Science Without Numbers
 The cover for the first edition, published by Princeton University Press | |
| Author | Hartry Field |
|---|---|
| Language | English |
| Subjects | Philosophy of mathematics |
| Publisher |
|
Publication date | 1980 |
| Publication place | United States |
| Pages | 130 |
| ISBN | 978-0-631-12672-0 |
| OCLC | 967261539 |
| Website | Oxford Academic |
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, 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
[edit]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.[1] 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.[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.
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".[3] This means, for example, that a theory of electrons should be based on the properties of electrons without referencing unrelated entities, including numbers and sets.
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.[4] 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.[5]
Summary
[edit]Science Without Numbers starts with some preliminary remarks in which Field clarifies his aims for the book.[6] 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.[7]
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.[8] Field develops this instrumentalist idea in more technical detail using the idea that mathematics is conservative.[9] 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.[10] Therefore, the predictive success of the theory can be fully explained by the truth of the nominalist portions of science, excluding any mathematics.[11]
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.[12] For example, although basic arithmetic can be reproduced non-numerically in first-order logic, the derivations this produces are far more longwinded.[a] 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.[13]
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.[14] 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.[15]
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.[16] Extending ideas from the previous sections of the book, Field produces nominalist versions of the concepts of continuity, products, derivatives, gradients, Laplacian and vector calculus.[17] Using these nominalist reconstructions, Field shows how to reformulate both the field equation of Newtonian gravity (Poisson's equation) and its equation of motion.[18] 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.
Notes
[edit]- ^ For the specific example of arithmetic, see Science Without Numbers, Ch. 2. "First Illustration of Why Mathematical Entities are Useful: Arithmetic".
References
[edit]Citations
[edit]- ^ Irvine 1990, pp. ix–xi.
- ^ Irvine 1990, p. xi; Burgess 1990, p. 2.
- ^ Colyvan 2001, p. 69.
- ^ Field 2016, Preface to Second Edition, P-1.
- ^ Hellman & Leng 2019, p. 1.
- ^ Buijsman 2017, p. 508.
- ^ Colyvan 2001, pp. 67–68.
- ^ Farrell 1981, p. 236; Malament 1982, p. 523; Meyer 2009, p. 273.
- ^ Chihara 2004, pp. 108–111.
- ^ Colyvan 2001, p. 71; Paseau & Baker 2023, p. 14.
- ^ Leng 2010, p. 46.
- ^ Malament 1982, p. 523.
- ^ MacBride 1999, pp. 434–435.
- ^ Colyvan 2001, pp. 72–73; Farrell 1981, pp. 236–237.
- ^ MacBride 1999, p. 436.
- ^ Clendinnen 1982, pp. 286–287; Meyer 2009, pp. 284–285.
- ^ Clendinnen 1982, p. 287; Manders 1984, p. 304.
- ^ Clendinnen 1982, p. 287.
Sources
[edit]- Buijsman, Stefan (2017). "The role of mathematics in science". Metascience. 26 (3): 507–509. doi:10.1007/s11016-017-0228-4. ISSN 1467-9981.
- Burgess, John P. (1990). "Epistemology and Nominalism". In Irvine, A. D. (ed.). Physicalism in Mathematics. Kluwer Academic Publishers. pp. 1–15. ISBN 978-94-010-7348-6.
- Chihara, Charles (2004). A Structural Account of Mathematics. Oxford University Press. ISBN 978-0-19-926753-8.
- Clendinnen, John F. (1982). "Review of Science without Numbers". Synthese. 51 (2): 283–291. ISSN 0039-7857. JSTOR 20115754.
- Colyvan, Mark (2001). The Indispensability of Mathematics. Oxford University Press. ISBN 978-0-19-516661-3.
- Farrell, Robert (1981). "Review of Science Without Numbers: A Defence of Nominalism". Australasian Journal of Philosophy. 59 (2): 235–237. doi:10.1080/00048408112340191.
- Field, Hartry (2016). Science Without Numbers: A Defense of Nominalism (2nd ed.). Oxford University Press. ISBN 978-0-19-877791-5.
- Hellman, Geoffrey; Leng, Mary (2019). "Review of Science Without Numbers: A Defense of Nominalism 2nd ed". Philosophia Mathematica. 27 (1): 139–148. doi:10.1093/philmat/nky022. ISSN 1744-6406.
- Irvine, A. D. (1990). "Nominalism, Realism & Physicalism in Mathematics: An Introduction to the Issues". In Irvine, A. D. (ed.). Physicalism in Mathematics. Kluwer Academic Publishers. pp. ix–xxvi. ISBN 978-94-010-7348-6.
- Leng, Mary (2010). Mathematics and Reality. Oxford University Press. ISBN 978-0-19-928079-7.
- MacBride, Fraser (1999). "Listening to Fictions: A Study of Fieldian Nominalism". The British Journal for the Philosophy of Science. 50 (3): 431–455. doi:10.1093/bjps/50.3.431. ISSN 0007-0882.
- Malament, David (1982). "Review of Science Without Numbers: A Defense of Nominalism". The Journal of Philosophy. 79 (9): 523–534. doi:10.2307/2026384. ISSN 0022-362X.
- Manders, Kenneth L. (1984). "Review of Science without numbers. A defence of nominalism". The Journal of Symbolic Logic. 49 (1): 303–306. doi:10.2307/2274113. ISSN 0022-4812.
- Meyer, Glen (2009). "Extending Hartry Field's Instrumental Account of Applied Mathematics to Statistical Mechanics". Philosophia Mathematica. 17 (3): 273–312. doi:10.1093/philmat/nkn026. ISSN 0031-8019.
- Paseau, Alexander C.; Baker, Alan (2023). Indispensability. Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press. ISBN 978-1-009-09685-0.