Гильбертовская концепция «существования в математике» и ее моделирование формальной аксиоматической теорией Ф+∃, рассматривающей существование не как квантор, а как модальность (на англ.)

Владимир Олегович Лобовиков

doi:10.47850/RL.2024.5.1.16-50

Авторы

Владимир Олегович Лобовиков Институт философии и права Уральского отделения РАН (г. Екатеринбург)

DOI:

https://doi.org/10.47850/RL.2024.5.1.16-50

Аннотация

Предмет исследования – принцип Гильберта, состоящий из двух частей: (a) утверждение об эквивалентности непротиворечивости и истины в математике; (b) утверждение об эквивалентности непротиворечивости и существования в математике. Цель – уточнение и обоснование упомянутого принципа. Научная новизна: для достижения этой цели, (1) построена некая пока неизвестная логически формализованная мультимодальная аксиоматическая система онтологии и эпистемологии, названная Ф+∃ ; (2) впервые с помощью Ф+∃ дано точное аксиоматическое определение понятия «существование как модальность»; (3) на искусственном языке Ф+∃ точно сформулирован принцип эквивалентности непротиворечивости и существования в математике; (4) впервые публикуются формальные дедуктивные выводы (в формальной теории Ф+∃ ) тех формул, которые вместе образуют (в соответствующей интерпретации) принцип Гильберта.

Биография автора

Владимир Олегович Лобовиков, Институт философии и права Уральского отделения РАН (г. Екатеринбург)

доктор философских наук, профессор, главный научный сотрудник

Библиографические ссылки

Айер, А. Дж. (2010). Язык, истина и логика. М.: «Канон+» РООИ «Реабилитация».

Ayer, A. J. (2010). Language, Truth and Logic. Moscow. (In Russ.)

Ершов, Ю. Л., Самохвалов, К. Ф. (2007). Современная философия математики: недомогание и лечение. Новосибирск: Параллель.

Ershov, Yu. L., Samokhvalov, K. F. (2007). Contemporary Philosophy of Mathematics: Ailment and Treatment. Novosibirsk. (In Russ.)

Лобовиков, В. О. (2014). Логический квадрат и гексагон эпистемических понятий. Эпистемы: сборник научных статей. Вып. 9: Аспекты аналитической традиции. Екатеринбург: Ажур. C. 57-68.

Lobovikov, V. O. (2014). Logical Square and Hexagon of Epistemic Notions. In Epistemes. Collection of Scientific Papers. Iss. 9. Aspects of the Analytic Tradition. Yekaterinburg. Pp. 57-68. (In Russ.)

Лобовиков, В. О. (2016). Уточнение формулировки и вариант решения проблемы эквивалентности деонтических, алетических и аксиологических модальностей с помощью аксиоматической системы философской эпистемологии. Эпистемы: сборник научных статей. Вып. 11: Ценности и нормы в структуре рационального познания. Екатеринбург: Макс-Инфо. C. 46-65.

Lobovikov, V. O. (2016). An Explication of Formulation and an Option of Solution of the Problem of Equivalence of of Deontic, Alethic and Axiological Modalities by Means of an Axiomatic System of Philosophical Epistemology. In Epistemes. Collection of Scientific Papers. Iss. 11. Values and Norms in Structure of Rational Knowledge. Yekaterinburg. Pp. 46-65. (In Russ.)

Лобовиков, В. О. (2023a). Искусственный интеллект и учение И. Канта о предписывании разумом законов природе (с точки зрения мультимодальной аксиоматической эпистемологии). Трансцендентальный поворот в современной философии – 8: метафизика, эпистемология, когнитивистика и искусственный интеллект: сборник тезисов международной научной конференции. Москва, 20–22 апреля 2023 г. Oтв. ред. С. Л. Катречко, А. А. Шиян. М.: Изд-во: РГГУ. С. 108-111.

Lobovikov, V. O. (2023a). Artificial Intelligence and I. Kant’s Doctrine of Reason’s Prescribing Laws to Nature (from a Multimodal Axiomatic Epistemology Viewpoint). In Katrechko, S. L., Shiyan, A. A. (eds.). Transcendental Turn in Contemporary Philosophy – 8. Metaphysics, Epistemology, Transcendental Cognitive Science and Artificial Intelligence. Proceedings of the International Workshop. Moscow, April 20–22, 2023. Pp. 108-111. Moscow. (In Russ.)

Лобовиков, В. О. (2023б). Знания, онтологии, и мультимодальные аксиоматические теории (Дедуктивное доказательство непротиворечивости формальной теории Ф0, синтезирующей формальную аксиологию, универсальную эпистемологию, и философскую онтологию). Материалы IX международной конференции «Знания – Онтологии – Теории» (ЗОНТ-2023). Новосибирск, 2–6 октября 2023 г. Математический центр в Академгородке, Российская ассоциация искусственного интеллекта, Российская инженерная академия, Institute of Electrical and Electronics Engineers (IEEE). Новосибирск: Ин-т математики им. С. Л. Соболева СО РАН, НГУ. С. 202-211.

Lobovikov, V. O. (2023b). Knowledges, Ontologies, and Multimodal Axiomatic Theories (A Deductive Proof of Consistency of a Formal Theory Ф0, Synthesizing Formal Axiology, Universal Epistemology, and Philosophical Ontology). In Materials of IX International Conference “Knowledges – Ontologies – Theories”. Novosibirsk, October 2–6, 2023. Mathematics Center in Akademgorodok, Russian Association of Artificial Intelligence, Russian Engineering Academy, Institute of Electrical and Electronics Engineers (IEEE). Novosibirsk. Pp. 202-211. (In Russ.)

Поппер, К. (1983). Логика и рост научного знания. М.: Прогресс.

Popper, К. (1983). Logic and Growth of Scientific Knowledge. Moscow. (in Russ.)

Тарский, А. (1948). Введение в логику и методологию дедуктивных наук. М: Гос. изд-во иностр. лит-ры.

Tarski, A. (1948). Introduction to Logic and to the Methodology of Deductive Sciences. Moscow. (In Russ.)

Целищев, В. В. (1978). Понятие объекта в модальной логике. Новосибирск: Наука.

Tselishchev, V. V. (1978). The Notion of Object in Modal Logic. Novosibirsk. (In Russ.)

Целищев, В. В. (2003). Онтология математики: объекты и структуры. Новосибирск: Нонпарель.

Tselishchev, V. V. (2003). Ontology of Mathematics: Objects and Structures. Novosibirsk. (In Russ.).

Целищев, В. В. (2004а). Нормативный характер дедуктивных теорий и значение математических терминов. Философия науки. Т. 20. № 1. С. 71-82.

Tselishchev, V. V. (2004a). The normative character of deductive theories and the meaning of mathematical terms. Philosophy of Science. Vol. 20. No. 1. Pp. 71-82. (In Russ.)

Целищев, В. В. (2004б). Нормативность дедуктивного дискурса: феноменология логических констант. Новосибирск: Нонпарель.

Tselishchev, V. V. (2004b). Normative-ness of Deductive Discourse: Phenomenology of Logic Constants. Novosibirsk. (In Russ.)

Целищев, В. В. (2005). Непротиворечивость и полнота как нормы дедуктивного мышления в свете теорем Гёделя о неполноте арифметики. Философия науки. Т. 25. № 2. С. 34-52.

Tselishchev, V. V. (2005). Consistency and completeness as the standards of deductive thinking from the standpoint of Gödel’s theorems about incompleteness of arithmetic. Philosophy of Science. Vol. 25. No. 2. Pp. 34-52. (In Russ.)

Целищев, В. В. (2010). Логика существования. М.: КРАСАНД.

Tselishchev, V. V. (2010). Logic of Existence. Moscow. (In Russ.)

Юм, Д. (1965). Сочинения. В 2 т. Т. 2. М.: Мысль.

Hume, D. (1965). Works. In 2 vols. Vol. 2. Moscow. (In Russ.)

Aristotle. (1994). The Works of Aristotle. Vol. 1. In Adler, M. J. (ed.). Great Books of the Western World. Vol. 7. Encyclopedia Britannica, Inc., Chicago; London.

Bader, R. M. (2021). Kantian Meta-Ontology. [Online]. Academic library – free online college e textbooks. The Routledge Handbook of Metametaphysics. Routledge. Pp. 23-31. Available at: https://ebrary.net/255633/sociology/existence_modality#161 (Accessed: 10 December 2023).

Berto, F. (2012). Existence as a Real Property. Dordrecht. Springer.

Berto, F. and Priest, G. (2014). Modal Meinongianism and Characterization: Reply to Kroon, Grazer. Philosophische Studien. Vol. 90. Pp. 183-200.

Blanchette, P. A. (1996). Frege and Hilbert on Consistency. Journal of Philosophy. Vol. 93. No. 7. Pp. 317-336. DOI: 10.2307/2941124

Blanchette, P. A. (2007). Frege on Consistency and Conceptual Analysis. Philosophia Mathematica. Vol. 15. No. 3. Pp. 321-346. DOI:10.1093/philmat/nkm028

Blanchette, P. A. (2018). The Frege-Hilbert Controversy. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Fall 2018 Edition). E. N. Zalta (ed.). Available at: https://plato.stanford.edu/archives/fall2018/entries/frege-hilbert/ (Accessed: 11 November 2023).

Blecher, I. (2012). Kant and the Meaning of Existence: A Modal Account. PhD Dissertation. Pittsburgh. University of Pittsburgh.

Bradie, M., Harms, W. (2020). Evolutionary Epistemology. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Spring 2020 Edition). E. N. Zalta (ed.). Available at: https://plato.stanford.edu/archives/spr2020/entries/epistemology-evolutionary/ (Accessed: 11 November 2023).

Chihara, Ch. (1990). Constructibility and Mathematical Existence. Oxford. The University Press.

Corrigan, K., Harrington, L. M. (2023). Pseudo-Dionysius the Areopagite. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Summer 2023 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/sum2023/entries/pseudo-dionysius-areopagite/ (Accessed: 11 November 2023).

Descartes, R. (1970). Meditations on First Philosophy, translated by E. Haldane and G. Ross. In The Philosophical Works of Descartes. Vol. 1. London. Cambridge University Press.

Doherty, F. T. (2017). Hilbert on Consistency as a Guide to Mathematical Reality. Logique et Analyse. No. 237. Pp. 107-128.

Fine, K. (1984). Critical Review of Parsons. Nonexistent Objects. Philosophical Studies. Vol. 45. Pp. 94-142.

Fine, K. (1985). Reasoning With Arbitrary Objects. Oxford. Blackwell.

Frege, G. (1971). On the Foundations of Geometry and Formal Theories of Arithmetic.

Kluge, E.-H. W. (transl.). New Haven, CT. Yale University Press.

Frege, G. (1980). Philosophical and Mathematical Correspondence. Gabriel, G., Hermes, H., Kambartel, F., Thiel, Ch., Veraart, A., McGuinness, B., Kaal, H. (eds.) Oxford. Blackwell Publishers.

Frege, G. (1984). Collected Papers on Mathematics, Logic and Philosophy. McGuinness, B. F. (ed.). Oxford. Blackwell Publishers.

Haaparanta, L. (1985). Frege's Doctrine of Being. Acta Philosophica Fennica. Vol. 39. Helsinki. Societas Philosophica Fennica. 182 p.

Haaparanta, L. (1986). On Frege’s Concept of Being. In Knuuttila, S., Hintikka, J. (eds.). The Logic of Being. Dordrecht. D. Reidel. Pp. 269-290.

Hartshorne, C. (1962). The Logic of Perfection. LaSalle, IL. Open Court.

Hartshorne, C. (1965). Anselm’s Discovery. LaSalle, IL. Open Court.

Hilbert, D. (1990). Foundations of Geometry. Unger, L. (transl.) (2nd English ed.). La Salle, IL. Open Court Publishing Co.

Hilbert, D. (1996a). Axiomatic thought. In Ewald, W. B. (ed.). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2. Oxford, New York. Oxford University Press Inc. Pp. 1105-1115.

Hilbert, D. (1996b). The new grounding of mathematics: First report. In Ewald, W. B. (ed.). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2. Oxford, New York. Oxford University Press Inc. Pp. 1115-1134.

Hilbert, D. (1996c). The logical foundations of mathematics. In Ewald, W. B. (Ed.). From Kant to Hilbert: A Source Book in the Foundations of Mathematics. Vol. 2. Oxford, New York. Oxford University Press Inc. Pp. 1134-1147.

Hintikka, J. (1962). Knowledge and Belief: An Introduction to the Logic of the Two Notions. Ithaca, NY. Cornell University Press.

Hintikka, J. (1969). Models for Modalities. Selected Essays. Dordrecht. D. Reidel Publishing Co.

Hintikka, J. (1974). Knowledge and the Known: Historical Perspectives in Epistemology. Dordrecht. D. Reidel Publishing Co.

Hintikka, J. (1984). Are There Nonexistent Objects? Why Not? But Where Are They? Synthese. Vol. 60. Pp. 451-458.

Hintikka, J. (1986). Kant, Existence, Predication and the Ontological Argument. In Knuuttila, S., Hintikka, J. (eds.). The Logic of Being. Dordrecht. D. Reidel. Pp. 249-268.

Jacquette, D. (1996). On Defoliating Meinong's Jungle. Axiomathes. No. 1-2. Pp. 17-42.

Jacquette, D. (1997). Meinongian Logic. The Semantics of Existence and Nonexistence. Berlin. New York. Walter de Gruyter.

Jacquette, D. (2015). Alexius Meinong. The Shepherd of Non-Being. Vol. 360. [Online]. Springer Link. Synthese Library. Available at: https://link.springer.com/book/10.1007/978-3-319-18075-5 (Accessed: 11 November 2023).

Janowitz, N. (1991). Theories of Divine Names in Origen and Pseudo-Dionysius. History of Religions. Vol. 30, Pp. 359-372.

Kant, I. (1994). The Critique of Pure Reason. Fundamental Principles of the Metaphysics of Morals. The Critique of Practical Reason. Preface and Introduction to the Metaphysical Elements of Ethics. General Introduction to the Metaphysics of Morals. The Science of Right. The Critique of Judgement. In Adler, M. J. (ed.). Great Books of the Western World. V. 39. Kant. Chicago. London. Encyclopedia Britannica, Inc.

Kant, I. (1996). Prolegomena to any future metaphysics: in focus. London. New York. Routledge.

Kneale, W. (1962). Modality de Dicto and de Re. In Nagel, E., Suppes, P. and Tarski, A. (eds.). Logic, Methodology and Philosophy of Science. Stanford. Stanford University Press. Pp. 622-633.

Leibniz, G. W. (1969). Philosophical Papers and Letters. Dordrecht, Netherlands. D. Reidel.

Leibniz, G. W. (1981). New Essays on Human Understanding. Cambridge, UK. Cambridge University Press.

Lobovikov, V. O. (2020). Knowledge Logic and Algebra of Formal Axiology: A Formal Axiomatic Epistemology Theory Sigma Used for Precise Defining the Exotic Condition under which Hume-and-Moore Doctrine of Logically Unbridgeable Gap between Statements of Being and Statements of Value is Falsified. Antinomies. Vol. 20. No. 4. Pp. 7-23. DOI: 10.24411/2686-7206-2020-10401

Lobovikov, V. O. (2021). A Logically Formalized Axiomatic Epistemology System Σ + C and Philosophical Grounding Mathematics as a Self-Sufficing System. Mathematics. Vol. 9. No. 16. Pp. 1859. DOI: 10.3390/math9161859

Lobovikov, V. O. (2022). Axiomatizing Philosophical Epistemology, a Formal Theory “Sigma + 2C”, and Philosophical Foundations of Mathematics. Аналитическая философия: траектории истории и векторы развития: сборник научных трудов Международной научной конференции, посвященной 80-летию научного руководителя Института философии и права СО РАН В. В. Целищева, Новосибирск, 25–26 февраля 2022 г. А. В. Хлебалин (ред.). Новосибирск: Офсет ТМ. С. 43-52.

Lobovikov, V. O. (2022). Axiomatizing Philosophical Epistemology, a Formal Theory “Sigma + 2C”, and Philosophical Foundations of Mathematics. In Hlebalin, A. V. (ed.). Analytical Philosophy: History Trajectories and Development Vectors. Proceedings of the International Scientific Conference Devoted to 80th Anniversary of V. V. Tselischev – Scientific Leader of the Institute of Philosophy and Law of Siberian Branch of Russian Academy of Sciences. Novosibirsk, February 25–26, 2022. Novosibirsk. Pp. 43-52.

Lobovikov, V. O. (2023). Combining Universal Epistemology with Formal Axiology in a Multimodal Formal Axiomatic Theory “Sigma+2C”, and Philosophical Foundations of Mathematics. Respublica Literaria. Vol. 4. No. 4. Pp. 88-113. DOI: 10.47850/RL.2023.4.4.88-113

Logan, I. (2009). Reading Anselm’s Proslogion: The History of Anselm’s Argument and its Significance Today. Aldershot. Ashgate.

Lutskanov, R. (2010). Hilbert’s Program: the Transcendental Roots of Mathematical Knowledge. Balcan Journal of Philosophy. Vol. 2. Iss. 2. Pp. 121-126.

Mancosu, P. (ed.). (1998). From Brouwer to Hilbert. The Debate on the Foundations of Mathematics in the 1920s. Oxford. Oxford University Press.

Marek, J. (2022). Alexius Meinong. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Fall 2022 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/fall2022/entries/meinong/ (Accessed: 11 November 2023).

Meinong, A. (1960). On Object Theory. In Chisholm, R. (ed.). Realism and the Background of Phenomenology. Glencoe. The Free Press.

Montague, R. (1960). Logical Necessity, Physical Necessity, Ethics, and Quantifiers. Inquiry: An Interdisciplinary Journal of Philosophy. Vol. 3. Pp. 259-269. DOI: 10.1080/00201746008601312

Moschovakis, J. (2007). Intuitionistic Logic. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Spring 2007 Edition). E. N. Zalta (ed.). Available at: https://plato.stanford.edu/archives/spr2007/entries/logic-intuitionistic/ (Accessed: 11 November 2023).

Murawski, R. (2002). Leibniz's and Kant's Philosophical Ideas and the Development of Hilbert's Programme. Logique et Analyse. Vol. 45. No. 179/180. Pp. 421-437.

Nelson, M. (2022). Existence. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Winter 2022 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/win2022/entries/existence/ (Accessed: 11 November 2023).

Neurath, O. (1982). Pseudorationalism of Falsification. In Philosophical Papers, 1913-1946. Vienna Circle collection. Vol. 16. Dordrecht, Holland. D. Reidel. Pp. 121-131.

Nolan, L. (2021). Descartes’ Ontological Argument. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Spring 2021 Edition). Edward N. Zalta (ed.). Available at: https://plato.stanford.edu/archives/spr2021/entries/descartes-ontological/ (Accessed: 4 January 2024).

Oppenheimer, P., Zalta, E. (1991). On the Logic of the Ontological Argument. In Tomberlin J. (ed.). Philosophical Perspectives. 5: The Philosophy of Religion. Atascadero. Ridgeview. Pp. 509-529.

Oppy, G. (2023). Ontological Arguments. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Fall 2023 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/spr2023/entries/ontological-arguments/ (Accessed: 4 January 2024).

Parsons, T. (1980). Nonexistent Objects. New Haven. Yale University Press.

Parsons, T. (1982). Are There Nonexistent Objects? American Philosophical Quarterly. Vol. 19. Pp. 365-371.

Perszyk, K. J. (1993). Nonexistent Objects. Meinong and Contemporary Philosophy. Dordrecht. Kluwer.

Plato. (1994). The Dialogues of Plato. In Adler, M. J. (ed.). Great Books of the Western World. Vol. 6: Plato. Chicago. London. Encyclopedia Britannica, Inc. Pp. 1-799.

Poincaré, H. (1912). The Latest Efforts of the Logisticians. The Monist. Vol. 22. No. 4. Pp. 524 539. [Online]. JSTOR. Available at: https://www.jstor.org/stable/27900395 (Accessed: 02 February 2024).

Priest, G. (2005). Towards Non-Being. The Logic and Metaphysics of Intentionality. Oxford. Clarendon.

Prior, A. N. (1961). Non-Entities. Philosophical Review. Vol. 70. Pp. 120-132.

Pseudo-Dionysius Areopagite. (1980). The Divine Names and Mystical Theology. Jones, J. (transl.). Milwaukee, Wisconsin. Marquette University Press.

Resher, N. (1959). On Logic of Existence and Denotation. Philosophical Review. Vol. 68. Pp. 157-180.

Quine, W. (1969). Ontological Relativity and Other Essays. New York, NY. Columbia University Press.

Quine, W. (1980). From a Logical Point of View. Nine Logico-Philosophical Essays. Cambridge, MA. Harvard University Press.

Reicher, M. (2022). Nonexistent Objects. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Winter 2022 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/win2022/entries/nonexistent-objects/ (Accessed: 4 January 2024).

Rendsvig, R., Symons, J., Wang, Y. (2023). Epistemic Logic. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Winter 2023 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/win2023/entries/logic-epistemic/ (Accessed: 4 January 2024).

Rosefeldt, T. (2020). Kant’s Logic of Existence. Journal of the History of Philosophy. Vol. 58. No. 3. Pp. 521-548.

Routley, R. (1980). Exploring Meinong’s Jungle and Beyond. Canberra. Australian National University.

Russell, B. (1905). On Denoting. Mind. Vol. 14. No. 56. Pp. 479-493.

Russell, B. (1941). An Inquiry into Meaning and Truth. New York. W. W. Norton & Company.

Russell, B. (1992). Theory of Knowledge: the 1913 Manuscript. In Eames, E. R. and Blackwell, K. (eds.). Collected Papers of Bertrand Russell. London and New York. Routledge.

Shanker, S. G. (1988). Wittgenstein’s remarks on the significance of Gödel’s theorem. In Shanker, S. G. (ed.) Gödel’s theorem in focus. London: Routledge, pp. 155-257.

Smith, J. F. (1985). The Russell-Meinong Debate. Philosophy and Phenomenological Research. Vol. 45. No. 3. Pp. 305-350.

Thomason, R. H. (ed.). (1974). Formal Philosophy. Selected Papers of Richard Montague. New Haven. Yale University Press.

Williams, T. (2023). Anselm of Canterbury. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Fall 2023 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/fall2023/entries/anselm/. (Accessed: 4 January 2024).

Williamson, T. (2013). Modal Logic as Metaphysics. Oxford. Oxford University Press.

Wolenski, J. (2016). Truth as Modality. Rzeszow, Poland. University of Information, Technology and Management. [Online]. Available at: http://ojs.philosophy.spbu.ru/

index.php/lphs/article/viewFile/370/377 (Accessed: 4 January 2024).

Wright, G. H. (1951). An Essay in Modal Logic (Studies in Logic and the Foundations of Mathematics). Amsterdam. North-Holland Publishing Company.

Wright, G. H. (1996). Truth-Logics. In Georg Henrik von Wright. Six Essays in Philosophical Logic. Acta Philosophica Fennica. Vol. 60. Pp. 71-91. Helsinki. Societas Philosophica Fennica.

Zach, R. (2023). Hilbert’s Program. [Online]. The Stanford Encyclopedia of Philosophy Archive. (Winter 2023 Edition). E. N. Zalta & U. Nodelman (eds.). Available at: https://plato.stanford.edu/archives/win2023/entries/hilbert-program/ (Accessed: 4 January 2024).

Zalta, E. N. (1983). Abstract Objects: An Introduction to Axiomatic Metaphysics. Dordrecht. D. Reidel.