Number and space
2.1. Set theory and the first two relation frames
Numerical relations and the theory of groups
2.3. The opening up of the numerical relation frame
2.5. The spatial relation frame
2.6. The spatial subject-object relation
2.7. Spatial relations
2.8. Objectivity in the choice of coordinate systems
2.9. The disclosure of the spatial relation
Set theory and
the first two relation frames
Time and again is mostly concerned with an analysis of the foundations of physics. Such an analysis would be quite impossible, however, without taking into account the quantitative and spatial relation frames. In chapter 2 we shall discuss
these, though not as extensively as our discussion of the kinetic and the physical aspects in subsequent chapters. This chapter should not be taken out of the context of this book. My only intention is to investigate the quantitative and the spatial modal
aspects insofar as they are relevant to physics. The mutual irreducibility of these aspects will be discussed later on. In the present section I shall give a provisional outline of their meaning, and discuss their relation to set theory. The reader should
keep in mind the mutual orthogonality of the distinction of law and subject, and that of the various relation frames.
The concept of a set
Plato and Aristotle introduced the traditional view that mathematics is concerned with numbers and with space. Since the end of the 19th century, many people thought that the theory of sets would provide mathematics with its foundations. Since the middle of the 20th century, the emphasis is more on structures and
Numbers constitute the relation frame for all sets and their relations. A set consists of a number of elements, varying from zero to infinity,
whether denumerable or not, but there are sets of numbers as well. What was the first, the natural number or the set? Just as in the case of the chicken and the egg, an empiricist may wonder whether this is a meaningful question. We have only one
reality available, to be studied from within. In the cosmos, we find chickens as well as eggs, sets as well as numbers. Of course, we have to start our investigations somewhere, but the choice of the starting point is relatively arbitrary. Rejecting the view
that mathematics is part of logics, I shall treat sets and numbers in an empirical way, as phenomena occurring in the cosmos.
At first sight, the concept of a set is rather trivial, in particular if the number of elements is finite. Then the set is denumerable and countable; we can number and count the elements. It becomes more intricate
if the number of elements is not finite yet denumerable (e.g., the set of integers), or infinite and non-denumerable (e.g., the set of real numbers).The numerical modal aspect of discrete quantity, as a universal mode of being, presupposes that every created
thing is a unity, and that there exists a multitude of such unities. The numerical modal aspect is universal since there is nothing in the creation which is not subjected to numerical order. This order can be described as the order of before and after, both
in its original meaning of more and less, and in its analogical meaning of smaller and larger in magnitude.
The spatial modal aspect of continuous extension explains why a unique ordering of everything created is impossible solely with the numerical order of before and after. Thus different sets may have the same number of elements, and different
things may have the same size. The spatial order of simultaneous coexistence (on the law side of the spatial modal aspect) makes possible the original spatial relation of relative position (on the subject side of the spatial modal aspect). This spatial modal
order also involves the analogical concept of equivalence with respect to some property, thereby allowing things to share this property in different degrees. The spatial modal order is only universal if it is considered together with the numerical order. Although
the order of simultaneity does not apply to everything created, one can account for all static relations if this order and the order of before and after are taken together.
Set theory is nowadays generally considered to be the basis of the theory of number. Later, in chapter 8, I shall discuss the concept of probability and argue that it refers
to the law-subject relation for individuality structures. Since the theories of probability and sets are closely related, I view the idea of a set as giving expression to the law-subject relation. Sets are always determined by some law. This is even the case
with examples like ‘the set of all books in my room’, for this refers to the law defining ‘books’. In this context the set of all things on my desk is ill defined without further specification of a ‘thing’. In general, classes
are not identifiable, or even imaginable, unless they are defined by a set of laws, and these laws are usually not of a mathematical kind. It is not strictly correct to say that a set is determined by a law. I prefer to say that a set has a law side and a
subject side. The idea of a set cannot be reduced, either to the law side, or to the subject side.
Numbers and sets
concept of number cannot be studied without the idea of sets. Both Baruch de Spinoza and Gottlob Frege observed that one cannot ascribe a number to things, unless these are grasped under a genus.
If in the realm of concrete things and events the post-numerical modal aspects are ignored, there is still the possibility of taking some of them together in a collection. After this process of abstraction, all that remains to be said is that concrete things
belong to classes of things. The common property of all finite collections is that they can be counted, regardless of the spatial, kinetic, physical, etc., properties of their elements. Thus all finite collections are related either directly by a one-to-one
correspondence, or by a one-to-one correspondence between one collection and a proper sub-set of another one. In the latter case the first collection is called smaller than the second one. Because this property is universal, one can now abstract from concrete
sets, discovering an abstract and unique collection of natural numbers, serving as a universal reference system for all finite collections.
On the other hand one cannot talk about a set without having a previous idea of a plurality of concrete things and events, nor can one dispense with the individual unity of its members. Aristotle considered the individuality of things as their only property relevant to arithmetic. For Aristotle individuality meant the identity of a thing
with itself and its being distinct from other things. Arithmetic had to abstract from all other properties of real things. For example, the universal law of addition demands that if a collection of m members is added to a collection of n members, one
always arrives at a collection of (m+n) members, whatever the character of the two collections, provided they have no member in common. This implies that each member has its own subjective identity.
Space and sets
The concept of space cannot be studied without the idea of sets either. A spatial figure is characterized
by being connected and having parts. At the same time we have to consider it as an uncountable set of points, though we cannot define it as such. The fact that we can consider each spatial figure as a collection of connected and nevertheless disjoint parts
is the necessary basis for the introduction of spatial magnitude.
On the other hand,
the idea of a set always has a spatial aspect. In a set we have a number of coexisting members. Members can simultaneously belong to different sets. The notion of sub-sets of a set refers to the simultaneous existence of a whole and its parts. Also the concepts
of ‘union’ and ‘intersection’ of sets refer clearly to the spatial modal aspect. In order to make the transition of all finite collections to the set of natural numbers, one often makes use of the concept of ‘equivalence class’.
The numerical order of more and less is not directly applicable to sets, but only to equivalence classes of sets, each equivalence class uniting all sets with the same number of elements. This also shows that the spatial as well as the numerical orders are
presupposed in this attempt to base a theory of numbers on set theory. In
fact, even if we talk about the set of natural numbers, we already refer to simultaneity.
Without the introduction of numerical and spatial orders, the sub-set of a set can only be partially ordered. In order to arrive at a universal order of sets, we have to introduce the more abstract orders of seriality and spatial simultaneity. For Aristotle,
the number of a set was a concrete property. Frege was one of the first to recognize the abstract character of the cardinal numbers: there is only one number six, regardless of how many sixtuples of concrete things exist.
Even Russell’s definition of the number of a class as ‘the class of all classes which are equivalent to that class’
presupposes the abstraction of all properties of sixtuples, except of being classes, and having six members. It especially presupposes the abstraction from the spatial order of simultaneity, for in this case, one abstracts from the fact that so many sextuples
It is not my intention to investigate the foundations of set
theory. The above arguments only serve to make clear the mutual orthogonality of the law-subject distinction, which finds its mathematical expression in the theory of sets, and the distinction of the various modal aspects, which we intend to study in this
and the subsequent chapters.
2.2. Numerical relations
theory of groups
The numbers form an abstract reference system for any serial
order. Having no concrete existence, their meaning is purely modal. They are numerical modal subjects, being subject to numerical modal laws only. The different number systems which are relevant to physics will be investigated briefly: the natural, integral,
rational, real, and complex numbers, as well as vectors. This will be done in a quasi-formal way, using a group-theoretic approach, because of the relevance of group-theory to present-day physics, and to our analysis of it. As will be seen later (10.5), groups
are typical structures with a numerical character, to be used as instruments in the analysis of the numerical, spatial and kinetic relation frames.
On the law side of the numerical relation frame time expresses itself as the serial order of before and after.
The number 2 is earlier than the number 3, because the latter can be generated from the former by addition of the number 1.
the subject side, the numerical difference is correlated to this temporal order. Obviously, the statement that some number is later than another one gives rise to the question: ‘How much later?’ Indeed, the numerical difference between two numbers
is related to their temporal order of earlier and later: the difference is positive or negative depending on this order (if a>b, then a–b>0,
This serial order forms the basis of Giuseppe Peano’s axioms formulating the laws for the sequence N of the natural numbers. The axioms apply the concepts of sequence, successor and first number, but do not apply the concept of equivalence. According to Peano, the concept of a successor is characteristic for the
1. N contains a natural number, indicated by 0.
2. Each natural number a is uniquely joined by a natural number a+, the
successor of a.
3. There is no natural number a such that a+=0.
4. From a+=b+
5. If a subset M of N contains the element 0, and besides each element
a its successor a+ as well, then M=N.
The transitive relation ‘larger than’ is now applicable
to the natural numbers.
The character of the natural numbers expressed by Peano’s axioms is primarily quantitatively characterized. It has no secondary foundation for lack of a relation frame preceding the
quantitative one. As a tertiary characteristic, the set of natural numbers has the disposition to expand itself into other sets of numbers.
The laws of addition, multiplication, and raising powers are derivable from Peano’s axioms. The class of natural numbers is complete with respect to these operations. If a and b are natural numbers, then a+b, a.b en ab are natural numbers as well. This does not always apply to subtraction, division
or taking roots, and the laws for these inverse operations do not belong to the character of natural numbers.
The set of natural
numbers is the oldest and best-known set of numbers. Yet it is still subject to active mathematical research, resulting in newly discovered regularities, making arithmetic an empirical science. Some theorems relate to prime numbers. Euclid proved that the number of primes is unlimited. An arithmetical law says that each natural number is the product of a unique set of primes. Several other theorems concerning primes are proved or conjectured.
In many ways, the set of primes is notoriously irregular. There is no law to generate them. If one wants to find all prime numbers less than
an arbitrarily chosen number n, this is only possible with the help of an empirical elimination procedure,
known as Eratosthenes’ sieve.
The whole-part relation
It is very important to distinguish a set from its members. The relation of a set to its elements
is a numerical law-subject relation, for a set is a number of elements. By contrast, the relation of a set to its subsets is a whole-part relation that can be projected on a spatial figure having parts. A subset is not an element of the set, not even a subset
having only one element. A set may be a member of another set. For instance,
the numerical equivalence class [n] is a set of sets. However,
the set of all subsets of a given set A (the ‘power set of A’) should not be confused with the set A itself.
Overlapping sets have one or more elements in common. The intersection AÇB of
two sets is the set of all elements that A and B have in common. The empty set or zero set Æ is the intersection of two sets having no elements in common. Hence, there is only one zero set. It is a subset
of all sets. If a set is considered a subset of itself, each set has trivially two subsets. (An exception is the zero set, having only itself as a subset).
The union AÈB of two sets looks more like a spatial than a numerical operation. Only if two sets have no elements in common,
the total number of elements is equal to the sum of the numbers of elements of the two sets apart. Otherwise, the sum is less.
Hence, even for denumerable sets the numerical relation frame is not sufficient. At least a projection on the spatial relation frame is
needed. This is even more true for non-denumerable sets.
Some sets are really spatial, like the set of points in a plane contained
within a closed curve. As its magnitude, one does not consider the number of points in the set, but the area enclosed by the curve. The set has an infinite number of elements, but a finite spatial measure.
A measure is a magnitude referring to but not reducible to the numerical relation frame. It is a number with a unit, a proportion.
This measure does not deliver a numerical relation between a set and its elements. It is not
a measure of the number of elements in the set. A measure is a quantitative relation between sets,
e.g., between a set and its subsets. If two plane spatial figures do not overlap but have a boundary in common, the intersection of the two point sets is not zero, but its measure is zero. The area of the common boundary is zero. For a spatial set, only subsets
having the same dimension as the set itself have a non-zero measure. Integral calculus is a means to determine the measure of a spatial figure, its length, area or volume.
For each determination of a measure, each measurement, real numbers are needed. That is remarkable, for an actual measurement can only yield a rational
The number 2 is natural, but it is an integer, a fraction, a real number and a complex number as well. Precisely formulated:
the number 2 is an element of the sets of natural numbers, integers, fractions, real, and complex numbers. This leads to the conjecture that the character of natural numbers does not determine a class of things, but a class of relations. The meaning of a number depends on its relation to all other numbers and the disposition of numbers to generate other numbers.
The natural numbers constitute a universal relation frame for all denumerable sets. Peano’s formulation characterizes the natural numbers by a sequence, that is a relation as well. The
integers, the rational, real, and complex numbers are definable as relations as well. Therefore, it is not strange that the number 2 answers different types of relations. A quantitative character determines a set of numbers, and a number may belong to several sets.
Since the addition of two numbers yields a number, and the difference between any two numbers is a number, some sets of numbers may form a group. In 1831 Évariste Galois introduced
the concept of a group in mathematics as a set of elements satisfying the following four axioms.
A group is a collection of distinct elements A, B, C, … on which a combination procedure is defined, such that for any pair of elements A, B an element AB can be generated, according to the following rules:
(a) If A
and B are elements of the group, then the combination AB is also
(b) (AB)C=A(BC)=ABC – the group operation is associative
(c) the group contains one element I, called the identity element, such that
for each element A of
the group, AI=IA=A.
(d) to each element A corresponds an inverse element A’, such that AA’=A’A=I.
Here, the equality sign (=) must be understood as ‘is the same as’, ‘is equal to’, ‘cannot be distinguished from’, or ‘can always be substituted for’. There is no intrinsic way to
distinguish the element AA’ from the element I, for instance. The extrinsic lingual distinction only accounts for the different possibilities of generating the same element.
These four rules form the generic character of a group (10.5). They do not fully determine a group, however. As to the law side, one has to specify the group operation, and as to the subject
side, one has to indicate the members of the group, by stipulating some members as a set of generators. The other members are dynamically generated by application of the group operation. Several different groups (i.e., having different members, and eventually
a different group operation) may have the same group structure. In that case the groups are called different isomorphic models or representations of the same group structure. An isomorphism consists on the subject side of a one-to-one correspondence between
the members of the two groups, and on the law side of a parallelism between the respective group operations. If the members A, B, C in one group correspond with the members K, L, M in the other group,
and if AB=C, then KL=M. Thus the law does not define its subjects: the subject side cannot be reduced to the law side. Isomorphism plays an important
part in finding objective relations, e.g. by projecting physical relations on mathematical ones.
As a character, a group is
qualified by the numerical relation frame. It has no foundation in a preceding frame (because there is no one), and it has the disposition of being applied in the numerical and later frames, in particular in the study of characters qualified by the physical,
kinetic, and spatial modal aspects.
Groups may be finite or infinite. The smallest groups contain just one element – evidently
the identity element. For example, the number 1 forms a multiplication group, and the number 0 an addition group. These two groups are even isomorphic. The number 1 and -1 also form a multiplication group, consisting of just two members. Finite groups are
very important in the physics of typical structures, but infinite groups are more interesting for the extension of the set of natural numbers.
Negative and rational numbers as relations
The set of natural numbers does not form a group, though if addition is taken as the group operation, the natural numbers satisfy rules (a) and (b). But there are no inverse elements, which
means that within the set of natural numbers, subtraction is not always defined. However, by including the number zero and the negative integers, one arrives at a group. The integers are generated as members of the smallest addition group, which includes among
its members the natural numbers. The group operation is addition, the inverse of a positive integer is a negative integer, and vice versa. To show that one has to specify some members of the group, it should be observed that the addition group of integers
is isomorphic to the addition group of even integers, of triples, etc. In this approach the positive integers are identified with the natural numbers.
Within the group structure, the element AB’ can be considered as expressing
the intrinsic relation between two elements A and B (for short, I shall say that AB’ is the relation between A and B). The relation between two integers is their numerical difference. The reverse relation is BA’. The relation of an element to itself is AA’=I, the identity. Because
AI=IA=A, the relation of an element to the identity element
is identical with the element itself. Therefore, the numerical difference between two numbers, as the basic numerical subject-subject relation, is a numerical modal subject itself.
Difference is not the only conceivable numerical relation. From addition we can derive the operation of multiplication of two natural numbers (as an abstraction of the repeated addition of equally numbered collections). If we introduce multiplication as a group operation, we generate the positive rational numbers as the members of the smallest multiplication group, whose members include the natural numbers. For the group of positive rational numbers the identity element is the number 1, the inverse of a rational number is a fraction, and the group relation is the ratio between two rational numbers.
The set of all rational numbers (positive, negative, and zero) is then defined as the addition group, whose elements include the positive rational numbers. It cannot be defined as a multiplication group, because the number 0 has no inverse for multiplication.
For the introduction of the rational numbers two group operations are required. This leads to the idea of a field, another ‘algebra’. A field is a collection of subjects in which two operations are defined
(e.g., addition and multiplication), each satisfying the same rules as for groups, except that the identity element for one operation has no inverse with respect to the second operation. The two operations are connected via the distributive law: (A+B)xC=(AxC)+(BxC). Examples are the fields of rational numbers, of real numbers, and of complex numbers. (There are finite fields as well.) They have the usual addition and multiplication as operations, whereas
dividing by zero is not defined.
and dense sets
The group structure does not specify an order between
the elements. The groups discussed so far can be ordered according to the law mentioned at the beginning of this section. A>B, if A–B>0, where ‘larger than zero’ means ‘being positive’.
A set is called discrete in a certain order, if in that order each element has just one successor. Every finite collection is discrete, and so are the sets of natural and integral numbers. In a series the natural numbers (acting as ‘ordinal
numbers’) serve as indices. A set is called denumerable if its members can be put in such a series, i.e., if there is a one-to-one correspondence between the members of this set and the natural numbers. The order of this series is extrinsic, while given
by the indices. An intrinsic numerical order is determined by the numerical values of the set’s members themselves.
consider the set of the rational numbers, which can be arranged in a series, as is shown in any textbook on number theory. In this series, in which a member is not necessarily larger than all preceding members, the members are arranged in an extrinsic numerical order (of the indices). The rational numbers in
their intrinsic numerical order of smaller and larger do not form a discrete series, but a dense set.
This means, in any interval there is at least one rational number, and therefore, an infinitude of rational numbers in any interval, and there is no empty interval, however small.
With the concept of a dense set, the limit is reached of the closed numerical modal aspect. It is the starting point for the opening up of this aspect, anticipating later modal aspects, as will be seen presently.
2.3. The development of the numerical relation frame
The road from the natural numbers to the real ones proceeds via the rational numbers. A set is denumerable if its elements can be put in a sequence. Georg Cantor demonstrated that all denumerable infinite sets
are numerically equivalent, such that they can be projected on the set of natural numbers. Therefore, he accorded them the same cardinal number, called Ào, aleph-zero, after the first letter of the Hebrew alphabet. Cantor assumed this ‘transfinite’ number to be the first in a sequence, Ào, À1, À2, … , where each is defined as the ‘power
set’ of its predecessor, i.e., the set of all its subsets.
The rational numbers are denumerable, at least if put in a
somewhat artificial order. The infinite sequence 1/1; 1/2,2/1; 1/3,2/3,3/1,3/2; 1/4,2/4,3/4,4/1,4/2,4/3; 1/5, … including all positive fractions is denumerable. In this order it has the cardinal number of Ào. However, this sequence is not ordered according to increasing magnitude.
In their natural (quantitative) order of increasing magnitude, the fractions lay close to each other, forming a dense set. This means that no rational number has a unique successor. Between each pair of rational numbers a and b there are infinitely many others. In their natural order, rational numbers are not denumerable, although they are denumerable in a different order. Contrary to a finite set, whether an infinite set is countable may depend
on the order of its elements.
Though the set of fractions in their natural order is dense, it is still possible to put other numbers between them. These are the irrational
numbers, like Ö2 and p. According to the tradition, Pythagoras or one of his disciples discovered that he could not express the ratio of the diagonal and the side of a square by a fraction of natural numbers. Observe the ambiguity of the word ‘rational’
in this context, meaning ‘proportional’ as well as ‘reasonable’. The Pythagoreans considered something reasonably understandable, if they could express it as a proportion. They were deeply shocked by their discovery that the ratio of
a diagonal to the side of a square is not rational. The set of all rational and irrational numbers, called the set of real numbers, turns out to be non-denumerable. I shall argue presently that the set of real numbers is continuous, meaning that no holes are
left to be filled.
Only in the 19th century, the distinction between a dense and a continuous set became clear. Before, continuity was often defined as infinite divisibility, not only of space. For ages, people have discussed about the question whether matter would be continuous or atomic. Could one go on dividing matter,
or does it consist of indivisible atoms? They overlooked a third possibility, namely that matter would be dense.
Even the division
of space can be interpreted in two ways. The first was applied by Zeno when he divided a line segment by halving it, then halving each part, etc. This is a quantitative way of division, not leading to continuity but to density. Each part has a rational proportion to the original line segment. Another way of dividing a line is by intersecting it by one or
more other lines. Now it is not difficult to imagine situations in which the proportion of two lines segments is irrational. (For instance, think of the diagonal of a square.) This spatial division shows the existence of points on the line that quantitative division cannot reach.
By his famous diagonal method, Cantor proved in 1892 that the set of real numbers is not denumerable. Cantor indicated the infinite amount of real numbers by the cardinal number C. He posed the problem of whether C equals À1, the transfinite number succeeding À0. At the end of the 20th century, this problem was still unsolved.
states that each irrational number is the limit of an infinite sequence or series of rational numbers, e.g., an infinite decimal fraction. This seems to prove that the set of real numbers can be reduced to the set of rational numbers, like the rational numbers are reducible
to the natural ones, but that may be questioned. Any procedure to find these limits cannot be done in a countable way, not consecutively. This would only lead to a denumerable (even if infinite) amount of real numbers. To arrive at the set of all real numbers requires a non-denumerable
procedure. But then we would use a property of the real numbers (not shared by the rational numbers) to make this reduction possible. And this appears to result in circular reasoning.
Suppose one wants to number the points on a straight or curved line, would the set of rational numbers be sufficient? Clearly not, because of the existence of spatial proportions
like that between the diagonal and the side of a square, or between the circumference and the diameter of a circle. Conversely, is it possible to project the set of rational numbers on a straight line? The answer is positive, but then many holes are left.
By plugging the holes, we get the real numbers, in the following empirical way.
Consider a continuous line segment AB. We want to mark the position of each point by a number giving the distance to one of the ends. These numbers include the set of infinite decimal fractions that Cantor proved to be non-denumerable. Hence, the set of points on AB is not denumerable. If we mark the point A by 0 and B by 1, each point of AB gets a number between 0 and 1. This is possible in many ways, but one of them is highly significant, because it uses the rational
numbers to introduce a metric, assigning the number 0.5 to the point halfway between A and B, and analogously for each rational number between 0 and 1. (This is possible in a denumerable procedure). Now the real numbers between 0 and 1 are defined as numbers corresponding one-to-one to the points on AB. These include the rational numbers between 0 and 1, as well as numbers like p/4 and other limits of infinite sequences or series.
The irrational numbers are surrounded by rational numbers (forming a dense set) providing the metric for the set of real numbers between 0 and 1.
The set of line segments on a straight line having a common end point is also a group. The group operation is the spatial addition of two line segments, the inverse is a line segment in the opposite direction, the identity element is a line
segment of length zero, and the group relation is a line segment equal in length to that between the non-common terminal points of two line segments. In the present context, the notions of line segment, congruence, and spatial addition are irreducible concepts:
they belong to the spatial modal aspect.
Now the real numbers are introduced as elements of the group (a) whose elements include the rational numbers; (b) which has arithmetical addition as its group operation; and (c) which is isomorphic to the former group of line segments. In order to make the one-to-one correspondence between the elements of the two groups definite, an arbitrary unit segment must be chosen. This shows that the set of real
numbers is not identical with the set of all segments with one common end point. In contrast, the set of all points on a line does not form a group. The reference (a) to the rational numbers is necessary to give the reals the character of numbers. Condition (c) is not sufficient for this purpose. A set is called continuous if its elements correspond one-to-one to the points on a line segment. There is no one-to-one correspondence possible between the elements of a denumerable group and those of a continuous group. A continuous set cannot be reduced to a denumerable one. The number
of elements in a continuous group is always infinite. On the one hand, the continuity of the set of real numbers anticipates the continuity of the set of points on a line. On the other hand, it allows of the possibility to project spatial relations on the
quantitative relation frame.
The introduction of the set of real numbers as an isomorphic copy of a spatial group already indicates
that the meaning of the real numbers is not originally numerical. Their meaning anticipates the spatial modal aspect. This means that the concept of isomorphy is a mathematical expression of the philosophical idea of projection. In contrast, the negative integers
and the rational numbers may be considered expressing modal numerical relations between natural numbers and among themselves, and thus as modal abstractions between discreet collections. So the modal meaning of negative and rational numbers remains completely
within the closed numerical modal aspect of discrete quantity.
Because the set of rational numbers is dense, it contains Cauchy
sequences: infinite sequences of elements An, given according to some law, such that for any positive number ε (however small) there is a number N, such that if n>N and m>N, then |Am–An|<ε. It may be observed that the existence of this limit does not depend on an actual completed infinitude of the series as a totality: an infinite discrete set does not have a last member.
It may occur that the limit A of a Cauchy sequence is not a member of the set. There are Cauchy sequences of rational numbers whose limits are not rational numbers themselves. The set consisting of all Cauchy sequences of rational numbers
is the set of all real numbers. The inclusion of these limits completes the dense set of rational numbers, making it a continuous set of real numbers.
However, the real numbers cannot be defined in this way. For instance, it is already presupposed that the limit A of a Cauchy sequence of rational numbers is a number, because otherwise the numerical difference |A–An| would have
no meaning. However, for the same reason, it is objectionable to say that this limit is not a number. It is an assumption to state that the limits of Cauchy sequences of rational numbers are (real) numbers,
and one has to show that this assumption is warranted.
The quantitative meaning of numbers
According to Dooyeweerd,
rational and real numbers must be considered mere functions of numbers, the only original numbers being the natural numbers.
For a similar reason some mathematicians introduced the integer and rational
numbers as equivalence classes of differences or ratios between natural numbers. Thus the integer 2 is the equivalence class of all differences (2+b)–b, where b ranges over all natural numbers. In this view the positive integers should
not be identified with the natural numbers, as I did, and, depending on the context, the symbol ‘2’ may stand for a natural number, an integer, a rational number, and eventually for a real or complex number. This view is understandable if one considers
the numbers as logically definable. In my view, numbers are discovered and are modal subjects under a law. Therefore I have no difficulty in identifying the number 2 as being the same member in different sets.
I agree that the natural numbers are primitives, whereas the existence of rational and real numbers depends on the existence of natural numbers. Nevertheless it is meaningful to speak of numbers,
also in the case of negative, rational and real numbers, as modal subjects to numerical laws. In order to see this, one has to recall that the mutual relationship of law and subject implies that there are no laws without subjects, or subjects without laws.
It may be imagined that mankind first discovered certain subjects (e.g., the natural numbers) and some laws (the laws of addition and multiplication) to which these are subjected. Afterwards, other laws were found (subtraction, division) pertaining to the
same subjects. But then one also discovered other subjects (negative and rational numbers) to the same laws. In my view there is no reason to call these newly discovered subjects mere functions of the already known primitive subjects. The real numbers are also subjected to the same laws of addition, multiplication, subtraction, and division as the rational numbers are. Thus these numerical predicates of infinite sets of rational numbers behave as subjects to numerical laws.
the meaning of the negative and rational numbers remains completely within the closed numerical modal aspect, because they denote numerical relations between discrete collections. The set of all real numbers turns out to be non-denumerable, i.e., it is impossible
to find a one-to-one correspondence of this set with the set of natural numbers. The meaning of a non-denumerable set cannot be found in the closed numerical modal aspect. But this meaning is found with the discovery of the one-to-one correspondence between
the set of all real numbers and the set of line segments introduced above. Hence, the meaning of the set of real numbers anticipates the spatial modal aspect. It requires the dynamic development of the numerical relation frame.
This is also the case with the meaning of individual real numbers. Real numbers objectify magnitudes, first of all spatial magnitudes: lengths, areas, volumes.
It was the great discovery of the Pythagorean school, that the rational numbers are insufficient for the numerical objectification of spatial magnitudes (T&E, 3.1). The diagonal in a unit square has a length of √2, and it can easily be shown that
this is not a rational number. In order to represent such magnitudes, one needs the real numbers.
Therefore the meaning of the real numbers anticipates the later modal aspects. The limit of an infinite series is never actualized, but
in the retrocipatory direction, real numbers become actual magnitudes. The length of a line segment is an actual, real magnitude. When the numerical relation frame is developed into the quantitative one, it original meaning is deepened and relativized, from
numerical to quantitative. The deepening means that not only discrete sets, but also magnitudes can be numerically ordered. With real numbers, non-numerical subjects can be ordered according to their magnitude without gaps or holes. This relativization of
modal meaning entails the loss of the discrete or denumerable character of numbers which they have in the numerical relation frame.
The temporal order in the numerical relation frame is that of earlier and later, and two numbers are called equal if they have the same
position in this order. Therefore only one number 2 should be allowed, whether understood as a natural number, an integer, a rational, or a real number. However, if the order of smaller and larger is applied to concrete subjects or collections, several subjects
may be equivalent with respect to some property.
In that case there will be at least one other property with respect to which
they will be different. In many cases it will be possible to order a set of subjects according to two or more independent properties. Thus there are series with two, three, or more indices. Discrete series can always be ordered in a single numerical order,
but this is not always desirable. It might also be that two independent properties have a continuous spectrum, in which case a unequivocal single numerical order is impossible. This notion of independence anticipates the spatial order of simultaneity, and
therefore discloses the numerical relation frame on the law side.
Magnitudes are non-numerical relations which can be objectified by real numbers. There are non-numerical relations which can only be ordered in a serial order of smaller and larger, if they are decomposed into components, which simultaneously determine
these relations. This applies in the first place to spatial position, but also to force, velocity, or the physical state of a system. Such relations are not objectified by a single real number, but by a multiplet of real numbers, called a vector. The minimum
amount of real numbers needed for an objectification of a property or relation is called the latter’s dimension. The corresponding vector has an equal number of independent components, which is, therefore, sometimes called the vector’s dimension.
By way of example, and because of their relevance to physics, the present section reviews the theories of vectors, of complex numbers,
and of Hilbert space.
A number vector is defined as an n-tuple of n real numbers, written bold-faced as a=(a1,a2,a3,…,an)
and being subjected to some well-known rules. The vectors with the
same number n of components form a group with vector addition as group operation, and the zero vector (0,0,0,…,0) as its identity element. The inverse of a vector a is –a=(-1)a.
It is easily verified that the set of real numbers is isomorphic to the set of one-component vectors. The independence of the components is not changed by addition.
Next the scalar
product is defined, a functional of two vectors having the same number of components, as the real number a.b=a1b1+a2b2+…+anbn.
Because the result is a number, not a vector, this product does not define a group. The norm |a| of a vector a is defined by |a|2=a.a=a12+a22+…+an2
One may wonder whether there exists an operation analogous to multiplication that gives rise
to a field of vectors. This is indeed the case with the two-component vectors called complex numbers, often written as a1+a2i=a1(1,0)+a2(0,1)=(a1,a2).
Here the vector (1,0) is identified with the real number 1, and the vector (0,1)=i is the so-called imaginary unit. The addition of complex
numbers is defined above. We call a*=a1–a2i the complex conjugate of a=a1+a2i.
The complex conjugate of a ‘real number’ (a1,0) is identical with itself. The product of two complex numbers is defined as the complex number (a1,a2)(b1,b2)=(a1b1–a2b2,a2b1+a1b2).
Together with the addition, this defines a field. The unit vector is (1,0), and the multiplicative inverse of a is a*/a.a*. We see that i2=-1, according to the popular definition of i.
The solutions of many problems concerning functions of real numbers are only possible, or more easily obtained, if the latter are considered as vectors (a,0) – i.e., if we consider those functions as functions of complex numbers.
This shows that the full meaning of disclosed modal subjects (real numbers) becomes clear only if the law side is also opened up (by the introduction of vectors). Besides vectors, there are other structures, like tensors and matrices, in which each component
has two or more indices. They anticipate more complicated spatial or non-spatial relations than vectors are capable of doing. With the introduction of real and complex numbers it is also possible to anticipate the kinetic and later modal aspects, as in integral
and differential calculus.
The concept of a vector can be further developed into vectors with complex components and functions of real or complex
variables. Quantum physics makes use of a so-called Hilbert space (chapter 9), which is not a space (there are no spatial subjects in it), but a set of complex functions, anticipating the spatial and later modal aspects.
Here it is not immediately necessary to define the scalar product (which can be different for different cases), if only the functions belonging to the set and the scalar product conform some quite general rules.
The possibility of mapping a Hilbert space on a set of vectors means that all Hilbert spaces with the same value for m are isomorphic to each other. This number m, the dimension
of the set, may be finite (as assumed above), infinite, and even non-denumerable.
2.5. The spatial relation frame
In 1899, David Hilbert formulated the foundations of projective geometry as relations between points, straight lines and planes, without defining these. Gottlob Frege thought that Hilbert referred to known subjects, but Hilbert denied this. He was only concerned with the relations between things, leaving aside their nature. According to Paul Bernays, geometry is not concerned with the nature of things, but with ‘a system of conditions
for what might be called a relational structure’. Inevitably, structuralism influenced the later emphasis on structures.
Topological, projective, and affine geometries are no more metric than the theory of graphs. They deal with spatial relations without considering the quantitative relation frame. I shall not discuss these non-metric geometries. The 19th- and 20th-century views about metric spaces and mathematical structures turn out
to be much more important to modern physics.
Mathematics studies inter aliaspatially qualified characters (10.5). Because these
are interlaced with kinetic, physical, or biotic characters, spatial characters are equally important to science. This also applies to spatial relations concerning the position and posture of one figure with respect to another one. A characteristic point,
like the centre of a circle or a triangle, represents the position of a figure objectively. The distance between these characteristic points objectifies the relative position of the circle and the triangle. It remains to stipulate the posture of the circle
and the triangle, for instance with respect to the line connecting the two characteristic points. A coordinate system is an expedient to establish spatial positions by means of numbers.
The metric of objective magnitudes
Spatial relations are rendered quantitatively by means of magnitudes like distance, length, area, volume, and angle. These objective properties of spatial
subjects and their relations refer directly (as a subject) to numerical laws and indirectly (as an object) to spatial laws.
and technology prefer to define magnitudes that satisfy quantitative laws. To make calculations with a spatial magnitude requires its projection on a suitable set of numbers (integral, rational, or real), such that spatial operations are isomorphic to arithmetical operations like addition or multiplication. This is only possible if a metric is available, a law to find magnitudes and their combinations.
For many magnitudes, the isomorphic projection
on a group turns out to be possible. For magnitudes having only positive values (e.g., length, area, or volume), a multiplication group is suitable. For magnitudes having both positive and negative values (e.g., position), a combined addition and multiplication
group is feasible. For a continuously variable magnitude, this concerns a group of real numbers. For a digital magnitude like electric charge, the addition group of integers may be preferred. It would express the fact that charge is an integral multiple of
the electron’s charge, functioning as a unit.
Every metric needs an arbitrarily chosen unit. Each magnitude has its own
metric, but various metrics are interconnected. The metrics for area and volume are reducible to the metric for length. The metric for speed is composed from the metrics of length and time. Connected metrics form a metric system.
The dynamic development of various metrics is not only indispensable for the natural sciences. If a metric system is available, cooperating governments or the
scientific community may decide to prescribe a metric to become a norm, for the benefit of technology,
traffic, and commerce. Processing and communicating of experimental and theoretical results requires the use of a metric system.
A point has no dimensions and could have been considered a spatial object if extension were essential for spatial subjects. However, a relation frame is not characterized by any essence
like continuous extension, but by laws for relations. Two points are spatially related by having a relative distance. The argument ‘a point has no extension, hence it is not a subject’ reminds of Aristotle and his adherents. They abhorred
nothingness, including the vacuum and the number zero as a natural number. Roman numerals do not include a zero, and Europeans did not recognize it until the end of the Middle Ages. Galileo Galilei taught his Aristotelian contemporaries that there is no fundamental
difference between a state of rest (the speed equals zero) and a state of motion (the speed is not zero).
It is correct that the property length does not apply to a point, any more than area can be ascribed to a line, or volume to a triangle.
The difference between two line segments is a segment having a certain length. The difference between two equal segments is a segment with zero length, but a zero segment is not a point. A line is a set having points as its elements, and each segment of the line is a subset. A subset with zero elements
or only one element is still a subset, not an element. A segment has length, being zero if the segment contains only one point. A point has no length, not even zero length: the concept of length is not applicable to points. Dimensionality implies that a part
of a spatial figure has the same dimension as the figure itself. A three-dimensional figure has only three-dimensional parts. We can neither divide a line into points, nor a circle into its diameters. A spatial relation of a whole and its parts is not a subject-object
relation, but a subject-subject relation.
Whether a point is a subject or an object depends on the nomic context, on the relevant laws. The relative position of the ends of a line
segment determines in one context a subject-subject relation (to wit, the distance between two points), in another context a subject-object relation (the objective length of the segment). Likewise, the sides of a triangle, having length but not area, determine
subjectively the triangle’s circumference, and objectively its area.
of numbers can be projected on a line, ordering its points numerically. To order all points on a line or line segment the natural, integral or even rational numbers are not sufficient. It requires the complete set of real numbers. The spatial order of equivalence
or co-existence presents itself to full advantage only in a more-dimensional space. In a three-dimensional space, all points in a plane perpendicular to the x-axiscorrespond simultaneously to a single point on that axis. With respect to the numerical
order on the x-axis, these points are equivalent. To lay down the position of a point completely requires several numbers (x,y,z,…) simultaneously, as many as the number of dimensions. Such an ordered set of numbers
constitutes a number vector (2.4).
For the character of a spatial figure too, the number of dimensions is a dominant characteristic.
The number of dimensions belongs to the laws constituting the character. A plane figure has length and width. A three-dimensional figure has length, width and height as mutually independent measures. The character of a two-dimensional figure like a triangle
may be interlaced with the character of a three-dimensional figure like a tetrahedron. Hence, dimensionality leads to a hierarchy of spatial figures. The base of the hierarchy is formed by one-dimensional spatial vectors.
Numerical and spatial vectors
Contrary to a number vector, a spatial vector is localized and oriented in a metrical space. Localization and orientation are
spatial concepts, irreducible to numerical ones. A spatial vector marks the relative position of two points. By means of vectors, each point is connected to all other points in space. Vectors having one point in common form an addition group. After the choice
of a unit of length, this group is isomorphic to the group of number vectors having the same dimension. Besides spatial addition, a scalar product is defined.
The group’s identity element is the vector with zero length. Its base is a set of orthonormal vectors, i.e., the mutually perpendicular unit vectors having a common origin. Each vector starting from that origin is a linear combination of the unit vectors.
So far, there is not much difference with the number vectors.
However, whereas the base of a group of number vectors is rather
unique, in a group of spatial vectors the base can be chosen arbitrarily. For instance, one can rotate a spatial base about the origin. It is both localized and oriented. The set of all bases with a common origin is a rotation group. The set of all bases having
the same orientation but different origins is a translation group. It is isomorphic both to the addition group of spatial vectors having the same origin and to the addition group of number vectors.
Besides a relative position, a spatial vector represents a displacement, the result of a motion. This is a disposition, a tertiary characteristic of spatial vectors.
Euclidean and non-Euclidean metrics
Euclidean space is homogeneous (similar at all positions) and isotropic (similar in all directions). Combining spatial translations, rotations, reflections with respect to a line or a plane and inversions with
respect to a point leads to the Euclidean group. It reflects the symmetry of Euclidean space. Symmetry points to a transformation keeping certain relations invariant.
At each operation of the Euclidean group, several quantities and relations remain invariant, for instance, the distance between two points, the angle between two lines, the shape and the area of a triangle, and the scalar product of two vectors.
In Euclidean geometry, the relative position of points is found with the help of a Cartesian coordinate system, allowing to represent each
spatial point by a vector (x,y,z,…). Having two points characterized by the vectors (x1,y1,z1,…) and (x2,y2,z2,…), the difference vector (x1-x2,y1-y2,z1–z2,…) characterizes the relative position of the two points. The distance of the two points is the norm d of this vector, determined by d2=(x1-x2)2+(y1-y2)2+(z1–z2)2+…
This expression is
called the metric of Euclidean space. A metric is a law according to which a numerical value can be
assigned to a non-numerical property or relation. The above formula is an objective representation of this law for the determination of lengths and distances in Euclidean space.
The metric depends on the symmetry of space. In an Euclidean space, Pythagoras’ law determines the metric. Since the beginning of the 19th century, mathematics acknowledges non-Euclidean spaces as well. (Long before, it was known that on a sphere the Euclidean metric is only applicable to distances small compared with the radius.) Preceded by Carl Friedrich Gauss, in 1854 Bernhard Riemann
formulated the general metric for an infinitesimal small distance in a multidimensional space.
For a non-Euclidean space, the coefficients in the metric depend on the position. To calculate a finite displacement requires the application of integral calculus. The result depends on the choice of the path of integration. The distance between two points is the smallest value of these paths. On the surface of a sphere,
the distance between two points corresponds to the path along a circle whose centre coincides with the centre of the sphere.
metric is determined by the structure and eventually the symmetry of the space. This space has the disposition to be interlaced with the character of kinetic space or with the physical character of a field. A well-known example is the general theory of relativity,
being the relativistic theory of the gravitational field (4.8).
A non-Euclidean space is less symmetrical than an Euclidean one having the same number of dimensions. Motion as well as physical interaction
may cause a break of symmetry in spatial relations.
2.6. Spatial subject-object relations
The distinction of subjects and objects as made in the philosophy of the cosmonomic idea (1.6) can best be illustrated with respect to spatial objects and objective magnitudes. The proper
parts of a spatial subject cannot have more or less dimensions than the subject itself. A two-dimensional subject can only have two-dimensional parts. Just as collections can only be added if they have no members in common, magnitudes of spatial subjects can
only be added if they have no common parts. But they may have common boundaries, because the boundaries are not parts of the subject. A boundary of a spatial subject always has a lower dimension than the subject itself, and, therefore, its subjective extension
(with respect to the magnitude of the subject) is zero (it has ‘measure’ zero). Spatial boundaries have an objective meaning within the spatial modal aspect. They delimit the objective magnitude of the subjects, and they allow the introduction
of numerical ordering within the spatial aspect.
The simplest spatial objects are points, having zero spatial extension. Points
have an important spatial meaning as boundaries of a line segment. Spatial points serve to determine its length, the objective magnitude of the line segment. Similarly, in a two-dimensional space, a line segment can only function objectively, as a boundary
of a triangle, e.g., by determining its area, which is again an objective spatial magnitude referring back to the numerical modal aspect. In this way the spatial relation frame is the first aspect to have objects as well as subjects.
It is of no use to define a line, a plane, or a space as a collection of points, lines, or planes, respectively. Although a line contains a continuous, non-denumerable collection of points, this cannot serve as a constitutive definition of a line. Rather the line constitutes the collection of points. Collections of this kind have
a dependent meaning. This becomes apparent if one tries to assign a number to a collection of points on a line segment. It can easily be proved that there exists a one-to-one correspondence between the points of this line segment and the points of any other
line segment, regardless of their relative length. Therefore, length, as an objective magnitude of the line segment, has no relation whatsoever to the number of points on the line segment.
2.7. Subject-subject relations
There is a spatial relation between two subjects if they are bound together in a common spatial manifold. Thus the spatial order is coexistence, static simultaneity, or equivalence,
and the corresponding subject-subject relation is relative spatial position. In the kinematic modal aspect simultaneity has only a limited, analogical meaning, as is shown in the theory of relativity, whereas in the numerical order of before and after simultaneity
is absent. Consider an (n–1)-dimensional boundary in an n-dimensional space, described by a continuous function f(r)=0, where r denotes the vector, ranging over all points in the n-dimensional
space. All points on one side of the boundary are characterized by f(r)>0, and all points on the other side by f(r)<0. This shows once more that the concept of a boundary (a spatial object) refers to
the numerical order of smaller and larger. With respect to this quasi-serial order, all points with vector r, such that f(r)=a, are equivalent. They simultaneously lie in the same (n–1)-dimensional
manifold objectified by this equation.
Just as numerical relations are subjected to a serial order (2.2), spatial relations
are subjected to an order of equivalence. A relation R(A,B) over a set is an equivalence relation
if for any two elements A and B of the set either R(A,B) or not, and if R(A,B) is reflexive, symmetric, and transitive.
All elements which are equivalent with a certain element A
constitute the equivalence class of A. It is a sub-set of the whole set over which the equivalence relation R is defined.
It can be shown that if this is the case there must be some property by which different equivalence classes in the same set can be distinguished. For instance, the equivalence classes of parallel lines in an Euclidean space can be distinguished by their relative
Consider a simple spatial problem: in which ways can spatial figures differ or be equivalent?
Generally speaking, by their shape, their magnitude, and their relative position. If two subjects have the same shape are called similar. If they also have the same magnitude (area or volume) they are called congruent. The concept of magnitude refers back
the numerical modal aspect and, more specifically, to the operation of addition: if we take two disjoint subjects together, we have to add their magnitudes. The concept of similarity is an equivalence relation, but it clearly does not lead to a universal ordering
of spatial subjects. The concept of magnitude allows us to find such an order, but this has a numerical, not a spatial character. Only spatial position can be qualified as an irreducible, universal, spatial subject-subject relation.
If two subjects are congruent, they can only differ in their position because otherwise they must be identical. Two subjects may have parts in common, they may
have nothing more than a boundary in common, or they may be completely disjoint. Otherwise, it is difficult to use the concept of relative position (although it is probably intuitively clear) without an objective description – namely, the distance and
relative orientation of the two subjects. The shape of a subject is also determined by the relative position of its boundaries, just as its magnitude. Relative position is subjected to the order of equivalence: the subjects considered should have the same
dimension, and must be in the same manifold – these are equivalence relations.
Spatial figures can be objectified by their
boundaries, in the simplest case by spatial points – for instance, a triangle by its vertices. If the shape of a subject is given, n points are needed to objectify the position of an n-dimensional
subject in an n-dimensional manifold. As a consequence, the relative position of two subjects is objectified
by the distances of the corresponding pairs of such points. This determines the relative distance as well as the relative orientation of the subjects. Thus the distance of two spatial points (besides the angle between two lines) is an objective, spatial relation.
2.8. Objectivity in
the choice of coordinate systems
The Euclidean metric defined above is independent of the
choice of the Cartesian coordinate system. It is not affected by any translation (or displacement), rotation, or inversion of the latter. I shall discuss this statement because the natural sciences claim to be objective, and because its relevance is called
into question by modern and postmodern conventionalist authors.
The possibility of assigning real numbers to points on a straight
line depends on the one-to-one correspondence between the numerical addition group of real numbers and the spatial addition group of line segments on a straight line. This correspondence is not unique in two senses: one is free to choose a unit, as well as
to choose the common end point of the set of line segments. Objectivity requires that the distance between two points (the objective relation between two spatial subjects) be independent of this arbitrary choice. This is expressed by saying that the distance
is invariant under the translations of the coordinate system: the space is homogeneous. All possible
displacements form a group, isomorphic to the group of all spatial difference vectors.
When a zero point has been chosen, one
is still free to choose a point to which to assign the number 1. This arbitrariness is limited by the requirement that the distance between two spatial points be independent of rotations of the coordinate system around any axis and about any angle. This is
called the isotropy of space. This implies that the unit be the same along all coordinate axes.
The set of all possible rotations in a plane forms a commutative group. Rotations around different axes in more-than-two-dimensional space form a non-commutative group.
Having chosen a set of coordinate axes and a unit, one is still free to assign the plus and minus directions on each axis. This results in inversion symmetry, the operation under which the distance must be invariant.
The rotations together with the reflections form the full orthogonal group. Each finite translation or rotation can be obtained as the result of a continuous motion. However, this is not the case with inversion, which refers back to the numerical order of
before and after. This implies that it will not always be possible to bring congruent spatial figures to coincide merely by a combination of translations and rotations. For example, the right- and left-hand gloves of a pair cannot replace each other.
By changing the unit, all distances are changed in the same ratio. All possible transformations of the unit form a multiplication group
which is isomorphic to the multiplication group of positive real numbers. Therefore, by changing the unit, all distance ratios must remain the same. Distances should be geometrically independent of the choice of the unit of length, but this cannot be accounted
for by a numerical analysis alone. In the theory of number vectors there is nothing of this kind: units do not occur in number theory. The meaning of the spatial subject-subject relation is determined by the irreducible meaning of the spatial relation frame,
and cannot be reduced completely to the numerical relations which objectify spatial relations. From an arithmetical point of view, the replacement of the metre by the centimetre as a unit of length causes all distances to become a hundred times larger. Transformations
of this kind are sometimes called trivial, but they are not, since they express the mutual irreducibility of the numerical and the spatial modal aspects.
These invariance properties are not only relevant to distances, but also clarify the concepts of congruence and similarity. Two spatial
figures (irrespective of their relative position) are congruent if the one can be transformed into the other by an operation belonging to the full group of translations, rotations, and inversion. Two figures are similar (having the same shape) if besides such
an operation all linear dimensions of one figure must be multiplied by a real number in order to arrive at the same result. This implies that if two figures are congruent or similar, they remain so under any transformation of the coordinate system of the types
The standard Euclidean metric is invariant under translations, rotations, and the inversion
of the coordinate system. In contrast, one can show that any other metric singles out a particular point, line, plane, or direction. Thus we can say that the standard metric represents the isotropy and homogeneity of space, which are assumed here because only
spatial relations between subjects are relevant, and not the ‘absolute position’ of any subject.
is only dependent on the choice of the unit. This arbitrariness reflects the amorphousness of space, by which we mean that we cannot assign a certain amount of points to a certain line segment. In fact, a one-to-one correspondence is possible between the points
of any pair of intervals, irrespective of their relative lengths. Therefore, the length of an interval as expressed by a certain number, is not an intrinsic spatial property. This is properly stressed by Adolf Grünbaum in his extensive studies on the
alleged conventionality of the metric. Grünbaum is the main 20th-century (though moderate) proponent of conventionalism. He repeatedly refers to Henri Poincaré and Bernhard Riemann, but, in fact, conventionalism
is merely a modern form of nominalism, which has its roots in the late Middle Ages and was defended by George Berkeley in the 18th and Ernst Mach in the 19th century. Grünbaum uses the amorphousness of space as an argument for the equivalence of all conceivable coordinate systems, but does not admit that some coordinate systems should be preferred
if they express the symmetry properties of space.
In the non-standard metric of a semiplane discussed by Grünbaum, the
distance is not invariant under a translation of the coordinate system along the y-axis. The non-standard metric which he discusses elsewhere is not invariant under rotations of the coordinate system. As Grünbaum rightly observes, the assignment of real numbers to spatial points only effects a coordinatization, not a metrization
of the manifold. However, his non-standard metrizations do not define proper spatial subject-subject relations. When a third spatial subject (the coordinate system) is used to objectify the spatial relations between two subjects, a metrization is required which keeps this spatial relation independent
of the position of that third subject. This is a requirement of objectivity which presupposes the
homogeneity and isotropy of space, that is, rejection of any absoluteness of space with respect to position or direction.
This does not mean that other metrizations should be rejected in all circumstances. Often they are very useful (e.g., polar coordinates
for spherical-symmetric problems). This actually reverses the argument. Instead of agreeing with Grünbaum that Cartesian coordinate systems are only used because they are often more convenient than others, non-standard metrics are only applied if it is
convenient in certain circumstances. A unique property of the standard metric is its invariance under translation, rotation, and inversion. This is not the case because of some convention, but follows from the homogeneity and isotropy of space. Grünbaum
has paid too much attention to the amorphousness of space, which implies the arbitrariness of the unit, and has neglected the symmetry properties inherent to Euclidean geometry reflecting those of space.
Grünbaum’s remarks could be accepted if they were related to topology, in which, e.g., one does not distinguish between a sphere and an ellipsoid, or a rectangle and a parallelogram.
Topology differs from metrical geometry because it lacks a metric. The theorems of topology hold for a figure regardless of how it is deformed in homogeneous strain. Grünbaum, however, directs his conventionalist views to metrical space.
2.9. The dynamic development
of the spatial relation frame
The metric depends on the symmetry of space. In an Euclidean space, Pythagoras’ law determines
the metric. Since the beginning of the 19th century, mathematics acknowledges non-Euclidean spaces as well (2.5). Preceded by Carl Friedrich Gauss, in 1854 Bernhard Riemann formulated the general metric for an infinitesimal small distance in a multidimensional
For a non-Euclidean space, the coefficients in the metric depend on the position. To calculate a finite displacement
requires the application of integral calculus. The result depends on the choice of the path of integration. The distance between two points is the smallest value of these paths. On the surface of a sphere, the distance between two points corresponds to the
path along a circle whose centre coincides with the centre of the sphere.
The metric is determined by the symmetry of the space,
even if it is developed into kinetic space as in the theory of relativity, or into the physical space called a field. A well-known example is the general theory of relativity, being the relativistic theory of the gravitational field.
The above criticism of Grünbaum’s conventionalist views also pertains to non-Euclidean manifolds. Non-Euclidean manifolds are in general less symmetric
than Euclidean ones. Grünbaum seems to overlook this. Only by tacitly assuming that the said requirement of objectivity (i.e., that the relative position of two subjects be independent of the choice of the reference system) is satisfied is it possible
to describe the nature of a manifold by its metric. This requirement is satisfied in Euclidean space by the rotation, translation, and inversion invariance of its metric. In non-Euclidean space one must either have similar intrinsic symmetries (as in the case
of a spherical surface), or refer to some extrinsic instance – for example, to an Euclidean space of higher dimension, or to a rigid body, or to kinematic motion, or to gravity, as is done in relativity theory.
In Gauss’ theory of curved manifolds, showing that the metric can be derived without reference to an outside system, he tacitly assumed that the unit in the orthogonal directions and at different positions is the same. The metric, and thus the
Gaussian curvature depend on the method of measuring lengths adopted on the manifold. Thus one can either start with the symmetries of the manifold, and require that the metric be invariant under the allowed symmetry operations, as is the case for Euclidean or spherical geometry,
or start with a rigid definition of length in order to investigate the structure of that manifold. One cannot have it both ways.
manifolds can be understood in two ways: as an (n–1)-dimensional boundary of an n-dimensional spatial subject (e.g., a spherical surface), or as a manifold whose metric is determined by kinematical or physical
laws (as e.g. in relativity theory). In the latter case the homogeneity and isotropy of space are relativized by those non-spatial laws. Motion as well as physical interaction causes a break of symmetry in spatial relations. In the former case they are relativized
by the n-dimensional subject whose (n-1)-dimensional boundary functions as a manifold. In both cases the spatial relations between subjects bounded to such a manifold become non-Euclidean because of some restriction,
like a boundary condition. This relativization is characteristic for the dynamic development of a relation frame. In kinematics or in physics, one speaks of a field as soon as the spatial isotropy and/or homogeneity is lost. A field may either be homogeneous,
if it is not isotropic, or it may be neither homogeneous nor isotropic.
Hence Euclidean geometry may be considered as having
an original spatial meaning, whereas the meaning of non-Euclidean geometry is found by reference either to the numerical modal aspect (in the concept of a boundary), or to the kinematic and the physical aspects.
Multiply connected manifolds
The spatial modal aspect can also be developed on the law side by the introduction of multiply connected manifolds. In the simplest case, a linear manifold
is open if, for three points, there is one and only one point which lies between the other two. This is the case, for example, with a straight line or a parabola. A linear manifold may also be closed (a circle) or self-intersecting (a lemniscate). Two-dimensional
manifolds may be simply connected (e.g., a plane) or multiply connected (e.g., a plane with a hole, a sphere, or a torus). In this case a criterion for being simply connected is given by the concept of contraction. A two-dimensional manifold is called simply
connected if any point and any closed curve meet the following two-part criterion: one can uniquely determine whether the point lies inside the curve, and if that is the case, whether the curve can be continuously contracted without leaving the manifold. The
surface of a sphere is not simply connected because it fails the first part of the criterion. The surface of a torus does not meet either part of the criterion. In a similar way simply-connectedness can be established for higher-dimensional manifolds, i.e.,
with the help of the concept of a boundary. Therefore these criteria of connectedness have an objective character.
connected manifolds are not irrelevant to physics. The gravitational fields and electric fields are simply connected, but the magnetic field around a current bearing conductor is multiply connected. As a consequence, a static electric field can be described
by a potential, but a magnetic field cannot.
 For instance Zermelo in 1908, quoted by
Quine 1963, 4: ‘Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions of ‘number’, ‘order’, and ‘function’ taking them in their pristine, simple form, and to develop
thereby the logical foundations of all of arithmetic and analysis.’ See Putnam 1975, chapter 2.
Shapiro 1997, 98: ‘Mathematics is the deductive study of structures’.
This reference system cannot be finite because of its abstract and universal character.
Dooyeweerd NC, II 79ff; Cassirer 1910, 47-54.
Cf. Beth 1944a, 61, 67, 68.
See, for instance, Beth 1944a and Russell 1919.
Dooyeweerd 1940, 167, 168; NC II, 79.
Russell 1919, 15; Carnap 1939, 38ff.
Peano took 1 to be the first natural number. Nowadays one usually starts with 0, to indicate the number of elements in the zero set. Starting from its element 0, the set of integral numbers can also be defined by stating that each element a has a
unique successor a+ as well as a unique predecessor a-, if (a+)- = a, see Quine 1963, 101.
 In the decimal system 0+=1,
1+=2, 2+=3, etc., in the binary system 0+=1, 1+=10, 10+=11, 11+= 100, etc. From axiom 2 it follows that N has no last number.
The fifth axiom states that the set of natural numbers is unique. The sequence of even numbers satisfies the first four axioms but not the fifth one. On the axioms rests the method of proof by complete induction: if P(n) is a proposition defined for
each natural number n³a, and P(a) is true, and P(n+) is true if P(n) is true, then P(n) is true for any n³a.
For each a, a+>a. If a>b and b>c, then a>c, for each trio a, b, c.
 Because the first relation
frame does not have objects, it makes no sense to introduce an ensemble of possibilities besides any numerical character class.
 Quine 1963, 107-116.
In 1931, Gödel (see Gödel 1962) proved that any system of axioms for the natural numbers allows of unprovable statements. This means that Peano’s axiom system is not logically complete.
Putnam 1975, xi: ‘… the differences between mathematics and empirical science have been vastly exaggerated.’ Barrow 1992, 137: ‘Even arithmetic contains randomness. Some of its truths can only be ascertained by experimental investigation.
Seen in this light it begins to resemble an experimental science.’ See Shapiro 1997, 109-112; Brown 1999, 182-191.
 Goldbach’s conjecture, saying that each even number can be written as the
sum of two primes in at least one way, dates from 1742, but is at the end of the 20th century neither proved nor disproved.
 From the set of natural numbers 1 to n, starting from 3 the sieve
eliminates all even numbers, all triples, all quintets except 5, (the quartets and sixtuplets have already been eliminated), all numbers divisible by 7 except 7 itself, etc., until one reaches the first number larger than Ön. Then all primes
smaller than n remain on the sieve. For very large prime numbers, this method consumes so much time that the resolution of a very large number into its factors is used as a key in cryptography. There are much more sequences of natural numbers subject
to a characteristic law or prescription. An example is the sequence of Fibonacci (Leonardo of Pisa, circa 1200). Starting from the numbers 1 and 2, each member is the sum of the two preceding ones: 1, 2, 3, 5, 8, 13, … This sequence plays a part in
the description of several natural processes and structures, see Amundson 1994, 102-106
Quine 1963, 30-32 assumes there is no objection to consider an individual to be a class with only one element, but I think that such an equivocation is liable to lead to misunderstandings.
 A well-known paradox
arises if a set itself satisfies its prescription, being an instance of self-reference. The standard example is the set of all sets that do not contain themselves as an element. According to Brown 1999, 19, 22-23 restricting the prescription to the elements
of the set may preclude such a paradox. This means that a set cannot be a member of itself, not even if the elements are sets themselves.
 The number of subsets is always larger than the number of elements,
a set of n elements having 2n subsets. A set contains an infinite number of elements if it is numerically equivalent to one of its subsets. For instance, the set of natural numbers is numerically equivalent to the set of even numbers
and is therefore infinite.
This is a consequence of the axiom stating that two sets are identical if they have the same elements.
If n(A) is the number of elements of A, then n(AÈB)=n(A)+n(B)–n(AÇB).
In mathematics, the theory of groups became an important part of Felix Klein’s Erlanger programm (1872) on the foundations of geometry.In physics, groups were first applied in relativity theory, and since 1925 in quantum physics and solid state
physics. Not to everyone’s delight, however, see e.g. Slater 1975, 60-62: about the ‘Gruppenpest’: ‘… it was obvious that a great many other physicists were as disgusted as I had been with the group-theoretical approach to the
In general, AB≠BA. If AB=BA, the group is called commutative or Abelean (after N.H. Abel).
 Cassirer 1910, 55, 56.
For the introduction of the set of natural numbers or the group of integers, we only need to specify one member, the number 1. All other integers are generated according to the group operation of addition. For the introduction of the multiplication group of
positive rational numbers, we have to rely on the set of prime numbers, and hence on the full set of natural numbers (which can only be defined with the addition as a group operation), because of the theorem that the number of prime numbers is infinite.
If a, b, c, and d are integers, the group-theoretical approach demands that a/1 = a, etc. Hence, the addition of the rational numbers must be defined as a/b+c/d=(ad+bc)/bd,
in order to arrive at the result that a/1+b/1=a+b.
It can be proved that the sum, the difference, the product and the quotient of two rational numbers (excluding division by 0) always gives a rational number. Hence, the set of rational numbers is complete or closed with respect to these operations.
Courant 1934, 59, 60. Although there exists a one-to-one correspondence between the integers and the rational numbers, their groups are not isomorphic.
 If a<b then a<a+c(b-a)<b,
for each rational value of c with 0<c<1.
Philosophers do not generally recognize the importance of dense sets for the transition of rational numbers to real numbers.
A sequence is an ordered set of numbers (a, b, c, …). Sometimes an infinite sequence has a limit, for instance, the sequence 1/2, 1/4, 1/8, … converges to 0. A series is the sum of a set of numbers (a+b+c+…).
An infinite series too may have a limit. For instance, the series 1/2+1/4+1/8+… converges to 1.
By multiplying a single irrational number like p, with all rational numbers, one finds already an infinite, even dense, yet denumerable subset of the set of real numbers. Also the introduction of real numbers by means of ‘Cauchy-sequences’ only
results in a denumerable subset of real numbers.
This procedure differs from the standard treatment of real numbers, see e.g. Quine 1963, chapter VI.
According to the axiom of Cantor-Dedekind, there is a one-to-one relation between the points on a line and the real numbers.
 It is not difficult to prove that the points on two different line segments
correspond one-to-one to each other.
Courant 1934, 39, 40, 60.
Up till the end of the 19th century, the distinction between denseness and continuity was not clearly recognized, see Grünbaum 1968, 13. In the past, continuity was sometimes defined as ‘infinite divisibility’, but this leads only to denseness.
Boyer 1939, 284-290. To avoid this pitfall the modern approaches of Weierstrass, Cantor, Dedekind, and Russell have been institutionalized.
 Dooyeweerd NC, II 79, 88, 170ff, 383; see also Strauss
1970-1971. In fact, this is not quite a new view: for some time, the negative numbers were called ‘numeri absurdi’, ‘aestimationes falsae’ or ‘fictae’, the irrational numbers ‘numeri surdi’, and the complex numbers
are still called ‘imaginary’; cf. Beth 1944b, 72, 73.
Beth 1948, 34ff; Russell 1919, chapter 7.
Cf. Beth 1944b, 50ff.
Beth 1950, 77ff; 1944, 23ff.
(a) The sum of two vectors is a vector defined as a+b = (a1+b1,a2+b2,…,an+bn).
(b) The product of a vector with a real number c is a vector defined as ca = (ca1,ca2,…can).
(c) Introducing the n unit vectors (1,0,0,…0), (0,1,0,…0), … (0,0,0,…1), any vector can be written as a = a1(1,0,0,
… 0) + a2(0,1,0,…0) + … + an(0,0,0,…1).
Because of its relevance to physics it may be recalled that the complex numbers can also be represented in other ways by a pair of real numbers. The most important is the representation in terms of sine and cosine functions, or equivalent, as an exponential
function. If a=p.cos x, and b=p.sin x, then a+bi=(a,b)=p.cos x+pi.sin x=p.exp.ix.
The norm of this complex number is |p|, and x is called the phase of the complex number. For any integer n, p.exp i(x+n.2π)=p.exp.ix.This representation is especially convenient with respect to multiplication:
(p.exp ix)(q.exp iy)=pq.exp i(x+y).
The quantum mechanical state space is called after David Hilbert, but invented by John von Neumann, in 1927.
(1) If a and b are arbitrary complex numbers, and f1, f2, and f3 are arbitrary members of the set, then g=af1+bf2 is also a member
of the set, which is therefore a group under addition.
(2) There exists a functional (f1,f2) called the scalar product, which is
a finite complex number, such that:
(d) (f1,f2+f3)=(f1,f2)+(f1,f3): the scalar product is a linear functional.
The norm ||f|| of the function f is a real non-negative number defined by
If (f1,f2)=0 we call f1 and f2 orthogonal, which implies that they are mutually independent. There
exists a maximum number m of mutually independent and normalized functions n1,n2,…,nm, such that (ni,ni)=1 for i=1,2,3,…,m,
and that (ni ,nj)=0 if i≠j for i,j=1,2,3,…,m. This implies that any function f in the set can be written as f=a1n1+a2n2+…+amnm,
where a1,a2,…am are complex numbers, ai = (f,ni).
respect to the basis (the set n1,n2,…,nm) f can be written as the vector f=(a1,a2,a3,…am).
The basis is not unique. In fact, the number of possible bases for a Hilbert space is infinite.
Since the beginning of the 19th century, projective geometry is developed as a generalization of Euclidean geometry.
Shapiro 1997, 158; Torretti 1999, 408-410.
e.g. Bourbaki, pseudonym for a group of French mathematicians. See Barrow 1992, 129-134; Shapiro 1997, chapter 5; Torretti 1999, 412.
A graph is a two- or more-dimensional discrete set of points connected by line stretches.
This is not the case with all applications of numbers. Numbers of houses project a spatial order on a numerical one, but hardly allow of calculations. Lacking a metric, neither Mohs’ scale of hardness nor Richter’s scale for earthquakes leads
Galileo 1632, 20-22.
In a quantitative sense a triangle as well as a line segment is a set of points, and the side of a triangle is a subset of the triangle. But in a spatial sense, the side is not a part of the triangle.
 In an Euclidean space,
the scalar product of two vectors a and b equals a.b=ab cos a. Herein a=Öa.a is the length of a and a is the angle between a and b.
If two vectors are perpendicular to each other, their scalar product is zero.
Van Fraassen 1989, 262.
If the coordinates of two points are given by (x1,y1,z1) and (x2,y2,z2), and if we call Dx=x2–x1
etc., then the distance Dr is the square root of Dr2=Dx2+Dy2+Dz2. This is the Euclidean metric.
 Non-Euclidean geometries were discovered
independently by Lobachevski (first publication, 1829-30), Bolyai and Gauss, later supplemented by Klein. Significant is to omit Euclides’ fifth postulate, corresponding to the axiom that one and only one line parallel to a given line can be drawn through
a point outside that line.
Riemann’s metric is dr2=gxxdx2+gyydy2+gxydxdy+gyxdydx+… Mark the occurrence of mixed terms besides quadratic
terms. In the Euclidean metric gxx=gyy=1, gxy=gyx=0, and Δx and Δy are not necessarily infinitesimal. See Jammer 1954, 150-166; Sklar 1974, 13-54. According to Riemann, a multiply extended
magnitude allows of various metric relations, meaning that the theorems of geometry cannot be reduced to quantitative ones, see Torretti 1999, 157.
 If i and j indicate x or y, the gij’s, are components
of a tensor. In the two-dimensional case gij is a second derivative (like d2r/dxdy). For a more-dimensional space it is a partial derivative, meaning that other variables remain constant.
In the general theory of relativity, the co-efficients for the four-dimensional space-time manifold form a symmetrical tensor, i.e., gij=gji for each combination of i and j. Hence, among the sixteen components of the tensor ten are independent.
An electromagnetic field is also described by a tensor having sixteen components. Its symmetry demands that gij=-gji for each combination of i and j, hence the components of the quadratic terms are zero. This leaves six independent components,
three for the electric vector and three for the magnetic pseudovector. Gravity having a different symmetry than electromagnetism is related to the fact that mass is definitely positive and that gravity is an attractive force. In contrast, electric charge can
be positive or negative and the electric Coulomb force may be attractive or repulsive. A positive charge attracts a negative one, two positive charges (as well as two negative charges) repel each other.
 Dooyeweerd NC,
II 383ff; Dooyeweerd’s statement that an object in some modal aspect cannot be a subject in the same modal aspect is obviously wrong.
Cp. Suppes 1972, 310.
Dooyeweerd 1940, 166; NC II, 85; Leibniz already considered space and time as orders of coexisting and successive things or phenomena. Cf. Jammer 1954, 4, 115; Whiteman 1967, 383; Čapek 1961, 15ff.
for all A; if R(A,B), then R(B,A); if R(A,B) and R(B,C), then R(A,C).
 The arbitrariness of the choice of the unit, sometimes called
‘gauge invariance’ must not be confused with the so-called ‘magnitude invariance’, according to which many properties of, e.g., spatial figures only depend on their shape and not on their magnitude. The former invariance is universally
valid while the latter has a far more limited validity. In particular, it is false for typical relations, such as the size of atoms. See Čapek 1961, 21-26.
 Grünbaum 1968, 12, 13.
 See Kolakowski 1966,
chapter 2 and 6. For a critique of conventionalism, see Popper 1959, 78ff, 144ff; Friedman 1972.
Grünbaum 1963, 18ff; 1968 16ff.
Grünbaum 1963, 98ff; Grünbaum, in Henkin et al. (eds.), 204-222.
Grünbaum 1963, 16; 1968, 34.
It should be noted that my critique is not quite appropriate to Grünbaum’s alternative metrization mentioned above. His semi-plane is only symmetric with respect to translations along the x-axis, and reflections with respect to the y-axis.
His non-standard metric reflects these two symmetries just as well as the standard metric does. But then a semi-plane is not a very interesting example, in particular not for Grünbaum’s purposes.
 In the general theory
of relativity, the co-efficients for the four-dimensional space-time manifold form a symmetrical tensor, i.e., gij=gji for each combination of i and j. Hence, among the sixteen components of the tensor ten are independent. An electromagnetic
field is also described by a tensor having sixteen components. Its symmetry demands that gij=-gji for each combination of i and j, hence the components of the quadratic terms are zero. This leaves six independent components, three for
the electric vector and three for the magnetic pseudovector. Gravity having a different symmetry than electromagnetism is related to the fact that mass is definitely positive and that gravity is an attractive force. In contrast, electric charge can be positive
or negative and the electric Coulomb force may be attractive or repulsive. A positive charge attracts a negative one, two positive charges (as well as two negative charges) repel each other.
 Grünbaum 1963, 8ff; Beth 1950, 71.
Nagel 1961, 244, 246.