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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:
Tarski’s Intuitive Notion of Set
"... Abstract. Tarski did research on set theory and also used set theory in many of his emblematic writings. Yet his notion of set from the philosophical viewpoint was almost unknown. By studying mostly the posthumously published evidence, his still unpublished materials, and the testimonies of some of ..."
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Abstract. Tarski did research on set theory and also used set theory in many of his emblematic writings. Yet his notion of set from the philosophical viewpoint was almost unknown. By studying mostly the posthumously published evidence, his still unpublished materials, and the testimonies of some of his collaborators, I try to offer here a first, global picture of that intuitive notion, together with a philosophical interpretation of it. This is made by using several notions of universal languages as framework, and by taking into consideration the evolution of Tarski’s thoughts about set theory and its relationship with logic and mathematics. As a result, his difficulties to reconcile nominalism and methodological Platonism are precisely located, described and much better understood. “I represent this very rude kind of antiPlatonism, one thing which I could describe as materialism, or nominalism with some materialistic taint, and it is very difficult for a man to live his whole life with this philosophical attitude, especially if he is a mathematician, especially if for some reasons he has a hobby which is called set theory, and worse –very difficult” (Tarski, Chicago, 1965) Tarski made important contributions to set theory, especially in the first years of his long and highly productive career. Also, it is usually accepted that set theory was the main instrument used by Tarski in his most significant contributions which had philosophical implications and presuppositions. In this connection we may mention the four definitions which are usually cited as supplying some sort of “conceptual analysis”, both methodologically (the first one) and from the point of view of the results obtained (the rest): (i) definable sets of real numbers; (ii) truth; (iii) logical consequence and (iv) logical notions. So we could reasonably conclude that for Tarski set theory was reliable as a working instrument, then presumably as a conceptual ground. As we shall see, there are some signs that the reason for this preference might have been its simple ontological structure as a theory, at least excluding the upper levels, the highest infinite. Nevertheless, very little was known about Tarski’s conception of set theory from the philosophical viewpoint, apart from some comments he made to his closest friends and collaborators, and the conjectures which could perhaps be