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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
The Practice of Finitism: Epsilon Calculus and Consistency Proofs in Hilbert's Program
, 2001
"... . After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the ..."
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. After a brief flirtation with logicism in 19171920, David Hilbert proposed his own program in the foundations of mathematics in 1920 and developed it, in concert with collaborators such as Paul Bernays and Wilhelm Ackermann, throughout the 1920s. The two technical pillars of the project were the development of axiomatic systems for ever stronger and more comprehensive areas of mathematics and finitistic proofs of consistency of these systems. Early advances in these areas were made by Hilbert (and Bernays) in a series of lecture courses at the University of Gttingen between 1917 and 1923, and notably in Ackermann 's dissertation of 1924. The main innovation was the invention of the ecalculus, on which Hilbert's axiom systems were based, and the development of the esubstitution method as a basis for consistency proofs. The paper traces the development of the "simultaneous development of logic and mathematics" through the enotation and provides an analysis of Ackermann's consisten...
UNIVERSITE ́ LAVAL
"... Une étude philosophique sur la logique et les mathématiques ..."
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To appear in Synthese.
"... of this paper, and for many helpful conversations. A version of this material was presented at the Boston Colloquium for Philosophy of Science. Thanks to my audience there for valuable questions and comments. A common objection to hierarchical approaches to truth is that they fragment the concept of ..."
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of this paper, and for many helpful conversations. A version of this material was presented at the Boston Colloquium for Philosophy of Science. Thanks to my audience there for valuable questions and comments. A common objection to hierarchical approaches to truth is that they fragment the concept of truth. This paper defends hierarchical approaches in general against the objection of fragmentation. It argues that the fragmentation required is familiar and unproblematic, via a comparison with mathematical proof. Furthermore, it offers an explanation of the source and nature of the fragmentation of truth. Fragmentation arises because the concept exhibits a kind of failure of closure under reflection. This paper offers a more precise characterization of the reflection involved, first in the setting of formal theories of truth, and then in a more general setting. truthprooffinal.tex: January 27, 2003 (12:03) It is often noted that Tarski’s [1935] hierarchy of languages and metalanguages fragments the concept of truth. Instead of one concept, we have
On Ontology and Realism in Mathematics * Outline
"... The paper is concerned with the way in which “ontology ” and “realism ” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possib ..."
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The paper is concerned with the way in which “ontology ” and “realism ” are to be interpreted and applied so as to give us a deeper philosophical understanding of mathematical theories and practice. Rather than argue for or against some particular realistic position, I shall be concerned with possible coherent positions, their strengths and weaknesses. I shall also discuss related but different aspects of these problems. The terms in the title are the common thread that connects the various sections. The discussed topics range widely. Certain themes repeat in different sections, yet, for most part the sections (and sometimes the subsections) can be read separately Section 1 however states the basic position that informs the whole paper and, as such, should be read (it is not too long). The required technical knowledge varies, depending on the matter discussed. Aiming at a broader audience I have tried to keep the technical requirements at a minimum, or to supply a short overview. Material, which is elementary for some readers, may be far from elementary for others (my apologies to both). I also tried to supply information that may be of interest to everyone interested in these subjects.
WHAT FINITISM COULD NOT BE ∗
"... chief difficulty for everyone who wishes to understand Hilbert’s conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentiall ..."
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chief difficulty for everyone who wishes to understand Hilbert’s conception of finitist mathematics is this: to specify the sense of the provability of general statements about the natural numbers without presupposing infinite totalities. Tait further argues that all finitist reasoning is essentially primitive recursive. In this paper, we attempt to show that his thesis “The finitist functions are precisely the primitive recursive functions ” is disputable and that another, likewise defended by him, is untenable. The second thesis is that the finitist theorems are precisely the universal closures of the equations that can be proved in PRA. KEY WORDS: finitist functions, primitive recursive functions, infinite totalities, finitist proof of the universal closure of an equation RESUMEN: En su artículo “Finitism ” (1981), W.W. Tait sostiene que la dificultad principal para quien quiere comprender la concepción hilbertiana de la matemática finitista es ésta: especificar el sentido de la demostrabilidad de enunciados generales sobre los números naturales sin presuponer totalidades
Numbers And Functions In Hilbert's Finitism
, 1998
"... or concrete? Some of the most fruitful sources on the topic of Hilbert's conception of finitism are his 1922 and 1926 papers, his collaborator Bernays's exchange with Mller (Mller 1923, Bernays 1923), as well as the relevant sections in Hilbert and Bernays (1934, 1939). In 1905, Hilbert gi ..."
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or concrete? Some of the most fruitful sources on the topic of Hilbert's conception of finitism are his 1922 and 1926 papers, his collaborator Bernays's exchange with Mller (Mller 1923, Bernays 1923), as well as the relevant sections in Hilbert and Bernays (1934, 1939). In 1905, Hilbert gives a first account of finitistic number theory in terms of strokes and equality signs. We note here that no identification of certain (sequences of) signs with numbers is made, rather, the sequences of 1's and ='s are divided into two classes, the class of entities (these are the sequences of the form "1...1 = 1...1" with equal numbers of 1's on the left and right) and the class of nonentities; the former are the true propositions. Hence we have here a finitistic account, not of numbers, but of numerical truth.