Philosophy Of Mathematics Research Articles

What is an experiment in mathematical practice? New evidence from mining the Mathematical Reviews

From a purely formalist viewpoint on the philosophy of mathematics, experiments cannot (and should not) play a role in warranting mathematical statements but must be confined to heuristics. Yet, due to the incorporation of new mathematical methods such as computer-assisted experimentation in mathematical practice, experiments are now conducted and used in a much broader range of epistemic practices such as concept formation, validation, and communication. In this article, we combine corpus studies and qualitative analyses to assess and categorize the epistemic roles experiments are seen—by mathematicians—to have in actual mathematical practice. We do so by text-mining a corpus of reviews from the Mathematical Reviews, which include the indicator word “experiment”. Our qualitative, grounded classification of samples from this corpus allows us to explore the various roles played by experiments. We thus identify instances where experiments function as references to established knowledge, as tools for heuristics or exploration, as epistemic warrants, as communication or pedagogy, and instances simply proposing experiments. Focusing on the role of experiments as epistemic warrants, we show through additional sampling that in some fields of mathematics, experiments can warrant theorems as well as methods. We also show that the expressed lack of experiments by reviewers suggests concordant views that experiments could have provided epistemic warrants. Thus, our combination of corpus studies and qualitative analyses has added a typology of roles of experiments in mathematical practice and shown that experiments can and do play roles as epistemic warrants depending on the mathematical field.

Philosophy of Mathematics in Primary Education Mathematics Learning: Ontological, Epistemological, and Methodological

The purpose of this study is to examine the relationship, position, and role of the philosophy of mathematics in learning mathematics in basic education through the perspective of the scope of philosophy, namely ontological mathematics, epistemological mathematics, and methodological mathematics. This research is a qualitative descriptive research with a literature study approach. Data collection techniques are carried out through the documentation method of articles originating from the Google Scholar database, as well as books related to the philosophy of mathematics. Data analysis techniques consist of data reduction, data presentation, and conclusion. The results highlight three main aspects within the scope of the philosophy of mathematics: ontological, epistemological, and methodological. Ontological mathematics addresses the ontological foundations of mathematics for teaching, including the role of mathematics in various contexts. The epistemology of mathematics emphasizes the importance of understanding the discipline from multiple perspectives to enhance learning and curriculum design. Meanwhile, the methodology of mathematics highlights the role of specific methods in determining mathematical solutions. In conclusion, understanding the philosophy of mathematics in these three aspects is important for educators to improve the learning process of mathematics at the primary level comprehensively and optimally.

Was Wittgenstein a radical conventionalist?

This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett in Philos Rev 68(3), 1959, p. 329). On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. Mathematical truths are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. The goal of the paper is not merely exegetical, however, and argues that radical conventionalism is able to withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics.

Components of arithmetic theory acceptance

This paper ties together three threads of discussion about the following question: in accepting a system of axioms S, what else are we thereby warranted in accepting, on the basis of accepting S? First, certain foundational positions in the philosophy of mathematics are said to be epistemically stable, in that there exists a coherent rationale for accepting a corresponding system of axioms of arithmetic, which does not entail or otherwise rationally oblige the foundationalist to accept statements beyond the logical consequences of those axioms. Second, epistemic stability is said to be incompatible with the implicit commitment thesis, according to which accepting a system of axioms implicitly commits the foundationalist to accept additional statements not immediately available in that theory. Third, epistemic stability stands in tension with the idea that in accepting a system of axioms S, one thereby also accepts soundness principles for S. We offer a framework for analysis of sets of implicit commitment which reconciles epistemic stability with the latter two notions, and argue that all three ideas are in fact compatible.

The Interest of Philosophy of Mathematics (Education)

The Interest of Philosophy of Mathematics (Education)

The Relation Between Mathematics Philosophy And Culture-Based Learning Model

The aim of this study is to study the correlation of mathematical philosophy with cultural-based learning models. The method of research used is the method of library research, i.e. conducting studies related to mathematical philosophy, philosophical mathematics learning, culture-based mathematic learning, the connection of mathematically philosopher with ethnomatematics, the relationship of mathematic philosophies with cultural-based learning models. The results of the research have been obtained that the philosofy of maths, the model of learning based on culture, and ethnomathematics have close connections in the context of understanding and teaching of Mathematics. Mathematical philosophy deals with the foundations of mathematical epistemology and ontology, while a culture-based learning model emphasizes the integration of values and cultural contexts in mathematics learning. Ethnomatematics refers to the study of mathematical ways of thinking in a particular culture. By understanding the cultures, traditions, and world views of a particular society, ethnomathematics can help develop more culturally relevant and meaningful learning models. Therefore, the interrelationship between mathematical philosophy, culture-based learning models, and ethnomathematics can create a more holistic and contextual approach to mathematics learning.
 Keywords: Mathematical Philosophy, Cultural Based Learning Model, Ethnomathematics

Epistemological Perspective of Mathematical Philosophy On Technology-Based Learning

This research is a literature study (library research) that uses a qualitative approach. The aim of this research is to examine the extent to which technology plays a role in mathematics learning from the perspective of the epistemology of mathematical philosophy. The study of the epistemology of the philosophy of mathematics that is being discussed is how does the development of the philosophy of mathematics influence technology? What is the relationship between the philosophy of mathematics and technology-based learning? As well as examples of technology-based mathematics learning models. The results and discussion of this article can be concluded: Philosophy is a science that has a close relationship in human life.The philosophy of mathematics influences the development of technology in general. Technological developments influence the development of learning. From a learning perspective, technology is not only something related to machines, but also related to people, ideas and activities in learning. The use of technology in the form of hardware and software can improve student learning outcomes. Currently, there are many software tools available for use in learning, such as GeoGebra. Choosing the right learning model and mathematics learning technology can make learning more effective

The Box and Muller Technique for Generating Neutrosophic Random Variables Follow a Normal Distribution

The focus of operations research is the existence of a problem that requires making an appropriate decision that helps reduce risk and achieves a good level of performance. Operations research methods depend on formulating realistic issues through mathematical models consisting of a goal function and constraints, and the optimal solution is the ideal decision, despite the multiplicity of these methods. However, we encounter many complex issues that cannot be represented mathematically, or many issues that cannot be studied directly. Here comes the importance of the simulation process in all branches of science, as it depends on applying the study to systems similar to real systems and then projecting this. The results if they fit on the real system. So simulation is the process of building, testing, and running models that simulate complex phenomena or systems using specific mathematical models. The simulation process depends on generating a series of random numbers subject to a regular probability distribution in the field [0, 1], and then converting these random numbers into random variables subject to the distribution law. Probability, according to which the system to be simulated operates, using appropriate techniques for both the probability density function and the cumulative distribution function. Classical studies have provided many techniques that are used during the simulation process, and to keep pace with the great scientific development witnessed by our contemporary world, we found that a new vision must be presented for this. Techniques A vision based on the concepts of neutrosophics, the science founded by the American mathematical philosopher Florentin Smarandache. The year 1995, in which new concepts of probabilities and probability distributions are used, as we presented in previous research some techniques from a neutrosophic perspective, and as an extension of what we presented previously, we present in this research a neutrosophic vision of the Box and Muller technique used to generate random variables that follow a normal distribution.

Kurt Gödel’in Edmund Husserl Okuması: Matematiğin Temellerini Fenomenolojinin Işığında Aramak

In his later works, the great logician and mathematician Kurt Gödel concentrates his focus on the philosophical problems such as the implications of set theory, the grammar and philosophy of language, objectivity and relativity, the ontological proof of God’s existence, and phenomenology as an exact method. This essay explores how Gödel reads the philosophy (of logic and mathematics) of his time and why he turns his attention to Husserl’s phenomenology for describing the foundations of mathematics. To begin with, Gödel employs Husserl’s significant distinction between Weltanschauung (worldview) philosophy and philosophy as rigorous science: According to the Weltanschauung philosophy, the spirit of time constantly changes so that the ideas discussed and goals attempted are meant to be temporal, and not for the sake of eternal truths, but for that of their own perfection; philosophy as rigorous science, on the other hand, is supratemporal so that its aim is to discover absolute and timeless values. As for the worldview of his time, Gödel sees the development of philosophy and mathematics leaned toward skepticism, pessimism, and positivism. The antinomies of set theory, for instance shaked the grounds on which mathematics and logic are founded. Gödel, too, uses these paradoxes in his incompleteness theorems in order to prove that there are some statements which can neither be proved nor disproved within a system. That also means that arithmetic is not eligible to prove its own consistency. From this, however, Gödel does not come to a conclusion for a nihilism in mathematics and logic: These mere antinomies of set theory do not “necessarily” lead us to logical positivism, and neither to such a materialism, nor to any kind of pessimistic theory of knowledge. The incompleteness theorems assert that there are arithmetical propositions that are true but neither provable nor unprovable within its own calculus, so that arithmetic is intrinsically incomplete. However, instead of Alfred Tarski’s pathological view of examining the detections within the faulty system and then reforming the system all together, Gödel holds that we need to change our methods to find new patterns that describe the antinomies pointing to the unrecoverable reality of the mathematical world. Thus, Gödel does not follow any variation of the Weltanschauung philosophy of his time, either attempting to reduce mathematical realities to mathematical proofs in order to get rid of antinomies, or endeavoring to rescue a complete system of truths by a closed formal system, both Weltanschauung philosophies fail to set forth a realistic method. In this context, Gödel finds the task of phenomenology analogous to what he pursues in terms of a systematic framework for the foundations of mathematics. Husserl’s phenomenology, in Gödel’s account, proliferates the intuition of (mathematical) essences and provides a clarification of meaning of undefinable concepts, such as the antinomies of set theory. Applying the phenomenological reduction to the objective reality of the mathematical world, Gödel believes one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly put, what Gödel finds in Husserl’s phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method giving us mathematical essences back again.

Abstraction and grounding

AbstractThe idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one: . The principal aim of this article is to use the notion of grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of metaphysical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege's Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction.

Jerónimo Muñoz’s Reception of Proclus’ In Euclidem: Philosophy of Mathematics and an Attempt to Prove the Parallel Postulate

Abstract The goal of this paper is to show – by way of a case study – how the contents of Proclus’ Commentary on the First Book of Euclid’s Elements were incorporated into university teaching in the sixteenth century. I analyse the impact of Proclus on the works of the Spanish mathematician and university professor Jerónimo Muñoz (ca. 1520–1591). In order to do so, I examine two manuscripts: Adnotationes in commentaria Procli super Euclidem (MS Vat. Lat. 6996), and Astrologicarum et geographicarum institutionum libri sex (MS Vat. Lat. 6998). I show that the contents of Proclus’ commentary pervade Muñoz’s mathematical writings and influence his mathematical ontology, his classification of mathematical disciplines, and the history and terminology of geometry that he adopts. Moreover, I expound on how Proclus’ text inspired Muñoz to maintain that the fifth postulate was a theorem, leading him to attempt a demonstration of it which pre-dates knowledge in the Latin West of Naṣīr ad-Dīn’s (1201–1274) previous attempt.

Epistemological and Ethical Philosophy of Mathematics In 21st Century Mathematics Learning Practices

21st century mathematics education has made use of science and technology while maintaining the philosophy of mathematics from an ethical and epistemological standpoint. This paper examines the goals of the philosophy of mathematics from an epistemological and ethical perspective. It also investigates the relationship between these perspectives and the practice of studying mathematics in the twenty-first century. This article was written using the Publish or Perish software and a systematic literature review approach. 41 initial articles were found by inputting keywords. Following review, fifteen articles were found to be in agreement. This study reveals that the goal of mathematics philosophy is to understand truth, meaning, certainty, and the nature and sources of mathematical knowledge, as well as how to interpret mathematics, from an epistemological perspective. Analyze and reflect on the ethical implications of mathematical practices. Its epistemological application in 21st century learning practices guides the way concepts are positioned, logical arguments are made, and mathematical problems are solved. Next, ethical considerations include maintaining student privacy, establishing learning agreements in every class, and avoiding creating group distinctions. by utilizing an interactive, student-centered learning paradigm like PBL, inquiry, contextual learning, use of technology, STEM, and others. Such behaviors will subsequently cultivate the values of critical, communicative, creative, and cooperative character. Therefore, we must uphold the fundamentals of learning in the 21st century, both in terms of epistemology and ethics.

A new role of mathematics in science: Measurement normativity

The applicability of mathematics in science has received renewed attention in recent years.The most influential recent approachesfocus on how mathematical structures relate to empirical structures. I argue that these accounts leave an important normative aspect of mathematics, which is key for scientific measurement, unaddressed. Building on the later Wittgenstein’s philosophy of mathematics, Icharacterize how mathematical models have both an empirical and a normative aspect: sometimes they are falsified or refined, but on other occasions they are used as rules of measurement. Thus, the normative aspect of mathematics contributes to its applicability in science in practice without precluding (but at the expense of) its empirical aspect. I illustrate this dual character with two simple examples first and then with a more substantial case study in neuroscience. Whilethe normative aspect of mathematics isoften desirable, the epistemic circularity that it entails may be epistemically harmful in some scenarios.

Hypothetical Inquiry in Plato's Timaeus

This paper re-constructs Plato's ‘philosophy of geometry’ by arguing that he uses a geometrical method of hypothesis in his account of the cosmos’ generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between ( metaxu) Forms and sensible particulars ( Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions of mathematical ontology. My key passage of interest is Timaeus’ account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth ( Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies’ origin by hypothesising two right-angled triangles as their starting-point ( arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world.

Spinoza, Emanation, and Formal Causation

abstract: Some recent scholars have argued that Spinoza's conception of causation should be understood in terms of the Aristotelian notion of a formal cause. I argue that while they are right to identify causation in Spinoza as a relation of entailment from an essence, they are mistaken about its philosophical pedigree. I examine three suggested lines of influence: (a) the late scholastic conception of emanation; (b) early modern philosophy of mathematics; and (c) Descartes's notion of the causa sui . In each case, the evidence indicates that causation in Spinoza should be categorized in Aristotelian terms as efficient and not formal.

Ancient Philosophy of Mathematics and Its Tradition

Ancient Philosophy of Mathematics and Its Tradition

Wittgenstein and Stenlund on Mathematical Symbolism

In recent work, Sören Stenlund (2015) contextualizes Wittgenstein’s philosophy of mathematics as aligned with the tradition of symbolic mathematics. In the early modern era, mathematicians began using purely formal methods disconnected from any obvious empirical applications, transforming their subject into a symbolic discipline. With this, Stenlund argues, they were freeing themselves of ancient ontological presuppositions and discovering the ultimately autonomous nature of mathematical symbolism, which eventually formed the basis for Wittgenstein’s thinking. A crucial premise of Wittgenstein’s philosophy of mathematics, on this view, is that the development of mathematical concepts is independent of any ontological implications and occurs in principle without normative connections to empirical applicability. This paper critically examines this narrative and arrives at the conclusion that Stenlund’s view of mathematical progress is in stark contrast to the later Wittgenstein’s writing, which emphasizes links between symbolisms and their applications.

Kantian Motives in Work of Ludwig Wittgenstein

It is proved that the basic framework of the premises and reasoning of Wittgenstein's “Tractatus Logico-philosophicus” corresponds quite well to the transcendental method (as formulated by H. Cohen). Whereas Kant’s philosophy proceeds from the fact of existence of mathematics and mathematised natural science and investigates their conditions of possibility, Wittgenstein proceeds from the fact that propositions of language describe reality and reveals the conditions of possibility of such descriptions. Kant, answering the question about the conditions of possibility of the named sciences, comes to the idea of the transcendental subject and the distinction between the world of phenomena and the thing in itself. Wittgenstein's investigation of the conditions of possibility that the world is described by propositions leads to the assertion that both the world and language are together in logical space. The latter constitutes the a priori and transcendental condition of the possibility that language “reaches out” to reality. For both the theories - in the “Critique of Pure Reason” and in the “Tractatus Logico-Philosophicus” - the idea of the boundary is important. In the “Critique of Pure Reason” it is the boundary of possible experience and cognition, while in the “Tractatus Logico-Philosophicus” it is the boundary of what can be thought and expressed by meaningful propositions. Related to the different definitions of the boundary is the difference in the treatment of mathematised natural science. For “The Critique of Pure Reason” was created in the era of unconditional acceptance of Newtonian mechanics. And the “Tractatus Logico-Philosophicus” was created at the time of the crisis of the Newtonian paradigm and its replacement by other notions of time and space. However, the idea of boundary, which is present in both doctrines, determines closeness in the attitude towards metaphysics between the author of “The Critique of Pure Reason” and the author of “The Tractatus Logico-Philosophicus”. The study also shows that Wittgenstein did not follow logicism in his philosophy of mathematics. For him, both mathematical objects and propositions of logic are constructions. The conviction about the constructive character of mathematical and logical objects shows an affinity with the Kantian tradition in the philosophy of mathematics.

Realism and the point at infinity: The end of the line?

Philosophers of mathematics often rely on the historical progress of mathematics in support of mathematical realism. These histories typically build on formal semantic tools to evaluate the changes in mathematics, and on these bases present later mathematical concepts as refined versions of earlier concepts which are taken to be vague. Claiming that this view does not apply to mathematical concepts in general, we present a case-study concerning projective geometry, for which we apply the tools of cognitive linguistics to analyse the developmental trajectory of the domain. On the basis of this analysis, we argue for the existence of two conceptually incompatible inferential structures, occurring at distinct moments in history, both of which yield the same projective geometric theorems; the first invoked by the French mathematicians Girard Desargues (1591–1661) and Jean-Victor Poncelet (1788–1867), and the second characterising a specific modern mode. We demonstrate that neither of these inferential structures can be considered as a refinement of the other. This case of conceptual development presents an issue to the standard account of progress and its bearing on mathematical realism. Our analysis suggests that the features that distinguish the underlying conceptually incompatible inferential structures are invisible to the standard application of the tools of formal semantics. Thus this case-study stands as an example of the manner and necessity of linguistics—specifically cognitive linguistics—to inform the philosophy of mathematics.

Internalism and the Determinacy of Mathematics

Abstract A major challenge in the philosophy of mathematics is to explain how mathematical language can pick out unique structures and acquire determinate content. In recent work, Button and Walsh have introduced a view they call ‘internalism’, according to which mathematical content is explained by internal categoricity results formulated and proven in second-order logic. In this paper, we critically examine the internalist response to the challenge and discuss the philosophical significance of internal categoricity results. Surprisingly, as we argue, while internalism arguably explains how we pick out unique mathematical structures, this does not suffice to account for the determinacy of mathematical discourse.