A Priority and Application: Philosophy of Mathematics in the Modern Period

2009 ◽  
Author(s):  
Lisa Shabel
Keyword(s):  
Early Modern ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Natural World ◽  
Pure Mathematics ◽  
Philosophical Account ◽  
Mathematical Ontology ◽  
To Come ◽  
Modern Mathematics

The state of modern mathematical practice called for a modern philosopher of mathematics to answer two interrelated questions. Given that mathematical ontology includes quantifiable empirical objects, how to explain the paradigmatic features of pure mathematical reasoning: universality, certainty, necessity. And, without giving up the special status of pure mathematical reasoning, how to explain the ability of pure mathematics to come into contact with and describe the empirically accessible natural world. The first question comes to a demand for apriority: a viable philosophical account of early modern mathematics must explain the apriority of mathematical reasoning. The second question comes to a demand for applicability: a viable philosophical account of early modern mathematics must explain the applicability of mathematical reasoning. This article begins by providing a brief account of a relevant aspect of early modern mathematical practice, in order to situate philosophers in their historical and mathematical context.

Mathematical Reasoning

2020 ◽  
Vol 57 (4) ◽  
pp. 74-86
Author(s):  
Vitaly V. Tselishchev ◽  
Keyword(s):  
Cognitive Abilities ◽  
Mathematical Reasoning ◽  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Formal System ◽  
Logical Structure ◽  
Formal Proof ◽  
Formal Proofs ◽  
Logical Conclusion

The article is devoted to the comparison of two types of proofs in mathematical practice, the methodological differences of which go back to the difference in the understanding of the nature of mathematics by Descartes and Leibniz. In modern philosophy of mathematics, we talk about conceptual and formal proofs in connection with the so-called Hilbert Thesis, according to which every proof can be transformed into a logical conclusion in a suitable formal system. The analysis of the arguments of the proponents and opponents of the Thesis, “conceptualists” and “formalists”, is presented respectively by the two main antagonists – Y. Rav and J. Azzouni. The focus is on the possibility of reproducing the proof of “interesting” mathematical theorems in the form of a strict logical conclusion, in principle feasible by a mechanical procedure. The argument of conceptualists is based on pointing out the importance of other aspects of the proof besides the logical conclusion, namely, in introducing new concepts, methods, and establishing connections between different sections of meaningful mathematics, which is often illustrated by the case of proving Fermat’s Last Theorem (Y. Rav). Formalists say that a conceptual proof “points” to the formal logical structure of the proof (J. Azzouni). The article shows that the disagreement is based on the assumption of asymmetry of mutual translation of syntactic and semantic structures of the language, as a result of which the formal proof loses important semantic factors of proof. In favor of a formal proof, the program of univalent foundations of mathematics In. Vojevodski, according to which the future of mathematical proofs is associated with the availability of computer verification programs. In favor of conceptual proofs, it is stated (A. Pelc) that the number of steps in the supposed formal logical conclusion when proving an “interesting” theorem exceeds the cognitive abilities of a person. The latter circumstance leads the controversy beyond the actual topic of mathematical proof into the epistemological sphere of discussions of “mentalists” and “mechanists” on the question of the supposed superiority of human intelligence over the machine, initiated by R. Penrose in his interpretation of the Second Theorem of Goedel, among whose supporters, as it turned out, was Goedel himself.

scholarly journals Theological Underpinnings of the Modern Philosophy of Mathematics.

2016 ◽  
Vol 44 (1) ◽  
pp. 31-54
Author(s):  
Vladislav Shaposhnikov
Keyword(s):  
Nineteenth Century ◽  
Modern Philosophy ◽  
Philosophy Of Mathematics ◽  
Main Concern ◽  
Cultural Expectation ◽  
Divine Knowledge ◽  
Pure Mathematics ◽  
And Mathematics ◽  
Modern Mathematics

Abstract The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern is nineteenth-century mathematics. Theology was present in modern mathematics not through its objects or methods, but mainly through popular philosophy, which absolutized mathematics. Moreover, modern pure mathematics was treated as a sort of quasi-theology; a long-standing alliance between theology and mathematics made it habitual to view mathematics as a divine knowledge, so when theology was discarded, mathematics naturally took its place at the top of the system of knowledge. It was that cultural expectation aimed at mathematics that was substantially responsible for a great resonance made by set-theoretic paradoxes, and, finally, the whole picture of modern mathematics.

Philosophy of Mathematics

Philosophy ◽  
2010 ◽  
Author(s):  
Otávio Bueno
Keyword(s):  
Philosophy Of Science ◽  
Set Theory ◽  
Category Theory ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Mathematical Explanation ◽  
Excellent Work ◽  
Novel Approaches

Philosophy of mathematics is arguably one of the oldest branches of philosophy, and one that bears significant connections with core philosophical areas, particularly metaphysics, epistemology, and (more recently) the philosophy of science. This entry focuses on contemporary developments, which have yielded novel approaches (such as new forms of Platonism and nominalism, structuralism, neo-Fregeanism, empiricism, and naturalism) as well as several new issues (such as the significance of the application of mathematics, the role of visualization in mathematical reasoning, particular attention to mathematical practice and to the nature of mathematical explanation). Excellent work has also been done on particular philosophical issues that arise in the context of specific branches of mathematics, such as algebra, analysis, and geometry, as well as particular mathematical theories, such as set theory and category theory. Due to limitations of space, this work goes beyond the scope of the present entry.

Apocalypticism and Globalization in the Early Modern World: A Medievalist’s Perspective

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jay Rubenstein
Keyword(s):  
Early Modern ◽  
Middle Ages ◽  
Belief Systems ◽  
World History ◽  
Natural World ◽  
Seventh Century ◽  
Modern World ◽  
Early Modernity ◽  
Age Of Discovery ◽  
The Middle Ages

Abstract The apocalyptic belief systems from early modernity discussed in this series of articles to varying degrees have precursors in the Middle Ages. The drive to map the globe for purposes both geographic and symbolic, finds expression in explicitly apocalyptic manuscripts produced throughout the Middle Ages. An apocalyptic political discourse, especially centered on themes of empire and Islam, developed in the seventh century and reached extraordinary popularity during the Crusades. Speculation about the end of world history among medieval intellectuals led them not to reject the natural world but to study it more closely, in ways that set the stage for the later Age of Discovery. These broad continuities between the medieval and early modern, and indeed into modernity, demonstrate the imperative of viewing apocalypticism not as an esoteric fringe movement but as a constructive force in cultural creation.

scholarly journals Showing Mathematical Flies the Way Out of Foundational Bottles: The Later Wittgenstein as a Forerunner of Lakatos and the Philosophy of Mathematical Practice

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
José Antonio Pérez-Escobar
Keyword(s):  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
History Of ◽  
Philosophy Of Mathematical Practice ◽  
The Way

Abstract This work explores the later Wittgenstein’s philosophy of mathematics in relation to Lakatos’ philosophy of mathematics and the philosophy of mathematical practice. I argue that, while the philosophy of mathematical practice typically identifies Lakatos as its earliest of predecessors, the later Wittgenstein already developed key ideas for this community a few decades before. However, for a variety of reasons, most of this work on philosophy of mathematics has gone relatively unnoticed. Some of these ideas and their significance as precursors for the philosophy of mathematical practice will be presented here, including a brief reconstruction of Lakatos’ considerations on Euler’s conjecture for polyhedra from the lens of late Wittgensteinian philosophy. Overall, this article aims to challenge the received view of the history of the philosophy of mathematical practice and inspire further work in this community drawing from Wittgenstein’s late philosophy.

Teleology

2008 ◽  
Author(s):  
D.M. Walsh
Keyword(s):  
Early Modern ◽  
Adaptive Systems ◽  
Natural World ◽  
Current Understanding ◽  
Living Thing ◽  
The World ◽  
Modern Thinking ◽  
Self Organizing

Teleology is a mode of explanation in which the presence, occurrence, or nature of some phenomenon is explained by the end to which it contributes. The model of explanation is “pure mechanism” which holds that there is a single kind of stuff in the world-”matter” that exhibits a single kind of change, motion. It falls into three classes: the argument from nonactuality, the argument from intentionality, and the argument from normativity. These objections are because of early modern thinking about the natural world. These arguments rely on the Platonic model of transcendent teleology. Aristotelian teleology complements our current understanding of goal-directed, self-organizing, adaptive systems. The success of development can be explained by plasticity which is a goal-directed capacity of organisms to produce and maintain a stable, well-functioning living thing. The understanding of how evolution can be adaptive requires us to incorporate teleology.

The New Entities of Abbacus and Renaissance Algebra

2017 ◽  
Author(s):  
Roi Wagner
Keyword(s):  
Fifteenth Century ◽  
Middle Ages ◽  
Natural Science ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Northern Italy ◽  
Historical Narrative ◽  
Negative Numbers ◽  
Late Middle Ages ◽  
Mathematical Signs

This chapter offers a historical narrative of some elements of the new algebra that was developed in the fourteenth to sixteenth centuries in northern Italy in order to show how competing philosophical approaches find an intertwining expression in mathematical practice. It examines some of the important mathematical developments of the period in terms of a “Yes, please!” philosophy of mathematics. It describes economical-mathematical practice with algebraic signs and subtracted numbers in the abbaco tradition of the Italian late Middle Ages and Renaissance. The chapter first considers where the practice of using letters and ligatures to represent unknown quantities come from by analyzing Benedetto's fifteenth-century manuscript before discussing mathematics as abstraction from natural science observations that emerges from the realm of economy. It also explores the arithmetic of debited values, the formation of negative numbers, and the principle of fluidity of mathematical signs.

Afterword

2020 ◽  
Author(s):  
Kathleen Long
Keyword(s):  
Early Modern ◽  
Natural World ◽  
Knowledge Systems ◽  
The Self ◽  
Modern World ◽  
Production Of Knowledge ◽  
The World ◽  
Human Understanding ◽  
Modern Era

In the early modern world, exceptional bodies are linked to knowledge, not as the production of knowledge of the self through the scrutiny of those who have been ‘othered’, but as a means of inducing self-scrutiny and awareness of the limitations of human understanding. Exceptional beings and phenomena entice us to consider the world beyond that which is familiar to us and raise questions concerning our knowledge systems based on notions of what is natural or, in our modern era, normal. Rather than reacting with horror, disgust or pity, we can learn to respect the variety, mobility and resilience of the natural world in our contemplation of that which we see as exceptional.

Type-theoretical checking and philosophy of Mathematics

1998 ◽  
Author(s):  
Nicolaas Govert de Bruijn
Keyword(s):  
Set Theory ◽  
Subject Matter ◽  
Mutual Influence ◽  
Mathematical Reasoning ◽  
Philosophy Of Mathematics ◽  
General Information ◽  
Point Of View ◽  
De Bruijn ◽  
Level Of Understanding ◽  
The Way

After millennia of mathematics we have reached a level of understanding that can be represented physically. Humankind has managed to disentangle the intricate mixture of language, metalanguage and interpretation, isolating a body of formal, abstract mathematics that can be completely verified by machines. Systems for computer-aided verification have philosophical aspects. The design and usage of such systems are influenced by the way we think about mathematics, but it also works the other way. A number of aspects of this mutual influence will be discussed in this paper. In particular, attention will be given to philosophical aspects of type-theoretical systems. These definitely call for new attitudes: throughout the twentieth century most mathematicians had been trained to think in terms of untyped sets. The word “philosophy” will be used lightheartedly. It does not refer to serious professional philosophy, but just to meditation about the way one does one’s job. What used to be called philosophy of mathematics in the past was for a large part subject oriented. Most people characterized mathematics by its subject matter, classifying it as the science of space and number. From the verification system’s point of view, however, subject matter is irrelevant. Verification is involved with the rules of mathematical reasoning, not with the subject. The picture may be a bit confused, however, by the fact that so many people consider set theory, in particular untyped set theory, as part of the language and foundation of mathematics, rather than as a particular subject treated by mathematics. The views expressed in this paper are quite personal, and can mainly be carried back to the author’s design of the Automath system in the late 1960s, where the way to look upon the meaning (philosophy) of mathematics is inspired by the usage of the unification system and vice versa. See de Bruijn 1994b for various philosophical items concerning Automath, and Nederpelt et al. 1994, de Bruin 1980, de Bruijn 1991a for general information about the Automath project. Some of the points of view given in this paper are matters of taste, but most of them were imposed by the task of letting a machine follow what we say, a machine without any knowledge of our mathematical culture and without any knowledge of physical laws.

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