Beyond the Axioms: the Question of Objectivity in Mathematics
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"There Is No Right to Be Stupid": a Voegelinian Analysis of Islamist
"There is No Right to Be Stupid": A Voegelinian Analysis of Islamist Terrorism with Reference to the "Elements of Reality" Copyright 2005 Celestino Perez, Jr. In the following essay I make three claims. First, in agreement with Eric Voegelin and Barry Cooper, I illustrate why a comprehensive study of politics requires that attention be paid to the symbols "God and man," "the Good", and "the soul" (Section I). I then briefly wonder if Voegelin and Cooper, despite offering cogent analyses of pernicious ideologies, ask too much of their readers? If so, what are the prospects for a competent political science and a healthy democratic populace (Section II)? In Section III, I suggest a different way of applying Voegelin's theory to an analysis of Islamist ideology, wherein I lay out ten "elements of reality" and five distilled "fundamental truths" derived from these elements. To the degree that someone distorts or neglects these elements of reality, he is a pneumopath. He may be very smart. But, alas, he is a "philosophical and religious ignoramus,"1 [1] and hence stupid. And, as Voegelin reminds us in Hitler and the Germans, "There is no right to be stupid." To be sure, if we allow ourselves to submit to stupidity, the gangster animals will be sure to mockingly let us know: 1 [1] Barry Cooper, New Political Religions, or An Analysis of Modern Terrorism ( Columbia : University of Missouri Press , 2004), xii. Cooper describes Sayyid Qutb's and other terrorists' "dogmatic certainty" as resulting from their being "philosophical and religious ignoramuses." "Yoohoo, silly ass!"2 [2] In the Concluding Notes, I lay out what I consider to be the gist of Cooper's argument, and I briefly comment on his Appendix in New Political Religions. -
The Metamathematics of Putnam's Model-Theoretic Arguments
The Metamathematics of Putnam's Model-Theoretic Arguments Tim Button Abstract. Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted mod- els. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I exam- ine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to meta- mathematical challenges. Copyright notice. This paper is due to appear in Erkenntnis. This is a pre-print, and may be subject to minor changes. The authoritative version should be obtained from Erkenntnis, once it has been published. Hilary Putnam famously attempted to use model theory to draw metaphys- ical conclusions. Specifically, he attacked metaphysical realism, a position characterised by the following credo: [T]he world consists of a fixed totality of mind-independent objects. (Putnam 1981, p. 49; cf. 1978, p. 125). Truth involves some sort of correspondence relation between words or thought-signs and external things and sets of things. (1981, p. 49; cf. 1989, p. 214) [W]hat is epistemically most justifiable to believe may nonetheless be false. (1980, p. 473; cf. 1978, p. 125) To sum up these claims, Putnam characterised metaphysical realism as an \externalist perspective" whose \favorite point of view is a God's Eye point of view" (1981, p. 49). Putnam sought to show that this externalist perspective is deeply untenable. To this end, he treated correspondence in terms of model-theoretic satisfaction. -
Forcopy-Editing Only
i i “Abstractionism_OUP_for_copyediting” — 2016/4/20 — 11:40 — page 1 — #10 i i 1 Introduction to Abstractionism Philip A. Ebert and Marcus Rossberg 1.1 WHAT IS ABSTRACTIONISM? Abstractionism in philosophy of mathematics has its origins in Gottlob Frege’s logicism—a position Frege developed in the late nineteenth and early twenti- eth century. Frege’s main aim wasfor to reduce copy-editing arithmetic and analysis to only logic in order to provide a secure foundation for mathematical knowledge. As is well known, Frege’s development of logicism failed. The infamous Basic Law V— one of the six basic laws of logic— Frege proposed in his magnum opus Grund- gesetze der Arithmetik—is subject to Russell’s Paradox. The striking feature of Frege’s Basic Law V is that it takes the form of an abstraction principle. The general form of an abstraction principle can by symbolised like this:1 ↵ = β ↵ β §( ) §( ) $ ⇠ where ‘ ’ is a term-forming operator applicable to expression of the type of ↵ § and β, and is an equivalence relation on entities denoted by expressions of ⇠ that type. Accordingly, abstraction principles are biconditionals that feature an equivalence relation on the right-hand side and an identity statement on the left-hand side. The abstracta denoted by the terms featuring in the identity statement on the left are taken to be introduced, in some sense, by the ab- straction principle, giving the equivalence on the right-hand side conceptual priority over them. More on this below. Frege’s ill-fated Basic Law V, involves co-extentionality (of functions) as the relevant equivalence relation on the right-hand side, introducing, what Frege – termed value-ranges, "' " , on the left:2 ( ) 1Here and below, we will omit prefixed universal quantifiers in abstraction principles. -
The Logic of Provability
The Logic of Provability Notes by R.J. Buehler Based on The Logic of Provability by George Boolos September 16, 2014 ii Contents Preface v 1 GL and Modal Logic1 1.1 Introduction..........................................1 1.2 Natural Deduction......................................2 1.2.1 ...........................................2 1.2.2 3 ...........................................2 1.3 Definitions and Terms....................................3 1.4 Normality...........................................4 1.5 Refining The System.....................................6 2 Peano Arithmetic 9 2.1 Introduction..........................................9 2.2 Basic Model Theory..................................... 10 2.3 The Theorems of PA..................................... 10 2.3.1 P-Terms........................................ 10 2.3.2Σ,Π, and∆ Formulas................................ 11 2.3.3 G¨odelNumbering................................... 12 2.3.4 Bew(x)........................................ 12 2.4 On the Choice of PA..................................... 13 3 The Box as Bew(x) 15 3.1 Realizations and Translations................................ 15 3.2 The Generalized Diagonal Lemma............................. 16 3.3 L¨ob'sTheorem........................................ 17 3.4 K4LR............................................. 19 3.5 The Box as Provability.................................... 20 3.6 GLS.............................................. 21 4 Model Theory for GL 23 5 Completeness & Decidability of GL 25 6 Canonical Models 27 7 On -
Yesterday's Algorithm: Penrose on the Gödel Argument
Yesterday’s Algorithm: Penrose and the Gödel Argument §1. The Gödel Argument. Roger Penrose is justly famous for his work in physics and mathematics but he is notorious for his endorsement of the Gödel argument (see his 1989, 1994, 1997). This argument, first advanced by J. R. Lucas (in 1961), attempts to show that Gödel’s (first) incompleteness theorem can be seen to reveal that the human mind transcends all algorithmic models of it1. Penrose's version of the argument has been seen to fall victim to the original objections raised against Lucas (see Boolos (1990) and for a particularly intemperate review, Putnam (1994)). Yet I believe that more can and should be said about the argument. Only a brief review is necessary here although I wish to present the argument in a somewhat peculiar form. Let us suppose that the human cognitive abilities that underlie our propensities to acquire mathematical beliefs on the basis of what we call proofs can be given a classical cognitive psychological or computational explanation (henceforth I will often use ‘belief’ as short for ‘belief on the basis of proof’). These propensities are of at least two types: the ability to appreciate chains of logical reasoning but also the ability to recognise ‘elementary proofs’ or, it would perhaps be better to say, elementary truths, such as if a = b and b = c then a = c, without any need or, often, any possibility of proof in the first sense. On this supposition, these propensities are the realization of a certain definite algorithm or program somehow implemented by the neural hardware of the brain. -
Church's Thesis As an Empirical Hypothesis
Pobrane z czasopisma Annales AI- Informatica http://ai.annales.umcs.pl Data: 27/02/2019 11:52:41 Annales UMCS Annales UMCS Informatica AI 3 (2005) 119-130 Informatica Lublin-Polonia Sectio AI http://www.annales.umcs.lublin.pl/ Church’s thesis as an empirical hypothesis Adam Olszewski Pontifical Academy of Theology, Kanonicza 25, 31-002 Kraków, Poland The studies of Church’s thesis (further denoted as CT) are slowly gaining their intensity. Since different formulations of the thesis are being proposed, it is necessary to state how it should be understood. The following is the formulation given by Church: [CT] The concept of the effectively calculable function1 is identical with the concept of the recursive function. The conviction that the above formulation is correct, finds its confirmation in Church’s own words:2 We now define the notion, […] of an effectively calculable function of positive integers by identifying it with the notion of a recursive function of positive integers […]. Frege thought that the material equivalence of concepts is the approximation of their identity. For any concepts F, G: Concept FUMCS is identical with concept G iff ∀x (Fx ≡ Gx) 3. The symbol Fx should be read as “object x falls under the concept F”. The contemporary understanding of the right side of the above equation was dominated by the first order logic and set theory. 1Unless stated otherwise, the term function in this work always means function defined on the set natural numbers, that is, on natural numbers including zero. I would rather use the term concept than the term notion. -
Mechanized Metatheory for the Masses: the POPLMARK Challenge
University of Pennsylvania ScholarlyCommons Departmental Papers (CIS) Department of Computer & Information Science August 2005 Mechanized Metatheory for the Masses: The POPLMARK Challenge Brian E. Aydemir University of Pennsylvania Aaron Bohannon University of Pennsylvania Matthew Fairbairn University of Cambridge J. Nathan Foster University of Pennsylvania Benjamin C. Pierce University of Pennsylvania, [email protected] See next page for additional authors Follow this and additional works at: https://repository.upenn.edu/cis_papers Recommended Citation Brian E. Aydemir, Aaron Bohannon, Matthew Fairbairn, J. Nathan Foster, Benjamin C. Pierce, Peter Sewell, Dimitrios Vytiniotis, Geoffrey Washburn, Stephanie C. Weirich, and Stephan A. Zdancewic, "Mechanized Metatheory for the Masses: The POPLMARK Challenge", . August 2005. Postprint version. Published in Lecture Notes in Computer Science, Volume 3603, Theorem Proving in Higher Order Logics (TPHOLs 2005), pages 50-65. Publisher URL: http://dx.doi.org/10.1007/11541868_4 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_papers/235 For more information, please contact [email protected]. Mechanized Metatheory for the Masses: The POPLMARK Challenge Abstract How close are we to a world where every paper on programming languages is accompanied by an electronic appendix with machinechecked proofs? We propose an initial set of benchmarks for measuring progress in this area. Based on the metatheory of System F, a typed lambda-calculus with second-order polymorphism, subtyping, and records, these benchmarks embody many aspects of programming languages that are challenging to formalize: variable binding at both the term and type levels, syntactic forms with variable numbers of components (including binders), and proofs demanding complex induction principles. -
9 Alfred Tarski (1901–1983), Alonzo Church (1903–1995), and Kurt Gödel (1906–1978)
9 Alfred Tarski (1901–1983), Alonzo Church (1903–1995), and Kurt Gödel (1906–1978) C. ANTHONY ANDERSON Alfred Tarski Tarski, born in Poland, received his doctorate at the University of Warsaw under Stanislaw Lesniewski. In 1942, he was given a position with the Department of Mathematics at the University of California at Berkeley, where he taught until 1968. Undoubtedly Tarski’s most important philosophical contribution is his famous “semantical” definition of truth. Traditional attempts to define truth did not use this terminology and it is not easy to give a precise characterization of the idea. The under- lying conception is that semantics concerns meaning as a relation between a linguistic expression and what it expresses, represents, or stands for. Thus “denotes,” “desig- nates,” “names,” and “refers to” are semantical terms, as is “expresses.” The term “sat- isfies” is less familiar but also plausibly belongs in this category. For example, the number 2 is said to satisfy the equation “x2 = 4,” and by analogy we might say that Aristotle satisfies (or satisfied) the formula “x is a student of Plato.” It is not quite obvious that there is a meaning of “true” which makes it a semanti- cal term. If we think of truth as a property of sentences, as distinguished from the more traditional conception of it as a property of beliefs or propositions, it turns out to be closely related to satisfaction. In fact, Tarski found that he could define truth in this sense in terms of satisfaction. The goal which Tarski set himself (Tarski 1944, Woodger 1956) was to find a “mate- rially adequate” and formally correct definition of the concept of truth as it applies to sentences. -
Theories of Truth
time 7 Theories of truth r A summary sketchl The object of this section is to sketch the main kinds of ~ theories of truth which have been proposed, and to indicate how they C relate to each other. (Subsequent sections will discuss some theories S'" p. in detail.) o Coherence theories take truth to consist in relations of coherence ~ among a set of beliefs. Coherence theories were proposed e.g. by '" ""C Bradley r9r4, and also by some positivist opponents of idealism, such o u as Neurath r932; more recently, Rescher r973 and Dauer r974 have C'" defended this kind of approach. Correspondence theories take the :l 0::'" truth of a proposition to consist, not in its relations to other proposi .= tions, but in its relation to the world, its cOllespondence to the facts. Theories of this kind were held by both Russell r9r8 and Wittgenstein r922, during the period of their adherence to logical atomism; Austin defended a version of the cOllespondence theory in r950' The prag matist theory, developed in the works of Peirce (see e.g. r877), Dewey (see e.g. r9or) and James (see e.g. r909) has affinities with both coherence and correspondence theories, allowing that the truth of a belief derives from its correspondence with reality, but stressing also that it is manifested by the beliefs' survival of test by experience, its coherence with other beliefs; the account of truth proposed in Dummett r959 has, in turn, quite strong affinities with the pragmatist VIew. 1 Proponents of the theories I shall discuss take different views about what kinds of items are truth-bearers. -
Critique of the Church-Turing Theorem
CRITIQUE OF THE CHURCH-TURING THEOREM TIMM LAMPERT Humboldt University Berlin, Unter den Linden 6, D-10099 Berlin e-mail address: lampertt@staff.hu-berlin.de Abstract. This paper first criticizes Church's and Turing's proofs of the undecidability of FOL. It identifies assumptions of Church's and Turing's proofs to be rejected, justifies their rejection and argues for a new discussion of the decision problem. Contents 1. Introduction 1 2. Critique of Church's Proof 2 2.1. Church's Proof 2 2.2. Consistency of Q 6 2.3. Church's Thesis 7 2.4. The Problem of Translation 9 3. Critique of Turing's Proof 19 References 22 1. Introduction First-order logic without names, functions or identity (FOL) is the simplest first-order language to which the Church-Turing theorem applies. I have implemented a decision procedure, the FOL-Decider, that is designed to refute this theorem. The implemented decision procedure requires the application of nothing but well-known rules of FOL. No assumption is made that extends beyond equivalence transformation within FOL. In this respect, the decision procedure is independent of any controversy regarding foundational matters. However, this paper explains why it is reasonable to work out such a decision procedure despite the fact that the Church-Turing theorem is widely acknowledged. It provides a cri- tique of modern versions of Church's and Turing's proofs. I identify the assumption of each proof that I question, and I explain why I question it. I also identify many assumptions that I do not question to clarify where exactly my reasoning deviates and to show that my critique is not born from philosophical scepticism. -
Modal Realism and Metaphysical Nihilism Gonzalo Rodriguez-Pereyra
Mind, 2004, 113 (452), pp. 683-704. Modal Realism and Metaphysical Nihilism Gonzalo Rodriguez-Pereyra 1. Modal Realism is an ontological doctrine whose most characteristic thesis is that there exist non-actual possible individuals which are of a kind with actual individuals. That is, there are non-actual chairs, tables, donkeys, people and stars. As developed by David Lewis, Modal Realism is accompanied by a cluster of theses, for instance that all possible worlds (i.e. actual and non-actual possible worlds) exist, that all possible worlds are of a kind, that possible worlds are maximal sums of spatiotemporally related objects, and that a sentence like ‘it is possible that p’ is true just in case there is a possible world where p. Modal Realism has, among its theoretical benefits, a reductive account, within limits, of modality. Among its costs, it counts clashing with several intuitive views. One of these is the view that it is possible that nothing exists, that is, that there could have been nothing. Lewis saw that his Modal Realism is incompatible with this view (Lewis 1986, p. 73 and Lewis 1991, p. 13, footnote 6). Another closely related intuitive view with which Lewis’s Modal Realism is incompatible is what has recently been called Metaphysical Nihilism, namely that it is possible that nothing concrete exists, that is, that there could have been nothing concrete. Metaphysical Nihilism is not only intuitive, there are persuasive arguments in its favour. So, other things being equal, to be compatible with Metaphysical Nihilism is a theoretical virtue. In this paper I shall argue that Modal Realism can be modified so as to be compatible with Metaphysical Nihilism. -
The Criterion of Truth
THE CRITERION OF TRUTH An article published in the Theosophical movement magazine, February 2017 THE STATED object of science is to discover truths about the natural world through strictly objective observation, experimental verification and theoretical explanations, free from all subjective biases. Testability and reproducibility of laws so discovered are the criterion of truth so far as modern science is concerned. Though science has made remarkable progress in unravelling laws of the physical world by empirical and inductive methods, and in their practical applications, leading scientists however confess that they have failed to develop a unified science with a single set of assumptions and terms to explain all observations as aspects of one coherent whole. In other words, they are still in search of one single universal principle, theoretical formulation of which will reconcile many different specialized observations resolvable into one grand synthetic whole which explains all the mysteries of the natural world. They admit that such a universal unifying principle alone is truth and that its discovery, by means of inductive method they adhere to, is still a far cry. Each one of the many sects of the world religions, resting on “Revelation,” lays claim to be the sole possessor of truth. Inasmuch as there is no other fertile source of mutual hatred and strife in the world than differences between the various religious sects, it is evident that none of them has the whole truth. Whatever truth they had originally possessed has been so mixed up with human error and superstitions over the centuries that they either repel and cause the reasoning and thinking portions of humanity to fall into agnosticism, or induce blind faith and superstitions in unthinking portions of the public who willingly submit to priestly authority.