7	Synthetic Physics and Nominalist Realism

John P. Burgess

Princeton University

‘Words do not reflect the world, not because there is no world, but because words are not mirrors.’

—Roger Shattuck

In mathematical as opposed to philosophical usage, geometry in the style of Euclid and physics in the style of Archimedes are called synthetic, whereas geometry in the style of Descartes and physics in the style of Laplace are called analytic. Analytically formulated theories differ from synthetically formulated theories in two respects. First, they involve the representation of spatial and temporal position by real-number coordinates and of extensive and intensive magnitudes by real-number measures of extent and intensity. Second, they involve the application of (algebra and) analysis, theorems and techniques pertaining to operations on (real numbers and) functions from and to real numbers.

These mathematical methods were creations of the seventeenth century, the first being associated with the names of Descartes and Fermat, the second with the names of Newton and Leibnitz. Analytic methods were important if not essential to the progress of physics in the seventeenth century and to Newton’s development of classical gravitational theory. However, when writing up his results for publication in the Principia, Newton attempted an exposition in a more synthetic style. This was perhaps in part because he himself viewed the older methods as more elegant, and was surely in part because he wished to avoid controversy with those who viewed the newer methods as less legitimate. Whatever its motives Newton’s procedure was not followed by his most important successors. Since the eighteenth century, modern physics has been analytic physics.

Whether it is formulated in synthetic or in analytic style, classical gravitational theory requires implicit or explicit use of notions pertaining to limits. No rigorous treatment of such notions, whether of the limiting position of a figure or of the limiting value of a function, was available in the seventeenth or eighteenth century. It is not for their rigor that the works of Newton and Laplace are admired. By now a rigorous formulation of classical gravitational theory in analytic style has long been available. To make available a rigorous formulation of classical gravitational theory in synthetic style, to do for the Principia what has long since been done for the Mécanique Céleste, was the goal of the recent book of Hartry Field (1980).¹

A general method for transforming analytic formulations into synthetic reformulations, applicable to any theory of a type common in pregeneral-relativistic, prequantum physics (any theory of one or more scalar- and/or vector-fields on a flat space-time) is implicit in Field’s book, and a modification thereof was made explicit in a recent paper of mine (Burgess, 1984).² In the section on Synthetic Physics in this chapter, a further modification thereof will be outlined (without proofs).

In this version, the process of transforming an analytic formulation into a synthetic reformulation is decomposed into three stages. In the first section, it is indicated how to formalize (using only elementary logic) a preformal analytic physical theory together with the pure analysis applied therein. In the section Invariantizing Ideology, it is indicated how to achieve invariance in the “ideology” of the theory, how to transform it so as to eliminate arbitrariness of the kind involved in fixing a frame of coordinates or scale of measurement. In the section Denumericalizing Ontology, it is indicated how to achieve non-numericality in the “ontology” of the theory, how to further transform it so as to eliminate numbers, functions from and to numbers, and sets of numbers. At both stages, the method involves coding: When objects of one sort are systematically related to objects of another sort, so that a p-tuple of the latter suffices to determine uniquely one of the former, then quantification over the former can be eliminated in favor of p-fold quantification over the latter.

For Field, denumericalization was the primary technical goal, and invariantization was a secondary technical goal. The two goals are independent and can be achieved in either order, although the order adopted here, invariantizing before denumericalizing, is perhaps the easier.³ In any case, neither technical goal is very difficult to achieve. However, Field had also the intuitive goal of achieving these technical goals in an attractively natural manner. In the third section, it is indicated how to yet further transform the theory so as to obtain a less repellently artificial list of primitives and postulates.

At this stage, the method draws on a traditional interpretation of algebra and analysis in classical synthetic geometry (briefly geometric algebra), on which real numbers are identified not with elements of a set-theoretic structure but rather with ratios of magnitudes. (Contrast the contemporary interpretation of classical synthetic geometry in algebra and analysis (briefly algebraic geometry), on which points are identified not with minimal parts of physical space but rather with p-tuples of real numbers.) In the seventeenth century, geometric algebra was viewed as foundationally significant, because the foundations of classical synthetic geometry were viewed as more secure than the foundations of algebra and analysis. It was advocated (somewhat obscurely and confusedly) in the Geometrie of Descartes and (quite clearly and distinctly) in the Universal Arith-metick of Newton. In the nineteenth century, with the introduction of nonclassical geometries and of set theory, there may have been less confidence in its foundational significance, but there was more work on its rigorous development. How much had been achieved by 1900 can be seen from the work cited by Field in this connection, Hubert’s Grundlagen der Geometrie.

Frege (1960), in Grundlagen der Arithmetik, criticized geometric algebra for presupposing and not explicating such notions as the following:

1.   There exist only finitely many Ls.

2.   There exist exactly as many Ls as gs.

Frege proposed to define such notions in terms of classes or concepts. In his book, Field proposed to take them as primitives, and this invocation of logical primitives outside, above, and beyond those of elementary logic has occasioned much philosophical controversy and some technical confusion in the literature. In the fourth section, it is explained why no such auxiliary apparatus as Frege’s classes or concepts or Field’s nonelementary logical primitives is needed.⁴

My hope in offering a new version of an old method is to settle questions as to the technical possibility of producing synthetic alternatives to analytic originals in the case of theories from turn-of-the-century physics. Then assuming for the sake of argument that the method can be adapted to theories from up-to-date physics,⁵ discussion can proceed to questions of its philosophical significance.

Field’s view on such philosophical questions was roughly as follows.⁶ He wished to defend nominalism, in the sense of disbelief in numbers, functions, and sets, against W. V. Quine. However, unlike some other nominalists, he agreed with Quine on several issues. He conceded that disbelieving in numbers, functions, and sets obliges one to disbelieve current analytic physics. He conceded that the use of current analytic physics is in practice indispensable, and he conceded that it is impermissible to use current analytic physics in practice while professing to disbelieve it in principle, unless one believes some alternative physics whose truth would explain the utility of current analytic physics—hence, the significance for Field of producing a synthetic alternative physics.

Field’s view was that, if a synthetic alternative can be produced, then belief in current analytic physics will be unjustifiable. In another recent paper (Burgess, 1983), I argued that such a view cannot be justified by appeal to any recognized or recognizably scientific criteria for the justification of belief but rather must rest on some nonscientific or antiscientific philosophical criterion.⁷ In the section Nominalist Realism of the present chapter, I attempt to uncover what this underlying philosophical criterion may be. I suggest that it may be not an indiscriminate preference for eliminating whatever is eliminable but rather a more discriminating preference for eliminating whatever seems to be “merely contributed by language” and not “a genuine reflection of reality.” It may be realism in the sense in which it is understood and opposed by Quine.

SYNTHETIC PHYSICS

Analysis and Analytic Physics

The pure analysis applied in physics can be formalized (in elementary logic) as follows. The language L₀ has the following two sorts of variables:

It has the following primitives:

The theory A₀ has the following postulates:

A1.	Number-Order, Sum, Product Axioms: The numbers under <, ⊕, ⊗ form an ordered field.
A2.	Set-Extensionality Axiom: Sets having exactly the same numbers as elements are identical.
A3.	Set-Comprehension Scheme: For each formula, the following axiom: There exists a set having exactly those points satisfying the formula as elements.
A4.	Set-Completeness Axiom: Any set contained in an interval (i.e., having lower and upper bounds) is contained in a minimal interval (i.e., has a greatest lower and a least upper bound).

An analytical physical theory of m scalar- and n vector-fields on a flat space-time can be formalized (in elementary logic) as follows. The language L adds to L₀ two further sorts of variables:

x, y, … for points of space-time

X, Y, … for regions of space-time

and the following further primitives,

z ⊳ ζ	z lies in ζ
〈z, ζ〉; (i = 1,2,3)	ζ is the ith spatial coordinate of z
〈z, ζ〉4	ζ is the temporal coordinate of z
|z, ζ|j (l ≤ j ≤ m)	ζ is the measure of the intensity of the jth scalar-field at z
|z, ζ|ki (1 ≤ k ≤ n)	ζ is the measure of the intensity in the direction of the ith spatial coordinate axis of the kth vector-field at z

The theory A adds to A₀ the following further postulates:

A5.	Region-Extensionality Axiom: Regions having exactly the same points lying in them are identical.
A6.	Region-Comprehension Scheme: For each formula, the following axiom: There exists a region having exactly the points satisfying the formula lying in it.
A7.	Coordinate Axioms: Trivial axioms, the same for all theories in the present format, about 〈 〉: the existence and uniqueness of the coordinates of any given point, and of the point having any given numbers as coordinates.
A8.	Intensity Axioms: Trivial axioms, the same for all theories in the present format, about | |: the existence and uniqueness of the measure of intensity of any given scalar-field at any given point, and the existence for any given scalar-field of points at which these measures of intensity are distinct and similarly for vector-fields.
A9.	Special Axioms: Nontrivial axioms, different for each theory in the present format, typically formalizations of differential equations.

Henceforth, to simplify exposition, it will be pretended that m = 1 and n = 0, and measure of intensity of the scalar-field will be abbreviated intensity; also it will be assumed that the theory takes space-time to be nonrelativistic (rather than special-relativistic) and the scalar-field to be of interval type (rather than of ratio type or some other type).

Invariantizing Ideology

Informally the goal of the present section and the strategy for achieving it may be described as follows: The symbols 〈 〉 and | | must be read as “coordinate” and “intensity” relative to some one frame of coordinates and scale of intensity, unspecified but fixed. However, for each theory in the present format, there will be many admissible frames and scales, frames and scales on which the postulates are satisfied. (Which frames and scales will be admissible depends on the mathematical form of the special axioms, which distinguishes the nonrelativistic from the special-relativistic, and the interval- from the ratio-type cases.) The choice of any one out of the many admissible frames and scales is arbitrary. The goal is to eliminate such arbitrary choices.

A formula with ℘ p ≥ 0 free variables will be termed invariant, if and only if, for any p objects and any two admissible frames or scales, the objects either satisfy the formula on both frames or scales, or else they satisfy the formula on neither of the frames or scales: Whether or not the objects satisfy the formula is independent of which frame or scale is fixed. The goal is to produce a language L’ and a theory A’ such that the following hold:

11.   The primitives of L’ are definable in L and are invariant.

12.   For any invariant formula of L, there is a formula of L’ such that the equivalence of the two formulas is deducible in A.

13.   The postulates of A’ are deducible in A.

14.   Any sentence of L’ that is deducible in A is deducible in A’.

Thus, L’ and A are to capture the invariant content of L and A.

The strategy will be to replace postulates asserting that certain conditions hold on some one, arbitrarily chosen frame or scale by postulates asserting that these conditions hold on any admissible frame or scale. Frames and scales cannot be quantified over directly in the formalism; therefore, they must be coded by objects … that can be.⁸ However, then the coding must be such that the following are expressible by invariant formulas:

↓(...)	... code an admissible frame
↑(· · ·)	... code an admissible scale
〈z, ζ, … 〉’	ζ is the ith coordinate of z on the frame coded by …
|z, ζ, … |’	ζ is the intensity at z on the scale coded by …

Formally a change of coordinates from the original, arbitrarily chosen frame to an alternative frame can be represented by a 5-by-4 matrix ξ = (ξ_ij) (0 ≤ i ≤ 4; 1 ≤ j ≤ 4) such that

1.   The submatrix (ξ_ij) (1 ≤ i, j ≤ 4) is nonsingular.

If the x’_i. are the coordinates of x on the original frame, then the coordinates of x on the alternative frame will be the X_j given by the following:

In the nonrelativistic case, the change of coordinates will be admissible if and only if the Galilean conditions hold:

2.   (a) ξ₁₄ = ξ₂₄ = ξ₃₄ = ξ₄₁ = ξ₄₂ = ξ₄₃ = 0

(b) ξ₄₄ ≠ 0, and the submatrix (ξ_ij) (1 ≤ i, j ≤ 3) is orthogonal.

A change of (measure of the) intensity (of the scalar-field) from the original, arbitrarily chosen scale to an alternative scale can be represented by a 2-by-l matrix ν = (v_k) (k = 0,1) such that

3.   ν₁ ≠ 0

If η’ is the intensity at y on the original scale, then the intensity at y on the alternative scale will be η, given by the following:

η = υ₀+υ₁·η’

In the interval-type case, the change of intensity will be admissible if and only if

4.   ν₁ > 0

The coding of a frame or scale by a matrix ξ or ν only indicates how it is obtained from the original, arbitrarily chosen frame or scale. Thus, the following cannot be expressed by invariant formulas:

ζ is the ith coordinate of z on the frame coded by ξ.

ζ is the intensity at z on the scale coded by ν

However, another coding is available, based on the observation that a frame or scale can be determined uniquely by the matrix χ or η of coordinates or intensities at a quintuple χ or pair y of suitable points. Writing out quantification over p-by-q matrices as pq-fold quantification over numbers, the following can be expressed by invariant formulas:

Let £’ have the following primitives:

For each sentence P of l, if 〈 〉 ’, and | |’ do not occur in P let P’ = . Otherwise, let P”(x,x,y,n) be the result of substituting throughout 〈…, χ, χ〈’, and |…, y, η I’ for 〈 … 〉 and |…|let 2P’ be the following:

Let A’ have as postulates the P’ for P a postulate of si plus the following:

Then 11–14 can be proved.

Denumericalizing Ontology

Informally the goal of the present section and the strategy for achieving it may be described as follows: Call a formula nonnumerical, if no number- or set-variables occur free in it. The goal is to produce a language L’* with only point- and region-variables and a theory A’* such that

N1.  The primitives of L’* are definable in L’ and are nonnumerical.

N2.  For every nonnumerical formula of L’, there exists a formula of L’* such that the equivalence of the two formula is deducible in A’.

N3.  The postulates of A’* are deducible in A’.

N4.  Any sentence of L’* that is deducible in A’ is deducible in A’*.

Thus, L’* and A’* are to capture the nonnumerical content of L’ and A’.

The strategy is the trivial one of coding numbers and sets by points and regions. Any triple of points (x₀,x₁,x) satisfying

◊(x₀,x₁,x) x₀, x₁ are distinct, and x₀,x₁, x are collinear

determines a unique real number ξ, satisfying

[x0, x1, x,ξ]

ξ is the ratio of the distance between x0 and χ to that between x0 and x1

(Here sign is determined by orientation: The sign of ξ is positive or negative according as x, and χ lie on the same or opposite side of x₀.) Every real number is determined by such a triple of points, and two such triples (x₀, x,, x) and (y₀, y_x, y) determine the same real number if and only if they satisfy the following:

x0,x1,x ~ y0,y1,y

The ratio of the distance between x0 and χ to that between x0 and xx equals the ratio of the distance between y0 and y to that between y0 and yl.

On the other hand, in the traditional symbolism of proportionality:

x₀x:x₀x, :: y₀y:y₀y₁

If this is understood as implying or presupposing ◊(x₀, x,, x) and ◊(y₀, y,, y), then ◊ is definable in terms of ~:

◊(x₀,x,x) ⇔ (x₀,x,x ~ x₀,x_l,x).

Formally the following are expressible by invariant, nonnumerical formula of X, hence, by nonnumerical formulas of X’:

[χ0,χρχ,ξ]	For each (‘, the ith coordinate of χ equals the ith coordinate of x0 plus ξ times the ith coordinate of x,.
x0, xj, x ~ y0, y,, y	∃ξ∃v([x0, x1, x,ξ] Λ [y0, y1, y, v]) Λ νξνναχ[(χρχ,ξ] Λ ·£ ξ = v).

Similarly we have $* for every other primitive $ of ?’.

Let ?’* have the following primitives:

For each formula 9¹ of if no number- or set-variables occur free or bound in P let P”* = 9”. Otherwise associate with each such variable ξ or ξ distinct variables x₀, x_x, x or x₀, x_x, X not already occurring in 9*, and let ty* be the result of substituting throughout as follows:

and similarly for other primitives $ of L’. Let A’* have as postulates:

AO*. P* for P· a (logical) identity postulate of L’

plus P* for P* a (nonlogical) postulate of si’. Then N1–N4 can be proved.

Synthetic Physics By Means of Geometric Algebra

The goal of the present section is to produce a language and a theory with a more attractively natural list of primitives and postulates.

The following are expressible by invariant, nonnumerical formulas of L hence by nonnumerical formulas of L’, hence by formulas of L’:

Bxyz	y lies between χ and z.
Cxyrs	χ and y and r and s are simultaneous, and the distance between χ and y equals that between r and s.
Dxyzrst	The ratio of the difference in intensity between χ and ζ to that between χ and y equals the ratio of the distance between r and t to that between r and s.
Exy	The intensity at x is less than that at y.

Let L’ z^# have as primitives the following:

B, C, D, E, L

(Here, C, E are appropriate to the nonrelativistic, interval-type case; in other cases, other primitives would be used.) Using (slight extensions of well-known) results in geometric algebra, it can be proved that the primitives of L’*^# are definable from the primitives of L’* and conversely, in the converse direction proceeding as follows:

For each sentence P of L’* let P^# be the result of substituting throughout for the primitives of L’* their definitions in terms of the primitives of L’*^# Let A‘^*# have the following postulates:

S1.  Basic Space-Time or Point-Order Axioms: The points under β form a four-dimensional affine space.

S2.  Region-Extensionality Axiom: A5.

S3.  Region-Comprehension Scheme: A6.

S4.  Region-Completeness Axiom: Every region contained in an interval is contained in a minimal interval.

S5.  Further Space-Time Axioms: In the nonrelativistic case, simultaneity is a relation of equivalence, and the points equivalent to any given point under B, C form a three-dimensional Euclidean space.

S6.  Basic Intensity Axioms: Trivial axioms, the same for all theories in the present format, about D.

S7.  Further Intensity Axioms: Trivial axioms, differing in the interval-type case from the ratio-type case but the same for all theories in the interval-type case, about E.

S8.  Special Axioms: A9’*^#.

Using (slight extensions of well-known) results on geometric algebra, it can be proved that, for any sentence P of L’*, P is deducible in A‘* if and only if, P^# is deducible in A‘*^#; in deducing P^# for P, a postulate of A‘* from the postulates of A‘^*# proceeding as follows:

In the literature of geometric algebra, the correspondence between affine space axioms and ordered field axioms is worked out in great detail (e.g., Desargues’ Theorem corresponds to the distributive law).

L’^*# and A‘^*# provide a formalization (in elementary logic) of a synthetic alternative to the original analytic physical theory formalized as in L and A.

Why Non-Elementary Logic Is Not Needed

There are two quite distinct ways in which one might suspect that auxiliary apparatus beyond the primitives of L’^*# and postulates of A‘^*# would be needed in synthetic physics.

On the one hand, one might suspect that some such auxiliary apparatus would be needed at some stage for the sake of the transformation of analytic into synthetic physics. One might suspect that, without it, the definition of some notion or the deduction of some result could not be carried out (contrary to what has been asserted in the previous sections). Such suspicions might be suggested by reading Frege (1960, pp. 25–26), who wrote:

Newton proposes to understand by number not so much a set of units as the relation in the abstract between any given magnitude and another magnitude of the same kind taken as unity. It may be granted that this is an apt description of number in the wider sense, in which it includes also fractions and irrational numbers; but it presupposes the concepts of magnitude and of relation in respect of magnitude. This should presumably mean that a definition of number in the narrower sense, or cardinal Number, will still be needed; for Euclid, in order to define the identity of two ratios between lengths, makes use of the concept of equimultiples, and equimultiples bring us back once again to numerical identity.

One might suspect that, in order to define notions pertaining to “identity of ratios” (like ~ and D above) from attractively natural primitives, the notion of “equimultiples,” or auxiliary apparatus in terms of which that notion can be defined (such as Frege’s classes or concepts, or Field’s nonelementary logicalprimitives) would be needed. Indeed Field invoked his auxiliary apparatus in just such a connection. Moreover, the invocation of some such auxiliary apparatus seems unavoidable, if one follows Euclid (who follows Eudoxus) in the treatment of identity of ratios.⁹

However, one need not follow Eudoxus (as Hubert in effect does not). As for ~, it can be defined in elementary logic from the notions of collinearity, parallelism, and intersection, which (as Field in effect noted) can be defined in elementary logic from the attractively natural primitive B. On one such definition, the general case is merely a combination of various special cases, each of which is readily understood on drawing a figure. For example, if x₀ = y₀ but is not collinear with x₁ and y₁, then x₀, x₁, x ~ y₀, y₁, y, if and only if the lines through x₁ and y₁ and through x and y are parallel; whereas, if the lines through x₀ and x₁ and through y₀ and y_x are parallel, then x₀, x,, x ~ y₀, y₁, y, if and only if the lines through x₀ and y₀, through x₁ and y₁ and through x and y intersect in a common point.

As for D, because it seems sufficiently attractively natural itself, the simplest procedure (although not the only possible procedure) is to admit it as a primitive. It expresses an idealization of what is established in the course of operationally determining the measure of the intensity of a scalar field using an instrument. For example, in determining the Celsius temperature of a sample using a mercury thermometer, what is established is roughly that Dxyzrst holds, where the following hold:

r is at the 0° mark on the thermometer.

s is at the 100° mark on the thermometer.

t is at the top of the column of mercury,

x is in a sample of freezing water.

y is in a sample of boiling water.

z is at the bottom of the thermometer, in the sample where Celsius temperature is being determined.

So much for suspicions of the first kind.

On the other hand, one might suspect that some such auxiliary apparatus would be needed for its own sake in analytic physics, and would need to be preserved in transforming analytic physics into synthetic physics. One might suspect that, without it, L and A would be inadequate for the formalization of preformal analytic physics (contrary to what has been asserted in the first section). Such suspicions might also be suggested by reading Frege, who continued with the following:

However, let it be, as it may be, the case that identity of ratios between lengths can in fact be defined without any reference to the concept of number. Even so, we should still remain in doubt as to how the number defined geometrically in this way is related to the number of ordinary life, which would then be entirely cut off from science… . [T]he question arises whether arithmetic itself can make do with a geometrical concept of number, when we think of some of the notions that occur in it, such as the Number of roots of an equation. … On the other hand, the number which gives the answer to the question How many? can answer among other things how many units are contained in a length.

Indeed, in analytic physics, one needs to be able to count, to answer “how many” points, regions, numbers, or sets satisfy some given condition.

However, notions and results pertaining to arithmetic and counting, to natural numbers (nonnegative integers, finite cardinals) can be developed in X and si without auxiliary apparatus using coding. Such use of coding might be illegitimate, if the goal were the Fregean one of providing an exegesis or explication of “the number of ordinary life.” However, the Fieldian goal is a different one. In outline, one procedure for developing arithmetic and counting is as follows:

First, the usual definition of ξ is a natural number, namely ξ belongs to every set containing zero and closed under adding one can readily be formalized. Next, using it, notions and results pertaining to the decimal expansion of real numbers can be formalized. These include notions and results pertaining to the usual coding of ordered pairs of real numbers by real numbers, which involves shuffling. For example, if the following:

ξ = 3.14159 …

ν = 2.71828

then (ξ,ν) is coded by

ζ = 32.1741185298… .

A function from real numbers to sets of real numbers can be coded by the set of ordered pairs (ξ,ν), with ν an element of the value of the function for argument ξ; thus, such functions can now be coded by sets of real numbers. Thus, the usual definition as follows:

There exist exactly ξ-many ξ such that ? (ξ).

Namely ξ is a natural number, and there exists a bijective function from the coding. Notions and results pertaining to the counting of sets can be formalized. Counting of numbers can be treated similarly (or reduced to counting of singleton sets). Finally, using the coordinate representation of points by ordered quadruples of real numbers and the coding of ordered quadruple real numbers by real numbers, counting of points and regions can be reduced to counting of numbers and sets. Field’s quantifiers can be defined and, thus, are not needed as primitives. So much for suspicions of the second kind.

NOMINALIST REALISM

The label realism is one that is often used in connection with disagreements like that between Field and Quine. Unfortunately it is also one that has been used in many different senses in traditional and recent philosophy. Precedents could be cited for attaching it to any one of a large number of theses, each susceptible to any one of a large number of interpretations. Towards sorting out some of these senses, it may be well to begin with a quotation from the Pragmatism of William James, which Morris Kline (1985) commented on.¹⁰ It describes an attitude towards the theories then current held by many from the time of Plato to the time of Newton:

When the first mathematical, logical and natural uniformities, the first laws, were discovered, men were so carried away by the clearness, beauty and simplification that resulted that they believed themselves to have deciphered authentically the external thoughts of the Almighty. His mind also thundered and reverberated in syllogisms. He also thought in conic sections, squares and roots and ratios, and geometrized like Euclid. He made Kepler’s laws for the planets to follow; he made velocity increase proportionally to the time in falling bodies; … and when we rediscover any one of these wondrous institutions, we seize his mind in its very literal intention.

The attitude thus described may be summarized as follows:

(0)  Current theory is an image in the human mind of the creative design in the Divine Mind.

Historically (0) seems to have been held in conjunction with most or all of the following:

(1)  The world is a projection of the creative design in the Divine Mind.

(2)  Current theory is a reflection in the human mind of the world.

(3)  One is justified in believing a theory, if and only if (or if and only if one is justified in believing that) it is a reflection in the human mind of the world.

(4)  One is justified in believing current theory.

Logically (1) and (2) together imply (0); whereas, given (3), (2) and (4) imply each other. On one interpretation, namely on an interpretation of the metaphor of reflection emphasizing the causal connection between any element of the image inside the mirror and the corresponding element of the scene outside the mirror, (3) implies the following:

(5)  One is justified in believing a theory involving the hypothesis that objects of a given sort exist, only if the presence of that hypothesis in the theory is caused by the presence of objects of that given in the world.

Field and Quine agreed in roughly identifying nominalism with the denial and antinominalism with the affirmation of the following:

(6)  (One is justified in believing (theories involving the hypothesis) that) numbers, functions, and sets exist.¹¹

Field is a nominalist, Quine an antinominalist. To see the pertinence of (0)-(5) to their disagreement over (6), note that they seem to agree as follows:

(A)  Current theory involves the hypothesis that numbers, functions, and sets exist.

(B)  Numbers, functions, and sets are not physical and do not physically cause belief (in theories involving the hypothesis) that they exist.

(C)  No objects aphysically cause any beliefs.

Field, unlike some nominalists, rejected reinterpretations of current theory denying (A).¹² Quine, unlike some antinominalists, would view as confused anyone who denied (B) and took numbers to be physical (or for that matter, mental), and as superstitious anyone who denied (C) and took beliefs to be effects of ecto-physical, hyperphysical, or paraphysical causes, or of extrasensory, suprasenso-ry, or praetersensory perceptions. Now given (A), (4) implies (6); whereas, given (B) and (C), (5) implies not-(6).

Quine roughly identified realism with the affirmation and antirealism with the denial of (3). Quine is an antirealist. He wrote (1963, pp. 78–79):

The fundamental-sounding philosophical question, How much of our science is merely contributed by language and how much is a genuine reflection of reality? is perhaps a spurious question which itself arises wholly from a certain particular type of language. Certainly we are in a predicament if we try to answer the question; for to answer the question we must talk about the world as well as about language, and to talk about the world we must already impose upon the world some conceptual scheme peculiar to our own special language. … We can improve our conceptual scheme, our philosophy, bit by bit while continuing to depend on it for support; but we cannot detach ourselves from it and compare it objectively with an unconceptualized reality. Hence it is meaningless, I suggest, to inquire into the absolute correctness of a conceptual scheme as a mirror of reality. Our standard for appraising basic changes of conceptual scheme must be, not a realistic standard of correspondence to reality, but a pragmatic standard. Concepts are language, and the purpose of concepts and of language is efficacy in communication and in prediction. Such is the ultimate duty of language, science, and philosophy, and it is in relation to that duty that a conceptual scheme has finally to be appraised.

Indeed Quine’s antinominalism is a consequence of his antirealism. He recognized that current theory, involving the hypothesis that numbers, functions and sets exist, is extremely efficacious; and it is because he recognized no criterion as more important for science than that of efficacy in communication and prediction, and recognized no criteria exterior, superior, or ulterior to those of science, that he viewed belief in the existence of numbers, functions, and sets as justified.

I suggest that similarly Field’s nominalism may be a consequence of realism. (Realism in Quine’s sense; that is, in Field’s own usage, realism amounts to little more than antinominalism, and antirealism to little more than nominalism.) In a recent paper, Field described what he took to be “probably the main ground for suspicion” about numbers, functions, and sets:¹³

According to the Platonist picture, the truth-values of our mathematical assertions depend on facts involving Platonic entities that reside in a realm outside of space-time. There are no causal connections between the entities in the Platonic realm and ourselves; how then can we know what is going on in that realm? … It seems as if to answer these questions one is going to have to postulate some aphysical connection, some mysterious mental grasping, between ourselves and the elements of this Platonic realm.

Changing rhetorical questions to forthright denials and suppressing the term of abuse “Platonic” and the technical jargon “truth conditions,” this seems to amount to an argument from overt premises, like (B) and (C) to a conclusion like not-(6). Such an argument requires a covert premise, like the causal interpretation (5) of the realist criterion (3) for justification of belief. Presumably, for Field, both conclusion and covert premise would carry a proviso: One is not justified in believing certain theories unless no suitable alternative theories can be produced. Producing such suitable alternative theories is for Field the main part of the argument for not-(6).

In a footnote to the quoted passage, Field cited a well-known paper of Paul Benacerraf. Benacerraf propounded (without pretending to be able to resolve) the following dilemma: There seem to be initially plausible arguments both for the negative conclusion that we cannot have knowledge of numbers and also for the positive conclusion that we do have knowledge of numbers. Field in effect paraphrased the argument for the negative horn of Benacerraf’s dilemma. In his paper, Benacerraf (1983) described this argument as resting on a causal criterion for knowledge, and, writing with Hilary Putnam in the editorial introduction to the new edition of their well-known anthology (Benacerraf & Putnam, 1983), Benacerraf described much opposition to Quine as resting on realist assumptions.¹⁴ What I am suggesting is that the causal criterion is itself a realist assumption and that Field is one of the opponents of Quine whose opposition rests on realist assumptions.

The disagreement between Quine’s overt antirealism and what seems to be Field’s covert realism is broad. It arises not only in philosophy of mathematics but also in philosophy of semantics, where the Quinean disquotational theory of truth stands opposed to the Fieldian reductionist theory of truth.¹⁵ The disagreement between Quinean antirealism and what seems to be Fieldian realism is also deep. Quineans reject the goal of reflection not merely as unimportant in comparison to the goal of efficacy but rather as unintelligible.

In his book, Field advocated achieving economy or parsimony in ontology and ideology through extravagance and prodigality in logic. To Quineans, the suggestion that theory would better reflect the world, if its ontology and ideology where deflated and its logic inflated, is unintelligible. For Quineans, the very distinction ontology/ideology/logic is merely a “recapitulation” of the distinction noun/verb/sentence, which is only a feature of the “philology” of a particular, special kind of language and not something imposed on us by the extra-linguistic world.

Quine’s favorite example of a language lacking noun phrases (including pronouns or variables) and, hence, presumably lacking ontology, is predicate-functor language. Quine has told us that “variables [can be] explained away,” and that “to be is to be the value of a variable.” Hence, presumably being can be explained away; having this sort of ontology rather than that sort of ontology cannot be something imposed on us by the extralinguistic world, because having an ontology at all is not something so imposed.)

I suggest that the disagreement between Quine and Field is so broad and so deep that it cannot be settled by any mere technical trick—not even by one that out-Newtons Newton.

ACKNOWLEDGMENTS

Over the past several years, through the kindness of several hosts, I have had the opportunity of presenting several versions of several sections of the present paper under several titles to several audiences, and have benefited much from the ensuing discussion. Among the many persons to whom I have thus become indebted, C. Wade Savage and Michael Kremer merit special mention as having inspired (or provoked) the inclusion of the final two sections of this chapter.

For my understanding of most of the issues surrounding the work of Hartry Field I am indebted to Saul Kripke—so much so that I have omitted discussion of many of these issues, considering that they should best be left to Kripke himself, whose long-awaited commentary on Field’s work I hope will soon be forthcoming.

REFERENCES

Benacerraf, P. (1983). Mathematical truth. Reprinted in (Benacerraf & Putnam, pp. 403–420.

Benacerraf, P., & Putnam, H. (Eds.). (1983). Philosophy of mathematics (2nd ed.). Cambridge: Cambridge University Press.

Burgess, J. P. (1983). Why I am not a nominalist. Notre Dame Journal of Formal Logic, 24, 93–105.

Burgess, J. P. (1984). Synthetic mechanics. Journal of Philosophical Logic, 13, 379–395.

Field, H. (1972). Tarski’s theory of truth. J. Phil., 69, 347–375.

Field, H. (1980). Science without numbers. Princeton: Princeton University Press.

Field, H. (1983). Realism and anti-realism about mathematics. Philosophical Topics, 13, 45–69.

Frege, G. (1960). Foundations of arithmetic. (J. L. Austin, Trans.) New York: Harper & Row.

Kline, M. (1985). Mathematics and the search for knowledge. Oxford: Oxford University Press.

Quine, W. V. O. (1963). Identity, ostension, and hypostasis. Reprinted in From a logical point of view (2nd ed.). New York: Harper & Row.

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¹Field wrote of “platonist” versus “nominalist” rather than “analytic” versus “synthetic” physics, but alluded to the analogy with analytic versus synthetic geometry on p. 44.

²More proofs are provided there than here.

³For the importance Field attaches to invariance (and the closely related feature of intrinsicness), see the many passages listed in the index to his book. He alluded to the possibility of invariantizing without denumericalizing on p. 49.

⁴The above (finite) cardinality-comparison quantifiers are by no means the only items of nonelementary logical apparatus invoked by Field, but they are the only ones that will be discussed in the present chapter. Field himself contemplated dropping them in his book (chapter 9, Part II).

⁵Field offered some hints for adaptation to the general-relativistic case in his book (p. 123).

⁶See his book, “Preliminary Remarks,” pp. 1–6.

⁷Especially Part 4, “Revolutionary Nominalism.” The judgment that theories of one type are superior to theories of another type by scientific criteria has revisionary implications for the practice of (education and research in pure) science: In branches where a theory of the type judged superior have been found, it will be taught; in branches where a theory of the type judged superior has not yet been found, one will be sought. Field seems to have disclaimed revisionary implications for the practice of science in his book (p. 90)

⁸The strategy is, thus, to introduce parameters (… coding frames and scales) and then quantify out. Field made occasional use of this strategy in his book (e.g. bottom p. 76 to top p. 77-where the parameters a.b.c and a’.b’, c’ in effect determine a frame).

⁹Field followed Eudoxus in defining <_{st, Scal} in terms of his primitive S-Cong in his book (chapter 8, Part B, pp. 64–67). In the notation of the present chapter, this amounts roughly to defining D above in terms of:

Fxyrs The difference in intensity between x and y equals that between r and s.

He used both his (finite) cardinality-comparison quantifiers and also a continuity axiom. The latter is needed, the former are not.

¹⁰Chapter XII contains many quotations from many authors, all in a similar vein, with references and commentary.

¹¹Both Field and Quine tended to “ascend” from questions of the status of numbers, functions, and sets to questions of the status of theories—hence, the parenthetical clauses. Field tended to write of “belief,” Quine of “assertion;” but the possibility of insincerity or assertion-without-belief did not seem to be an issue between them.

¹²See his (unfavorable) remarks on Charles Chihara’s strategy of reinterpretation in several passages (listed in the index) in his book.

¹³I am indebted to Field for providing me with a copy of this paper prior to publication. I am informed that it has subsequently appeared (1983).

¹⁴In the editorial introduction, one finds:

But why should the simplest … theory … have any tendency to be true? Quine, good pragmatist that he is, tends to pooh-pooh such questions; but more realistically minded philosophers are sure to be troubled, (p 35)

The question “Why should one believe that there is a preestablished harmony between our feelings of simplicity … and truth?” . . . assumes a “realist” notion of truth, (p. 37)

On Quine’s disquotational theory of truth, belief in the hypothesis that a theory is true is not importantly different from belief in the theory itself and is justified by pragmatic criteria of efficacy, including simplicity.

¹⁵See Field (1972). The literature to which this paper has given rise cannot be adequately summarized here. Roughly Field viewed the presence of the phrase “is true” in current language as illegitimate pending the production of a reductive definition that would show, say, an utterance or inscription of “white” being true of snow in English to amount to air or ink, snow, and the nervous tissue of English-speakers being physically related in some special way. Roughly Quine (1963) denied that the function of the phrase is to reflect any physical or aphysical relation among any physical or aphysical objects in the extralinguistic world; rather the function of the phrase is the intralinguistic one of disquotation, which contributes to efficacy in communication and, thus, justifies the presence of the phrase in current language. The disagreement between Quine and Field over “is true” seems to me exactly analogous to their disagreement over “two.”