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.

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.

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.

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 Three Roles of Empirical Information in Philosophy: Intuitions on Mathematics do Not Come for Free

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Deborah Kant ◽  
José Antonio Pérez-Escobar ◽  
Deniz Sarikaya
Keyword(s):  
Set Theory ◽  
Philosophy Of Mind ◽  
Experimental Philosophy ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
The Other ◽  
Empirical Information ◽  
Philosophy Of Mathematical Practice ◽  
Armchair Philosophy

Abstract This work gives a new argument for ‘Empirical Philosophy of Mathematical Practice’. It analyses different modalities on how empirical information can influence philosophical endeavours. We evoke the classical dichotomy between “armchair” philosophy and empirical/experimental philosophy, and claim that the latter should in turn be subdivided in three distinct styles: Apostate speculator, Informed analyst, and Freeway explorer. This is a shift of focus from the source of the information towards its use by philosophers. We present several examples from philosophy of mind/science and ethics on one side and a case study from philosophy of mathematics on the other. We argue that empirically informed philosophy of mathematics is different from the rest in a way that encourages a Freeway explorer approach, because intuitions about mathematical objects are often unavailable for non-mathematicians (since they are sometimes hard to grasp even for mathematicians). This consideration is supported by a case study in set theory.

Language Use in Mathematical Practice: An Ethnographic Perspective

2018 ◽  
Vol 4 (2) ◽  
pp. 208-232
Author(s):  
Saumya Malviya
Keyword(s):  
South Asian ◽  
Social Practice ◽  
Language Use ◽  
Philosophy Of Mathematics ◽  
Mathematical Practice ◽  
Two Dimensions ◽  
Mathematical Language ◽  
Form Of Life ◽  
Action Knowledge

Mathematics is often seen as an epitome of cold objectivity and astounding infallibility. Particularly for the outsiders, it comes across as an extremely rigid and closed system which seems impenetrable owing to its very specific and technical language. This article problematises these assumptions and seeks to study mathematics as a social practice with insights drawn from an anthropology of language and concepts, Wittgenstein’s philosophy of mathematics and semiotics. Using the anthropological insight that a language is always embedded in a form of life, this article shows how mathematical practice generates its own conventions and forms of language use. In particular, two dimensions of language use in mathematics are delineated and their consequences for further research are drawn out. In the first part of the article, the role of concepts in the discourse of mathematics is explored and in the second it is shown how applying a rigid distinction between syntax and semantics to mathematical language obstructs our understanding of its fluid and dynamic character. The argument unfolds through an analyses of interviews, texts and classroom sessions and shows how mathematical practice is heavily context bound and mathematicians often display an ethnographic attentiveness towards their work. The general tenor of the description is such that it attempts to trace the ethical dimension latent in mathematical practice and suggests a possibility of exploring it as a form of life. Connected to this thought is the argument that like any other practice, mathematical practice generates its own forms of reflections which cannot simply be assimilated to philosophical/theoretical knowledge. The question whether this action knowledge regarding mathematics has some relation to the South Asian location where the ethnography unfolds is also tentatively explored.

Skolem and pessimism about proof in mathematics

2005 ◽  
Vol 363 (1835) ◽  
pp. 2407-2418 ◽  
Author(s):  
Paul J Cohen
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Mathematical Reasoning ◽  
Critical Role ◽  
Good Health ◽  
Axiom System ◽  
Great Success ◽  
To Come ◽  
Axiomatic Systems

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 prescient paper, he even sketches the possibility of interesting new models for set theory itself, something later realized by the method of forcing. This is in contrast to Hilbert's belief that mathematics could resolve all its questions. We discuss the role of new axioms for set theory, questions in set theory itself, and their relevance for number theory. We then look in detail at what the methods of the predicate calculus, i.e. mathematical reasoning, really entail. The conclusion is that there is no reasonable basis for Hilbert's assumption. The vast majority of questions even in elementary number theory, of reasonable complexity, are beyond the reach of any such reasoning. Of course this cannot be proved and we present only plausibility arguments. The great success of mathematics comes from considering ‘natural problems’, those which are related to previous work and offer a good chance of being solved. The great glories of human reasoning, beginning with the Greek discovery of geometry, are in no way diminished by this pessimistic view. We end by wishing good health to present-day mathematics and the mathematics of many centuries to come.

The Modern Aristotle: Michael Polanyi’s Search for Truth against Nihilism

2018 ◽  
Author(s):  
Ronald Hoinski ◽  
Ronald Polansky
Keyword(s):  
Philosophy Of Science ◽  
Modern Science ◽  
Michael Polanyi ◽  
Contemporary Philosophy ◽  
The Face ◽  
Human Desire ◽  
The Mind ◽  
Theoretical Science

David Hoinski and Ronald Polansky’s “The Modern Aristotle: Michael Polanyi’s Search for Truth against Nihilism” shows how the general tendencies of contemporary philosophy of science disclose a return to the Aristotelian emphasis on both the formation of dispositions to know and the role of the mind in theoretical science. Focusing on a comparison of Michael Polanyi and Aristotle, Hoinski and Polansky investigate to what degree Aristotelian thought retains its purchase on reality in the face of the changes wrought by modern science. Polanyi’s approach relies on several Aristotelian assumptions, including the naturalness of the human desire to know, the institutional and personal basis for the accumulation of knowledge, and the endorsement of realism against objectivism. Hoinski and Polansky emphasize the promise of Polanyi’s neo-Aristotelian framework, which argues that science is won through reflection on reality.

Novel Approaches in the Treatment of Antipsychotic-Induced Hyperprolactinemia:The Role of Partial Agonists of D2 Dopaminergic Receptors

2010 ◽  
Vol 4 (3) ◽  
pp. 230-240 ◽  
Author(s):  
Domenico De Berardis ◽  
Tiziano Acciavatti ◽  
Daniela Campanella ◽  
Nicola Serroni ◽  
Luigi Olivieri ◽  
...  
Keyword(s):  
Dopaminergic Receptors ◽  
Partial Agonists ◽  
Novel Approaches
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