Platonism and Mathematical Intuition in Kurt Gödel's Thought

1995 ◽  
Vol 1 (1) ◽  
pp. 44-74 ◽  
Author(s):  
Charles Parsons
Keyword(s):  
Set Theory ◽  
Real World ◽  
Mathematical Knowledge ◽  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Mathematical Objects ◽  
The Real ◽  
Robust Realism ◽  
Mathematical Intuition

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.On this Gödel commented:Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish.

Wittgenstein on Mathematical Proof

1990 ◽  
Vol 28 ◽  
pp. 79-99 ◽  
Author(s):  
Crispin Wright
Keyword(s):  
Mathematical Proof ◽  
Philosophy Of Mathematics ◽  
Short Paper ◽  
Foundations Of Mathematics ◽  
The Real ◽  
Philosophical Investigations

To be asked to provide a short paper on Wittgenstein's views on mathematical proof is to be given a tall order (especially if little or no familiarity either with mathematics or with Wittgenstein's philosophy is to be presupposed!). Close to one half of Wittgenstein's writings after 1929 concerned mathematics, and the roots of his discussions, which contain a bewildering variety of underdeveloped and sometimes conflicting suggestions, go deep to some of the most basic and difficult ideas in his later philosophy. So my aims in what follows are forced to be modest. I shall sketch an intuitively attractive philosophy of mathematics and illustrate Wittgenstein's opposition to it. I shall explain why, contrary to what is often supposed, that opposition cannot be fully satisfactorily explained by tracing it back to the discussions of following a rule in the Philosophical Investigations and Remarks on the Foundations of Mathematics. Finally, I shall try to indicate very briefly something of the real motivation for Wittgenstein's more strikingly deflationary suggestions about mathematical proof, and canvass a reason why it may not in the end be possible to uphold them.

Projects: Developing Modelers and Modeling Opportunities: The Stepping Stones Project

1997 ◽  
Vol 90 (8) ◽  
pp. 686-688
Keyword(s):  
Mathematical Modeling ◽  
Mathematics Education ◽  
Real World ◽  
Mathematical Knowledge ◽  
School Mathematics ◽  
Stepping Stones ◽  
Evaluation Standards ◽  
The Real ◽  
World Modeling

Mathematical modeling is an emerging theme in mathematics education. In addition to giving students a knowledge of the applications of mathematics and a process for applying mathematics in the “real” world, modeling offers teachers an excellent vehicle for introducing and developing students' mathematical knowledge. For these reasons, modeling occupies a prominent place in the recommendations of the Curriculum and Evaluation Standards for School Mathematics (NCTM 1989).

Order types of ordinals in models of set theory

1976 ◽  
Vol 41 (2) ◽  
pp. 489-502 ◽  
Author(s):  
John E. Hutchinson
Keyword(s):  
Set Theory ◽  
Real World ◽  
The Real ◽  
Order Types ◽  
Models Of Set Theory

AbstractAn ordinal in a model of set theory is truly countable if its set of predecessors is countable in the real world. We classify the order types of the sets of truly countable ordinals. Models with indiscernibles and other related results are discussed.

scholarly journals Stanisław Leśniewski: Rethinking the Philosophy of Mathematics

2015 ◽  
Vol 23 (1) ◽  
pp. 125-138
Author(s):  
Rafal Urbaniak
Keyword(s):  
Nineteenth Century ◽  
Set Theory ◽  
Philosophy Of Mathematics ◽  
Foundations Of Mathematics ◽  
Mathematical Research ◽  
Main Candidate

Near the end of the nineteenth century, a part of mathematical research was focused on unification: the goal was to find ‘one sort of thing’ that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterprise hit a rough patch with Frege’s failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. In parallel to this development was the work of Stanislaw Leśniewski (1886–1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Leśniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology – a theory which, instead of sets and elements, spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Leśniewski’s project and to briefly comment on its contemporary relevance.

Hegel’s ‘Bad Infinity’ as a Logical Problem

2016 ◽  
Vol 37 (2) ◽  
pp. 258-280 ◽  
Author(s):  
Vojtěch Kolman
Keyword(s):  
Set Theory ◽  
Philosophy Of Mathematics ◽  
Main Line ◽  
Foundations Of Mathematics ◽  
Modern Logic ◽  
Infinitesimal Calculus ◽  
Logical Problem ◽  
Philosophical System

AbstractThe paper analyses the concept of ‘bad infinity’ in connection with Hegel’s critique of infinitesimal calculus and with the belittling of Hegel’s mathematical notions by the representatives of modern logic and the foundations of mathematics. The main line of argument draws on the observation that Hegel’s difference is only derivatively a mathematical one and is primarily of a broadly logico-epistemological nature. Because of this, the concept of bad infinity can be fruitfully utilized, by way of inversion, in an analysis of the conceptual shortcomings of the most prominent foundational attempts at dealing with infinite quanta, such as Cantor’s set theory and Hilbert’s axiomatism. As such, the paper is an attempt at reconstructing Hegel’s philosophy of mathematics and its role in his philosophical system and, more importantly, as a contribution to logic in the more general and radical sense of the word.

scholarly journals Signo e sentido interno na filosofia da matemática pré-crítica

DoisPontos ◽  
2005 ◽  
Vol 2 (2) ◽  
Author(s):  
Ernesto Maria Giusti
Keyword(s):  
Critical Theory ◽  
Mathematical Knowledge ◽  
Natural Theology ◽  
Philosophy Of Mathematics ◽  
Critical Philosophy ◽  
Mathematical Objects ◽  
Inner Sense ◽  
Internal Sense

Este artigo defende que, na Investigação sobre os princípios da teologia natural e da Moral, o conceito de “sentido interno” é central à reflexão kantiana sobre a matemática. Ele não deve ser entendido em sua acepção crítica, mas corresponde antes àquela derivada de Locke, pelo intermédio de Crusius, e se resume a uma reflexão mental sobre figuras e conceitos matemáticos. Ao incluir no sentido interno a dimensão simbólica do conhecimento matemático, Kant podia ainda ignorar um dos problemas centrais de sua reflexão madura sobre a matemática, a saber, como a inspeção de objetos matemáticos particulares permite formular provas e leis de caráter universal. Isto acentua diferenças, muitas vezes ignoradas, entre a filosofia da matemática crítica e pré-crítica. Sign and Internal Sense in the pre-critical philosophy of mathematics Abstract In this article it is argued that, in the Inquiry concerning the Principles of natural theology and of morals, the concept of “inner sense” is central to Kant´s theses about mathematics. It must not, however, be taken in its critical meaning, corresponding rather to a loosely lockean influx, trough Crusius, indicating only that mental reflexion about mathematical objects or concepts is involved in mathematical knowledge. The concept of inner sense incorporates, in particular, the symbolic aspect of mathematical knowledge, and, in so doing, allows Kant to ignore what would become one of the central problems of his mature tought, namely how, in mathematics, the inspection of particulars provides us with proofs and laws which are general in character. This indicates to some relevant differences between the critical and pre-critical theory of mathematics which have been regularly overlooked.

scholarly journals Generalized operations in soft set theory via relaxed conditions on parameters

Filomat ◽  
2017 ◽  
Vol 31 (19) ◽  
pp. 5955-5964
Author(s):  
Mujahid Abbas ◽  
Muhammad Ali ◽  
Salvador Romaguera
Keyword(s):  
Set Theory ◽  
Real World ◽  
Soft Set ◽  
Mathematical Tool ◽  
Soft Sets ◽  
Real World Data ◽  
World Data ◽  
The Real ◽  
New Concepts

Soft set theory has been evolved as a very useful mathematical tool to handle uncertainty and ambiguity associated with the real world data based structures. Parameters with certain conditions have been used to classify the data with the help of suitable functions. The aim of this paper is to relax conditions on parameters which lead us to propose some new concepts that consequently generalize existing comparable notions. We introduce the concepts of generalized finite soft equality (gf-soft equality), generalized finite soft union (gf-soft union) and generalized finite soft intersection (gf-soft intersection) of two soft sets. We prove results involving operations introduced herein. Moreover, with the help of examples, it is shown that these operations are proper generalizations of existing comparable operations.

scholarly journals Lower secondary students’ encounters with mathematical literacy

2021 ◽  
Author(s):  
Oda Heidi Bolstad
Keyword(s):  
Secondary Students ◽  
Real World ◽  
Mathematical Knowledge ◽  
School Context ◽  
Modern Society ◽  
Work Life ◽  
Mathematical Literacy ◽  
The Real

AbstractWorldwide, there has been an increased emphasis on enabling students to recognise the real-world significance of mathematics. Mathematical literacy is a notion used to define the competencies required to meet the demands of life in modern society. In this article, students’ encounters with mathematical literacy are investigated. The data comprises interviews with 22 students and observations of 16 mathematics lessons in three grade 9 classes in Norway. The analysis shows that students’ encounters with mathematical literacy concern specific mathematical topics and contexts from personal and work life. Students’ encounters with ML in school is characterised by an emphasis on developing mathematical knowledge within the school context.

The Problem of Conceptual Understanding

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Real Number ◽  
Set Theory ◽  
Conceptual Understanding ◽  
Philosophy Of Mathematics ◽  
Axiom System ◽  
The Real ◽  
Iterative Conception ◽  
Universe Of Sets ◽  
The Universe ◽  
The Web

This chapter considers one of the most intriguing questions that philosophy of mathematics in practice must, sooner or later, confront: how understanding of mathematics is obtained. In particular, it examines how issues of meaning and understanding in relation to practice and use relate to the question of the acceptability of “classical” or postulational mathematics, a question usually formulated in terms of consistency. The chapter begins with a discussion of the iterative conception of the universe of sets and its presuppositions, analyzing it from the standpoint of the web of practices. It then addresses the issue of conceptual understanding in mathematics, as exemplifid by the theory Zermelo–Fraenkel axiom system (ZFC). Finally, it looks at arguments based on the idea of the real-number continuum as a source of justification for the axioms of set theory.

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