On Tarski's foundations of the geometry of solids

The paper (Tarski:Lesfondementsdelageometriedescorps,AnnalesdelaSoci´´ Polonaise de Math´ ematiques, pp. 29-34, 1929) is in many ways remarkable. We address three historico-philosophical issues that force themselves upon the reader. First we argue that in this paper Tarski did not live up to his own methodological ideals, but displayed instead a much more pragmatic approach. Second we show that Lephilosophy and systems do not play the significant role that one may be tempted to assign to them at first glance. Especially the role of background logic must be at least partially allocated to Russell's systems of Principia mathematica. This analysis leads us, third, to a threefold distinction of the technical ways in which the domain of discourse comes to be embodied in a theory. Having all of this in place, we discuss why we have to reject the argument in (Gruszczyand Pietruszczak: Full development of Tarski's Geometry of Solids, The Bulletin of Symbolic Logic, vol. 4 (2008), no. 4, pp. 481-540) according to which Tarski has made a certain mistake.

[1]  Hourya Sinaceur,et al.  Alfred Tarski: Semantic Shift, Heuristic Shift In Metamathematics , 2004, Synthese.

[2]  P. Simons.  Parts: A Study in Ontology , 1991 .

[3]  James T. Smith.  Definitions and Nondefinability in Geometry , 2010, Am. Math. Mon..

[4]  T. D. Laguna.  Point, Line, and Surface, as Sets of Solids , 1922 .

[5]  Michael Friedman,et al.  Bertrand Russell's The Analysis of Matter: Its Historical Context and Contemporary Interest , 1985, Philosophy of Science.

[6]  R. Cooper.  Process and Reality , 2014 .

[7]  Elena Anne Marchisotto,et al.  The legacy of Mario Pieri in geometry and arithmetic , 2007 .

[8]  Bertrand Russell,et al.  La géométrie dans le monde sensible , 1924 .

[9]  Zbigniew Stachniak,et al.  S. Leśniewski's lecture notes in logic , 1988 .

[10]  Jagdish Mehra,et al.  Our knowledge of the External World , 1974 .

[11]  Solomon Feferman,et al.  Alfred Tarski: Life and Logic , 2004 .

[12]  Paolo Mancosu.  Tarski on models and logical consequence , 2006 .

[13]  Johan van Benthem,et al.  What is Spatial Logic? , 2007, Handbook of Spatial Logics.

[14]  Rafal Gruszczynski,et al.  Full Development of Tarski's Geometry of Solids , 2008, Bulletin of Symbolic Logic.

[15]  L. W. Szczerba.  Tarski and Geometry , 1986, J. Symb. Log..

[16]  Alfred Tajtelbaum-Tarski.  On the Primitive Term of Logistic , 1998 .

[17]  C. Kuratowski.  Sur l'opération Ā de l'Analysis Situs , 1922 .

[18]  Iris Loeb,et al.  From Mereology to Boolean Algebra: The Role of Regular Open Sets in Alfred Tarski’s Work , 2014 .

[19]  John Etchemendy,et al.  Tarski on truth and logical consequence , 1988, Journal of Symbolic Logic.

[20]  C. Allen,et al.  Stanford Encyclopedia of Philosophy , 2011 .

[21]  G. Hicks.  An Enquiry concerning the Principles of Natural Knowledge , 1920, Nature.

[22]  Thierry Coquand,et al.  Constructive Topology and Combinatorics , 1992, Constructivity in Computer Science.

[23]  Eugene C. Luschei,et al.  The logical systems of Lesniewski , 1962 .

[24]  Boleslaw Sobocinski.  Atomistic mereology. I , 1971, Notre Dame J. Formal Log..

[25]  Alfred Tarski,et al.  Der Wahrheitsbegriff in den formalisierten Sprachen , 1935 .

[26]  C. Lejewski.  A note on Leśniewski's axiom system for the mereological notion of ingredient or element , 1983 .

[27]  C. H. Langford.  Some Theorems on Deducibility , 1926 .

[28]  Alfred Tarski.  The completeness of elementary algebra and geometry , 1967 .

[29]  Paola Cantù,et al.  Aristotle’s prohibition rule on kind-crossing and the definition of mathematics as a science of quantities , 2010, Synthese.

[30]  Douglas Patterson,et al.  New essays on Tarski and philosophy , 2008 .

[31]  E. V. Huntington.  A set of postulates for abstract geometry, expressed in terms of the simple relation of inclusion , 1913 .

[32]  Steven Givant,et al.  Unifying threads in alfred tarski’s work , 1999 .

[33]  M. H. A. Newman,et al.  I.—MR. RUSSELL'S “CAUSAL THEORY OF PERCEPTION” , 1928 .

[34]  L. Birch,et al.  The concept of nature , 1992 .

[35]  Robert S. Lubarsky.  Geometric spaces with no points , 2010, J. Log. Anal..

[36]  Andrew Arana,et al.  The architecture of modern mathematics , 2008 .

[37]  Arianna Betti,et al.  The Classical Model of Science: a millennia-old model of scientific rationality , 2010, Synthese.

[38]  Johan van Benthem,et al.  Handbook of Spatial Logics , 2007 .

[39]  Stanisław Jaśkowski.  Une modification des définitions fondamentales de la géométrie des corps de M. A. Tarski , 1948 .

[40]  Philippe Devaux.  Expérience et formalisme: A propos de "Géométrie dans le monde sensible", de Jean Nicod, Paris, Alcan, 1924 , 1930 .

[41]  Hans Hermes,et al.  Introduction to mathematical logic , 1973, Universitext.

[42]  Paul Deheuvels.  La recherche scientifique , 1990 .

[43]  Alfred North Whitehead,et al.  An enquiry concerning the principles of natural knowledge , 1919 .

[44]  Timothy Williamson,et al.  Parts. A Study in Ontology , 1990 .

[45]  D. Bostock.  Whitehead and Russell on Points , 2010 .

[46]  Paolo Mancosu.  Fixed‐ versus Variable‐domain Interpretations of Tarski’s Account of Logical Consequence , 2010 .