Science Without Numbers

''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.

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.

Independently of the appeal of nominalism, Field was motivated by a desire to formulate scientific explanations "in terms of the intrinsic features of hesystem, without invoking extrinsic entities". For Field, numbers are extrinsic to physics since they are causally irrelevant to the behaviour of physical systems. He argued that things intrinsic to physical theories, like physical objects and spacetime, should be preferred when constructing explanations in science.

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. The forms of nominalism popular at the time were revisionist in that they aimed to reinterpret mathematical sentences so that they were not about mathematical objects. In contrast, Field's fictionalism accepts that mathematics is committed to the existence of mathematical objects, but argues that mathematics is simply untrue.

Field adopts an instrumentalist account of mathematics, arguing that mathematics does not have to be true to be useful. Field argues that, unlike theoretical entities like electrons and quarks, mathematical objects do not allow theories to predict anything new. Instead, mathematics' role in science is simply to aid in the derivation of empirical conclusions from other empirical claims, which could theoretically occur without using mathematics at all. Field develops this instrumentalist idea in more technical detail using the idea 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. Therefore, the predictive success of the theory can be fully explained by the truth of the nominalist portions of science, excluding any mathematics.

Field takes the conservativeness of mathematics to explain why it is acceptable for mathematics to be used in science. He further argues that its usefulness is due to it simplifying the derivation of empirical conclusions. For example, although basic arithmetic can be reproduced non-numerically in first-order logic, the derivations this produces are far more longwinded. Field shows how mathematics can skip these derivations through the use of bridge laws, which can connect nominalistic statements to mathematical ones, allowing derivations to proceed efficiently within mathematics before returning to the nominalistic theory.

Field's reformulation of physics is based on Hilbert's axiomatization of geometry, in which numerical distances are replaced with relations between spacetime points like ''betweenness'' and ''congruence''. Hilbert proved a representation theorem showing that these relations between spacetime points are homomorphic to numerical distance relations. This notion of a representation theorem serves as the bridge law in Field's approach, allowing mathematical reasoning to be associated with nominalistic counterparts in a strictly structure-preserving way.

In addition to Hilbert's treatment of geometry, Field's reformulation takes similar ideas from measurement theory to nominalize scalar physical quantities like temperature and gravitational potential. Field again uses relational concepts (like temperature-betweenness and temperature-congruence) to recover various features of scalar fields in physics. Extending ideas from the previous sections of the book, Field produces nominalist versions of the concepts of continuity, products, derivatives, gradients, Laplacians and vector calculus. Using these nominalist reconstructions, Field shows how to reformulate both the field equation of Newtonian gravity (Poisson's equation) and its equation of motion. Besides the technical contents of the book, ''Science Without Numbers'' also includes discussions on the philosophical viability of Field's approach, including the benefits of intrinsic explanations and the challenges of its prolific use of spacetime points and second-order logic.

Technical details and analysis

Conservativeness

The conservativeness of mathematics claims that for any nominalist theory ''N'' and mathematical theory ''M'', everything that is a logical consequence of ''N + M'' must also be a logical consequence of just ''N''. However, the concept of logical consequence is ambiguous. It can be thought of semantically; in which case, it is put in terms of the logical impossibility of the theory being true and the entailed statement being false. Or it can be thought of syntactically; that is, put in terms of the derivability of the entailed statement from the theory. In ''Science Without Numbers'', Field included proofs in first-order logic that mathematics is both syntactically and semantically conservative. However, for his full nominalization of Newtonian gravitational theory, which relies on second-order logic, he only showed that mathematics is semantically conservative.

A prominent area of discussion on ''Science Without Numbers'' is the problems that arise from these two ideas of logical consequence. According to Stewart Shapiro, the project within ''Science Without Numbers'' is best understood when assuming a syntactic version of conservativeness. Throughout ''Science Without Numbers'', conservativeness is explained in terms of derivability, and the semantic interpretation is potentially problematic for nominalism because it relies on the existence of things like models or possibilities. On the other hand, a version of Gödel's incompleteness theorems holds for Field's nominalization. This means that there are some facts about spacetime that cannot be derived from Field's nominalist theory and, therefore, the syntactic conservativeness result does not hold for Field's full second-order theory.

A related issue concerns Field's use of metalogic. His proof of semantic conservativeness was a model-theoretic proof using set theory and his proof of syntactic conservativeness was proof-theoretic using standard proof theory. These proofs are metalogical because they are about the properties of logical systems and define logical terms like logical consequence. One argument against Field is that his use of metalogic is not acceptable because his proofs include mathematical objects like models and proofs, but he had not provided a nominalization of metalogic.

In ''Science Without Numbers'', Field stated that his use of mathematical objects was valid because his argument was merely a reductio ad absurdum; an argument that assuming mathematics to be true leaves it in "an unstable position: it entails its own unjustifiability". However, some analyses of the work criticized this justification, claiming that conservativeness was used by Field to explain why it is acceptable for mathematics to be used in science, which goes beyond a reductio argument. In papers released after ''Science Without Numbers'' in response to these objections, Field attempted to give a nominalist interpretation of metalogic by taking modal operators as primitive and using these to define a semantic version of logical consequence.

Dispensability and attractiveness

''Science Without Numbers'' attempts to show the dispensability of mathematics to science. However, Field did not understand dispensability merely as the ability to eliminate mathematics from science; he further required that the elimination result in an "attractive" theory. Technically, any class of entities is eliminable from a theory so long as it can be separated out from the rest of the theory, according to Craig's theorem. However, Field rejected this approach to eliminating entities as uninformative since it does not result in a theory based on "a small number of basic principles".

In ''Science Without Numbers'', Field argued that his nominalist theory was attractive because it offers intrinsic explanations of physical facts. Field does not precisely define intrinsicality but he does say that extrinsic entities are those "whose properties are irrelevant to the behaviour of the system being explained". He also states that extrinsic explanations tend to be arbitrary because they rely on arbitrary choices about units of measurement like inches or metres. He argues that intrinsic theories can remove arbitrariness and even explain the arbitrariness found in other formulations. For example, a uniqueness theorem for Hilbert's axioms shows that the rules of geometry are invariant under a multiplicative factor on distance; for Field, this explains why different units of measurement are equally valid and it does so in terms of the intrinsic structure of spacetime.

One criticism of Field's approach contends that Field has ignored theoretical virtues beyond intrinsicality such as unification and simplicity. According to this line of thought, mathematical scientific theories are more attractive than nominalist theories precisely because mathematics unifies and simplifies the theory. Field's method for nominalizing science, by contrast, is necessarily a piecemeal approach, in that it must proceed theory by theory and will not necessarily provide an overarching framework like mathematics does.

Legacy

''Science Without Numbers'' jointly won the 1986 Lakatos Prize, an award given to "outstanding contributions to the philosophy of science" by the London School of Economics, with Bas van Fraassen's ''The Scientific Image''.

A workshop called ''Science Without Numbers, 40 Years Later'' was held remotely in November 2020. It had been scheduled as a symposium session for the American Philosophical Association but was cancelled due to the COVID-19 pandemic, leading to a remote workshop instead. The workshop's website said the book had "become one of the most influential works in the philosophy of mathematics" and that its impact had extended into several other areas of philosophy.

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External links

*
Science Without Numbers
' at Oxford Academic
* ''Science Without Numbers'' at Internet Archive

Books about philosophy of mathematics
Books about philosophy of physics
Princeton University Press books
Oxford University Press books
1980 non-fiction books
Philosophy of science books