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

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

[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!

[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.