MATH WITHOUT MYTH

SUBVERSIVE FOUNDATION THEORY

Whether it is possible to operate mathematics without

invoking things that do not and cannot exist, I do not know.

The task I have set myself is simply to determine what

exists and what does not. Which among the familiar items

which appear in mathematical discourse can be said to exist

and which are mere fictions.

CONTENTS

1. Sets 2

2. Number 9

3. The Nature of Number 11

4. The Varieties of Number 17

5. The Infinite 21

6. A Brief History of the Infinite 26

1

7. Space and Time 31

8. Hypermyth 37

9. Formalism 45

10. Conclusion 51

1. Sets

One kind of item with a strong claim to exist is the

set. Sets were almost invisible before Cantor put them on

the map in the late nineteenth century, but they now occupy

a central position in the mathematical universe. Are there

such things? Can we get rid of them? We may have doubts

about the existence of sets, because they are clearly not

the same sorts of things as the concrete physical

individuals which occupy a central position in our universe.

Early on in our cognitive career, we learn to refer to the

things, persons and events revealed to us in perception,

2

perhaps bestowing names upon them. How do we get from this

to the introduction of sets?

The key to understanding this move is to recognize that

we are not restricted to a singular reference to each

individual object. We also have a capacity for plural

reference. We can say: “Fido is coming”: but we can also

say: “Dogs are coming.” Our language contains plural as

well as singular forms to allow us to handle this function

of plural reference. Sometimes, a plural reference can be

replaced by a string of singular references. If we know the

names of the dogs that are coming, we can replace “Dogs are

coming” by “Fido is coming and Rover is coming and Spot is

coming.” Unless we are prepared to stipulate that these are

all the dogs that are on their way, we must allow the

possibility of adding other names to the list. This

stipulation changes the corresponding statement which

doesn’t use names into the more precise: “Exactly three dogs

are coming.”

3

Even if there are cases of plural reference which can

be handled through string theory, there are other cases

which are more resistant. Traditional logicians

distinguished between the distributive and collective senses

of universal statements in the form “All S are P.” This is

because some universal statements assign properties which

can be distributed to every member of the group, whereas

others indicate a property of the group as a whole. “All

persons in the elevator weigh over 2000 pounds.” This

statement indicates their combined weight, which is

important if we are to find out whether laws have been

broken or cables are likely to break. Such a statement

clearly cannot be analyzed as a string of singular

statements. The people in the elevator have been

amalgamated into a single thing with a property of its own.

Sets must be brought in when our purpose is not to

weigh the people in the elevator, but to count them. The

number of people in the elevator is not a property of the

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separate individuals, but of the group as a whole. It is

not, however, a physical property of an amalgamated object,

like the combined weight. We may well ask what kind of

property it is and what kind of object it characterizes.

It is clear that the group or set is very different

from its members: it is not itself a person in the elevator

or any kind of physical thing. We may say that the set is

introduced through the combination of its members. This is

not enough, however, because it does not specify the kind of

combination involved. It is not the kind of combination

through which we make a gin and tonic by mixing the gin and

the tonic in a glass. It is not even the kind of

combination where we put together the people in the elevator

in order to get their combined weight. When we do this, we

can ignore the plurality of people and focus on the total

weight. But when we form a set, we cannot ignore the

plurality of its members, since this is essential to its

existence as a set. The set is a totality, or the unity of

5

a plurality.

I am talking as if sets are somehow introduced through

the act of a cognitive subject. This is certainly true,

even if one also believes that in forming sets we are merely

recognizing the existence of something in some sense already

there, which is the Platonic view. I myself find such

metaphysical speculations hard to credit, but I do not want

to make a fight about it. Instead, I shall focus on

unpacking what is involved when we either recognize or

construct the sets which we introduce.

The observer may form the set of persons in the

elevator when the doors open, revealing who is inside. To

form the set, the observer requires a criterion to determine

what to include and what to exclude. In this case the

criterion is the concept of a human being, which allows the

observer to include Mrs Smith and to exclude Fido, the dog

she is taking for a walk. We need more than a criterion,

however: we also require an array to which we can apply the

6

criterion. In this case, the array is presented to the

observer in perceptual experience.

We may form a subset of the set of persons in the

elevator, such as the set of women in the elevator, by

applying a more stringent criterion.1 We may also form a

wider set by relaxing the criterion to let in Fido, the dog.

Further relaxation will bring in more things. Can we keep

going until we form a widest set incorporating absolutely

everything in the elevator? This is more problematic,

because it depends on our having a well-defined concept of

what will count as a thing in the elevator, which we do not

appear to possess. The challenge to count the things in the

elevator is even more fearsome than the challenge to count

the islands in an archipelago, which can be met by adopting

some arbitrary condition determining what will count as a

distinct island.

1 This subset includes only members of the original set, but it is not necessarily what is called a proper subset which excludes at least one of the original members.

7

It is an even more giant leap of faith to posit a set

incorporating absolutely everything. We may call this the

omnium gatherum, where the dubious Latin reflects the

dubious status of the entity it describes. The urge to

posit such an unmanageable entity may have its source in an

axiom from orthodox set theory. This is the Axiom of

Separation, according to which we can form a set of objects

which satisfy some condition, only as a subset of some set

already introduced. This may be a method for forming

subsets2, but it must not be elevated into a necessary

condition for set formation. So long as we have available

an array of items from which we can draw the elements which

we wish to include in the sets we do form, it is not

necessary to combine this amorphous plurality of items in a

well-formed set. Indeed, such a set does not seem even

possible. A well-formed set must have a definite number of

2 This is not, indeed, the only way, since we can pick at random elements from the original set to form their own little group.

8

members, even if we do not know what that number is. Two

sets with different members are different sets, and two sets

with a different number of members must have different

members. The supposed all-comprehensive set cannot have a

definite number of members without a criterion of what will

count as a single item, which we do not appear to have. I

can separate from everything I am given the cows in the

field and count them up. If I am asked to count the things

in the field, I am baffled. Thus, a set of everything

available seems neither necessary nor possible.

Many sets we form are homogeneous, where we use a

criterion to determine what is in and what is out. Other

sets, however, may be heterogeneous, when we form the set by

making a list of the members we want to include. I have no

problem about either method of set formation. Normally, the

sets we form have at least two members: often, well over

two. Mathematicians, however, recognize unit sets with

exactly one member. How can we distinguish the unit set

9

from the single element which it contains? There is no

problem if we are using a concept to extract a specific set

from the given array. It may very well happen that there is

only one object in the plurality we are facing which

conforms to the concept. The unit set is clearly different

from its single member, since the criterion associated with

the set allows the possibility of other members. Things are

trickier, if we begin to form a list and stop after the

first item. Do we have a set with only one member?

The null set, the set with no members, is even more

problematic. We can certainly talk about sets with no

members, such as the set of unicorns and the set of zombies.

We have the idea of possible sets, but do we form an actual

set, if nothing is found that qualifies for membership? One

implication of the orthodox theory is that all null sets are

identical. Empty sets cannot have different members, since

they do not have any members. This forces us to say that all

unicorns are zombies and all zombies are unicorns. Modern

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logic may have a way to digest these paradoxical claims, but

they remain intuitively unpalatable. Since empty sets are

identifiable only through their criteria for membership,

different criteria introduce different possible sets, even

if suitable objects are not available to realize these

possibilities. Axiomatic set theory may find it difficult

to function, deprived of the null set, but this need not

destroy mathematics, which operated well enough before set

theory appeared in the nineteenth century.

Whether or not we can get rid of the null set, what is

certainly true is that the null set is the foundation stone

upon which modern axiomatic set theory has been built. It

might seem that the more logical course was to begin with

the set that includes absolutely everything (omnium gatherum)

and form subsets. In fact, modern theorists begin at the

other end with the set that includes absolutely nothing.

One way to move forward is to form the unit set that

includes the null set as its only member, then form the pair

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that includes this set and the null set, then form the

triple including the pair, the unit set and the null set,

and so on.

Another vexing question is whether it is coherent to

form a set that is a member of itself. Naively, the set of

all sets would appear to be a member of itself, since it is

itself a set. Sets that are governed by a criterion can

certainly satisfy their own criterion. But to form a set we

require more than a criterion. We also require an array of

objects from which we can select the elements that satisfy

this criterion. But the set we form by selection cannot

itself participate in this array! If it did, it would have

to exist before it existed! This would be a vicious circle

of the kind that exercised thinkers such as Bertrand Russell

and Henri Poincare at the beginning of the twentieth

century. The only way to escape the circle would be to fall

back on a dogmatic Platonism that assigns co-existence to

everything that can be introduced in mathematics at any

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point.

A final category of problematic set is the infinite

set. The fundamental difficulty is that a set is the

completed totality of its members, whereas an infinite

series, for instance, is a series that cannot be completed.

This is a complicated and controversial topic, and I shall

postpone a full discussion. In the meantime I shall sum up

the results reached so far by assigning grades to the

various categories of sets distinguished so far.

Grade A sets: sets with two or more members.

Grade B sets: unit sets.

Grade C set: the null set

Grade F sets: the all-comprehensive set: sets that are

members of themselves: infinite sets.

Sets that obtain a passing grade may be described as

objects in a wide sense in so far as they are designated by

count nouns. We can answer questions about how many sets

13

there are which satisfy certain conditions. If we know how

many chairs there are in the room, we can give an exact

answer to the question about how many subsets there are for

the set of chairs in the room. This distinguishes sets from

things like water and gold which are designated by mass

terms. Nevertheless, sets are clearly very different kinds

of things from the physical objects and people usually

designated by count nouns. If we like, we can call them

abstract objects, but this is merely to paper over our

ignorance with a word. For the time being, I shall content

myself with this ignorance, since the metaphysical

discussion of the various modes of being and the

interconnections among them is a formidable project. My

conclusion is that well-formed sets have a good claim on

existence in some sense.

2. Numbers

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Another good claim is made by the natural numbers. There is

such a thing as the number “four”. Like sets, natural

numbers are mathematical objects, because they can be

counted. There is an exact answer to the question: “How

many prime numbers are there between twenty and thirty?”

Natural numbers satisfying a certain condition can be

counted, because they can be formed into a set. This will

endorse their claim to be treated as objects, since the

distinct elements forming a set are objects of one kind or

another.

In Section 2, the sets discussed were formed from

empirical objects, falling under empirical concepts. Many

cases are clear enough, but there can be messy borderline

cases where it is not certain whether or not an available

object conforms to the defining concept. This will damage

the determinate boundaries of the set, required by the

15

ideal. It is therefore useful to have cases where the

available objects are well-defined natural numbers. Sets of

natural numbers are essential for the organization of

arithmetic, whether we use the standard base ten or base

twelve (my favourite) or base two (favoured by computer

science.) Originally, the series of natural numbers comes

as a long string, generated through the iteration of the

successor function, beginning with either zero or one. The

string goes on and on, since there is no limit to the future

time in which we imagine the continuation of the series.

Before long the string will become unmanageable, unless

something is done. What must be done is the introduction of

what is called a “base” which groups numbers in sets of the

same size. Standard base ten arithmetic uses sets with ten

members. The iteration of the successor function provides

the manifold for unification, but the act of unification

presupposes the concept associated with the specific base.

For base ten arithmetic, the concept is the concept of the

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decade. To form a group to which this concept may be

correctly applied, it is necessary to form the successor of

the number nine, but it is not enough to simply add a tenth

member to the plurality of natural numbers constructed so

far. We also require the combination of the plurality under

the aegis of the concept of the decade. Only in this way

can we introduce the mathematical object which is the set of

the first ten natural numbers. This mathematical object is

not the same as the natural number “ten”, and does not

arrive automatically when the number “ten” is created. It

arrives only for the cognitive subject combining natural

numbers in accordance with the concept of the decade: it

would not arrive for cognitive subjects using a different

base, employing, for instance, the concept of the dozen.

Once the first decade is in place, we can use the same

concept with other numbers to form a second decade. This is

possible because the generality of the concept allows it to

control different acts of combination involving different

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manifolds. Working systematically, we can form up the first

ten decades in the natural number series, and we can use the

concept of the decade once again to unify these ten decades

to form a decade of decades, also known as a “century”.

Since they are sets, the original decades are mathematical

objects which can be elements in higher order sets.3

3. The Nature of Number

We have been affirming the existence of the natural numbers,

but so far have offered no explanation of the kind of

existence they enjoy, except to say that they are abstract

3 Although he did not have available the idea of a set, Immanuel Kant explains the essence of this procedure in the Critique of Pure Reason: “our counting, as is easily seen in the case of larger numbers,is a synthesis according to concepts, because it is executed in accordance with a common ground of unity, as for instance, the decade.” A78 B104 (translated by Norman Kemp Smith).

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objects which do not exist in the same way as concrete

physical things like horses and carts. One suggestion is

that they are properties. They are not, indeed, properties

like colours, which are properties of visible objects - they

are properties of sets. Any set with four members will have

a property shared by every set with four members. This

means that all such sets will conform to the same concept

which can be predicated of each of them.

To treat natural numbers as properties of sets is

indeed to introduce a third ontological category in addition

to concrete objects and sets. Some have been tempted to use

Occam’s razor to reduce the category of properties to the

category of sets. Instead of saying that a certain pair has

the property of twoness, we may say that the pair is an

element in the set of all pairs. For a real economy, of

course, we must replace all properties with sets. When an

apple is red, we must replace the claim that this apple has

the property of redness with the claim that it is an element

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in the set of all red things. This move has its

attractions, if it can find a way to overcome the challenge

of circularity. How can we form a set of red objects

without a concept to determine which objects are red and

which objects are not? Without the concept of such a

property, we cannot determine which objects are to be

included in the set.

If we are not content to say simply that numbers are

properties of sets and leave it at that, there is one idea

which may help. We can explain what it is for sets to be

the same size, to have the same number of members, by using

the idea of one-to-one correlation. If the members of the

first set can be paired up with the members of the second

set, without leaving uncorrelated items in either set, the

sets are the same size. (One-to-one correlation requires,

of course, that each pair contains one element from each

set, and no element from either set can appear in more than

one pair.)

20

There are sometimes natural pairings which allow us to

determine that sets are the same size without having to

count them. When the troop of Irish dragoons came into

Fyvie, we could say that the set of riders was the same size

as the set of horses, so long as no horse was observed

without a rider and there was no dragoon without a horse.

It is not necessary to determine the number of dragoons and

horses to recognize that these sets are the same size. It

is also possible to set up artificial pairings to discover

whether sets are the same size. To show that I have more

apples than oranges, I can set out my fruit in two rows,

with an orange opposite each apple. If there are apples

left over, then the set of apples is more numerous than the

set of oranges. There is no need to count the sets,

although this could certainly be done.

A more elaborate method can be used to determine

whether a neighbouring farmer has more sheep than I have. I

pass my sheep one by one through a narrow gate and make a

21

notch in my stick every time a sheep passes through. I then

take my stick over to the other farm, and as the other sheep

pass through a gate, I move my finger along the notches. If

I run out of notches before the neighbouring farmer runs out

of sheep, then he has more sheep than I do. My notched

stick is a primitive tool for counting sheep!4 I can

improve my tool by inscribing a symbol beside each notch

with a different symbol each time: I call this a “numeral”.

If several notched sticks are manufactured and the same

numerals inscribed in the same order on the different

sticks, it will be possible to line up the results reached

by the different instruments. Once the list of numerals has

been memorized, one can actually dispense with the sticks

and simply mark down the signs or express them through

vocalization.

This method of counting sheep has the sheep passing

4 The abacus is another more sophisticated tool of the samekind.

22

through a gate one by one with the first sheep followed by

the second sheep, followed by the third sheep and so on.

The sheep are counted through the incision of successive

notches on the stick. The number of sheep in the flock,

however, is not tied to any particular order in which the

sheep are passed through the gate, and it may be thought

that the cardinal number of sheep in the flock is more

fundamental than the ordinal numbers through which we assign

a spot in the series to each sheep as it passes through.

. One may, however, take the opposite view and regard

the ordinal numbering as more fundamental. Perhaps the idea

of the number of elements in a set is unintelligible apart

from the idea of counting the elements one by one. Since

there are many different orders in which the elements can be

counted, we reach the idea of the cardinal number of the set

by abstracting from the diversity of the ways of counting.

In the same way, we can measure the distance from A to B,

only by measuring either the trip from A to B or the trip

23

from B to A. We form the concept of distance by abstracting

from the direction of the trip.

It may be argued on the other side that if the number

of elements in a group can be recognized immediately without

any palaver, if it is small enough. We recognize

immediately that a pair of shoes is a group of two without

counting: “Left shoe, right shoe” or “Right shoe, left

shoe.” No need for sticks and notches. Even quite large

square numbers can be grasped directly, if the objects being

counted are neatly organized in a square in Pythagorean

fashion. This might suggest that cardinal numbers are more

fundamental, even if we must appeal for help to the ordinal

system in order to determine the cardinality of larger

sets.5.

We may call these two views of the nature of number the

static view and the dynamic view. The static view may be

5 In a similar way, we may estimate the distance between two objects seen in the distance without envisaging a trip in either direction.

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associated with a Platonic conception of mathematical

objects as having their being in an eternal domain: the

dynamic view is closer to the Kantian position that the

manifold is combined through a successive synthesis, possible

in virtue of a pure intuition of an unlimited future time.

Bringing in these two great names suggests that this is not

an issue to be easily decided.

In any event, we can certainly agree that if two sets

are the same size and have the same number of members, there

is at least one way in which the elements of the first set

can be correlated one-to-one with the elements in the second

set. The interesting question is whether the converse

holds. If we have a scheme of one-to-one correlation, does

this introduce two sets of the same size? The natural

numbers can be correlated one-to-one with the even numbers.

We correlate each natural number with the even number which

is twice its size. Does this introduce a set of even

numbers which is the same size as the set of natural

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numbers? One obstacle is that the set of even numbers

appears to be a proper subset of the set of natural numbers.

Every even number is also a natural number, but not every

natural number is an even number. This would normally mean

that the set of natural numbers is larger than the set of

even numbers! We have a dramatic conflict between the

criteria normally used to determine whether sets are the

same size. The set of even numbers and the set of natural

numbers must be the same size, because of the one-to-one

correlation: but they cannot be the same size, because the

first is a subset of the second.

Cantor decided to jump one way, followed by the whole

tribe of mathematicians. He stipulated that one-to-one

correlation was to be the deciding criterion. He did indeed

recognize that cases in which the subset criterion was

overruled were cases of a special kind, where the sets

involved are infinite sets. This raises a further problem

for the standard system, since we have given infinite sets a

26

failing grade on the ground that they cannot form completed

totalities. What we correlate one-to-one in the case of the

natural and the even numbers are not infinite sets but

infinite series. There will be a further investigation of

the mysteries of the infinite beginning in Section 5.

4. The Varieties of Number

Because mathematical objects like sets and natural

numbers are so different from physical objects, there must

be some uncertainty about their mode of existence. This

uncertainty about the status of genuine mathematical

entities will, of course, make it easier to slip in spurious

items. We can imagine unicorns, but rule them out because we

can never find them. If we posit an unusual mathematical

object, what would count as not being able to find it?

27

There are some things whose non-existence is generally

recognized. There is no such thing as the largest natural

number, because one can always form a larger number by

adding one to any candidate. For similar reasons, there is

no such thing as the largest prime number. One can always

form a larger prime number than any candidate by multiplying

together the candidate with all the smaller prime numbers

and then adding one. Since there are certainly sets which

are not members of themselves (the set of horses is not a

horse), it is plausible to think that one can form up a set

comprising all such sets. Bertrand Russell, however,

offered a proof that there can be no such thing. If the set

of all sets not members of themselves is a member of itself,

then it is not a member of itself, and if it is not a member

of itself, then it is a member of itself.

Natural numbers, however, are generally recognized.

Everyone agrees that there are exactly two prime numbers

greater than twenty and less than thirty, no more and no

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less. But what about the negative numbers? Taking away six

from four, the standard answer is “minus two”: but there is

another answer, which is that you cannot take away six from

four. If there are four chairs in the room and you take

away four, that is the end. There is no more taking away

that can be done. If one is using numbers to count the

members of a set, the smallest number one can use is zero.

There is no such thing as a set with minus two members.

To use this fact as a reason to rule out the legitimacy

of negative numbers would be very extreme, since negative

numbers clearly have their uses. To operate with negative

numbers we do, indeed, need a more complex conceptual scheme

than is required for counting the members of sets. One can

use negative numbers in a system in which one is counting

steps, up or down, forwards or backwards, where the system

has an origin which one can go below, where one can end up

below the origin if one takes more steps down than up. A

negative number can be used to indicate the number of steps

29

one finishes below the origin. In counting the number of

backward steps or the number of steps one ends up below the

line, the numbers used are, of course, standard natural

numbers.

It may be argued that the number one is in a sense more

basic than the number zero. When we count the number of

sheep in a field, we begin “one, two, three...” and not

“zero, one, two, three...” Nevertheless, if we want to

operate a system which uses negative numbers, we cannot do

without the number zero to represent the origin. The number

one represents one step ahead or one step back, but not the

here and now.

By introducing the notion of an origin with steps up

and down (or earlier and later), we can make sense of the

idea of subtracting a larger number from a smaller. Can we

also make sense of the idea of dividing a smaller number by a

larger? If we have four people and eight apples, we can

divide the apples evenly among them by giving them two each.

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But what if we have four people and only three apples?

Easy: we divide the apples into quarters and give everyone

three quarters! We can have this kind of division when we

have a whole which can be divided into parts. We imagine

that the natural numbers are whole numbers that can be

broken up into parts or fractions. When the dividing number

is larger, we get what are called proper fractions less than

one. When the dividing number is smaller, but does not

divide evenly into the larger number, we get an improper

fraction.

There is another mathematical notion closely associated with

fractions: this is the ratio. The ratio is essentially a

comparative relation between two natural numbers: for

example, one number may be double another. Four is twice

two and six is twice three. The ratio of four to two is the

same as the ration of six to three. We can use ratios to

define fractions. When we introduce the half, the whole is

double the part. The ratio of the whole to the part is two

31

to one (or four to two or six to three).

Ratios are usually called rational numbers, but they

are never the same as natural numbers, since they

necessarily involve two components. There is clearly no

natural number to identify with the rational number 2/3, but

it is tempting to identify the rational number 2/1 with the

natural number 2. This would be a mistake, since the one

thing is a simple number and the other is a ratio. This

would be to identify the number “two” with the relation

“twice”.

This does not seem at all mysterious and the hope is

that a similar strategy can be used when there is reason to

doubt the existence of a familiar mathematical entity, such

as the square root of two. The hard line is that there is

no square root of two, since there is no number which when

multiplied by itself gives us exactly two. Candidates turn

out to be either too large or too small. One can indeed try

to form up two sets, one containing the too large and the

32

other containing the too small. (One must realize, of

course, that the set of the too small does not contain a

largest member, and the set of the too large does not

contain a smallest member, any more than there is such a

thing as the largest natural number.) This means that one

can define a “cut” in the series of rational numbers,

organized in increasing size, as Dedekind (1831-1916) has

suggested. The rational numbers on one side of the cut will

belong to the class of the too small: the rational numbers

on the other side of the cut will belong to the class of the

too large. But there is no rational number that sits

exactly on the cut.

What one can try is to define the square root of two in

terms of either the class of the too small or the class of

the too large. The class of the too small is the usual

choice. The trouble is that this is not the square root of

two that we are looking for, but some kind of substitute.

If we multiply the class of the too small by itself

33

(whatever that would mean!), we do not get the natural

number two.

The usual solution is to introduce a separate domain of

so-called real numbers, in which everything familiar is

strangely different, as Alice found when she passed through

the looking glass. In this domain there is a home for the

square root of two, for pi and for other strange entities,

but we find that our familiar numbers have been utterly

transformed. We still have a number 2, but it is no longer

the number which introduced segments of Sesame Street and it is

no longer the number 2 whose square root we originally

sought. It has been turned into the class of all rational

numbers which are less than 2.

The creation of this surrealistic domain is softened by

arguing that there are actually three types of numbers -

natural numbers, rational numbers and real numbers. This

tactic diverts attention from the radical move through the

looking glass into the world of real numbers. The natural

34

number 2 is different from the rational number 2/1 or 8/4.

In the same way, the real number 2 is said to be different

from both the rational number 2 and the natural number 2.

5. The Infinite

The new entity introduced by mathematicians is certainly not

the square root of 2, but is the thing even logically

coherent? The class of rational numbers whose square is

less than 2 must be an infinite set: but are infinite sets

really possible? We have already awarded infinite sets a

failing grade on the ground that we must conceive a set as a

completed totality, whereas the combination of an infinite

plurality cannot be completed. Some mathematicians known as

“Finitists” take this seriously, but most are not deterred

and forge ahead. To rule out infinite sets without mercy is

certainly harsh and very destructive. Even the set of

35

natural numbers would have to go! This set is a central

assumption in mathematics and it has been given its own name

“N”. Is there any way that the set of natural numbers can be

saved?

There is certainly something which looks like the set

of natural numbers: this is the series of natural numbers.

This series is an infinite progression. Beginning with

either 0 or 1, we apply the successor function and generate

the next member in the series. We then apply the successor

function again to the new number generated, and continue the

process for as long as we please. There is no limit to how

far we can go. The question is whether there is a

legitimate totality comprising all the members in the

series. We can certainly refer to members in the series

through the device of plural reference. When Tarzan saw the

elephants coming, he is reputed to have said: “Elephants are

coming!” No doubt the elephants coming constituted a herd

which was either large or small, but Tarzan was referring to

36

the elephants and not the herd. In the same way, in a

single thought I can think of several or many things without

thinking of a totality constituted by these things.

The elephants coming can certainly form a set, but can

we say the same about the natural numbers. Certainly, we

can refer to the series of natural numbers and the many

items which it involves. We can assign properties to any

and every member of the series, for instance, that it can be

given a successor. We can even endorse a principle of

mathematical induction according to which if a property

belongs to the number one (or zero) and belongs to a natural

number if it belongs to its predecessor, then that property

belongs to any natural number.

Although it is not itself a set, the series of natural

numbers can, indeed, belong to a set, such as the set formed

from the series of natural numbers, the series of even

numbers, and the series of odd numbers. We cannot stipulate

that the elements in sets must be sets, because this would

37

rule out the set of elephants. To recognize the natural

numbers as forming a series that is not a set will certainly

torpedo the project of reformulating the whole of

mathematics in terms of a pure axiomatic set theory,

involving nothing but sets, perhaps based on pure logic.

The program of logicism enshrined in Principia Mathematica6 will

collapse: but this is not the collapse of mathematics and

perhaps no great mischief!

Can the series of natural numbers take over the

legitimate functions of the set of natural numbers? The

difference is that the series not an infinite totality but

an unlimited plurality. In the Critique of Pure Reason Immanuel

Kant distinguishes three categories of quantity. We have

unity, plurality, and totality. (A80 B106) The three

categories in the triad are not unconnected. The third

category “arises from the combination of the second category

6 Bertrand Russell and A.N. Whitehead, Principia Mathematica, 3 volumes (Cambridge, 1910-13.)

38

with the first.” (B 110) Totality is just plurality

considered as unity.7 Although the category of totality

(the Kantian equivalent of the concept of set) presupposes

in this way the other categories of unity and plurality, for

Kant it does not emerge automatically from these other

categories, but requires a special act of the understanding.

Kant goes on to argue that this special act may not always

be possible, even when we are in possession of the ideas of

unity and plurality. His example connects his argument

directly to our present investigation. “Thus the concept of

a number (which belongs to the category of totality) is not

always possible simply on the presence of concepts of

plurality and unity (for instance, in the representation of

the infinite).” (B 111)8 Although there is an unlimited

plurality of natural numbers, any attempt to form natural

numbers into a totality or set will always leave out other

7 This conception perhaps explains why Grade A sets must have atleast two members8 As translated by Norman Kemp Smith.

39

natural numbers.

The classical organization of the natural number system

was carried out in the 19th century by the Italian

mathematician Giuseppe Peano, who uses three primitive

ideas: “one”, “number”, and “successor”.9 “One” is a proper

name designating the number one: the successor of n is the

number derived from n through the use of the successor

function: and “number” is a general term covering the number

one and any successor of a natural number. No mention here

of big N. The set of natural numbers N may be lurking in the

background, although in mathematics things are not supposed

to lurk, but it does not appear up front.

At first sight it does not appear that Peano’s set of

primitive propositions is complete. He includes: “The

successor of any number is a number”, but this does not

9 Peano’s primitive idea one was later changed to zero for mathematical reasons. I shall stick with one because of the basic function of the concept of the unit and its connection withthe Kantian category of unity. My main argument, however, would not be affected, if one were replaced by zero.

40

entail that a number (other than one) must be the successor

of some number, a proposition that is equally primitive. We

can, indeed, cover both principles by simply defining a

natural number as either one or the successor of some

natural number, but there is another principle omitted that

is, perhaps, the most central of all. This is the principle

that each natural number has a successor.10 This is

universally accepted and is the very essence of the natural

number system, but it should be explicitly stated,

particularly since it involves an important presupposition.

It involves the assumption of an unlimited future in which

we can imagine the unlimited iteration of the successor

function.

10 Bertrand Russell covers up the vital distinction by claiming that in virtue of the Peano principle: “The successor of a numberis a number” we can assert: “Every number that we reach will havea successor.” This is a big mistake, since what the Peano proposition asserts is merely that if a number has a successor, then that successor will also be a number. It is not even asserted that any natural number has a successor. See Introduction to Mathematical Philosophy (London, 1919), 5-6.

41

It is agreed that there can be no finite set of natural

numbers that can incorporate every natural number, since

there will always be at least one member of the set with a

successor that is not an element of the set. Many

mathematicians, however, believe that there is an infinite set

of natural numbers, even although this is not mandated by

the fundamental principle. To understand where such a

belief could come from and to evaluate its validity, we must

explore the origins of the idea of the infinite.

6. A Brief History of the Infinite11

Worry about the infinite has been with us for a very long

time, certainly since the time of the pre-Socratics. It

appears that the world contains a large number of physical

11 This brief history is largely based on material from a book by Ivor Leclerc The Nature of Physical Existence (London andNew York, 1972). More detail can be found in Part One of this book

42

things which have a bodily existence because they are made

of matter. Matter is a necessary condition of physical

existence, but it cannot explain the plurality of physical

things. We require an empty space to separate one thing

from another: without this, the matter of the one would run

into the matter of the other and they would not be distinct.

Descartes, who was anxious to identify space and

material substance, claiming that the essence of material

substance was extension, actually found a way around this

difficulty with his theory of vortices. Imagine a sphere

surrounded by a ring, a bit like the rings of Saturn. If

the sphere is matter rotating in one direction and the ring

is matter rotating in a different direction, we can separate

them in thought, even if there is no space between them.

The Greeks never thought of this, but it is not in any

event nearly as plausible as the idea that physical things

are separated by the empty space which lies between their

boundaries. Since each physical object is determinate,

43

definite, finite and bounded, the empty space, or the void,

will be indeterminate and infinite. Now, the physical is

what is and the void is what is not, so that the infinite

will be a property of what is not. As Parmenides pointed

out, what is not, is not and is beneath notice.

The infinite is not a property of a thing that exists,

because what exists is a finite being. Nor can it be a

property of the plurality of the things that exist, because

that will also be finite. Therefore, the infinite must be a

property of what does not exist. But how can this be? The

dialectic in early Greek thought is more complex and more

sophisticated than this, but this will give an idea of the

difficulties which led Aristotle to propose a completely

different account of the infinite.

Aristotle’s revolutionary idea is that the infinite is

neither a thing nor a property of a thing but describes a

process. When we think, for instance, of the series of

natural numbers, we are thinking of a process through which

44

the natural numbers are gradually produced by the repeated

application of the successor function. This process is said

to be infinite, because it can never come to an end. To

think of the infinity of the series is to think of the

possibility of continuing the series beyond the point reached

so far, however far along this may be. We may call this the

potential infinite and contrast it with the actual infinite,

which would be the property of some object. If there were a

set of natural numbers, it would be an infinite mathematical

object. Infinity would actually be a property of the thing.

But it appears that there can be no such thing as the set of

natural numbers. To constitute all the members of such a

set would be to complete an infinite series, and an infinite

series is one that cannot be completed.

Another example of the potential infinite is the

infinite divisibility of extension. Any finite extension

can be divided into parts, and these parts can in turn be

divided into smaller parts. This process can be repeated

45

ad infinitum. To say this is to envisage the possibility of

continuing the process without ever stopping. It is not to

imagine the happy day when our finite extension will finally

be divided out into an infinite number of points.

Aristotle’s account is so logical and so persuasive

that it is difficult to understand why anyone should ever

want to bring back the actual infinite. It is not

mathematicians, but theologians, who are to blame.

Mathematics may be able to make do without the set of

natural numbers: theology cannot manage without an infinite

God. For theology, God is no mere potentiality, but an

actual, infinite Being: so such a thing is possible.

This was not a problem for Aristotle who did not

believe that God was infinite: for Aristotle, God was perfect.

The fundamental distinction in Aristotle’s philosophy was

between matter and form. An individual thing is a composite

of matter and form. It is an individual existence with a

definite character. It is a this something (tode ti). The

46

form contributes its nature: the matter is what makes it this

individual. Matter in itself, lacking all form, is

completely indeterminate and hence infinite. But this does

not mean that it is an absolute zero.12 It has potentiality

- the potentiality of acquiring form and becoming a universe

of real things.

In this system to call God “infinite” is no compliment:

it is to think of God as a kind of blob, with even less

definiteness than your usual blob! For Aristotle, God

differs from ordinary things, not because He lacks form and

is therefore infinite, but because He contains no matter.

God is pure form, pure actuality containing nothing that it

is merely potential. It is a mistake to think of God as

everlasting, since this refers to a development through time

- the home of the potential. God is out of time and

eternal, like the Forms of Plato.

Today the Aristotelian conception of God is strange and

12 For one thing, it is denoted by a mass term.

47

unfamiliar, because we are accustomed to a very different

conception of God as a person. For Christians, Muslims and

Jews, this notion presumably derives from primitive Jewish

ideas.13 The trouble is to make a sufficient distinction

between the Divine Person and ordinary human persons. Since

human persons are finite, we make a big enough gap by

conceiving God as Infinite.

This move, however, requires a different conception of

the infinite from the traditional Greek idea. The infinite

for Aristotle is what is below the level of finite beings:

it is a mere potentiality actualized when imbued with form:

the Divine Infinite is somehow above. For us, both

infinities are equally unknown: we have knowledge only of

definite beings through access to their form. We may have a

13 We may perhaps go back even further to the idea of a Sun God. The sun is real enough and has an enormous influence on our life on earth. To turn the sun into the Sun God, requiring supplication and propitiation, is to suppose that the sun is a person with the same sort of beliefs and desires that we have. The main purpose of the argument from analogy is not to prod us into accepting other persons like ourselves, but to put a stop to moves like this.

48

negative knowledge of the divine infinity as that which is

not finite, but this negative description would also apply

to Aristotelian prime matter. A positive description is

also possible if we are allowed the idea of the perfect. As

a person, God will have knowledge, but His knowledge will be

perfect knowledge, unlike the imperfect knowledge which we

enjoy. We can also say that God has infinite knowledge, but

this is to use the negation of the finite. When we use the

notion of perfection, it is our imperfect knowledge which is

described through the negative term. Nicholas of Cusa

solidified the positive concept of the infinite with his

notion of the maximum. The maximum is that than which

nothing can be greater. Thus, it is not enough to say that

God is Great: God is the Greatest!

If we are allowed to conceive God as an actual infinite

being, the way may be open for other actual infinities.

Augustine believed that God could grasp the infinite

totality of the natural numbers and that there must

49

therefore be the actual totality of the natural numbers

which Aristotle denied. For Bruno the physical universe was

also infinite. If the universe is infinite, it makes no

sense to say that it has a centre and no sense to say that

the earth is at the centre. To claim that the sun goes

round the earth will not help.

7. Space and Time

It is a bit disconcerting to discover that the actual

infinite of modern mathematics has its origin in a belief in

the actual infinity of God. Because Bruno’s theory that the

physical universe was also infinite failed to make a

sufficient distinction between the natural world and a

transcendent Deity, the authorities considered it heretical,

with serious consequences for Bruno14. There was wide14 Giordano Bruno was burned at the stake in 1600. The

50

support, however, for Augustine’s idea that a God with

infinite power was able to grasp the entire totality of the

natural numbers. With Divine help, we may gain some support

for the idea of the set of natural numbers, but even if

there is an all-powerful Deity, the matter is not settled.

There are logical limits to the power of God. Not even God

could create a stone so heavy that he couldn’t lift it! If

it is logically impossible to complete an infinite series,

not even God can complete it.

There is, however, another source of the idea of the

infinite that does not require the support of the Deity.

This is the infinity of space and time. When we think of

the future, we necessarily imagine a future without limits.

We cannot coherently think of the end of time: to think of

the end of anything is to imagine a time after the end

within which the end is defined. If we assume an end to

belief that the universe was infinite was not, indeed, the only heretical idea with which he annoyed the authorities.

51

time, we must imagine a time beyond the end of time, which

was therefore not the end of time after all.

If we take space and time to be real things, then we

are bringing in the actual infinite through the back door.

Immanuel Kant refused to accept this. He denied that space

and time were actual, infinite objects and interpreted them

as forms of sensibility which “have their seat in the

subject.” It is not necessary to go this far. We can deny

that space and time are real things, not because they are

not real, but because they are not things. We can make this

intelligible by treating ‘space’ and ‘time’ as mass terms,

like ‘water’ and ‘gold’. The idea of mass terms has become

more prominent in recent years15, challenging the

pretensions of the system of modern logic formulated by

Russell and Whitehead. But the idea has been around for a

long time: the elements of Empedocles - earth, air, fire and

15 See, for instance, Henry Laycock, Words Without Objects (Oxford, 2006).

52

water - were all denoted by mass terms.

The key idea in the conceptual scheme of space and time

is probably the idea of the future. The future is the

primary domain of the possible. Unless we assume that

everything has already been sewn up through a rigid

determinism, we may imagine possibilities in the future

which may or may not come to pass. We must also imagine an

unlimited succession of future states in which these

possibilities may be enacted.

It is this openness of the future which makes possible

and intelligible the infinite series of natural numbers.

This series is defined through the successor function, which

is the form of the act of adding one to a given natural

number. But in addition to the successor function, the

representation of the series presupposes the possibility of

the unlimited iteration of this function, which in turn

presupposes the representation of an unlimited future time.

We can structure the future by imagining a series of

53

stages where A is before B, which is before C, which is

before D, etc. This structure is asymmetrical (If A is

before B, B is not before A) and transitive (If A is before

B, and B is before C, then A is before C). We can assign a

similar structure to space if we imagine a trip in which we

start from here and gradually get further and further away

from our starting point. The structuring of time and space will

develop into the measurement of time and space, if we can

introduce into our progression, not just successive steps, but

equal steps. This was taken literally by Roman soldiers who

measured out a mile as a thousand paces.16 This could

provide not just a measure of distance, but also a measure

of time, with the unit of time being the time taken to cover

a mile at a steady pace. This measure of time, of course,

was never used, because other more convenient ways of

16 In case you think that the Romans either had very long legs or very short miles, the Roman pace (passus) was a double step measured, e.g. from right heel to right heel

54

measuring time were available, e.g. by using the succession

of night and day.

If we can measure a path from A to B, we may be able to

imagine a more direct route which is shorter. The Roman

road from London to York presumably curved up and around the

hills that were in the way. We can imagine a shorter route

which we could construct if we had the resources to dig

through these hills. This brings in the idea of the

shortest possible route or the shortest distance17 between

two points, which we call the straight line. We now have

the key concept required by the geometry of Euclid.

In working out their geometrical system, the early

Greeks soon made a horrifying discovery. They discovered

that the diagonals and sides of a square are not

commensurable. By the Theorem of Pythagoras, to specify

the ratio between the diagonal and the side of a square, we

17 Strictly, we should not talk about longer or shorter distances between A and B. Rather, the distance between A and B is defined as the shortest path from A to B.

55

must introduce the square root of two. I have already

argued that two has no square root, but it now seems that we

need it!

There is only one way to avoid bringing back the square

root of two, and that is by denying the existence of the

straight line! The straight line is the shortest line

connecting two points. That is to say, a straight line is

the line between two points than which no other can be

shorter. This is strangely reminiscent of the definition of

the maximum provided by Nicolas of Cusa, for whom the

maximum is that than which nothing can be greater. The

straight line is the minimum distance between two points.

In mathematics we do not always have a maximum: there is no

greatest number in the series of natural numbers. Perhaps

we do not always have a minimum. There may be no such thing

as the minimum distance between two points. If there is no

such thing as this minimum distance, we do not require an

irrational number to provide its measure. Certainly, we can

56

imagine shorter and shorter paths between the opposite

vertices of the square, but any such path can be assigned a

rational number. Only the absolutely shortest path would

require an irrational number, but there is no such path: it

is merely an idea of reason!

For similar reasons, we can remove the need for pi by

denying the existence of the perfect circle. We can

coherently posit better and better approximations with an

associated representation of the ratio between the

circumference and the diameter, but each of these

approximations can be designated through a regular rational

number. It is widely recognized that we cannot produce a

perfect circle, due to the limitation of our powers. I am

arguing that we cannot coherently imagine a perfect circle.

If there is no such thing as the exact distance between two

points, then it is not possible to have a point which moves

so as always to be at the same distance from a fixed point

which is the centre of the circle!

57

We are now ready for an even more radical move.

Euclidean space is represented as a mathematically dense

extensive continuum.18 This means that between any two

distinct points another point can always be inserted. The

continuity of space has been a source of problems, beginning

in ancient times with the paradoxes of Zeno. Perhaps these

difficulties have their origin in the very idea of a spatial

point. The spatial point may be another myth waiting to be

exposed. The spatial point is not a part of space reached

when we complete the infinite process of subdividing a given

space into its constituents. As we have seen, such an

infinite process can never be completed, so that the spatial

point is an ignis fatuus. Even if a line could be analyzed

into an infinite number of points of zero magnitude, the

line could never be reconstituted from these components. No

matter how many zeroes are added together, we never get

anything more than zero. If we are to find a role for the

18 This is also true for standard non-Euclidean spaces.

58

idea of a point, it is not as a part of a line, but as a cut

between adjacent parts of a line. This means, incidentally,

that the recent practice of construing a line as a set of

points will have to be given up, since a set of points

presupposes the points, whereas a point must be defined in

terms of the parts of a line that are themselves lines.

There is a similar difficulty with the familiar concept

of a moment or instant. Since we represent time as an

extensive continuum, any period of time can be subdivided

into earlier and later stages. There is no limit to how

often this process of subdivision can be carried out. The

moment or instant is what we would reach after the

completion of this infinite process of subdivision. But an

infinite process is one that cannot be completed. This

means that temporal moments are not to be conceived as parts

of time from which we can reconstitute temporal durations,

if we have a sufficient number. The best use for the concept

of the moment is as the interface between adjacent parts of

59

time, and as a cut in the temporal continuum, it cannot be

regarded as a part of that continuum.

It is worth pointing out that there is no way in which

the present of immediate experience can be compressed into a

single moment that is the interface between past and future.

For one thing, the content of immediate experience clearly

incorporates motion and change. The very idea of the

present moment is actually incoherent, since it puts

together the idea of the present with its source in the

McTaggart A-series with the notion of a moment that belongs

to the B-series, structured by the relation of before and

after. Further discussion, however, would take us into new

and difficult territory, and I shall go no further.

8. Hypermyth

In this section I shall enter the territory of the

60

Hypermyth. One very strange object encountered fairly early

in a mathematical career is the square root of minus one,

which is needed in trigonometry. Although a negative number

multiplied by a positive number will yield a negative

number, two negative numbers multiplied together will yield

a positive number, so that even a negative number multiplied

by itself will yield a positive number. This means that

there can be no such thing as the square root of minus one.

I do not need to spend time on this case, since the mythical

status of this entity is generally recognized by calling it

an imaginary number.

Instead, I shall now devote my time to the mythical

domain, sometimes called Cantor’s paradise. Cantor believed

that not only are there different infinite sets, but also

that among these infinite sets, some are more infinite than

others.19 His key notion is one-to-one correspondence. Two

19 This seems dangerously close to the idea advanced by George Orwell in Animal Farm that all animals are equal, but some are moreequal than others.

61

finite sets are the same size, if and only if the members of

the one can be put into one-to-one correspondence with the

members of the other. Cantor wishes to extend this idea of

one-to-one correspondence beyond the domain of finite sets.

Although the series of even numbers is different from the

series of natural numbers, since it leaves out all the odd

numbers, it is easy to arrange a one-to-one correspondence.

Simply correlate each natural number with the even number

which doubles it. (We can use the function y=2x to map from

the series of natural numbers to the series of even

numbers.) Since an infinite series can never run out of

unused elements available for correlation with any member of

some other infinite series, it would appear that all

infinite series are the same size, using this criterion.

I have been explaining the point by referring to the

series of natural numbers and the series of even numbers.

Cantor is, however, invested in positing the set of natural

numbers and the set of even numbers. This immediately brings

62

in a conflict with another criterion governing the size of

sets which I explained at the end of section 3. This other

criterion is that a set cannot be the same size as a proper

subset that excludes elements which belong to the original

set. According to the first criterion, the set of natural

numbers and the set of even numbers are the same size:

according to the second criterion, the set of natural

numbers is larger than the set of even numbers. This has

the flavour of an Antinomy of Pure Reason, as featured by

Immanuel Kant in the Critique of Pure Reason.

The antinomy is to be dissolved in Kantian fashion by

denying the existence of the completed totalities which are

the source of the problem. If there is no such thing as the

set of natural numbers or the set of even numbers, then it

is meaningless to say that the one is a proper subset of the

other, and the second criterion loses application. We are

still left, in a sense, with the first criterion, although

it is no longer a criterion of the size of sets. We are

63

left with a function with which to correlate natural numbers

with even numbers.

Cantor has also devised a complicated and ingenious

system through which to correlate the rational numbers one-

to-one with the natural numbers. He takes this to establish

that the set of rational numbers is the same size as the set

of natural numbers, but I cannot go along with this, since I

deny the existence of such sets. I have no objection,

however, to the scheme of correlation itself which links

each and every rational number with a corresponding natural

number.

Cantor next turned his attention to the real numbers.

Is there a scheme of correlation through which we can

connect every real number with a partner among the natural

numbers? Since there are no such things as the infinite

sets of rational numbers with which Cantor identifies real

numbers, we are already in the domain of myth, but I shall

let this pass. We might expect Cantor to come up with an

64

even more ingenious scheme of correlation, but instead he

turns around and denies that there can be any such scheme.

He begins by supposing that we have found such a scheme, and

then argues that any such scheme will inevitably leave out

some real number which has no partner among the natural

numbers!

Cantor wishes to show that even the set of real numbers

less than one is larger than the set of natural numbers.

Any such real number can be expressed in the form of a non-

terminating decimal. If the set of real numbers less than

one is the same size as the set of natural numbers, we must

be able to line up each such real number with a

corresponding natural number.

We can display a possible fragment of the required

table as follows:

1: 0.54229.....2: 0.89767.....3: 0.31584.....4: 0.12345.....5: 0.38412......

65

It is easy to begin the construction of a non-terminating

decimal which cannot be on this list. If the first digit of

the first decimal is five, write six: otherwise, write five.

If the second digit of the second decimal is five, write six:

otherwise, write five. Keep going in this fashion. In this

way any decimal that begins 0.65655.... will not be on the

fragment of the list constructed so far.

The answer is, of course, that the decimal

corresponding to the sixth natural number might begin in

this way. This can be countered by continuing the

construction along the diagonal so that the special decimal

has a sixth digit different from the sixth digit of the

decimal corresponding to number six. Whenever the condition

specified through the construction of the diagonal number so

far is satisfied by adding another decimal to the list, one

can always defeat this move by adding another digit to the

diagonal number.

66

Cantor would like to leave it at that, but there is

another side to the story which seems equally valid.

Whenever the new digit has been added to the diagonal

decimal, one can always defeat this move by adding another

decimal to the list corresponding to the natural numbers,

since the supply of natural numbers is inexhaustible. The

crucial question is what happens at infinity when all the

non-terminating decimals have terminated? The crucial

question is clearly incoherent, but if one does indulge in

speculative eschatology, why should one suppose that the

last act is one in which we have a diagonal number which

cannot be added to the list, rather than a list which does

not permit the further extension of the diagonal decimal.

To see why both ideas are equally incoherent, consider

the attempt to use the dialectic of Zeno to prevent the

chicken from getting across the road. The chicken cannot

get to the other side without first reaching the midway

point, so we imagine that the chicken does just that. We

67

now face a further condition. To reach the midpoint, the

chicken must first get half way to the midpoint. We may

suppose that the chicken gets there too. We now face the

intimidating prospect of an infinite series of conditions

and satisfactions and with a wave of the hand we conclude

that not only will the chicken not get across the road, but

it cannot even get started! But why should we suppose that

the last thing is a condition which the chicken does not

satisfy rather than a satisfaction for which no further

condition is produced? Is it not more reasonable to say

that the chicken must have satisfied every condition put in

its way when it actually reaches the other side!

. Cantor has, however, another argument to show that

some infinite numbers are more infinite than others, usually

known as Cantor’s Theorem. This theorem makes use of the

idea of the power set of a given set. The power set is the

set of all subsets of the given set. In the case of finite

sets, the power set is always larger and contains more

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elements than can be correlated one to one with the elements

in the original set. For a finite set of a given size, one

can actually calculate the number of elements in the power

set. Cantor’s theorem purports to provide a general proof

which is not specifically restricted to finite sets. Thus

the power set of the set of natural numbers will have a

cardinality greater than N and the power set of this power

set will have a cardinality greater still. This introduces a

series, indeed an infinite series, of what are called

Transfinite Cardinals, each of which is larger than the one

before.

Cantor’s argument takes the form of a reductio ad absurdum

and begins by assuming a scheme of one-to-one correlation

connecting the elements in the given set with the elements

in the correlated power set. In such a scheme, each item in

the original set is either an element in its correlate in

the power set or it is not. Let us collect together all the

items which are not elements in their correlates in the

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power set. This collection which contains elements drawn

from the original set will be a subset of the original set

and hence an element in the power set. Let us call this

collection K. Since we are assuming a one-to-one

correlation between the elements in the power set and the

elements in the given set, K must itself have a correlate in

the original set which we shall call k. As an element of

the given set, small k must either be an element of its

correlate big K or it is not. Now, if small k is an element

of big K, then it is an element of its correlate and so

cannot belong to big K: but if small k is not an element of

big K, then it is not an element of its correlate and so

must belong to big K! This is a contradiction so that the

original assumption of a one-to-one correlation between the

given set and its power set must be abandoned.

The argument appears watertight and has indeed been

widely accepted. It does, however, have a weak spot. In

addition to the official assumption of the correlation which

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is rejected when the contradiction is discovered, there is

also an unacknowledged assumption which may be the source of

the trouble. If we call the original set S and its power

set P, this is the assumption that there is such a thing as

the set of all elements in S which are not elements of their

correlates in P. This is a definite assumption and it is an

assumption of which one may be reasonably suspicious. It

looks rather like the assumption which is the source of

Russell’s paradox, the assumption that there is such a thing

as the set of all sets not members of themselves.

Indeed, there is a special case in which we find a

connection much closer than merely a similar appearance.

Let us consider the set of all sets. Such a set is an

immediate challenge to the conclusion of Cantor’s theorem.

There can be no set of sets which is bigger than the set of

all sets. No power set, which is a set of sets, can be

bigger than the set of all sets. Not even the power set of

the set of all sets can be bigger than the set of all sets!

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This directly contradicts Cantor’s theorem.

It is easy to see why Cantor’s Theorem must collapse in

this special case. If S is the set of all sets, then every

element in P, the power set of S, since it is a set, will

also be an element in S itself. We can therefore correlate

every element in the power set with itself as element in S.

The set K constructed by Cantor, which is the set of all

elements in S which are not elements of their correlates in

P, will therefore be the set of all sets which are not

elements of themselves! Russell’s paradox shows that we

cannot construct such a set.

One may try to escape the special case by denying the

existence of the set of all sets. Unless the denial is

unprincipled, arbitrary and ad hoc, one must have a reason.

The criterion for membership in the set of all sets, being a

set, seems above suspicion. It is also legitimate to use

the concept of set in a plural reference to an indefinite

manifold of sets without any fixed limit to the number of

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sets covered by such a plural reference. One may, however,

draw the line against the combination of all sets that could

be incorporated in a plural reference into a single

totality, which might then be an element in other sets.

Also, the set of all sets would fall victim to the vicious

circle principle, since it would have to be one of the many

that it combines.

The fate of the set of all sets is a warning that it is

not enough to have a coherent criterion to determine what

will and what will not be in the set. One must also have

available a given manifold to which the criterion may be

applied. Moreover, to construct a set, it is necessary to

combine the elements in the plurality that satisfy the

criterion in a single total. I have argued earlier that if

we are dealing with a plurality that is not limited, such as

the series of natural numbers, we cannot form a set. We may

also refer to pluralities not organized as series, such as

the plurality of sets not members of themselves, without

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turning this indefinite plurality into a totality or set.

If there is no such thing as an infinite set, then Cantor’s

theorem cannot be extended to cover infinite sets, and the

whole parade of transfinite cardinals will vanish. Even if

one is not prepared to go as far as a total denial of

infinite sets, Cantor’s argument is seriously weakened when

one recognizes that it is not enough to have a coherent

criterion in order to posit a set of all elements satisfying

that criterion. Thus, the contradiction that emerges from

Cantor’s theorem may be due to the unjustified positing of

the set of elements in S that are not elements of their

correlates in the power set, and not to the original posit

of the one-to-one correlation.

9. Formalism

One can evade the whole issue of what exists and what does

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not by opting for a thoroughgoing Formalism, where nothing

exists except strings of symbols, together with formation

rules, which determine which symbols are permissible and

which strings are well-formed, a nd transformation rules,

which determine which well-formed formulae can be derived

from the axioms of the system, preserving the distinguished

value that is accorded to these axioms.

There is no doubt that formal systems can be

constructed and operated. They must, however, be carefully

distinguished from their interpretations and non-formal

counterparts. To take an example from logic, we must make a

distinction between the propositional calculus and the

formal system which sets out its skeleton. The logical

constants in the propositional calculus have a definite

meaning, more or less the same meaning as some familiar

words in ordinary language. The sign for conjunction in the

propositional calculus has roughly the same meaning as the

word “and”, belonging to the natural language. The logical

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constants in the propositional calculus have, indeed, a

stripped down truth-functional meaning which ignores the

subtleties of ordinary language. The order of the conjuncts

that makes no difference in the logical system can make a

difference in the natural language. “They got married and

had a baby” does not mean the same as “They had a baby and

got married.”

Even the minimal meaning of the logical constants in

the propositional calculus has to be drained from the

corresponding symbols in the formal system. Moreover, the

variables in the propositional calculus are propositional

variables, which are designed to be replaced by

propositions, bearing truth values, when the expressions in

the calculus are turned into molecular propositions through

appropriate substitutions for the variables. There is no

such restriction in the formal .calculus20.

20 There are, indeed, other interpretations of the same formal system, such as an interpretation using classes set out by P.F. Strawson in the fourth chapter of his Introduction to Logical

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Although we can use a formal system to set out the

skeleton of the propositional calculus, it is the logical

system and not the formal skeleton which controls the

construction of all formal systems. One cannot have one

rule which allows a certain formula as well-formed and

another rule which rejects the very same formula. This is

prohibited by the logical law of contradiction which belongs

to one special interpretation of the formal calculus, not to

the formal calculus itself.

There is, indeed, a particular kind of formal system,

said to be inconsistent, in which the transformation rules

permit the derivation from the axioms of every well-formed

formula. Such systems are definitely problematic, since the

transformation rules fail to carry information. In a system

in which the axioms have a distinguished value which the

transformation rules transfer to some, but not all, well-

formed formulae of the system, we have information carried

Theory (London, 1952).

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by the rules of derivation. Where there is no such

distinction, there is no information, since information

requires the decision of alternatives in an information

space.21

Such systems are said to be inconsistent, on the ground

that inconsistent propositions are also thought to entail every

proposition.22 But even if the formal counterpart of an

inconsistent logical system is a system in which every well-

formed formula has the distinguished value, not all such

21 An older version of this fundamental principle in information theory is the dictum of Spinoza: “Omnis determinatio est negatio.”

22 This can be shown using the resources of the propositional calculus. From a contradictionin the form “p and not p” we can derive “p” by conjunctive simplification. Now derive “p or q” by disjunctive addition, where “q” is any proposition. Now extract the second conjunct “not p”, again by conjunctive simplification. Using disjunctive argument, we now deduce “q” asour final conclusion. I am notmyself completely happy with this argument, since it involves insulating the affirmation in the initial premiss from the rulesof inference employed in the argument. If we assume p and not p, does not this destroy the whole basis for a deductive argument of any kind?

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formal systems have a sign which functions like negation in

thought and language. Therefore, for a formal system to be

inconsistent in the sense defined does not involve anything

like contradiction in the ordinary sense. Even if it is

true that whenever we have a formal system whose

interpretation in the domain of propositions involves

contradictory axioms, we also have a system in which all

well-formed formulae can be derived from the set of axioms,

the converse does not follow. It does not follow that all

formal systems in which we lose the distinction between

expressions which do and do not have a distinguished value

involve a contradiction. This would be the traditional

fallacy of the simple conversion of a universal affirmative

proposition!

In any event, we certainly have a distinction between

the propositional calculus proper and its formal

counterpart. Which of these systems is the more

fundamental? A case can be made for the formal system,

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because it is more general and has a wider scope. The

propositional calculus is a special interpretation of the

general abstract system and other interpretations are

possible, as Strawson has shown. On the other hand, I have

argued that the logical system must control the construction

of any and every formal system. We may think of our logic

as the operating system presupposed by all thought and

language, including the discourse in which formal systems

are introduced. We create the formal counterpart of the

propositional calculus by deliberately removing the special

meaning of the logical constants in order to reveal a

structure which may be shared by other systems.

Perhaps the answer to the conundrum is to simply say:

“It all depends on what you mean by ‘fundamental’”. Since

the considerations on each side are reasonably clear, the

problem must lie with an unclarity in the concept of the

fundamental. In this context, the term “fundamental” may

cause more trouble than it is worth, and should therefore be

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dropped.

We face in mathematics exactly the same sort of issue

we encountered in logic. Peano provided an axiomatization of

arithmetic: in his system, the ideas “(natural) number”,

“successor”, and “zero” have a definite arithmetical

meaning. We can, however, construct the formal counterpart

of the Peano system by removing the original arithmetical

meaning from its expressions, leaving behind only the formal

structure. The move to the formal counterpart makes

possible other interpretations of the formal system, such as

the series of even numbers and the series of rational

numbers, organized in Cantor’s fashion. Another

interpretation takes zero as the null set, the successor

function as the operation that forms from a given set the

unit set that contains exactly the original set, with the

concept corresponding to number covering sets formed in this

way.

One bizarre move is to take this interpretation to

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replace the standard arithmetical axiom set promoted by

Peano. Both systems are certainly connected, since they both

interpret the same formal structure, but the concepts used

in the two interpretations are completely different. The

purpose of the move is no doubt to bring arithmetic under

the aegis of set theory, but the move is just as strange and

no more permissible than would be the replacement of the

propositional calculus by the set theoretical interpretation

of the associated formal system outlined by Strawson.

Just as logical ideas control the formal counterpart of

the propositional calculus, and every possible

interpretation thereof, so also arithmetical ideas are

involved in every possible interpretation of the abstract

counterpart of the Peano structure. The fourth member of

the series of even numbers may not be the number “four”, but

it is the fourth member! Although the move to the formal

counterpart may look like a move back to basics, it is the

arithmetical system itself which is more firmly rooted in

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our cognitive practice. A grasp of ordinal numbering is

built into our very representation of formal arithmetic.

Both the original Peano system in which the number one

is a primitive term and the usual system in which the

primitive term is zero are different interpretations of the

same formal structure. In the original system, the natural

number four is the fourth element in the structure: in the

usual system it is the fifth element; but this discrepancy

should not confuse anyone who knows what is going on.

Provided a distinction is made between traditional

mathematics and the construction of formal systems, I have

no objection to formal systems that may be operated as a

useful mathematical tool. So long as a formal system is

consistent, in the formalist sense, preserving a distinction

between expressions which have the distinguished value and

those which do not, the expressions in the system can be

used to carry information. Formal systems cannot be a

source of myth, because they do not require us to believe in

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anything. They are therefore perfectly compatible with my

project of constructing a math without myth.

10. Conclusion

I am sorry to say that my purpose in this paper has been

largely destructive. I have tried to drive out of

mathematics mythical entities that have no right to be

there. I have not tackled difficult metaphysical questions

about the mode of being of the entities I have been prepared

to leave standing. It may have emerged that my sympathies

are not with Parmenides and Plato, but I do not claim to

have settled this issue. Nor have I tried to provide

patches to repair any damage I may have done to the fabric

of mathematics, although my hope is that the excision of

illegitimate objects has not produced gaps that cannot be

closed.

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.

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