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Articles 31 - 40 of 40
Full-Text Articles in Logic and Foundations of Mathematics
Empty Souls: Confession And Forgiveness In Hegel And Dostoevsky, Ryan Johnson
Empty Souls: Confession And Forgiveness In Hegel And Dostoevsky, Ryan Johnson
Sophia and Philosophia
“Towards the end of a sultry afternoon early in July a young man came out of his little room in Stolyarny Lane and turned and in the direction of Kameny Bridge in central St. Petersburg.”[1] Right then, this young man, a former law student named Rodion Raskolnikov, is caught in an agonizing conversation with himself over whether or not to commit the ultimate crime: to murder an innocent person. Exasperated, wondering what to do with such a weighty decision, he cried aloud, “that’s why I don’t act, because I am always talking. Or perhaps I talk so much just because …
Toward A Kripkean Concept Of Number, Oliver R. Marshall
Toward A Kripkean Concept Of Number, Oliver R. Marshall
Dissertations, Theses, and Capstone Projects
Saul Kripke once remarked to me that natural numbers cannot be posits inferred from their indispensability to science, since we’ve always had them. This left me wondering whether numbers are objects of Russellian acquaintance, or accessible by analysis, being implied by known general principles about how to reason correctly, or both. To answer this question, I discuss some recent (and not so recent) work on our concepts of number and of particular numbers, by leading psychologists and philosophers. Special attention is paid to Kripke’s theory that numbers possess structural features of the numerical systems that stand for them, and to …
Wabi-Sabi Mathematics, Jean-Francois Maheux
Wabi-Sabi Mathematics, Jean-Francois Maheux
Journal of Humanistic Mathematics
Mathematics and aesthetics have a long history in common. In this relation however, the aesthetic dimension of mathematics largely refers to concepts such as purity, absoluteness, symmetry, and so on. In stark contrast to such a nexus of ideas, the Japanese aesthetic of wabi-sabi values imperfections, temporality, incompleteness, earthly crudeness, and even contradiction. In this paper, I discuss the possibilities of “wabi-sabi mathematics” by showing (1) how wabi-sabi mathematics is conceivable; (2) how wabi-sabi mathematics is observable; and (3) why we should bother about wabi-sabi mathematics
Explanatory Proofs And Beautiful Proofs, Marc Lange
Explanatory Proofs And Beautiful Proofs, Marc Lange
Journal of Humanistic Mathematics
This paper concerns the relation between a proof’s beauty and its explanatory power – that is, its capacity to go beyond proving a given theorem to explaining why that theorem holds. Explanatory power and beauty are among the many virtues that mathematicians value and seek in various proofs, and it is important to come to a better understanding of the relations among these virtues. Mathematical practice has long recognized that certain proofs but not others have explanatory power, and this paper offers an account of what makes a proof explanatory. This account is motivated by a wide range of examples …
Abstraction And Epistemic Economy, Marco Panza
Abstraction And Epistemic Economy, Marco Panza
MPP Published Research
Most of the arguments usually appealed to in order to support the view that some abstraction principles are analytic depend on ascribing to them some sort of existential parsimony or ontological neutrality, whereas the opposite arguments, aiming to deny this view, contend this ascription. As a result, other virtues that these principles might have are often overlooked. Among them, there is an epistemic virtue which I take these principles to have, when regarded in the appropriate settings, and which I suggest to call ‘epistemic economy’. My purpose is to isolate and clarify this notion by appealing to some examples concerning …
What Do We Mean By Logical Consequence?, Jesse Endo Jenks
What Do We Mean By Logical Consequence?, Jesse Endo Jenks
Summer Research
In the beginning of the 20th century, many prominent logicians and mathematicians, such as Frege, Russell, Hilbert, and many others, felt that mathematics needed a very rigorous foundation in logic. Many results of the time were motivated by questions about logical truth and logical consequence. The standard approach in the early part of the 20th century was to use a syntactic or proof-theoretic definition of logical consequence. This says that "for one sentence to be a logical consequence of [a set of premises] is simply for that sentence to be derivable from [them] by means of some standard system of …
The Philosophy Of Mathematics: A Study Of Indispensability And Inconsistency, Hannah C. Thornhill
The Philosophy Of Mathematics: A Study Of Indispensability And Inconsistency, Hannah C. Thornhill
Scripps Senior Theses
This thesis examines possible philosophies to account for the practice of mathematics, exploring the metaphysical, ontological, and epistemological outcomes of each possible theory. Through a study of the two most probable ideas, mathematical platonism and fictionalism, I focus on the compelling argument for platonism given by an appeal to the sciences. The Indispensability Argument establishes the power of explanation seen in the relationship between mathematics and empirical science. Cases of this explanatory power illustrate how we might have reason to believe in the existence of mathematical entities present within our best scientific theories. The second half of this discussion surveys …
Ante Rem Structuralism And The No-Naming Constraint, Teresa Kouri
Ante Rem Structuralism And The No-Naming Constraint, Teresa Kouri
Philosophy Faculty Publications
Tim Räz has presented what he takes to be a new objection to Stewart Shapiro's ante rem structuralism (ARS). Räz claims that ARS conflicts with mathematical practice. I will explain why this is similar to an old problem, posed originally by John Burgess in 1999 and Jukka Keränen in 2001, and show that Shapiro can use the solution to the original problem in Räz's case. Additionally, I will suggest that Räz's proposed treatment of the situation does not provide an argument for the in re over the ante rem approach.
Restall's Proof-Theoretic Pluralism And Relevance Logic, Teresa Kouri
Restall's Proof-Theoretic Pluralism And Relevance Logic, Teresa Kouri
Philosophy Faculty Publications
Restall (Erkenntnis 79(2):279–291, 2014) proposes a new, proof-theoretic, logical pluralism. This is in contrast to the model-theoretic pluralism he and Beall proposed in Beall and Restall (Aust J Philos 78(4):475–493, 2000) and in Beall and Restall (Logical pluralism, Oxford University Press, Oxford, 2006). What I will show is that Restall has not described the conditions on being admissible to the proof-theoretic logical pluralism in such a way that relevance logic is one of the admissible logics. Though relevance logic is not hard to add formally, one critical component of Restall’s pluralism is that the relevance logic that gets added must …
A New Interpretation Of Carnap's Logical Pluralism, Teresa Kouri
A New Interpretation Of Carnap's Logical Pluralism, Teresa Kouri
Philosophy Faculty Publications
Rudolf Carnap’s logical pluralism is often held to be one in which corresponding connectives in different logics have different meanings. This paper presents an alternative view of Carnap’s position, in which connectives can and do share their meaning in some (though not all) contexts. This re-interpretation depends crucially on extending Carnap’s linguistic framework system to include meta-linguistic frameworks, those frameworks which we use to talk about linguistic frameworks. I provide an example that shows how this is possible, and give some textual evidence that Carnap would agree with this interpretation. Additionally, I show how this interpretation puts the Carnapian position …