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Hilbert’s twentyfourth problem
 American Mathematical Monthly
, 2001
"... 1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Cong ..."
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1. INTRODUCTION. For geometers, Hilbert’s influential work on the foundations of geometry is important. For analysts, Hilbert’s theory of integral equations is just as important. But the address “Mathematische Probleme ” [37] that David Hilbert (1862– 1943) delivered at the second International Congress of Mathematicians (ICM) in Paris has tremendous importance for all mathematicians. Moreover, a substantial part of
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
Constructing Cardinals from below
"... this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a secondorder variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The fo ..."
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this paper are all formulated in terms of a formula #(X), with only X free. For now, the formula is one in the language of basic set theory and X is a secondorder variable. The corresponding condition is that #(A) is true in R(#) for some A and, for no # # is #(A R(#)) true in R(#). The formal expression that this condition is an existence condition is the axiom #X[#(X) # ### R(#))] (1) (X) is the result of restricting the first and secondorder bound variables in #(X) to R(#) and R(# + 1), respectively. Axioms of this form have been called reflection principles, because they express the fact that R(#)'s possession of a certain property is reflected by R(#)'s possession of it for some # #
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, 1974], motiva ..."
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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
Hierarchies Ontological and Ideological
"... Abstract: Gödel claimed that ZermeloFraenkel set theory is ‘what becomes of the theory of types if certain superfluous restrictions are removed’. The aim of this paper is to develop a clearer understanding of Gödel’s remark, and of the surrounding philosophical terrain. In connection with this, we ..."
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Abstract: Gödel claimed that ZermeloFraenkel set theory is ‘what becomes of the theory of types if certain superfluous restrictions are removed’. The aim of this paper is to develop a clearer understanding of Gödel’s remark, and of the surrounding philosophical terrain. In connection with this, we discuss some technical issues concerning infinitary type theories and the programme of developing the semantics for higherorder languages in other higherorder languages. 1
Conceptions of the Continuum
"... Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question ..."
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Abstract: A number of conceptions of the continuum are examined from the perspective of conceptual structuralism, a view of the nature of mathematics according to which mathematics emerges from humanly constructed, intersubjectively established, basic structural conceptions. This puts into question the idea from current set theory that the continuum is somehow a uniquely determined concept. Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions. 1. What is the continuum? On the face of it, there are several distinct forms of the continuum as a mathematical concept: in geometry, as a straight line, in analysis as the real number system (characterized in one of several ways), and in set theory as the power set of the natural numbers and, alternatively, as the set of all infinite sequences of zeros and ones. Since it is common to refer to the continuum, in what sense are these all instances of the same concept? When one speaks of the continuum in current settheoretical
• allow relations F n and G n (n ≥ 1) to be distinct even when The Theory of Relations, Complex Terms, and a Connection Between λ and ɛ Calculi
"... This paper introduces a new method of interpreting complex relation terms in a secondorder quantified modal language. We develop a completely general secondorder modal language with two kinds of complex terms: one kind for denoting individuals and one kind for denoting nplace relations. Several i ..."
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This paper introduces a new method of interpreting complex relation terms in a secondorder quantified modal language. We develop a completely general secondorder modal language with two kinds of complex terms: one kind for denoting individuals and one kind for denoting nplace relations. Several issues arise in connection with previous, algebraic methods for interpreting the relation terms. The new method of interpreting these terms described here addresses those issues while establishing an interesting connection between λ and ɛ calculi. The resulting semantics provides a precise understanding of the theory of relations. Consider a secondorder language with quantification over both individuals and nplace relations (n ≥ 0), a modal operator, definite descriptions (i.e., complex individual terms, interpreted rigidly for simplicity), and λexpressions (i.e., complex nplace relation terms). The
This is a preliminary version of a review which will appear in Research in the History of Economic Thought and Methodology, Volume 24A (2006), edited by
"... formalist revolution or what happened to orthodox economics after World War II?”, the specter of mathematician David Hilbert has haunted economists’s discussions of formalization and axiomatization. Briefly, if one looks upon formalized economics, or formalism, with a loathing built on fear (or a fe ..."
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formalist revolution or what happened to orthodox economics after World War II?”, the specter of mathematician David Hilbert has haunted economists’s discussions of formalization and axiomatization. Briefly, if one looks upon formalized economics, or formalism, with a loathing built on fear (or a fear based on loathing), one demonizes Hilbert since the philosophical notion of formalism, in the history of metamathematics, is usually associated with Hilbert. That is, in the history of philosophy of mathematics, there appears to be a distinction between formalists and empiricists, on the nature of mathematical objects. The ontological reflections in that arcane literature has Hilbert holding the position that mathematics is simply a formal system, and its symbols are simply marks on paper. It is an easy step then to look for traces of David Hilbert in the development of mathematical economics in the 20 th century. Seek, and ye shall find, and critics of mainstream economics have found Hilbertian connections in Vienna with Menger’s seminar. As a result Hilbert gets caught up in the origin stories of general equilibrium theory which lead all the way to von Neumann and the development of game theory. From Vienna and general equilibrium theory it is a short step, though a false step, to have Hilbert as the spiritual advisor to the Cowles Commission and thence to ArrowDebreu. From there of course one can launch tirades about the formalist revolution in economics and have Hilbert bearing some of the blame for a misguided economics.
The cognitive basis of arithmetic
"... Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities ..."
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Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities