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Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
Anxiety and Abstraction in NineteenthCentury Mathematics
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For guidance on citations see FAQs. c ○ [not recorded] Version: [not recorded] Link(s) to article on publisher’s website:
DOI: 10.1017/S0269889704000043 Printed in the United Kingdom Anxiety and Abstraction in NineteenthCentury
"... The first part of this paper surveys the current literature in the history of nineteenthcentury mathematics in order to show that the question “Did the increasing abstraction of mathematics lead to a sense of anxiety? ” is a new and valid question. I argue that the mathematics of the nineteenth cen ..."
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The first part of this paper surveys the current literature in the history of nineteenthcentury mathematics in order to show that the question “Did the increasing abstraction of mathematics lead to a sense of anxiety? ” is a new and valid question. I argue that the mathematics of the nineteenth century is marked by a growing appreciation of error leading to a note of anxiety, hesitant at first but persistent by 1900. This mounting disquiet about so many aspects of mathematics after 1850 is seldom discussed. The second part explores the issue of anxiety in mathematical life through an interesting account of an address made by a mathematician in 1911, Oscar Perron. The third and final part ventures some conclusions about the value of anxiety as a question for historians of mathematics to pursue. 1. A Brief Survey of the Current Literature A “standard model: ” the nineteenth century as a century of progress Historians of mathematics like to portray the growth of mathematics in the nineteenth century as a success story. They start with the reforms of French education begun during the French Revolution. They note with pleasure such themes as the rigorization of the calculus, 1 the rediscovery of projective geometry, 2 the discovery of NonEuclidean