The present paper submits a result of applying a hitherto unknown logically formalized axiomatic axiology-and-epistemology theory “Sigma+V” to the relativity principle formulated by Galileo Galilei. By this application, the author has continued checking the remarkable (paradigm-breaking) hypothesis that formal-axiological interpreting strictly universal laws of classical theoretical mechanics could have a heuristic value for the theory proper. Along with systematical studying proper algebraic structure of formal axiology of nature, the axiomatic (hypothetic-deductive) method is used in this research as well. The investigation accomplishments are the followings. Galileo Galilei principle of relativity of motion has been represented in a two-valued algebraic system of formal axiology by a wonderful formal-axiological equation which could be called a “formal-axiological analog of Galileo relativity principle”. A precise definition of that algebraic system is given. The remarkable formal-axiological equation has been created (and checked) in that algebraic system by attentive computing relevant compositions of evaluation-functions. Precise definitions of the relevant evaluation-functions are accomplished by tables. The remarkable formula modeling Galileo Galilei principle of relativity of motion (given the appropriate interpretation of the formal theory) has been formally-logically inferred within Sigma+V from a couple of nontrivial assumptions, namely, 1) a precisely defined assumption of a-priori-ness of knowledge, 2) the above-mentioned formal-axiological analog of the relativity principle by Galileo Galilei. A not-manifest but quite exact axiomatic definition of “a-priori-ness of knowledge” is provided. The formal-logical inference is performed in perfect accordance with the mathematical rigor norms formulated within the formalism doctrine by D. Hilbert, therefore, examining the formal deductive inference submitted in the paper can be accomplished easily. Being a nontrivial scientific novelty for proper theoretical physics, hitherto the formal-logical derivation has not been published and discussed elsewhere.
References
[1]
Lovelace, A.A. (1844) A Letter to Andrew Crosse. https://mathshistory.st-andrews.ac.uk/Extras/Lovelace_letter
[2]
Hilbert, D. (1990) Foundations of Geometry [Grundlagen der Geometrie]. Open Court Publishing Co., La Salle.
[3]
Hilbert, D. (1996) Axiomatic Thought. In: Ewald, W.B., Ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 2, Oxford University Press, Oxford, 1105-1115.
[4]
Hilbert, D. (1996) 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 University Press, Oxford, 1115-1134.
[5]
Hilbert, D. (1996) The Logical Foundations of Mathematics. In: Ewald, W.B., Ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, Oxford, 1134-1147.
[6]
Hilbert, D. (1996) Logic and the Knowledge of Nature. In: Ewald, W.B., Ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, Oxford, 1157-1165.
[7]
Hilbert, D. (1996) On the Infinite. In: Ewald, W.B., Ed., From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, Oxford, 367-392.
[8]
Carnap, R. (1931) Overcoming Metaphysics by Logical Analysis of Language [überwindung der Metaphysik durch logische Analyse der Sprache]. Erkenntnis, 2, 219-241. https://doi.org/10.1007/BF02028153
[9]
Carnap, R. (1935) The Rejection of Metaphysics. In: Weitz, M., Ed., 20th-Century Philosophy: The Analytic Tradition, Free Press, New York, 206-220.
[10]
Carnap, R. (1956) Meaning and Necessity: A Study in Semantics and Modal Logic. University of Chicago Press, Chicago and London.
[11]
Carnap, R. (1966) Philosophical Foundations of Physics: An Introduction to the Philosophy of Science. Basic Books, New York.
[12]
Carnap, R. (1967) The Logical Structure of the World. Pseudo-Problems in Philosophy. University of California Press, Berkeley and Los Angeles.
[13]
Mach, E. (1914) The Analysis of Sensations, and the Relation of the Physical to the Psychical. Court Publishing Co., Chicago, London.
[14]
Mach, E. (1919) The Science of Mechanics. A Critical and Historical Account of Its Development. Open Court Publishing Co., Chicago, London.
[15]
Mach, E. (2006) Measurement and Representation of Sensations. Lawrence Erlbaum Associates, Inc., Mahwah.
[16]
Neurath, O. (1983) Pseudorationalism of Falsification. In: Cohen, R.S., Neurath, M. and Neurath, O., Eds., Philosophical Papers 1913-1946, D. Reidel, Dordrecht, 121-131. https://doi.org/10.1007/978-94-009-6995-7_10
[17]
Reichenbach, H. (1951) The Rise of Scientific Philosophy. University of California Press, Berkeley and Los Angeles. https://doi.org/10.1525/9780520341760
[18]
Reichenbach, H. (1957) The Philosophy of Space and Time. Dover Publications, New York.
[19]
Reichenbach, H. (1965) The Theory of Relativity and a Priori Knowledge. University of California Press, Berkeley.
[20]
Reichenbach, H. (1969) Axiomatization of the Theory of Relativity. University of California Press, Berkeley.
[21]
Reinchenbach, H. (1978) The Theory of Motion According to Newton, Leibniz, and Huyghens. In: Reichenbach, M. and Cohen, R.S., Eds., Hans Reichenbach Selected Writings 1909-1953, Vienna Circle Collection Book Series (VICC, Volume 4b), Springer, Dordrecht, 48-68. https://doi.org/10.1007/978-94-009-9855-1_2
[22]
Reichenbach, H. (1980) From Copernicus to Einstein. Dover Publications, New York.
[23]
Russell, B. (1914) Our Knowledge of the External World as a Field for Scientific Method in Philosophy [The Lowell Lectures, 1914]. Open Court Publ. Co., London.
[24]
Russell, B. (1940) An Inquiry into Meaning and Truth. George Allen and Unwin, London.
[25]
Russell, B. (1948) Human Knowledge: Its Scope and Limits. George Allen and Unwin, London.
[26]
Russell, B. (1956) Logic and Knowledge. George Allen and Unwin, London.
[27]
Russell, B. (1986) Is There an Absolute Good? Russel: The Journal of Bertrand Russell Studies, 6, 144-149. https://doi.org/10.15173/russell.v6i2.1679
[28]
Schlick, M. (1974) General Theory of Knowledge. Springer-Verlag, Wien. https://doi.org/10.1007/978-3-7091-3099-5
[29]
Schlick, M. (1979) Philosophical Papers (Volume I). D. Reidel, Dordrecht.
[30]
Schlick, M. (1979) Philosophical Papers (Volume II). D. Reidel, Dordrecht.
[31]
Wittgenstein, L. (1992) Tractatus Logico-Philosophicus. Routledge & K. Paul, London, New York.
[32]
Newton, I (1994) Mathematical Principles of Natural Philosophy. In: Adler, M.J., Ed., Great Books of the Western World, Vol. 32: Newton. Huygens, Encyclopedia Britannica, Inc., Chicago, London, 1-372.
[33]
Newton, I. (2004) Newton: Philosophical Writings. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511809293
[34]
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, Encyclopedia Britannica, Inc., Chicago, London, 1-613.
[35]
Kant, I. (1996) Prolegomena to Any Future Metaphysics: in Focus. Routledge, London, New York. https://doi.org/10.1017/CBO9781139164061
[36]
Kant, I. (2004) Metaphysical Foundations of Natural Science. Cambridge University Press, Cambridge, New York. https://doi.org/10.1017/CBO9780511809613
[37]
Schopenhauer, A. (2000) New Paralipomena [Novye paralipomeny]. In: Schopenhauer, A., Ed., Introduction to Philosophy. New Paralipomena. Of Interesting: A Collection of Writings [Vvedenie v filosofiyu. Novye paralipomeny. Ob interesnom: Sbornik], Popurri, Minsk, 55-389. (In Russian)
[38]
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, 4, 7-23.
[39]
Lobovikov, V.O. (2021) A Logically Formalized Axiomatic Epistemology System Σ + C and Philosophical Grounding Mathematics as a Self-Sufficing System. Mathematics, 9, 1859. https://doi.org/10.3390/math9161859
[40]
Lobovikov, V.O. (2021) A Formal Deductive Inference of the Law of Inertia in a Logically Formalized Axiomatic Epistemology System Sigma from the Assumption of Knowledge A-Priori-Ness. Journal of Applied Mathematics and Physics, 9, 441-467. https://doi.org/10.4236/jamp.2021.93031
[41]
Lobovikov, V.O. (2021) Formal Inferring the Law of Conservation of Energy from Assuming A-Priori-Ness of Knowledge in a Formal Axiomatic Epistemology System Sigma. Journal of Applied Mathematics and Physics, 9, 1011-1040. https://doi.org/10.4236/jamp.2021.95070
[42]
Lobovikov, V.O. (2022) Formally Deriving the Third Newton’s Law from a Pair of Nontrivial Assumptions in a Formal Axiomatic Theory “Sigma-V”. Journal of Applied Mathematics and Physics, 10, 1561-1586. https://doi.org/10.4236/jamp.2022.105109
[43]
Galilei, G. (1967) Dialogue Concerning the Two Chief World Systems—Ptolemaic and Copernican. University of California Press, Berkeley. https://doi.org/10.1525/9780520342941
[44]
Galilei, G. (1994) Dialogues Concerning the Two Sciences. In: Adler, M.J., Ed., Great Books of the Western World, Vol. 26: Gilbert. Galileo. Harvey, Encyclopedia Britannica, Inc., Chicago, London, 129-260.
[45]
Drake, S. (1978) Galileo at Work: His Scientific Biography. University of Chicago Press, Chicago.
[46]
Lobovikov, V.O. (2017) A Vector-Definition of Implication and A Vector-Definition of the Notion of “Law of Contraposition of Binary Operation” (A Structural-Functional Analogy between Logic and Pure-A-Priori Knowledge of Nature Exemplified by the Principle of Relativity of Velocity of Movement Discovered by Galileo Galilei) [Vektornoe opredelenie implikacii i vektornaya definiciya ponyatiya zakon kontrapozicii binarnoj operacii (Strukturno-funkcional’naya analogiya mezhdu logikoj i chistym estestvoznaniem a priori na primere otkrytogo Galileo Galileem principa otnositel’nosti skorosti dvizheniya)]. Discourse-P, 26, 43-60. (In Russian) https://doi.org/10.17506/dipi.2017.26.1.4360
[47]
Lobovikov, V.O. (2007) Mathematical Ethics, Metaphysics, and the Natural Law (Algebra of Metaphysics as Algebra of Formal Axiology) [Matematicheskaya etika, metafizika i estestvennoe pravo (Algebra metafiziki kak algebra formal’noi aksiologii)]. Institute of Philosophy and Law of the Ural Branch of Russian Academy of Sciences [Institut filosofii i prava Ural’skogo otdeleniya RAN], Yekaterinburg. (In Russian)
[48]
Lobovikov, V.O. (2021) New Ideas in Philosophical Ontology and a Dual Code for Representing It in Artificial Intelligence Systems. Discrete Mathematical Modelling the Theory of Relativity of Space and Time by a System of Equations of Two-Valued Algebra of Formal Axiology [Novye idei v filosofskoj ontologii i dvoichnyj kod dlya ee predstavleniya v iskusstvennyh intellektual’nyh sistemah. Diskretnoe matematicheskoe modelirovanie teorii otnositel’nosti prostranstva i vremeni sistemoj uravnenij dvuznachnoj algebry formal’noj aksiologii]. New Ideas in Philosophy [Novye idei v filosofii], 8, 11-20. (In Russian)
[49]
Lobovikov, V.O. (2020) New Ideas in Philosophy of Dialectical Materialism: Justifying an Equivalence of “Being” and “Inner Contradictoriness of Matter” by “Computing” Corresponding Evaluation-Functions in Algebra of Formal Axiology [Novye idei v filosofii dialekticheskogo materializma: Obosnovanie ekvivalentnosti «bytiya» i «vnutrennej protivorechivosti materii» putem «vychisleniya» sootvetstvuyushchih cennostnyh funkcij v algebre formal’noj aksiologii]. New Ideas in Philosophy [Novye idei v filosofii], 7, 17-28. (In Russian)
[50]
Lobovikov, V.O. (2019) A Mathematical Model of “Dialectical Logic”, and Relatively Autonomous Cognitive Robots [Matematicheskaya model’ «dialekticheskoj logiki» i otnositel’no avtonomnye poznayushchie roboty]. Antinomies [Antinomii], 19, 29-48. (In Russian) https://doi.org/10.17506/aipl.2019.19.1.2948
[51]
Lobovikov, V.O. (2018) A Discrete Mathematical Model of the Dialectical Principle of Universal Interconnection [Diskretnaya matematicheskaya model’ dialekticheskogo principa vseobshchej vzaimosvyazi]. Discourse-P [Diskurs-Pi], 30, 118-128. (In Russian) https://doi.org/10.17506/dipi.2018.30.1.118128
[52]
Corry, L. (2004) David Hilbert and the Axiomatization of Physics (1898-1918): From Grundlagen der Geometrie to Grundlagen der Physik (Archimedes: New Studies in the History and Philosophy of Science and Technology, Vol. 10). Springer Science + Business Media, B.V., Dordrecht, 530 p.
[53]
Quinn, F. (2012) A Revolution in Mathematics? What Really Happened a Century Ago, and Why It Matters Today. Notices of the American Mathematical Society, 1, 1-11. https://doi.org/10.1090/noti858
[54]
Tarski, A., Henkin, L. and Suppes, P. (1959) The Axiomatic Method with Special Reference to Geometry and Physics. North-Holland Publ. Co., Amsterdam.
[55]
Lobovikov, V.O. (2018) Proofs of Logic Consistency of a Formal Axiomatic Epistemology Theory Ξ, and Demonstrations of Improvability of the Formulae \"(Kq → q)\" and (q → q) in It. Journal of Applied Mathematics and Computation, 2, 483-495. https://doi.org/10.26855/jamc.2018.10.004
[56]
Ivin, A.A. (1970) Foundations of Evaluation Logic [Osnovanija logiki ocenok]. Izdatel’stvo Moskovskogo universiteta, Moscow. (In Russian)
[57]
Mendelson, E. (1966) Introduction to Mathematical Logic. Second (Corrected) Edition, D. van Nostrand Co., Inc., Princeton.
[58]
Hume, D. (1874) A Treatise of Human Nature Being an Attempt to Introduce the Experimental Method of Reasoning into Moral Subjects. In: Green, T.H. and Grose, T.H., Eds., The Philosophical Works of David Hume in Four Volumes, Vol. 2, Longmans, Green, and Co., London, 1-374.
Guthrie, W.K.C. (1962) A History of Greek Philosophy. Vol. I: The Earlier Pre-Socratics and the Pythagoreans. The University Press, Cambridge.
[61]
Makovel’skij, A. (1999) Pre-Socratics. Pre-Eleatic and Eleatic periods [Makovel’skij Dosokratiki. Doeleatovskij i eleatovskij periody]. Harvest, Minsk. (In Russian)
[62]
Aristotle (1994) The Works of Aristotle. Vol. I. In: Adler, M.J., Ed., Great Books of the Western World, Vol. 7, Encyclopedia Britannica, Inc., Chicago, 1-726.
[63]
Hume, D. (1994) An Enquiry Concerning Human Understanding. In: Adler, M.J., Ed., Great Books of the Western World, V. 33, Encyclopedia Britannica, Inc., Chicago, 451-509.
[64]
Hume, D. (1998) An Enquiry Concerning the Principles of Morals. Oxford University Press, Oxford.
[65]
Einstein, A., Lorentz, H.A., Minkowski, H. and Weyl, H. (1952) The Principle of Relativity. W. Perrett and G.B. Jeffery, Trans., Dover Books, New York.
[66]
Einstein, A. (1994) Relativity: The Special and the General Theory. In: Adler, M.J., Ed., Great Books of the Western World, Vol. 56: 20th Century Natural Science, Encyclopedia Britannica, Inc., Chicago, 191-243.
[67]
Guthrie, W.K.C. (1965) A History of Greek Philosophy. Vol. II: The Presocratic Tradition from Parmenides to Democritus. Cambridge University Press, Cambridge.
[68]
Plato (1994) The Dialogues of Plato. In: Adler, M.J., Ed., Great Books of the Western World, Vol. 6: Plato, Encyclopedia Britannica, Chicago, 1-799.
[69]
Plotinus (1991) The Enneads. Translated by Stephen MacKenna, Abridged and Edited by John Dillon, Penguin Books, London.
[70]
Guthrie, W.K.C. (1981) A History of Greek Philosophy. Vol. VI: Aristotle an Encounter. The University Press, Cambridge.
[71]
Guthrie, W.K.C. (1975) A History of Greek Philosophy. Vol. IV: Plato: The Man and His Dialogues: Earlier Period. The University Press, Cambridge.
[72]
Guthrie, W.K.C. (1978) A History of Greek Philosophy. Vol. V: The Later Plato and the Academy. The University Press, Cambridge.
[73]
Leibniz, G.W. (1982) Essays in Four Volumes. V. 1 [Sochineniya v chetyrekh tomah. T. 1]. Mysl’, Moscow. (In Russian)
[74]
Lobovikov, V.O. (2021) A Logically Formalized Axiomatic Epistemology System Ksi Modeling Kant’s Extraordinary Statement of Physicist’s Prescribing A-Priori Laws to Nature. Discourse-P [Diskurs-Pi], 18, 142-157. https://doi.org/10.17506/18179568_2021_18_2_142