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Eric Snyder's book, "Semantics and the Ontology of Number," is a valuable resource for those interested in the philosophy of mathematics. The book explores the relationship between language and the existence of numbers, focusing on the "easy argument" for their existence. Snyder argues that linguistic considerations are crucial in understanding the ontology of numbers and supports a cautious form of referentialism. The book also discusses the use of type-shifting operations in semantics to analyze the meaning of number expressions. Overall, the book offers insights into the philosophy of mathematics and the semantics of number expressions. [Extracted from the article]

This article discusses the historical approach to the philosophy of mathematics, with a focus on the influence of philosopher Imre Lakatos. The author, Donald Gillies, argues that Lakatos played a significant role in introducing historical approaches to the philosophy of mathematics, although he was not the first to do so. Gillies examines various works that cite Lakatos and explores the impact of the historical approach on the field. He concludes that historical approaches to the philosophy of mathematics have produced valuable insights and should be considered a progressive research program. [Extracted from the article]

Jean W. Rioux's book, "Thomas Aquinas' Mathematical Realism," explores Aquinas' philosophy of mathematics and its relevance to contemporary problems in the field. Rioux contrasts Aquinas' views with those of Aristotle, highlighting their complementarity. Aquinas' philosophy emphasizes the conditions that mathematical objects must fulfill to be considered legitimate, rather than simply questioning their existence. Rioux also examines Aquinas' views on the construction of mathematical objects and their relationship to proof construction. The book provides insights into the nature of mathematics and its various aspects, making it a valuable resource for scholars in medieval philosophy and the philosophy of mathematics. [Extracted from the article]

The article "Indispensability" in the journal Philosophia Mathematica discusses the development of Indispensability Arguments in the philosophy of mathematics. These arguments support mathematical realism by asserting the indispensability of mathematics to our best scientific theories and explanations. The article explores the traditional Quine-Putnam version of the argument and addresses objections to each premise. It also introduces the enhanced indispensability argument, which focuses on the indispensability of mathematics for scientific explanations. The authors provide strategies for defending Indispensability Arguments against objections and offer a comprehensive overview of the topic. Overall, the article presents a concise and convincing defense of Indispensability Arguments, making it a valuable resource for those interested in the philosophy of mathematics. [Extracted from the article]

The online version of this paper has been temporarily withdrawn due to problems with the display of certain elements. A corrected version will be made available as soon as these issues have been resolved. The correct PDF version of the paper remains available via the PDF button under the paper's title. The publisher apologizes to readers and to the author for these problems.By Neil TennantReported by Author [Extracted from the article]

Dominique Pradelle's book, "Être et genèse des idéalités: Un ciel sans éternité," explores the nature of mathematical knowledge and the paradox of mathematical objects being both ideal and timeless. Pradelle examines the views of philosophers such as Thomas Aquinas, Bernard Bolzano, and Edmund Husserl, who argue that idealities exist independently of subjective mental acts and have intersubjective and omnitemporal validity. The book also discusses the phenomenological constitution of mathematical objects and their dependence on a horizon structure, as well as the role of axiomatization and formalization in changing the nature of mathematical objects. Pradelle's work challenges naive realism and supports a nuanced understanding of mathematical objects. [Extracted from the article]

"Joel D. Hamkins.Lectures on the Philosophy of Mathematics" is an introductory textbook written by a prominent set theorist. The book covers both mathematical and philosophical material, combining interesting mathematical material with discussions of philosophical issues. While the book may cover too much mathematical material for some readers, it could be of great interest to students studying a combination of math and philosophy. The book is divided into chapters on Number, Rigor, Infinity, Geometry, Proof, Computability, Incompleteness, and Set theory, with the chapters on Number and Set theory being the longest. The author often adopts a lecturing attitude, introducing key ideas but leaving the issues open for further exploration. The book also discusses the ontological controversy between the universe and multiverse views in set theory, with the author presenting arguments for the multiverse view. Overall, the book provides a wide and ambitious coverage of the philosophy of mathematics, although some readers may find certain topics could have been presented in a different light or more carefully. [Extracted from the article]

The text discusses two books on the philosophy of mathematics education. The first book, "Connecting Humans to Equations," explores the human dimensions of mathematics and covers topics such as the ontological and epistemological aspects of mathematics. The second book, "Critical Mathematics Education," presents the Critical Mathematics Education (CME) paradigm, which aims to empower students to think critically about mathematics and its societal implications. Both books offer valuable insights into the philosophy of mathematics education and are recommended for philosophers and mathematics teachers. [Extracted from the article]

Even the claim that mathematicians do not examine their hypotheses, which Plato's Socrates makes explicitly and repeatedly as a contrast between mathematics and dialectic, is valuable only when you face both its falsity and its truth. She is less interested in Plato knowing Eudoxus's completed theory or other advanced mathematical facts, than she is in Plato's awareness of mathematicians striving - in some ways like dialecticians but in other ways unlike them - along their way to creating such theories. That is, Plato believed mathematicians establish that their hypotheses solve problems and that is all the mathematicians seek, and Plato approved of that. (p. 7, compare the retrospective summary on p. 40) 2 PLATO IN ENGLISH: TEXT AND TRANSLATION Landry's default translation is [8] (Landry p. 1n4). [Extracted from the article]

I use my reading of Plato to develop what I call as-ifism, the view that, in mathematics, we treat our hypotheses as if they were first principles and we do this with the purpose of solving mathematical problems. I then extend this view to modern mathematics showing that when we shift our focus from the method of philosophy to the method of mathematics, we see that an as-if methodological interpretation of mathematical structuralism can be used to provide an account of the practice and the applicability of mathematics while avoiding the conflation of metaphysical considerations with mathematical ones. [ABSTRACT FROM AUTHOR]

Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 14 Linnebo, Øystein [ 2013 ]: ' The potential hierarchy of sets ', The Review of Symbolic Logic, 6, 205 - 228. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 5 Dummett, M. [ 1991 ]: Frege: Philosophy of Mathematics. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 10 Hellman, G. [ 2021a ]: Mathematics and Its Logics: Philosophical Essays. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 18 Simpson, S.G. [ 2009 ]: Subsystems of Second Order Arithmetic. [Extracted from the article]

In any case, I do not see how definiteness in any sense Husserl might have conceived it, by being erected as an ideal, not only of mathematics, but of physics could in Husserl's mind account for the many and different applications he knew mathematics has in physics. Unlike "foundationalist" approaches to the philosophy of mathematics, such as logicism, formalism, and constructivism, which are clearly aimed at reconstructing and sometimes even somehow reforming mathematics, Husserl's approach, as Hartimo remarks, is faithful to mathematics I as it is practised i . It has been some time now since the philosophical community has learned to appreciate Husserl's contribution to the philosophies of logic, mathematics, and science in general, despite still some prejudices and misinterpretations in certain academic circles incapable of reading Husserl beyond the incompetent and malicious review which Frege wrote in 1894 of his I Philosophie der Arithmetik i ( I PA i ) [[8]], hereafter I Hua i XII. The mathematics that he submits to phenomenological transcendental analyses is mathematics as he finds it, a fact Hartimo acknowledges: 'in his approach to mathematics, Husserl interestingly combines this kind of transcendental description with an attempt to describe mathematical practice as it is' (p. 19). [Extracted from the article]

THE LOGIC OF PLURALS 2.0: CRITICAL PLURAL LOGIC The existence of non-standard plurality-based Henkin models for plural logic calls into question some of the main reasons for assuming TPL with its standard models. First, if TPL is assumed, a set-based model theory will not verify the instance of plural comprehension introducing the universal plurality, and, secondly, by Plural Cantor there will be more pluralities than sets to represent them. Florio and Linnebo argue that those considerations concern the retaining of absolute generality, which rules out set-based model theory, and two I parity constraints i (based on type preservation in semantics and the modal profile of pluralities vs concepts), which rule out intensional interpretations of plural logic (§§7.7-7.8). On the one hand, plural logic should be so constrained that an account of sets in terms of pluralities should be possible: the pluralities indeed forming sets should be "properly circumscribed", and so the "set of" operation should be applied to well-defined objects. [Extracted from the article]

Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 4 Hellman, Geoffrey [ 1989 ]: Mathematics Without Numbers: Towards a Modal-Structural Interpretation. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 3 Field, Hartry [ 1989 ]: Realism, Mathematics, and Modality. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 2 Field, Hartry [ 1980 ]: Science Without Numbers. [Extracted from the article]

I Mathematics and Its Logics i is a collection of Geoffrey Hellman's papers. According to Hellman, there are principles statable in the language of his modal stage theory - essentially, that of modal, plural logic with vocabulary for stages and sets - that are at once significantly more self-evident than the potentialist's axiom of infinity and that nevertheless imply it as a consequence. Saliently, Boolos abandons the axiom of replacement altogether; more subtly, in order to secure infinity, he has to write in the existence of an infinite stage as a new axiom in the stage theory. On potentialist theories, in general, sets are possibilia, and so the relevant "axiom of infinity" does not assert that infinite sets I exist i , but rather only the "potentialist translation" of the standard axiom, that infinite sets I are possible i . [Extracted from the article]

This article surveys recent literature by Parsons, McGee, Shapiro and otherson the significance of categoricity arguments in the philosophy of mathematics.After discussing whether categoricity arguments are sufficient to securereference to mathematical structures up to isomorphism, we assess what exactlyis achieved by recent `internal' renditions of the famous categoricityarguments for arithmetic and set theory.

Chapter 6 worries about how conditional claims, such as |$[\alpha]$| , and modal claims more generally, could be true, given non-factualism about the entities involved. For [1], Balaguer argues (p. 126) that we have no idea what the existence of abstract objects consists in, since there is nothing that can be done to clarify the notion of such existence: one can say only what abstract existence is I not i , not what it I is i . It seems to me that Balaguer effectively recognises this point in Chapter 10, where he says ( I e.g. i , p. 271) that statements like |$[\alpha]$| are part of his "world view". As hardly needs to be said, the history of philosophy, Ancient, Medieval, and Modern, East and West, is littered with disputes about whether entities of certain kinds exist. [Extracted from the article]

While Juliet Floyd's recent book - the newest entry in the Cambridge Elements of Philosophy of Mathematics series - comes in at a brisk 74 pages, the work punches well above its weight. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC 5 Wittgenstein, Ludwig [ 1976 ]: Wittgenstein's Lectures on the Foundations of Mathematics. Instead, the motivation behind what Wittgenstein says is, according to Floyd, much more subtle: "By a proof (or calculation) being "surveyable" Wittgenstein construes it as a I picture i of what I can i (or cannot) be done. For Floyd's Wittgenstein, logic represents the study of form series of sentences, while mathematics goes beyond this to study their aspects, that is, their "ways of seeing forms of possible structurings I of i objects and procedures" (p. 16). [Extracted from the article]

The advent of Homotopy Type Theory/Univalent Foundations (HoTT/UF) in recent years has given a new impetus to the ever ongoing philosophical discussion about foundations of mathematics. [Ternullo] addresses a specific currently much debated issue in between philosophy of set theory and set theory itself, the concept of multiverses, while [Welch] is a rather advanced account of using reflection principles for elucidating questions of independence in set theory. The papers devoted to set theory [Dzamonja/Kant], [Fontanella], [Ternullo], [Welch] should be accessible for any philosopher of mathematics, although they require a good understanding of classical set theory including models, independence results, and descriptive set theory for a deeper appreciation. However, structural set theory is arguably closer to mathematical practice outside of set theory, where the intricate |$\in$| -structure of material set theory is irrelevant and objects need only be defined up to isomorphism.". [Extracted from the article]

Carl Posy's I Mathematical Intuitionism i is an installment in the increasingly impressive Cambridge Elements in the Philosophy of Math series, edited by Penelope Rush and Stewart Shapiro. Posy motivates the development of intuitionistic logic (of which Brouwer disapproved) as follows: Brouwer doggedly refused to adopt even the simplest logical notation. Posy begins with the observation that, for Brouwer, mathematics is concerned with the construction of mathematical objects in pure intuition, and, as a result, intuitionistic mathematics is synthetic I a priori i (the influence of Kant is of course clear here). [Extracted from the article]

So "the number of the I Iliad" i is not well-defined and hence number is not a property of the I Iliad i . Hossack presents a clearly argued case that numbers (cardinals, ordinals, and ratios) are not objects (as Platonists think), nor properties of objects, but properties of quantities. [Extracted from the article]

Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism , which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all (known) properties of the numbers, achieving a key neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. [ABSTRACT FROM AUTHOR]

Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude by arguing that versions of OSR that do not rely on mathematical frameworks for representational purposes need not be vulnerable to Benacerraf's second problem. [ABSTRACT FROM AUTHOR]

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e. , how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g. , about the notion of feasibility in mathematics, the |$\mathbf{P} \neq \mathbf{NP}$| problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. [ABSTRACT FROM AUTHOR]

This paper reveals David Hilbert's position in the philosophy of mathematics, circa 1900, to be a form of non-eliminative structuralism , predating his formalism. I argue that Hilbert withstands the pressing objections put to him by Frege in the course of the Frege-Hilbert controversy in virtue of this early structuralist approach. To demonstrate that this historical position deserves contemporary attention I show that Hilbertian structuralism avoids a recent wave of objections against non-eliminative structuralists to the effect that they cannot distinguish between structurally identical but importantly distinct mathematical objects, such as the complex roots of |$-1$|⁠. [ABSTRACT FROM AUTHOR]

Platonism and the proto-ontology of mathematics: Learning from the axiom of choice, pp. 99-134. Ordinals vs. cardinals in HT ht and beyond, 193-225. Explanation and realism: Interwoven themes in the philosophy of mathematics, pp. 41-58. [Extracted from the article]

According to some scholars, such as Rodych and Steiner, Wittgenstein objects to Gödel's undecidability proof of his formula $$G$$, arguing that given a proof of $$G$$, one could relinquish the meta-mathematical interpretation of $$G$$ instead of relinquishing the assumption that Principia Mathematica is correct (or $$\omega$$ -consistent). Most scholars agree that such an objection, be it Wittgenstein's or not, rests on an inadequate understanding of Gödel's proof. In this paper, I argue that there is a possible reading of such an objection that is, in fact, reasonable and related to Gödel's proof. [ABSTRACT FROM AUTHOR]

Certain mathematical problems prove very hard to solve because some of their intuitive features have not been assimilated or cannot be assimilated by the available mathematical resources. This state of affairs triggers an interesting dynamic whereby the introduction of novel conceptual resources converts the intuitive features into further mathematical determinations in light of which a solution to the original problem is made accessible. I illustrate this phenomenon through a study of Bertrand's paradox. [ABSTRACT FROM AUTHOR]

Jody Azzouni argues that whilst it is indeterminate what the criteria for existence are, there is a criterion that has been collectively adopted to use 'exist' that we can employ to argue for positions in ontology. I raise and defend a novel objection to Azzouni: his view has the counterintuitive consequence that the facts regarding what exists can and will change when users of the word 'exist' change what criteria they associate with its usage. Considering three responses, I argue Azzouni has best reason to take one that ultimately renders unsuccessful his arguments against mathematical abstracta. [ABSTRACT FROM AUTHOR]

Although mathematicians often use it, mathematical beauty is a philosophically challenging concept. How can abstract objects be evaluated as beautiful? Is this related to their visualisations? Using an example from graph theory (the highly symmetric Petersen graph), this paper argues that, in making aesthetic judgements, mathematicians may be responding to a combination of perceptual properties of visual representations and mathematical properties of abstract structures; the latter seem to carry greater weight. Mathematical beauty thus primarily involves mathematicians' sensitivity to aesthetics of the abstract. [ABSTRACT FROM AUTHOR]

This paper explores the role of aesthetic judgements in mathematics by focussing on the relationship between the epistemic and aesthetic criteria employed in such judgements, and on the nature of the psychological experiences underpinning them. I claim that aesthetic judgements in mathematics are plausibly understood as expressions of what I will call 'aesthetic-epistemic feelings' that serve a genuine cognitive and epistemic function. I will then propose a naturalistic account of these feelings in terms of sub-personal processes of representing and assessing the relation between cognitive processes and certain properties of the stimuli at which they are directed. [ABSTRACT FROM AUTHOR]

Mathematicians frequently use aesthetic vocabulary and sometimes even describe themselves as engaged in producing art. Yet aestheticians, in so far as they have discussed this at all, have often downplayed the ascriptions of aesthetic properties as metaphorical. In this paper I argue firstly that the aesthetic talk should be taken literally, and secondly that it is at least reasonable to classify some mathematics as art. [ABSTRACT FROM AUTHOR]

Recent work in the philosophy of mathematics has suggested that mathematical structuralism is not committed to a strong form of the Identity of Indiscernibles (II). Jos'e Berm'udez demurs, and argues that a strong form of II can be warranted on structuralist grounds by countenancing identity properties, or haecceities, as legitimately structural. Typically, structuralists dismiss such properties as obviously non-structural. I will argue to the contrary that haecceities can be viewed as structural but that this concession does not warrant Berm'udez's version of II but, rather, another easily falsified version. I close with some reflections on reference vis-à-vis structurally indiscernible objects. [ABSTRACT FROM AUTHOR]

I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the role that cognitive science should play in the philosophy of mathematics. [ABSTRACT FROM AUTHOR]

This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between (i) the conceptions that treat the whole continuum as prior to its parts, and (ii) those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity (i) and those that are more readily available for mathematico-scientific use (ii). [ABSTRACT FROM AUTHOR]

A mathematical matrix is usually defined as a two-dimensional array of scalars. And yet, as I explain, matrices are not in fact two-dimensional arrays. So are we to conclude that matrices do not exist? I show how to resolve the puzzle, for both contemporary and older mathematics. The solution generalises to the interpretation of all mathematical discourse. The paper as a whole attempts to reinforce mathematical structuralism by reflecting on how best to interpret mathematics. [ABSTRACT FROM AUTHOR]

Among the most interesting features of Homotopy Type Theory (HoTT) is the way it treats identity, which has various unusual characteristics. We examine the formal features of "identity types" in HoTT, and how they relate to its other features including intensionality, constructive logic, the interpretation of types as concepts, and the Univalence Axiom. The unusual behaviour of identity types might suggest that they be reinterpreted as representing indiscernibility. We explore this by defining indiscernibility in HoTT and examine its relationship with identity. We argue that identity types are a primitive component of HoTT and cannot be reduced to indiscernibility. [ABSTRACT FROM AUTHOR]

According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, I argue that what is picked out by the mathematics are structural facts that go beyond any specific physical facts. [ABSTRACT FROM AUTHOR]

Linsky and Zalta try to explain how we can refer to mathematical objects by saying that this happens through definite descriptions which may appeal to mathematical theories. I present two issues for their account. First, there is a problem of finding appropriate pre-conditions to reference, which are currently difficult to satisfy. Second, there is a problem of ensuring the stability of the resulting reference. Slight changes in the properties ascribed to a mathematical object can result in a shift of reference and this leads to various problems, e.g., it makes inferring knowledge much harder than it is. [ABSTRACT FROM AUTHOR]

Aesthetics in philosophy of mathematics is too narrowly construed. Beauty is not the only feature in mathematics that is arguably aesthetic. While not the highest aesthetic value, being interesting is a sine qua non for publishability. Of the many ways to be interesting, being explanatory has recently been discussed. The motivational power of what is interesting is important for both directing research and stimulating education. The scientific satisfaction of curiosity and the artistic desire for beautiful results are complementary but both aesthetic. [ABSTRACT FROM AUTHOR]

Fine and Antonelli introduce two generalizations of permutation invariance -- internal invariance and simple/double invariance respectively. After sketching reasons why a solution to the Bad Company problem might require that abstraction principles be invariant in one or both senses, I identify the most finegrained abstraction principle that is invariant in each sense. Hume's Principle is the most fine-grained abstraction principle invariant in both senses. I conclude by suggesting that this partially explains the success of Hume's Principle, and the comparative lack of success in reconstructing areas of mathematics other than arithmetic based on non-invariant abstraction principles. [ABSTRACT FROM AUTHOR]