**Relations.**
Having considered the properties that substances have in a spatiomaterial
world, the next step in demonstrating necessary truths about the world from
these ontological assumptions is to determine the kinds of relations that
substances have in the world. Relations are different from properties only
in that relations hold of (or are true of) more than one substance at once.
Thus, relations will be explained ontologically as aspects that hold of more
than one substance, just as properties were explained as aspects of substances
taken separately. In short, relations are aspects of the world.

It is because of how substances exist together as a world that there are relational aspects of substances. But relations have been introduced in two different ways. The relations among points (that is, the parts of space) are part of the essential nature of space as postulated by this ontology. And there are relations among bits of matter, because each coincides with some part of space or other. (Spatial relations among bits of matter is one of the basic aspect of the natural world that was used as evidence for spatiomaterialism.) Both kinds of relations are part of our ontology, and both will be used to explain other kinds of relations.

Ontological philosophy proves that propositions about relations are true by deriving them from spatiomaterialism. That is, it shows how the relations are constituted by its basic substances, space and matter, given their essential natures and their basic relationship as parts of the same world. Propositions that follow from the best ontological explanation of the natural world are ontologically necessary and, thus, prior to what we know about what actually happens in the world from experience . That is what it means to say that they are "necessary truths," according to ontological philosophy.

These ontologically necessary propositions about the basic relations in a spatiomaterial world include what is usually called mathematics. That is, the basic relations that hold among points (or that can hold among bits of matter at any moment) are, as we shall see, the subject matter of mathematics. There are other ontologically necessary relations in the world, such as those that derive from substances being in time and from further aspects of the essential natures of matter and space. They are merely complications of these basic relations, which will be taken up in the next chapter, Change, which is the subject matter of science.

The ontological explanation of the truth of mathematics and science involves a different set of necessary truths from those already discussed. Unlike the truths about the intrinsic and extrinsic essential natures of substances, these further truths depend on substantivalism about space. The relations among points are part of the essential nature of space. Nor would there be any relations among bits of matter without space to help constitute them.

Explaining relations as aspects of a world constituted by space and matter is straightforward enough, but it is not the traditional way of explaining the truth of mathematics. Epistemological philosophy takes relations to be objects of knowledge, and obstacles to explaining how the basic relations are known to give rise to philosophical problems about the nature of mathematical truth. But the critique of epistemological philosophy is a consequence of ontological philosophy, and so let us begin by considering what can be said about relations as aspects of a spatiomaterial world.

**Relations
as aspects of substances.** In a spatiomaterial world, the relations that hold among particular
substances are of two kinds, the relations that hold among points (or parts
of space) as part of the essential nature of space, and the relations that
hold among bits of matter because each coincides with some part of space.
Since the ontological foundation of geometry is space, let us consider what
holds simply because of its nature before we see what that implies about the
relations among bits of matter contained by space. That will put us in a position
to take up the ontological explanation of arithmetic.

**Geometry.**
Geometry describes the structure of space. Space, as we have assumed,
is made up of many particular substances whose essential natures include their
being related to one another in the way described by three-dimensional (Euclidean)
geometry.

The parts of space are particular substances, according to spatiomaterialism, but in geometry, they are called points. Points are identified by their locations in space, since that is how they differ from one another, and they are recognized to be simple, that is, without length, width, breadth, or any possibility of being divided into parts.

The propositions of geometry include the following: Any two points determines a straight line, where a straight line is the path of the shortest distance between them. Any straight line can be extended continuously in a straight line. A straight line and any point not on it determines a plane. Intersecting lines have only one point in common, and when the angles determined by them are equal, the angles are "right angles" and the lines are perpendicular. Through any point, there are exactly three mutually perpendicular straight lines. There is a metric to the distances between points, so that things equal to the same thing are also equal to one another. And so on . . . There is no need to state all the propositions of geometry here, since they are well known.

Since geometry has been used to help define the essential nature of one of the two basic substances postulated by spatiomaterialism, ontological philosophy can explain why geometry is true of the parts of space by the correspondence between geometrical propositions and space as a substance. Those propositions describe an order among the parts of space, and since space is homogeneous, the order is universal and holds in every region. Or as we assumed (provisionally) in the foundation, each part of space has the same kinds of relations to all the other parts of space as every other part of space has to parts others than itself. But it is relevant to notice that explaining the truth of geometry by its correspondence to space does not depend on geometry being stated as an axiom system.

*Geometry as an axiom system.* The propositions of geometry
can be stated as a system in which some are treated as assumptions, and all
the rest are all deduced from them (and definitions of terms introduced to
simplify the statement of geometrical propositions). The former propositions
are called "axioms," and the latter are called "theorems."
This way of organizing geometrical propositions was discovered by the ancient
Greeks. It was worked out in some detail by Euclid. It aims at an optimal
arrangement among the proposition in which some of the simplest and most intuitive
propositions are singled out and used to generate all the rest, that is, producing
the most in the way of consequences using the least in the way of premises.
Geometry lends itself to axiomatization because it describes a simple structure
that contains implicitly many complex relations. The relations among the parts
of space is a kind of order that makes the whole uniquely simple, and when
the axioms describe certain basic aspects of that structure, it is possible
to combine those relations in ways that describe all the other relations that
must also hold among points, lines, angles, and the like. Such constructions
from simpler truths are the derivation of theorems in geometry.

The significance of this deductive arrangement among the propositions of geometry has long been understood epistemologically, that is, as a way of knowing that geometrical propositions are true. Deductive inferences preserve the truth of the premises, and since the axioms of geometry seem to be self-evidently true, it seemed that deriving them from the axioms would prove that they are also true.

This epistemological approach became less attractive, however, as two facts about such axioms systems became known.

The first was that there are different ways of axiomatizing geometry. That is, different geometrical propositions can be used as axioms, and still all the rest follow logically. Thus, there is no necessary order by which some should be taken as implying others.

Second, and more importantly, it became clear that the deductive relationship cannot, by itself, establish any truth about the world. The truth of the theorems depends on the truth of the axioms. But the truth of the axioms cannot be shown within the deductive system. The axioms contain terms which are not defined within the system, or so-called "primitive terms," and thus, the truth of the axioms depends on what those terms refer to. And there are other objects that will make the axioms of geometry true (the set of whole numbers, if nothing else, according to the Löwenheim-Skolem theorem). The deducibility of the theorems from the axioms means that the theorems will be true of whatever objects make the axioms true, but unless the primitive terms in the axioms refer to points and their relations, the theorems of geometry will have nothing to do with the structure of space.

Thus, even though it is possible to *come to know *that
some geometrical propositions are true by deriving them from others that are
true, that does not explain *why they are true*. It merely shows that
they are true, if the premises are true. Hence, the truth of both depends
on how the premises are true. Ontological philosophy is not bothered by the
aforementioned discoveries, because it explains why both kinds of geometrical
propositions are true in the same way, that is, by virtue of their correspondence
to the world.

If geometry is formulated as an axiom system, then the primitive terms, which are not defined within the system, are taken as referring to the substances it postulates or to aspects of them. The axioms are, therefore, descriptions of the essential nature of one of the two basic substances postulated by spatiomaterialism. But so are the theorems derived from them. They are also descriptions of the essential nature of space. Apart from being entailed by the axioms, what makes the theorems different is that they can be stated without introducing any new basic terms (that is, any terms that are not defined by those used in the axioms).

*Euclidean Geometry.* In the nineteenth century, however,
the deductive view of the truth of geometry suffered another blow, because
it was discovered that several axiom systems can be constructed for geometry
that are alike in making the most out of the least even though they differ
from one another in one of the axioms, the so-called parallel axiom. Euclid’s
fifth postulate holds, in effect, that through any point not on a line, one,
and only one, parallel line can be drawn in the same plane as the first line.
But Lobatchevsky and Bolyai showed that this axiom could be replaced by one
holding that more than one line through such a point could be extended infinitely
in the plane without intersecting the first line and the resulting geometry
would be just as rich in implications. Later Riemann showed that the axiom
could be replaced by one holding that there are no parallel lines at all,
because any line drawn in the plane through a point not on the same line will
intersect with the first line in two points. Both of these new geometries
were just as rich in theorems as Euclid’s.

The existence of such non-Euclidean geometries shows that it
is *possible *that space is curved (that is, that geometry is consistent
even with carious artificial, new distance functions). But that is not of
much consequence to ontological philosophy, for it explains how geometry is
true, not by the deducibility of theorems from the axioms of geometry, but
rather by the correspondence of the axioms (and, thus, the theorems) to the
structure of space.

The correspondence theory of truth does, of course, force us to decide which geometry describes the space we are postulating. And that depends on the nature of the space that we find in the world, for we are following the empirical method in deciding which ontology to believe. That is, we choose the simplest ontological explanation that will explain the basic features of the world. Since the simplest is obviously Euclidean geometry, the space we postulate has a three-dimensional Euclidean structure.

To be sure, since it is an empirical claim, it could turn out that space is not Euclidean. In that case, ontological philosophy would have to start over again with non-Euclidean space of some kind — or else give up spatiomaterialism and go back to epistemological philosophy. But as it turns out, there is no good reason to doubt that space is Euclidean.

What has led naturalists to give up Euclidean space is Einsteinian
relativity. Einstein’s general theory of relativity holds that spacetime is
curved, and that means that it is not Euclidean. But the *curvature of spacetime
*is quite a different thing from the *curvature of space *as a substance
enduring through time, and as we have promised, spatiomaterialism offers a
perfectly intelligible interpretation of what Einstein’s general theory calls
"curved spacetime" on the assumption that substantival space is
Euclidean. That removes any empirical reason for doubting that space is Euclidean,
and thus, we are free to believe the simplest geometry that explains the categorical
features of what we find in the world.

*What geometry corresponds to. *Geometry holds of space in
a spatiomaterial world, because the space it postulates is a substance whose
essential nature is defined as making geometry true of it. The relations among
points, that is, the simplest parts of space, are geometrical. But given how
we explain the spatial relations among bits of matter, geometry also most
hold of them (except for limitations that may be imposed by bits of matter
having a finite sizes in space), because they coincide with parts of space.
Thus, the propositions of geometry are true not only of the relations among
parts of space, but also of the relations among bits of matter.

In both cases, geometry is ontologically necessary, because it is part of the ontology that we are taking to describe the basic nature of existence. That means that it is prior to what is known about what happen in the world by experience, and that is the sense in which ontology if prior to science and other ordinary ways of knowing about the world.

However, this proof the the ontological necessity of geometry involves a genuine ontological explanation only when its propositions are taken as applying to bits of matter. In that case, they describe facts about the world that depend on both ontological causes, space and matter. There is no genuine ontological explanation of why geometry holds of space itself, because its geometrical nature is what is assumed about just one of the basic substances being used as an ontological cause.

**Arithmetic.**
Besides the relations among points and bits of matter that describe
the structure of space, bits of matter and points have a more abstract relationship
to one another. They are all parts of a single world in way that allows them
to be picked out individually and, thus, to be grouped together. Space is
also an ontological cause of this more abstract relationship, for it comes
from particular substances having spatial relations that all fit together
geometrically. Thus, arithmetic is no less ontologically necessary than the
relations that make geometry true.

Arithmetic is, basically, the theory of numbers. The basic numbers are whole numbers, or integers, and arithmetic includes the laws governing their addition, multiplication, subtraction and division. Arithmetic can be taken broadly as including all the propositions about the numbers (except those that have to do with what numbers refer to and how propositions about them are true).

Given the arithmetic of whole numbers, it is possible to construct rational numbers, negative numbers, irrational numbers, and complex numbers and to show that these numbers also obey the laws of addition, multiplication, subtraction, and division. With the use of set theory, transfinite number can also be introduced, though special laws govern operations on them. Taken broadly, therefore, arithmetic includes algebra, the calculus, and analysis.

Even geometry can be included, for its propositions can be generated by way of analytic geometry, or the "algebra of geometry," as Descartes showed. The contemporary attitude is to take arithmetic as more basic than geometry, though that is to reverse the ancient Greek assumption.

* Set
theory.* It is possible to give an ontological explanation of the truth of
all these propositions at once, because they can all be derived from set theory.
Set theory provides the foundation that mathematicians currently use to prove
the truth of arithmetical propositions, taken broadly. But there are various
ways of axiomatizing set theory, just as there are for geometry. The most
widely used by mathematicians is the Zermelo-Fraenkel system, and its axioms
will be used here to show how the truth of arithmetic (and mathematics generally)
can be explained ontologically. (A similar argument could be constructed for
other axiomatizations of set theory.)

Set theory is a formal system in which the axioms are simply
assumed to be true. Though its axioms describe the nature of sets, "set"
is a primitive term, and so the axioms are an implicit definition of that
term. Thus, *if we can show that the substances that constitute the spatiomaterial
world satisfy the axioms of set theory, that will show that all the propositions
of arithmetic are true of them*. Furthermore, since nothing exists in a
spatiomaterial world but those substances, it will also show that this interpretation
of set theory includes all possible interpretations of its axioms, and thus,
that it includes all the ways that set theory can be true by virtue of corresponding
to the world. Thus, this is, in effect, to derive the truth of mathematics
from the spatiomaterialist ontology, which shows that mathematics is a necessary
truth of ontological philosophy.

Let us consider, therefore, whether the substances in a spatiomaterial world satisfy the axioms of Zermelo-Fraenkel set theory.

*Axiom 1.* The first axiom defines "sets," in effect,
by holding that *two sets are identical when they have the same members*.
To explain its truth ontologically, we must say what the members of sets are
and what the sets themselves are.

Sets can be members of sets, but unless there is something else
the most basic sets are sets of, only the empty set can exist. Set theory
says nothing about the nature of the *ultimate members *of sets except
to assume that they are all distinct and can be distinguished from one another.
But in a spatiomaterial world, nothing exists at any moment except all the
parts of space and all the bits of matter, which it contains. Hence, those
substances and what they constitute are the only possible ultimate members
of sets that exist wholly at any moment. (We will see how arithmetic can be
extended to cover different moments in time in Change.)
Particular points in space can be picked out by their locations, and so can
particular lines, figures, and other geometrical constructs, since they are
constituted by such points. Likewise, let us assume that bits of matter can
also be picked out by their locations in space, though we will not explain
the sense in which it is true until we take up the concrete nature of matter
(in Change).
And if ordinary material objects are constituted by elementary bits of matter
and parts of space, as spatiomaterialism holds, they can be picked out in
a similar way. Indeed, any collection of points in space and/or bits of matter
can be picked out as an individual in such a way. These are all the substances,
elementary and compound, that can exist at any moment in a spatiomaterial
world, and thus, they include all possible ultimate members of sets in such
a world.

The sets of such members are, however, distinct from the substances,
which are their ultimate members, and in order to explain ontologically how
the axioms of set theory are true, there must also be something to which the
term "set" refers. What explains the existence of sets in a spatiomaterial
world is the fact that all its substances have spatial relations to one another.
That is the aspect of the world that makes it *possible *to pick our
particular substances and group them together. Since their possibility is
entailed by the essential nature of a spatiomaterialist world, every possible
set actually exists as an a distinct aspect of the world.

To be sure, sets would not be recognized to exist without rational beings like us to pick out their members and actually group them together. And we shall see how rational beings (with the spatiotemporal and rational imagination required to construct such sets) come to exist in a spatiomaterial world. But rational subjects are not essential to the existence of sets, since sets are aspects of the world (though I may refer to sets by saying that rational beings pick individuals out and group them together).

Substances may be grouped together in many different ways, by
using various properties to define them, but *every *such class can,
in principle, be constructed by the spatial relations of the substances making
it up. (They must have spatial relations, since every substance is constituted
by a set of basic substances, according to ontological philosophy.) Spatial
relations make it possible not only to pick out each substance as distinct
from all the rest, but also to group any substances together. Space is a whole
of which they are all already parts, and being parts of it, substances can
be parts of lesser wholes.

To be sure, merely being parts of the same world also makes them part of a single whole. But that does not make it possible to group them together, because if "the world" is defined as merely all the substances that exist, it would not even be possible to distinguish among particular substances (of the same kind), much less to relate some of them to one another in a way that others are not related. But having spatial relations means that each substance has a unique relationship to all the others and, at the same time, that each is part of a single whole, three dimensional space with them. (Though a bit of matter and the part of space containing it have the same spatial relations to every other substance in the world, they can be distinguished from one another by the kind of substances they are.)

Thus, space is an ontological cause of every set, for it is the wholeness of space that explains the existence of sets. Thus, groups constructed by grouping substances (elementary or composite) together can be taken as the basic sets of Zermelo-Frankel set theory.

The first axiom of Zermelo-Fraenkel set theory holds that *two
sets are identical if they have the same members*. It is true of sets in
a spatiomaterial world, given this ontological interpretation of sets and
their ultimate members. It is true of the basic sets, because the substances
that wind up together in a set do not depend on how they are grouped together,
but on which substances they are, for that is the aspect of the world that
constitutes the existence of the set. Sets with the same members will be constituted
by the same substances. And it holds of sets of sets, because if sets are
constructed by grouping substances in this way, sets of sets are just groups
of groups formed in this way, and two groups of the same groups will be constituted
by the same groups of substances. There is no ontological difference between
the two sets.

*Axiom 2.* The second axiom holds that *the empty set exists*.
The empty set does exist in a spatiomaterial world in the same sense as any
set. The same aspect of the world that makes it possible to group substances
together also makes it possible to form a group without any members. Whether
or not it has any members, the grouping itself depends on how space makes
the world whole, that is, on how space itself is whole and how everything
contained by space is related in its three dimensions. That aspect of the
world is not constituted by substances taken separately, but by how they exist
together as a world, and that aspect is what explains the existence of the
empty set.

*Axiom 3.* The third axiom holds that *if *x *and
*y *are sets, then the unordered pair {*x,y*} is a set*. That
is to say that sets can be members of sets as well as basic substances, and
the truth of this axiom has already been explained.

Sets exist in the sense that spatial relations allow substances to be grouped in all possible ways. But sets that exist in that sense can themselves be grouped in a similar way into groups. For the same reason, it is possible to group sets of sets into sets, and sets of sets of sets into sets, and so on.

*Axiom 4.* The fourth axiom holds that *the union of a
set of sets is a set*, that is, that a set can be formed from all the distinct
substances that are members of at least one set included in the set of sets.
That axiom is true in a spatiomaterial world, because sets are just groups
of substances. Any substance can be picked out by its spatial relations. And
if a substance is a member of more than one of the member sets, it will not
become two substances in the union of the sets, because its identity with
a substance in the other sets can be determined by its spatial relations.

*Axiom 5.* The fifth axiom holds that *the infinite set
exists*, including transfinite cardinals. The obstacle to taking the axiom
of infinity to be a truth about the natural world has been doubts about the
bits of matter in the world being infinite in number. Even if spatiomaterialism
did not (yet) take a stance on that issue, it would entail the existence of
infinite sets, including transfinite cardinals, because it takes space as
well as matter to be a substance.

Space may not be infinite in extent, but since any finite line
is infinitely divisible, there are infinite sets of points (for example, the
points determined by cutting a line in half, cutting the half-line in half,
cutting the quarter-line in half, etc.). Such sets are denumerably infinite,
because they can be put in a one-to-one relation with whole numbers. And if
the world *is *infinite, the bits of matter in the world can also be
put in one-to-one relations with the whole numbers.

But substantivalism about space also entails the existence of transfinite sets of substances, for the number of points on a finite line is indenumerably infinite.

*Axiom 6.* The sixth axiom of Zermelo-Frankel set theory
is that *any property that can be formalized in the language of the theory
can be used to define a set*. The truth of this axiom is entailed by this
ontological explanation of the world, because properties are aspects of substances
and all properties are explained by showing how they are constituted by substances.
Since properties can all be explained by the substances whose aspects they
are, it holds for all the properties that can be formalized in the language
of the theory.

*Axiom 7.* The seventh axiom holds that, *for any set,
the power set can be formed*; that is, that the collection of all subsets
of any given set is a set. This follows from our ontological explanation of
the existence of sets, for it implies that all sets that can be formed of
the particular substances in the world exist, and that includes all the subsets
of any set formed, that is, its power set. (What makes this axiom so important
is that the power set is itself a set, and another set can be formed of its
subsets, over and over again indefinitely.)

*Axiom 8.* The eighth axiom is the so-called "axiom
of choice," which holds that *from any collection of non-empty, non-overlapping
sets, a new set can be formed by selecting one member from each set*. This
axiom is clearly true, if sets are all ultimately made up of substances as
members (that is, are complex substances), because substances exist.

Despite being used in many mathematical proofs, this axiom has not been considered self-evident, because there seems to be no way to assure that it is possible to pick out a particular member of every set. However, it is always possible, given the ontological explanation of the truth of this axiom. Since the ultimate members of every set are points in space, bits of matter, or determinate combinations of basic substances, it is possible to pick out a specific member of each set by its spatial relations. For example, select the particular substance from each set which is closest to a given point, or in cases of ties, the first in an ordered set of directions in three dimensions from a given point.

*Axiom 9.* The ninth axiom holds that *no set is a member
of itself*. This axiom avoids certain paradoxes that can arise from taking
sets to be members of themselves, for example, Russell’s paradox about whether
the set of sets that are not members of themselves is a member of itself.
(If it is not a member of itself, it must be a member of the set; but if it
is a member to the set as defined, it is a member of itself.) But this is
not just a device to avoid paradoxes. It is a fact about sets, if sets are
formed by grouping substances or groups ultimately made up of substances together,
because it is not possible to include the group one is currently constructing
as a member of the group. It does not yet exist, and so rational beings having
nothing to group together with the members. Thus, no set is a member of itself.

These are the axioms of Zermelo-Fraenkel set theory, and as we have seen, they are true of a spatiomaterial world, if the ultimate members of sets are substances and sets exist in the sense that substances (and groups of them) can be grouped together. Since deduction preserves the truth of its premises, all of mathematics that can be derived from them (including arithmetic, algebra, the calculus, and analysis) is also true of the natural world, if spatiomaterialism is true. Hence, the truths of arithmetic are not only true, but also ontologically necessary, that is, prior to empirical science.

* Solutions
of puzzles about set theory. *There are further advantages of the ontological explanation of the
truth of arithmetic, because it solves several puzzles that have cast doubt
on mathematics in the twentieth century.

*Totality.*
It is remarkable that all the truths of arithmetic can be generated
by Zermelo-Fraenkel set theory without countenancing the all-inclusive set,
that is, the set of all sets. That was required in order to avoid paradoxes,
because the all-inclusive set would be a member of itself. But in terms of
set theory itself, it is puzzling how sets could exist without all the sets
being a set, for they are all parts of the same world.

On this ontological explanation of the truth of set theory, however, there is no puzzle. All the sets do exist together, because they are aspects of a single world, in the sense that they can all be constructed by grouping substances or groups of substances together. That explains how all of the sets can exist without there being a set of all sets. The totality is the world itself. And the set of all sets cannot be formed. As we have seen, it is not possible for a rational subject to group the set he is constructing as a member of the set he is constructing, for it does not yet exist.

*Consistency.*
This ontological explanation of the truth of set theory and the arithmetic
theorems that follow from it proves that they are consistent. That is important,
because mathematicians want assurance that their deductions will not generate
paradoxes, that is, contradictions. In 1931, Kurt Gödel (1906-1978) showed
that any formal system that is complex enough to generate the propositions
of arithmetic cannot be shown to be consistent on the basis of set theory
or logic alone. The inability to prove the consistency of arithmetic has been
a source of embarrassment and consternation, because mathematicians now look
to formalizations, such as set theory, as the foundation for their mathematical
proof.

It is, however, possible to show the consistency of a formal system by giving an interpretation (or model) of it that is assumed to be consistent. That is how the consistency of non-Euclidean geometries was demonstrated. The axioms of Lobachevskian and Riemannian geometry were shown to hold of geometrical objects that were constructed within Euclidean geometry, and that proved that those non-Euclidean geometries were both consistent, because Euclidean geometry was assumed to be consistent.

Although the consistency of arithmetic cannot be shown by logical means, it can be shown ontologically. The reason no one doubted the consistency of Euclidean geometry is that it holds of the structure of the world and the world actually exists. There cannot be any contradiction in propositions that merely describe the nature of something that actually exists. That was an ontological proof of the consistency of Euclidean geometry, and that is the kind of proof that spatiomaterialism gives of the consistency of arithmetic. If set theory is understood as a description of the groups that can be formed of substances in a spatiomaterial world ((by rational beings in that world), then the existence of that world shows that set theory and all the theorems that follow from it are consistent. There can be no paradoxes.

*Completeness.*
Another embarrassment to basing arithmetic on set theory was also
contained in Gödel’s 1931 paper, namely, his incompleteness theorem. He showed
that there are propositions in arithmetic that cannot be proved. (And what
is more, he showed by further, less formal, means that those propositions
are true.) That is, Gödel proved by the use of arithmetic that, if any formal
system that is complex enough to include arithmetic is consistent, then it
is incomplete.

His proof depended on using numbers (Gödel numbers) to represent
not only propositions in arithmetic, but also *propositions about logical
relations *among arithmetic propositions. By representing both arithmetic
and a formal system for describing logical relations in arithmetic by numbers,
Gödel was able to construct a sentence within arithmetic that says, when interpreted,
"This sentence is not provable."

Now, is this sentence provable in arithmetic? If it is not provable, it is true. But it must be true, if arithmetic is consistent, because if it were provable, it would be false, and arithmetic would not be consistent. Hence, there is a true statement in arithmetic that cannot be proved.

What Gödel showed was the *logical incompleteness *of arithmetic
and set theory. But that does not necessarily mean that the propositions of
arithmetic are not a complete set of truths about the numbers and their properties.
That is true only if mathematical truth is taken to be mere provability within
set theory (or any other formal system). But that is what ontological philosophy
denies. It explains the truth of arithmetic ontologically, that is, as correspondence
to the world. And there is no reason to doubt that arithmetic, founded on
set theory, is ontologically complete.

That is, Gödel’s incompleteness theory does not give us any
reason to believe that there are true arithmetic propositions *about the
world* that are not provable in arithmetic. The statement Gödel constructed,
which said, in effect, "This statement is not provable," depended
on interpreting the numbers in terms of the symbols used in arithmetic and
in a formal system for describing logical relations among propositions in
arithmetic. That is not a reference to substances in the world, but a reflective
reference to formal systems as they are understood by the rational beings
using them, and as we shall see when we explain the nature of reason (in Change:
Stage 9), a far more complex ontological explanation is required
to spell out the nature of formal systems in terms of the substances constituting
the natural world.)

Far from being a puzzle about mathematical truth, therefore, Gödel’s incompleteness theorem is a reason for believing that the truth of mathematics should be explained ontologically. There is no reason to doubt the ontological necessity of mathematical truth, that is, its priority to what is known by empirical science about the world on the basis of experience of what happens there.

*Determinacy
of reference.* Determinacy of reference. A further puzzle was posed by the Löwenheim-Skolem
theorem. It holds that a formal system constructed to generate propositions
about one kind of mathematical object can always be given another interpretation
in which they are true of an entirely different set of objects. For example,
any consistent set of axioms constructed to generate all the theorems about
real numbers, which are non-denumerable, can be given another interpretation
in which they are true of sets which are denumerable, such as the integers.
Likewise, axioms designed to derive all the theorems about the whole numbers
can be given an interpretation in which they are true of non-denumerable sets.
Indeed, every consistent set of axioms has a countable model.

No puzzles are posed by the Löwenheim-Skolem theorem, however, if the truth of mathematics is explained ontologically. Indeed, such a theorem is just what just what should be expected, if mathematics is true because of its correspondence to the world. A formal system, such as set theory, has primitive terms, which are not defined in the system, and what makes it possible to give other interpretations in which those axioms are true is assigning different referents to those primitive terms. But when the truth of arithmetic propositions is explained as correspondence to the world, the primitive terms of the axioms of set theory are introduced as references to substances and the groups that can be formed of substances in a spatiomaterial world, and there is no possibility of another interpretation. All of mathematics that follows from set theory refers to certain aspects of the world.

And we must distinguish between geometry generated as analytic geometry and geometry as explained above, because the correspondence to the world in the latter restricts the interpretation of such terms as "line," "angle," and the like to only certain possible sets in the world.

*The usefulness of mathematics in science.* This ontological
explanation of the truth of arithmetic and geometry may also make it possible
to solve other problems (for example, by showing that there is no good reason
to believe that the continuum hypothesis is true), but enough has been said
to illustrate its significance. There is, however, one final consequence that
is worth noting, though it is as much a problem about science as about mathematics.

The assumption that the truth of mathematics comes down to provability within a formal system has made it seem puzzling that mathematics should be so useful in science. Indeed, that is the most unsettling puzzle about mathematics in the view of contemporary philosophers, who take these puzzles as casting doubt on mathematics as the model of true knowledge. But it is not at all puzzling, given this ontological interpretation of the truth of mathematics.

It is not puzzling that mathematics is so useful in science, when its propositions are understood to be about the most basic aspects of the world, namely, how the world is made up of many distinct, particular substances and how, being related to one another spatially, they can be grouped together in all possible ways. Such sets include all the quantitative aspects of substances, from distances and times to masses and forces. Thus, it is hardly surprising that sets in that sense and the ontologically necessary propositions that hold of them because they are substances in a spatiomaterial world are relevant in explaining what happens in the world. Their relevance will become even more clear in Change when we take time into consideration and describe the concrete nature of matter and space. The basic laws of physics describe quantitatively precise regularities about how bits of matter move and interact, and since mathematics holds of the sets picked out for those purposes, there is no wonder that mathematics describes relations that are relevant in those descriptions.

It is not easy for contemporary physicists to see this, however, because the twentieth century revolutions in physics have forced them to abandon the expectation of an intuitive understanding of what their highly mathematical theories are about. Though the intelligibility of scientific theories in terms of spatial imagination was taken for granted in classical physics, it is now generally assumed that it is beyond our grasp. But the ontological explanation of the truth of contemporary physics will show that that is not necessarily the case.