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Journal articles on the topic 'Foundations of mathematics'

Author: Grafiati

Published: 4 June 2021

Last updated: 17 June 2025

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1

Mancosu, Paolo. "Between Russell and Hilbert: Behmann on the Foundations of Mathematics." Bulletin of Symbolic Logic 5, no. 3 (1999): 303–30. http://dx.doi.org/10.2307/421183.

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AbstractAfter giving a brief overview of the renewal of interest in logic and the foundations of mathematics in Göttingen in the period 1914-1921, I give a detailed presentation of the approach to the foundations of mathematics found in Behmann's doctoral dissertation of 1918, Die Antinomie der transfiniten Zahl und ihre Auflösung durch die Theorie von Russell und Whitehead. The dissertation was written under the guidance of David Hilbert and was primarily intended to give a clear exposition of the solution to the antinomies as found in Principia Mathematica. In the process of explaining the theory of Principia, Behmann also presented an original approach to the foundations of mathematics which saw in sense perception of concrete individuals the Archimedean point for a secure foundation of mathematical knowledge. The last part of the paper points out an important numbers of connections between Behmann's work and Hilbert's foundational thought.

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2

IANCU, MIHNEA, and FLORIAN RABE. "Formalising foundations of mathematics." Mathematical Structures in Computer Science 21, no. 4 (2011): 883–911. http://dx.doi.org/10.1017/s0960129511000144.

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Over recent decades there has been a trend towards formalised mathematics, and a number of sophisticated systems have been developed both to support the formalisation process and to verify the results mechanically. However, each tool is based on a specific foundation of mathematics, and formalisations in different systems are not necessarily compatible. Therefore, the integration of these foundations has received growing interest. We contribute to this goal by using LF as a foundational framework in which the mathematical foundations themselves can be formalised and therefore also the relations between them. We represent three of the most important foundations – Isabelle/HOL, Mizar and ZFC set theory – as well as relations between them. The relations are formalised in such a way that the framework permits the extraction of translation functions, which are guaranteed to be well defined and sound. Our work provides the starting point for a systematic study of formalised foundations in order to compare, relate and integrate them.

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3

Moldovan, Angelo-Vlad. "Between Pathology and Well-Behaviour – A Possible Foundation for Tame Mathematics." Studia Universitatis Babeș-Bolyai Philosophia 67, Special Issue (2022): 67–81. http://dx.doi.org/10.24193/subbphil.2022.sp.iss.04.

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"An in-depth examination of the foundations of mathematics reveals how its treatment is centered around the topic of “unique foundation vs. no need for a foundation” in a traditional setting. In this paper, I show that by applying Shelah’s stability procedures to mathematics, we confine ourselves to a certain section that manages to escape the Gödel phenomenon and can be classified. We concentrate our attention on this mainly because of its tame nature. This result makes way for a new approach in foundations through model-theoretic methods. We then cover Penelope Maddy’s “foundational virtues” and what it means for a theory to be foundational. Having explored what a tame foundation can amount to, we argue that it can fulfil some of Maddy’s foundational qualities. In the last part, we will examine the consequences of this new paradigm – some philosophical in nature – on topics like philosophy of mathematical practice, the incompleteness theorems and others. Keywords: foundations of mathematics, tame mathematics, clarity-based knowledge, philosophy of mathematical practice, incompleteness theorems "

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4

Woleński, Jan. "Foundations of Mathematics and Mathematical Practice. The Case of Polish Mathematical School." Studia Historiae Scientiarum 21 (August 26, 2022): 237–57. https://doi.org/10.4467/2543702XSHS.22.007.15973.

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The foundations of mathematics cover mathematical as well as philosophical problems. At the turn of the 20th century logicism, formalism and intuitionism, main foundational schools were developed. A natural problem arose, namely of how much the foundations of mathematics influence the real practice of mathematicians. Although mathematics was and still is declared to be independent of philosophy, various foundational controversies concerned some mathematical axioms, e.g. the axiom of choice, or methods of proof (particularly, nonconstructive ones) and sometimes qualified them as admissible (or not) in mathematical practice, relatively to their philosophical (and foundational) content. Polish Mathematical School was established in the years 1915–1920. Its research program was outlined by Zygmunt Janiszewski (the Janiszewski program) and suggested that Polish mathematicians should concentrate on special branches of studies, including set theory, topology and mathematical logic. In this way, the foundations of mathematics became a legitimate part of mathematics. In particular, the foundational investigations should be conducted independently of philosophical assumptions and apply all mathematically accepted methods, finitary or not, and the same concerns other branches of mathematics. This scientific ideology contributed essentially to the remarkable development of logic, set theory and topology in Poland.

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5

Bagaria, Joan. "On Turing’s legacy in mathematical logic and the foundations of mathematics." Arbor 189, no. 764 (2013): a079. http://dx.doi.org/10.3989/arbor.2013.764n6002.

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6

Khudaikulova, Saida, and Shahabbas Ruzikulov. "QUANTUM MATHEMATICS AND PHYSICS: STUDYING MATHEMATICAL FOUNDATIONS AND APPLICATIONS." MEDICINE, PEDAGOGY AND TECHNOLOGY: THEORY AND PRACTICE 2, no. 12 (2024): 293–95. https://doi.org/10.5281/zenodo.14549617.

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Quantum mechanics and quantum physics have revolutionized our understanding of the fundamental nature of reality. At the core of this revolution lies quantum mathematics, which provides the mathematical foundation for describing the motion of particles at microscopic scales. This article explores the fundamental mathematical structures of quantum mechanics, including Hilbert spaces, operators, and wave functions, as well as their applications in modeling physical systems. The research also examines how quantum physics contrasts with classical physics concepts and offers new insights into topics such as quantum entanglement, superposition, and quantum computing. By analyzing the mathematical foundations of quantum theories, the article aims to shed light on the intersection of mathematics and physics, offering a deeper understanding of how mathematical formulas help predict and explain quantum phenomena. Furthermore, it discusses the potential implications of quantum mathematics in emerging fields such as quantum computing and cryptography.

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7

Mcavaney, K. L., Albert D. Polimeni, and H. Joseph Straight. "Foundations of Discrete Mathematics." Mathematical Gazette 76, no. 477 (1992): 433. http://dx.doi.org/10.2307/3618422.

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8

Shiu, P., and Paul Taylor. "Practical Foundations of Mathematics." Mathematical Gazette 84, no. 499 (2000): 175. http://dx.doi.org/10.2307/3621547.

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9

Bottia, Martha Cecilia, Stephanie Moller, Roslyn Arlin Mickelson, and Elizabeth Stearns. "Foundations of Mathematics Achievement." Elementary School Journal 115, no. 1 (2014): 124–50. http://dx.doi.org/10.1086/676950.

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10

Streicher, Thomas. "Practical Foundations of Mathematics." Science of Computer Programming 38, no. 1-3 (2000): 155–57. http://dx.doi.org/10.1016/s0167-6423(00)00009-5.

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11

Wagner, Roi. "Wronski's Foundations of Mathematics." Science in Context 29, no. 3 (2016): 241–71. http://dx.doi.org/10.1017/s0269889716000077.

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ArgumentThis paper reconstructs Wronski's philosophical foundations of mathematics. It uses his critique of Lagrange's algebraic analysis as a vignette to introduce the problems that he raised, and argues that these problems have not been properly appreciated by his contemporaries and subsequent commentators. The paper goes on to reconstruct Wronski's mathematical law of creation and his notions of theory and techne, in order to put his objections to Lagrange in their philosophical context. Finally, Wronski's proof of his universal law (the expansion of a given function by any series of functions) is reviewed in terms of the above reconstruction. I argue that Wronski's philosophical approach poses an alternative to the views of his contemporary mainstream mathematicians, which brings up the contingency of their choices, and bridges the foundational concerns of early modernity with those of the twentieth-century foundations crisis. I also argue that Wronski's views may be useful to contemporary philosophy of mathematical practice, if they are read against their metaphysical grain.

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12

Reed, Dwight W. "Foundations of kinship mathematics." Mathematical Social Sciences 21, no. 2 (1991): 197–200. http://dx.doi.org/10.1016/0165-4896(91)90085-6.

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13

Shaposhnikov, V. A. "The myth of the three crises in foundations of mathematics Part 1." Philosophy of Science and Technology 26, no. 2 (2021): 64–77. http://dx.doi.org/10.21146/2413-9084-2021-26-1-64-77.

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The story of “the three crises in foundations of mathematics” is widely popular in Russian publications on the philosophy of mathematics. This paper aims at evaluating this story against the background of the contemporary scholarship in the history of mathematics. The conclusion is that it should be considered as a specimen of modern myth-making activity brought to the fore by an unconscious tendency to model the whole history of mathematics on the pattern of the foundational crisis of the first decades of the 20th century. What is more, the consideration of the specific role and character of the foundations in both early Greek mathematics and 18th-century mathematics gives an occasion to raise a more general question regarding the true meaning of the historicity of mathematics. The first part of this paper deals with the point whether there was a foundational crisis in pre-Euclidean Greek mathematics caused by the discovery of incommensurable magnitudes and Zeno’s paradoxes. The result is negative: we have no direct historical evidence of such a crisis; as for secondary considerations, they also mainly count against it. The idea of the first crisis in foundations of mathematics has emerged as a result of the unjustified transference of the modern grasp of foundational issues and the modern “mentalité de crise” to the ancient past.

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14

Kroon, Frederick W., William S. Hatcher, and William Hatcher's. "The Logical Foundations of Mathematics." Journal of Symbolic Logic 51, no. 2 (1986): 467. http://dx.doi.org/10.2307/2274073.

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15

Joseph, George Ghevarughese. "Foundations of Eurocentrism in mathematics." Race & Class 28, no. 3 (1987): 13–28. http://dx.doi.org/10.1177/030639688702800302.

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16

Krieger, Martin H. "The ethnomethodological foundations of mathematics." Advances in Mathematics 64, no. 3 (1987): 326–32. http://dx.doi.org/10.1016/0001-8708(87)90012-0.

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17

Stout, Lawrence Neff. "Upsetting the Foundations for Mathematics." Philosophia Scientae, no. 9-2 (November 1, 2005): 5–21. http://dx.doi.org/10.4000/philosophiascientiae.515.

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18

Bloor, David. "The Living Foundations of Mathematics." Social Studies of Science 17, no. 2 (1987): 337–58. http://dx.doi.org/10.1177/030631287017002009.

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19

Wang, Hongguang, and Guoping Du. "Chinese Research on Mathematical Logic and the Foundations of Mathematics." Asian Studies 10, no. 2 (2022): 243–66. http://dx.doi.org/10.4312/as.2022.10.2.243-266.

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This paper outlines the Chinese research on mathematical logic and the foundations of mathematics. Firstly, it presents the introduction and spread of mathematical logic in China, especially the teaching and translation of mathematical logic initiated by Bertrand Russell’s lectures in the country. Secondly, it outlines the Chinese research on mathematical logic after the founding of the People’s Republic of China. The research in this period experienced a short revival under the criticism of the Soviet Union, explorations under the heavy influence of the Cultural Revolution, and the vigorous development of mathematical logic teaching and research after the period of “Reform and Opening Up” that started in the late 1970s, and the full integration of Chinese mathematical logic research into the international academic circle in the new century after 2000. In the third part, it focuses on the unique and original results of the Chinese mathematical logic research teams from the following three aspects: medium logic, lattice implication algebras and their lattice-valued systems of logic, and Chinese notation of logical constants, which can be used as a substantive supplement to the relevant literature on the history of mathematical logic in China. The last part is a reflection on the shortcomings of contemporary Chinese research on mathematical logic and the foundations of mathematics.

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20

Shaposhnikov, V. A. "The myth of the three crises in foundations of mathematics Part 2." Philosophy of Science and Technology 26, no. 2 (2021): 81–95. http://dx.doi.org/10.21146/2413-9084-2021-26-2-81-95.

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The story of “the three crises in foundations of mathematics” is widely popular in Russian publications on the philosophy of mathematics. This paper aims at evaluating this story against the background of the contemporary scholarship in the history of mathematics. The conclusion is that it should be considered as a specimen of modern myth-making activity brought to the fore by an unconscious tendency to model the whole history of mathematics on the pattern of the foundational crisis of the first decades of the 20th century. What is more, the consideration of the specific role and character of the foundations in both early Greek mathematics and 18th-century mathematics gives an occasion to raise a more general question regarding the true meaning of the historicity of mathematics. In the first part of this paper, it has been demonstrated that there is no evidence to recognize the so-called “first foundational crisis” in pre-Euclidean Greek mathematics caused by the discovery of incommensurable magnitudes. The second part is mainly devoted to the historical situation of the 18th century and the beginning of the 19th century, the alleged epoch of “the second foundational crisis” caused by the obscurity of the basic notions of the calculus. Despite the lack of clarity as far as the foundations of mathematical analysis are concerned, there are no signs of a foundational crisis in 18th-century mathematics. On the contrary, in the 19th century, at the time of “the second scientific revolution”, there are striking illustrations of crisis awareness. The cultural processes that engendered this awareness are shown to be identical for the 19th century and the beginning of the 20th century, hence the second and the third foundational crises are proved to be the different stages of the same unique crisis. Finally, we have only one foundational crisis instead of three.

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21

MARQUIS, JEAN-PIERRE. "CATEGORICAL FOUNDATIONS OF MATHEMATICS OR HOW TO PROVIDE FOUNDATIONS FORABSTRACTMATHEMATICS." Review of Symbolic Logic 6, no. 1 (2012): 51–75. http://dx.doi.org/10.1017/s1755020312000147.

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AbstractFeferman’s argument presented in 1977 seemed to block any possibility for category theory to become a serious contender in the foundational game. According to Feferman, two obstacles stand in the way: one logical and the other psychological. We address both obstacles in this paper, arguing that although Feferman’s argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

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22

MCLARTY, COLIN. "FOUNDATIONS AS TRUTHS WHICH ORGANIZE MATHEMATICS." Review of Symbolic Logic 6, no. 1 (2012): 76–86. http://dx.doi.org/10.1017/s1755020312000159.

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The article looks briefly at Feferman’s most sweeping claims about categorical foundations, focuses on narrower points raised in Berkeley, and asks some questions about Feferman’s own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.

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23

Carvajalino, Juan. "EDWIN B. WILSON, MORE THAN A CATALYTIC INFLUENCE FOR PAUL SAMUELSON’S FOUNDATIONS OF ECONOMIC ANALYSIS." Journal of the History of Economic Thought 41, no. 1 (2019): 1–25. http://dx.doi.org/10.1017/s105383721800038x.

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This paper is an exploration of the genesis of Paul Samuelson’s Foundations of Economic Analysis (1947) from the perspective of his commitment to Edwin B. Wilson’s mathematics. The paper sheds new light on Samuelson’s Foundations at two levels. First, Wilson’s foundational ideas, embodied in maxims that abound in Samuelson’s book, such as “Mathematics is a Language” or “operationally meaningful theorems,” unified the chapters of Foundations and gave a sense of unity to Samuelson’s economics. Second, Wilson influenced certain theoretical concerns of Samuelson’s economics. Particularly, Samuelson adopted Wilson’s definition of a stable equilibrium position of a system in terms of discrete inequalities. Following Wilson, Samuelson developed correspondences between the continuous and the discrete in order to translate the mathematics of the continuous of neoclassical economics into formulas of discrete magnitudes. In Foundations, the local and the discrete provided the best way of operationalizing marginal and differential calculus. The discrete resonated intuitively with data; the continuous did not.

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24

Perminov, Vasilii Ya. "Metaphysics and the Foundations of Mathematics." Russian Studies in Philosophy 50, no. 4 (2012): 24–42. http://dx.doi.org/10.2753/rsp1061-1967500402.

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25

MORTENSEN, CHRIS, and LESLEY ROBERTS. "Semiotics and the foundations of mathematics." Semiotica 115, no. 1-2 (1997): 1–26. http://dx.doi.org/10.1515/semi.1997.115.1-2.1.

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26

Silva, Walzi C. S. da. "On the Living Foundations of Mathematics." Ciência e filosofia, no. 5 (December 1, 1996): 107. http://dx.doi.org/10.11606/issn.2447-9799.cienciaefilosofi.1996.105313.

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27

Romney, A. Kimball. ": Foundations of Kinship Mathematics . Pinhsiung Liu." American Anthropologist 89, no. 2 (1987): 473–74. http://dx.doi.org/10.1525/aa.1987.89.2.02a00440.

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28

Watt, Helen M. G., and Merrilyn Goos. "Theoretical foundations of engagement in mathematics." Mathematics Education Research Journal 29, no. 2 (2017): 133–42. http://dx.doi.org/10.1007/s13394-017-0206-6.

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29

Laborde, Colette. "Towards theoretical foundations of mathematics education." ZDM 39, no. 1-2 (2007): 137–44. http://dx.doi.org/10.1007/s11858-006-0015-y.

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30

Shapiro, Stewart. "Foundations of Mathematics: Metaphysics, Epistemology, Structure." Philosophical Quarterly 54, no. 214 (2004): 16–37. http://dx.doi.org/10.1111/j.0031-8094.2004.00340.x.

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31

Plotnitsky, Arkady. "“In Mathematical Language”: On Mathematical Foundations of Quantum Foundations." Entropy 26, no. 11 (2024): 989. http://dx.doi.org/10.3390/e26110989.

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The argument of this article is threefold. First, the article argues that from its rise in the sixteenth century to our own time, the advancement of modern physics as mathematical-experimental science has been defined by the invention of new mathematical structures. Second, the article argues that quantum theory, especially following quantum mechanics, gives this thesis a radically new meaning by virtue of the following two features: on the one hand, quantum phenomena are defined as essentially different from those found in all previous physics by purely physical features; and on the other, quantum mechanics and quantum field theory are defined by purely mathematical postulates, which connect them to quantum phenomena strictly in terms of probabilities, without, as in all previous physics, representing or otherwise relating to how these phenomena physically come about. While these two features may appear discordant, if not inconsistent, I argue that they are in accord with each other, at least in certain interpretations (including the one adopted here), designated as “reality without realism”, RWR, interpretations. This argument also allows this article to offer a new perspective on a thorny problem of the relationships between continuity and discontinuity in quantum physics. In particular, rather than being concerned only with the discreteness and continuity of quantum objects or phenomena, quantum mechanics and quantum field theory relate their continuous mathematics to the irreducibly discrete quantum phenomena in terms of probabilistic predictions while, at least in RWR interpretations, precluding a representation or even conception of how these phenomena come about. This subject is rarely, if ever, discussed apart from previous work by the present author. It is, however, given a new dimension in this article which introduces, as one of its main contributions, a new principle: the mathematical complexity principle.

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32

RICHTER, MICHAEL M., and AGNAR AAMODT. "Case-based reasoning foundations." Knowledge Engineering Review 20, no. 3 (2005): 203–7. http://dx.doi.org/10.1017/s0269888906000695.

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A basic observation is that case-based reasoning has roots in different disciplines: cognitive science, knowledge representation and processing, machine learning and mathematics. As a consequence, there are foundational aspects from each of these areas. We briefly discuss them and comment on the relations between these types of foundations.

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33

TUCKER, DUSTIN. "INTENSIONALITY AND PARADOXES IN RAMSEY’S ‘THE FOUNDATIONS OF MATHEMATICS’." Review of Symbolic Logic 3, no. 1 (2010): 1–25. http://dx.doi.org/10.1017/s1755020309990359.

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In ‘The Foundations of Mathematics’, Frank Ramsey separates paradoxes into two groups, now taken to be the logical and the semantical. But he also revises the logical system developed in Whitehead and Russell’s Principia Mathematica, and in particular attempts to provide an alternate resolution of the semantical paradoxes. I reconstruct the logic that he develops for this purpose, and argue that it falls well short of his goals. I then argue that the two groups of paradoxes that Ramsey identifies are not properly thought of as the logical and semantical, and that in particular, the group normally taken to be the semantical paradoxes includes other paradoxes—the intensional paradoxes—which are not resolved by the standard metalinguistic approaches to the semantical paradoxes. It thus seems that if we are to take Ramsey’s interest in these problems seriously, then the intensional paradoxes deserve more widespread attention than they have historically received.

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34

Dyckhoff, Roy. "PRACTICAL FOUNDATIONS OF MATHEMATICS (Cambridge Studies in Advanced Mathematics 59)." Bulletin of the London Mathematical Society 32, no. 5 (2000): 619–22. http://dx.doi.org/10.1112/s0024609300217372.

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35

Miranker, Willard L., and Gregg J. Zuckerman. "Mathematical foundations of consciousness." Journal of Applied Logic 7, no. 4 (2009): 421–40. http://dx.doi.org/10.1016/j.jal.2008.05.002.

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36

Rozimorotovna, Saidova Nilufar, Norkulova Zebiniso Khamza kizi, and Jumaqulova Zilola Shuhrat kizi. "PEDAGOGICAL FOUNDATIONS FOR THE DEVELOPMENT OF COMMUNICATIVE SKILLS OF MATHEMATICS TEACHERS." International Journal of Advance Scientific Research 4, no. 11 (2024): 27–30. http://dx.doi.org/10.37547/ijasr-04-11-05.

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The article presents an overview of the theoretical and pedagogical foundations for developing the communicative competence of future teachers. The author also analyzes the content of the special subject "Communicative competence of a teacher" for developing the communicative competence of a future teacher. Keywords

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37

Melkikh, Alexey V. "The Brain and the New Foundations of Mathematics." Symmetry 13, no. 6 (2021): 1002. http://dx.doi.org/10.3390/sym13061002.

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Many concepts in mathematics are not fully defined, and their properties are implicit, which leads to paradoxes. New foundations of mathematics were formulated based on the concept of innate programs of behavior and thinking. The basic axiom of mathematics is proposed, according to which any mathematical object has a physical carrier. This carrier can store and process only a finite amount of information. As a result of the D-procedure (encoding of any mathematical objects and operations on them in the form of qubits), a mathematical object is digitized. As a consequence, the basis of mathematics is the interaction of brain qubits, which can only implement arithmetic operations on numbers. A proof in mathematics is an algorithm for finding the correct statement from a list of already-existing statements. Some mathematical paradoxes (e.g., Banach–Tarski and Russell) and Smale’s 18th problem are solved by means of the D-procedure. The axiom of choice is a consequence of the equivalence of physical states, the choice among which can be made randomly. The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized. The consistency of mathematics is due to directed evolution, which results in effective structures. Computing with qubits is based on the nontrivial quantum effects of biologically important molecules in neurons and the brain.

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38

Drake, F. R. "How recent work in mathematical logic relates to the foundations of mathematics." Mathematical Intelligencer 7, no. 4 (1985): 27–35. http://dx.doi.org/10.1007/bf03024483.

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39

Radu, Mircea. "Justus Grassmann's Contributions to the Foundations of Mathematics: Mathematical and Philosophical Aspects." Historia Mathematica 27, no. 1 (2000): 4–35. http://dx.doi.org/10.1006/hmat.1999.2266.

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40

Potter, Michael. "Intuitive and Regressive Justifications†." Philosophia Mathematica 28, no. 3 (2020): 385–94. http://dx.doi.org/10.1093/philmat/nkaa034.

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Abstract In his recent book, Quine, New Foundations, and the Philosophy of Set Theory (2018), Sean Morris attempts to rehabilitate Quine’s NF as a possible foundation for mathematics. I explain why he does not succeed.

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41

Orey, Daniel Clark, and Milton Rosa. "REVISITING THE THEORETICAL FOUNDATIONS OF ETHNOMODELLING." Educação Por Escrito 14, no. 1 (2023): e45054. http://dx.doi.org/10.15448/2179-8435.2023.1.45054.

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By including mathematical activities from outside of the school environment, the process of modelling shows us that mathematics is more than the manipulation of mathematical symbols, procedures, and practices. The application of ethnomathematical techniques along with the tools of modelling allows us to see a holistic reality to mathematics. From this perspective, one pedagogical approach that connects the cultural features of mathematics with its school/academic aspects is named ethnomodelling, which is a process of translation and elaboration of problems and questions taken from systems that are part of the reality of the members of any cultural group. In this article we offer an alternative goal for educational research, which is the acquisition of both emic (local) and etic (global) approaches for the implementation of ethnomodelling in the classrooms. We also discuss a third approach on ethnomodelling, which is the dialogical (glocal) approach, which combines both emic and etic approach bases. Finally, we define ethnomodelling as the study of mathematical phenomena within a culture because it is a social construct and is culturally bound, which adds the cultural characteristics of mathematics into the modelling process.

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42

Marek, V. Wiktor, and Jan Mycielski. "Foundations of Mathematics in the Twentieth Century." American Mathematical Monthly 108, no. 5 (2001): 449. http://dx.doi.org/10.2307/2695803.

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43

Mancosu, Paolo, and Mathieu Marion. "Wittgenstein, Finitism, and the Foundations of Mathematics." Philosophical Review 110, no. 2 (2001): 286. http://dx.doi.org/10.2307/2693685.

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44

Väänänen, Jouko. "Second-Order Logic and Foundations of Mathematics." Bulletin of Symbolic Logic 7, no. 4 (2001): 504–20. http://dx.doi.org/10.2307/2687796.

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AbstractWe discuss the differences between first-order set theory and second-order logic as a foundation for mathematics. We analyse these languages in terms of two levels of formalization. The analysis shows that if second-order logic is understood in its full semantics capable of characterizing categorically central mathematical concepts, it relies entirely on informal reasoning. On the other hand, if it is given a weak semantics, it loses its power in expressing concepts categorically. First-order set theory and second-order logic are not radically different: the latter is a major fragment of the former.

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45

Седов, Ю. Г. "Remarks Concerning the Phenomenological Foundations of Mathematics." Logical Investigations 22, no. 1 (2016): 136–44. http://dx.doi.org/10.21146/2074-1472-2016-22-1-136-144.

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In this paper I investigate the phenomenological approach to foundations of mathematics. Phenomenological reflection plays the certain role in extension of mathematical knowledge by clarification of meanings. The phenomenological technique pays our attention to our own acts in the use of the abstract concepts. Mathematical constructions must not be considered as passive objects, but as categories are given in theoretical acts, in categorical experiences and in our senses. Phenomenology moves like a category theory from formal components of knowledge to the dynamics of constitutive process.

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46

Knott, Libby. "Problem Posing from the Foundations of Mathematics." Mathematics Enthusiast 7, no. 2-3 (2010): 413–32. http://dx.doi.org/10.54870/1551-3440.1198.

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Mancosu, P. "WITTGENSTEIN, FINITISM, AND THE FOUNDATIONS OF MATHEMATICS." Philosophical Review 110, no. 2 (2001): 286–89. http://dx.doi.org/10.1215/00318108-110-2-286.

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Beran, Michael J. "The Evolutionary and Developmental Foundations of Mathematics." PLoS Biology 6, no. 2 (2008): e19. http://dx.doi.org/10.1371/journal.pbio.0060019.

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Tatt, W. W. "Godel's Unpublished Papers on Foundations of Mathematics†." Philosophia Mathematica 9, no. 1 (2001): 87–126. http://dx.doi.org/10.1093/philmat/9.1.87.

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Chaitin, Gregory. "Computers, Paradoxes and the Foundations of Mathematics." American Scientist 90, no. 2 (2002): 164. http://dx.doi.org/10.1511/2002.2.164.

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