R
elations.
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.
R
elations
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.
G
eometry.
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.
A
rithmetic.
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.
S
et
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.
S
olutions
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.
T
otality.
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.
C
onsistency.
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.
C
ompleteness.
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.
D
eterminacy
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.
R
elations
as objects of knowledge. Ontological philosophy explains
relations as aspects of the world that exist because of the essential
nature of space and how space contains bits of matter at any moment,
and correspondence to them explains, as we have seen, how
mathematical propositions are true. That means that mathematics is
prior to empirical science in the sense of being ontologically
necessary. However, necessity in the sense of being certain
is what has traditionally been thought to make mathematics different
from empirical science. Certainty is what is relevant about
mathematics when the project is justifying belief in certain
propositions by how they are related to what is known in other ways.
Thus, epistemological philosophy approaches mathematical objects as
objects of knowledge, rather than as aspects of the world, and it is
not obvious that what mathematics is about are the most basic
relations that hold in the world.
T
he
problem of mathematical knowledge. When the certainty of
mathematics is taken for granted, the problem of mathematical
knowledge is to explain how such certainty is possible, that is, why
it is more certain than what is known by ordinary experience of what
happens in the world.i
It
is somewhat misleading to think of the certainty of mathematics only
as a problem, for in the beginning, that is what inspired belief in
epistemological philosophy. In ancient Greece, mathematics was taken
as an example to show the possibility of philosophy as a superior
kind of knowledge of the world, one that revealed necessary truths.
In the
Meno,
for example, Plato describes Socrates as asking a slave boy a series
of questions about some lines he draws in the sand which lead the boy
to recognize the truth of a special case of Pythagoras’ theorem
(that the square built on the diagonal of a square is twice the area
of the first square). That put the slave boy in a position to defend
what he knew rationally, and Plato used that story to illustrate how
knowledge is different from true belief.
Beliefs
about whose truth one can be certain are what philosophy pursues out
of its love of wisdom, according to Plato. Above the entrance to
Plato’s Academy, the first university, it was written that no one
should enter who does not know mathematics.
It
is hard to overstate how important mathematics has been to the
credibility of philosophy’s claim to provide a kind of knowledge of
that is superior to our ordinary ways of knowing what happens in the
world through experience. But given its role in epistemological
philosophy, the issue about how the certainty of mathematical
knowledge is possible becomes the issue of how realism is possible.
T
heories
of mathematical knowledge. To set the stage for
considering the received explanations of the certainty of
mathematics, let us consider briefly what ontological philosophy
implies about the knowledge of mathematics. We will then take
up the epistemological theories.
O
ntological
theories of mathematical knowledge. We have explained why
mathematics is true by showing how its propositions correspond to
relations as basic aspects of a spatiomaterial world. Geometry
corresponds to the structure that space has as (part of) its
essential nature as a substance, and that explains why the
propositions of geometry hold of bits of matter in space as well as
points. Arithmetic holds of the particular substances postulated by
spatiomaterialism, because they all have spatial relations to one
another, making it possible to pick out particular substances and to
group them together in sets. But that does not explain how it is
possible for rational beings like us to know that these propositions
are true — and to know that they are true in a way that makes them
certain in comparison to empirical science and other ordinary ways of
knowing about the world.
The
short answer is that mathematics is not certain, but merely prior to
empirical science. Mathematical propositions are among the necessary
truths proved by ontological philosophy. They are ontologically
necessary, because they are entailed by the best ontological
explanation of the natural world, namely, spatiomaterialism. That is
the foundation that ontological philosophy uses to prove that
propositions are necessarily true about the world, and mathematical
propositions are among them, because they correspond to basic aspects
of any spatiomaterial world. But to prove that propositions are
ontologically necessary is not necessarily to prove that they are
certain, that is, epistemologically necessary. Since
spatiomaterialism itself is an empirical truth, the justification of
what follows from it is ultimately empirical and, thus, falsifiable
by experience. It is nevertheless prior to empirical science, because
ontological explanations are prior to efficient-cause explanations.
What follows from spatiomaterialism could be false, because
spatiomaterialism could be false. But if what follow from it is
false, we must give up our otherwise empirically well-founded belief
about the basic nature of existence and deny that the world is
constituted by its two, opposite kinds of basic substances.
Nevertheless,
mathematics seems
to
be certain. It was not without reason that traditional philosophy
took an epistemological approach to necessary truths. And the long
answer to the question about why beings like us believe that
mathematics is true and believe that it is more certain than science
has to do with the nature of reason. Reason is a cognitive capacity
that evolves in certain animals, and as we shall see (in Change:
Evolutionary stage 9
and
following), reason has an ontologically necessary nature which
involves two forms of imagination. But it will be easier to explain
the received, epistemological philosophies of mathematics if we
anticipate that explanation with a brief account of them here.
Animals
become rational as they evolve the use of language, and in a world of
space and matter in time, it is plausible to suppose that those
animals already have a spatial imagination by which they can
understand the structure of space.
By "spatial
imagination", I mean a brain mechanism (a system of
representation in what will be called the "animal behavior
guidance system") that uses spatial images of objects and
temporal sequences of them to represent objects, their spatial
relations to one another in three dimensions, and how their spatial
relations change as a result of motion or being manipulated. At its
core, it is a memory mechanism that records the locations of objects
by lining up images of them in the order they would appear as a
result of locomotion in each direction in space, and since "covert
locomotion," that is, motor commands for moving the body that
are not actually executed, can call up those images in sequence, it
serves as a form of imagination that gives animals a nonlinguistic
way of thinking about the basic geometrical structure of space and
the effects of motion on their relations. (Spatial imagination is
this brain mechanism that makes it possible for computers to generate
what is called "virtual reality.")
Furthermore,
in animals that can manipulate objects, such as primates, spatial
imagination also includes an ability to think about the geometrical
structures of objects and how they interact when being manipulated.
Acts of imagination call up spatial images of objects in sequences
that represent the effects of manipulating them in various ways.ii
Spatial
imagination gives even nonlinguistic animals an intuitive way of
understanding the structure of space, that is, that spatial relations
among objects. And as suggested in the last chapter, since such brain
activity involves a form of matter whose intrinsic nature registers
what is happening throughout the forebrain, spatial imagination is
what makes it appear that sensory qualia are located in phenomenal
space. That is, its structure is what gives rise to complex
phenomenal properties and what we are calling the unity of
consciousness. In this context, however, it phenomenal appearance
explains the faculty of intuition on which epistemological philosophy
typically bases its theory about the nature of reason.
It
is not surprising that such a cognitive faculty evolves in a
spatiomaterial world, given that animals acquire food by ingesting
other objects in space, for it gives them more power over objects in
space. Indeed, we shall see that its evolution is inevitable in
worlds where evolution can occur at all. But this nonlinguistic
understanding of the spatial and temporal aspect of the world is
inherited by animals in which language evolves, and in such animals,
spatial imagination comes under the control of verbal behavior.
In order to
understand a sentence about objects in space, users of language must
construct its meaning in imagination. Spatial imagination makes it
possible to connect words to particular material objects in space,
and thus, learning the meanings of words involves the development of
"abstract images," which correspond to properties and
relations, or the aspects of objects in space that are called
"abstract objects." (As we shall see, they develop in the
brain as states that represent many different particular objects of
certain kinds indifferently.)
Furthermore,
learning to combine such words grammatically involves the development
of complex representations, in which properties are related to the
objects that have them and states of affairs are represented. Thus,
language is a second system of representation. The capacity of
language to represent basic aspects of a spatiomaterial world derives
therefore, from the spatial imagination of the (mammalian) animal
system of representation. (This is the role of what I will later call
"natural sentences.")
However,
rational imagination, as I will call it, depends on another kind of
linguistic representation, in addition to the linguistic
representations based on spatial imagination (or "natural
sentences') and the representations of spatial imagination itself.
The use of language, as we shall see, eventually makes the animals in
which it evolves reflective. (This further stage in the evolution of
language introduces what I will call "psychological sentences.")
The ability
to use more complex sentences enables language-using animals to
represent to themselves the (brain) states (such as perceptions,
memories, beliefs, desires, and intentions) that occur in the process
of perceiving and thinking about the natural world and to think about
the roles that such states play in causing behavior and beliefs.
Thus, these animals can reflect on the causes of their beliefs and
behavior.
But in
reflective animals, such reflective (brain) states can themselves be
causes of the conclusions they draw about how to behave or what to
do, and thus, they have earned a special name. They are called
"reasons." In other words, reasons are basically just
causes of conclusions that are represented as causes as an essential
part of the process of causing such conclusions. Considering how
language depends on spatial imagination to connect words to objects
in the world, the control that language has over spatial imagination
transforms the animal faculty of imagination into rational
imagination, a capacity to think about the possible reasons for
certain conclusions.
These
three elements — the animal's spatial imagination, how it connects
linguistic representations to the world, and how language eventually
enables the animals in whom it evolves to reflect on the reasons for
their beliefs and intentions — are essential to reason, and they
explain why it seems that mathematical truths are certain. Spatial
imagination is an intuitive way of understanding the structure of
space, and thus, if spatial relations among substances are the basic
subject matter of mathematics, it is an intuitive understanding of
mathematical propositions.
There
is, of course, a longer story to be told about how reflection on its
operations evolves into explicit knowledge of geometry and
arithmetic. But for now, let us simply notice that, as rational
beings with such knowledge reflect on the causes of their beliefs, a
difference between mathematics and empirical science will inevitably
appear. Though it is possible to know the propositions of geometry
and arithmetic by perception, in the same way as other facts about
nature, it is eventually noticed that they have reasons for believing
mathematical propositions that do not depend on perceiving what
actually happens in the world. They seem forced to believe, for
example, that a straight line is the shortest distance between two
points and that two plus three is five by their very understanding of
those propositions. Those beliefs seem especially compelling, because
those facts about the world are built into the structure of their
spatial imagination.
Thus, when
epistemological philosophers reflect on how they know that
mathematical propositions are true, the first hypothesis is that
geometrical objects and numbers are objects of a special kind which
are revealed only to rational intuition (or what ontological
philosophy explains as the subjective, phenomenal appearance of
rational imagination). That is basically Platonism about mathematics.
Given how
theorems about geometrical figures and numbers can be derived from
axioms, however, another possible hypothesis is that mathematical
propositions are a result of logic or reasoning. That leads to forms
of anti-realism about mathematics.
In either
case, however, there there seem to be reasons for believing
mathematical propositions that are sufficient, but which do not
depend on perceiving what happens in the natural world. That explains
the apparent certainty of mathematics. It can be known in a way that
does not seem to be vulnerable to what is learned about the world
through perception.
These
epistemological theories do not lead to errors in mathematics,
because what seems certain in this way actually holds universally in
a spatiomaterial world. That is, what spatial imagination corresponds
to is the basic aspect of the world in which rational beings find
themselves.
It is
interesting to notice that, since that basic aspect of the world is
its spatial structure, or the aspect of the world that, more than
anything else, makes the world whole, mathematics is a way of knowing
about the wholeness of the world. And since it is known by subjects
who are part of that world, mathematics is the part's knowledge of
the basic nature of the whole of which it is part.
This
ontological explanation of the apparent certainty of mathematical
knowledge is the foundation for its critique of epistemological
philosophy of mathematics.
E
pistemological
theories of mathematical knowledge. The approach of
epistemological philosophy is just opposite to ontological
philosophy. Instead of starting with ontology and showing that
mathematical truths are ontologically necessary, epistemological
philosophy starts by reflecting on how we know about the truth of
mathematical propositions and tries to show that they are necessary
in the sense of being certain, or epistemologically necessary. The
basic form of success in epistemological philosophy of mathematics is
realism about entities beyond what is known by ordinary experience of
the natural world, and as we have seen, the fate of epistemological
philosophy is sealed, because its realism involves metaphysical
dualism. The problems of metaphysical dualism eventually leads to
anti-realism.
As
the example of Socrates and the slave boy in the Meno suggests,
mathematical knowledge was the original inspiration for philosophy’s
claim to have a superior way of knowing about the world. It was the
first way philosophy ever claimed to prove there are necessary
truths. Since epistemological philosophy began with Plato's use of
the certainty of mathematics to illustrate the success of realism,
realism in the philosophy of mathematics is now called "Platonism."
Given the fate of epistemological philosophy, Platonism eventually
leads to anti-realism. But in the case of mathematics, even most
anti-realists affirm the certainty of its propositions. There is,
however, a form of anti-realism that denies the certainty of
mathematics by assimilating it to empirical science, that is, by
denying that there is any basic difference between mathematics and
science.
A
brief account of the history of epistemological philosophy of
mathematics follows, and having seen how ontological philosophy can
explain why mathematics appears to be certain to those who reflect on
how they know it, I will use the ontological theory of reason to show
not only what is true and false in the traditional theories of
mathematics, but also how the philosophical problems caused by the
epistemological approach are solved by ontological philosophy.
R
ealism:
Platonism about mathematics. For philosophers who argue from how
we know to what can be known, success comes from showing that we have
knowledge of the real existence of entities of some kind beyond a
kind of knowledge that is taken for granted, that is, knowledge of
reality beyond appearance. In the philosophy of mathematics, realism
is called "Platonism," after its founder. But Platonism
takes different forms in the ancient and modern worlds.
A
ncient
Platonism about mathematics. Plato’s explanation of what the
slave boy learned from Socrates is that beings like us have a faculty
of reason that makes us aware of objects that are fundamentally
different from the objects of perception. That is how all genuine
knowledge (as opposed to mere belief) was explained by Plato, and it
is the model for Platonism in mathematics. Numbers and geometrical
objects are part of a reality that Platonists believe lies beyond
appearances in natural world.
According
to Plato, the Forms in the realm of Being are different from natural
objects, which are known by perception (that is, empirical
knowledge), because the Forms do not change and never appear
differently from what they really are. What enables us to know about
them is rational intuition, which Plato repeatedly contrasted to
perception, as knowledge to mere belief. But it is the difference in
the natures of the objects being cognized that was supposed explain
the certainty and necessity of mathematical truths.
Rational
intuition of mathematical objects does involve appearances, according
to ontological philosophy, for there is a faculty of rational
imagination in the brain and its activity has an appearance to the
subject by way of phenomenal properties (by generating bits of matter
whose intrinsic natures register brain activity). But that is not the
appearance of objects that are outside space and time, and the belief
that the objects being grasped are Platonic Forms involves an
insuperable problem, namely, Platonic ontological dualism.
Mathematical
propositions hold of objects found in the natural world, and in order
for Platonists to explain how our knowledge of such propositions is
certain, they need a way to explain why truths about abstract
entities in a realm beyond nature reveal something about objects that
exists in nature. The main problem with platonic realism, as Plato
himself recognized, is that there is no way to explain how objects
outside space and time can have any effect on objects in the natural
world.iii
Ontological
philosophy avoids the problems of Platonic realism by taking
mathematical objects to be aspects of the natural world, rather than
abstract entities that exist in a transcendent realm. But it can also
explain why they appear to be abstract entities. In both cases,
abstract entities are reifications of concepts based on spatial
imagination.iv
Though
geometrical structures are always concrete parts of space, they seem
to be universal, because space exists everywhere with the same
three-dimensional structure. Since reflective subjects with spatial
imagination recognize such geometrical structures in many different
particulars, it is not surprising that they think of them as
universals or abstract entities.
Likewise,
though the material objects they count are concrete particular
substances existing independently of one another, their spatial
relations are what makes it possible for them to be grouped together,
and since that makes the results of arithmetic operations the same
everywhere, numbers seem to abstract entities.
M
odern
realism (Platonism) about mathematics. With the rise of modern
philosophy, the problems with Platonism about mathematics were
transformed, but not solved. Plato was a naive realist about both
perception and reason. He believed that the objects of both forms of
intuition (that is, perception and rational imagination) exist
independently of the subject, but are nevertheless immediately
present to the subject. The modern period began with the recognition
that perception is mediated by appearances (or "ideas’) that
are part of the mind, and that meant that rational intuition is
likewise just another kind of appearance in the mind (what Descartes
called clear and distinct ideas). That eliminated the problems caused
by Plato's attempt to explain the relationship between the objects of
perception and the objects of reason as the relationship between
Forms in a realm of Being and visible objects in the realm of
Becoming. But modern philosophers were still Platonists, in effect,
because they believed that what makes knowledge of mathematics
certain, in contrast to empirical knowledge, is that it is about
abstract objects that exist independently of both the subject and the
natural world. But instead of existing in a realm of Being,
mathematical objects were taken to exist as ideas in the mind of God.
In short, as an offspring of the marriage of Platonism and
Christianity in the medieval period, the modern era had inherited a
rationalistic theology in which God played the role of the realm of
Being.
Modern
philosophy still had to explain, however, why mathematics is true of
the natural world. Indeed, the question was even more pressing,
because the new science discovered laws of nature that are highly
mathematical. Those laws described precise quantitative relationships
among properties, such as distance, mass, time, and velocity, and
since relations among different quantities of the same property are
arithmetical, those physical descriptions required the truth of
mathematical propositions.
Modern
philosophy had, however, a ready solution, at least, until doubts
about theistic supernaturalism late in the eighteenth century,
because the objects of mathematics were assumed to be ideas in God’s
mind. God created the natural world according to a rational plan, and
since God had used mathematics to create a world governed by natural
laws, the discovery of those laws was basically seeing into God’s
mind. In The Assayer (1610), for example, Galileo described
nature as a book that God had written in the language of mathematics.
And Descartes used God to prove that our clear and distinct ideas
about geometry corresponded to extension, the essential nature of the
bodily substance. In other words, it was possible for rational beings
to recognize the truth of mathematical propositions, because
rationality comes from their being created in God’s image.
A
nti-realism
about mathematics. The problems of supernaturalism eventually
made Platonism in either its ancient or modern form untenable. There
is simply no way to prove the existence of entities existing beyond
the natural world. But anti-realism about mathematics takes two
fundamentally different forms, because mathematics still seems to be
certain, even if realism is doubtful.
One form
continues to accept the certainty of mathematics and tries to explain
how there can be such self-evident truths without having to prove the
existence of entities beyond what is given to ordinary experience of
the world.
The other
form takes the denial of the existence of platonic entities beyond
the natural world to mean that mathematics must be about the natural
world, and by assimilating mathematics to empirical science, denies
that mathematics has the kind of certainty that is taken for granted
by realists and other anti-realists.
A
nti-realism
that affirms the certainty of mathematics. The nineteenth
century was a transitional period in the history of mathematics. Not
only did the rise of naturalism made Platonism less attractive, but
developments in mathematics itself also made it less plausible that
mathematics describes the essential nature of a reality beyond the
subject, natural or supernatural.
Though
Euclidean geometry had once made it seem obvious that one kind of
rational certainty somehow reveals the inherent nature of the natural
world, the discovery of non-Euclidean geometry cast doubt on that
assumption. The certainty of geometry seemed to depend more on the
deducibility of theorems from axioms. And recognition that the
arguments (about infinity) on which the calculus had been based were
logically faulty focused mathematicians on the project of making
mathematical proofs more rigorous. Though physics undoubtedly
required mathematics for its spectacularly successful descriptions of
regularities in the natural world, it was, by the beginning of the
twentieth century, plausible to hold that the certainty of
mathematics does not come from knowing a special kind of object that
exists independently of the subject. Instead, it seemed possible to
explain its special certainty as deriving from the nature of the
rational subjects themselves.
Various
theories of the certainty of mathematical truth have been proposed in
the twentieth century, and disputes among them tend to be technical.
But a rough sketch of two opposite approaches and their problems will
put us in a position to see why naturalists now seem to have little
choice but to treat mathematics as a species of empirical, scientific
truth. Both of the following views give up the belief in
independently existing, abstract entities, and both explain its
certainty by holding that it is a kind of truth that is discovered
within the mind. And both are just what would be expected of
epistemological philosophers, given how ontological philosophy
explains the ability of rational beings to know the truth of
mathematical propositions. One takes account of the role of spatial
imagination and attempts to reduce all of mathematics to objects of
rational intuition, and the other takes account of the role of
language in expressing those intuitions and attempts to reduce
mathematics to logic or the structure of language.
I
ntuitionism.
Intuitionism derives historically from Kant, and it
reflects the assumptions of modern philosophy. Kant argued that
mathematics is a priori knowledge about the natural world
because it describes the structure of the forms of intuition (space
and time) in which nature itself is presented in experience. Proofs
of mathematical propositions involve the construction of mathematical
objects in imagination, and thus, they must conform to the mind’s
pure forms of intuition, space and time. But that means that
mathematical truth hold necessarily and universally in experience of
the natural world, because the two forms of intuition are also
conditions of possible experience.
Kant
was describing the process by which rational beings do actually come
to accept the certainty of mathematical propositions, according to
our ontological explanation of reason. The role that Kant ascribed to
space and time as forms of intuition in understanding mathematics is
explained by spatial imagination, and that accounts for knowledge of
geometrical and arithmetical propositions.
The use of
a language to control and to reflect on the structure of spatial
imagination gives one a nonlinguistic understanding of what is meant
by such concepts as "point", "line", "plane",
and "sphere," and thus, one can "see" that there
is a shortest distance between two points and that a line and a point
not on it determines a plane. One can also recognize the truth of
simple propositions, such as that exactly three lines intersecting at
a point can be mutually perpendicular, that three planes can be
mutually perpendicular, and that any closed plane figure with just
three internal angles has three sides.
Ontological
philosophy confirms Kant's theory of arithmetical knowledge in a
similar way. Spatial imagination enable reflective subjects to think
about the operations of singling objects out, combining them as
groups, adding and subtracting members, and the like, and thus, they
can recognize the truth of arithmetic axioms and construct theorems
of arithmetic in imagination. That makes it seem that such truths
about the world can be known prior to discovering their truth by
perception, because what makes arithmetic true is the way in which
space makes different bits of matter parts of the same world and that
aspect of the world is represented in spatial imagination.
It also
seems, in a similar way, that they can know the truth of theorems in
other mathematical systems constructed from arithmetic and geometry,
such as calculus, prior to discovering their truth by perception.
Though
Kant did not develop his constructivist approach to mathematical
propositions in much detail, intuitionism was taken up by many
mathematicians in the twentieth century (including Henri Poincaré,
1854-1912) and given a detailed defense by L. E. J. Brouwer
(1881-1966).
In the end,
however, intuitionism was not acceptable to most mathematicians,
because the requirement that all mathematical objects be constructed
in imagination required giving up too much of mathematics. (Brouwer
rejected the axiom of choice, actually infinite sets, Cantor’s
transfinite numbers, and any arguments for the existence of
mathematical objects based on the law of excluded middle.)
Even if it
were possible for intuitionists to construct all of mathematics,
however, this way of explaining the certainty of mathematics implies
that its truth comes from the structure of thought. Kant believed
that nature is just the phenomenal world, which is "in the
mind," so to speak, and though he never doubted there is a
noumenal world (or things in themselves) beyond the phenomenal world,
he denied that mathematical truths hold of it (or them). That may be
plausible to Kantians, but it is not plausible to naturalists.
Naturalists believe that what exists independently of the subject is
a world of material objects with spatial relations that change over
time, and intuitionism does not imply that mathematics is true of
that world.
L
ogicism
and formalism. The other way of explaining the certainty
of mathematics in epistemological philosophy without accepting
Platonism tries, in effect, to reduce mathematics to language.
Historically, it has taken two main forms, logicism and formalism.
Logicism
holds that all of mathematics is derivable from logic. Its
philosophical roots are in Leibniz, but it was developed more
rigorously by Gottlob Frege (1848-1925) and Bertrand Russell
(1872-1970) around the turn of the twentieth century.
It turned
out, however, that the laws of logic needed to generate number theory
involved several axioms that hardly seemed to be laws of logic at
all. They included, for example, the axiom of reducibility (which
holds that propositions about higher types, or sets of sets, could be
reduced to propositions about first order members of sets), the axiom
of infinity (which affirms the existence of infinite sets), and the
axiom of choice (which says that from any set of non-empty,
non-overlapping sets, it is possible to form a set of one member from
each). If one were to insist that these are laws of logic and that
they are known by rational intuition of some kind, logicism would be
a kind of Platonism in which the laws of logic, rather than the
mathematical objects themselves, have an independent existence as
abstract objects outside of space and time. But to many, these axioms
seemed more doubtful than the propositions about numbers that were
derived from them.
"Formalism"
is the name of the project pursed by David Hilbert (1862-1943) in
order avoid the problems of logicism. He did not believe that
mathematics could be reduced to the laws of logic. He held that each
branch of mathematics requires its own axioms and rules of inference.
But he believed that logicism was on the right track in taking
logical entailments to be what explains mathematical truth. Thus,
Hilbert set out to prove the certainty of mathematics by
reconstructing each branch of mathematics as a formal system with its
own axioms, rules of inference, and theorems. But these statements
were to be stripped of any meaning outside the formal system and
treated as meaningless symbols, mere marks on paper, which were
written down in sequence according to strict rules. Each formal
system would include all the propositions in some branch of
mathematics, and the rigor of these symbolic manipulations was
supposed to prove the certainty of its theorems.
The
formalists’ explanation of mathematical certainty, however,
required systems constructed in this way to be free of
contradictions, and so Hilbert saw the main challenge as
demonstrating their consistency. For this purpose, he developed a
special formal system for describing formal systems, a
"metamathematics," which was supposed to be beyond
reproach. In the end, however, it was not possible to demonstrate the
consistency of arithmetic, or even of set theory, as Gödel showed.
(This is the origin of the puzzles encountered by set theory that
were solved in the ontological explanation of the truth of arithmetic
in Relations:
Solutions to puzzles.)
What
logicists and formalists are getting at can be understood from our
sketch of the nature of rational subjects. They are also
epistemological philosophers reflecting on how we know the truth of
mathematical propositions. But by contrast to intuitionists, they
abstract from spatial imagination and focus on the use of language.
They identify brain states by the linguistic representations they
involve, and they use logical relations to keep track of the causal
roles that brain states play in drawing conclusions about what to
believe. By focusing exclusively on the formal relations among brain
states identified in that way, whole systems of mathematical proofs
can be reconstructed as formal deductive systems. The logical
structure of language represents the elements in such reasoning
completely enough that there are formal tests of the validity of
those inferences, making it seem that their truth can explained by
their deducibility from certain axioms and definitions.
Though
the validity of deductive relationships in formal systems does afford
a certain concept of certainty, it does not explain how mathematics
is true. Even logicists complained that formalism cannot account for
the truth of the axioms or the usefulness of the definitions that are
assumed. But neither do the axioms used in geometry and arithmetic
follow from the laws of logic. Indeed, any consistent set of
sentences could be used as axioms and definitions, because as far as
formal logic is concerned, deductive systems are just rule-governed
ways of transforming assumptions as inscriptions that preserve their
truth. Formalism has no explanation of why the axioms used in
mathematics should be singled out as true. Nor does it explain why
they, or the theorems derived from them, should be useful in
describing the natural world.
A
nti-realism
that denies the certainty of mathematics. Applicability to the
natural world is, however, as crucial to the nature of mathematics as
its apparent certainty, and since neither the intuitionists nor
logicists/formalists were able to explain its certainty in a way that
would also explain its applicability to nature, naturalists could not
help being attracted to the view that mathematical objects are
somehow part of the natural world. That would be, like Platonism, a
kind of realism about mathematical objects. But since our way of
knowing about the natural world is perception, it would be more like
scientific realism, for there would be no basic difference between
mathematics and empirical science. And it would have to deny the
certainty of mathematics.
The
view that mathematics is a form of empirical knowledge was first
defended by John Stuart Mill in the nineteenth century, but it was
renewed in 1983 by Philip Kitcher.v
Kitcher
rejected what he called "apriorism", the belief that the
certainty of mathematical knowledge comes from its being
epistemologically prior to experience of nature, and proceeded to
explain mathematics as a species of scientific knowledge. Kitcher
bases knowledge of mathematics on perception, by thinking of
arithmetic operations as "idealizations" of publicly
observable manipulations of natural objects.vi
The price
of explaining how mathematics is true about nature seems to be giving
up the belief that it has a certainty that is basically different
from natural science. Kitcher explains the appearance of certainty by
the extremely general character of the regularities described by
mathematical hypotheses. But since there is no essential difference
between mathematics and scientific hypotheses, he agrees that they
are confirmed in basically the same way.
Ontological
philosophy agrees with Kitcher in rejecting apriorism. It also takes
mathematics to be a form of empirical knowledge in the end. But the
end does not come so quickly as it does for Kitcher, because
ontological philosophy recognizes two levels of explanations
(ontological-cause explanations and efficient-cause explanations)
and, accordingly, two levels of empirical truths (empirical ontology
and empirical science). In other words, instead of taking mathematics
to be knowledge of very general regularities about what happens in
the world, it sees mathematics as knowledge about the most basic (or
categorical) features of what exists in the world, namely, how space
makes the world whole. That means that mathematics is still prior to
empirical science in a philosophically relevant way. But the priority
is ontological rather than epistemological.
Furthermore,
when this explanation of mathematical truth as ontologically
necessary is combined with what ontological philosophy holds about
the nature of reason, there is even a sense in which mathematics is
epistemologically more certain than empirical science. As we
have seen, it holds that mathematical knowledge is not merely a
correspondence of linguistic representations to the world, but also
involves a correspondence of representations in the brain’s spatial
imagination to the world. Thus, unlike Kitcher’s theory, it can
explain the role that constructions in imagination play in proving
mathematical truths according to intuitionists as well as the role of
formal deductive relationships among sentences that logicists and
formalists take to be basic.vii
But as we
have seen, what explains the truth of both geometry and arithmetic
ontologically are the spatial relations that particular substances
have in a spatiomaterial world. Thus, since the rational subject is
part of the world, mathematical knowledge involves a relationship
between subject and object that is a correspondence between the
structure of spatial imagination in a part of the world and the basic
structure of the whole world of which he is part. It is within that
basic correspondence that rational being discover what happens in the
world by perception, and thus, if this is a spatiomaterial world,
mathematics is not only ontologically necessary, but
epistemologically certain.
iFor
an accessible discussion of the problem of certainty in the
philosophy of mathematics, see Kline
(1980). A somewhat more technical, but still readable discussion of
issues about infinity is Lavine
(1994).
iiCovert
manipulation also makes it possible to combine images of effects of
motion from locomotion imagination into a single geometrical
structure in manipulative imagination to think about all the
relations of the objects in some territory at once, like a map.
iiiIn
a much discussed paper, Benacerraf
(1973) argues against Platonism on the ground that it is not
compatible with a causal theory of mathematical knowledge. Bigelow
(1988) nevertheless takes universals to be the objects of
mathematics, and he avoids this problem with abstract entities by
following Armstrong
(1983) and assuming, in effect, that universals are just spatial
relations of bits of matter, which are always instantiated (that is,
as so-called "tropes").
ivThis
is similar to the "skeptical fictionalist" view of
mathematical truth defended by naturalists like Field
(1980) and Papineau
(1993, pp. 193-197). They think that they must deny the existence of
mathematical objects because such objects are abstract. But that is
because they do not recognize the ontological role that space plays
in making arithmetic true. They are implicitly materialists (taking
space to be just spatial relations) who are nominalists about the
concepts used in science, and though they recognize that mathematics
can facilitate complex inferences about the natural world, they
believe that all those inferences could, in principle, be made
without referring to numbers or geometrical figures as abstract
entities. Thus, they take such numbers and geometrical figures to be
useful fictions and are skeptical about their existence. But that
makes it just as puzzling why mathematics holds of the natural world
as Platonism does. However, if space as a substance containing all
the bits of matter is the ontological cause of geometrical figures
and the groupings of material objects called numbers, there is an
alternative, non-fictionalist defense of geometry and arithmetic
truth. Though there is every reason to be skeptical about the
existence of mathematical objects that are abstract entities, that
is no reason to believe that numbers are just useful fictions. They
could describe something concrete -- a very general, ontological
effect of the structure of space on the bits of matter it contains.
vKitcher's
approach is endorsed by other philosophers of mathematics, such as
J. Bigelow
(1988, p. 3). Bigelow holds that mathematics is about universals,
but he follows Armstrong's
(1983) "a posteriori realism" in taking universals to be
physical, and thus, in the terms used here, he is a materialist.
Indeed, the subtitle of his book is "A Physicalist's Philosophy
of Mathematics".
viThe
traditional theory about mathematical truth that comes closest to
Kitcher's is abstractionism, the view originally defended by
Aristotle that mathematical objects are abstractions from perceived
objects. See Körner
(1960, pp. 18ff). Kitcher has a more sophisticated theory about how
mathematics is derived from perception than Aristotle. According to
his "evolutionary theory of mathematical knowledge" (p.
92), the abstractions come from idealizing the operations of
arithmetic, though Kitcher insists that this is compatible with
saying that "arithmetic describes the structure of reality"
(p. 109).
viiThe
assumption that the use of language makes spatio-temporal
imagination a cognitive capacity of reflective subjects enables
spatiomaterialists to answer objections that Kitcher
(1983, pp. 50ff) raises to intuitionism (or "constructivism').
Contrary to Kitcher, it is possible to distinguish essential
properties from those that are accidental, because imagination is
not just "pictures in the mind", but images that
reflective subjects construct and manipulate within its structure.
When a geometrical figure, such as a triangle, is defined, it is
constructed in imagination, and thus, assuming that language-using
subjects can reflect on what they are doing, they can see the
effects of varying triangles in all possible ways on their
inferences about them. Second, although Kitcher is right to insist
that infinite sequences of operations, such the division of a line,
cannot be carried out in practice, subjects who can reflect on what
they do in imagination and its effects can come to see that what
will happen each time is limited in a certain way and, thus, infer
what would, and would not, happen if the operations were taken to
infinity. Finally, the problems about exactness that may arise with
Kitcher's "mental pictures" do not arise with
spatio-temporal imagination. For example, imagined straight lines
cannot be crooked, for they are constructed according to the
understanding of the structure of space that is built into the
structure of spatio-temporal imagination, that is, as the path of
the shortest distance between two points. In short, since
ontological philosophers postulate space as a substance containing
all the matter in the world, they need only recognize the basic role
that a spatio-temporal imagination would play in the reflective
subject's knowledge of the world to explain how reflective subjects
have a priori knowledge of mathematical truth, because its structure
corresponds to the structure of space. Indeed, without the capacity
to see what is given in perception against the background of what
imagination tells us is possible in three dimensional space, it is
hard to see how we could perceive a line as straight, a set of three
lines as a triangle, or anything as a mathematical object.