Klein, Hilbert, and the Gottingen Mathematical Tradition

Osiris ◽  
1989 ◽  
Vol 5 ◽  
pp. 186-213 ◽  
Author(s):  
David E. Rowe
Keyword(s):  
Mathematical Tradition

Mathematical Knowledge and the Interplay of Practices

2015 ◽  
Author(s):  
José Ferreirós
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
Philosophy Of Mathematics ◽  
Advanced Mathematics ◽  
New Approach ◽  
Agent Based ◽  
Epistemology Of Mathematics ◽  
Actual Experience ◽  
Elementary Math ◽  
Mathematical Tradition

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, the book uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, the book shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. It argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. It demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, the book challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

Visual Information and Valid Reasoning

1996 ◽  
Author(s):  
Jon Barwise ◽  
John Etchemendy
Keyword(s):  
Visual Information ◽  
Formal Proof ◽  
Theory And Practice ◽  
Visual Images ◽  
Standard View ◽  
Mathematical Problems ◽  
Mathematical Tradition ◽  
Cognitive Activities ◽  
The Relationship ◽  
Mathematical Discovery

Psychologists have long been interested in the relationship between visualization and the mechanisms of human reasoning. Mathematicians have been aware of the value of diagrams and other visual tools both for teaching and as heuristics for mathematical discovery. As the chapters in this volume show, such tools are gaining even greater value, thanks in large part to the graphical potential of modern computers. But despite the obvious importance of visual images in human cognitive activities, visual representation remains a second-class citizen in both the theory and practice of mathematics. In particular, we are all taught to look askance at proofs that make crucial use of diagrams, graphs, or other nonlinguistic forms of representation, and we pass on this disdain to our students. In this chapter, we claim that visual forms of representation can be important, not just as heuristic and pedagogic tools, but as legitimate elements of mathematical proofs. As logicians, we recognize that this is a heretical claim, running counter to centuries of logical and mathematical tradition. This tradition finds its roots in the use of diagrams in geometry. The modern attitude is that diagrams are at best a heuristic in aid of finding a real, formal proof of a theorem of geometry, and at worst a breeding ground for fallacious inferences. For example, in a recent article, the logician Neil Tennant endorses this standard view: . . . [The diagram] is only an heuristic to prompt certain trains of inference; . . . it is dispensable as a proof-theoretic device; indeed, . . . it has no proper place in the proof as such. For the proof is a syntactic object consisting only of sentences arranged in a finite and inspectable array (Tennant [1984]). . . . It is this dogma that we want to challenge. We are by no means the first to question, directly or indirectly, the logocentricity of mathematics arid logic. The mathematicians Euler and Venn are well known for their development of diagrammatic tools for solving mathematical problems, and the logician C. S. Peirce developed an extensive diagrammatic calculus, which he intended as a general reasoning tool.

A survey of the mathematical tradition of a subcontinent

Metascience ◽  
2011 ◽  
Vol 21 (2) ◽  
pp. 309-311
Author(s):  
Toke Knudsen
Keyword(s):  
Mathematical Tradition

scholarly journals Kant's Mathematical World

2021 ◽  
Author(s):  
Daniel Sutherland
Keyword(s):  
Nineteenth Century ◽  
Philosophy Of Mathematics ◽  
Mathematical Cognition ◽  
Critical Philosophy ◽  
Cognitive Capacities ◽  
Mathematical World ◽  
The World ◽  
Mathematical Tradition ◽  
Euclid’S Elements ◽  
Way Of Thinking

Kant's Mathematical World aims to transform our understanding of Kant's philosophy of mathematics and his account of the mathematical character of the world. Daniel Sutherland reconstructs Kant's project of explaining both mathematical cognition and our cognition of the world in terms of our most basic cognitive capacities. He situates Kant in a long mathematical tradition with roots in Euclid's Elements, and thereby recovers the very different way of thinking about mathematics which existed prior to its 'arithmetization' in the nineteenth century. He shows that Kant thought of mathematics as a science of magnitudes and their measurement, and all objects of experience as extensive magnitudes whose real properties have intensive magnitudes, thus tying mathematics directly to the world. His book will appeal to anyone interested in Kant's critical philosophy -- either his account of the world of experience, or his philosophy of mathematics, or how the two inform each other.

Italian and Dutch Developments of Science

2020 ◽  
Vol 9 (2) ◽  
pp. 71-86
Author(s):  
Andrea Bergamini
Keyword(s):  
Early Modernity ◽  
Italian Culture ◽  
Mathematical Tradition

This article illustrates how during early modernity Italian and Dutch cultures and particularly artistic traditions contributed differently to both the theoretical and practical developments of science. To achieve this goal, it will firstly compare the two forms of detextualization of space operated by Italian artists and by Dutch artists. Finally, it will indicate how each detextualization allowed for the development within the science of the mathematical tradition by the Italian Culture and the experimental tradition by the Dutch culture.

Situation-Theoretic Account of Valid Reasoning with Venn Diagrams

1996 ◽  
Author(s):  
Sun-Joo Shin
Keyword(s):  
Semantic Analysis ◽  
Formal System ◽  
Venn Diagram ◽  
Formal Proofs ◽  
Representation System ◽  
Venn Diagrams ◽  
Linguistic Representation ◽  
Deductive Systems ◽  
Representation Systems ◽  
Mathematical Tradition

Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. Since elementary school we have been taught how to draw Venn diagrams for a problem, how to manipulate them, how to interpret the resulting diagrams, and so on. However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs. This is just a reflection of a general prejudice against visualization which resides in the mathematical tradition. With this bias for linguistic representation systems, little attempt has been made to analyze any nonlinguistic representation system despite the fact that many forms of visualization are used to help our reasoning. The purpose of this chapter is to give a semantic analysis for a visual representation system—the Venn diagram representation system. We were mainly motivated to undertake this project by the discussion of multiple forms of representation presented in Chapter I More specifically, we will clarify the following passage in that chapter, by presenting Venn diagrams as a formal system of representations equipped with its own syntax and semantics:. . . As the preceding demonstration illustrated, Venn diagrams provide us with a formalism that consists of a standardized system of representations, together with rules of manipulating them. . . . We think it should be possible to give an informationtheoretic analysis of this system, . . . . In the following, the formal system of Venn diagrams is named VENN. The analysis of VENN will lead to interesting issues which have their ana logues in other deductive systems. An interesting point is that VENN, whose primitive objects are diagrammatic, not linguistic, casts these issues in a different light from linguistic representation systems. Accordingly, this VENN system helps us to realize what we take for granted in other more familiar deductive systems. Through comparison with symbolic logic, we hope the presentation of VENN contributes some support to the idea that valid reasoning should be thought of in terms of manipulation of information, not just in terms of manipulation of linguistic symbols.

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