FOM: real numbers and the real world

[Allen Hazen]

If I may expand a bit on Robert Black's (entirely correct, I think) account
of Hartry Field's views in his 1980 book "Science without Numbers":
   Field believed that arbitrary regions of space-time count as genuine
physical objects.  (This view, though not beyond controversy, has been
accepted by many writers in the philosophical tradition Field represents.
Even Tarski, in his "Wahrheitsbegriff," appealed to it in defense of the
ontological assumptions of syntax (e.g. that there are formulas in the
formal language of arbitrary finite length), suggesting that they were
plausible if inscription-shaped regions of space were accepted as being
inscriptions even if not inked in.)  Field extended this to include
space-time POINTS.  (Again, the view that dimensionless points are real
physical objects is widespread, though other philosophical tendencies have
criticized it: Whitehead, for example, attempted a reductive analysis of
points by his "method of extensive abstraction.)  Finally, Field accepted
an unrestricted mereology ("complete logic of Goodmanian sums"): for any
class of spatial regions-- including points-- there is a region of which
they are parts (and of which nothing is a part that isn't "composed" of
them).  Given the (classical-- Field's worked example was from classical
phuysics) assumption that space and time are continua, this means that he
in essence believed in (accepted in his ontology, to use jargon) a full
(monadic) second-order theory of (objects corresponding to) quadruples of
real numbers: given a few geometrical notions as primitives, Field was
working in a theory that allows quite strong classical theories to be
interpreted in it.
   As anybody acquainted with the disputacious habits of philosophers could
predict <grin>, Field's views have been criticized.  One criticism is that
the spatio-temporal entities he took to be physical-- points, for example--
are better thought of as abstractions.  My recollection is that David
Malament's review, in "Journal of Philosophy" 79 (1982), pp.523-534, was
critical and also a helpful, thought-provoking, contribution to
philosophical clarity.
   Field's book is, I think, of some mathematical interest: the
constructions used in giving his interpretation are ingenious.  Most of
them, I think, are familiar to geometers, but they are presented clearly
and with good motivation, which might make the book a useful supplementary
text for students of geometry.  Field himself stressed that the mere
possibility of interpreting the classical theories in his framework was not
the deciding issue, but claimed that methodological advantages made his
philosophy attractive: that reformulating physics in his "nominalistic" way
made physical explanations more perspicuous.
--
Allen Hazen
Philosophy Department
University of Melbourne