Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-18T08:44:19.671Z Has data issue: false hasContentIssue false
  • English
  • Français

On Tarski's Foundations of the Geometry of Solids

Published online by Cambridge University Press:  15 January 2014

Arianna Betti  and
Iris Loeb
Arianna Betti
Affiliation:
Faculty of Philosophy, Vu University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The NetherlandsE-mail: ariannabetti@gmail.com, E-mail: i.loeb@vu.nl
Iris Loeb
Affiliation:
Faculty of Philosophy, Vu University Amsterdam, De Boelelaan 1105, 1081 HV Amsterdam, The NetherlandsE-mail: ariannabetti@gmail.com, E-mail: i.loeb@vu.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper [Tarski: Les fondements de la géométrie des corps, Annales de la Société Polonaise de Mathématiques, 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 Leśniewski's philosophy 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 [Gruszczyński and Pietruszczak: Full development ofTarski'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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 18 , Issue 2 , June 2012 , pp. 230 - 260
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[2007] Aiello, Marco, Pratt-Hartmann, Ian, and van Benthem, Johan, What is spatial logic?, Handbook of spatial logics (Aiello, Marco, Pratt-Hartmann, Ian, and van Benthem, Johan, editors), Springer, 2007, pp. 1–11.CrossRefGoogle Scholar
[1935] Aroukiewicz, Kazimierz et al., Biblio-biographische Notizen zum Vorkonferenzbericht, Erkenntnis, vol. 5 (1935), no. 1, pp. 186–195.Google Scholar
[2008] Betti, Arianna, Polish axiomatics and its truth: On Tarski's LeSniewskian background and the Ajdukiewicz connection, New essays on Tarski and philosophy (Patterson, Douglas, editor), Oxford University Press, 2008, pp. 44–71.CrossRefGoogle Scholar
[2009] Bostock, David, Whitehead and Russell on points, Philosophia Mathematica, vol. 18 (2009), no. 1, pp. 1–52.Google Scholar
[2010] Cantù, P, Aristotle's prohibition rule on kind-crossing and the definition of mathematics as a science of quantities, Synthese, vol. 174 (2010), no. 2, pp. 225–235.CrossRefGoogle Scholar
[1956] Church, Alonzo, Introduction to mathematical logic, Princeton, Princeton University Press, 1956.Google Scholar
[1991] Coquand, Thierry, Constructive topology and combinatorics, Constructivity in computer science (Myers, J. Paul Jr. and O'Donnell, Michael J., editors), Lecture Notes in Computer Science, vol. 613, Springer, 1991, pp. 159–164.Google Scholar
[2010] de Jong, Willem R. and Betti, Arianna, The classical model of science: amillenniaold model of scientific rationality, Synthese, vol. 174 (2010), no. 2, pp. 185–203.Google Scholar
[1922] Laguna, Theodore de, Point, line, and surface, as sets of solids, Journal of Philosophy, vol. 19 (1922), pp. 449–461.CrossRefGoogle Scholar
[1985] Demopoulos, William and Friedman, Michael, Bertrand Russell's The Analysis of Matter: Its historical context and contemporary interest, Philosophy of Science, vol. 52 (1985), no. 4, pp. 621–639.Google Scholar
[1988] Etchemendy, John, Tarski on truth andlogical consequence, The Journal of Symbolic Logic, vol. 53 (1988), no. 1, pp. 51–79.CrossRefGoogle Scholar
[2004] Feferman, Anita Burdman and Feferman, Solomon, Alfred Tarski – Life and Logic, Cambridge University Press, Cambridge, 2004.Google Scholar
[1999] Givant, Steven, Unifying threads in Alfred Tarski's work, The Mathematical Intelligencer, vol. 21 (1999), no. 1, pp. 47–58.CrossRefGoogle Scholar
[1986a] Givant, Steven R. and Mackenzie, Ralph (editors), Alfred Tarski: Collected papers, vol. 1, Birkhäuser, 1986.Google Scholar
[1986b] Givant, Steven R. and Mackenzie, Ralph (editors), Alfred Tarski: Collected papers, vol. 4, Birkhäuser, 1986.Google Scholar
[2008] Gruszczyńso, Rafał and Pietruszczak, Andrzej, Full development of Tarski's geometry of solids, this Bulletin, vol. 4 (2008), no. 4, pp. 481–540.Google Scholar
[1913] Huntington, E. v., A set ofpostulates for abstract geometry, expressed in terms of the simple relation of inclusion, Mathematische Annalen, vol. 73 (1913), no. 4, pp. 522–559.Google Scholar
[1948] Jaśkowski, S., Une modification des definitions fondamentales de la géometrie des corps de M. A. Tarski, Annales de la Société Polonaise de Mathématiques, vol. 21 (1948), pp. 298–301.Google Scholar
[1949] Jaśkowski, S., Quelques problèmes actuels concernant les fondements des mathématique, Časopis pro Pěstování Matematiky a Fysiky, vol. 74 (1949), pp. 74–78.Google Scholar
[1922] Kuratowski, Casimir, Sur l'opération A de l'Analysis Situs, Fundamenta Mathematicae, vol. 3 (1922), pp. 182–199.CrossRefGoogle Scholar
[1926] Langford, C. H., Some theorems on deducibility, The Annals of Mathematics, vol. 28 (1926), no. 1/4, pp. 16–40.Google Scholar
[1983] Lejewski, C., A note on Leśniewski's axiom system for the mereological notion of ingredient or element, Topoi, vol. 2 (1983), pp. 63–71.CrossRefGoogle Scholar
[1927, 1928, 1929, 1930, 1931] Leśniewski, S., O podstawach matematyki (On the foundations of mathematics), Przegl d Filozoficzny, vol. xxx, xxxi, xxxii, xxxiii, xxxiv (1927, 1928, 1929, 1930, 1931), pp. 164-206, 261-291, 60-105, 77-105, 142–170, translated and reprinted in [Surma, Srzednicki, Barnett, and Rickey, 1991].Google Scholar
[1928] Leśniewski, S., O podstawach matematyki (On the foundations of mathematics), Przegl d Filozoficzny, vol. XXXI (1928), pp. 261–291, translated and reprinted in [Surma, Srzednicki, Barnett, and Rickey, 1991].Google Scholar
[1988a] Leśniewski, S., Class theory, S. Lésniewski's lecture notes in logic (Srzednicki, Jan T. J. and Stachniak, Zbigniew, editors), Kluwer Academic Publishers, 1988, pp. 60–125.Google Scholar
[1988b] Leśniewski, S., Definitions and theses of Lésniewski's ontology, S. Lésniewski's lecture notes in logic (Srzednicki, Jan T. J. and Stachniak, Zbigniew, editors), Kluwer Academic Publishers, 1988, pp. 29–59.Google Scholar
[2011] Loeb, Iris, From Mereology to Boolean algebra: The role of regular open sets in Alfred Tarski's work, submitted, 2011.Google Scholar
[2010] Lubarsky, Robert S., Geometric spaces with no points, Journal of Logic and Analysis, vol. 2 (2010), no. 6, pp. 1–10.Google Scholar
[1962] Luschei, Eugene C., The logical systems of Leśniewski, North-Holland Publishing Company, Amsterdam, 1962.Google Scholar
[2006] Mancosu, Paolo, Tarski on models and logical consequence, The architecture of modern mathematics (Ferreirós, J. and Gray, J. J., editors), Oxford: Oxford University Press, 2006, pp. 209–237.CrossRefGoogle Scholar
[2010] Mancosu, Paolo, Fixedversus variable-domain interpretations of Tarski's account of logical consequence, Philosophy Compass, vol. 5 (2010), no. 9, pp. 745–759.Google Scholar
[2007] Marchisotto, Elena Anne and Smith, James T., The legacy of Mario Pieri in geometry and arithmetic, Springer, 2007.Google Scholar
[1928] Newman, M. H. A., Mr. Russell's “Causal theory of perception”, Mind, vol. 37 (1928), no. 146, pp. 137–148.CrossRefGoogle Scholar
[1923] Nicod, J., La géométrie dans le monde sensible, Ph.D. thesis , Univ. de Paris, 1923.Google Scholar
[1908] Pieri, Mario, La geometria elementare istituita sulle nozioni di “punto” e “sfera”, Memorie di matematica e di fisica della Societá Italiana della Scienze, vol. 15 (1908), pp. 345–450.Google Scholar
[1914] Russell, B., Our knowledge of the external world, Chicago and London: Open Court, 1914.Google Scholar
[1987] Simons, Peter M., Parts: A study in ontology, Oxford University Press, 1987.Google Scholar
[2001] Sinaceur, Hourya, Alfred Tarski: Semantic shift, heuristic shift in metamathematics, Synthese, vol. 126 (2001), pp. 49–65.CrossRefGoogle Scholar
[2010] Smith, James T., Definitions andnondefinability in geometry, American Mathematical Monthly, vol. 117 (2010), pp. 475–489(15).Google Scholar
[1956] Sobociński, Bolesław, On well-constructed axiom systems, Yearbook of the Polish Society of Arts and Sciences Abroad/Rocznik Towarzystwa Polskiego na Obczyźnie, vol. 6 (1956), pp. 54–65.Google Scholar
[1971] Sobociński, Bolesław, Atomistic Mereology. I, Notre Dame Journal of Formal Logic, vol. 12 (1971), no. 1, pp. 89–103.Google Scholar
[2003] Sundholm, Göran, Tarski and Lesniewski on languages with meaning versus languages without use, In search of the Polish tradition—essays in honor of Jan Wolenski on the occasion of his 60th birthday (Hintikka, J., Czarnecki, T, Kijania-Placek, K., Placek, T., and Rojczczak, A., editors), Dordrecht: Kluwer Academic Publishers, 2003, pp. 109–127.Google Scholar
[1991] Surma, S. J., Srzednicki, J. T., Barnett, D. I., and Rickey, V. F. (editors), Stanisław Leśniewski: Collected works, vol. I, Polish Scientific Publishers/Kluwer Academic Publishers, 1991.Google Scholar
[1986] Szczerba, L. W., Tarski and geometry, The Journal of Symbolic Logic, vol. 51 (1986), no. 4, pp. 907–912.Google Scholar
[1974] Szmielew, W., The role of the Pasch axiom in the foundations of Euclidean geometry, Tarski symposium, 1974, pp. 123–132.Google Scholar
[1923] Tarski, A., O wyrazie pierwotnym logistyki (On the primitive term of logistic), Przegl d Filosficzny, vol. 26 (1923), pp. 68–89.Google Scholar
[1929] Tarski, A., Les fondements de la géométrie des corps, Annales de la Société Polonaise e Mathématiques, (1929), pp. 29–34, reprinted in [Givant and Mackenzie, 1986a].Google Scholar
[1930] Tarski, A., Bio-bibliographie, unpublished; Vienna Circle Foundation, 1930.Google Scholar
[1935] Tarski, A., Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, vol. 1 (1935), pp. 261–405.Google Scholar
[1937] Tarski, A., Sur la méthode déductive, Traveaux du ixe congrés international de philosophie, tome 6, Actualités Scientifiques et Industrielles, vol. 535, Herman et Cie, Paris, 1937, pp. 95–103.Google Scholar
[1956a] Tarski, A., Foundations of the geometry of solids, Logic, semantics, metamathematics, Oxford: Oxford University Press, 1956, pp. 24–29.Google Scholar
[1956b] Tarski, A., Logic, semantics, metamathematics, Oxford: Oxford University Press, 1956.Google Scholar
[1956c] Tarski, A., On the foundations of Boolean algebra, Logic, semantics, metamathematics, Oxford: Oxford University Press, 1956, pp. 320–341.Google Scholar
[1959] Tarski, A., What is elementary geometry?, The axiomatic method (with special reference to geometry an physics) (Henkin, Leon, Suppes, Patrick, and Tarski, Alfred, editors), North-Holland Publishing Company, Amsterdam, 1959, reprinted in [Givant and Mackenzie, 1986b], pp. 16–29.Google Scholar
[1967] Tarski, A., The completeness of elementary algebra and geometry, Paris, Centre National De La Recherche Scientifique, Institute Blaise Pascal, 1967.Google Scholar
[1999] Tarso, Alfred and Givant, Steven, Tarski's systems of geometry, this Bulletin, vol. 5 (1999), no. 2, pp. 175–214.Google Scholar
[2008] Varzi, Achille, Boundary, The Stanford Encyclopedia of Philosophy (Zalta, Edward N., editor), Fall ed., 2008.Google Scholar
[1919] Whitehead, Alfred North, An enquiry concerning the principles of natural knowledge, Cambridge University Press, 1919.Google Scholar
[1920] Whitehead, Alfred North, The concept of nature, Cambridge University Press, 1920.Google Scholar
[1929] Whitehead, Alfred North, Process and reality, New York: Macmillan, 1929.Google Scholar