Mathematical Logic: Proof Theory, Constructive Mathematics

2021 ◽  
Vol 17 (4) ◽  
pp. 1693-1757
Author(s):  
Samuel Buss ◽  
Rosalie Iemhoff ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen
2008 ◽  
pp. 907-952
Author(s):  
Samuel Buss ◽  
Helmut Schwichtenberg ◽  
Ulrich Kohlenbach

2011 ◽  
Vol 8 (4) ◽  
pp. 2963-3002 ◽  
Author(s):  
Samuel Buss ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen

2005 ◽  
pp. 779-813
Author(s):  
Helmut Schwichtenberg ◽  
Vladimir Keilis-Borok ◽  
Samuel Buss

2018 ◽  
Vol 14 (4) ◽  
pp. 3121-3183
Author(s):  
Samuel Buss ◽  
Rosalie Iemhoff ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen

2014 ◽  
Vol 11 (4) ◽  
pp. 2933-2986 ◽  
Author(s):  
Samuel Buss ◽  
Ulrich Kohlenbach ◽  
Michael Rathjen

1999 ◽  
Vol 5 (1) ◽  
pp. 1-44 ◽  
Author(s):  
Wilfried Sieg

AbstractHilbert's finitist program was not created at the beginning of the twenties solely to counteract Brouwer's intuitionism, but rather emerged out of broad philosophical reflections on the foundations of mathematics and out of detailed logical work; that is evident from notes of lecture courses that were given by Hilbert and prepared in collaboration with Bernays during the period from 1917 to 1922. These notes reveal a dialectic progression from a critical logicism through a radical constructivism toward finitism; the progression has to be seen against the background of the stunning presentation of mathematical logic in the lectures given during the winter term 1917/18. In this paper, I sketch the connection of Hilbert's considerations to issues in the foundations of mathematics during the second half of the 19th century, describe the work that laid the basis of modern mathematical logic, and analyze the first steps in the new subject of proof theory. A revision of the standard view of Hilbert's and Bernays's contributions to the foundational discussion in our century has long been overdue. It is almost scandalous that their carefully worked out notes have not been used yet to understand more accurately the evolution of modern logic in general and of Hilbert's Program in particular. One conclusion will be obvious: the dogmatic formalist Hilbert is a figment of historical (de)construction! Indeed, the study and analysis of these lectures reveal a depth of mathematical-logical achievement and of philosophical reflection that is remarkable. In the course of my presentation many questions are raised and many more can be explored; thus, I hope this paper will stimulate interest for new historical and systematic work.


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