scholarly journals Conceptions of Infinity and Set in Lorenzen’s Operationist System

2021 ◽  
pp. 23-46
Author(s):  
Carolin Antos
Keyword(s):  
Constructive Mathematics ◽  
Dialogical Logic ◽  
Actual Infinite ◽  
Potential Infinity ◽  
And Mathematics

AbstractIn the late 1940s and early 1950s, Lorenzen developed his operative logic and mathematics, a form of constructive mathematics. Nowadays this is mostly seen as a precursor of the better-known dialogical logic (Notable exceptions are the works of Schroeder-Heister 2008; Coquand and Neuwirth 2017; Kahle and Oitavem 2020.), and one might assume that the same philosophical motivations were present in both works. However, we want to show that this is not everywhere the case. In particular, we claim that Lorenzen’s well-known rejection of the actual infinite, as stated in Lorenzen (1957), was not a major motivation for operative logic and mathematics. Rather, we argue that a shift happened in Lorenzen’s treatment of the infinite from the early to the late 1950s. His early motivation for the development of operationism is concerned with a critique of the Cantorian notion of set and with related questions about the notions of countability and uncountability; it is only later that his motivation switches to focusing on the concept of infinity and the debate about actual and potential infinity.

3. Historical views of infinity

2017 ◽  
pp. 32-53
Author(s):  
Ian Stewart
Keyword(s):  
Immanuel Kant ◽  
Physical Reality ◽  
Human Mind ◽  
Space Time ◽  
Time And Motion ◽  
Potential Infinity ◽  
Zeno's Paradoxes ◽  
And Mathematics

‘Historical views of infinity’ focuses on historical attitudes to infinity in philosophy, religion, and mathematics, including Zeno’s famous paradoxes. Infinity is not a thing, but a concept, related to the default workings of the human mind. Zeno’s paradoxes appear to be about physical reality, but they mainly address how we think about space, time, and motion. A central (but possibly dated) contribution was Aristotle’s distinction between actual and potential infinity. Theologians, from Origen to Aquinas, sharpened the debate, and philosophers such as Immanuel Kant took up the challenge. Mathematicians made radical advances, often against resistance from philosophers.

scholarly journals De la determinación del infinito a la inaccesibilidad en los cardinales transfinitos

1994 ◽  
Vol 26 (78) ◽  
pp. 27-71
Author(s):  
Carlos Álvarez J.
Keyword(s):  
Point Of View ◽  
Inaccessible Cardinal ◽  
General View ◽  
Very Old ◽  
Cardinal Numbers ◽  
Actual Infinite ◽  
Potential Infinity

In this paper I deal with two problems in mathematical philosophy: the (very old) question about the nature of infinity, and the possible answer to this question after Cantor’s theory of transfinite numbers. Cantor was the first to consider that his transfinite numbers theory allows to speak, within mathematics, of an actual infinite and allows to leave behind the Aristotelian statement that infinity exists only as potential infinity. In the first part of this paper I discuss Cantorian theory of transfinite numbers and his particular point of view about this matter. But the development of the theory of transfinite numbers, specially the theory of transfinite cardinal numbers, has reached with the inaccessible cardinal numbers a new dilemma which makes us think that Aristotelian characterization of the infinity as potential is again a possible answer. The second part gives a general view of this development and of the theory of the inaccessible cardinal numbers in order to make clear my point of view concerning Aristotelian potential infinity.

scholarly journals The Immanence of Truths and the Absolutely Infinite in Spinoza, Cantor, and Badiou

2020 ◽  
Vol 41 (2) ◽  
Author(s):  
Jana Ndiaye Berankova
Keyword(s):  
Mathematical Theory ◽  
Deductive Reasoning ◽  
Large Cardinals ◽  
Georg Cantor ◽  
Mathematical Concepts ◽  
Potential Infinity ◽  
Mathematical Ontology ◽  
The Absolute ◽  
And Mathematics ◽  
The Relationship

The following article compares the notion of the absolute in the work of Georg Cantor and in Alain Badiou’s third volume of Being and Event: The Immanence of Truths and proposes an interpretation of mathematical concepts used in the book. By describing the absolute as a universe or a place in line with the mathematical theory of large cardinals, Badiou avoided some of the paradoxes related to Cantor’s notion of the “absolutely infinite” or the set of all that is thinkable in mathematics W: namely the idea that W would be a potential infinity. The article provides an elucidation of the putative criticism of the statement “mathematics is ontology” which Badiou presented at the conference Thinking the Infinite in Prague. It emphasizes the role that philosophical decision plays in the construction of Badiou’s system of mathematical ontology and portrays the relationship between philosophy and mathematics on the basis of an inductive not deductive reasoning.

Intuitionism vs. Classicism

2015 ◽  
Author(s):  
Nick Haverkamp
Keyword(s):  
Twentieth Century ◽  
Classical Logic ◽  
Constructive Mathematics ◽  
Mathematical Methods ◽  
Common Assumption ◽  
Michael Dummett ◽  
Mathematical Expressions ◽  
The Common ◽  
And Mathematics ◽  
Dutch Mathematician

In the early twentieth century, the Dutch mathematician L.E.J. Brouwer launched a powerful attack on the prevailing mathematical methods and theories. He developed a new kind of constructive mathematics, called intuitionism, which seems to allow for a rigorous refutation of widely accepted mathematical assumptions including fundamental principles of classical logic. Following an intense mathematical debate esp. in the 1920s, Brouwer's revolutionary criticism became a central philosophical concern in the 1970s, when Michael Dummett tried to substantiate it with meaning-theoretic considerations. Since that time, the debate between intuitionists and classicists has remained a central philosophical dispute with far-reaching implications for mathematics, logic, epistemology, and semantics. In this book, Nick Haverkamp presents a detailed analysis of the intuitionistic criticism of classical logic and mathematics. The common assumption that intuitionism and classicism are equally legitimate enterprises corresponding to different understandings of logical or mathematical expressions is investigated and rejected, and the major intuitionistic arguments against classical logic are scrutinised and repudiated. Haverkamp argues that the disagreement between intuitionism and classicism is a fundamental logical and mathematical dispute which cannot be resolved by means of meta-mathematical, epistemological, or semantic considerations.

Models and mathematics. Tools of the mathematical biologist

JAMA ◽  
1965 ◽  
Vol 194 (3) ◽  
pp. 269-272
Author(s):  
J. T. Apter
Keyword(s):  
And Mathematics

The Role of Test Anticipation in the Link Between Performance-Approach Goals and Academic Achievement

2016 ◽  
Vol 75 (3) ◽  
pp. 123-132 ◽  
Author(s):  
Marie Crouzevialle ◽  
Fabrizio Butera
Keyword(s):  
Learning Strategies ◽  
Test Performance ◽  
Longitudinal Design ◽  
Study Strategies ◽  
High Achievers ◽  
Course Content ◽  
And Mathematics ◽  
Set Up ◽  
Approach Goals ◽  
Goal Endorsement

Abstract. Performance-approach goals (i.e., the desire to outperform others) have been found to be positive predictors of test performance, but research has also revealed that they predict surface learning strategies. The present research investigates whether the high academic performance of students who strongly adopt performance-approach goals stems from test anticipation and preparation, which most educational settings render possible since examinations are often scheduled in advance. We set up a longitudinal design for an experiment conducted in high-school classrooms within the context of two science, technology, engineering, and mathematics (STEM) disciplines, namely, physics and chemistry. First, we measured performance-approach goals. Then we asked students to take a test that had either been announced a week in advance (enabling strategic preparation) or not. The expected interaction between performance-approach goal endorsement and test anticipation was moderated by the students’ initial level: The interaction appeared only among low achievers for whom the pursuit of performance-approach goals predicted greater performance – but only when the test had been scheduled. Conversely, high achievers appeared to have adopted a regular and steady process of course content learning whatever their normative goal endorsement. This suggests that normative strivings differentially influence the study strategies of low and high achievers.

Working Memory and Number Sense as Predictors of Mathematical (Dis-)Ability

2015 ◽  
Vol 223 (2) ◽  
pp. 102-109 ◽  
Author(s):  
Evelyn H. Kroesbergen ◽  
Marloes van Dijk
Keyword(s):  
Working Memory ◽  
Learning Disabilities ◽  
Spatial Working Memory ◽  
Number Sense ◽  
Mathematical Learning ◽  
Visual Spatial ◽  
Mathematical Learning Disabilities ◽  
Visual Spatial Working Memory ◽  
And Mathematics

Recent research has pointed to two possible causes of mathematical (dis-)ability: working memory and number sense, although only few studies have compared the relations between working memory and mathematics and between number sense and mathematics. In this study, both constructs were studied in relation to mathematics in general, and to mathematical learning disabilities (MLD) in particular. The sample consisted of 154 children aged between 6 and 10 years, including 26 children with MLD. Children performing low on either number sense or visual-spatial working memory scored lower on math tests than children without such a weakness. Children with a double weakness scored the lowest. These results confirm the important role of both visual-spatial working memory and number sense in mathematical development.

Work Engagement and Research Output Among Female and Male Scientists

2016 ◽  
Vol 15 (2) ◽  
pp. 55-65 ◽  
Author(s):  
Lonneke Dubbelt ◽  
Sonja Rispens ◽  
Evangelia Demerouti
Keyword(s):  
Work Engagement ◽  
Research Output ◽  
Research Performance ◽  
Daily Work ◽  
Time Control ◽  
Science Technology ◽  
And Performance ◽  
And Mathematics ◽  
Number Of Publications ◽  
The Relationship

Abstract. Women have a minority position within science, technology, engineering, and mathematics and, consequently, are likely to face more adversities at work. This diary study takes a look at a facilitating factor for women’s research performance within academia: daily work engagement. We examined the moderating effect of gender on the relationship between two behaviors (i.e., daily networking and time control) and daily work engagement, as well as its effect on the relationship between daily work engagement and performance measures (i.e., number of publications). Results suggest that daily networking and time control cultivate men’s work engagement, but daily work engagement is beneficial for the number of publications of women. The findings highlight the importance of work engagement in facilitating the performance of women in minority positions.

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