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Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Choice Sequences: a Retrospect
, 1996
"... Introduction The topic of this talk will be the lasting interest of L.E.J. Brouwer's notion of choice sequence for the philosophy of mathematics. In the past here has been done a good deal of work on choice sequences, but in the last decade the subject is a bit out of fashion, for several reas ..."
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Introduction The topic of this talk will be the lasting interest of L.E.J. Brouwer's notion of choice sequence for the philosophy of mathematics. In the past here has been done a good deal of work on choice sequences, but in the last decade the subject is a bit out of fashion, for several reasons, which I shall not go into here. In this retrospective I want to take a look with you at a special aspect of choice sequences, namely their interest as an important "casestudy" in the philosophy of mathematics. How does mathematics arrive at its concepts, and discover the principles holding for those concepts? This is a typically philosophical question, more easily posed than answered. A procedure which certainly has played a role and still plays a role might be described as informally rigorous analysis of a concept That is to say,  given an informally described, but intuitively clear concept,  one analyzes the concept as carefully as possibl
Concepts and Axioms
, 1998
"... The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic lo ..."
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The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...