Abstract
The Formal Axiomatic Method has been proposed by Hilbert about a century ago and it is appropriate to ask how it performed during the past century. It appears to me that its impact is somewhat controversial. On the one hand, during this time period the Formal Axiomatic Method was and still remains the standard method of theory-building in eyes of logicians and logically-minded mathematicians, physicists, biologists and philosophers. On the same side of the scale I put the progress in the logico-mathematical investigations (some of which use the title of foundations of mathematics), which apply this method in some form.
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Notes
- 1.
Zermelo’s principal motivation for axiomatizing set theory was saving Cantor’s so-called “naive” set theory from paradoxes (Peckhaus 2008).
- 2.
The Continuum Hypothesis conjectured by Cantor states that there is no cardinal number strictly bigger than the minimal infinite cardinal number ℵ0 (which can be described as the “number of all natural numbers”) and strictly smaller that the cardinal number \({2}^{\aleph _{0}}\) of the set of all subsets of some set having the cardinal number ω (for example, the set of all series of natural numbers, including infinite series). Number \({2}^{\aleph _{0}}\) has been identified by Cantor with the number of points on a given continuous line or surface; hence the name of this conjecture.
- 3.
Remind from Sect. 3.4 that Hilbert’s distinction between real and ideal mathematical objects translates into Hilbert’s distinction between (contentual) mathematics and metamathematics as follows: mathematics studies ideal objects with a help of real syntactic constructions; metamathematics studies real syntactic constructions without using anything ideal. This, remind, was Hilbert’s original idea supposed to help mathematics to “get real” without leaving the ideal “Cantor’s Paradise”. When in the light of Gödel’s incompleteness theorems and other developments it became clear that metamathematics cannot do solely with finitary means, some limited “ideal content” – i.e., some more advanced mathematical content – was allowed in it. The title of “The Mathematics of Metamathematics” appeared in 1969 (Rasiowa and Sikorski 1963) perfectly illustrates this shift, which has relaxed the boundary between mathematics and metamathematics.
- 4.
A similar view is developed by Shapiro in (1971).
- 5.
The original French title is Eléments de mathématique, which uses the unusual singular form “mathématique” (while the usual French word for mathematics is “mathématiques”). So a more accurate English translation of the title is Elements of Mathematic. This unusual singular form of the word is supposed to stressed Bourbaki’s aim of the unification of mathematics.
- 6.
As Bourbaki notices here GT can be defined through different axioms. What determines the identity of this theory is its set of true propositions (including both axioms and theorems inferred from these axioms) but not a given set of axioms.
- 7.
If one asks what is specifically “philosophical” about the multiple formal systems presented in this Handbook then, I think, the answer is twofold: all these formal systems are designed with certain philosophical motivations and/or used for treating some philosophical problems. The relevant notion of being philosophical derives from a particular notion of philosophy, which can be roughly identified with the Analytic Philosophy.
- 8.
Although this second degree of freedom (which adds to the free choice of axioms) has been not previewed by Hilbert himself I don’t qualify its discovery as a modification of Hilbert’s Axiomatics Method. However this new degree of freedom undermines Hilbert’s suggested epistemic justification of his method and thus creates new epistemological problems. I shall discuss these problems and suggest a solution in Chap. 10.
Bibliography
G.J. Allman, Greek Geometry from Thales to Euclid (Hodges, Figgis, Dublin, 1889)
D.W. Anderson, Fibrations and geometric realizations. Bull. Am. Math. Soc. 84, 765–786 (1978)
K. Andersen, The Geometry of an Art. The History of the Mathematical Theory of Perspective from Alberti to Monge (Springer, New York/London, 2007)
A. Arnauld, Nouveaux Eléments de Géométrie (Guillaume Desprez, Paris, 1683)
S. Awodey, An answer to G. Hellman’s question does category theory provide a framework for mathematical structuralism? Philos. Math. 12(3), 54–64 (2004)
S. Awodey, Type theory and homotopy (2010). arXiv:1010.1810v1 (math.CT)
S. Awodey, A. Bauer, Introduction to Categorical Logics (draft 2009) (2009), http://www.docstoc.com/docs/12818008/Introduction-to-Categorical-Logics
S. Awodey, M. Warren, Homotopy theoretic models of identity types. Math. Proc. Camb. Philos. Soc. 74, 45–55 (2009). arXiv:0709.0248v1 (math.LO)
J. Baez, Quantum quandaries: a category-theoretic perspective (2004). arXiv:quant-ph/0404040
J. Baez, J. Dolan, Categorification, in Higher Category Theory, ed. by E. Getzler, M.M. Kapranov. Contemporary Mathematics, vol. 230 (American Mathematical Society, Providence, 1998), pp. 1–36. arXiv:math.QA/9802029
M. Balaguer, Platonism and Anti-Platonism in Mathematics (Oxford University Press, New York, 1998)
Y. Bar-Hillel, A. Fraenkel, A. Levy, Foundations of Set Theory (North-Holland, Amsterdam, 1973)
I. Barrow, in Geometrical Lectures, ed. by J.M. Child (Dover, Mineola, 2006)
J. Bell, The development of categorical logic, in The Handbook of Philosophical Logic, vol. 12, ed. by D.M. Gabbay (Kluwer, Dordrecht/Boston, 2005)
J. Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, ed. by J. Bénabou et al. Lecture Notes in Mathematics, vol. 47 (Springer, Berlin/New York, 1967), pp. 1–77
J. Bénabou, Fibred categories and the foundation of naive category theory. J. Symb. Log. 50(1), 10–37 (1985)
P. Benacerraf, What numbers could not be. Philos. Rev. 74, 47–73 (1965)
P. Bernays, Axiomatic Set Theory (North-Holland, Amsterdam, 1958)
D. Bonnay, Logicality and invariance. Bull. Symb. Log. 14(1), 29–68 (2006)
R. Bonola, Non-Euclidean Geometry (Dover, New York, 1955)
G.A. Borelli, Euclides Restitutus (Pisa, 1658)
N. Bourbaki, Éléments de la théorie des ensembles. Prolégomènes sur la notion de théorie mathématique (état 1). Unpublished manuscript available at online Bourbaki Archive at http://mathdoc.emath.fr/archives-bourbaki/PDF/050_iecnr_059.pdf (1935–1939)
N. Bourbaki, Eléments de Mathématique, 10 vols (Hermann, Paris, 1939–1988)
N. Bourbaki, Architecture of mathematics. Am. Math. Mon. 57(4), 221–232 (1950)
N. Bourbaki, Theory of Sets (Hermann, Paris, 1968)
F.H. Bradley, Principles of Logic (Oxford University Press, London, 1922)
N.G. Bruijn, On the roles of types in mathematics. Cahiers du Centre de Logique 8, 27–54 (1995)
L. Brunschvicg, Les étapes de la philosophie mathématique (Alcan, Paris, 1912)
F. Cajori, A History of Mathematical Notation (Open Court, Chicago, 1929)
G. Cantor, Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen (Teubner, Leipzig, 1883a)
G. Cantor, Über unendliche, lineare punktmannichfaltigkeiten. 5. fortsetzung. Math. Ann. 21, 545–591 (1883b)
G. Cantor, Beitraege zur begrundung der transfiniten mengenlehre. Math. Ann. 46, 481–512 (1885)
F. Cardone, J.R. Hindley, History of lambda-calculus and combinatory logic, in Handbook of the History of Logic, vol. 5, ed. by D.R. Gabbay, J. Woods (Elsevier/North Holland, Amsterdam/Boston/Heidelberg, 2009), pp. 723–817
E. Cassirer, Kant und die moderne mathematik. Kant-Studien 12, 1–40 (1907)
E. Cassirer, Philosophie der symbolischen Formen (in zwei Banden) (Bruno Cassirer, Berlin, 1923–1925)
P.J. Cohen, The independence of the continuum hypothesis. Proc. Natl. Acad. Sci. U.S.A. 50, 1143–1148 (1963)
L. Corry, The empiricist root of hilbert’s axiomatic approach, in Proof Theory: History and Philosophical Significance, ed. by V. Hendricks et al. Synthese Library, vol. 292 (Kluwer, Dordrecht/Boston, 2000), pp. 35–54
L. Corry, Modern Algebra and the Rise of Mathematical Structures (Birkhäuser, Basel/Boston, 2004)
L. Corry, The origins of Hilbert’s axiomatic method, in The Genesis of General Relativity, Vol. 4 Theories of Gravitation in the Twilight of Classical Physics: The Promise of Mathematics and the Dream of a Unified Theory, ed. by J. Renn (Springer, New York, 2006), pp. 139–236
E. Dean, J. Avigad, J. Mumma, A formal system for euclid’s elements. Rev. Symb. Log. 2(4), 700–768 (2009)
T.A. Debs, M.L.G. Redhead, Objectivity, Invariance, and Convention: Symmetry in Physical Science (Harvard University Press, Cambridge, Massachusetts, 2007)
G. Deleuze, Différence et répétition (Presses universitaires de France, Paris, 1968)
R. Descartes, La Géométrie (Guillaume Desprez, Paris, 1886)
R. Descartes, La géométrie (Hermann, Paris, 2008)
M. Detlefsen, Poincaré against logicians. Synthese90(3), 349–378 (1992)
H. Deutsch, Relative identity (entry in stanford encyclopedia of philosophy) (2002), http://plato.stanford.edu/entries/identity-relative/
F. Dosse, History of Structuralism, Vols. I (The Rising Sign, 1945–1966) and II (The Sign Sets, 1967-Present) (trans. by D. Glassman) (The University of Minnesota Press, Minneapolis, 1997)
H.M. Edwards, The genesis of ideal theory. Arch. Hist. Exact Sci. 23, 321–378 (1980)
S. Eilenberg, S. MacLane, General theory of natural equivalences. Trans. Am. Math. Soc. 58(2), 231–294 (1945)
A. Einstein, Geometry and Experience, Ideas and Opinions, Bonanza Books, New York, p. 232–246 (1954)
Euclid, The optics of Euclid (English trans. by H.E. Burton). J. Opt. Soc. Am. 35(5), 357–372 (1945)
Euclid, Elements English trans. by (R. Fitzpatrick Ausitn, through lulu.com, 2011)
Euclides, in Euclidis Opera Omnia, ed. by J.L. Heiberg (Teubner, Lipsiae, 1883–1886)
S. Feferman, Categorical foundations and foundations of category theory, in Logic, Foundations of Mathematics and the Computability Theory, ed. by R. Butts, J. Hintikka (Reidel, Dordrecht/Boston, 1977), pp. 149–169
J. Ferreiros, Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Science Networks. Historical Studies, vol. 23 (Birkhauser, Basel/Boston, 1999)
R. Feys, H. Curry, W. Craig, Combinatory Logic, Vol. 1 (North Holland, Amsterdam, 1958)
J.N. Findlay, Plato: The Written and Unwritten Doctrines (Routledge, London, 1974)
M. Fitting, Intensional logic (entry in stanford encyclopedia of philosophy) (2006–2011), http://plato.stanford.edu/entries/logic-intensional/
B.C. Fraassen, The Scientific Image (Clarendon, Oxford, 1980)
G. Frege, Die Grundlagen der Arithmetik (W. Koebner, Breslau, 1884)
G. Frege, Über sinn und bedeutung. Zeitschrift für Philosophie und philosophische Kritik 100, 25–50 (1892)
G. Frege, Grundgesetze der Arithmetik, Bd. II (Verlag Hermann Pohle, Jena, 1903)
G. Frege, On sense and reference, in Translations from the Philosophical Writings of Gottlob Frege, ed. by P. Geach, M. Black. (Basil Blackwell, Oxford, 1952), pp. 56–78
G. Frege, Grundgesetze der Arithmetik (Hildesheim, Olms, 1962)
G. Frege, The Basic Laws of Arithmetic. Exposition of the System (translated and edited, with an introduction, by M. Furth) (California Press, Berkeley, 1964)
G. Frege, On the Foundations of Geometry and Formal Theories of Arithmetic (Yale University Press, New Haven, 1971)
S. French, Quantum physics and the identity of indiscernibles. Br. J. Philos. Sci. 39, 233–246 (1988)
S. French, D. Krause, Identity in Physics: A Historical, Philosophical, and Formal Analysis (Oxford University Press, Oxford/New York, 2006)
H. Freudental, The main trends in the foundations of geometry in the 19th century, in Proceedings of the 1960 International Congress on Logic, Methodology and Philosophy of Science, Standford 1962, (1960), pp. 613–621
P. Freyd, Aspects of topoi. Bull. Aust. Math. Soc. 7, 1–76 (1972)
M. Friedman, Kant and the Exact Sciences (Harvard University Press, Cambridge, 1992)
D.M. Gabbay (ed.), What Is a Logical System? (Oxford University Press, New York, 1994)
D.M. Gabbay, F. Guenthner (eds.), Handbook of Philosophical Logic (Springer, Berlin/New York, 2001)
G. Galilei, [Discourses on the] Two New Sciences (trans. S. Drake) (Wall and Emerson, Toronto, 1974)
C.F. Gauss, Disquisitiones generales circa superficies curvas (Dieterich, Gottinga, 1828)
P.T. Geach, Logic Matters (Basil Blackwell, Oxford, 1972)
S.I. Gelfand, Yu.I. Manin, Methods of Homological Algebra (Springer, Berlin/New York, 2003)
G. Gentzen, über die existenz unabhängiger axiomensysteme zu unendlichen satzsystemen. Math. Ann. 107, 329–350 (1932)
G. Gentzen, Untersuchungen über das logische schliessen i. Math. Z. 39(2), 176–210 (1934)
G. Gentzen, Untersuchungen über das logische schlieβen ii. Math. Z. 39(3), 405–431 (1935)
K. Gödel, The consistency of the axiom of choice and of the generalized continuum hypothesis. Proc. Natl. Acad. Sci. U.S.A. 24, 556–557 (1938)
M. Grandis, Directed Homotopy Theory (2001). arXiv:math.AT/0111048
J.G. Granstrom, Treatise on Intuitionistic Type Theory (Springer, Dordrecht, 2011)
I. Grattan-Guinness, An unpublished paper by georg cantor: Principien einer theorie der ordnungstypen. Acta Math. 124, 65–107 (1970)
J. Gray (ed.), The Symbolic Universe: Geometry and Physics 1890–1930 (Oxford University Press, Oxford/New York, 1999)
M.J. Greenberg, Euclidean and Non-Euclidean Geometries (W.H. Freeman, San Francisco, 1993)
T.L. Heath, The Thirteen Books of Euclid’s Elements Translated from the Text of Heiberg with Introduction and Commentary (Cambridge University Press, Cambridge, 1926)
T.L. Heath, Mathematics in Aristotle (Oxford University Press, Oxford, 1949)
T.L. Heath, A History of Greek Mathematics (Dover, New York, 1981). (First published 1921)
T.L. Heath, A Manual of Greek Mathematics (Dover, Mineola, 2003). (First published 1931)
G.W.F. Hegel, in Science of Logic, trans. by A.V. Miller (George Allen and Unwin, London, 1969)
G. Heinzmann (ed.), Poincaré, Russell, Zermelo et Peano. Textes de la discussion (1906–1912) sur les fondements des mathématiques: des antinomies à la prédicativité (Blanchard, Paris, 1986)
J. Heis, The fact of modern mathematics: geometry, logic, and concept formation in Kant and Cassirer, Ph.D. thesis, University of Pittsburgh, 2007
G. Hellman, Three varieties of mathematical structuralism. Philos. Math. 2(9), 184–211 (2001)
G. Hellman, Does category theory provide a framework for mathematical structuralism? Philos. Math. 2(11), 129–157 (2003)
G. Hellman, Structuralism, mathematical, in Encyclopedia of Philosophy, vol. 9, 2nd edn., ed. by D.M. Borchert (Macmillan, Detroit, 2006), pp. 270–273
A. Heyting, Intuitionism: An Introduction (North-Holland, Amsterdam, 1956)
D. Hilbert, Grundlagen der Geometrie (B.G. Teubner, Leipzig, 1899)
D. Hilbert, On the concept of number, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. 2, ed. by W. Ewald (Clarendon, Oxford, 1996), pp. 1089–1096
D. Hilbert, Mathematical problems. Bull. Am. Math. Soc. 8(10), 437–479 (1902)
D. Hilbert, Grundlagen der Geometrie (zweite Auflage) (Teubner, Leipzig, 1903)
D. Hilbert, Axiomatic thought, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. 2, ed. by W. Ewald (Clarendon, Oxford, 1996), pp. 1105–1115
D. Hilbert, The new grounding of mathematics, in From Kant to Hilbert: A Source Book in the Foundations of Mathematics, vol. 2, ed. by W. Ewald (Clarendon, Oxford, 1996), pp. 1115–1134
D. Hilbert, Foundations of mathematics, in From Frege to Gödel: A Source Book in the Mathematical Logic, vol. 2, ed. by J. van Heijenoort (Harvard University Press, Cambridge, Massachusetts, 1967), pp. 464–480
D. Hilbert, The Foundations of Geometry (Open Court, LaiSalle, 1950)
D. Hilbert, W. Ackermann, Grundzüge der theoretischen Logik (Springer, Berlin/Heidelberg, 1928)
D. Hilbert, W. Ackermann, Principles of Mathematical Logic (Chelsea, New York, 1950)
D. Hilbert, P. Bernays, Grundlagen der Mathematik (in 2 vols) (Springer, Berlin, 1934–1939)
D. Hilbert, P. Bernays, Foundations of Mathematics 1. Commented translation by C.-P. Wirth (College publications, London, 2011)
D. Hilbert, S. Chon-Vossen, Anschauliche Geometrie (Springer, Berlin, 1932)
R. Hilpinen, Logic as calculus and logic as language. Synthese 17, 324–330 (1967)
R. Hilpinen, Peirce’s logic, in Handbook of the History of Logic: The Rise of Modern Logic from Leibniz to Frege, vol. 3, ed. by D.M. Gabbay et al. (Elsevier, Amsterdam/London, 2004) pp. 611–658
J. Hintikka, Hilbert vindicated? Synthese 110(1), 15–36 (1997a)
J. Hintikka, Lingua Universalis vs. Calculus Ratiocinator. An Ultimate Presupposition of Twentieth-Century Philosophy. Collected Papers, vol. 2 (Kluwer, Dordrecht/Boston, 1997b)
J. Hintikka, U. Remes, The Method of Analysis. Its Geometrical Origin and Its General Significance (D. Reidel, Dordrecht/Boston, 1974)
J. Hintikka, U. Remes, Ancient geometrical analysis and modern logic, in Essays in Memory of Imre Lakatos, ed. by R.S. Cohen, P.K. Feyerabend, M.W. Wartofsky. Boston Studies in the Philosophy of Science, vol. 39 (D. Reidel, Dordrecht/Boston, 1976)
M. Hofmann, T. Streicher, A groupoid model refutes uniqueness of identity proofs, in Proceedings of the 9th Symposium on Logic in Computer Science (LICS), Paris, 1994
M. Hofmann, T. Streicher, The groupoid interpretation of type theory, in Twenty-Five Years of Constructive Type Theory, ed. by G. Sambin, J. Smith (Oxford University Press, New York, 1998), pp. 83–111
W. Horward, The formulae-as-types notion of construction, in To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, ed. by J.P. Seldin, J.R. Hindley (Academic, London/New York, 1980), pp. 479–490
D. Hume, in A Treatise of Human Nature, ed. L.A. Selby (Oxford University Press, New York, 1978)
P. Hylton, Russell, Idealism, and the Emergence of Analytic Philosophy (Oxford University Press, New York, 1990)
J.R. Isbell, Review of Lawvere:1966. Math. Rev. 34 (1967) #7332
B. Jacobs, Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141 (Elsevier, Amsterdam/New York, 1999)
P. Johnstone, The point of pointless topology. Bull. Am. Math. Soc. (N.S.) 8(1), 41–53 (1983)
P. Johnstone, Sketches of an Elephant: A Topos Theory Compendium, Vols. 1–2. (Oxford University Press, Oxford/New York 2002)
A. Joyal, Remarks on homotopical logic. Mathematisches Forschungsinstitut Oberwolfach, Mini-Workshop: The Homotopy Interpretation of Constructive Type Theory, Report N 11/2011, 2011, pp. 19–22
D.M. Kan, Adjoint functors. Trans. Am. Math. Soc. 87, 294–329 (1958)
I. Kant, Kritik der reinen Vernunft (Meiner Verlag, Hamburg, 1998)
Im. Kant, Critique of Pure Reason (Translated into English by P. Guyer) (Cambridge University Press, Cambridge/New York, 1999)
M.M. Kapranov, V.A. Voevodsky, ∞-groupoids and homotopy types. Cahiers de Topologie et Géometrie Différentielle Catégoriques32(1), 29–46 (1991)
J. Keranen, The identity problem for realist structuralism. Br. J. Philos. Sci. 9(3), 308–330 (2001)
F. Klein, über die sogenannte nicht-euklidische geometrie. Math. Ann. 4, 573–625 (1871)
F. Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (A. Deichert, Erlangen, 1872)
F. Klein, über die sogenannte nicht-euklidische geometrie (zweiter aufsatz). Math. Ann. 6, 112–145 (1873)
A.N. Kolmogorov, S.V. Fomin, Elementi teorii funkcii i funkcionalnogo analysa (in Russian). (Nauka, Moscow, 1967)
E.E. Kummer, Über die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren. J. für Math. (Crelle) 35(1847), 327–367
R. Kromer, Tool and Object: A History and Philosophy of Category Theory (Birkhauser, Basel/Boston, 2007)
K. Kunen, Set Theory: An Introduction to Independence Proofs (Elsevier, Amsterdam, 1980)
K. Kuratowski, Topologie I-II (Warszawa-Lwów, 1948–1950)
J. Ladyman, Structural realism. Stanford Encyclopedia of Philosophy (2009), http://plato.stanford.edu/entries/structural-realism/
J. Lambek, Deductive systems and categories. (i) Syntactic calculus and residuated categories. Math. Syst. Theory 2, 287–318 (1968)
J. Lambek, Deductive systems and categories. (ii) Standard constructions and closed categories, in Category Theory, Homology Theory, and Their Applications, ed. by P.J. Hilton. Volume 86 of Lecture Notes in Mathematics Springer, vol. 1 (Springer, Berlin, 1969), pp. 76–122
J. Lambek, Deductive systems and categories. (iii) Cartesian closed categories, intuitionistic propositional calculus, and combinatory logic, in Toposes, Algebraic Geometry and Logic, ed. by F.W. Lawvere. Volume 274 of Lecture Notes in Mathematics (Springer, Berlin/New York, 1972), pp. 57–82
F.W. Lawvere, Functorial semantics of algebraic theories. Ph.D. Columbia University, 1963
F.W. Lawvere, An elementary theory of the category of sets. Proc. Natl. Acad. Sci. U.S.A. 52, 1506–1511 (1964)
F.W. Lawvere, The category of categories as a foundation for mathematics. In: S. Eilenberg et al. (eds.), Proceedings of the Conference on Categorical Algebra, La Jolla, 1965 (Springer, New York, 1966a), pp. 1–21
F.W. Lawvere, Functorial semantics of elementary theories. J. Symb. Log. 31, 294–295 (1966b)
F.W. Lawvere, Theories as categories and the completeness theorem. J. Symb. Log. 32, 562 (1967)
F.W. Lawvere, Adjointness in foundations. Dialectica 23, 281–296 (1969a)
F.W. Lawvere, Diagonal arguments and cartesian closed categories, in Category Theory, Homology Theory and Their Applications II, ed. by P.J. Hilton. Volume 92 of Lecture Notes in Mathematics, vol. 2 (Springer, Berlin, 1969b) pp. 134–145
F.W. Lawvere, Equality in hyperdoctrines and comprehension schema as an adjoint functor, in Applications of Categorical Algebra. Proceedings of the Symposoia in Pure Mathematics, New York, 1968, vol. XVII, (1970a), pp. 1–14
F.W. Lawvere, Quantifiers and sheaves, in Actes du congres international des mathematiciens, Nice, 1970, ed. by M. Berger, J. Dieudonne et al. (1970b), pp. 329–334
F.W. Lawvere, Some thoughts on the future of category theory, in Category Theory, Proceedings, Como, 1990. Springer Lecture Notes in Mathematics, vol. 1488 (Springer, 1991), pp. 1–13
F.W. Lawvere, Categories of space and quantity, in The Space of Mathematics, ed. by J. Echeverria et al. (de Gruyter, Berlin/New York, 1992), pp. 14–30
F.W. Lawvere, Coherent toposes and cantor’s ‘lauter einsen’. Philos. Math. 2, 5–15 (1994a)
F.W. Lawvere, Tools for the advancement of objective logic: closed categories and toposes, in The Logical Foundations of Cognition, ed. by J. Macnamara, G.E. Reyes. Proceedings of the February 1991 Vancouver Conference “Logic and Cognition” (Oxford University Press, 1994b), pp. 43–56
F.W. Lawvere, Adjoints in and among bicategories, in Logic and Algebra, ed. by A. Ursini, P. Agliano (Marcel Dekker, New York, 1996), pp. 181–189
F.W. Lawvere, Foundations and applications: axiomatization and education. Bull. Symb. Log. 9(2), 213–224 (2003)
F.W. Lawvere, Functorial semantics of algebraic theories and some algebraic problems in the context of functorial semantics of algebraic theories (reprint with the author’s comments). Repr. Theory Appl. Categ. 5, 1–121 (2004)
F.W. Lawvere, An elementary theory of the category of sets (long version) with the author’s commentary. Repr. Theory Appl. Categ. 11, 1–35 (2005a)
F.W. Lawvere, Taking categories seriously. Repr. Theory Appl. Categ. 2, 1–24 (2005b)
F.W. Lawvere, Adjointness in foundations with the author’s commentary. Repr. Theory Appl. Categ. 16, 1–16 (2006)
F.W. Lawvere, R. Rosebrugh, Sets for Mathematics (Cambridge University Press, Cambridge/New York, 2003)
F.W. Lawvere, S.H. Schanuel, Conceptual Mathematics (Cambridge University Press, Cambridge/New York, 1997)
T. Leinster, A survey of denitions of n-category. Theory Appl. Categ. 10, 1–70 (2002)
T. Leinster, Higher Operads, Higher Categories. London Mathematical Society Lecture Note Series, vol. 298 (Cambridge University Press, Cambridge/New York, 2004). arXiv:math/0305049
C. Levi-Strauss, The Elementary Structures of Kinship (Taylor and Francis, London, 1969)
J. Lambek, P.J. Scott, Higher Order Categorical Logic (Cambridge University Press, Cambridge/New York, 1986)
N.I. Lobachevsky, Geometrische Untersuchungen zur Theorie der Parallellinien (F. Fincke, Berlin, 1840)
N.I. Lobachevsky, New foundations of geometry, in Selected Works in Geometry (Academy of Sciences of USSR Publishing, Moscow, 1956), pp. 73–312 (in Russian)
L. Lobachevsky, Geometrical Researches on the Theory of Parallels (trans. by G.B. Halsted) (Cosimo, New York, 2007)
J.R. Lucas, Treatise on Time and Space (Methuen, London, 1973)
J. Lurie, Higher Topos Theory. Annals of Mathematics Studies, vol. 170 (Princeton University Press, Princeton, 2009)
S. MacLane, Structure in mathematics. Philos. Math. 4(2), 174–183 (1996)
S. MacLane, Categories for the Working Mathematician, 2nd edn. (Springer, New York, 1998)
S. MacLane, I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer, New York, 1992)
M. Makkai, Towards a categorical foundation of mathematics. Lect. Notes Log. 11, 153–190 (1998)
Yu.I. Manin, Lectures on Algebraic Geometry (in Russian) (Moscow State University, Moscow, 1970)
Yu. Manin, Georg cantor and his heritage (talk at the meeting of the german mathematical society and the cantor medal award ceremony) (2002). arXiv:math.AG/02009244 v1/
J. Mayberry, The Foundations of Mathematics in the Theory of Sets (Cambridge University Press, Cambridge/New York, 2000)
J.-P. Marquis, From a Geometrical Point of View: A Study of the History and Philosophy of Category Theory (Springer, Dordrecht/London, 2009)
J.-P. Marquis, Reyes G.E, The history of categorical logic: 1963–1977, in Handbook of the History of Logic: Sets and Extensions in the Twentieth Century, vol. 6, ed. by A. Kanamori (Elsevier, Amsterdam/Boston, 2012), pp. 689–800
P. Martin-Löf, Intuitionistic Type Theory. Notes by Giovanni Sambin of a Series of Lectures Given in Padua, June 1980 (BIBLIOPOLIS, Napoli, 1984)
C. McLarty, Axiomatizing a category of categories. J. Symb. Log. 56(4), 1243–1260 (1991)
C. McLarty, Elementary Categories, Elementary Toposes (Clarendon, Oxford, 1992)
C. McLarty, Exploring categorical structuralism. Philos. Math. 12(1), 37–53 (2004)
B. Mélès, Pratique mathématique et lecture de hegel de jean cavaillès à william lawvere. Philos. Sci. 16(1), 153–182 (2012)
C.U. Moulines, W. Balzer, J.D. Sneed, An Architectonic for Science: The Structuralist Approach (Reidel, Dordrecht, 1987)
P. Nahin, An Imaginary Tale (Princeton University Press, Princeton, 1998)
P. Natorp, Kant und die Marbuger Schule. Kant-Studien 17, 193–221 (1912)
R. Nozick, Invariances. The Structure of Objective World (Harvard University Press, Cambridge-Massachusetts, 2001)
C. Parsons, Mathematical Thought and Its Objects (Cambridge University Press, New York, 2008)
M. Pasch, Vorlesungen über neueren Geometrie (Teubner, Leipzig, 1882)
V. Peckhaus, Pro and contra Hilbert: Zermelo’s set theories. Philos. Sci. (2008), http://philosophiascientiae.revues.org/390
P. Pritchard, Plato’s Philosophy of Mathematics. International Plato Studies, vol. 5 (Academia, Sankt Augustin, 1995)
Proclus, A Commentary on the First Book of Euclid’s Elements (trans. by G.R. Morrow) (Princeton University Press, Princeton, 1970)
D. Quillen, Homotopical Algebra. Lecture Notes in Mathematics, vol. 43 (Springer, Berlin/New York, 1967)
W.O. Quine, From a Logical Point of View (Harvard University Press, Cambridge, Massachusetts, 1953)
W.O. Quine, Ontological Relativity and Other Essays (Columbia University Press, New York, 1969)
D. Rabouin, Structuralisme et comparatisme en sciences humaines et en mathématiques: un malentendu? in Le moment philosophique des années 1960 en France, ed. by P. Maniglier (Presses universitaires de France, Paris, 2011), pp. 37–57
P.K. Rashevskii, On the dogma of the natural numbers. Russ. Math. Surv. 28, 143–148 (1973)
H. Rasiowa, R. Sikorski, The Mathematics of Metamathematics (Naukowe, Państwowe Wydawn, 1963)
G.E. Reyes, L.P. Reyes, H. Zolfaghari, Generic Figures and Their Gluings (Polimetrica, Milano, 2004)
H.-J. Rheinberger, On Historicizing Epistemology (Stanford University Press, Stanford, 2010)
B. Riemann, über die hypothesen, welche der geometrie zugrunde liegen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13(1867), 133–152 (1854)
B. Riemann, über die hypothesen, welche der geometrie zu grunde liegen (habilitationsvortrag of 1854). Abhandlungen der Kgl. Gesellschaft der Wissenschaften zu Göttingen 13, 133–152 (1867)
A. Rodin, Events and Intensional Sets, in Logica’2002, Proceedings of the International Symposium, ed. by T. Childers, O. Majer (Czech Academy of Science, Prague, 2003), pp. 221–232
A. Rodin, Identity and categorification. Philos. Sci. 11(2), 27–65 (2007)
A. Rodin, How mathematical concepts get their bodies. Topoi 29(1), 53–60 (2010)
D. Rowe, Klein, hilbert, and the göttingen mathematical tradition. Osiris 5, 186–213 (1989)
D. Rowe, The calm before the storm: Hilbert’s early views on foundations, in Proof Theory: History and Philosophical Significance, ed. by V. Hendricks et al. Synthese Library, vol. 292 (Kluwer, Dordrecht/Boston, 2000), pp. 55–88
B. Russell, An Essay on the Foundations of Geometry (Cambridge University Press, Cambridge, 1897)
B. Russell, An Critical Exposition of the Philosophy of Leibniz (Cambridge University Press, Cambridge, 1900)
B. Russell, Principles of Mathematics (Allen and Unwin, London, 1903)
B. Russell, Les paradoxes de la logique. Revue de métaphysique et de morale 14, 627–650 (1906a). Reprinted in Heinzmann (1986)
B. Russell, On some difficulties in the theory of transfinite numbers and order types. Proc. Lond. Math. Soc. 4(Ser. 2), 29–53 (1906b). Reprinted in Heinzmann (1986)
B. Russell, Mathematical logic as based on the theory of types. Am. J. Math. 28, 222–262 (1908). Reprinted in Heinzmann (1986).
B. Russell, The philosophy of logical atomism. Monist 28, 495–527 (1918)
B. Russell, The classification of relations, in Collected Papers, vol. 2 (Unwin Hyman, London, 1990), pp. 138–146
B. Russell, A. Whitehead, Principia Mathematica, 3 vols (Cambridge University Press, Cambridge, 1910–1913)
G.G. Saccheri, Euclide Ab Omni Naevo Vindicatus (Typis P. A. Montani, Milano, 1733)
H. Scholz, H. Schweitzer, Die sogenannten Definitionen durch Abstraktion (Felix Meiner, Leipzig, 1935)
M. Schönfinkel, Über die bausteine der mathematischen logik. Math. Ann. 92, 305–316 (1924a)
M. Schönfinkel, On the building blocks of mathematical logic, in From Frege to Gödel: A Source Book in the Mathematical Logic, ed. by J. van Heijenoort (Harvard University Press, Cambridge, 1967), pp. 355–366
D. Scott, Constructive validity, in Symposium on Automatic Demonstration, Versailles, 1968, ed. by M. Laudet et al. Lecture Notes on Mathematics, vol. 125, (1970), pp. 237–275
R.A.G. Seely, Locally cartesian closed categories and type theory. Math. Proc. Camb. Math. Soc. 95 33–48 (1984)
J.P. Seldin, The logic of Church and Curry, Handbook of the History of Logic, vol. 5, ed. by D.R. Gabbay, J. Woods (Elsevier/North Holland, Amsterdam/Boston, 2009), pp. 819–894
M. Serfati, La revolution symbolique: la Constitution de l’Ecriture Symbolique Mathématique (Editions Petra, Paris, 2005)
S. Shapiro, Foundations without Foundationalism: A Case for Second-order Logic (Oxford University Press, New York, 1991)
A. Shenitzer, The cinderella career of projective geometry. Math. Intell. 13(2), 50–55 (1991)
J.D. Sneed, The Logical Structure of Mathematical Physics (Reidel, Dordrecht, 1971)
G. Sommaruga, History and Philosophy of Constructive Type Theory. Synthese Library vol. 290 (Springer, Berlin/New York, 2000)
T. Streicher, Investigations into intensional type theory. Habilitationsschrift. Ludwig-Maximilians-Universität München, 1993
F. Suppe, The Structure of Scientific Theories: Symposium, 1969, Urbana. Outgrowth with a Critical Introduction and an Afterword by Frederick Suppe (University of Illinois Press, Urbana, 1977)
F. Suzuki. Homotopy and path integrals (2011). arXiv:1107.1459
A. Tarski, Introduction to Logic and to the Methodology of Deductive Sciences (Oxford University Press, New York, 1941)
A. Tarski. Sentential calculus and topology, in Logic, Semantics, Metamathematics: Papers from 1923 to 1938 (trans. by J.H. Woodger) (Clarendon, Oxford, 1956), pp. 421–454
A. Tarski, What are logical notions? Hist. Philos. Log. 7, 143–154 (1986)
M. Toepell, On the origins of david hilbert’s “grundlagen der geometrie”. Arch. Hist. Exact Sci. 35(4), 329–344 (1986)
A. Trendelenburg, Logische Untersuchungen (Verlag von S. Hirzel, Leipzig, 1862)
S. Unguru, On the need to rewrite the history of greek mathematics. Arch. Exact Sci. 15, 67–114 (1975)
D. van Dalen, Internal type theory, in Types for Proofs and Programs, Torino, 1995. Lecture Notes in Computer Science, vol. 1158 (Springer, Berlin/New York, 1996), pp. 120–134
D. van Dalen, The development of Brouwer’s intuitionism, in Proof Theory: History and Philosophical Significance, ed. by V. Hendricks et al. Synthese Library, vol. 292 (Kluwer, Dordrecht/Boston, 2000), pp. 117–152
B. van der Waerden, Moderne Algebra (Springer, Berlin, 1930–1931)
V.L. Vasyukov, Mathematical pluralism (in Russian), in Proceedings of International Conference on Philosophy, Mathematics, Linguistics: Aspects of Interaction, Saint-Petersburg, 20–22 Nov 2009, ed. by O. Prosorov et al. (2009), pp. 222–227
O. Veblen, J.H.C. Whitehead, The Foundations of Differential Geometry. Cambridge Tracts in Mathematics and Mathematical Physics, vol. 29 (Cambridge University Press, Cambridge, 1932)
J. Venn, Symbolic Logic (MacMillan, London, 1881)
V.A. Voevodsky, Introduction to the univalent foundations of mathematics (video recording of lecture given at princeton institute of advanced studies 10 Dec 2010, www.math.ias.edu/node/2778
V.A. Voevodsky, Univalent foundations. Mathematisches Forschungsinstitut Oberwolfach, mini-workshop: the homotopy interpretation of constructive type theory, Report N 11/2011, 2011, pp. 7–10
V.A. Voevodsky et al., Homotopy Type Theory: Univalent Foundations of Mathematics, Institute for Advanced Study (Princeton), The Univalent Foundations Program, 2013. http://homotopytypetheory.org/book/
M.L. Wantzel, Recherches sur les moyens de reconnaître si un problème de géométrie peut se résoudre avec la règle et le compas. Journal de Mathématiques Pures et Appliquées 2(1), 366–372 (1837)
M.A. Warren, Homotopy theoretic aspects of constructive type theory, Ph.D. thesis, 2008, www.andrew.cmu.edu/user/awodey/students/warren.pdf
Ch. Wells, Sketches: Outline with references, Deparment of Computer Science, Katholieke Universiteit Leuven, 1994
H. Weyl, Space-Time-Matter (translated into English by H.L. Brose) (Dover, Mineola, 1950)
D. Wiggins, Sameness and substance (Blackwell, Oxford, 1980)
E. Wigner, The unreasonable effectiveness of mathematics in the natural sciences. Commun. Pure Appl. Math. 13, 1–14 (1960)
H. Wussing, The Genesis of the Abstract Group Concept: A Contribution to the History of the Origin of Abstract Group Theory (Dover, New York, 2007)
R. Zach, Hilbert’s program, in Stanford Encyclopedia of Philosophy, ed. by E.N. Zalta et al. (Stanford University, Stanford, 2003), http://plato.stanford.edu/
E. Zermelo, Untersuchungen über die Grundlagen der Mengenlehre. Math. Ann. 59, 261–281 (1908). Reprinted in Heinzmann (1986).
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Rodin, A. (2014). Formal Axiomatic Method and the Twentieth Century Mathematics. In: Axiomatic Method and Category Theory. Synthese Library, vol 364. Springer, Cham. https://doi.org/10.1007/978-3-319-00404-4_4
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