Remarks on the notion of standard non-isomorphic natural number series

Isles, David J.

doi:10.1007/bfb0090731

Cited by 10 publications

(3 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Complexity theory seeks to provide a mathematical framework for making such contrasts precise. 24 See, e.g., (Isles, 1981) for a partial reconstruction. These include not only the application of deductive rules such as modus ponens, but also the treatment of substitution, unification, and deciding the identity of terms which must be carried out algorithmically in a computational implementation of a proof system for first-order logic.…”

Section: Ind(lmentioning

confidence: 99%

Strict Finitism, Feasibility, and the Sorites

Dean

2018

The Review of Symbolic Logic

View full text Add to dashboard Cite

This paper bears on three di↵erent topics: observational predicates and phenomenal properties; vagueness; and strict finitism as a philosophy of mathematics. Of these three, only the last requires any preliminary comment. Dummett (1975), p. 302 Why are mathematicians so convinced that exponentiation is total? Because they believe in the existence of abstract objects called numbers. What is a number? Originally, sequences of tally marks were used to count things. Then positional notation -the most powerful achievement of mathematics -was invented.Nelson (1986), p. 173Abstract. This paper bears on four topics: observational predicates and phenomenal properties, vagueness, strict finitism as a philosophy of mathematics, and the analysis of feasible computability. It is argued that reactions to strict finitism point towards a semantics for vague predicates in the form of nonstandard models of weak arithmetical theories of the sort originally introduced to characterize the notion of feasibility as understood in computational complexity theory. The approach described eschews the use of non-classical logic and related devices like degrees of truth or supervaluation. Like epistemic approaches to vagueness, it may thus be smoothly integrated with the use of classical model theory as widely employed in natural language semantics. But unlike epistemicism, the described approach fails to imply either the existence of sharp boundaries or the failure of tolerance for soritical predicates. Applications of measurement theory (in the sense of Krantz et al. 1971) to vagueness in the nonstandard setting are also explored.

show abstract

Section: Ind(lmentioning

confidence: 99%

Strict Finitism, Feasibility, and the Sorites

Dean

2018

The Review of Symbolic Logic

View full text Add to dashboard Cite

show abstract

“…Later on, the works of van Dantzig and Esenin-Vol'pin complemented this interpretation and explicitly put forward a position that has provided much of the basis for further developments in this area. These include the work of van Bendegem (1985); the formalization and analysis of parts of Esenin-Vol'pin's work carried out by Geiser (1974) and Isles (1981), and the papers by Isles (1992Isles ( , 1997. A formal analysis of feasibility is described in (Parikh 1971), and by different means in (Simon 1977).…”

mentioning

confidence: 97%

Strict finitism and feasibility

Cardone

1995

Lecture Notes in Computer Science

View full text Add to dashboard Cite

Abstract. We examine the relation between predicative recurrence and strictly finitistic tenets in the philosophy of mathematics, primarily by focusing on the r61e of numerical notations in computing. After an overview of Wittgenstein's ideas on the "surveyability" of notations, we analyze a subtle form of circularity in the usual justification of the primitive recursive definition of exponentiation (Isles 1992), and suggest connections with recent works on predicative recurrence (Leivant 1993b, Bellantoni & Cook 1993.A long-standing thread in the foundations of mathematics questions the ontology of numeric terms and the absoluteness of the informal notion of finiteness. This paper is a preliminary exploration of some of the ideas (and puzzles) underlying these "strictly finitistic" approaches, especially in the light of recent works that relate predicative formalisms to computational complexity (Nelson 1986, Buss 1986, Leivant 1993a. One would expect investigations in these areas to lead to a unified perspective, thereby contributing to an assessment of the Karp-Cook Thesis on identifying feasible computability with poly-time (see (Davis 1982); this identification goes back to Edmonds 1965).Large finite collections, and the natural numbers that count them, are the main source of perplexity for strict finitists: their acceptance makes the ontological status of constructions requiring an arbitrarily large but finite number of steps indistinguishable from that of the objects of platonistic mathematics. We shall * Address: Dipartimento di Scienze dell'Informazione, via Comelico 39/41, 1-20135 Milano, Italy. E-mail: felice~imiucca, csi.unimi.it. This is an expanded version of my talk for the Logic and Computational Complexity Meeting, and is part of a larger project analyzing the confluence of mathematical results and philosophical arguments i,i the analysis of feasible computations that will be described in a future joint paper with Daniel Leivant, to whom I am greatly indebted for stimulating my interest in this field and for prompting me to write this preliminary account. It will be apparent from the text that his views, and also those of David Isles, have been quite influential on the present paper (of course they are not at all responsible for any of my mistakes). I am grateful also to Stephen Bellantoni, Paolo Boldi, Gabriele Lolli, Diego Marconi, Piergiorgio Odifreddi and Nicoletta Sabadini for criticisms and suggestions, and to Roberta Mari for playing both as Proponent and Opponent in the design of the games described in the last section. Finally, I would like to thank my friend, Miss Claudia Bonino, for betting long ago that this paper would eventually have been written. The research was supported by CNR Cooperation Project "Linguaggi Applicativi e Dimostrazioni Costruttive" and MURST 40%.

show abstract

“…Or in the case of DavidIsles (1981Isles ( , 2004) that arithmetical functions such as multiplication or exponentiation do not obviously lead to places on the list of numbers we can count to by 1's.…”

mentioning

confidence: 99%

The Argument of Mathematics

Aberdein¹,

Dove²

2013

View full text Add to dashboard Cite

Remarks on the notion of standard non-isomorphic natural number series

Cited by 10 publications

References 2 publications

Strict Finitism, Feasibility, and the Sorites

Strict Finitism, Feasibility, and the Sorites

Strict finitism and feasibility

The Argument of Mathematics

Contact Info

Product

Resources

About