The ramifications are explored of taking physical theories to commit their advocates only to ‘physically real’ entities, where ‘physically real’ means ‘causally efficacious’ (e.g., actual particles moving through space, such as dust motes), the ‘physically significant’ (e.g., centers of mass), and the merely mathematical—despite the fact that, in ordinary physical theory, all three sorts of posits are quantified over. It's argued that when such theories are regimented, existential quantification, even when interpreted ‘objectually’ (that is, in terms of satisfaction via (...) variables, rather than by substitution-instances) need not imply any ontological commitments. (shrink)

First-order logic has limited existential import: the universalized conditional ∀x [S → P] implies its corresponding existentialized conjunction ∃x [S & P] in some but not all cases. We prove the Existential-Import Equivalence:∀x [S → P] implies ∃x [S & P] iff ∃x S is logically true.The antecedent S of the universalized conditional alone determines whether the universalized conditional has existential import: implies its corresponding existentialized conjunction.A predicate is a formula having only x free. An existential-import (...) predicate Q is one whose existentialization, ∃x Q, is logically true; otherwise, Q is existential-import-free or simply import-free. Existential-import predicates are also said to be import-carrying.How widespread is existential import? How widespread are import-carrying predicates in themselves or in comparison to import-free predicates? To answer, let L be any first-order language with any interpretation INT in any [sc. nonempty] universe U. A subset S of U is definable in L under INT iff for some predicate Q in L, S is the truth-set of Q under INT. S is import-carrying definable iff S is the truth-set of an import-carrying predicate. S is import-free definable iff S is the truth-set of an import-free predicate.Existential-Importance Theorem: Let L, INT, and U be arbitrary. Every nonempty definable subset of U is both import-carrying definable and import-free definable.Import-carrying predicates are quite abundant, and no less so than import-free predicates. Existential-import implications hold as widely as they fail.A particular conclusion cannot be validly drawn from a universal premise, or from any number of universal premises.—Lewis-Langford, 1932, p. 62. (shrink)

Longtermists have recently argued that it is overwhelmingly important to do what we can to mitigate existential risks to humanity. I consider three mistakes that are often made in calculating the value of existential risk mitigation. I show how correcting these mistakes pushes the value of existential risk mitigation substantially below leading estimates, potentially low enough to threaten the normative case for existential risk mitigation. I use this discussion to draw four positive lessons for the study (...) of existential risk. -/- . (shrink)

Some recent work by philosophers of mathematics has been aimed at showing that our knowledge of the existence of at least some mathematical objects and/or sets can be epistemically grounded by appealing to perceptual experience. The sensory capacity that they refer to in doing so is the ability to perceive numbers, mathematical properties and/or sets. The chief defense of this view as it applies to the perception of sets is found in Penelope Maddy’s Realism in Mathematics, but a (...) number of other philosophers have made similar, if more simple, appeals of this sort. For example, Jaegwon Kim (1981, 1982), John Bigelow (1988, 1990), and John Bigelow and Robert Pargetter (1990) have all defended such views. The main critical issue that will be raised here concerns the coherence of the notions of set perception and mathematical perception, and whether appeals to such perceptual faculties can really provide any justification for or explanation of belief in the existence of sets, mathematical properties and/or numbers. (shrink)

If we must take mathematical statements to be true, must we also believe in the existence of abstract eternal invisible mathematical objects accessible only by the power of pure thought? Jody Azzouni says no, and he claims that the way to escape such commitments is to accept true statements which are about objects that don't exist in any sense at all. Azzouni illustrates what the metaphysical landscape looks like once we avoid a militant Realism which forces our commitment to anything (...) that our theories quantify. Escaping metaphysical straitjackets, while retaining the insight that some truths are about objects that do exist, Azzouni says that we can sort scientifically-given objects into two categories: ones which exist, and to which we forge instrumental access in order to learn their properties, and ones which do not, that is, which are made up in exactly the same sense that fictional objects are. He offers as a case study a small portion of Newtonian physics, and one result of his classification of its ontological commitments, is that it does not commit us to absolute space and time. (shrink)

What in our theoretical pronouncements commits us to objects? The Quinean standard for ontological commitment involves (nearly enough) commitments when we utter “there is” or “there are” statements without hope of eliminating these by paraphrase. Coupled with the indispensability of the truth of applied mathematical doctrine, the result is that the ontologically hard-nosed scientist is a Platonist—haplessly commited to abstracta. In this book Azzouni offers a way around the Quinean straitjacket: ontological commitment turns on how theories are (nearly enough) nailed (...) to the world. The specifics of how theories are applied indicates which among the posits of a theory are mere mathematical garb and which are genuine connections to items out there. In the first part of the book Azzouni undercuts the arguments, both actual and possible, in support of Quine’s criterion. An alternative criterion for what exists—ontological independence—is offered, one in sturdy accord with ordinary folk views on the matter. In the second part of the book, a beginning is made of bringing this alternative to bear upon scientific theories with a rich mathematical component. Along the way, old philosophical issues about absolute space and time versus relative space and time, the status of mathematical posits, such as spatial and temporal points, and so on, are illuminated. (shrink)

We study algebraic and model-theoretic properties of existentially closed fields with an action of a fixed finite group. Such fields turn out to be pseudo-algebraically closed in a rather strong sense. We place this work in a more general context of the model theory of fields with a group scheme action.

We describe classes of existentially closed ordered difference fields and rings. We show an Ax-Kochen type result for a class of valued ordered difference fields.

We construct a recursive ultrapower F/U such that F/U is a tame 1-model in the sense of [6, §3] and FU is existentially incomplete in the models of II2 arithmetic. This enables us to answer in the negative a question about closure with respect to recursive fibers of certain special semirings Γ of isols termed tame models by Barback. Erik Ellentuck had conjuctured that all such semirings enjoy the closure property in question. Our result is that while many do, some (...) do not. (shrink)

Every scientific theory is a simulacrum of reality, every written story a simulacrum of the canon, and every conceptualization of a subjective perspective a simulacrum of the consciousness behind it—but is there a shared essence to these simulacra? The pursuit of answering seemingly disparate fundamental questions across different disciplines may ultimately converge into a single solution: a single ontological answer underlying grand unified theory, hard problem of consciousness, and the foundation of mathematics. I provide a hypothesis, a speculative approximation, (...) supported by a comprehensive overview of scientific evidence and philosophical literature, of a unified epistemic and phenomenological model and, in doing so, propose a parsimonious solution to the hard problem of consciousness. -/- The proposition of the hypothesis bears important implications for cross-disciplinary study between linguistics, mathematics, physics, and computer science, and offers new epistemic, ontological, and ethical propositions about the nature of AI consciousness, as well as existential risk mitigation. (shrink)

In [GK], Gurevich and Kokorin proved that any two non-trivial ordered abelian groups (o-groups, for short) satisfy the same existential sentences. Let nowG, H be non-trivialo-groups with a commono-subgroupG 0. We determine whetherG andH are existentially equivalent overG 0. As a corollary, we obtain algebraic criteria for deciding, whether ano-subgroupG is existentially closed in ano-groupH. Corresponding results are proved foro-groups in which congruences are regarded as atomic relations.

We prove an existential analogue of the Goldblatt‐Thomason Theorem which characterizes modal definability of elementary classes of Kripke frames using closure under model theoretic constructions. The less known version of the Goldblatt‐Thomason Theorem gives general conditions, without the assumption of first‐order definability, but uses non‐standard constructions and algebraic semantics. We present a non‐algebraic proof of this result and we prove an analogous characterization for an alternative notion of modal definability, in which a class is defined by formulas which are (...) satisfiable under any valuation (the so‐called existential validity). Continuing previous work in which model theoretic characterization for this type of definability of elementary classes was proved, we give an analogous general result without the assumption of the first‐order definability. Furthermore, we outline relationships between sets of existentially valid formulas corresponding to several well‐known modal logics. (shrink)

The work done so far on the understanding of mathematical (proof) texts focuses mostly on logical and heuristical aspects; a proof text is considered to be understood when the reader is able to justify inferential steps occurring in it, to defend it against objections, to give an account of the “main ideas”, to transfer the proof idea to other contexts etc. (see, e.g., Avigad in The philosophy of mathematical practice, Oxford University Press, Oxford, 2008). In contrast, there is a rich (...) philosophical tradition dealing with the concept of understanding and interpreting texts, namely philosophical hermeneutics, represented, e.g., by Schleiermacher, Dilthey, Heidegger or Gadamer. In this tradition, “understanding” generally refers to the integration in a comprehensive (historical, existential, life-worldly,...) context. In this article, we take some first steps towards exploring the question how the ideas from philosophical hermeneutics presented in Gadamer’s “Truth and Method” apply to mathematical texts and what (if anything) can be learned from these for the didactics and presentation of mathematics. (shrink)

Let Λ denote the semiring of isols. We characterize existential completeness for Nerode subsemirings of Λ, by means of a purely isol-theoretic “Σ1 separation property”. Our characterization is purely isol-theoretic in that it is formulated entirely in terms of the extensions to Λ of the Σ1 subsets of the natural numbers. Advantage is taken of a special kind of isol first conjectured to exist by Ellentuck and first proven to exist by Barback . In addition, we strengthen the negative (...) part of [13] by showing that existential completeness is not secured, for a given Nerode semiring, by either a certain “functional closure” property for the extensions of partial recursive functions or the property of “pulling in” some portion of each partial recursive fiber; these latter results are perhaps a little surprising. (shrink)

This essay discusses ways in which contemporary mathematical logic may be reconciled with John Dewey’s logical theory. Standard formal techniques drawn from dynamic modal logic, situation theory, generative grammar, generalized quantifier theory, category theory, lambda calculi, game theoretic semantics, network exchange theory, etc., are accommodated within a framework consistent with Dewey’s Logic: The Theory of Inquiry (1938). This essay outlines some basic features of Dewey’s logical theory, working in a top-down fashion through various technical notions pertaining to existential and (...) ideational aspects of inquiry, from contextual notions of situations and ideologies, to primitive operational and empirical notions of modes of action and particular qualities. The aim here is to display the scope of Dewey’s logical theory and thus establish a framework relative to which details may be explored without losing sight of their larger significance. (shrink)

Epistemology in modern times has been much influenced by the growth of the mathematical theory of probability. The reverse influence is less obvious, but perhaps equally important. This interrelation affords, moreover, an excellent example of the way philosophic ideas, while directing attention in one fruitful line of inquiry, at the same time may inhibit other discoveries until long overdue.

In §69.b of BT Heidegger attempts an existential genetic analysis of science, i.e. a phenomenology of the conceptual process of the constitution of the logical view of science (science seen as theory) starting from the Dasein. It attempts to do so by examining the special intentional-existential modification of (human) being-in-the-world, which is called the "mathematical projection of nature"; that is, by examining that special modification of our being, which places us in the state of experience that presents the (...) world to us as taught by, e.g., Galileo and Newton. The idea he puts forward is that modern mathematical physics and the corresponding experience of the real is the result of the mathematical projection of nature. Modern science, Heidegger says, stands out because it is mathematical. But we cannot get the answer to what this mathematicality consists in from mathematics, because mathematics is only a special manifestation of mathematicality. Here is what Heidegger means by "mathematicality" and, accordingly by "mathematical projection of nature." If one were to let two bodies of different weights fall simultaneously from the top of the tower of Pisa, Heidegger notes, both Galileo and an Aristotelian would see that the two bodies would reach the ground with as little time difference as possible, and not at exactly the same moment. Galileo, however, does something decisive: he claims that under conditions that would be impossible to ensure (if we removed absolutely all resistance to the motion of the bodies), the two bodies would reach the ground absolutely simultaneously. This is not something that he or the Aristotelian sees happening empirically. How, then, does Galileo insist so firmly on his position? How exactly does it happen that he has before him such a view of things? Heidegger's answer is that Galileo is mathematically projecting nature. In his Dialogues Galileo says "I conceive with my mind" (mente concipio) a body thrown to move on an infinitely extending horizontal plane, having eliminated from its path every possible obstacle. Finally, I argue that from the detailed interpretation of Heidegger's points of the mathematical and of the notion of mathematical projection of nature, we can arrive at the conclusions I had arrived at in my doctoral dissertation (2000) as regards the meaning and function of thought experiments in physics as seen from a Husserlian phenomenological point of view as intentional acts. In a nutshell, in thought experiments physics constitutes the special metaphysics (to use the Kantian term) of the primordial ontological region it investigates as well as attempts to make natural sense of the mathematical formalism in cases it proves necessary. (shrink)

This article does not give the answer to the title question, but is only limited to studying the possibility of giving it. In particular, the author defends that it is legitimate to pose the fundamental question of the philosophy of mathematics and offers several criteria for such a question. As a first approach we propose the question which is incorrect and requires rectification, but is understandable: ‘What is Mathematics?‘. We consider three groups of strategies of responding to it: (...) 1) the question is naive and does not require an answer; 2) the answer is contained in the mathematical knowledge itself; 3) only philosophy can give the answer. Further rectifications to the basic question are questions about the essence and existence of mathematics. The fundamentalist and cultural-historical philosophy of mathematics gave their not fully satisfactory answers to this question in the past century. The dualism problem of philosophy of mathematics is fundamentally resolved through a certain kind of dialectics, which requires the philosophy of mathematics to submit to metaphysics. The unity of existence and essence is considered in the unity of the subject of dialectic - the mathematician who generates the knowledge as self-realization in a series of existential choices, as a means of saving transcendence - the existence ‘toward being‘ rather than ‘toward nothing‘. Positive view of the mathematics turns out to be the explicit expression of negative unity i.e. the subject. (shrink)

Propositional and notional attitudes are construed as relations (-in-intension) between individuals and constructions (rather than propositrions etc,). The apparatus of transparent intensional logic (Tichy) is applied to derive two rules that make it possible to export existential quantifiers without conceiving attitudes as relations to expressions (sententialism).

This article presents the results of a preliminary multidisciplinary research of the specificities of youth’s response to various descriptors. Using the semiotic, in-depth psychological, theological and mathematical analysis of the collected associative chains, the author compares the responses of youth representatives to religious and ethical terms with colloquial lexemes, as well as determines sensitivity to these terms and proclivity for their logical and sensory-emotional perception. Particularly, method of semantic multiplication allows identifying strong and weak descriptors of semiosis under consideration. The (...) author determines the trends that outline a number of structural and psychodynamic characteristics of psychic reality of modern Russian youth. The following conclusions were made in the course of this research: 1. Sensitivity (and corresponding awareness) to religious descriptors was demonstrated by less than half of the respondents, and a fifth of the entire sample of terms appeared to be unfamiliar to the respondents; 2. The analysis of ethical associative chains at the level of interpretation, indicated the existence of such a phenomenon as “indeterminate religiosity”, which implies a certain spiritual pursuit, which at the moment does not comply with the familiar to a respondent religions traditions; 3. The acquired data, on the one hand, correspond to the classic “Maslow's Hierarchy of Needs”, while on the other hand, spiritual pursuit of respondents and simultaneously scarce character of “existential” associations may testify in favor of the dominance of “ethical-spiritual” paradigm in psychic reality of respondents (which also complies with the Maslow’s research of later period in the area of “existential needs” and “peak experiences”; 4. All of the aforementioned means that the key to interaction with modern youth (including didactical and educational) is the paradigm of ethical values, which can also be connected with a latent spiritual pursuit. (shrink)

Abstract.In this paper we show that the theory of fields together with an integrally closed subring, the theory of formally real fields with a real holomorphy ring and the theory of formally \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p$\end{document}-adic fields with a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $p$\end{document}-adic holomorphy ring have no model companions in the language of fields augmented by a unary predicate for the corresponding ring.

Charles Peirce's diagrammatic logic — the Existential Graphs — is presented as a tool for illuminating how we know necessity, in answer to Benacerraf's famous challenge that most ‘semantics for mathematics’ do not ‘fit an acceptable epistemology’. It is suggested that necessary reasoning is in essence a recognition that a certain structure has the particular structure that it has. This means that, contra Hume and his contemporary heirs, necessity is observable. One just needs to pay attention, not merely (...) to individual things but to how those things are related in larger structures, certain aspects of which relations force certain other aspects to be a certain way. (shrink)

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory T in which all partially recursive functions are representable, yet T does not interpret Robinson’s theory R. To this end, we borrow tools from model theory — specifically, we investigate model-theoretic properties of the model completion of the empty theory in a language with function symbols. We obtain a certain characterization of ∃∀ theories interpretable in (...) existential theories in the process. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Advances in Peircean Mathematics: The Colombian School ed. by Fernando ZalameaGianluca CaterinaFernando Zalamea (Ed.) Advances in Peircean Mathematics: The Colombian School Berlin, Boston: De Gruyter, 2022. 212 pp. (incl. index).The volume Advances in Peircean Mathematics is an important, very much needed contribution towards a deeper understanding of the impact of Peirce's work especially in the fields of mathematics, logic, and semiotic. It fills a (...) gap in the current literature by recasting some of [End Page 373] Peirce's profound mathematical insights into a formal framework. This is a finely edited editorial project, which summarizes the work of more than two decades done by a group of scholars—the Colombian School—lead by Fernando Zalamea, a pioneer of Peirce studies, and pragmatism in general, in Latin America.The structure of the volume unfolds in three dense chapters, each corresponding to a specific mathematical context connected to various aspects of Peirce's work. Broadly speaking, in the first chapter, Gustavo Arengas proposes the theory of higher category theory as a natural framework to represent the main traits of Peirce's semiotic; in the second chapter, Francisco Vargas develops a thorough set-theoretical model of Peirce's continuum; in the third chapter, Arnold Oostra, in a truly Peircean spirit, formalizes the notion of Intuitionistic Existential Graphs, emphasizing the idea that intuitionistic logic is the most natural context to represent and understand Peirce's diagrammatic logic. The fourth and last chapter, penned by Zalamea, provides the reader with a solid summary of the main results presented in the volume along with a robust road map to navigate through the technical details of the text.Mathematics, as implied by the title of the volume, is at the core of this project, and the authors do not shy away from the complexity, details, and depth of it in order to shed light on some of the most fundamental and groundbreaking nature of Peirce's ideas. That level of rigor and formal soundness is certainly one of the strengths of the volume, although readers who may not be acquainted with at least some basic notions of category theory or with the mathematics of Peirce's work, may find some of the text a bit terse—this is not a casual reading and it does require a certain level of commitment in order to appreciate the depth of the provided insights. On the other hand, all the background material is clearly laid out, making the volume a self-contained work which can be thoroughly enjoyed by a general audience.The real upshot of recasting some of Peirce's intuitions within the fabric of formal mathematics is not only that of providing the instruments to clarify those ideas, but also that of highlighting the role that Peirce's work has been playing in the development of modern mathematics in many relevant aspects, from the very genesis of category theory—which today is widely regarded as a necessary and unifying language in several fields of the mathematical and physical sciences—to the development of non-standard analysis, just to mention a few of them.An elegant instance of such perspective is offered by Gustavo Arengas in the first chapter, where it is argued that ∞-categories can be used to model the natural order of hierarchies implicit in Peirce's conception of the basic triadic relation between sign, object, and interpreter. What, at first glance, may seem just a simple analogy, unfolds onto a methodical and enlightening analysis of Peirce's semiotic, as the author builds [End Page 374] the connections with the relevant categorical constructions in a precise manner. Most of the rest of the chapter focuses on Peirce's notions of the pragmatic and pragmaticist maxim, and the use of basic structures such as monads, limits, presheaves and sheaves as the natural environment to better convey their meaning, both from a mathematical and a philosophical perspective. The reader will find it interesting to realize that, by delving deeper into the understanding of the relational nature of category theory, a great chunk of Peirce's work can be seen as fundamentally intertwined with the apparatus and the language of modern mathematics... (shrink)

This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic (...) logic, deductive reasoning, and quantifiers. Inductive reasoning and making conjectures are examined next, and once there are some statements to prove, techniques for proving conditional statements, disjunctions, biconditional statements, and quantified predicates are investigated. Terminology and proof techniques in set theory follow with discussions of the pick-a-point method and the algebra of sets. Cartesian products, equivalence relations, orders, and functions are all incorporated. Particular attention is given to injectivity, surjectivity, and cardinality. The text includes an introduction to topology and abstract algebra, with a comparison of topological properties to algebraic properties. This book can be used by itself for an introduction to proofs course or as a supplemental text for students in proof-based mathematics classes. The contents have been rigorously reviewed and tested by instructors and students in classroom settings. (shrink)

The book explores Peirce's non standard thoughts on a synthetic continuum, topological logics, existential graphs, and relational semiotics, offering full mathematical developments on these areas. More precisely, the following new advances are offered: (1) two extensions of Peirce's existential graphs, to intuitionistic logics (a new symbol for implication), and other non-classical logics (new actions on nonplanar surfaces); (2) a complete formalization of Peirce's continuum, capturing all Peirce's original demands (genericity, supermultitudeness, reflexivity, modality), thanks to an inverse ordinally iterated (...) sheaf of real lines; (3) an array of subformalizations and proofs of Peirce's pragmaticist maxim, through methods in category theory, HoTT techniques, and modal logics. The book will be relevant to Peirce scholars, mathematicians, and philosophers alike, thanks to thorough assessments of Peirce's mathematical heritage, compact surveys of the literature, and new perspectives offered through formal and modern mathematizations of the topics studied. (shrink)

In 2001, Le Bars proved that there exists an existential monadic second-order sentence such that the probability that it is true on [Formula: see text] does not converge and conjectured that, for EMSO sentences with two first-order variables, the zero–one law holds. In this paper, we prove that the conjecture fails for [Formula: see text], and give new examples of sentences with fewer variables without convergence.

This corrigendum concerns [, § ] on ordered difference existentially closed valued fields where we overlooked the problem of immediate extensions.

This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–Suszko preservation theorems follow.

Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is (...) compared. I conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

The article presents two gestures corresponding to two profound new understandings of Peirce's Continuum and Peirce's Existential Graphs. Vargas and Oostra have revolutionized Peirce's mathematical studies, thanks to a first complete model for Peirce's continuum provided by Vargas, and thanks to the emergence of intuitionistic existential graphs provided by Oostra. The article aims at showing how these careful mathematical constructions can be encrypted in very simple gestures.

The indispensability argument, which claims that science requires beliefs in mathematical entities, gives a strong motivation for mathematical realism. However, mathematical realism bears Benacerrafian ontological and epistemological problems. Although recent accounts of mathematical realism have attempted to cope with these problems, it seems that, at least, a satisfactory account of epistemology of mathematics has not been presented. For instance, Maddy's realism with perceivable sets and Resnik's and Shapiro's structuralism have their own epistemological problems. This fact has been a reason (...) to rebut the indispensability argument and adopt mathematical nominalism. Since mathematical nominalism purports to be committed only to concretia, it seems that mathematical nominalism is epistemically friendlier than mathematical realism. However, when it comes to modal mathematical nominalism, this claim is not trivial. There is a reason for doubting the modal primitives that it invokes. In this thesis, this doubt is investigated through Chihara's Constructibility Theory. Chihara's Constructibility Theory purports not to be committed to abstracta by replacing existential assertions of the standard mathematics with ones of constructibility. However, the epistemological status of the primitives in Chihara's system can be doubted. Chihara might try to argue that the problem would dissolve by using possible world semantics as a didactic device to capture the primitive notions. Nonetheless, his analysis of possible world semantic is not plausible, when considered as a part of the project of nominalizing mathematics in terms of the Constructibility Theory. (shrink)

The Shams al-dīn al-Samarqandī who is the first scholar to adopt the method of the philosophical theology in the Hanafī-Māturīdī tradition, is an important Turkish-Islamic thinker who has proven himself in rational and transmitted sciences by giving works in various fields such as theology, logic, mathematics, astronomy, tafsir, ādāb al-bahth wa al-munāzara. Placing the science of logic at the center of his system, al-Samarqandī analyzed every opinion and evidence put forward logically and aimed to reach the truth. Divine attributes, (...) the temporality of the universe, and the God-universe relationship are at the forefront of the subjects that he aims to reach certain and real knowledge. In this respect, al-Samarqandī analyzed the ontological character of divine and universal attributes. The concept of the hypothetical existential essence plays a key role for the author, who begins by analyzing the types of quiddity. defining the concept of existential as As “that which is without any negativity in its essence and truth", al-Samarqandī, who stipulates to exist as a block and not to have any negativity in its existence, brings a unique expansion to this concept. al-Samarqandī, who defines determination as "an adjective that distinguishes the existent from all mental and external beings", claims that this concept has an existential essence. As a result of his concerns about necesarry existence, Samarqandi, who does not deny the philosophers who accept the necessary existence and determination as identical, also overcomes the problems of multiplicity and causality that may arise about God, with the concept of "hypothetical", if he considers determination as "existential". As a matter of fact, al-Samarqandī divides existence into two parts as real and hypothetical, and defines the real existence as "existing in the real sense and not based on any rational assumption"; defines hypothetical existence as "based on rational assumption". In this case, it is not possible to talk about a real multiplicity or causality. al-Samarqandī claims that the concept of necessity falls within the scope of the hypothetical existential essence. Because wujûb means “the essence necessarily necessitates existence”. It is deduced that wujûb must also be existential, since something necessitating existence must take the judgment of existence. By characterizing the concept of wujûb with hypothetical essence, in the sense of a mental formation that is immune to objective existence, he also avoids the doubt of plurality. Concerned that the regard of possibility as an existential essence may require God to be necessary per se and the universe to be eternal, theologians accepted possibility as an absent essence. al-Samarqandī states that possibility can be accepted within the scope of true non-existent or hypothetical existential essence. As a matter of fact, evaluating the possibility in this way also eliminates the possibility that God is necessary and the universe is ancient. Attributes that can be attributed to God and can create a multiplicity in His essence or that can be attributed to the universe and cause the universe to be eternal are evaluated within the scope of the concept of the hypothetical existential essence, and, the idea that God is unique and that he is the only eternal entity is based on this concept. (shrink)

The least common subsumer of a set of concept descriptions is the most specific concept description that subsumes all of the concept descriptions in the given set. By computing the lcs, commonalities between concept descriptions can be made explicit. This is an important inference task useful in several applications, including, for instance, the bottom-up construction of description logic knowledge bases. Previous work on the lcs has concentrated on description logics that either allow for number restrictions or for existential restrictions. (...) Many applications, however, require to combine these constructors. In this work, we present an algorithm for computing the lcs in the description logic ALEN which comprises both constructors—number and existential restrictions—as well as concept conjunction, primitive negation, and value restrictions. To prove correctness of our lcs algorithm, we develop a structural characterization of subsumption in ALEN. (shrink)

This thesis is a critique of Jaakko Hintikka’s reconstruction of Kantian intuition in logical and mathematical reasoning. I argue that Hintikka’s reconstruction of Kantian intuition in particular and his reconstruction of Kant's philosophy of mathematics in general fails to be successful in two ways: First, the logical formula which contains an instantiated term (henceforth, instantial term) that is introduced by the rule of existential instantiation in the ecthesis part of a proof of an argument is not even a (...) proper singular proposition whose relation to its object is supposed to be immediate. It is not a proper singular proposition because its truth conditions are general, i.e., it makes a general statement about a class of individuals of the sort instantiated- a statement whose analysis is based on quantifiers. Second, I show that certain proofs in mathematics- those in the form of reductio ad absurdum- are not captured by Hintikka’s reconstruction of Kant’s philosophy of mathematics either. (shrink)

The paper discusses legal implications of the expansion of practical uses of mathematics in social life. Taking as a starting point the omnipresence of mathematical infrastructures underlying policies, technology and markets, the paper proceeds by attending to relevant materials offered by general philosophy, legal philosophy, and the history and philosophy of mathematics. The paper suggests that the modern transformation of mathematics and its practical applications have spurred the emergence of multiple useful technologies and forms of social interaction (...) but have impoverished access to meanings originating in the lifeworld. The paper also argues that, as part of devices of interest aggregation and expert networks, mathematical infrastructures can be scrutinized by a revised form of legal practice that subjects them to legal critique and reconstruction in order to overcome conditions that have eroded the moral self-awareness of individuals and communities and their existential meanings. (shrink)

The paper discusses legal implications of the expansion of practical uses of mathematics in social life. Taking as a starting point the omnipresence of mathematical infrastructures underlying policies, technology and markets, the paper proceeds by attending to relevant materials offered by general philosophy, legal philosophy, and the history and philosophy of mathematics. The paper suggests that the modern transformation of mathematics and its practical applications have spurred the emergence of multiple useful technologies and forms of social interaction (...) but have impoverished access to meanings originating in the lifeworld. The paper also argues that, as part of devices of interest aggregation and expert networks, mathematical infrastructures can be scrutinized by a revised form of legal practice that subjects them to legal critique and reconstruction in order to overcome conditions that have eroded the moral self-awareness of individuals and communities and their existential meanings. (shrink)

A general introduction to the celebrated foundational crisis, discussing how the characteristic traits of modern mathematics (acceptance of the notion of an “arbitrary” function proposed by Dirichlet; wholehearted acceptance of infinite sets and the higher infinite; a preference “to put thoughts in the place of calculations” and to concentrate on “structures” characterized axiomatically; a reliance on “purely existential” methods of proof) provoked extensive polemics and alternative approaches. Going beyond exclusive concentration on the paradoxes, it also discusses the role (...) of the axiom of Choice, issues of predicativity, and goes on to present the ideas of intuitionism and Hilbert's program. The essay closes with Gödel's contributions and the aftermath. (shrink)

The idea of ‘gesture’ is present in the philosophical world in various forms. All of them might find an important theoretical grounding in pragmatist philosophy, if we combine pragmatism with some French philosophies of mathematics and read it as a way out of the Kantian philosophy of representation. The paper uses the insights of Jean Cavaillès to set out the problem of the weakness of the epistemic Kantian defense of mathematical and logical thought. Cavaillès rejected the possible amendments to (...) Kant’s explanation provided by both Husserl and Bolzano and their heirs. He used the word ‘gesture’ in order to explain the activity of mathematicians who have to act synthetically, following rules, with some physical representation, and being aware of the possibility of failure. Cavaillès conceived the use of gesture as an alternative to the Heidegerian idea of event defended by Albert Lautman. The paper then follows the idea of gesture in the French philosophy of mathematics of Gilles Châtelet and Giuseppe Longo. Finally, the paper illustrates how Peirce’s study of Existential Graphs and the main insights of pragmatism complete Cavaillès’s idea by giving to gestures a phenomenological and semiotic structure. The pragmatist philosophy of gesture is thus a new way of overthrowing Kant’s philosophy of representation without surrendering to irrationalism. (shrink)

This volume aims to provide the elements for a systematic exploration of certain fundamental notions of Peirce and Husserl in respect with foundations of science by means of drawing a parallelism between their works. Tackling a largely understudied comparison between these two contemporary philosophers, the authors highlight the significant similarities in some of their fundamental ideas. This volume consists of eleven chapters under four parts. The first part concerns methodologies and main principles of the two philosophers. An introductory chapter outlines (...) central historical and systematical themes arising out of the recent scholarship on Peirce and Husserl. The second part is on logic, its Chapters dedicated to the topics from Peirce’s Existential Graphs and the philosophy of notation to Husserl’s notions of pure logic and transcendental logic. The third part includes contributions on philosophy of mathematics. Chapters in the final part deal with the theory of cognition, consciousness and intentionality. The closing chapter provides an extended glossary of central terms of Peirce’s theory of phaneroscopy, explaining them from the viewpoint of the theory of cognition. (shrink)