Online research in philosophy

This is a dialogue in which five characters are involved. Various issues in the philosophy of mathematics are discussed. Among those issues are these: numbers as abstract objects, our knowledge of numbers as abstract objects, a proof as showing a mathematical statement to be true as opposed to the statement being true in virtue of having a proof.

In an intrepid article entitled “Why Anselm's Proof in the Proslogion Is Not an Ontological Argument,”45 G.E.M. Anscombe takes issue with the traditional reading of Anselm's text. According to this reading Anselm's proof in Proslogion 2 depends upon the premise that existence is a perfection; and as a result of this dependency it has been given the label “ontological argument.” I In challenging the traditional reading, Anscombe proposes a corrected version of Anselm’s proof—a version which eliminates the premise that existence is a perfection and which thereby undermines the rationale for considering the proof to be an “ontological argument.” Her corrected version runs as follows: 26 Anscombe on Anselm..

Preface -- Introduction -- There is only one reality -- The ultimate perspective and the ultimate drama -- Proof #1: Science -- Proof #2: History -- Proof #3: Prophecy -- Proof #4: Supernatural -- Proof #5: Psychology -- Proof #6: Sociology -- Proof #7: Inerrancy -- Proof #8: Micro-science -- Proof #9: Logic -- Proof #10: The only provably -- Inerrant, complete system -- Why proof is important -- Personal iplications of proof.

Abstract:  Contrary to received opinion, Descartes' view on the merits of the ontological proof may actually agree with that of Thomas Aquinas, whose rejection of the a priori existence proof has stocked the armories of anti-Anselmians ever since. In a rarely noted passage of the First Replies, Descartes claims not to differ in any respect from Thomas on the proof, a claim that gains sense in light of recent work on the Fifth Meditation. That work in turn reveals a well-founded, if surprising, understanding of the Cartesian proof and of Cartesianism's true relation to Thomism.

In his "Ontological proof", Kurt Gödel introduces the notion of a second-order value property, the positive property P. The second axiom of the proof states that for any property φ: If φ is positive, its negation is not positive, and vice versa. I put forward that this concept of positiveness leads into a paradox when we apply it to the following self-reflexive sentences: (A) The truth value of A is not positive; (B) The truth value of B is positive. Given axiom 2, sentences A and B paradoxically cannot be both true or both false, and it is also impossible that one of the sentences is true whereas the other is false.

In this paper, I argue that commentators have missed a significant clue given by Descartes in coming to understand his 'ontological' proof for the existence of God. In both the analytic and synthetic presentations of the proof throughout his writings, Descartes notes that the proof works 'in the same way' as a particular geometrical proof. I explore the significance of such a parallel, and conclude that Descartes could not have intended readers to think that the argument consists of some kind of intuition. I argue that for Descartes the attribute of existence is a 'second-order' attribute that is demonstrated to belong to the idea of God on the basis of 'first-order' attributes. The proof, properly understood, is in fact a demonstration. Having brought to light the geometrical parallels between the ontological and geometrical proofs, we have new evidence to resolve the 'intuition versus demonstration' controversy that has characterized much of the discussion of Descartes's ontological argument.

This paper provides an interpretation and evaluation of Spinoza’s highly original version of the ontological proof in terms of the concept of substance instead of the concept of perfection in the first book of his Ethics. Taking the lead from Kant’s critique of ontological arguments in the Critique of Pure Reason, the paper explores the underlying ontological and epistemological presuppositions of Spinoza’s proof. The main topics of consideration are the nature of Spinoza’s definitions, the way he conceives of the relation between a substance and its essence, and his conception of existence. Once clarity is shed upon these fundamental issues, it becomes possible to address the proof in its own terms. It is then easy to see that Kant’s objections miss their target and that the same is true of those advanced by another of the ontological argument’s most famous critics, Bertrand Russell. Finally, several interpretations of Spinoza’s proof are proposed and critically evaluated; on all of them, the argument turns out to be either invalid or question-begging.

This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic.

Proof and perception : the context of the argumentum cartesianum -- Refutations of atheism : ontological arguments in English philosophy, 1652-1705 -- Being and intuition : Malebranche's appropriation of the argument -- An adequate conception : the argument in Spinoza's philosophy -- Ontological arguments in Leibniz and the German enlightenment -- Kant's systematic critique of the ontological argument -- Hegel's reconstruction of the argument.