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15
Tarski's System of Geometry
 Bulletin of Symbolic Logic
, 1999
"... . This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabh auser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the histo ..."
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. This paper is an edited form of a letter written by the two authors (in the name of Tarski) to Wolfram Schwabh auser around 1978. It contains extended remarks about Tarski's system of foundations for Euclidean geometry, in particular its distinctive features, its historical evolution, the history of specific axioms, the questions of independence of axioms and primitive notions, and versions of the system suitable for the development of 1dimensional geometry. In his 192627 lectures at the University of Warsaw, Alfred Tarski gave an axiomatic development of elementary Euclidean geometry, the part of plane Euclidean geometry that is not based upon settheoretical notions, or, in other words, the part that can be developed within the framework of firstorder logic. He proved, around 1930, that his system of geometry admits elimination of quantifiers: every formula is provably equivalent (on the basis of the axioms) to a Boolean combination of basic formulas. From this theorem he...
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
Step By Recursive Step: Church's Analysis Of Effective Calculability
 BULLETIN OF SYMBOLIC LOGIC
, 1997
"... Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Ch ..."
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Alonzo Church's mathematical work on computability and undecidability is wellknown indeed, and we seem to have an excellent understanding of the context in which it arose. The approach Church took to the underlying conceptual issues, by contrast, is less well understood. Why, for example, was "Church's Thesis" put forward publicly only in April 1935, when it had been formulated already in February/March 1934? Why did Church choose to formulate it then in terms of G odel's general recursiveness, not his own #definability as he had done in 1934? A number of letters were exchanged between Church and Paul Bernays during the period from December 1934 to August 1937; they throw light on critical developments in Princeton during that period and reveal novel aspects of Church's distinctive contribution to the analysis of the informal notion of e#ective calculability. In particular, they allow me to give informed, though still tentative answers to the questions I raised; the char...
Frege versus Cantor and Dedekind: On the Concept of Number
, 1997
"... This paper is in honor of my colleague and friend, Leonard Linsky, on the occasion of his retirement. I presented the earliest version in the Spring of 1992 to a reading group, the other members of which were Leonard Linsky, Steve Awodey, Andre Carus and Mike Price. I presented later versions in the ..."
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This paper is in honor of my colleague and friend, Leonard Linsky, on the occasion of his retirement. I presented the earliest version in the Spring of 1992 to a reading group, the other members of which were Leonard Linsky, Steve Awodey, Andre Carus and Mike Price. I presented later versions in the autumn of 1992 to the philosophy colloquium at McGill University and in the autumn of 1993 to the philosophy colloquium at CarnegieMellon University. The discussions following these presentations were valuable to me and I would especially like to acknowledge Emily Carson (for comments on the earliest draft), Michael Hallett, Kenneth Manders, Stephen Menn, G.E. Reyes, Teddy Seidenfeld, and Wilfrid Sieg and the members of the reading group for helpful comments. But, most of all, I would like to thank Howard Stein and Richard Heck, who read the penultimate draft of the paper and made extensive comments and corrections. Naturally, none of these scholars, except possibly Howard Stein, is respon
Russell’s Absolutism vs.(?)
"... Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And ..."
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Along with Frege, Russell maintained an absolutist stance regarding the subject matter of mathematics, revealed rather than imposed, or proposed, by logical analysis. The Fregean definition of cardinal number, for example, is viewed as (essentially) correct, not merely adequate for mathematics. And Dedekind’s “structuralist” views come in for criticism in the Principles. But, on reflection, Russell also flirted with views very close to a (different) version of structuralism. Main varieties of modern structuralism and their challenges are reviewed, taking account of Russell’s insights. Problems of absolutism plague some versions, and, interestingly, Russell’s critique of Dedekind can be extended to one of them, ante rem structuralism. This leaves modalstructuralism and a category theoretic approach as remaining nonabsolutist
What are numbers, and what is their meaning?:
, 1872
"... 1888 What are numbers, and what is their meaning? Let us recall that by 1850 the subject of analysis had been given a solid footing in the real numbers — infinitesimals had given way to small positive real numbers, the ε’s and δ’s. In 1858 Dedekind was in Zürich, lecturing on the differential calcu ..."
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1888 What are numbers, and what is their meaning? Let us recall that by 1850 the subject of analysis had been given a solid footing in the real numbers — infinitesimals had given way to small positive real numbers, the ε’s and δ’s. In 1858 Dedekind was in Zürich, lecturing on the differential calculus for the first time. He was concerned about his introduction of the real numbers, with crucial properties being dependent upon the intuitive understanding of a geometrical line. 1 In particular he was not satisfied with his geometrical explanation of why it was that a monotone increasing variable, which is bounded above, approaches a limit. By November of 1858 Dedekind had resolved the issue by showing how to obtain the real numbers (along with their ordering and arithmetical operations) from the rational numbers by means of cuts in the rationals — for then he could prove the above mentioned least upper bound property from simple facts about the rational numbers. Furthermore, he proved that applying cuts to the reals gave no further extension.
TeachingGeometry According to Euclid
"... itions, from the first printed edition of 1482 up to about 1900. Billingsley, in his preface to the first English translation of the Elements (1570) [1], writes, "Without the diligent studie of Euclides Elements, it is impossible to attaine unto the perfecte knowledge of Geometrie, and consequently ..."
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itions, from the first printed edition of 1482 up to about 1900. Billingsley, in his preface to the first English translation of the Elements (1570) [1], writes, "Without the diligent studie of Euclides Elements, it is impossible to attaine unto the perfecte knowledge of Geometrie, and consequently of any of the other Mathematical Sciences." Bonnycastle, in the preface to his edition of the Elements [4], says, "Of all the works of antiquity which have been transmitted to the present time, none are more universally and deservedly esteemed than the Elements of Geometry which go under the name of Euclid. In many other branches of science the moderns have far surpassed their masters; but, after a lapse of more than two thousand years, this performance still retains its original preeminence, and has even acquired additional celebrity for the fruitless attempts which have been made to establish a different system." Todhunter, in the preface to his edition [18]
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