Mathematical Entities Research Articles

Can the new indispensability argument be saved from Euclidean rescues?

The traditional formulation of the indispensability argument for the existence of mathematical entities (IA) has been criticised due to its reliance on confirmational holism. Recently a formulation of IA that works without appeal to confirmational holism has been defended. This recent formulation is meant to be superior to the traditional formulation in virtue of it not being subject to the kind of criticism that pertains to confirmational holism. I shall argue that a proponent of the version of IA that works without appeal to confirmational holism will struggle to answer a challenge readily answered by proponents of a version of IA that does appeal to confirmational holism. This challenge is to explain why mathematics applied in falsified scientific theories is not considered to be falsified along with the rest of the theory. In cases where mathematics seemingly ought to be falsified it is saved from falsification, by a so called ‘Euclidean rescue’. I consider a range of possible answers to this challenge and conclude that each answer fails.

A decision-making approach using point-cloud-based granular information

Real-life decision problems are often solved by using a very limited set of data points. The computational complexity of such decision problems can be handled easily by using a mathematical entity called stochastic point cloud (SPC). In general, SPC is a form of granular information and is simulated by a data-driven stochastic process. In this study, particular interests are given on normal-distribution-driven SPCs. When the parameters of a normally distributed variable is not clearly known due to the lack of information or due to the multiplicity of regression analysis, it forms a set of SPCs. Three types of such SPCs are described in detail in this study. The effectiveness of such SPCs in solving a real-life decision problem (i.e., how to minimize vehicle emissions in Abu Dhabi Emirate of United Arab Emirates) is also shown. To develop more realistic and man–machine-friendly decision support systems one can use SPCs.

Understanding the Infinite I: Niceness, Robustness, and Realism

This paper treats the situation where a single mathematical construction satisfies a multitude of interesting mathematical properties. The examples treated are all infinitely large entities. The clustering of properties is termed ‘niceness’ by the mathematician Michiel Hazewinkel, a concept we compare to the ‘robustness’ described by the philosopher of science William Wimsatt. In the final part of the paper, we bring our findings to bear on the question of realism which concerns not whether mathematical entities exist as abstract objects, but rather whether the choice of our concepts is forced upon us.

Three-dimensional surgical simulation

In this article, we discuss the development of methods for computer-aided jaw surgery, which allows us to incorporate the high level of precision necessary for transferring virtual plans into the operating room. We also present a complete computer-aided surgery system developed in close collaboration with surgeons. Surgery planning and simulation include construction of 3-dimensional surface models from cone-beam computed tomography, dynamic cephalometry, semiautomatic mirroring, interactive cutting of bone, and bony segment repositioning. A virtual setup can be used to manufacture positioning splints for intraoperative guidance. The system provides further intraoperative assistance with a computer display showing jaw positions and 3-dimensional positioning guides updated in real time during the surgical procedure. The computer-aided surgery system aids in dealing with complex cases with benefits for the patient, with surgical practice, and for orthodontic finishing. Advanced software tools for diagnosis and treatment planning allow preparation of detailed operative plans, osteotomy repositioning, bone reconstructions, surgical resident training, and assessing the difficulties of the surgical procedures before the surgery. Computer-aided surgery can make the elaboration of the surgical plan a more flexible process, increase the level of detail and accuracy of the plan, yield higher operative precision and control, and enhance documentation of cases.

Understanding Game Semantics Through Coherence Spaces

Game Semantics has successfully provided fully abstract models for a variety of programming languages not possible using other denotational approaches. Although it is a flexible and accurate way to give semantics to a language, its underlying mathematics is awkward. For example, the proofs that strategies compose associatively and maintain properties imposed on them such as innocence are intricate and require a lot of attention. This work aims at beginning to provide a more elegant and uniform mathematical ground for Game Semantics. Our quest is to find mathematical entities that will retain the properties that make games an accurate way to give semantics to programs, yet that are simple and familiar to work with. Our main result is a full, faithful strong monoidal embedding of a category of games into a category of coherence spaces, where composition is simple composition of relations.

Some Uses of Logic in Rigorous Philosophy

This paper is concerned with the use of logic to solve philosophical problems. Such use of logic goes counter to the prevailing empiricist tradition in analytic circles. Specifically, model-theoretic tools are applied to three fundamental issues in the philosophy of logic and mathematics, namely, to the issue of the existence of mathematical entities, to the dispute between first- and second-order logic and to the definition of analyticity.

Time as a Geometric Concept Involving Angular Relations in Classical Mechanics and Quantum Mechanics

The goal of this paper is to introduce the notion of a four-dimensional time in classical mechanics and in quantum mechanics as a natural concept related with the angular momentum. The four-dimensional time is a consequence of the geometrical relation in the particle in a given plane defined by the angular momentum. A quaternion is the mathematical entity that gives the correct direction to the four-dimensional time.

Informational Realism and World 3

This paper takes up a suggestion made by Floridi that the digital revolution is bringing about a profound change in our metaphysics. The paper aims to bring some older views from philosophy of mathematics to bear on this problem. The older views are concerned principally with mathematical realism—that is the claim that mathematical entities such as numbers exist. The new context for the discussion is informational realism, where the problem shifts to the question of the reality of information. Mathematical realism is perhaps a special case of informational realism. The older views concerned with mathematical realism are the various theories of World 3. The concept of World 3 was introduced by Frege, whose position was close to Plato’s original views. Popper developed the theory of World 3 in a different direction which is characterised as ‘constructive Platonism’. But how is World 3 constructed? This is explored by means of two analogies: the analogy with money, and the analogy with meaning, as explicated by the later Wittgenstein. This leads to the development of an account of informational realism as constructive Aristoteliansim. Finally, this version of informational realism is compared with the informational structural realism which Floridi develops in his 2008 and 2009 papers in Synthese. Similarities and differences between the two positions are noted.

A 3D finite element model for the vibration analysis of asymmetric rotating machines

This paper suggests a 3D finite element method based on the modal theory in order to analyse linear periodically time-varying systems. Presentation of the method is given through the particular case of asymmetric rotating machines. First, Hill governing equations of asymmetric rotating oscillators with two degrees of freedom are investigated. These differential equations with periodic coefficients are solved with classic Floquet theory leading to parametric quasimodes. These mathematical entities are found to have the same fundamental properties as classic eigenmodes, but contain several harmonics possibly responsible for parametric instabilities. Extension to the vibration analysis (stability, frequency spectrum) of asymmetric rotating machines with multiple degrees of freedom is achieved with a fully 3D finite element model including stator and rotor coupling. Due to Hill expansion, the usual degrees of freedom are duplicated and associated with the relevant harmonic of the Floquet solutions in the frequency domain. Parametric quasimodes as well as steady-state response of the whole system are ingeniously computed with a component-mode synthesis method. Finally, experimental investigations are performed on a test rig composed of an asymmetric rotor running on nonisotropic supports. Numerical and experimental results are compared to highlight the potential of the numerical method.

On introducing a set perspective in the learning of limits of real sequences

An account is made of the relationship between the convergence behaviour of a sequence and the accumulation points of the underlying set of the sequence. The aim is to provide students with opportunities to contrast two types of mathematical entities through their commonalities and differences in structure. The more set-oriented perspective that arises provides students with an alternative source for proving propositions concerning limiting properties of sequences. Some examples are given and comments are made on the cognitive level.

Charles Parsons: Mathematical Thought and its Objects

This volume by Prof. Parsons at Harvard University presents an overview of some of the most relevant themes in the philosophy of mathematics under the reading of the structuralist programme. The philosophical stand behind structuralism can be phrased as the idea that ‘‘mathematical objects do not have a richer ‘nature’ than is given by the basic relations of some structure in which they reside’’ (p. xi). A defence of structuralism passes therefore, in the first place, through a presentation of mathematical entities as pure abstract objects given in certain structures. Such a theory is formulated in Chap. 1, along with the related notions of intuition, being, existence and representation. The starting point is an appropriate formulation of categories of predication and reference, for which Parsons provides a large conceptual and historical background, commenting on Kant, Frege, Brouwer, Hilbert, Carnap and Quine. From the pre-theoretical assumption that identity is a basic requirement for the meaningful identification of objects, first-order logic is considered the appropriate universal framework to talk about (mathematical) entities. To weaken this certainty, three basic questions are formulated (cf. p. 12): first, is there any need for intensional operators? second, how do objects differ from other entities? third, does reference to objects imply their existence? The first problem will be addressed in Chap. 3 in relation to modal notions. The second and third issues occupy the remaining part of Chap. 1. The nature of non-objects is approached by generalization of predicates and functors by nominalization: this strategy fails to provide some instances of predicates, and similar arguments are given for the axioms of separation and replacement in set theory and for the method of semantic ascent. Regimentation of such a situation shows a need for a second-order structure (or a typing polymorphic structure). The reservations concerning first-order logic as the appropriate

On the ontology of fictional characters: A semiotic approach

Why are we deeply moved by the misfortune of Anna Karenina if we are fully aware that she is simply a fictional character who does not exist in our world? But what does it mean that fictional characters do not exist? The present article is concerned with the ontology of fictional characters. The author concludes that successful fictional characters become paramount examples of the ‘real’ human condition because they live in an incomplete world what we have cognitive access to but cannot influence in any way and where no deeds can be undone. Unlike all the other semiotic objects, which are culturally subject to revisions, and perhaps only similar to mathematical entities, the fictual characters will never change and will remain the actors of what they did once and forever.

Protein folding invariants

This article discusses the topological invariants associated with an augmented ribbon model of single domain protein. The model is a triple (S, G, J) in S3 where S is a 2-manifold with boundary, G is a circle-with-chords, and J is an arc. The surfaces satisfy an embedding condition called laundry. The invariants are necessary and sufficient conditions for two triples to be equivalent by ambient isotopy. The model describes the native state, the unfolded state, and a unique folding pathway as a single mathematical entity. This may help illuminate some of the remarkable properties of protein. Twist transitions are introduced that allow the surface to pass through itself. A new arithmetic involving the complex numbers is used to represent variable linking numbers.

Clavius, Proclus, and the Limits of Interpretation: Snapshot-Idealization Versus Projectionism

It is hard to overestimate the role played by Christopher Clavius (1538–1612), professor of mathematics at the Collegio Romano from 1565 until his death, in the establishment of mathematics as an incontestable part of the Jesuit curriculum and body of thought. Besides his direct influence on the constitution of the mathematical part(s) of the Ratio studiorum, Clavius’s mathematical writings, ranging from introductory textbooks to commentaries to polemical pamphlets, almost immediately began to serve as standard references, despite their — occasionally — controversial character. Today, Clavius’s contentious stand towards the older school at the Collegio embodied in the person of Benedictus Pereira, who held an extremely negative view on mathematics, is regarded as a decisive factor in the sixteenth-century renaissance of mathematics, both within and outside the Society. Nowadays, most scholars seem to agree that the philosophical framework of Clavius’s institutional struggle can be found in the Prolegomena to his edition of Euclid’s Elements, first published in Rome in 1574. According to some, the general outlines of this preface reflect a philosophy of mathematics that is largely indebted to the one presented by the Neoplatonist Proclus (410/412 – 485) in the double prologue of his commentary on the first book of Euclid’s Elements. Without wanting to obscure the importance of Clavius’s revolutionary project, I will try to sketch a different picture of Clavius’s philosophy of mathematics, both on the ontological and on the epistemological level. In this re-evaluation, the limits of both Proclus’s influence and the interpretation of his commentary on Euclid will be my primary focus. Clavius’s views on the nature of mathematical objects will be shown to be incompatible with the two conventional modes of interpretation found in the ancient commentators, namely the projectionism maintained by Proclus on the one hand, and the constructive abstractionism of writers such as Ammonius and Philoponus on the other. For Proclus, mathematical objects are extended and divisible projections of the unextended and indivisible ideas with which our understanding is equipped. The imagination acts both as the faculty that projects these ideas and as the imaginary screen on which they are projected. The constructive abstractionism read in Ammonius, for example, sees mathematical entities as existing only in the mind of the mathematician who in thought has separated, i.e. abstracted, them from matter. Clavius’s position will lay bare the need to invoke a third category, namely that of a ‘snapshot-idealization’: in mathematics, according to Clavius, sensible objects are indeed considered free from matter, but they are not attributed any properties that they do not perfectly instantiate as sensible objects. As in a photograph, certain

Creativity, Freedom, and Authority: A New Perspective On the Metaphysics of Mathematics

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type.

Chaslin, la clinique de la discordance et le réel en mathématique

Philippe Chaslin proposes a return to a pure clinic that excludes etiological hypotheses because the cosmography of his symptomatic complexes, lacking causal determination, cannot constitute a coherent ensemble. Therefore, he sets up a classification of clinical types, whose logic, following that of B. Russell, encounters multiplicity as the limit of its classification. He is thus lead to found a paradoxical class, that of discordance, which gives rise to the determination of a central disorder, specific to discordant insanity, in which P. Chaslin isolates a clinical type, “discordant verbal insanity”, characterized by psittacism, that is, the cleavage between language and intelligence. Whereas psittacism in the domain of the pathological presents itself as a manifestation of a dysfunction in the act of speech, within the domain of symbolic logic, it belongs to the order of mathematical invention. In his essay on “The psychological mechanism of pure mathematical operations”, P. Chaslin questions the enigma of the objectivation of mathematical entities made by the avant-garde mathematicians of his era. At the crossroads of a question oriented on the one hand by the clinical case approached using the signifying chain, and on the other by the mathematical invention in its littoral function, P. Chaslin circumscribes a real that is manifest within the symbolic in the form of a paradox. In so doing, he has come to the edge of an epistemological reversal.

Mathematical Explanation in Science

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss potential future directions of research for each side in the debate over the existence of abstract mathematical objects. 1. Introduction: Mathematical Explanation2. Indispensability and Explanation3. Is the Mathematics Indispensable to the Explanation? 3.1. Object-level arbitrariness3.2. Concept-level arbitrariness3.3. Theory-level arbitrariness4. Is the Explanandum ‘Purely Physical’?5. Is the Mathematics Explanatory in Its Own Right?6. Does Inference to the Best Explanation Apply to Mathematics? 6.1. Leng's first argument6.2. Leng's second argument6.3. Leng's third argument7. Conclusions

Mathematical Nominalism and Measurement

In this paper I defend mathematical nominalism by arguing that any reasonable account of scientific theories and scientific practice must make explicit the empirical non-mathematical grounds on which the application of mathematics is based. Once this is done, references to mathematical entities may be eliminated or explained away in terms of underlying empirical conditions. I provide evidence for this conclusion by presenting a detailed study of the applicability of mathematics to measurement. This study shows that mathematical nominalism may be regarded as a methodological approach to applicability, illuminating the use of mathematics in science.

The concept of nondeterminism

Nondeterminism is a fundamental concept in computer science that appears in various contexts such as automata theory, algorithms and concurrent computation. We present a taxonomy of the different ways that nondeterminism can be defined and used; the categories of the taxonomy are domain, nature, implementation, consistency, execution and semantics. An historical survey shows how the concept was developed from its inception by Rabin & Scott, Floyd and Dijkstra, as well the interplay between nondeterminism and concurrency. Computer science textbooks and pedagogical software are surveyed to determine how they present the concept; the results show that the treatment of nondeterminism is generally fragmentary and unsystematic. We conclude that the teaching of nondeterminism must be integrated through the computer science curriculum so that students learn to see nondeterminism both in terms of abstract mathematical entities and in terms of machines whose execution is unpredictable.

Uma Solução para o Problema de Benacerraf

The Benacerraf’s problem is a problem about how we can attain mathematical knowledge: mathematical entities are entities not located in space-time; we exist in spacetime; so, it does not seem that we could have a causal connection with mathematical entities in order to attain mathematical knowledge. In this paper, I propose a solution to the Benacerraf’s problem supported by the Quinean doctrines of naturalism, confirmational holism and postulation. I show that we have empirical knowledge of centres of mass and of entitiesoutside of our light cone and that these entities are inefficacious causality entities, at least, with us. At the end, I defend the existential knowledge of centres of mass and of entities outside of our light cone against the Eleatic principle of Cheyne, that we only could attain existential knowledge of entities by a causal connection.