Jill V. Buhay
Math 339 Paper 2
Due Wed May 8, 2002
Instrumentalism (i.e. the theoretical extreme) holds that scientific theories
and models of the universe are valid only in the sense that they are useful in
predicting events and explaining data consistently, while at the same time
making no claim that anything they describe actually exists. Field, author of
Science Without Numbers, attempts to nominalize the structure of physical
space, physics, Newtonian Space-Time, Quantities and the Laws affecting them,
Newtonian Gravitational Theory.
Regarding nominalism and the structure of physical space, Field argues
that Hilbert’s Theory (of the Euclidean Theory) not only dispenses with real
numbers, but it is genuinely a nomilnalistic theory of the structure of physical
space. Field raises several important issues. The first is the idea that Hilbert's
axiomation requires physical space to be isomorphic to the real numbers. Field
claims that there is not much of a difference between postulating a physical
space and the real numbers. Field acknowledges the fact that the nominalistic
objection arises when looking at the abstractness to using the real numbers. He
believes that even postulating one real number would be a violation of
nominalism. He goes on to say that even postulating even many physical entities
(uncountably many parts of a physical object) is not an objection to nominalism.
He does present the idea that using the term "physical entity" to describe
space-time points is a bit unsettling, but Field claims that space-time points are
not abstract identities. He continues by saying from a platonist perspective, our
knowledge of mathematical structures of abstract entities is a priori, whereas the
structure of physical space is empirical. Most platonists who believe the current
physical theory believe it because it cannot be proven incorrect. They also
believe that it is a high-level empirical hypothesis, that there are lines in space,
which are isomorphic to real numbers. Moreover they are not likely to identify the
real numbers with the point on any physical line. Choosing which line to use to
identify the real numbers and a point that on the line to identify some o would be
arbitrary. Doing so would mean that we would have to identify the real numbers
with know of only empirically.
Even if we chose to ignore these points, Field says that there is more
reason to believe that postulating physical space is not like postulating real
numbers. He states that the ideology that goes with the postulate of points of
space is less rich than the postulate of the real numbers. Operations of addition
and multiplication are included in the postulate of real numbers, but these
operations are not directly defined on space-time points in Hilbert's Theory. Field
says that these operations are not even implicitly definable since any introduction
of an addition or multiplication function on space-time points would have to rely
on an arbitrary choice of two distinct points. addition can be constructed within
Hilbert's theory, but in reference to intervals and not points. Multiplication cannot
be constructed in Hilbert's Theory either. To do so would require an arbitrary
choice of one interval to serve as the unit interval. Field says there is no such
idea within Hilbert's Theory.
In order to nominilize physics, Field proposes that the nominalistic
formulation must be an attractive one. Otherwise, he claims that nominalization
would be trivial. To show this, one would take as the axioms of the physical
theory all the nominalistically statable consequences of the platonic formulation
of the theory. Field's method is different. He introduces two different approaches,
the metric approach and the synthetic approach.
In the metric approach, Field takes a particular function d, which will
symbolize a specific mapping of pairs of points of space into the real numbers.
Then regard the mathematical laws of real numbers, functions, etc. as
independently given, we can use the function d to get a simple set of axioms for
the geometry. The synthetic approach is the one that Hilbert followed. This
approach symbolizes a specific mapping of pairs of points of space into the real
numbers without using real numbers, functions etc. Field believes that these
synthetic approaches to physical theory are advantageous, not only because it is
nominalstic, but also in comparison to the metric approach, in some ways it is
more illuminating. He claims that synthetic approaches explain what is going on
without appeal to extraneous, irrelevant entities. From a proper synthetic theory,
Field feels that one would be able to prove the equivalence of intrinsic and
extrinsic explanations.
When Field begins to nominalize Newtonian Space-Time, he states that
this version also requires being "attractive." Moreover, it needs to be a
formulation that does not appeal to arbitrarily chosen objects to serve as units of
length, a system of coordinates, or to any such thing. His first step is to give a
nominalistc treatment of space-time. Field admits that this will be a tricky task,
Newtonian space-time not only is four dimensional instead of three, but it also
does not a have a full Euclidean structure.
There are mainly two errors that arise from the fact that Newtonian space
does not have a full eucidean structure. The first is that there is not objective way
to compare spatial distance with temporal distance. Mainly there is no means of
comparison that naturally singled out by the laws of Newtonian mechanics. The
second method lacks a full Euclidean structure. One way to nominalize
Newtonian Space-Time would be to change physical theory so that the absolute
acceleration is not used. Another way, without having to change physical theory,
is to give a treatment of absolute acceleration without absolute rest. Although
these conclusions are limited, they show the possibility of a nominalistic account
of the structure of space-time.
Next, Field deals with the entities that exist within space-time, which
include scalar fields such as temperature or gravitational potential. An example
would be in dealing with temperature. In order to create coordinate-independent
treatment, one would need to introduce a continuum of temperature properties
where each one has a specific temperature. Then one could describe the
structure of the system of properties by using intrinsic relations instead of using
numbers.
Field's approach is to introduce temperature-betweenness and
temperature congruence relations among space-time points. He says that this
will give us a 3-place relation Temp-bet, with y Temp-Bet xz meaning that y is a
space-time point at a temperature between the temperature of points x and z.
Filed continues by saying that a 4-place relation Temp-Cong is with xy TempCong zw meaning that the temperature between x and y is equal in absolute
value to the temperature difference between the points z and w. If we want to
laws which in their extrinsic formulations are not invariant under temperature
reversal, we will need a 2-place predicate Temp-Less where x Temp-Less why
means that the point x is lower than or equal in temperature to point y.
Finally, Field discusses the nominalization of the Newtonian Gravitational
Theory. Regarding Newtonian Gravitational theory, Field sheds light on the
nominalizations of continuity, products and ratios, derivatives, higher derivatives,
Laplaceans, Poisson's equation, inner products, gradients, differentiation of
vector fields, and the law of motion saying that all can be nominalized.
To say that a function is T continuous nominalistcally, one cannot talk
about T at all. The sentence to say that T is continuous will quantify over spacetime p[points and space-time regions. We must show that for any model of the
joint axiom system for space-time and the scalar quantity in we are referring to,
and any representation functions phi and psi, the statement, t in continuous if
true in the model if and only if T is a continuous function. This will extend the
representation theorem to the larger axiom system that includes the new
continuity clam, that T is continuous. The uniqueness theorem will apply
automatically because the new continuity axiom will only involve the basic
predicated used in the original axiom system. The nominalistic continuity axiom
quantifies over space-time regions.
On products and ratios, Field comments that there is no hope multiplying
nominalistaclly unless conditions are met. The most we can say is that the result
of multiplying one spatial temporal interval with one temperature interval is less
than the result of multiplying another spatio-temporal interval with another
temperature interval. We can also hope to this if two spatio-temporal intervals are
themselves comparable, more specifically when they lay on a straight line.
The general strategy Field follows starts out by giving a joint axiom system
containing axioms for space-time, axioms for the gravitational potential, and
axioms for mass density. he then proves that for any model of this joint axiom
system, there would be a one to one function from the domain of the model to the
four-dimensional, and two functions phi and psi from the domain onto the reals.
All of which satisfy homomorphism conditions. He then showed that there were
further nominalistic statements expressible using the primitives of the joint axiom
system. Field claims that the nominalistic formulation of the physical theory in
conjunction with standard mathematic yields the usual platonistic formulation of
the theory. Conversely, the nominalistic formulation is a consequence of the
platonistic formulation, given standard mathematics. It then follows that any
statement in the nominalistic language proposed is a consequence of the
platonistic axioms. Moreover standard mathematics is a consequence of
nominalistic axioms unaided by mathematics. He continues to say that the
nominalistic formulation of physics and the platonistic formulation have the same
nominalistically statable consequences. So mathematical entitles are
theoretically dispensable in the theory of gravitation.
Field manages to nominalize the structure of physical space, physics,
Newtonian Space-Time, Quantities and the Laws affecting them, Newtonian
Gravitational Theory. This shows the reader that through carefully following the
steps that Field has taken, scientific theories and models of the universe can be
validated in the sense that they are useful in predicting events and explaining
data consistently, while at the same time making no claim that anything they
describe actually exist.