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QUANTUM MECHANICS AS A SPACETIME THEORY
, 2005
"... Abstract. We show how quantum mechanics can be understood as a spacetime theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects ..."
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Abstract. We show how quantum mechanics can be understood as a spacetime theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space entities. The geometry of atoms and subatomic objects differs from that of classical objects. The systems that are nonlocal when measured in the classical spacetime continuum may be localized in the quantum continuum. We compare this new description of spacetime with the Bohmian picture of quantum mechanics. 1. What is quantum spacetime? Both modern mathematics and modern physics underwent serious foundational crises during the 20th century. The crisis in mathematics occured at the beginning of the century and the main problem was to deal with certain infinities that are directly related to the concept of real number. Poincaré [31] explained this crisis in terms of different attitudes to infinity, related to Aristotle’s actual infinity and the potential infinity (the first attitude believes that the actual infinity exists, we begin with the collection in which we find the preexisting objects, the second holds that a collection is formed by successively adding new members, it is infinite because we can see no reason why this process should stop). It led finally to the emergence of new, nonstandard definitions of real numbers. The crisis in physics concerns the interpretation of the quantum theory, the measurement problem and the question of nonlocality. In previous works we showed how in principle certain paradoxes of the quantum theory can be explained provided we enlarge our conception of number [10].Our goal was to show how the basic axioms of quantum mechanics can be reformulated in terms of nonstandard real numbers that we call qrumbers. It is our goal in the present paper to analyze nonlocality and the concept of spacetime at the light of the new conceptual tools that we developed in the past.
The Logic of Brouwer and Heyting
, 2007
"... Intuitionistic logic consists of the principles of reasoning which were used informally by ..."
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Intuitionistic logic consists of the principles of reasoning which were used informally by
Cantor's Grundlagen and the Paradoxes of Set Theory
"... This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 197 ..."
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This paper was written in honor of Charles Parsons, from whom I have profited for many years in my study of the philosophy of mathematics and expect to continue profiting for many more years to come. In particular, listening to his lecture on "Sets and classes", published in [Parsons, 1974], motivated my first attempts to understand proper classes and the realm of transfinite numbers. I read a version of the paper at the APA Central Division meeting in Chicago in May, 1998. I thank Howard Stein, who provided valuable criticisms of an earlier draft, ranging from the correction of spelling mistakes, through important historical remarks, to the correction of a mathematical mistake, and Patricia Blanchette, who commented on the paper at the APA meeting and raised two challenging points which have led to improvements in this final version
Tous droits réservésDogmas and the Changing Images of Foundations
"... Le contenu de ce site relève de la législation française sur la propriété intellectuelle et est la propriété exclusive de l'éditeur. Les œuvres figurant sur ce site peuvent être consultées et reproduites sur un support papier ou numérique sous réserve qu'elles soient strictement réservées ..."
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Le contenu de ce site relève de la législation française sur la propriété intellectuelle et est la propriété exclusive de l'éditeur. Les œuvres figurant sur ce site peuvent être consultées et reproduites sur un support papier ou numérique sous réserve qu'elles soient strictement réservées à un usage soit personnel, soit scientifique ou pédagogique excluant toute exploitation commerciale. La reproduction devra obligatoirement mentionner l'éditeur, le nom de la revue, l'auteur et la référence du document. Toute autre reproduction est interdite sauf accord préalable de l'éditeur, en dehors des cas prévus par la législation en vigueur en France. Revues.org est un portail de revues en sciences humaines et sociales développé par le Cléo, Centre pour l'édition
THE CAUSAL STORY OF THE DOUBLE SLIT EXPERIMENT IN QUANTUM REAL NUMBERS
"... Abstract. A causal story of the double slit experiment for a massive scalar particle is told using quantum real numbers as the numerical values of the position and momentum of the particle. The quantum real number interpretation postulates an independent physical reality for the quantum particle. I ..."
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Abstract. A causal story of the double slit experiment for a massive scalar particle is told using quantum real numbers as the numerical values of the position and momentum of the particle. The quantum real number interpretation postulates an independent physical reality for the quantum particle. It provides an ontology for the particle in which its qualities have numerical values even when they have not been measured. It satisfies experimental tests to the same degree of accuracy as the standard quantum theory because the standard expectation values are infinitesimal quantum real numbers. Questions, unanswerable in the standard theories, concerning the behaviour of single particles in the experiment are answered. 1. Introduction to quantum real numbers In order to tell our causal story, we must temporarily replace the orthodox Copenhagen interpretation of quantum mechanics with a more
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
History
"... Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics— in arithmetic (number theory), analysis and set theory. Already in his famous “Mathematical problems ” of 1900 [Hilbert, 1900] he raised, as the seco ..."
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Proof theory was created early in the 20th century by David Hilbert to prove the consistency of the ordinary methods of reasoning used in mathematics— in arithmetic (number theory), analysis and set theory. Already in his famous “Mathematical problems ” of 1900 [Hilbert, 1900] he raised, as the second problem, that of proving the consistency of the arithmetic of the real numbers. In 1904, in “On the foundations of logic and arithmetic ” [Hilbert, 1905], he for the first time initiated his own program for proving consistency. 1.1 Consistency Whence his concern for consistency? The history of the concept of consistency in mathematics has yet to be written; but there are some things we can mention. There is of course a long tradition of skepticism. For example Descartes considered the idea we are deceived by a malicious god and that the simplest arithmetical truths might be inconsistent—but that was simply an empty skepticism. On the other hand, as we now know, in view of Gödel’s incom
Primitive Recursive Arithmetic and its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections In Honor of Per MartinLöf on the Occasion of His Retirement
"... We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal ..."
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We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics. 1. Skolem tells us in the Concluding Remark of his seminal paper on primitive recursive arithmetic (PRA), “The foundations of arithmetic established by means of the recursive mode of thought, without use of apparent variables ranging over infinite domains ” [1923], that the paper was written in 1919 after he had studied Whitehead and Russell’s Principia Mathematica and in reaction to that work. His specific complaint about the foundations of arithmetic (i.e. number theory) in that work was, as implied by his title, the essential role in it of logic and in particular quantification over infinite domains, even for the understanding of the most elementary propositions of arithmetic such as polynomial equations; and he set about to eliminate these infinitary quantifications by means of the “recursive mode of thought. ” On this ground, not only polynomial equations, but all primitive recursive formulas stand on their own feet without logical underpinning. 2. Skolem’s 1923 paper did not include a formal system of arithmetic, but as he noted in his 1946 address, “The development of recursive arithmetic” [1947], formalization of the methods used in that paper results in one of the many equivalent systems we refer to as PRA. Let me stop here and briefly describe one such system. ∗Is paper is loosely based on the Skolem Lecture that I gave at the University of Oslo in June, 2010. The present paper has profited, both with respect to what it now contains and with respect to what it no longer contains, from the discussion following that lecture. 1 We admit the following finitist types1 of objects: