Mathematical Truth Research Articles

An intuitionistic interpretation of Bishop’s philosophy

Abstract The constructive mathematics developed by Bishop in Foundations of Constructive Analysis succeeded in gaining the attention of mathematicians, but discussions of its underlying philosophy are still rare in the literature. Commentators seem to conclude, from Bishop’s rejection of choice sequences and his severe criticism of Brouwerian intuitionism, that he is not an intuitionist–broadly understood as someone who maintains that mathematics is a mental creation, mathematics is meaningful and eludes formalization, mathematical objects are mind-dependent constructions given in intuition, and mathematical truths are experienceable. This paper develops and defends an intuitionistic interpretation of Bishop’s philosophical views.

Pricean ignorance

ABSTRACT Richard Price’s moral epistemology provides a distinctive account, not only of the sources of our moral knowledge, but also of its limits – that is, of the moral truths that we do not and even cannot know. According to this moral epistemology, the fundamental moral truths are necessary rather than contingent; if they are knowable at all, they are knowable a priori. In general, fundamental moral truths are akin to mathematical truths. Specifically, these necessary moral truths are grounded in the essences of act-types, which are knowable through ‘intuition’ if knowable at all. However, Price firmly rejects utilitarianism, recognizing several different ‘branches of virtue’, which can conflict with each other. As he argues, we cannot know the truth about act-types that exemplify such conflicting branches of virtue if the competing considerations are too finely balanced. This view of moral epistemology is compared with those of several other philosophers, including W. D. Ross. It is argued that Price has a more unified view than Ross: for Price, all moral knowledge flows from a priori intuition, but this is compatible with our being ignorant of many of the fine details of the fundamental truths of morality.

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.

Delving into the continuum hypothesis: A thorough examination of set theory and the nuances of mathematical infinity

This essay delves deep into one of the most intriguing mathematical puzzles of all time: the Continuum Hypothesis. Beginning with a robust foundational exploration, it sheds light on the key concepts of cardinality and power sets, which are pivotal to the realm of set theory. These foundational ideas set the stage for a deeper investigation into the relationship that the Continuum Hypothesis shares with real numbers and natural numbers. Historically, the Continuum Hypothesis has tantalized mathematicians. This paper takes a journey through time, highlighting the various endeavors to either prove or refute this hypothesis. Some of the most brilliant minds have grappled with its complexities, leaving behind a rich tapestry of mathematical thought. Furthermore, a significant portion of our discussion is centered on situating the Continuum Hypothesis within the context of Zermelo-Fraenkel Set Theory (ZFC). The intricate interplay between the hypothesis and ZFC offers profound insights and raises thought-provoking questions about the very nature of mathematical truth.

Mapping Connectivity Patterns: Degree-Based Topological Indices of Corona Product Graphs

Graph theory (GT) is a mathematical field that involves the study of graphs or diagrams that contain points and lines to represent the representation of mathematical truth in a diagrammatic format. From simple graphs, complex network architectures can be built using graph operations. Topological indices (TI) are graph invariants that correlate the physicochemical and interesting properties of different graphs. TI deal with many properties of molecular structure as well. It is important to compute the TI of complex structures. The corona product (CP) of two graphs G and H gives us a new graph obtained by taking one copy of G and V G copies of H and joining the i th vertex of G to every vertex in the i th copy of H . In this paper, based on various CP graphs composed of paths, cycles, and complete graphs, the geometric index (GA) and atom bond connectivity (ABC) index are investigated. Particularly, we discussed the corona products P s ⨀ P t , C t ⨀ C s , K t ⊙ K s , K t ⊙ P s , and P s ⊙ K t and GA and ABC index. Moreover, a few molecular graphs and physicochemical features may be predicted by considering relevant mathematical findings supported by proofs.

For Mathematical Truth to Be Known: A Provability-based Condition Construction

One of the defining characteristics of mathematics is its emphasis on proof, which is the process of demonstrating the truth or falsehood of a mathematical claim through a rigorous and logical argument. However, while the concept of proof is central to mathematics, it is also a complex and multifaceted notion that raises a range of philosophical and practical questions. This article proposed a condition beyond the JTB framework for mathematical knowledge by using mathematics provability with a careful reflection on Gettier cases.

Later Wittgenstein on ‘Truth’ and Realism in Mathematics

AbstractI show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse.

Cantorův diagonální důkaz

Cantor's diagonal proof is sig­nificant both because the central method of proof used in it has been subsequently applied in a number of other proofs, and because it is considered to confirm the existence of infinite sets whose size fun­ damentally and by an order of magnitude exceeds the size of the "classical" infinite set represented by all natural numbers, while their size can theoretically exceed every conceivable limit. Although Can­tor's proof is generally accepted by the scientific community, some experts are somewhat reserved about it. The aim of this paper is to present Cantor's proof in an accessible way, while pointing out its (hidden) assumptions and possible problematic points, and pointing out that some of its underlying assumptions are not indisputable mathematical truths, but rather postulated propositions that may or may not be accepted.

Бенедетто Кроче. Вопрос о методе в истории философии

This publication presents a translation into Russian of Benedetto Croce’s essay from one of his later books “Philosophy and Historiography”. Here he raises the question of how the historian of philosophy should interpret those moments when the reasoning of a philosopher who is being studied is accidentally or deliberately not cleared up by him, or its development stops at a certain point. Considering the possible reasons for this, Croce touches on several themes to which his earlier writings were devoted and to which he returned throughout his scholarly life. Thus, he draws attention to the contradiction in the theory of knowledge of G. Vico, which consists in the application of the principle of verum-factum not only to history, but also to the mathematical sciences: on the one hand, only God can achieve truth in knowledge of nature, since he created nature, and on the other hand, truth in mathematics is available to man, since he “creates” mathematical elements and operations, and with them the method of natural science. Croce also points as an example to one of the contradictions in Hegel’s philosophy of nature, which consists in the confusion of empirical and speculative methods. It led to the endowment of the natural sciences with the status of “semi-philosophy”, to the untrue application of dialectics to the facts and concepts of the natural sciences. Convinced that the verum-factum formula can be combined with the teaching of absolute spiritualism, Croce warns against attributing such a connection to Vico himself, as this would mean incorporating later experience and a state of knowledge into his theory.

The causality-chance binomial: explanatory principle of scientific knowledge in medicine?

This essay questions mathematical truths as an explanatory principle of the medical scientific knowledge. It analyzes, in the first place, the current concept of normality based on a distribution of probabilistic values and its limitations and mistakes to capture the complexity of the human condition are highlighted. The closed systems (gambling) origin of the theory of probabilities and the binomial causality-chance are compared with open systems typical of the complexity of the vital process, and their extreme differences are argued. The nonsense of depositing in the causality-chance binomial the meaning of associations between events typical of the complexity of human life in health and disease is highlighted. The characteristics of mechanistic causality (punctual, homogeneous, linear, unidirectional and fixed), which equates the organism with a machine and is the only accepted scientific explanation of events of human life, are confronted with those of contextual causality (diffuse, heterogeneous, hierarchical, multidirectional and changing), which specifies various interacting causal orders that shape of human condition: the historical, the social, the political, the economical, the cultural or the biological that represents a scrutinizing and penetrating look at the complexity of human beings. It concludes the superiority of contextual causality over mechanistic causality that opens up explanatory possibilities of the vital events that are usually put away as "effects of chance". This integrative approach to the human complexity can enrich and strengthen the clinical method that is now degraded and at risk of extinction.

Mathematics First: Russell’s Methodological Response to Bradley

Abstract In this article I examine the dispute between F. H. Bradley and Bertrand Russell concerning the reality of relations. I show that Bradley’s objections to Russell’s view, that there are such things as relations which serve to effect the unity of complex items, were rooted in a methodological approach which Russell did not share. On Bradley’s view, one must be able to offer reductive analyses of the items one postulates in order that commitment to those items be justified. I argue that Russell expressly rejected this methodological principle of Bradley’s, and instead adopted the view that one may justifiably postulate entities if doing so aids in the illumination of mathematical truths. I show that the postulation of relations does, on Russell’s view, serve to provide that illumination. Russell held that the truths of mathematics constitute fixed data, and that philosophical positions may be judged as successful according to the extent that they possess explanatory power with respect to this data. I argue that Bradley’s and Russell’s exchanges in print, as well as in private correspondence, reflect Russell’s awareness of a fundamental difference in methodological approach. I conclude that Russell elected not to answer Bradley’s objections on their own terms, but rather rejected the methodological assumptions from which those objections emerged.

Sharon Berry.A Logical Foundation for Potentialist Set Theory

This book offers a foundation for mathematics grounded in a collection of axioms for logical possibility in a first-order language. The offered foundation is argued to have various epistemological benefits, in particular as regards our justification for believing certain axioms of set theory, and more generally as regards the ‘access’ problem for mathematical objects. The foundation for mathematics is provided in stages. First, Berry argues for a ‘potentialist’ interpretation of standard ZF set theory in terms of some axioms for logical possibility that are argued to ‘seem clearly true’. Then, she argues for a Carnap-inspired account of the semantics of ordinary mathematical language in support of the idea that all mathematical discourse as usually pursued can be interpreted faithfully in terms of the foundation provided by her potentialist set theory. The end result, it is argued, is something approximating a logicist foundation for mathematics: standard mathematical truths are interpreted in terms of a pure logic of logical possibility.

David Armstrong on the Metaphysics of Mathematics

This paper has two components. The first, longer component (sec. 1-6) is a critical exposition of Armstrong's views about the metaphysics of mathematics, as they are presented in Truth and Truthmakers and Sketch for a Systematic Metaphysics. In particular, I discuss Armstrong's views about the nature of the cardinal numbers, and his account of how modal truths are made true. In the second component of the paper (sec. 7), which is shorter and more tentative, I sketch an alternative account of the metaphysics of mathematics. I suggest we insist that mathematical truths have physical truthmakers, without insisting that mathematical objects themselves are part of the physical world.

Mind, Value, and Cosmos: On the Relational Nature of Ultimacy

Mind, Value, and Cosmos: On the Relational Nature of Ultimacy

Afterword: Some Illustrations

Afterword: Some Illustrations

Music and mathematics in Tom Johnson’s work: the composer’s view

ABSTRACT Tom Johnson is an interesting case of interdisciplinarity in that he makes music and mathematics coincide in a very obvious and literal way. He was led to this approach in the context of the American avant-garde of the 1970s, in which many artists tried to set artistic creation in the field of the impersonal. In using mathematics, Tom Johnson is in search for something that might allow the music to compose itself automatically. He establishes sequences of numbers and translates them literally into melodies, harmonies or rhythms. In doing so, he manages to make the mathematical background clearly appear. He seems to think of his music as a way of reflecting mathematics in the form of audible phenomena, and ultimately, he attaches some kind of mystical dimension to this experience. Mathematical truths, as he calls them, are unquestionable. Their laws do not depend on human will, they are deeply inscribed in nature, and a composition strictly based on them is necessarily linked to the absolute.

Inference to the best explanation as supporting the expansion of mathematicians’ ontological commitments

This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” (IBE) as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths.

The “Pythagorean” “Theorem” and the rant of racist and civilizational superiority – Part 2

Previously we saw that racist prejudice is supported by false history. The false history of the Greek origins of mathematics is reinforced by a bad philosophy of mathematics. There is no evidence for the existence of Euclid. The “Euclid” book does not contain a single axiomatic proof, as was exposed over a century ago. Such was never the intention of the actual author. The book was brazenly reinterpreted, since axiomatic proof was a church political requirement, and used in church rational theology adopted during the Crusades, as a counter to Islamic rational theology. Deductive proofs are MORE fallible than inductive or empirical proofs. Even a validly proved mathematical theorem, such as the “Pythagorean” theorem (based on Hilbert’s axioms), is invalid knowledge in the real world. There is no concept of approximate truth in formal mathematics. Nevertheless, the myth of “superior” axiomatic proofs in the “Euclid” book continues to be reiterated by Western historians, and colonial education teaches axiomatic mathematics. Actually, superior practical value comes from the two “Pythagorean” calculations well known to Indian/Egyptian tradition, but unknown to Greeks. The advantage of related decolonized courses in mathematics has been pedagogically demonstrated. But understanding and political will are needed to change colonial/church education.

Neutrality and Force in Field’s epistemological objection to platonism

ABSTRACT Field’s challenge to platonists is the challenge to explain the reliable match between mathematical truth and belief. The challenge grounds an objection claiming that platonists cannot provide such an explanation. This objection is often taken to be both neutral with respect to controversial epistemological assumptions, and a comparatively forceful objection against platonists. I argue that these two characteristics are in tension: no construal of the objection in the current literature realises both, and there are strong reasons to think that no version of Field’s epistemological objection which has both Neutrality and Force can be construed.

Guarantees for Existence of a Best Canonical Polyadic Approximation of a Noisy Low-Rank Tensor

The canonical polyadic decomposition (CPD) of a low-rank tensor plays a major role in data analysis and signal processing by allowing for unique recovery of underlying factors. However, it is well known that the low-rank CPD approximation problem is ill-posed. That is, a tensor may fail to have a best rank $R$ CPD approximation when $R>1$. This article gives deterministic bounds for the existence of best low-rank tensor approximations over ${\mathbb{K}}={\mathbb{R}}$ or ${\mathbb{K}}={\mathbb{C}}$. More precisely, given a tensor ${\mathcal T} \in {\mathbb{K}}^{I \times I \times I}$ of rank $R \leq I$, we compute the radius of a Frobenius norm ball centered at ${\mathcal T}$ in which best ${\mathbb{K}}$-rank $R$ approximations are guaranteed to exist. In addition we show that every ${\mathbb{K}}$-rank $R$ tensor inside of this ball has a unique canonical polyadic decomposition. This neighborhood may be interpreted as a neighborhood of “mathematical truth" in which CPD approximation and computation are well-posed. In pursuit of these bounds, we describe low-rank tensor decomposition as a “joint generalized eigenvalue" problem. Using this framework, we show that, under mild assumptions, a low-rank tensor which has rank strictly greater than border rank is defective in the sense of algebraic and geometric multiplicities for joint generalized eigenvalues. Bounds for existence of best low-rank approximations are then obtained by establishing perturbation theoretic results for the joint generalized eigenvalue problem. In this way we establish a connection between existence of best low-rank approximations and the tensor spectral norm. In addition we solve a “tensor Procrustes problem" which examines orthogonal compressions for pairs of tensors. The main results of the article are illustrated by a variety of numerical experiments.