Kurt Gödels onvolledigheidsstellingen en de grenzen van de kennis

2021 ◽  
Vol 113 (1) ◽  
pp. 157-182
Author(s):  
Jean Paul Van Bendegem
Keyword(s):  
Mathematical Knowledge ◽  
Intermediate Level ◽  
Human Knowledge ◽  
Alan Turing ◽  
Limits Of Knowledge ◽  
Incompleteness Theorems ◽  
Philosophical Importance ◽  
Gödel’S Incompleteness Theorems

Abstract Kurt Gödel’s incompleteness theorems and the limits of knowledgeIn this paper a presentation is given of Kurt Gödel’s pathbreaking results on the incompleteness of formal arithmetic. Some biographical details are provided but the main focus is on the analysis of the theorems themselves. An intermediate level between informal and formal has been sought that allows the reader to get a sufficient taste of the technicalities involved and not lose sight of the philosophical importance of the results. Connections are established with the work of Alan Turing and Hao Wang to show the present-day relevance of Gödel’s research and how it relates to the limitations of human knowledge, mathematical knowledge in particular.

scholarly journals Experience and Mathematical Knowledge

2017 ◽  
Vol 21 (2) ◽  
pp. 209-222
Author(s):  
Rodolfo Gaeta
Keyword(s):  
Mathematical Knowledge ◽  
A Priori ◽  
The Other ◽  
Human Knowledge ◽  
Human Beings ◽  
Common View ◽  
Abstract Entities ◽  
Selection Experiments ◽  
Main Tenet ◽  
Previous Learning

According to a very common view, the main tenet of empiricism is the conviction that all human knowledge derives from sensory experience. But classic philosophers representing empiricism hold that mathematical knowledge is a priori. Mill intended to demonstrate that the laws of arithmetic and geometry have inductive origins. But Frege and others authors showed that Mill’s arguments were wrong. Benacerraf held that, since mathematical objects are abstract entities, they could not have any causal relationship with human beings, so they cannot be known by us. On the other hand, biology and psychology show that in animals and human creatures we can find innate behaviours, in accordance with the theory on natural selection. Experiments performed by Wynn and by other psychologists strongly support that very young babies can determine the results of simple arithmetical operations without any previous learning. We conclude that there are convincing reasons to accept the rationalist thesis about the a priori character of mathematical knowledge.

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