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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 the argument, we should believe in mathematical objects 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.

Background
Science Without Numbers emerged during a period of renewed interest in the philosophy of mathematics following a number of influential papers by Paul Benacerraf, particularly his 1973 article "Mathematical Truth". In that paper, Benacerraf argued that it is unclear how the existence of non-physical mathematical objects such as numbers and sets can be reconciled with a scientifically acceptable epistemology. This argument was among Field's motivations for writing Science Without Numbers; he aimed to provide an account of mathematics that was compatible with a naturalistic view of the world.

The main goal of the book was to defend nominalism, the view that mathematical objects do not exist, and to undermine the motivations for platonism, the view that mathematical objects do exist. Field believed that the only good argument for platonism is the Quine–Putnam indispensability argument, which argues that we should believe in mathematical objects because mathematics is indispensable to science. A key motivation for the book was to undermine this argument by showing that mathematics is indeed dispensable to science. [cite all this - maybe Bob Hale chapter, also Buijsman source]

Independently of the appeal of nominalism, Field was motivated by a desire to formulate scientific explanations "in terms of the intrinsic features of [the] system, without invoking extrinsic entities". This means, for example, that we should formulate our theory of electrons in terms of the properties of electrons without referencing unrelated entities, including numbers and sets.[Find example like this to cite]

According to Field, he began work on the book in the winter of 1978, intending to write a long journal article. However, during the process of writing, it became too long to be feasibly published in a journal format. It was initially published in 1980 by Princeton University Press; a second edition was published in 2016 by Oxford University Press featuring minimal changes to the main text and a new preface.

Summary
Science Without Numbers starts with some preliminary remarks in which Field clarifies his aims for the book. He outlines that he is concerned mainly with defending nominalism from the strongest arguments for platonism—the indispensability argument in particular—and is less focused on putting forward a positive argument for his own view. He distinguishes the form of nominalism he aims to defend, fictionalism, from other types of nominalism that were more popular in the philosophy of mathematics at the time.[cite this] Such forms of nominalism are revisionist in that they aim to reinterpret mathematical sentences so that they are not about mathematical entities. In contrast, Field's fictionalism accepts that mathematics is committed to the existence of mathematical entities, but argues that such mathematics is simply untrue. [find better cite]

Field adopts an instrumentalist account of mathematics, arguing that mathematics does not have to be true to be useful. His view develops ideas in Hilary Putnam's 1967 paper "The thesis that mathematics is logic" and Carl Hempel's idea of a "theoretical juice extractor". Both Putnam and Hempel emphasized the idea that mathematics' role in science is simply to aid in the derivation of empirical conclusions from other empirical claims, which could occur without using mathematics at all. In Science Without Numbers Field produced a more precise formulation of this idea. He claimed that mathematics is conservative. This means that if a nominalistic statement is derivable from a scientific theory with the use of mathematics, then it is also derivable without the mathematics. More precisely, where $\phi$ is a nominalistic statement, $N$  is a purely nominalistic theory and $M$  is a mathematical theory:

$$\textrm{if} \ N + M \vDash \phi, \ \textrm{then} \ N \vDash \phi $$

[cite Hellman & Leng 2019, Chihara 2004 pp. 108-113] Explain conservativeness more / consequences of this (difference between mathematics and theoretical entities) / fact that Field provided a conservativeness proof

(Maybe relegate technical details to footnote? Then can comment on nuances and intricacies Field introduces to the simple definition)

Field illustrates how mathematics can be useful to science with the example of arithmetic. For Field, the usefulness of mathematics is in its ability to simplify the derivation of empirical conclusions from theories. This can be done by linking statements from the nominalistic theory to mathematical statements with the use of "bridge laws". [explain bridge laws] Derivations can then occur within mathematics before descending back to the nominalistic theory again with the use of the bridge laws. Field considers the example of the following nominalistic argument:


 * 1) There are exactly twenty-one aardvarks;
 * 2) On each aardvark there are exactly three bugs;
 * 3) Each bug is on exactly one aardvark; so
 * 4) There are exactly sixty-three bugs

The validity of this argument can be proved within a purely nominalistic theory. However, to do so will require complex logical restatements of the premises and conclusion. Using bridge laws to link the premises to mathematical counterparts and calculating the number of bugs using basic arithmetic is far simpler. [cite all this, introduce representation theorem as bridge law / link to mathematical counterparts, Hilbert]

Bridge laws, Representation theorems

Field then turns to Hilbert's axiomatization of geometry as an example of a theory that removes reference to mathematics to illustrate precisely how nominalistic theories can be linked to their mathematical counterparts. He appeals to the concept of representation theorems to show that...

Hilbert's axiomatization of geometry

Field's extensions

[Need to note Putnam came to disagree with 1967 paper due to indispensability of mathematics at some point]

Responses/counterarguments/ analysis
Modern physics (QM etc.)

Use of second order logic

Substantivalism

Reception
In a 1980 letter to Field, which was published in the second edition, W. V. Quine said that the book was "an impressive piece of work: reasonable, ingenious, learned, and as central philosophically as can be".

Legacy
Prior to Science Without Numbers, fictionalism was not a popular view in the philosophy of mathematics.

Field's work in Science Without Numbers and subsequent publications has been credited with a resurgence in interest in nominalism and fictionalism in contemporary philosophy of mathematics. [cite this]

A conference titled Science Without Numbers, 40 Years Later was held in November 2020. The conference website called the book "one of the most influential works in the philosophy of mathematics" and stated that its influence had extended to multiple subfields of philosophy, inspiring a sizeable secondary literature.