Science without Numbers

by Hartry Field

Science Without Numbers caused a stir in philosophy on its original publication in 1980, with its bold nominalist approach to the ontology of mathematics and science. Hartry Field argues that we can explain the utility of mathematics without assuming it true. Part of the argument is that good
mathematics has a special feature (conservativeness) that allows it to be applied to nominalistic claims (roughly, those neutral to the existence of mathematical entities) in a way that generates nominalistic consequences more easily without generating any new ones. Field goes on to argue that
we can axiomatize physical theories using nominalistic claims only, and that in fact this has advantages over the usual axiomatizations that are independent of nominalism. There has been much debate about the book since it first appeared. It is now reissued in a revised contains a substantial new
preface giving the author's current views on the original book and the issues that were raised in the subsequent discussion of it.

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

192 pages, Paperback

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

[Carson Phillips]

First of all, I am too uneducated for this book. It became apparent really quickly that I was attempting to jump into a conversation that had been going on for a while such that the language used was impenetrable.

That said, I enjoyed some of the parts that I did understand. This book is a bit of a modern classic on philosophy of math because of its ambitious aim to take down the Quine-Putnam Indispensability Argument for mathematical realism by nominalizing the mathematical apparatus of scientific theories to show that the criterion of ontological commitment fails to reify mathematical objects. Field takes Newtonian Gravitational Theory as his scientific theory of choice as a case study. Critiques, of course, say that the programme would be much more pressed to nominalize general relativity or quantum mechanics (and I’d have to agree it seems quite difficult), but attempts already exist in the literature to do just that. Field is effectively credited with opening this can of worms.

I’m partial to the fictionalist programme because I also think mathematic talk is strictly false but conventionally true. For example, a sentence like “wizards are magical” is strictly speaking false because there are no wizards that exist out there in the real world in which to ground that predicate, but the sentence is conventionally true based on the metaphorical meaning of the predicate. However, I’ve never felt that the Indispensability Argument ever held much weight. To me, it confuses descriptions and explanations. While Field’s project is to show that mathematical entities are in fact dispensable in scientific theories and therefore don’t have to be reified, I would just deny that the role of mathematical objects in scientific theories is explanatory and therefore doesn’t make sense to reify. I think explicating the difference in practical disposability and true disposability would be illuminating to that end as well.

[Harris Bolus]

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]

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]

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.

[é]

ambitious, even brilliant, but unreadable. rather, it reads like a very long paper: very technical and most of it is details you won't need or care about unless you're a working philosopher. there are a few good arguments in there for his position but you're probably better off just reading a summary (or better yet, i think he explains himself nicely in the preface). the preface to the second edition, published almost 40 years after the first and where he responds to various criticisms, is probably the best part of the book and well worth reading.

[Eric]

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.