetta on kokonainen elävien olioiden, eläinten, entelekhioiden ja sielujen maailma." Näin Leibnizin ajatus elämän jatkumosta sai uutta pontta. -/- Keksinnön vaikutus näkyy myös Leibnizin teoksessa Uusia esseitä inhimillisestä ymmärryksestä, jossa hän esittelee pienet perseptiot, joita voidaan pitää tietoteoreettisena vastineena pieneliöille. -/- Tarkastelen esitelmässäni Leibnizin reaktioita (...) mikroskooppiin ja sen antamaan uuteen kuvaan elämästä ja keskustelen joistakin Leibnizin filosofian piirteistä, joissa voidaan huomata uuden keksinnön vaikutusta. -/- An article on the continuum of life in Leibniz's philosophy. (shrink)
A number of general theories of physics provide a model for the fundamental rules that govern our universe, becoming a structural framework to which the new discoveries must conform. The theory of relativity is such a general theory. The theory of relativity is a complex theoretical framework that facilitates the understanding of the universal laws of physics. It is based on the curved space-time continuum fabric abstract concept, and it is well suited for interpreting cosmic events. More so, a (...) general theory based on abstract concepts and imagination facilitates the emergence of countless new extravagant theories. A new simplified theory of the natural world is necessary, a simple theory that provides a verifiable framework on which new discoveries can be integrated. The paper describes a view of the abstract time/space concept, and also a very simple model of our ever-changing universe. Views of physicists, mathematicians, chemists, engineers and of course philosophers have to be all in harmony with such a theory. We leave in a beautiful, uniform, and logical world. We live in a world where everything probable is possible. Contact email: gondork@yahoo.com . (shrink)
In order to explain Wittgenstein’s account of the reality of completed infinity in mathematics, a brief overview of Cantor’s initial injection of the idea into set- theory, its trajectory and the philosophic implications he attributed to it will be presented. Subsequently, we will first expound Wittgenstein’s grammatical critique of the use of the term ‘infinity’ in common parlance and its conversion into a notion of an actually existing infinite ‘set’. Secondly, we will delve into Wittgenstein’s technical critique of the concept (...) of ‘denumerability’ as it is presented in set theory as well as his philosophic refutation of Cantor’s Diagonal Argument and the implications of such a refutation onto the problems of the Continuum Hypothesis and Cantor’s Theorem. Throughout, the discussion will be placed within the historical and philosophical framework of the Grundlagenkrise der Mathematik and Hilbert’s problems. (shrink)
I prove both the mathematical conjectures P ≠ NP and the Continuum Hypothesis are eternally unprovable using the same fundamental idea. Starting with the Saunders Maclane idea that a proof is eternal or it is not a proof, I use the indeterminacy of human biological capabilities in the eternal future to show that since both conjectures are independent of Axioms and have definitions connected with human biological capabilities, it would be impossible to prove them eternally without the creation and (...) widespread acceptance of new axioms. I also show that the same fundamental concepts cannot be used to demonstrate the eternal unprovability of many other mathematical theorems and open conjectures. Finally I investigate the idea’s implications for the foundations of mathematics including its relation to Godel’s Incompleteness Theorem and Tarsky’s Undefinability Theorem. (shrink)
The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathemat- ical and moral disagreement is not as straightforward as those arguments present it. In particular, I (...) argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. (shrink)
A prominent objection against the logicality of second-order logic is the so-called Overgeneration Argument. However, it is far from clear how this argument is to be understood. In the first part of the article, we examine the argument and locate its main source, namely, the alleged entanglement of second-order logic and mathematics. We then identify various reasons why the entanglement may be thought to be problematic. In the second part of the article, we take a metatheoretic perspective on the matter. (...) We prove a number of results establishing that the entanglement is sensitive to the kind of semantics used for second-order logic. These results provide evidence that by moving from the standard set-theoretic semantics for second-order logic to a semantics which makes use of higher-order resources, the entanglement either disappears or may no longer be in conflict with the logicality of second-order logic. (shrink)
Set-theoretic pluralism is an increasingly influential position in the philosophy of set theory (Balaguer [1998], Linksy and Zalta [1995], Hamkins [2012]). There is considerable room for debate about how best to formulate set-theoretic pluralism, and even about whether the view is coherent. But there is widespread agreement as to what there is to recommend the view (given that it can be formulated coherently). Unlike set-theoretic universalism, set-theoretic pluralism affords an answer to Benacerraf’s epistemological challenge. The purpose of this paper is (...) to determine what Benacerraf’s challenge could be such that this view is warranted. I argue that it could not be any of the challenges with which it has been traditionally identified by its advocates, like of Benacerraf and Field. Not only are none of the challenges easier for the pluralist to meet. None satisfies a key constraint that has been placed on Benacerraf’s challenge. However, I argue that Benacerraf’s challenge could be the challenge to show that our set-theoretic beliefs are safe – i.e., to show that we could not have easily had false ones. Whether the pluralist is, in fact, better positioned to show that our set-theoretic beliefs are safe turns on a broadly empirical conjecture which is outstanding. If this conjecture proves to be false, then it is unclear what the epistemological argument for set-theoretic pluralism is supposed to be. (shrink)
For Anaxagoras, both before the beginning of the world and in the present, “all is together” and “everything is in everything.” Various modern interpretations abound regarding the identity of this “mixture.” It has been explained as an aggregation of particles or as a continuous “fusion” of different sorts of ingredients. However—even though they are not usually recognized as a distinct group—there are a number of other scholars who, without seemingly knowing each other, have offered a different interpreta- tion: Anaxagoras’ mixture (...) as an “interpenetration” of different ingredients, which are as far-extended as the whole mixture is. As a result, there are different entities occupying the same place at the same time. This explanation assigns to Anaxagoras the same model of mixture which was later used by the Stoics. A new book by Marmodoro helps us to clarify this position. (shrink)
I shall attempt in what follows to show how mereology, taken together with certain topological notions, can yield the basis for future investigations in formal ontology. I shall attempt to show also how the mereological framework here advanced can allow the direct and natural formulation of a series of theses – for example pertaining to the concept of boundary – which can be formulated only indirectly (if at all) in set-theoretic terms.
The philosophy of mathematics has been accused of paying insufficient attention to mathematical practice: one way to cope with the problem, the one we will follow in this paper on extensive magnitudes, is to combine the `history of ideas' and the `philosophy of models' in a logical and epistemological perspective. The history of ideas allows the reconstruction of the theory of extensive magnitudes as a theory of ordered algebraic structures; the philosophy of models allows an investigation into the way epistemology (...) might affect relevant mathematical notions. The article takes two historical examples as a starting point for the investigation of the role of numerical models in the construction of a system of non-Archimedean magnitudes. A brief exposition of the theories developed by Giuseppe Veronese and by Rodolfo Bettazzi at the end of the 19th century will throw new light on the role played by magnitudes and numbers in the development of the concept of a non-Archimedean order. Different ways of introducing non-Archimedean models will be compared and the influence of epistemological models will be evaluated. Particular attention will be devoted to the comparison between the models that oriented Veronese's and Bettazzi's works and the mathematical theories they developed, but also to the analysis of the way epistemological beliefs affected the concepts of continuity and measurement. (shrink)
At Republic 435c-d and again at 504b-e, Plato has Socrates object to the city/soul analogy and declare that a “longer way” is necessary for gaining a more “exact grasp” of the soul. I argue that it is in the Philebus, in Socrates’ presentation of the “god-given” method of dialectic and in his distinctions of the kinds of pleasure and knowledge, that Plato offers the resources for reaching this alternative account. To show this, I explore (1) the limitations of the tripartition (...) of the soul that Socrates’ own objections in the Republic suggest, (2) the route of the “longer way” through the Eleatic dialogues to the Philebus, (3) the procedures that constitute the “god-given” method and the structure of the eidetic field it discloses, and (4) the resources that, considered in light of the method, Socrates’ distinctions of the kinds of pleasure and knowledge provide for the more “exact grasp” of the soul. (shrink)
A study of the significance of Plato's resumption of the simile of model and likeness in the Timaeus, with attention to the place of the Timaeus in the "longer way" that Plato has Socrates announce in the Republic. The reader embarked on the "longer way," I argue, will find in the accounts of the elements and of the kinds of animals unannounced but detailed exhibitions of the "god-given" method of dialectic that Plato has Socrates announce in the Philebus.
Must the interpreter of the Platonic dialogues choose between the so-called "unwritten teachings" reported by Aristotle in Metaphysics A6 and the dialogues? I argue, on the contrary, that a reading of the dialogues that is sensitive to their pedagogical irony will find the "unwritten teachings" exhibited in them. I identify the key teachings in Metaphysics A6, show how the Parmenides and the Philebus point to them, and explicate a full exhibition of them in the Statesman.
This monography provides an overview of the conceptual developments that leads from the traditional views of infinite (and their paradoxes) to the contemporary view in which those old paradoxes are solved but new problems arise. Also a particular insight in the problem of continuity is given, followed by applications in theory of computability.
Aristotle describes time as continuous (cf. Phys. 219a 10-15). We argue here, first, that the time's continuity and magnitude's continuity differ, even though time's continuity depends on magnitude: actually a magnitude can be divided, that is, can fail to be continuous, but not time: time can be never actually divided, because an actual division in time would imply something like a real point in which time is denied, and that is impossible according to Aristotle (cf. Phys. 251b 10-28). The only (...) divisions that time as continuous would admit are those made by the soul, i. e. only logical divisions. Secondly, we indicate that time could be considered from two perspectives: a logical –or, in a way, mathematical–, and other in the strict sense real. These two perspectives would be suggested by the distinction of meanings that Aristotle makes about the now (cf. Phys. 219b 10-13). The ontological perspective would show certain nexus between the Aristotle's analysis of time with those of contemporary Physics. (shrink)
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