:Science Without Numbers

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.{{Sfn|Irvine|1990|pp=ix–xi}} 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.{{Sfnm|1a1=Irvine|1y=1990|1p=xi|2a1=Burgess|2y=1990|2p=2}}

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.{{Sfnm|1a1=Buijsman|1y=2017|1p=507|2a1=Clendinnen|2y=1982|2p=283}}{{Efn|For a more explicit definition of platonism and nominalism, see for example {{harvnb|Colyvan|2012|pp=8–9}}.}}

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".{{Sfn|Colyvan|2001|p=69}} 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.{{Sfnm|1a1=Eddon|1y=2014|1p=271|2a1=Marcus|2y=2013|2pp=166–168}}

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.{{Sfn|Field|2016|loc=Preface to Second Edition, P-1}} 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.{{Sfn|Hellman|Leng|2019|p=1}}

Science Without Numbers starts with some preliminary remarks in which Field clarifies his aims for the book.{{Sfn|Buijsman|2017|p=508}} 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.{{Sfn|Clendinnen|1982|p=283}} 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.{{Sfn|Colyvan|2001|pp=67–68}}

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.{{Sfnm|1a1=Farrell|1y=1981|1p=236|2a1=Malament|2y=1982|2p=523|3a1=Meyer|3y=2009|3p=273}} Field develops this instrumentalist idea in more technical detail using the idea that mathematics is conservative.{{Sfn|Chihara|2004|pp=108–111}} 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.{{Sfnm|1a1=Colyvan|1y=2001|1p=71|2a1=Paseau|2a2=Baker|2y=2023|2p=14}} Therefore, the predictive success of the theory can be fully explained by the truth of the nominalist portions of science, excluding any mathematics.{{Sfn|Leng|2010|p=46}}

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.{{Sfn|Malament|1982|p=523}} For example, although basic arithmetic can be reproduced non-numerically in first-order logic, the derivations this produces are far more longwinded.{{Efn|For the specific example of arithmetic, see Science Without Numbers, Ch. 2. "First Illustration of Why Mathematical Entities are Useful: Arithmetic".}} 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.{{Sfn|MacBride|1999|pp=434–435}}

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.{{Sfnm|1a1=Colyvan|1y=2001|1pp=72–73|2a1=Farrell|2y=1981|2pp=236–237}} 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.{{Sfn|MacBride|1999|p=436}}

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.{{Sfnm|1a1=Clendinnen|1y=1982|1pp=286–287|2a1=Meyer|2y=2009|2pp=284–285}} 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.{{Sfnm|1a1=Clendinnen|1y=1982|1p=287|2a1=Manders|2y=1984|2p=304}} Using these nominalist reconstructions, Field shows how to reformulate both the field equation of Newtonian gravity (Poisson's equation) and its equation of motion.{{Sfn|Clendinnen|1982|p=287}} 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.{{Sfnm|1a1=Buijsman|1y=2017|1p=509|2a1=Friedman|2y=1981|2p=506}}

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.{{efn|Technically, this statement of the conservativeness of mathematics is only valid if N is mathematically agnostic. In general, scientific theories will make claims that are not mathematically agnostic. For example, the statement "all objects obey Newton's laws" implies that mathematical objects do not exist because mathematical objects do not obey Newton's laws. If this is the case, then the combination of N + M is simply inconsistent; N implies that no mathematical objects exist, whilst M implies that some mathematical objects exist. For a more general statement of the conservativeness of mathematics, nominalistic statements and theories must first be rewritten in a mathematically agnostic form like "all objects that are not mathematical objects obey Newton's laws".{{sfn|Chihara|2004|pp=108–113}}}} 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.{{Sfn|Leng|2010|pp=48–49}} 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.{{Sfn|Shapiro|1983|p=525}}

A prominent area of discussion on Science Without Numbers is the problems that arise from these two ideas of logical consequence.{{Sfn|Mortensen|1998|p=183}} 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.{{Sfnm|1a1=Shapiro|1y=1983|1pp=525–526, 528|2a1=Leng|2y=2010|2pp=51–52}} 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.{{Sfn|Shapiro|1983|pp=526–527}}

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.{{Sfn|Chihara|2004|p=319}} 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.{{Sfn|MacBride|1999|p=442}}

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".{{Sfnm|1a1=Paseau|1a2=Baker|1y=2023|1p=16|2a1=Lockwood|2y=1982|2p=282}} 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.{{Sfnm|1a1=Hale|1y=1990|1p=123|2a1=Chihara|2y=1991|2p=162}} 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.{{Sfnm|1a1=Hale|1y=1990|1pp=124–125|2a1=Leng|2y=2010|2p=52}}{{Efn|Prominent objections to Field's nominalization of metalogic come from Bob Hale and Crispin Wright, focused in particular on a supposed tension between conservativeness and Field's characterization of mathematics as false but possibly true.{{sfn|MacBride|1999|p=443–444}} See, for example, {{harvnb|Hale|1990}} and {{harvnb|Wright|Hale|1992}}.}}

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".{{Sfnm|1a1=Colyvan|1y=2001|1pp=77, 88|2a1=Field|2y=2016|2p=8}}

In Science Without Numbers, Field argued that his nominalist theory was attractive because it offers intrinsic explanations of physical facts.{{Sfn|Colyvan|2001|p=88}} Field does not precisely define intrinsicality{{Sfn|Milne|1986|p=341}} but he does say that extrinsic entities are those "whose properties are irrelevant to the behaviour of the system being explained".{{Sfn|Colyvan|2001|p=69}} He also states that extrinsic explanations tend to be arbitrary because they rely on arbitrary choices about units of measurement like inches or metres.{{Sfn|Milne|1986|p=341}} 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.{{Sfnm|1a1=Milne|1y=1986|1p=342|2a1=Eddon|2y=2014|2p=281|3a1=Colyvan|3y=2001|3pp=73–74}}

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.{{Sfnm|1a1=Marcus|1y=2013|1pp=172–173|2a1=Paseau|2a2=Baker|2y=2023|2pp=18–19}}

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.{{Cite web |last= |first= |date=1987-09-15 |title=1986 Lakatos Award |url=https://www.lse.ac.uk/philosophy/blog/1987/09/15/1986-lakatos-award-bas-van-fraassen-and-hartry-field/ |access-date=2025-06-08 |publisher=London School of Economics, Department of Philosophy, Logic and Scientific Method |language=en-GB}}

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.{{Cite web |date=November 2020 |title=Science Without Numbers, 40 Years Later |url=https://sites.google.com/view/swn40years/home |access-date=2025-06-10 |publisher=University of California, San Diego |at=Introduction}}

{{Notelist}}

{{Reflist}}

{{refbegin|30em}}

• {{cite journal |last=Buijsman |first=Stefan |date=2017 |title=The Role of Mathematics in Science |journal=Metascience |volume=26 |issue=3 |pages=507–509 |doi=10.1007/s11016-017-0228-4 |issn=1467-9981}}
• {{Cite book |last=Burgess |first=John P. |author-link=John P. Burgess |title=Physicalism in Mathematics |date=1990 |publisher=Kluwer Academic Publishers |isbn=978-94-010-7348-6 |editor-last=Irvine |editor-first=A. D. |editor-link=Andrew David Irvine |pages=1–15 |chapter=Epistemology and Nominalism}}
• {{Cite book |last=Chihara |first=Charles |author-link=Charles Chihara |title=Constructibility and Mathematical Existence |publisher=Oxford University Press |year=1991 |isbn=0-19-823975-0}}
• {{Cite book |last=Chihara |first=Charles |author-link=Charles Chihara |title=A Structural Account of Mathematics |publisher=Oxford University Press |year=2004 |isbn=978-0-19-926753-8}}
• {{cite journal |last=Clendinnen |first=John F. |date=1982 |title=Review of Science Without Numbers |journal=Synthese |volume=51 |issue=2 |pages=283–291 |doi=10.1007/BF00413829 |issn=0039-7857 |jstor=20115754}}
• {{Cite book |last=Colyvan |first=Mark |author-link=Mark Colyvan |title=The Indispensability of Mathematics |publisher=Oxford University Press |year=2001 |isbn=978-0-19-516661-3 |language=en}}
• {{Cite book |last=Colyvan |first=Mark |author-link=Mark Colyvan |title=An Introduction to the Philosophy of Mathematics |publisher=Cambridge University Press |year=2012 |isbn=978-0-521-82602-0}}
• {{Cite book |last=Eddon |first=Maya |title=Companion to Intrinsic Properties |publisher=De Gruyter |year=2014 |isbn=978-3-11-029086-8 |editor-last=Francescotti |editor-first=Robert M. |pages=271–290 |chapter=Intrinsic Explanations and Numerical Representations}}
• {{cite journal |last=Farrell |first=Robert |date=1981 |title=Review of Science Without Numbers: A Defence of Nominalism |journal=Australasian Journal of Philosophy |volume=59 |issue=2 |pages=235–237 |doi=10.1080/00048408112340191}}
• {{Cite book |last=Field |first=Hartry |author-link=Hartry Field |title=Science Without Numbers: A Defense of Nominalism |date=2016 |publisher=Oxford University Press |isbn=978-0-19-877791-5 |edition=2nd}}
• {{cite journal |last=Friedman |first=Michael |author-link=Michael Friedman (philosopher) |date=1981 |title=Review of Science Without Numbers: A Defense of Nominalism |journal=Philosophy of Science |volume=48 |issue=3 |pages=505–506 |doi=10.1086/289018 |issn=0031-8248}}
• {{Cite book |last=Hale |first=Bob |author-link=Bob Hale (philosopher) |chapter=Nominalism |date=1990 |title=Physicalism in Mathematics |pages=121–144 |editor-last=Irvine |editor-first=A. D. |editor-link=Andrew David Irvine |publisher=Kluwer Academic Publishers |isbn=978-94-010-7348-6}}
• {{cite journal |last1=Hellman |first1=Geoffrey |author-link=Geoffrey Hellman |last2=Leng |first2=Mary |author-link2=Mary Leng |date=2019 |title=Review of Science Without Numbers: A Defense of Nominalism 2nd ed |journal=Philosophia Mathematica |volume=27 |issue=1 |pages=139–148 |doi=10.1093/philmat/nky022 |issn=1744-6406}}
• {{Cite book |last=Irvine |first=A. D. |author-link=Andrew David Irvine |title=Physicalism in Mathematics |date=1990 |publisher=Kluwer Academic Publishers |isbn=978-94-010-7348-6 |editor-last=Irvine |editor-first=A. D. |pages=ix–xxvi |chapter=Nominalism, Realism & Physicalism in Mathematics: An Introduction to the Issues}}
• {{Cite book |last=Leng |first=Mary |author-link=Mary Leng |title=Mathematics and Reality |date=2010 |publisher=Oxford University Press |isbn=978-0-19-928079-7}}
• {{cite journal |last=Lockwood |first=Michael |date=1982 |title=Review of Science Without Numbers: A Defence of Nominalism |journal=The Philosophical Quarterly |volume=32 |issue=128 |pages=281–283 |doi=10.2307/2219330 |issn=0031-8094}}
• {{Cite journal |last=MacBride |first=Fraser |date=1999 |title=Listening to Fictions: A Study of Fieldian Nominalism |journal=The British Journal for the Philosophy of Science |language=en |volume=50 |issue=3 |pages=431–455 |doi=10.1093/bjps/50.3.431 |issn=0007-0882}}
• {{cite journal |last=Malament |first=David |author-link=David Malament |date=1982 |title=Review of Science Without Numbers: A Defense of Nominalism |journal=The Journal of Philosophy |volume=79 |issue=9 |pages=523–534 |doi=10.2307/2026384 |jstor=2026384 |issn=0022-362X}}
• {{cite journal |last=Manders |first=Kenneth L. |date=1984 |title=Review of Science Without Numbers: A Defence of Nominalism |journal=The Journal of Symbolic Logic |volume=49 |issue=1 |pages=303–306 |doi=10.2307/2274113 |jstor=2274113 |issn=0022-4812}}
• {{Cite journal |last=Marcus |first=Russell |author-link=Russell Marcus |date=2013 |title=Intrinsic Explanation and Field's Dispensabilist Strategy |journal=International Journal of Philosophical Studies |language=en |volume=21 |issue=2 |pages=163–183 |doi=10.1080/09672559.2012.727011 |issn=0967-2559}}
• {{Cite journal |last=Meyer |first=Glen |date=2009 |title=Extending Hartry Field's Instrumental Account of Applied Mathematics to Statistical Mechanics |journal=Philosophia Mathematica |language=en |volume=17 |issue=3 |pages=273–312 |doi=10.1093/philmat/nkn026 |issn=0031-8019}}
• {{Cite journal |last=Milne |first=Peter |date=1986 |title=Hartry Field on Measurement and Intrinsic Explanation |journal=The British Journal for the Philosophy of Science |language=en |volume=37 |issue=3 |pages=340–346 |doi=10.1093/bjps/37.3.340 |issn=0007-0882}}
• {{cite journal |last=Mortensen |first=Chris |date=1998 |title=On the Possibility of Science Without Numbers |journal=Australasian Journal of Philosophy |volume=76 |issue=2 |pages=182–197 |doi=10.1080/00048409812348341 |issn=0004-8402}}
• {{Cite book |last1=Paseau |first1=Alexander C. |author-link=Alexander Paseau |title=Indispensability |last2=Baker |first2=Alan |author-link2=Alan Baker (philosopher) |date=2023 |publisher=Cambridge University Press |isbn=978-1-009-09685-0 |series=Cambridge Elements in the Philosophy of Mathematics}}
• {{cite journal |last=Shapiro |first=Stewart |date=1983 |title=Conservativeness and Incompleteness |journal=The Journal of Philosophy |volume=80 |issue=9 |pages=521–531 |doi=10.2307/2026112 |issn=0022-362X}}
• {{Cite journal |last=Wright |first=Crispin |author-link=Crispin Wright |last2=Hale |first2=Bob |author-link2=Bob Hale (philosopher) |date=1992 |title=Nominalism and the Contingency of Abstract Objects |journal=The Journal of Philosophy |volume=89 |issue=3 |pages=111–135 |doi=10.2307/2026789 |jstor=2026789 |issn=0022-362X}}

{{refend}}

• [https://academic.oup.com/book/26363 Science Without Numbers] at Oxford Academic
• Science Without Numbers at Internet Archive

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