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Hilbert’s Program Then and Now
, 2005
"... Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and els ..."
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Hilbert’s program is, in the first instance, a proposal and a research program in the philosophy and foundations of mathematics. It was formulated in the early 1920s by German mathematician David Hilbert (1862–1943), and was pursued by him and his collaborators at the University of Göttingen and elsewhere in the 1920s
Hilbert’s “Verunglückter Beweis,” the first epsilon theorem and consistency proofs. History and Philosophy of Logic
"... Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One propo ..."
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Abstract. On the face of it, Hilbert’s Program was concerned with proving consistency of mathematical systems in a finitary way. This was to be accomplished by showing that that these systems are conservative over finitistically interpretable and obviously sound quantifierfree subsystems. One proposed method of giving such proofs is Hilbert’s epsilonsubstitution method. There was, however, a second approach which was not refelected in the publications of the Hilbert school in the 1920s, and which is a direct precursor of Hilbert’s first epsilon theorem and a certain “general consistency result. ” An analysis of this socalled “failed proof ” lends further support to an interpretation of Hilbert according to which he was expressly concerned with conservatitvity proofs, even though his publications only mention consistency as the main question. §1. Introduction. The aim of Hilbert’s program for consistency proofs in the 1920s is well known: to formalize mathematics, and to give finitistic consistency proofs of these systems and thus to put mathematics on a “secure foundation.” What is perhaps less well known is exactly how Hilbert thought this should be carried out. Over ten years before Gentzen developed sequent calculus formalizations
Spacetime Geometry Translated into the Hegelian and Intuitionist Systems
"... Abstract: Kant noted the importance of spatial and temporal intuitions (synthetics) in geometric reasoning, but intuitions lend themselves to different interpretations and a more solid grounding may be sought in formality. In mathematics David Hilbert defended formality, while L. E. J. Brouwer cited ..."
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Abstract: Kant noted the importance of spatial and temporal intuitions (synthetics) in geometric reasoning, but intuitions lend themselves to different interpretations and a more solid grounding may be sought in formality. In mathematics David Hilbert defended formality, while L. E. J. Brouwer cited intuitions that remain unencompassed by formality. In this paper, the conflict between formality and intuition is again investigated, and it is found to impact on our interpretations of spacetime as translated into the language of geometry. It is argued that that language as a formal system works because of an auxiliary innateness that carries sentience, or feeling. Therefore, the formality is necessarily incomplete as sentience is beyond its reach. Specifically, it is argued that sentience is covertly connected to spacetime geometry when axioms of congruency are stipulated, essentially hiding in the formality what is sensecertain. Accordingly, geometry is constructed from primitive intuitions represented by onepointedness and routeinvariance. Geometry is recognized as a twosided language that permitted a Hegelian passage from Euclidean geometry to Riemannian geometry. The concepts of general relativity, quantum mechanics and entropyirreversibility are found to be the consequences of linguistic type reasoning, and perceived conflicts (e.g., the puzzle of quantum gravity) are conflicts only within formal linguistic systems. Therefore, the conflicts do not survive beyond the synthetics because what is felt relates to inexplicable feeling, and because the question of synthesis returns only to Hegel’s absolute Notion.
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
Gödel on Intuition and on Hilbert’s finitism
"... There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the con ..."
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There are some puzzles about Gödel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, Gödel’s writings represent a smooth evolution, with just one rather small doublereversal, of his view of finitism. He used the term “finit ” (in German) or “finitary ” or “finitistic ” primarily to refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [Gödel, 1938a] and the lecture notes for a lecture at Yale [Gödel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of firstorder number theory, P A; but starting in the Dialectica paper
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Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. It is a publisher's requirement to display the following notice: The documents distributed by this server have been provided by the contributing authors as a means to ensure timely dissemination of scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the authors or by other copyright holders, notwithstanding that they have offered their works here electronically. It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may not be reposted without the explicit permission of the copyright holder. In the case of Springer, it is the publisher’s requirement that the following note be added: “An author may selfarchive an authorcreated version of his/her article on his/her own website and his/her institution’s repository, including his/her final version; however he/she may not use the publisher’s PDF version which is posted on www.springerlink.com. Furthermore, the author may only post his/her version provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer’s website. The link must be accompanied by the following text: “The original publication is available at www.springerlink.com. ” Semantic Information and the Correctness Theory of Truth
Cohesiveness
 INTELLECTICA, 2009/1, 51, PP.
, 2009
"... It is characteristic of a continuum that it be “all of one piece”, in the sense of being inseparable into two (or more) disjoint nonempty parts. By taking “part ” to mean open (or closed) subset of the space, one obtains the usual topological concept of connectedness. Thus a space S is defined to b ..."
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It is characteristic of a continuum that it be “all of one piece”, in the sense of being inseparable into two (or more) disjoint nonempty parts. By taking “part ” to mean open (or closed) subset of the space, one obtains the usual topological concept of connectedness. Thus a space S is defined to be connected if it cannot be partitioned into two disjoint nonempty open (or closed) subsets – or equivalently, given any partition of S into two open (or closed) subsets, one of the members of the partition must be empty. This holds, for example, for the space R of real numbers and for all of its open or closed intervals. Now a truly radical condition results from taking the idea of being “all of one piece ” literally, that is, if it is taken to mean inseparability into any disjoint nonempty parts, or subsets, whatsoever. A space S satisfying this condition is called cohesive or indecomposable. While the law of excluded middle of classical logic reduces indecomposable spaces to the trivial empty space and onepoint spaces, the use of intuitionistic logic makes it possible not only for nontrivial cohesive spaces to exist, but for every connected space to be cohesive.In this paper I describe the philosophical background to cohesiveness as well as some of the ways in