Justification, Truth, and Belief http://www.jtb-forum.pl December 2001 TOMASZ BIGAJ Field’s Program: A Defense It has been over twenty years since Hartry Field’s famous book Science without Numbers was published, yet it still gives rise to controversies and discussions. In 1998 April’s issue of Analysis an interesting article appeared (Melia 1998), in which the author formulates several objections to the program of nominalization of science, not discussed extensively elsewhere. As it happened, I have worked extensively on Field’s program for some time, trying to overcome difficulties created by his approach and to generalize his particular results. I presented some outcomes of my investigations in my book (Bigaj 1997). Here I would like to concentrate on particular arguments against Field’s attempt, put forward in the aforementioned article. Because I happen to be a proponent (naturally, not without certain reservations) of Field’s general approach, I feel obliged to point out certain weaknesses of Melia’s arguments. Answering to Melia’s criticism I also want to report some of my earlier results to the effect that it is possible to strengthen substantially Field’s method of nominalization by showing that it can be applied to a wider range of physical theories without any additional proofs. I am assuming that the reader is familiar with the basic ideas of Field’s approach (in Melia’s article one can find a very clear and accurate 1 outline of this approach). 1. A nominalistic interpretation of irrational ratios between distances Melia in his article argues that Field’s program of eliminating reference to mathematical objects in science is incomplete. Usually critics of Field’s approach point to the fact that he had failed to show that all (or at least most of) interesting scientific theories, like for example quantum theory, can be nominalized along his line 2 . Field claims that he succeeded in nominalizing Newtonian gravitational theory, but this theory is still far away from modern physics. However, Melia goes even further in his criticism. He maintains that even in the simple case of Euclidean geometry Field’s way of nominalization is incomplete, because it does not give any method of interpreting sentences stating that the distance between certain points equals to some irrational number (e.g. the number e, the base of natural logarithms). In that way he creates the challenge for the nominalist, who would like 1 Maybe I should mention that I found in Melia’s paper one obvious mistake: the formula in the last paragraph on page 64 expresses not the fact that the space is 3-dimensional, but rather that it is 2- dimensional (it says that there are at least three non-collinear points). However, this mistake has no consequences in further analysis. 2 Cf. for example Malament 1982.