Work
Science Without Numbers
Overview
Science Without Numbers: A Defence of Nominalism is a 1980 book on the philosophy of mathematics by Hartry Field. In the book, Field defends nominalism, the view that mathematical objects such as numbers do not exist. The book was written broadly in response to an argument for the existence of mathematical objects called the indispensability argument. According to this argument, belief in mathematical objects is justified because mathematics is indispensable to science. The main project of the book is producing technical reconstructions of science that remove reference to mathematical entities, hence showing that mathematics is not indispensable to science.
Modelled on Hilbert's axiomatization of geometry, which eschews numerical distances in favor of primitive geometrical relationships, Field demonstrates an approach to reformulate Newton's theory of gravity without the need to reference numbers. According to Field's philosophical program, mathematics is used in science because it is useful, not because it is true. He supports this view with the idea that mathematics is conservative; that is, mathematics cannot be used to derive any physical facts beyond those already implied by the physical aspects of a theory. He further proves that statements in his nominalist reformulation can be systematically associated with mathematical statements, which he believes explains how mathematics can be used to legitimately derive physical facts from scientific theories.
Science Without Numbers was immediately influential upon release and generated much discussion in the philosophy of mathematics. Areas of debate included to what extent Field's reformulation of Newtonian gravitational theory was an attractive alternative to the original, mathematical theory, and whether its reliance on geometrical objects like spacetime points was acceptable. Technical issues were also raised concerning the definition of conservativeness and Field's use of metalogic and second-order logic. Following the release of the book, other philosophers worked to extend Field's account to areas such as quantum mechanics and general relativity.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Philosophers who discussed Science Without Numbers
Related ideas
- First-order logic
- Independence (mathematical logic)
- Nelson Goodman
- American Philosophical Association
- General relativity
- Princeton University Press
- Bas van Fraassen
- Paul Benacerraf
- Hartry Field
- Mathematical object
- Nominalism
- Reductio ad absurdum
- Simplicity
- Quantum mechanics
- Set (mathematics)
- Set theory
- Spacetime
- Crispin Wright
- Intrinsic and extrinsic properties
- Arithmetic
- Computably enumerable set
- Willard Van Orman Quine
- Second-order logic
- Philosophy of science
Place in the tradition
Part of Analytic philosophy.