Computability, Notation, and de re Knowledge of Numbers

Stewart Shapiro,Richard Samuels,Eric Snyder

doi:10.3390/philosophies7010020

Stewart Shapiro, Richard Samuels + Show 1 more

Open Access

https://doi.org/10.3390/philosophies7010020

Copy DOI

Journal: Philosophies	Publication Date: Feb 18, 2022
Citations: 1	License type: CC BY 4.0

Affiliation: The Ohio State University, Ashoka University

Abstract

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of which number is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of notation. The purpose of this article is to explore the relationship between the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects.

Highlights

It is generally recognized that context sensitivity is a fact of linguistic life, and a large number of tools have been developed to accommodate this feature of natural languages
Extreme Platonism aside, we have de re knowledge or belief about particular numbers only after we manage to represent them in language or thought
One option is to concede that the notion of computability, as applied to number-theoretic functions, is relative to a notation, a kind of context-sensitivity

Summary

Introduction

Our problem here is not exactly that, but rather how we manage to have de re knowledge of particular numbers, such as twelve, sixty-four, and four. The other vexed matter concerns de re propositional attitudes generally. O. Quine [5,6], among others, pointed out that, in general, such matters are highly context-sensitive, and, in particular, interest-relative. It is generally recognized that context sensitivity is a fact of linguistic life, and a large number of tools have been developed to accommodate this feature of natural languages. De re propositional attitude reports concern ways that ordinary objects and people are represented, in language or in thought (if those are different). Extreme (epistemic) Platonism aside, we have de re knowledge or belief about particular numbers only after we manage to represent them in language or thought.

Notation

De re Attitudes toward Numbers

Idealization

Connecting the Dots

Beyond the Natural Numbers