Philosophy of Mathematics

The physical foundations of mathematics in the theory of emergent space-time-matter were considered. It is shown that mathematics, including logic, is a consequence of equation which describes the fundamental field. If the most fundamental level were described not by mathematics, but something else, then instead of mathematics there would be consequences of this something else.

Modal structuralism promises an interpretation of set theory that avoids commitment to abstracta. This article investigates its underlying assumptions. In the first part, I start by highlighting some shortcomings of the standard axiomatisation of modal structuralism, and propose a new axiomatisation I call MSST (for Modal Structural Set Theory). The main theorem is that MSST interprets exactly Zermelo set theory plus the claim that every set is in some inaccessible rank of the cumulative hierarchy. In the second part of the (...) article, I look at the prospects for supplementing MSST with a modal structural reflection principle, as suggested in Hellman (2015). I show that Hellman’s principle is inconsistent (Theorem 5.32), and argue that modal structural reflection principles in general are either incompatible with modal structuralism or extremely weak. (shrink)

Inhaltsverzeichnis (vorläufig) 1 Einführung. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 -/- 2 Pythagorismus . . . . . . . . . . . . . . . . . . . . . . . . . (...) . . . . . . . . . . . . . . . . . . . . 3 -/- 2.1 Pythagoras und die Pythagoräer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 -/- 2.2 Zahlenmystik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 -/- 2.3 Mathematik als Grundlage der Wissenschaft . . . . . . . . . . . . . . . . . . . . . 3 -/- 2.4 Kritische Würdigung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 -/- 3 Platonismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 -/- 3.1 Historischer Hintergrund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 -/- 3.2 Die platonische Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 -/- 3.3 Argumente für einen Platonismus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 -/- 3.4 Die metaphysische Kluft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 -/- 3.5 Der Platonismus und die Kontinuums-Hypothese . . . . . . . . . . . . . . . . . 20 -/- 4 Rationalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 -/- 4.1 Historischer Hintergrund . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 -/- 4.2 Die rationalistische Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 -/- 4.3 Argumente für den Rationalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 -/- 4.4 Probleme des Rationalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 -/- 5 Kantianismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -/- 5.1 Imanuell Kant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -/- 5.2 Mathematik in der Kritik der reinen Vernunft . . . . . . . . . . . . . . . . . . . . 27 -/- 5.3 Kritik der Position Kant's . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 -/- 6 Logizismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.1 Die logizistische Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 -/- 6.2 Historische Entwicklung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 -/- 6.3 Einführung in die formale Logik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 -/- 6.4 Der Versuch der Reduktion der Arithmetik . . . . . . . . . . . . . . . . . . . . . . 37 -/- 6.5 Einschätzung des Logizistischen Vorhaben . . . . . . . . . . . . . . . . . . . . . . . 43 -/- 7 Intuitionismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 -/- 7.1 Luitzen Brouwer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 -/- 7.2 Die Metaphysik des Intuitionismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 -/- 7.3 Intuitionistische Logik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 -/- 7.4 Intuitionistische Mathematik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 -/- 7.5 Kritik des Intuitionismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 -/- 8 Formalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 -/- 8.1 Die formalistische Position . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 -/- 8.2 Wittengenstein und der Positivismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 -/- 8.3 Hilberts Formalismus und das Problem der Wiederspruchsfreiheit . . . 67 -/- 8.4 Moderne Spielarten des Formalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 -/- 8.5 Kritik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 -/- 9 Konstruktivismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 -/- 10 Strukturalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 -/- 10.1 Philosophischer Strukturalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 -/- 10.2 Strukturalismus in der Mathematik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 -/- 10.3 Kritik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 -/- 11 Naturalismus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 -/- 11.1 Die Entwicklung des Naturalismus in der Philosophie . . . . . . . . . . . . . 73 -/- 11.2 Naturalismus in Bezug auf die Mathematik . . . . . . . . . . . . . . . . . . . . . . 73 -/- 11.3 Kritik . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 -/- 12 Fiktionalismus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 -/- 13 Anhang . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Literaturverzeichnis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79. (shrink)

Even the ancient Greeks defined the Dream as a happy πόλις, Heraclitus - κόσμοπόλις, Socrates - ethical anthropology, Plato - Good, Hegel - absolute idea, Marx - communism... All of Humanity has made a lot of its survival experience for the realization of Dreams. Without any plan, to the touch to, only by the method of "trial and error" it aspired the Dream on unknown roads, which often stymied deadlocks. Among the many achieved results of Humanity by Plato's prompts, the (...) authors singled out "εἶδος", "reasonable" Numbers (1,2 ... 10), five of which (1,2 ... 5) were examples of Marx's formations. The found "εἶδος" lined up themselves and show the possibility of further building (11.12 ... 16) of the World Reason of God. This is Plato's "single right path", unambiguously leading to the universal Dream of Humanity – Truth of Good. (shrink)

Brief note explaining the content, importance, and historical context of my joint translation of Quine's The Significance of the New Logic with my single-authored historical-philosophical essay 'Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s'.

Albert Lautman (b. 1908–1944) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that: ‘in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe’ (Lautman 2011, 87). He goes on to characterize this (...) reality as an ‘ideal reality’ that ‘governs’ the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by a what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems. (shrink)

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study (...) of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them. (shrink)

The discrepancy between syntax and semantics is a painstaking issue that hinders a better comprehension of the underlying neuronal processes in the human brain. In order to tackle the issue, we at first describe a striking correlation between Wittgenstein's Tractatus, that assesses the syntactic relationships between language and world, and Perlovsky's joint language-cognitive computational model, that assesses the semantic relationships between emotions and “knowledge instinct”. Once established a correlation between a purely logical approach to the language and computable psychological activities, (...) we aim to find the neural correlates of syntax and semantics in the human brain. Starting from topological arguments, we suggest that the semantic properties of a proposition are processed in higher brain's functional dimensions than the syntactic ones. In a fully reversible process, the syntactic elements embedded in Broca's area project into multiple scattered semantic cortical zones. The presence of higher functional dimensions gives rise to the increase in informational content that takes place in semantic expressions. Therefore, diverse features of human language and cognitive world can be assessed in terms of both the logic armor described by the Tractatus, and the neurocomputational techniques at hand. One of our motivations is to build a neuro-computational framework able to provide a feasible explanation for brain's semantic processing, in preparation for novel computers with nodes built into higher dimensions. (shrink)

The analysis of long economic cycles allows us to understand long-term worldsystem dynamics, to develop forecasts, to explain crises of the past, as well as the current global economic crisis. The article offers a historical sketch of research on K-waves; it analyzes the nature of Kondratieff waves that are considered as a special form of cyclical dynamics that emerged in the industrial period of the World System history. It offers a historical and theoretical analysis of K-wave dynamics in the World (...) System framework; in particular, it studies the influence of the long wave dynamics on the changes of the world GDP growth rates during the last two centuries. Special attention is paid to the interaction between Kondratieff waves and Juglar cycles. The article is based on substantial statistical data, it extensively employs quantitative analysis, contains numerous tables and diagrams. On the basis of the proposed analysis it offers some forecasts of the world economic development in the next two decades. The article concludes with a section that presents a hypothesis that the change of K-wave upswing and downswing phases correlates significantly with the phases of fluctuations in the relationships between the World-System Core and Periphery, as well as with the World System Core changes. (shrink)

Introduction to mathematical logic, part 2.Textbook for students in mathematical logic and foundations of mathematics. Platonism, Intuition, Formalism. Axiomatic set theory. Around the Continuum Problem. Axiom of Determinacy. Large Cardinal Axioms. Ackermann's Set Theory. First order arithmetic. Hilbert's 10th problem. Incompleteness theorems. Consequences. Connected results: double incompleteness theorem, unsolvability of reasoning, theorem on the size of proofs, diophantine incompleteness, Loeb's theorem, consistent universal statements are provable, Berry's paradox, incompleteness and Chaitin's theorem. Around Ramsey's theorem.

To make sense of what Gilles Deleuze understands by a mathematical concept requires unpacking what he considers to be the conceptualizable character of a mathematical theory. For Deleuze, the mathematical problems to which theories are solutions retain their relevance to the theories not only as the conditions that govern their development, but also insofar as they can contribute to determining the conceptualizable character of those theories. Deleuze presents two examples of mathematical problems that operate in this way, which he considers (...) to be characteristic of a more general theory of mathematical problems. By providing an account of the historical development of this more general theory, which he traces drawing upon the work of Weierstrass, Poincaré, Riemann, and Weyl, and of its significance to the work of Deleuze, an account of what a mathematical concept is for Deleuze will be developed. (shrink)

W. V. Quine was one of the most influential figures of twentieth-century American analytic philosophy. Although he wrote predominantly in English, in Brazil in 1942 he gave a series of lectures on logic and its philosophy in Portuguese, subsequently published as the book O Sentido da Nova Lógica. The book has never before been fully translated into English, and this volume is the first to make its content accessible to Anglophone philosophers. Quine would go on to develop revolutionary ideas about (...) semantic holism and ontology, and this book provides a snapshot of his views on logic and language at a pivotal stage of his intellectual development. The volume also includes an essay on logic which Quine also published in Portuguese, together with an extensive historical-philosophical essay by Frederique Janssen-Lauret. The valuable and previously neglected works first translated in this volume will be essential for scholars of twentieth-century philosophy. (shrink)

W. V. Quine was one of the most influential figures of twentieth-century American analytic philosophy. Although he wrote predominantly in English, in Brazil in 1942 he gave a series of lectures on logic and its philosophy in Portuguese, subsequently published as the book O Sentido da Nova Lógica. The book has never before been fully translated into English, and this volume is the first to make its content accessible to Anglophone philosophers. Quine would go on to develop revolutionary ideas about (...) semantic holism and ontology, and this book provides a snapshot of his views on logic and language at a pivotal stage of his intellectual development. The volume also includes an essay on logic which Quine also published in Portuguese, together with an extensive historical-philosophical essay by Frederique Janssen-Lauret. The valuable and previously neglected works first translated in this volume will be essential for scholars of twentieth-century philosophy. (shrink)

As the digitization of information accelerates, the push to encode our surrounding numerically instead of linguistically increases. The role that language has traditionally played in the nomenclature of an integrative taxonomy is being replaced by the numeric identification of one or few quantitative characteristics. Nineteenth-century scientific systems of color identification divided, grouped, and named colors according to multiple characteristics. Now color identification relies on numeric values applied to spectrographic readings. This means of identification of color lacks the taxonomic rigor of (...) nineteenth century systems. Identifying color by numeric value instead of by grouping and naming them, strips color taxonomy of all but one quantitative aspect of a color. I use the case of color taxonomy to argue against a similar trend of numeric identification in the biological sciences. Unlike historically more integrative approaches to taxonomy in biology, genomic sequencing identifies one or few quantitative characteristics to encode an organism. If genomic sequencing becomes the primary means of identification in the biological sciences, just as in numeric systems of color identification, scientific taxonomy would suffer. Basing my analysis on theories of perception of division and on theories of language, I use the cases of color and species to argue for the advantages of an integrative taxonomic system of naming and categorizing over a method of identification, which encodes limited characteristics numerically. I hold that language is the most sophisticated tool for systematic taxonomy and that taxonomic nomenclature should be retained. (shrink)

In the past century the received view of definition in mathematics has been the stipulative conception, according to which a definition merely stipulates the meaning of a term in other terms which are supposed to be already well known. The stipulative conception has been so absolutely dominant and accepted as unproblematic that the nature of definition has not been much discussed, yet it is inadequate. This paper examines its shortcomings and proposes an alternative, the heuristic conception.

I argue that the construction of representation theorems is a powerful tool for creating novel objects and theories in mathematics, as the construction of a new representation introduces new pieces of information in a very specific way that enables a solution for a problem and a proof of a new theorem. In more detail I show how the work behind the proof of a representation theorem transforms a mathematical problem in a way that makes it tractable and introduces information into (...) it that it did not contain at the beginning of the process. (shrink)

Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this rea­ son, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the (...) concept of abstraction. In our work, I will try to explain the mathematical abstraction that Aristotle has developed to understand mathematical philosophy. (shrink)

Stewart Shapiro’s ante rem structuralism recognizes the structural or ‘algebraic’ aspects of mathematical practice while still offering a face-value semantics. Fictionalism, as a purely ‘algebraic’ approach, is held to be at a disadvantage, as compared with Shapiro’s structuralism, in not interpreting mathematics at face value. However, the face-value reading of mathematical singular terms has difficulty explaining how we can use such terms to pick out a unique referent in cases where the relevant mathematical structures admit non-trivial automorphisms. Shapiro offers a (...) solution to this difficulty, but his solution, I argue, evens the score between Shapiro’s structuralism and fictionalism. (shrink)

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. (shrink)

This paper attempts to be a contribution to the epistemological project of explaining complex conceptual structures departing from more basic ones. The central thesis of the paper is that there are what I call “functionally structured concepts”, these are non-harmonic concepts in Dummett’s sense that might be legitimized if there is a function that justifies the tie between the inferential connection the concept allows us to trace. Proving this requires enhancing the russellian existential analysis of definite descriptions to apply to (...) functions and using this in proving the legitimacy of such concepts. The utility of the proposal is shown for the case of thick ethical terms and an attempt is made to use it in explaining the development of natural numbers. This last move could allow us to go one step lower in explaining the genesis of natural numbers while maintaining the notion of abstract numbers as higher order entities. (shrink)

The object of this book is to present a radical novel conception of the ontological categories, their nature and epistemic importance. A conception that constitutes a challenge to the prevailing tenets, if not paradigms, of ontology today. The arguments and observations are given without addressing nor directly contesting the current theories on the subject. However, its author emphasises some of the main conclusions that entail from the new perspective, in particular regarding the role of philosophy among the sciences. Departing from (...) the novelty of considering distinctions to be the subject matter of thought and language -that is, of reference and meaning- it is observed that there are certain concepts, encompassed under the notion of “Being”, that are each “all” comprising categories. It is explained that these categories are conformed by certain ontological relations, which seem to stand for the structure of reality in-itself, and cannot be, in any manner, denied cognitive content nor objective existence in any possible world. Following this, it is argued that they constitute the primary premises of judgment and ultimate explanatory resources, and, thus, the fundaments of logic and mathematics. Moreover, language is shown to be structured according to them, and it is likewise explained that, as primary premises of all our judgments, they cannot be but determined a priori, standing for aspects of mind objective reality of a non-sensible nature, which are essential elements of cognition. It is shown, that it is they that enable to bridge the gap between the mind and the world, but set a limit to our possible knowledge of reality, which forces to presuppose the existence of higher or hyper-orders of reality. The importance of this work is, that from a naturalist stance, its observations and arguments constitute a strong case against established and well-rooted tenets in contemporary philosophy, while point to the need of focussing the field of the discipline to the study of the cognitive content of our innately determined a priori concepts. (shrink)

In this chapter, one considers finance at its very foundations, namely, at the place where assumptions are being made about the ways to measure the two key ingredients of finance: risk and return. It is well known that returns for a large class of assets display a number of stylized facts that cannot be squared with the traditional views of 1960s financial economics (normality and continuity assumptions, i.e. Brownian representation of market dynamics). Despite the empirical counterevidence, normality and continuity assumptions (...) were part and parcel of financial theory and practice, embedded in all financial practices and beliefs. Our aim is to build on this puzzle for extracting some clues revealing the use of one research strategy in academic community, model tinkering defined as a particular research habit. We choose to focus on one specific moment of the scientific controversies in academic finance: the ‘leptokurtic crisis’ opened by Mandelbrot in 1962. The profoundness of the crisis came from the angle of the Mandelbrot’s attack: not only he emphasized an empirical inadequacy of the Brownian representation, but also he argued for an inadequate grounding of this representation. We give some insights in this crisis and display the model tinkering strategies of the financial academic community in the 1970s and the 1980s. (shrink)

Conversational exculpature is a pragmatic process whereby information is subtracted from, rather than added to, what the speaker literally says. This pragmatic content subtraction explains why we can say “Rob is six feet tall” without implying that Rob is between 5'0.99" and 6'0.01" tall, and why we can say “Ellen has a hat like the one Sherlock Holmes always wears” without implying Holmes exists or has a hat. This article presents a simple formalism for understanding this pragmatic mechanism, specifying how, (...) in context, the result of such subtractions is determined. And it shows how the resulting theory of conversational exculpature accounts for a varied range of linguistic phenomena. A distinctive feature of the approach is the crucial role played by the question under discussion in determining the result of a given exculpature. (shrink)

Language exists because human subjects define themselves in the circumstance they are in. This is possible because they are able to know, not directly through their senses only, but adding something new to the construct they create in their conscience. The main thing they add to the construct created is categories, something invented or fabricated by the human subject at the moment of speaking.