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Relevant bibliographies by topics / Hilbert's axioms

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Academic literature on the topic 'Hilbert's axioms'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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Journal articles on the topic "Hilbert's axioms"

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Negri, Sara, and Jan von Plato. "From mathematical axioms to mathematical rules of proof: recent developments in proof analysis." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180037. http://dx.doi.org/10.1098/rsta.2018.0037.

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A short text in the hand of David Hilbert, discovered in Göttingen a century after it was written, shows that Hilbert had considered adding a 24th problem to his famous list of mathematical problems of the year 1900. The problem he had in mind was to find criteria for the simplicity of proofs and to develop a general theory of methods of proof in mathematics. In this paper, it is discussed to what extent proof theory has achieved the second of these aims. This article is part of the theme issue ‘The notion of ‘simple proof’ - Hilbert's 24th problem’.

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Pambuccian, Victor. "Prolegomena to any theory of proof simplicity." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180035. http://dx.doi.org/10.1098/rsta.2018.0035.

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By looking at concrete examples from elementary geometry, we analyse the manner in which the simplicity of proofs could be defined. We first find that, when presented with two proofs coming from mutually incompatible sets of assumptions, the decision regarding which one is simplest can be made, if at all, only on the basis of reasoning outside of the formal aspects of the axiom systems involved. We then show that, if the axiom system is fixed, a measure of proof simplicity can be defined based on the number of uses of axioms deemed to be deep or valuable, and prove a number of new results rega

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Cohen, Paul J. "Skolem and pessimism about proof in mathematics." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 363, no. 1835 (2005): 2407–18. http://dx.doi.org/10.1098/rsta.2005.1661.

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Attitudes towards formalization and proof have gone through large swings during the last 150 years. We sketch the development from Frege's first formalization, to the debates over intuitionism and other schools, through Hilbert's program and the decisive blow of the Gödel Incompleteness Theorem. A critical role is played by the Skolem–Lowenheim Theorem, which showed that no first-order axiom system can characterize a unique infinite model. Skolem himself regarded this as a body blow to the belief that mathematics can be reliably founded only on formal axiomatic systems. In a remarkably prescie

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Siekmann, J., and P. Szabó. "The undecidability of the DA-unification problem." Journal of Symbolic Logic 54, no. 2 (1989): 402–14. http://dx.doi.org/10.2307/2274856.

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AbstractWe show that the DA-uniflcation problem is undecidable. That is, given two binary function symbols ⊕ and ⊗, variables and constants, it is undecidable if two terms built from these symbols can be unified provided the following DA-axioms hold:Two terms are DA-unifiable (i.e. an equation is solvable in DA) if there exist terms to be substituted for their variables such that the resulting terms are equal in the equational theory DA.This is the smallest currently known axiomatic subset of Hilbert's tenth problem for which an undecidability result has been obtained.

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Vavilov, Nikolai. "Reshaping the metaphor of proof." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 377, no. 2140 (2019): 20180279. http://dx.doi.org/10.1098/rsta.2018.0279.

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The simplistic view of Mathematics as a logical system of formal truths deduced from a limited set of axioms by a limited set of inference rules immediately shatters when confronted with the history of Mathematics, or current mathematical practice. To become useful, mathematical Philosophy should contemplate what Mathematics actually was, over centuries, and what it is now, rather than speculate what it should be according to the philosophical orthodoxy. The first dogma that must be completely revised is the idea of proof as a text, rather than what it is for mathematicians themselves: a proce

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CUMMINGS, JAMES, MATTHEW FOREMAN, and MENACHEM MAGIDOR. "SQUARES, SCALES AND STATIONARY REFLECTION." Journal of Mathematical Logic 01, no. 01 (2001): 35–98. http://dx.doi.org/10.1142/s021906130100003x.

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Since the work of Gödel and Cohen, which showed that Hilbert's First Problem (the Continuum Hypothesis) was independent of the usual assumptions of mathematics (axiomatized by Zermelo–Fraenkel Set Theory with the Axiom of Choice, ZFC), there have been a myriad of independence results in many areas of mathematics. These results have led to the systematic study of several combinatorial principles that have proven effective at settling many of the important independent statements. Among the most prominent of these are the principles diamond(♢) and square(□) discovered by Jensen. Simultaneously, a

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Rustemov, B. "AXIOMATIC THEORY OF EVERYTHIN - THE FUNDAMENTAL BASIS OF SUSTAINABLE DEVELOPMENT OF THE WORLD." Sciences of Europe, no. 153 (November 27, 2024): 84–92. https://doi.org/10.5281/zenodo.14227581.

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The article reports on the creation of an axiomatic theory claiming the status of "Theory of Everything". The scientific novelty and the main difference between this theory and its predecessors is the design of the laws of physics as causal laws. This approach made it possible to identify the metaphysical essence of Hilbert's sixth problem and solve it. Based on the accepted axioms, the basic metaphysical law was established, which made it possible to formalize a complete picture of the world in the form of an Euler-Venn diagram. This scientific picture of the world allowed us to establish the

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Zeman, Jan. "Hilbertova aritmetizace geometrie." FILOSOFIE DNES 10, no. 1 (2019): 45–63. http://dx.doi.org/10.26806/fd.v10i1.269.

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Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů

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Zeman, Jan. "Hilbertova aritmetizace geometrie." FILOSOFIE DNES 10, no. 1 (2019): 45–63. http://dx.doi.org/10.26806/fd.v10i1.415.

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Tato práce se podrobně věnuje způsobu, jakým David Hilbert (1862–1943) pojal aritmetizaci geometrie v knize Grundlagen der Geometrie z roku 1899. Nejprve stručně představíme Hilbertovy předchůdce z téže doby, kteří buď po změnách v založení geometrie volali, nebo je již sami prostřednictvím axiomaticko-deduktivní metody zapracovali. Neopomeneme přitom, co dílu předcházelo v dřívějších Hilbertových přednáškách. Následně se pokusíme nastínit obsah prvních dvou kapitol knihy a vysvětlit dobové i věcné souvislosti, nutné k jejich pochopení. Představíme způsob implicitních definic základních pojmů

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Richter, William, Adam Grabowski, and Jesse Alama. "Tarski Geometry Axioms." Formalized Mathematics 22, no. 2 (2014): 167–76. http://dx.doi.org/10.2478/forma-2014-0017.

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Summary This is the translation of the Mizar article containing readable Mizar proofs of some axiomatic geometry theorems formulated by the great Polish mathematician Alfred Tarski [8], and we hope to continue this work. The article is an extension and upgrading of the source code written by the first author with the help of miz3 tool; his primary goal was to use proof checkers to help teach rigorous axiomatic geometry in high school using Hilbert’s axioms. This is largely a Mizar port of Julien Narboux’s Coq pseudo-code [6]. We partially prove the theorem of [7] that Tarski’s (extremely weak!

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Dissertations / Theses on the topic "Hilbert's axioms"

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Ward, Peter James. "Euclid's Elements, from Hilbert's Axioms." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1354311965.

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Portela, Antonio Edilson Cardoso. "Noções de geometria projetiva." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25586.

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PORTELA, Antonio Edilson Cardoso. Noções de geometria projetiva. 2017. 58 f. Dissertação (Mestrado Profissional em Matemática em Rede Nacional) - Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.<br>Submitted by Jessyca Silva (jessyca@mat.ufc.br) on 2017-09-06T17:17:00Z No. of bitstreams: 1 2017_dis_aecportela.pdf: 1065928 bytes, checksum: 468c05aa35745f3fd2761f13aa26eff1 (MD5)<br>Rejected by Rocilda Sales (rocilda@ufc.br), reason: Boa tarde, Estou devolvendo a Dissertação de ANTONIO EDILSON CARDOSO PORTELA, para que o mesmo realize algumas correções na formatação do trabal

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SOUZA, Carlos Bino de. "Geometria hiperbólica : consistência do modelo de disco de Poincaré." Universidade Federal Rural de Pernambuco, 2015. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/6695.

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Submitted by (lucia.rodrigues@ufrpe.br) on 2017-03-28T14:00:56Z No. of bitstreams: 1 Carlos Bino de Souza.pdf: 2371603 bytes, checksum: d2f0bb2e430fc899161fe573fbae4e50 (MD5)<br>Made available in DSpace on 2017-03-28T14:00:56Z (GMT). No. of bitstreams: 1 Carlos Bino de Souza.pdf: 2371603 bytes, checksum: d2f0bb2e430fc899161fe573fbae4e50 (MD5) Previous issue date: 2015-08-26<br>Euclid wrote a book in 13 volumes called Elements where systematized all the mathematical knowledge of his time. In this work, the 5 postulates of Euclidean geometry were presented. For several years, the 5th Postula

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Books on the topic "Hilbert's axioms"

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Ludwig, Günther. An Axiomatic Basis for Quantum Mechanics: Volume 1 Derivation of Hilbert Space Structure. Springer Berlin Heidelberg, 1985.

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Ludwig, Günther. An Axiomatic Basis for Quantum Mechanics: Volume 2 Quantum Mechanics and Macrosystems. Springer Berlin Heidelberg, 1987.

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Ludwig, Günther, and Kurt Just. An Axiomatic Basis for Quantum Mechanics: Volume 2 Quantum Mechanics and Macrosystems. Springer, 2011.

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Hilbert: : En busca de los axiomas universales. RBA, 2017.

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Rau, Jochen. Quantum Theory. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780199595068.003.0002.

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From the outset statistical mechanics will be framed in the language of quantum theory. The typical macroscopic system is composed of multiple constituents, and hence described in some many-particle Hilbert space. In general, not much is known about such a system, certainly not the precise preparation of all its microscopic details. Thus, its description requires a more general notion of a quantum state, a so-called mixed state. This chapter begins with a brief review of the basic axioms of quantum theory regarding observables, pure states, measurements, and time evolution. Particular attentio

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Book chapters on the topic "Hilbert's axioms"

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Hartshorne, Robin. "Hilbert’s Axioms." In Undergraduate Texts in Mathematics. Springer New York, 2000. http://dx.doi.org/10.1007/978-0-387-22676-7_3.

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Sossinsky, A. "Hilbert’s axioms for plane geometry." In The Student Mathematical Library. American Mathematical Society, 2012. http://dx.doi.org/10.1090/stml/064/19.

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Anglin, W. S., and J. Lambek. "Non-Euclidean Geometry and Hilbert’s Axioms." In The Heritage of Thales. Springer New York, 1995. http://dx.doi.org/10.1007/978-1-4612-0803-7_18.

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Schuster, Peter, and Daniel Wessel. "Syntax for Semantics: Krull’s Maximal Ideal Theorem." In Paul Lorenzen -- Mathematician and Logician. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-65824-3_6.

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AbstractKrull’s Maximal Ideal Theorem (MIT) is one of the most prominent incarnations of the Axiom of Choice (AC) in ring theory. For many a consequence of AC, constructive counterparts are well within reach, provided attention is turned to the syntactical underpinning of the problem at hand. This is one of the viewpoints of the revised Hilbert Programme in commutative algebra, which will here be carried out for MIT and several related classical principles.

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Lellmann, Björn, and Dirk Pattinson. "Correspondence between Modal Hilbert Axioms and Sequent Rules with an Application to S5." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2013. http://dx.doi.org/10.1007/978-3-642-40537-2_19.

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Brading, Katherine A., and Thomas A. Ryckman. "Hilbert’s Axiomatic Method and His “Foundations of Physics”: Reconciling Causality with the Axiom of General Invariance." In Einstein and the Changing Worldviews of Physics. Birkhäuser Boston, 2011. http://dx.doi.org/10.1007/978-0-8176-4940-1_8.

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Giaquinto, M. "Hilbert’s Programme." In The Search for Certainty. Oxford University PressOxford, 2002. http://dx.doi.org/10.1093/oso/9780198752448.003.0018.

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Abstract Hilbert aimed to justify acceptance and use of arithmetic,analysis, and set theory. A natural way of approaching this task is to look for indubitable truths to serve as axioms for a comprehensive theory, along with inference rules that are indubitably truth-preserving. This may have been Frege’s approach, but following discovery of the contradiction, Russell concluded that some element of doubt about the truth of the axioms is unavoidable. The best one could hope for, on Russell’s view, was that the axioms were inductively well supported by their known consequences. Accordingly, Russe

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Lützen, Jesper. "Hilbert and Gödel on Axiomatization and Incompleteness." In A History of Mathematical Impossibility. Oxford University PressOxford, 2023. http://dx.doi.org/10.1093/oso/9780192867391.003.0017.

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Abstract Hilbert’s formal approach to mathematics called for the axiomatization of all branches of mathematics. According to Hilbert, a mathematical theory exists if its axioms are consistent. That raised the question of consistency of arithmetic. Hilbert also required that the axiom system should be complete in the sense that it would be possible to prove any well-formed statement in the theory or prove its negation. In his second Paris problem, he called for proofs of completeness and consistency of arithmetic. However, in 1930 Gödel proved that such proofs do not exist in the system. Morero

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Dasgupta, Subrata. "Entscheidungsproblem: What’s in a Word?" In It Began with Babbage. Oxford University Press, 2014. http://dx.doi.org/10.1093/oso/9780199309412.003.0008.

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In 1900, the celebrated German mathematician David Hilbert (1862–1943), professor of mathematics in the University of Göttingen, delivered a lecture at the International Mathematics Congress in Paris in which he listed 23 significant “open” (mathematicians’ jargon for “unsolved”) problems in mathematics. Hilbert’s second problem was: Can it be proved that the axioms of arithmetic are consistent? That is, that theorems in arithmetic, derived from these axioms, can never lead to contradictory results? To appreciate what Hilbert was asking, we must understand that in the fin de siècle world of ma

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Gratzer, Walter. "The limits of logic." In Eurekas and euphorias. Oxford University PressNew York, NY, 2002. http://dx.doi.org/10.1093/oso/9780192804037.003.0131.

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Abstract In 1931, an intellectual bombshell exploded in the sheltered world of mathematics. The perpetrator of the outrage was a young German, Kurt Godel (1906-78), and the most illustrious of the casualties David Hilbert [11], doyen of German mathematicians. The Hilbert Project, as it was called, had the aim of establishing a complete system of axioms from which all of mathematics could eventually be rigorously developed. (Remote from everyday reality as such a preoccupation might appear, this, like other researches of Hilbert’s and Gödel’s, proved to have a profound bearing on topics in scie

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Conference papers on the topic "Hilbert's axioms"

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Navarro, Juan F. "EL ARTE COMO AXIOMA DEL ARTE." In III Congreso Internacional de Investigación en Artes Visuales :: ANIAV 2017 :: GLOCAL. Universitat Politècnica València, 2017. http://dx.doi.org/10.4995/aniav.2017.4617.

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Desde que las primeras mujeres comenzaron a representar escenas de caza en las paredes de las cuevas, el Arte se ha valido de los métodos y procesos de la Ciencia. En la postmodernidad, su discurso se ha vuelto cada vez más complejo, autorreferencial y, en cierto sentido, axiomático: el Arte se presenta como un axioma del Arte. Expresado en términos análogos a los planteados en el teorema de incompletitud de Gödel, podemos afirmar que hay proposiciones del sistema del Arte que no son decidibles dentro del propio sistema. El objeto de AXIOMA es construir una teoría axiomática del sistema del Ar

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Zhang, Qimeng, and Wensheng Yu. "A Case Study in Formalizing Hilbert's Foundations of Geometry in Coq: Establishing Key Properties of Lines in Hilbert's Axiom System." In 2023 China Automation Congress (CAC). IEEE, 2023. http://dx.doi.org/10.1109/cac59555.2023.10451291.

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D’Ariano, Giacomo Mauro. "How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms." In QUANTUM MECHANICS: Are There Quantum Jumps? - and On the Present Status of Quantum Mechanics. AIP, 2006. http://dx.doi.org/10.1063/1.2219356.

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Gan, Chuli, Xiaojiang Zhan, Yi Ding, and Jiangtao Xi. "Suppressing the zero-frequency component of hologram with Hilbert-Huang transform in single-shot off-axis holography." In Digital Holography and Three-Dimensional Imaging. Optica Publishing Group, 2022. http://dx.doi.org/10.1364/dh.2022.w5a.11.

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This paper proposes a method based on Hilbert-Huang transform to suppress the zero-frequency component of holograms with only one shot. It can effectively improve the quality of reconstructed phase objects.

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Soulard, F. B., A. Purvis, R. McWilliam, et al. "Iterative zero-order suppression from an off-axis hologram based on the 2D Hilbert transform." In Digital Holography and Three-Dimensional Imaging. OSA, 2012. http://dx.doi.org/10.1364/dh.2012.dsu3c.4.

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Taylor, P. H., and B. A. Williams. "Wave Statistics for Intermediate Depth Water: New Waves and Symmetry." In ASME 2002 21st International Conference on Offshore Mechanics and Arctic Engineering. ASMEDC, 2002. http://dx.doi.org/10.1115/omae2002-28554.

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A study has been made into the average shape of large crests and troughs during several storms using wave elevation data from the WACSIS measurement programme. The analysis techniques adopted were data-driven at all times, in order to test whether 2nd order wave theory could reproduce important features in the field data. The sea surface displayed obvious non-linear behaviour, reflected in the fact that the shapes of crests were always sharper and larger than their trough equivalents. Assuming that the dominant non-linear correction is second order in the wave steepness (but without a knowledg

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