Science Without Numbers: The Defence of Nominalism

by Hartry Field

According to the doctrine of nominalism, abstract entities―such as numbers, functions, and sets―do not exist. The problem this normally poses for a description of the physical world is as any such description must include a physical theory, physical theories are assumed to require mathematics, and mathematics is replete with references to abstract entities. How, then, can nominalism reasonably be maintained? In answer, Hartry Field shows how abstract entities ultimately are dispensable in describing the physical world and that, indeed, we can "do science without numbers." The author also argues that despite the ultimate dispensability of mathematical entities, mathematics remains useful, and that its usefulness can be explained by the nominalist. The explanation of the utility of mathematics does not presuppose that mathematics is true, but only that it is consistent. The argument that the nominalist can freely use mathematics in certain contexts without assuming it to be true appears early on, and it first seems to license only a quite limited use of mathematics. But when combined with the later argument that abstract entities ultimately are dispensable in physical theories, the conclusion emerges that even the most sophisticated applications of mathematics depend only on the assumption that mathematics is consistent and not on the assumption that it is true. Originally published in 2050. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Genres: Philosophy, Mathematics, Science, Physics, Nonfiction, Metaphysics

144 pages, Hardcover

First published December 12, 1980

About the author

HARTRY FIELD (B.A., Wisconsin; M.A., Ph. D. Harvard), Silver Professor of Philosophy, specializes in metaphysics, philosophy of mathematics, philosophy of logic, and philosophy of science. He has had fellowships from the National Science Foundation, the National Endowment for the Humanities, and the Guggenheim Foundation. He is the author of Science Without Numbers (Blackwell 1980), which won the Lakatos Prize, of Realism, Mathematics and Modality (Blackwell 1989), and of Truth and the Absence of Fact (Oxford 2001). Current interests include objectivity and indeterminacy, a priori knowledge, causation, and the semantic and set-theoretic paradoxes.

Reviews

[Harris Bolus · Rating 5 out of 5]

One of the most technically challenging books I’ve ever read. Honestly, I can’t judge how much of it made sense. But I thoroughly enjoyed it, and I learned a lot about mathematical logic. Judging from what the internet has to say about it, it seems like it’s continually enjoyed a great reception, and I can see why. Highly recommend if you’re up for a challenge!

[HuloniousMonkgus · Rating 4 out of 5]

didn't read the whole book, but I got the gist of it. Field is trying to argue that platonism (i.e., the idea that abstract mathematical entities/forms exist) is not necessary to do applied mathematics, as (for example) in mechanical physics. His main question of interest can be stated as: "What sort of account is possible of how mathematics is applied to the physical world." Though he uses nominalistic methods, Field is not a nominalist, instead he wants proof that platonism itself is not necessary. In other words, this is mainly a philosophically negative project.
Additionally, Field argues that all that is necessary for mathematics to be practical (he's not only interested in the application of mathematics, but thinks that it's very important for mathematics to be applicable to "interesting insights" about the world; this was not a direct quote) is for it to be "consistent." This implies that mathematics need not hold any truth values , that is, whether mathematical propositions are true or not is irrelevant to their application. So, for example: the statement "2 + 2 = 4" may or may not be true, but as long as it necessarily follows that the addition of 2 with itself necessarily leads to 4, then the dichotomy is not of issue (i.e., whether "2 + 2 = 4" is actually true, becomes an irrelevant statement). More explicitly, Field wants to assume that mathematics is *conservative*, meaning that one can make any inference made using nominalistic premises to nominalist conclusions using mathematical, could be (technically) made without its use (this would be a main difference between using mathematical entities and using theoretical entities, in that theoretical entities do not hold these properties).
I would not say that truth values are *completely* irrelevant to Field, but thinking of the argument this way makes his ideas clearer. In fact, field has another book "Truth in the absence of fact" where he argues that nominalistic mathematical propositions/entities can in fact be true. That said, field argues that the only non-begging arguments for the view that mathematics is a body of truths all rest ultimately on the applicability of mathematics to the physical world. Importantly, the problem of what mathematical conclusions follow from what mathematical premises remains but this is logical knowledge, not mathematical knowledge). Following, Field uses logical methods in order to verify that the mathematics used are indeed consistent and applies to sequentially more complicated applications.
He begins with arithmetic, follows with geometry and then does mechanical physics: Physical space, Physics, Newtonian Space-Time, and Newtonian Gravitational Theory. I did not read any of the chapters on its application, I believe it had mainly to do with using Hilbert's axiomatization of Euclidian geometry to make nominalistic accounts of physical phenomena. He has a chapter where he discusses the relativity of ontological commitment to the underlying logic which I could get to and I am fairly disappointed about that (I will try to get to it in the future).
As to what do I think about this book, I don't think I am in a position to criticize the book entirely since I do not read it entirely. I would say that a general worry I had which was not talked about in the parts that I read, was the fact that (1) his ideas rely on the assumption that logic is objective and (2) his overreliance on physics and cosmology. Especially for (2), there are simply some physical concepts (like time) which at times (no pun intended) may be indeterminate or undefined.

[Kyle · Rating 5 out of 5]

Very interesting idea of formalizing a theory (such as Newtonian gravitation) without quantifying over abstract entities (such as real numbers). I will have to think about whether I ultimately find it convincing for a nominalist position, though I think it does succeed in making mathematical platonism dispensable for scientific theories.

This is not easy reading because it is essentially a reformulation of Newtonian gravitation, but the ideas and consequences are well written. Because it is a strange (at least to one used to "normal" mathematics) way of thinking about scientific theories, it requires a good bit of thought to see if it is convincing, and I think Field could have written out some of the ideas more completely. However, it is quite thought-stimulating.

[Eric · Rating 5 out of 5]

You need a bit of context about debates in the philosophy of mathematics and logic and acquaintance with real analysis, some basic metalogic, and Newtonian mechanics to engage with this book. Given that, however, it's very approachable, clear, and persuasive. I went in dead set against Field's position and I came out with a great appreciation for it's merits.