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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
The Undecidability of Propositional Adaptive Logic ∗
, 2005
"... p r e p r i n t s i n a n a l y t i c p h i l o s o p h y ..."
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p r e p r i n t s i n a n a l y t i c p h i l o s o p h y
Necessity in mathematics
"... Mathematical writing commonly uses a good deal of modal language. This needs an explanation, because the mathematical assumptions and arguments themselves normally have no modal content at all. We review the modal expressions in the first hundred pages of a wellknown algebra textbook, and find two ..."
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Mathematical writing commonly uses a good deal of modal language. This needs an explanation, because the mathematical assumptions and arguments themselves normally have no modal content at all. We review the modal expressions in the first hundred pages of a wellknown algebra textbook, and find two uses for the modal language there: (a) metaphors of human powers, used for colouring that helps the readability; (b) formatting expressions which highlight the structure of the reasoning. We note some ways in which historians and philosophers of mathematics might have been misled through taking these modal expressions to be part of the mathematical content. Facts A and B below seem at first sight to be inconsistent with each other. So we have a paradox. FACT A: Mathematics contains no modal notions. For example one sufficient condition for the correctness of a mathematical argument is that it should be formalisable as a proof in ZermeloFraenkel set theory. ZermeloFraenkel set theory has just two primitive notions, ‘set’ and ‘is a member of’. Neither of these notions is modal. Of course mathematics is full of necessary truths, for example this theorem of analysis: