A Geometric Strategy
A Geometric Strategy
Abstract and Keywords
Develops in some detail a strategy of nominalistic interpretation that assumes that points of spacetime are legitimate, concrete, physical entities. What makes the strategy possible is the fact that analytic geometry in the style of Descartes can be interpreted in synthetic geometry in the style of Euclid, using triples of points to represent real numbers (namely, as ratios of pairs of line segments connecting each of a pair of points with a third). This strategy can handle classical theories based on Euclidean space and special‐relativistic theories based on Minkowski space. The extension of the strategy to general relativity and quantum mechanics remains to be worked out, as does the treatment of the higher branch of geometry known as descriptive set theory.
Keywords: analytic geometry, Descartes, descriptive set theory, Euclid, general relativity, Minkowski, quantum mechanics, spacetime, special relativity, synthetic geometry
0. Overview
The geometric strategy of nominalistic reconstruction faces two main tasks, one technical, one philosophical. On the technical side, geometric nominalism seeks to eliminate numerical entities in favour of geometric entities. In traditional mathematical as opposed to philosophical usage, pure geometry in the Euclidean style is called synthetic, and coordinate geometry in the Cartesian style is called analytic. The technical task may thus be described as that of producing synthetic alternatives to standard analytic formulations of scientific theories. On the philosophical side, geometric nominalism seeks to persuade nominalists to be indulgent towards geometricalia, to admit them as concrete. In contemporary philosophical usage, substantival and relational views of space are understood as the acceptance and the rejection of such geometric entities as points and regions of space. The philosophical task may thus be described as that of defending substantivalism against relationalism against a background of nominalism. On both sides, geometric nominalism has deep historical roots in the work of early modern physicists and geometers, which can only be brief noted in the present chapter. (For more information, the reader is referred to the standard reference work Kline (1972) on the history of mathematics and the basic survey Sklar (1974) of the philosophy of space and time.) It also has extensive intellectual debts to more recent logicians and philosophers.
Presumably the alternatives to standard formulations of scientific theories that geometric nominalism seeks to provide should be more ‘elegant’ or ‘attractive’ than the instrumentalist ‘theory’ consisting of the bare assertion that concreta behave ‘as if’ abstracta existed and standard scientific theories were true. Just how much more ‘elegant’ the alternatives must be depends on just what the grounds for dissatisfaction with the ‘as if’ theory are supposed to be. Surely the theories for which geometric nominalism seeks to provide alternatives to the standard formulations should be ‘realistic’, not in any philosophical sense, but in the everyday sense of being based on the most up‐to‐date science. For a philosophical (p.98) thesis should not be based on a scientific falsehood. The main dilemma for geometrical nominalism is this: the case for accepting geometric entities as concrete draws on realistic, contemporary, twentieth‐century physics; but the most elegant elimination of numerical entities in favour of such geometric entities can be carried out only for unrealistic, classical, nineteenth‐century physics. It remains an open question how attractive a nominalistic alternative to up‐to‐date physics can be developed.
This chapter provides a brief discussion of background on syntheticism and substantivalism in section 1. An elegant treatment of unrealistic physics is exhibited in some detail in sections 2–4, both in order to indicate the kind of thing one might hope some day to achieve for more realistic physics, and in order to provide background to a discussion in article 5.a of the difficulties standing in the way of such an achievement at present, and in the optional semi‐technical appendix article 5.b of a conceivable further difficulty that might arise in the future.
1. Historical Background
Throughout the early modern period it was the consensus view that the basic objects of algebra, real numbers, are to be identified with ratios of lengths, areas, or volumes, and that the basic operations and laws of algebra are to be explained and justified in terms of geometric constructions and theorems. Even the pioneers who introduced coordinate methods, and thereby made the techniques of algebra and calculus available for application to problems of geometry and mechanics, still regarded the older pure geometry as providing the foundation for algebra. Such an attitude is hinted at, somewhat obscurely and confusingly, in the opening paragraphs of René Descartes's Géométrie, and is expounded, clearly and distinctly, in the opening pages of Isaac Newton's Universal Arithmetick. Contemporary geometric nominalism goes further than the early modern consensus by requiring that analytic methods, however useful in the context of discovery, are not to be mentioned in the context of justification. But even for this there is precedent in Newton's practice in his Principia, though his example was not imitated by many of his successors, such as Pierre Simon de Laplace in the Mécanique Céleste. Present‐day geometric nominalism, of course, would wish to conform to twentieth‐century standards of rigour, which are higher than were those of the seventeenth century. A more rigorous synthetic geometry and geometric algebra than that of Descartes or Newton is provided by the famous (p.99) monograph of their successor David Hilbert (Hilbert 1900). Further logical refinements can be found in the work of Alfred Tarski and his school, beginning with Tarski (1959).
Long before the time of Hilbert and Tarski, however, the foundational significance of the reduction of algebra to geometry had come to seem doubtful. For the discovery of non‐Euclidean geometry in the early 1800s undermined the early modern consensus according to which geometry was a foundation for algebra and analysis, doubts being expressed already by C. F. Gauss, one of the main co‐discoverers of the new geometries. Already with Gauss, and more explicitly and emphatically with Bernhard Riemann, one finds the view, which has become the later modern consensus, that one must distinguish mathematical geometry from physical geometry. One should perhaps speak rather of mathematical ‘geometries’ in the plural, for they are legion. These myriad geometries are not conflicting opinions about some one and the same object, space, but rather are definitions of various different classes of mathematical ‘spaces’. If one does speak after all of mathematical ‘geometry’ in the singular, it must be understood as the comparative studies of all these different ‘spaces’. (These mathematical ‘spaces’ include, besides those mathematical structures, such as ‘Riemannian manifolds’, that have at one time or another been proposed as images of physical space, many others, namely, all those, such as ‘Hilbert space’, that are sufficiently similar to allow them to be fruitfully investigated by similar methods, regardless of what the nature of the applications if any of these other structures may be.)
Inverting the earlier order of things, the ‘points’ of these ‘spaces’ are now standardly taken to be set‐theoretically generated out of real numbers, so that algebra and analysis become the foundation rather than the superstructure. In effect, a ‘point’ of a k‐dimensional mathematical ‘space’ is often simply identified with the k‐tuple real numbers that are its coordinates. This inversion was an important motive for the search for a new, non‐geometric foundation for the numerical side of mathematics, now standardly taken to be provided by the set‐theoretic generation of the real numbers out of the natural numbers, themselves set‐theoretically generated (a process brief outlined in article I.B.1.a). In so far as it seeks to restore geometry as a foundation for mathematics, geometric nominalism is not revolutionary but counter‐revolutionary.
According to the later modern consensus, mathematicians collect various different mathematical ‘spaces’, while physicists select a single one from among them as an image of physical space, the selection being made (p.100) on empirical grounds. However, physical geometry, by itself and without auxiliary hypotheses, makes no empirical predictions. For instance, an attempt to test the Euclidean vs. non‐Euclidean hypotheses about the sum of angles in a triangle by surveying some large triangle on earth or in the heavens, as was done by Gauss, never tests just these geometric hypotheses alone. For even the use of the most low‐tech surveying instruments, the sextant and the plumb, involves auxiliary hypotheses about light and weight: the hypotheses that light travels, and weights fall, in straight lines. Ultimately the selection of a physical geometry is inseparable from the choice of an electromagnetic theory and of a gravitational theory, a point much emphasized by Henri Poincaré. Thus while mathematical geometry becomes a branch of the theory of sets, physical geometry becomes a component of the theory of electromagnetism and gravitation.
But one perhaps should not use the term ‘geometry’—or any of the terms in the spatial or temporal column in the adjoining table. For whether it is acceptable to speak in physics of spatial ‘points’ and so forth depends on whether it is acceptable to speak of absolute rest, to speak of being at the same place on different occasions.
Terminology
Spatial 
Temporal 
Combined 
Neutral 

geometry 
chronometry 
geometry‐chronometry 
kinematics 
points 
instants 
point‐instants 
events 
space 
time 
space‐time 
world 
Now though absolutism vs. relativism was a live issue as regards rest around 1700, and as regards simultaneity was still a live issue around 1900, the victory of relativism since the work of Albert Einstein has been complete. On further thought, perhaps precisely because the defeat of absolutism has been so very complete there will no longer be any serious danger of confusion if one after all uses ‘geometry’ and other spatial and temporal terms, understanding them as colloquial abbreviations for the more cumbersome but more accurate expressions in the combined or neutral columns of the table.
(p.101) Contemporary geometric nominalism is committed to substantivalism as opposed to relationalism, to the acceptance as opposed to the rejection of such geometric entities as space, regions, and points. Centuries ago, when Newton and G. W. von Leibniz were debating these issues, substantivalism and relationalism were tightly intertwined with absolutism and relativism, respectively. Fortunately for geometric nominalism, in the course of the long history of such debates the two issues became disentangled. Indeed, present‐day substantivalist arguments, of which the best‐known is perhaps that in the position paper Earman (1970), are based specifically on relativistic considerations.
One consequence of the kind of division of labour instanced by the distinction between mathematical and physical geometry is that it permits mathematics to progress by addition, while physics has to progress by amendment. And there have been, since the nineteenth century, several successive amendments to the Newtonian physical geometry. First, there was non‐relativistic geometry, implicit in classical gravitational theory. Second, there was and is special‐relativistic geometry, implicit in classical electromagnetic theory—though it took quite a bit of work to make it explicit—and retained by the quantum theories that provide the best currently available accounts of electromagnetic and weak and strong nuclear forces. Third, there is general‐relativistic geometry, explicit in the best currently available theory of gravitation. Fourth, since the best currently available theories of electromagnetic and nuclear phenomena on the one hand and of gravitational phenomena on the other hand are each presumed to require amendment in order to take account of the other, there will eventually have to be a super‐relativistic geometry in the much hoped for ‘final theory of everything’. At present there is no consensus even as to how many dimensions the geometry of a theory unifying quantum mechanics and general relativity would ascribe to physical space. (Figures as high as 26 have been mentioned.)
Though the occurrence of amendments has seemed to many to argue, if not in favour of founding mathematics on set theory, at least against syntheticism, against founding mathematics on geometry, still the content of the amendments has seemed to many to argue in favour of substantivalism, in favour of regarding geometric‐chronometric entities as physical. One main consideration arises already in connection with the shift from the matter‐theoretic standpoint of classical, Newtonian gravitational theory to the field‐theoretic standpoint of classical, Maxwellian electromagnetic theory. In the latter, charged bodies do not act instantaneously at a distance by electromagnetism on other charged bodies, as in (p.102) the former massive bodies act instantaneously at a distance by gravity on other massive bodies. Rather, electromagnetic action propagates at a finite speed, that of light, and as a consequence, much of the energy of a physical system cannot be localized in material bodies, but must rather be ascribed to a force‐field between them extending throughout space. (When the theory is formulated special‐relativistically, the very distinction between mass and energy blurs.) According to many philosophical commentators, the force‐field must be considered to be a physical entity, and as the distinction between space and the force‐field may be considered to be merely verbal, space itself may be considered to be a physical entity.
The other main consideration arises in connection with the shift to general‐relativistic, Einsteinian gravitational theory. This blurs or abolishes, as regards gravity, the distinction between kinematics and dynamics, or between inertial and accelerated motion. The presence of massive bodies is not taken to exert a gravitational force deflecting the motion of other bodies from the straight path in which they would otherwise move. Rather, it is taken to deform space itself, making the straightest paths available for a body to move in more curved than they would have been in the absence of those massive bodies. According to many philosophical commentators, space may hence be considered a causal patient and agent, shaped by the presence of massive bodies, and constraining the motion of other bodies by the paths it makes available. However reactionary it may be in seeking to base algebra and analysis on physical geometry, geometric nominalism is thoroughly progressive in its substantivalism, in so far as that substantivalism is based on such considerations as the foregoing. It must be recognized, however, that just how much support such considerations can provide for the acceptance of geometricalia by nominalists depends in part on just what is supposed to be the objection to abstracta, the motivation for nominalism, in the first place.
2. Technical Background
At a very fundamental level of mathematical geometry comes affine basic coordinate plane geometry. Here plane (as opposed to solid or hyperspace) of course indicates a geometry of two (as opposed to three or four) dimensions, while coordinate (as opposed to pure) indicates an analytic theory involving both geometrical and numerical entities rather than a synthetic theory involving only the former and avoiding the latter. Here also basic (as opposed to intermediate or higher) indicates a theory in which the geometric entities are points (as opposed to special (p.103) regions or arbitrary regions), while affine indicates a kind of minimal theory common to Euclidean and several non‐Euclidean geometries.
The geometry with the above long name may be formalized as a two‐sorted theory T ^{†} in a two‐sorted language L ^{†}. The language L ^{†} will have variables x, y, z, . . . for plane points, and variables X, Y, Z, . . . for real numbers. It will have one primary primitive, for the order relation among points, ‘y lies between x and z’. It will have the three secondary primitives of analysis for the order, sum, and product relations among real numbers, but not for integrity. And it will have two mixed primitives for (preferred) coordinates:
The theory T ^{†} will have no primary axioms. It will have the secondary axioms of analysis (in the sense of article I.B.1.b) except those for integrity, the basic algebraic axioms for order, sum, and product, and the continuity scheme. It will have two mixed axioms of (preferential) coordination:
 X is the horizontal coordinate of x
 (on some preferred coordinate system)
 X is the vertical coordinate of x
 (on this preferred coordinate system)

(i) for every point x there exists a unique pair of real numbers X _{1}, X _{2} such that

X _{1}, X _{2} are the coordinates of x (on the preferred coordinate system), and

for every pair of real numbers X _{1}, X _{2} there exists a unique point x such that

X _{1}, X _{2} are the coordinates of x (on the preferred coordinate system)

(ii) for any points x, y, z with coordinates X _{1}, X _{2}, Y _{1}, Y _{2}, Z _{1}, Z _{2} (on the preferred coordinate system), y lies between x and z if and only if

there exists a real number U such that 0 ≤ U ≤ 1 and

Y _{1} = X _{1} · U + Z _{1} · (1 − U) and Y _{2} = X _{2} · U + Z _{2} · (1 − U)
In older formalizations of geometry, as in Hilbert's famous monograph already cited, other geometric entities beyond points, beginning with straight lines, are mentioned. There are variables ξ, υ, ζ, . . . for lines, and primitives for relationships between points and lines or among lines, such (p.104) as incidence, ‘x lies on ξ’ or ‘ξ goes through x’, and parallelism, ‘ξ goes parallel to υ’. But these are in a sense superfluous, since lines can be represented by or reduced to points, and since the relationships among points that are the counterparts of the incidence and parallelism relationships above, namely, the collinearity relation and the equidirectedness relation:
are already definable in terms of order, by the conditions:
 x lies aligned with x _{1} and x _{2}
 x _{2} lies with respect to x _{1} parallel to how y _{2} lies with respect to y _{1}
 x lies between x _{1} and x _{2}, or x _{1} lies between x _{2} and x, or x _{2} lies between x and x _{1}
 there exists no z such that z is collinear both with x _{1} and x _{2} and with y _{1} and y _{2}
A semi‐popular exposition of formalized geometry (Tarski 1959) puts it as follows:
Thus, in our formalization. . . only points are treated as individuals. . . . [Our] formalization does not provide for variables of higher orders and no symbols are available to. . . denote geometrical figures. . . . It should be clear that, nevertheless, we are able to express in our symbolism. . . results which can be found in textbooks and which are formulated there in terms referring to special classes of geometrical figures such as the straight lines. . . the segments, the triangles, the quadrilaterals, and more generally the polygons with a fixed number of vertices. . . This is primarily a consequence of the fact that, in each of the classes just mentioned, every geometrical figure is determined by a fixed finite number of points. For instance, [in the notation of the present work] instead of saying that a point z lies on the straight line through the points x and y, we can state that either x lies between y and z or y lies between z and x or z lies between x and y. . .
Using these notions, several important assertions of elementary geometry can be expressed. Notable among these is the version of Euclid's parallel postulate assertion known as Playfair's Postulate:
Also, the assertion known as Desargues's Theorem and illustrated in the adjoining figure: (p.105)
 for any x, y, z that are not collinear, there exists a point u
 such that no point is collinear both with x and y and with z and u;
 and if v is any other such point, then z is collinear with u and v
Also, the continuity scheme associated with the name of Dedekind, involving for any formula P(z) the assertion:
 for any distinct points o, x, y, z, x′, y′, z′, u, v, w,
 if o is collinear both with x and x′ and with y and y′ and with z and z′,
 and v is collinear both with x and y and with x′ and y′,
 and w is collinear both with y and z and with y′ and z′,
 and u is collinear both with z and x and with z′ and x′,
 then u and v and w are collinear
All these assertions can be fairly easily deduced in T ^{†} using coordinates—a miniature illustration of the usefulness of algebra (and analysis) in geometry (and mechanics).
 for every x and y
 if there is some z′ between x and y such that P(z′) and
 if there is some z″ between x and y such that not P(z″) and
 if for every z′ and z″ between x and y such that
 P(z′) and not P(z″),
 z′ is between x and z″ and z″ is between z′ and y
 then there is a w between x and y such that
 P(z) holds for all z between x and w,
 and P(z) does not hold for any z between w and y
Modern coordinate geometry differs from traditional pure geometry in not one but two philosophically relevant respects. Ontologically, it involves real numbers; ideologically, it involves arbitrary choices. A coordinate system may be called admissible if the axioms are true on it. There are many admissible coordinate systems, and the choice of one of them as the preferred coordinate system is arbitrary. Before considering how to denumericalize the theory, or eliminate real numbers, it will be desirable to consider how to invariantize the theory, or eliminate this arbitrary choice. For an alternative that is both invariantized and denumericalized would, other things being equal, be more elegant than an alternative that was merely denumericalized.
3. The Strategy for Classical Geometry
a. Generalization
An invariantization can be obtained in two stages, and an elegant denumericalization thereof in two more. For any admissible coordinate system there will be unique points u, v, w with coordinates (0, 0), (0, 1), (p.107) and (1, 0) on that system, by the coordination axiom (i) of section 2. These may be called the benchmarks of the coordinate system. (For collinear points u, v, w there will be no admissible coordinate system having them as benchmarks, by the coordination axiom (ii) of section 2.) It can be shown that for non‐collinear points u, v, w there will be a unique admissible coordinate system having them as benchmarks. The coordinates on this alternative system of any point x can be obtained algebraically, by what is called an ‘affine transformation’ and is explained in any good textbook on linear algebra, from the coordinates on the preferred system of the point x and of the points u, v, w.
One thus obtains a definitionally redundant extension T ^{#} in L ^{#} if one adds to T ^{†} in L ^{†} two new primitives for the generalized coordinate notions:
 [u, v, w are not collinear and]
 X is the horizontal coordinate of x with respect to
 (the coordinate system having as benchmarks) u, v, w
along with a defining axiom (iii) indicating how the generalized coordinates with respect to u, v, w of a point x are obtained algebraically from the preferred coordinates of the point x and of the points u, v, w. To any formula Q of L ^{#} one may associate a generalization Q ^{+}, obtained as follows: first prefix the formula by a universal quantification, ‘for any non‐collinear u, v, w. . .’, and then replace each occurrence of ‘. . . preferred coordinates. . .’ by ‘. . . generalized coordinates with respect to u, v, w. . . ’. It can be shown that the generalizations (i ^{+}), (ii ^{+}), (iii ^{+}) of the axioms (i), (ii), (iii) are obtainable algebraically from the latter axioms. One thus obtains an implicationally redundant extension T ^{‡} in L ^{‡} = L ^{#} if one adds to T ^{#} in L ^{#} the former axioms, along with the trivial axiom:
 [u, v, w are not collinear and]
 X is the vertical coordinate of x with respect to
 (the coordinate system having as benchmarks) u, v, w
(o^{+}) there exist non‐collinear points u, v, w
T ^{‡} in L ^{‡} is a merely redundant extension of T ^{†} in L ^{†}.
b. Invariantization
Consider now the restriction T ^{+} in L ^{+} of T ^{‡} in L ^{‡} obtained by deleting the primitives for preferred coordinates and the axioms involving them. It is trivially seen that every formula of L ^{+} is invariant, or independent of arbitrary choices, whereas it is equally trivially seen that not every formula (p.108) of L ^{‡} is invariant. Hence T ^{‡} in L ^{‡} will not be an expressively conservative extension of T ^{+} in L ^{+}. However, it can be shown that any formula Q of L ^{‡} that is invariant is equivalent to its generalization Q ^{+}, which is a formula of L ^{+}. Hence T ^{‡} in L ^{‡} will be what may be called an ‘invariantly expressively conservative’ extension of T ^{+} in L ^{+}. It can also easily be shown that generalization carries axioms of T ^{‡} to axioms of T ^{+}, preserves logical relations of implication, and leaves formulas of L ^{+} unchanged. Hence T ^{‡} in L ^{‡} is a deductively conservative extension of T ^{+} in L ^{+}. Thus T ^{+} in L ^{+} captures the invariant content of T ^{‡} in L ^{‡} and hence of the original T ^{†} in L ^{†}. It remains to obtain an elegant T ^{§} in L ^{§} that will capture the non‐numerical content of T ^{+} in L ^{+}.
c. Denumericalization
The strategy now is to apply the general method of Chapter I.B. To do so one must be able to define a notion:
and deduce the principles of uniqueness and existence:
 the k‐tuple of points x _{1}, . . ., x _{k} represents the real number X
 every k‐tuple of points represents at most one real number
 every real number is represented by at least one k‐tuple of points
This can be done by drawing on the traditional conception of real numbers as ratios of lengths. More formally, the required representation notion amounts to:
which can be expressed algebraically in terms of the coordinates X′_{0}, X″_{0}, X′_{1}, X″_{1}, X′, X″ of x _{0}, x _{1}, x with respect to some/any admissible u, v, w, as follows:x _{0}, x _{1}, x represent X if and only if x is collinear with x _{0} and x _{1}, and X is the ratio of the distance from x _{0} to x to the distance from x _{0} to x _{1}, (taken with a positive sign if x _{1} is between x _{0} and x or x is between x _{0} and x _{1} and with a negative sign if x _{0} is between x and x _{1})

X′ = X′_{0} + X · X′_{1} and X″ = X″_{0} + X · X″_{1}
(p.109) First, L ^{±} involves several new primary primitives. These include a primitive for the counterpart, which may be called the subproportionality, to the order primitive on numbers. It amounts to:
Similarly for the other algebraic primitives and the generalized coordinate primitives. Second, T ^{±} involves several new primary axioms. These are the counterparts of the basic algebraic axioms, of the continuity scheme for numbers, and of the generalized coordinate axioms as mentioned under article 3.b above. Except for the counterpart of the continuity scheme for numbers, which amounts to something very like the continuity scheme for points, these axioms have an artificial look from a geometric viewpoint, though they are counterparts of axioms that had a natural look from an algebraic viewpoint. Thus the ontological benefit (from a nominalist viewpoint) of eliminating numbers is accompanied by the ideological costs of artificial new primitives and artificial new axioms.
 the ratio of the distance from x _{0} to x to the distance from x _{0} to x _{1} is less than
 the ratio of the distance from y _{0} to y to the distance from y _{0} to y _{1}
d. Beautification
But the long tradition of geometric algebra, beginning with the later books of Euclid's Elements, continuing through the medieval Arabs and on to Descartes, Newton, and their contemporaries, further advanced in the nineteenth century to a culmination in Hilbert, now explained in any good textbook on the foundations of geometry, with final logical refinements by Tarski and his school, can be drawn upon to reduce these costs. First, it can be shown that no new primitives are needed: all the new primitives of L ^{±} are definable using elementary logic from the single primitive of order for points. Notably, subproportionality, mentioned under article 3.c above, can easily be defined in terms of proportionality:
together with order for points. And proportionality itself can less easily be defined in terms of order for points.
 x _{0} x : x _{0} x _{1} :: z _{0} z : z _{0} z _{1}, or
 the ratio of the distance from x _{0} to x to the distance from x _{0} to x _{1} equals
 the ratio of the distance from z _{0} to z to the distance from z _{0} to z _{1}
(p.110) Two special cases are indicated in the adjoining figures.
This purely geometrical definition of proportionality is one of the main improvements of nineteenth‐century over earlier synthetic geometry. The ancient definition, attributed to Eudoxus, and found in Euclid, presupposes the notions of natural number, counting, and so forth. Or at least, it makes use of finite comparative cardinality quantifiers: (p.111)
The present definition is by contrast purely geometrical, employing no arithmetical or numerical notions.
 [there are finitely many Fs and finitely many Gs and]
 there are as many Fs as there are Gs
For another instance, the counterpart of the first coordination primitive:
can be defined by the condition:
 the horizontal coordinate of x with respect to u, v, w equals
 the ratio of the distance from y _{0} to y to the distance from y _{0} to y _{1}
where the important notion of projection here can be defined by:
 uz : uv :: y _{0} y : y _{0} y _{1} where
 z is the projection of x in the direction of the line through u, w to the line through u, v
 z is the intersection with the line through u, v of
 the unique line parallel to the line through u, w and passing through x
Second, it can be shown that only natural new axioms are needed, resembling those illustrated in figures. All the new axioms of T ^{±} are deducible using elementary logic from the axioms of an affine basic pure plane geometry T ^{§} set forth by Tarski and Szczerba (1965). The distributive law, for instance, roughly speaking corresponds to Desargues's Theorem cited in section 2. This last theory T ^{§}, in the language L ^{§} = L ^{† °} having order on points as its only primitive, provides an elegant invariantization and denumericalization of the original T ^{†} in L ^{†}.
4. The Strategy for Classical Physics
a. From Algebra to Analysis
The adaptation of the method of section 3 from the case of a very fundamental level of mathematical geometry can be extended to the case of classical physics in several stages. It can be adapted to the case where one has available in the original, coordinate theory not just the apparatus of algebra but also the further apparatus of analysis. This further apparatus consists of one additional primitive (with attendant additional axioms), for integrity. The method just outlined provides an invariantization and denumericalization with some further apparatus, consisting of one additional primitive (with attendant additional axioms) for what again may be called integrity: (p.112)
As this new notion may to some tastes look artificial from a geometric standpoint, it may be desirable to mention alternatives.
 [x _{0}, x _{1}, x are collinear and x _{1} lies between x _{0} and x and]
 the ratio of the distance from x _{0} to x to the distance from x _{0} to x _{1} is integral
Note first that if x _{0} x : x _{0} x _{1} is a strictly positive integer, then the region η consisting of all points y between x _{0} and x inclusively such that x _{0} y : x _{0} x _{1} is a non‐negative integer contains x _{0}, x _{1}, and x and fulfills certain conditions. Namely, it is discrete in the sense that:
And it is evenly spaced in the sense that:
 for each y in η except x there is a y ^{+} in η
 that is closest to y in the direction of x
 for each y in η except x _{0} there is a y ^{+} in η
 that is closest to y in the direction of x _{0}
And finally, x _{0} ^{+} is just x _{1}. (It is a fairly easy and pleasant exercise to show that the restricted notion of equidistance used here, for two pairs of points with all four points involved collinear, can be defined in terms of parallelism and hence in terms of order.)
 for each y in η strictly between x _{0} and x,
 the distance from y to y ^{−} equals the distance from y to y ^{+}
Note second that, conversely, if there is a region η fulfilling all the conditions above, then x _{0} x : x _{0} x _{1} is a positive integer. It follows that introduction of the integrity primitive with attendant additional axioms could be avoided in favour of introduction of regions of points with the incidence primitive and appropriate additional axioms (the analogues of extensionality, comprehension, and choice for sets of points). This alternative may to some tastes seem more natural from a geometric standpoint.
The assumption of arbitrary regions, moreover, is not needed. One can make do with any of a number of classes of special regions. One such class is that of finite regions (which are all that are required for the above definition). Another such class is that of open regions (though this would require a slight modification of the above definition), and yet another is that of closed regions. Here a point x is in the interior of an interval ab if it is strictly between a and b; an open region X in the line is one such that for every point x in X, there are a, b such that x is in the interior of the interval ab and every point in the interior of ab is in X; a closed region in the line is one whose complement is open. (Similar definitions can be made for the plane, beginning with the definition that x is in the interior of a triangle abc if there is a point y in the interior of the interval (p.113) ab such that x is in the interior of the interval cy.) Conversely, it is known that the theories of various special classes of regions are equivalent to each other and to the theory of points with an integrity primitive, in the sense that any one can be reduced to any other by the method of Chapter I.B. Any one of these theories may be taken as the official version of affine intermediate pure plane geometry.
b. From Lower to Higher Dimensions
The method of section 3 can be adapted to any finite number of dimensions. In adapting the method from two‐dimensional plane geometry to three‐dimensional solid geometry and then four‐dimensional hyperspace kinematics, the number of points making up a sequence of benchmarks must be increased from three to four and then five.
c. From Affine to Euclidean
Also at a very basic level of mathematical geometry comes Euclidean basic coordinate plane geometry. This is an extension of affine basic coordinate plane geometry with one new primitive for the notion of equidistance (for any two pairs of points, not just for the case where all four points involved are collinear):
and with one new axiom:
 x lies from y as far as z lies from w, or
 the segment xy is congruent to the segment zw
The most important of several further notions expressible in terms of equidistance is perpendicularity:
 for any points x, y, z, w with coordinates
 X _{1}, X _{2}, Y _{1}, Y _{2}, Z _{1}, Z _{2}, W _{1}, W _{2} (on the preferred coordinate system),
 x lies from y as far as z lies from w if and only if
 (Y _{1} − X _{1})^{2} + (Y _{2} − X _{2})^{2} = (W _{1} − Z _{1})^{2} + (W _{2} − Z _{2})^{2}
expressible by:
 z lies from x right‐angled to how y lies from x, or
 the segment xz is orthogonal to the segment xy
 there is a w such that x lies between w and y and
 x lies from w as far as x lies from y and
 z lies from w as far as z lies from y
(p.114) The method of section 3 can be adapted from the affine to the Euclidean case (and the extension of the method from algebra to analysis, and from lower dimensions to higher dimensions, carries over as well). Indeed, no change in the method at all is needed at the stage of denumericalization. That a change is needed at the stage of invariantization follows from the fact that, there being more axioms, it will be harder for a coordinate system to be admissible in the sense that all the axioms are true on it, so there will be fewer admissible coordinate systems. To put the matter another way, there will be fewer admissible transformations of the preferred coordinate system: in the jargon used in textbooks of linear algebra, the ‘Euclidean group’ of transformations is smaller than the ‘affine group’. Or to put the matter yet another way, it will be harder for a triple of points u, v, w to be admissible in the sense of being the benchmarks of an admissible coordinate system. One still needs non‐collinearity. It can be shown that what will be needed in addition for admissibility is the following:
With this single change, however, the method of invariantization can be shown to carry over, appropriate axioms being set forth in the semi‐popular exposition of Tarski already cited.
 the segment xz is congruent and orthogonal to the segment xy
d. From Mathematical to Physical
The kinematics of pre‐relativistic gravitational theory is intermediate between affine and Euclidean. In the usual jargon, the ‘Galilean group’ is intermediate between the affine group and the Euclidean group. To put the matter in another way, more useful in the present context, one does not have a full equidistance notion, but one does have a partial equidistance notion:
with the axiom:
 x, y, z, w are simultaneous and
 x lies from y as far as z lies from w
The most important of several further notions expressible in terms of equidistance are simultaneity itself and partial perpendicularity:
 for any points x, y, z, w with coordinates
 X _{1}, X _{2}, X _{3}, X _{4}, Y _{1}, Y _{2}, Y _{3}, Y _{4}, Z _{1}, Z _{2}, Z _{3}, Z _{4}, W _{1}, W _{2}, W _{3}, W _{4},
 x, y, z, w are simultaneous and
 x lies from y as far as z lies from w if and only if
 X _{4} = Y _{4} = Z _{4} = W _{4} and
(p.115) (Y _{1} − X _{1})^{2} + (Y _{2} − X _{2})^{2} + (Y _{3} − X _{3})^{2} = (W _{1} − Z _{1})^{2} + (W _{2} − Z _{2})^{2} + (W _{3} − Z _{3})^{2}
As in the Euclidean case, the only important change needed from the affine case is in the definition of admissibility of benchmarks u _{0}, u _{1}, u _{2}, u _{3}, u _{4}. It can be shown that what will be needed in addition for admissibility is the following:
 x, y, z are simultaneous and
 z lies from x right‐angled to how y lies from x
 u _{0}, u _{1}, u _{2}, u _{3} are simultaneous and
 u _{0} u _{1}, u _{0} u _{2}, u _{0} u _{3} are pairwise congruent and orthogonal and
 u _{0}, u _{4} are non‐simultaneous
Otherwise the method of section 3 carries over unchanged. The appropriate axioms can be patched together from the affine and Euclidean cases. They include the assertions that all points constitute under the order relation a four‐dimensional affine space, that simultaneity is an equivalence relation, and that all points simultaneous with a given point constitute under the order relation and the partial equidistance relation a three‐dimensional Euclidean space.
e. From Kinematics to Dynamics
Classical gravitational theory adds to non‐relativistic kinematics certain dynamical notions. Perhaps the most elegant analytic formulation would involve two new notions, mass density and gravitational potential. These are straightforwardly presentable as, respectively, a scalar field or assignment of real numbers to points, and a vector‐field or assignment of triples of real numbers to points (or triple of assignments of real numbers to points). To put the matter another way, one would have, in an analytic formulation, four new primitives as follows:
 the mass density at x is X
 the gravitational potential at x in the i ^{th} axial direction is X (for i = 1, 2, 3)
An additional arbitrary choice, beyond that of coordinate system, is involved here, namely the choice of scale of measurement for mass. Hence there will be additional complications at the stage of invariantization. (p.116) These complications can be unravelled. In the course of invariantizing, the above primitives are replaced by:
In the course of denumericalizing, these primitives are replaced by:
 the ratio of the mass density at x to the mass density at y is Z
 the ratio of the gravitational potential at x
 in the direction from y to z
 to the gravitational potential at x′
 in the direction from y′ to z′ is W
Since with most instruments the operational determination of intensive magnitudes like mass reduces to the determination of a ratio of extensive magnitudes of length (the ratio of the distance from the null‐mark on the dial to the unit‐mark to the distance from the null‐mark to the location of the pointer), these should not seem too artificial or devious.
 the ratio of the mass density at x to the mass density at y
 is equal to z _{0} z : z _{0} z _{1}
 the ratio of the gravitational potential at x
 in the direction from y to z
 to the gravitational potential at x′
 in the direction from y′ to z′ is w _{0} w : w _{0} w _{1}
Otherwise the method of section 3 carries over unchanged, and there seems to be no obstacle to extending it to a more comprehensive scientific theory incorporating the classical, Newtonian theory as its fundamental physics. (Of course, in a more comprehensive scientific theory, including natural history in addition to fundamental physics, many particular places and occasions will be mentioned, and there will be no question of invariance for the theory as a whole: the requirement of invariance makes sense only for the fundamental physics.)
f. From Pre‐Relativistic to Special‐Relativistic
Nor does there seem to be any obstacle to adapting the method further from theories incorporating a non‐relativistic perspective to theories incorporating a special‐relativistic perspective, or in the usual jargon, involving the ‘Lorentz‐Poincaré group’ rather than the Galilean group. To be sure, in the former case, a pure or ‘synthetic’ geometry appropriate to the non‐relativistic case is obtainable by patching together pure or ‘synthetic’ geometries appropriate to the affine and Euclidean cases, and these were made available in the work beginning with Euclid's, and (p.117) hence long before the coordinate or ‘analytic’ geometry was made available by the work of Descartes and Pierre de Fermat. Whereas in the latter case, the coordinate or ‘analytic’ geometry appropriate to the special‐relativistic case came first, in the work of Hermann Minkowski, so that one speaks of Minkowski space. Nonetheless, a pure or ‘synthetic’ geometry appropriate to the special‐relativistic case, comparable in style to Euclid's (as rigorized by Hilbert) while agreeing in substance with Minkowski's, was made available shortly afterwards in the work of Alfred Robb, see Robb (1914). A more formalized version of this little‐known work of Robb could play the role in the special‐relativistic case that the better‐known work of Tarski and his school has played in the non‐relativistic case.
5. Obstacles to Extending the Strategy
a. Post‐Classical Physics
Obstacles that it is not known how to surmount arise in connection with quantum mechanics. The difficulty is not with the underlying geometry or kinematics, which is Minkowskian or special‐relativistic. Rather, the difficulty is with the additional physical or dynamical notions involved. The difficulty is that when quantum‐mechanical theories are presented in field‐theoretic form—and it is presumably this form, rather than the particle‐theoretic form, that one would want to consider—the mathematical objects that have to be assigned to points are much more complicated than scalars (single real numbers) or three‐ or four‐vectors (triples or quadruples of real numbers). It is known (and was indicated in article I.B.1.b) how to represent these more complicated mathematical objects by real numbers. But the known representation is devious and awkward, not straightforward and elegant.
Further obstacles arise in connection with general relativity, and these do arise from the underlying geometry, which is Riemannian or differential. At the stage of invariantization, there are difficulties including the following. In all of the cases considered previously, it was possible to avoid quantifying over coordinate systems, and so ascending to a higher level of abstraction, because coordinate systems were representable by their benchmark points. But this is not so in the more general coordinate systems of Riemannian or differential geometry, and the usual approach to invariantizing in this case does involve ascending to a higher level of abstraction, which would seem to create major obstacles to subsequent denumericalization. Even if these difficulties were surmounted, at the (p.118) stage of denumericalization there would be the further and major difficulty that there seems to be no pure or ‘synthetic’ Riemannian or differential geometry, no list of axioms that are natural‐looking from a geometric viewpoint and that could play the role in this case that was played by the lists of axioms provided by the work of Tarski and his school in most of the cases considered previously (or of Robb in the special‐relativistic case).
As for a physical theory unifying quantum mechanics and general relativity, such a theory could be expected to present both the difficulties presented by the former, and the difficulties presented by the latter, and also new difficulties of its own. But the most obvious obstacle to developing an elegant, synthetic, pure, natural‐looking, invariant, straightforward version of such a theory at present is the circumstance that so far no one has developed even an inelegant, analytic, coordinate, artificial‐looking, arbitrary‐choice‐dependent, devious version of such a theory. Thus a question mark hangs over the geometric nominalist strategy. But two more positive remarks are in order.
First, if one does care about realism, but does not care about elegance, one can probably get by just routinely applying the general method of Chapter I.B. Essentially all that is required is that there be some kind of ‘metric’ μ (x, y) or measure of separation between points x, y, and that the geometry assume that space is continuous rather than discrete. For this amounts to assuming that every real number can be represented as a ratio μ (x, y) : μ (u, v), which is enough to make the general method applicable. Second, to say that it is not known to be possible to provide elegant nominalistic versions of quantum mechanics and general relativity is not to say that it is known to be impossible. Rather, whether the obstacles enumerated can be surmounted is an open research problem. As a consequence of nominalism's being mainly a philosopher's concern, this open research problem is moreover one that has so far been investigated only by amateurs—philosophers and logicians—not professionals—geometers and physicists; and the failure of amateurs to surmount the obstacles is no strong grounds for pessimism about what could be achieved by professionals.
b. Non‐Empirical Physics
According to substantivalism, geometric entities are to be recognized as physical entities and geometric relationships as physical relationships, and hence presumably geometric questions as physical questions. Now, when it was claimed in article I.B.1.b that analysis without any higher (p.119) set theory provides enough mathematics for applications in physics, ‘physics’ was being understood in the ordinary sense rather than this substantivalist sense. So it must now be asked, first whether analysis without any higher set theory provides enough mathematics for applications to geometry; and second whether, if not, that fact poses a problem for the nominalistic strategy of this chapter. The first question is technical, the second philosophical.
To state the technical question more precisely, consider the kind of geometric theory T used in the strategy of this chapter, intermediate affine pure geometry as in article 3.a. (As mentioned there, it may be formulated either in terms of points and a special ‘integrity’ primitive, or in terms of points and certain special classes of regions. The latter formulation will be more appropriate to present purposes.) Let T ^{+} be the result of adding the apparatus of analysis to T, and T ^{‡} the result of adding the apparatus of set theory. The question whether analysis is sufficient for applications to geometry and hence for ‘physics’ in the substantivalist sense, or whether higher set theory is needed, amounts to the following question: is there any geometric assertion P, any assertion expressible in the language L of T, whether or not it would have obvious implications for ‘physics’ in the ordinary sense, that can be proved in T ^{‡} but that cannot be proved in T ^{+}? It has already been shown in this chapter that any assertion P expressible in L and provable in T ^{+} is in fact provable in T, since the apparatus of analysis can be reconstrued geometrically. Hence the question may be reworded: is there any conjecture P expressible in the language L of T but not provable in T, that becomes provable in T ^{‡}?
The answer is yes: by the Incompleteness Theorem of Kurt Gödel (together with what has been shown in this chapter, that the apparatus of analysis can be reconstrued geometrically), it follows that the assertion that the theory T is consistent can be reconstrued or coded geometrically as a conjecture P(T) expressible in the language L of T, and that it is not provable from the axioms of T. It is, however, provable from the axioms of T ^{‡}. However, three remarks are in order. First, as a coded assertion, P(T) is not at all natural as an assertion of geometry. Further, as a consistency assertion, P(T) is something that no one who genuinely accepts the theory T genuinely could doubt. Finally, the Gödel method applies to any theory, so for instance the assertion that the theory T ^{‡} is consistent can also be reconstrued or coded geometrically as a conjecture P(T ^{‡}) in the language L of T and it cannot be proved in T ^{‡}. These remarks might be summed up by saying that the Gödel example is one where (p.120) adding set theory to geometry makes only a ‘formal’ difference. There are, however, other examples where it makes a ‘material’ difference.
To recall the philosophical question, however, would even an example that makes a ‘material’ difference pose a problem for the nominalistic strategy of this chapter? This question may be reformulated: suppose there is some natural geometric question P that cannot be settled by proof or disproof in the nominalistically acceptable theory T, but can be settled in the nominalistically unacceptable theory T ^{‡} that results when the apparatus of set theory is added; is that a problem for the nominalist? The answer is, not unless and until the geometric theorem comes to play a role in physics in the ordinary sense of ‘physics’ and not just in the substantivalist sense of ‘physics’. For only if and when that happens is the nominalist obliged to concede that the question that has been settled by set theory has been settled the right way. And only if and when the nominalist is obliged to concede that set theory settles geometric questions that cannot be settled nominalistically and settles them the right way, does the nominalist's position come to resemble the instrumentalism that it is the aim of strategies like that of this chapter to avoid. And so far, examples where set theory makes a ‘material’ difference to geometry have not yet come to play a role in ‘physics’ in the ordinary sense. Still, it may be of interest to describe briefly one such example.
The first task in presenting examples is to introduce the special classes of regions, and the special properties of regions, involved. The definitions will be given for the line; the corresponding definitions for the plane are exactly analogous. The special classes of open regions, and their complements the closed regions, have already been defined. The next important special classes of regions are the F _{σ} regions, and their complements the G _{δ} regions. The usual analytic definition is that an F_{σ} region is one that is a union F _{0} ∪ F _{1} ∪ F _{2} ∪ . . . of countably many closed regions. It is important in the present context to note that a purely synthetic definition is available, using only geometric notions there has already been reason to introduce, such as that of evenly spaced points in a line or of projection of a point in the plane to the line. Derivatively, parallel lines in the plane are evenly spaced if their points of intersection with any transverse line are evenly spaced, and the projection of a region in the plane to the line is the linear region whose points are the projections of the points of the planar region. A region in a line X is F_{σ} if it is the projection to X of the intersection of two regions Y ∩ Z, where Y is a closed region in the plane, and Z is a region in the plane that is a union of evenly spaced lines parallel to X. The last important special classes of regions are the (p.121) analytic regions, the projections to the line of G_{δ} regions in the plane; the co‐analytic regions, their complements; and the Borel regions, the regions that are both analytic and co‐analytic. (The importance of the analogues of Borel regions of linear points, namely, Borel sets of real numbers, in the mathematical apparatus of sophisticated physics was mentioned in article I.B.1.b.)
The second task in presenting examples is to present the special properties of regions involved. The pertinent properties are the ones usually explained intuitively as follows. Consider a horizontal interval I in the line. For any subregion A of I one may imagine an infinite game for two players, IN and OUT. I consists of two halves, left and right. The game begins with IN picking one of these. It in turn consists of two halves. The game continues with OUT picking one of these. It in turn consists of two halves. The game continues with IN picking one of these. Alternating in this way, IN and OUT successively pick smaller and smaller intervals, which in the infinite limit narrow down to a single point. IN or OUT wins according as this point is in or out of A. A strategy (respectively, counter‐strategy) is a rule telling IN (respectively, OUT) what to pick at each stage, as a function of the opponent's previous picks. A strategy (respectively, counter‐strategy) is winning if when IN (respectively, OUT) picks according to the rule it provides, IN (respectively, OUT) always wins. The game is called determined and the set A is called determinate if there is either a winning strategy or a winning counter‐strategy. It is important in the present context to note that a purely synthetic definition is available. Consider any horizontal interval $\mathcal{F}$. It may be bisected into two equal subintervals $\mathcal{F}\prime \text{,}\mathcal{F}\prime \prime $. There is a rectangle R′ with upper horizontal side ℱ′ and with vertical side half as long as its horizontal side, and a similar rectangle R″″ for ℱ″″. Call these the dependent rectangles of $\mathcal{F}$. Consider now the square whose lower horizontal side is I. The primary and secondary subrectangles of this square are defined as follows. The rectangles dependent on the upper horizontal side of the square are primary, and called the initial rectangles. For any primary rectangle R, the dependent rectangles on its lower side are secondary rectangles, called the successor rectangles of R. For any secondary rectangle R, the dependent rectangles on its lower side are primary rectangles, called the successor rectangles of R. A strategic region is one that contains exactly one initial rectangle and that, whenever it contains a primary rectangle R, contains each successor of R and exactly one successor of each successor of R. A counter‐strategic region is analogously defined. Parts of such a set are illustrated in the adjoining figure, which invite comparison with the (p.122)
The intersection of a strategic region S′ and a counter‐strategic region S″ consists of a single large rectangle at the top, a single smaller one next below it, and a single smaller one next below that, and so on, and contains only one complete vertical interval of unit length. Its lower boundary point is a point in the interval I called the outcome of S′, S″. A subregion A of I is positively (respectively, negatively) determinate if there is a strategic S′ (respectively, a counter‐strategic S″) such that for any counter‐strategic S″ (respectively, any strategic S′), the outcome of S′, S″ is in (respectively, out of) X. X is determinate if it is either positively or negatively determinate.
The hypotheses GD, BD, and CD of open, Borel, and co‐analytic determinacy are respectively that every open, every Borel, or every co‐analytic region is determinate. The status of these geometric hypotheses in relation to set theory is as follows. Specialists in set theory have considered adding further axioms to the axioms ZFC accepted by mathematicians generally. Roughly speaking, most of those who favour adding new axioms favour something called the axiom of measurables, which produces a system ZFM; a few may favour something called the axiom of constructibility, which produces a system ZFL. (The two axioms, of measurables and of constructibility, are incompatible.) So besides the system T ^{‡} obtained by adding the apparatus of ZFC to the geometric theory T, one can consider the systems T ^{#} and T ^{±}, both stronger than T ^{‡}, and incompatible with each other, obtained by adding ZFM and ZFL. Then GD can be proved in the geometric theory T (Gale and Stewart). BD can be proved in T ^{‡} (D. A. Martin), but not in T (Harvey Friedman). CD can be proved in T ^{#}, but not in T ^{‡}, since it can actually be disproved in T ^{±} (Martin, Friedman again). Some further information is available in Burgess (1989). For a philosophically oriented account of these and related matters, the reader is referred to the work of Penelope Maddy, especially the two‐part paper Maddy (1988), which contains full references to the original technical literature, and which makes clear, as the present brief account cannot, the central role of determinacy in the qualitative (as opposed to quantitative) theory of regions of linear points, or in the more usual analytical terminology, the descriptive (as opposed to metric) theory of sets of real numbers, or descriptive set theory for short.