Russell: introduction to mathematical philosophy:
definition of number
Russell says – defining number should not be confused with defining plurality
number is what is characteristic of numbers – as man is what is characteristic of men
a plurality is not an instance of a number – but of some particular number
a trio of men is an instance of the number 3 – and the number 3 is an instance of number
the trio is not an instance of number
my view is that numbering is the act of marking instances of a plurality
we follow conventions in doing this – that is established practises – and that means an established terminology
the act of counting is repetition with specific identification of instances – it is a progressive act
each instant is marked as distinct i.e. by the marks ‘1’ ‘2’ ‘3’ etc.
these marks – are syntactical conventions - conventions not of the instances – but of the act
the act of counting is not particular to any circumstances – it is an act that has general application
as a matter of established practise we name these marks ‘numbers’
the term ‘number’ is thus a general term that refers to the marks made in the act of numbering
Russell goes on to say a number is something that characterizes certain collections – namely those that have that number
we can ask what is it to ‘have that number’?
how does a collection have a number?
firstly what is a collection?
clearly it is a conception
and I would say a conception is an ideal means of organizing individuals
anything can be ‘collected’ – anything came be a member of a collection
the reason for the organization – the collection - depends on other considerations
but essentially it is about what a collection is to be used for
when we speak abstractly about a collection – we refer to its members
an individual is a member of a collection – for the reason of that collection
that is the reason for the collection – what is it is designed for - is the reason for membership
on this view a number does not characterize a collection
the purpose of the collection is what characterizes it
the fact that one collection has 5 members and another has 5 actually tells us nothing of the character of the collections
that they are numerically the same is of no consequence
a numerical characterization of a collection simply gives us the number of its membership
it is really just a quantitative description of the collection
and yes trivially that does distinguish it from collections of another number
but it does not distinguish the collection in terms of its reason
Russell says a class or a collection can be defined in two ways – we may give an extensional definition – one that enumerates its members
and an intensional definition as when we mention a defining property
on the view that I have put above a class is not defined extensionally
that is to say enumerating the members of a class does not define the class
fred and jack and john – can be members of any number of collections
their membership is not what defines the class –
the class is defined by its reason for being – that is why it was constructed in the first place
now to intensional definition
can the class be defined by what the members have in common?
well they may have many things in common – but the reason for them being classified –made into a class – is not the fact they have something in common
it is the purpose that classification is to be put to
and that purpose – whatever it may be is outside of the classification – outside of the class – it is the reason for the class - and therefore is not internal to the class –
Russell goes on to say when we come to consider infinite class we find that enumeration is not even theoretically possible for beings who live for a finite time
we cannot enumerate all natural numbers
our knowledge of such collections must be derived from intensional definitions
I don’t see this last point at all
intensional definition – finding what is common to the members of a group – tells us nothing of the number of the group
so intensional definition is quite irrelevant to the question of infinite classes
the infinity of natural numbers derives not from there being an actual infinity of things called numbers
but from the fact that we can understand progressive (as in continuing in time) repetition as being in principle without end
that is the key to infinity is the concept of progression and the fact of repetition
Russell goes on to say
firstly numbers themselves form an infinite collection – and cannot therefore be defined by enumeration
secondly – collections having a given number of terms themselves – presumably form an infinite collection – e.g. there are an infinite number of trios in the world
and thirdly we wish to define ‘number’ in such a way that infinite numbers are possible
thus we must be able to speak of the number of terms in an infinite collection
and such collections must be defined by intension
it is clear here that for Russell intensional definition is the key to his idea of infinite number
but as I have pointed out intension has nothing to do with infinity or number
the notion of infinity comes down to that of repetitive action
numbering is the marking of any such action
as such the idea of infinite number – has no sense to it
the point is this – infinity is not an attribute – it is an operation
on such a view there seems to be no sense at all in speaking of infinite collections
that is to say the description ‘infinite’ is not applicable to collection
we cannot speak of infinite classes
again I say a collection – a class - is defined not by what is in it or its number but rather its reason – its purpose – its function
Russell says it is clear that number is a way of bringing together certain collections – those that have a given number of terms
we can suppose all couples in one bundle – all trios in another etc.
in this way we obtain various bundles of collections
each bundle consisting of all the collections that have a certain number of terms
each bundle is a class whose members are collections i.e. – classes
thus each is a class of classes
the bundle consisting of all couples e.g. is a class of classes –
each couple is a class with two members -
the whole bundle of couples is a class with an infinite number of members –
each of which is a class of two members
it is true that you can keep classifying – that you can classify within classification – there can be many good reasons for doing this
Russell says the whole bundle of couples is a class with an infinite number of members
‘infinite number of members’ as I have argued above makes no sense
what you can say here is that the whole bundle of couples is a class with an unknown number of members
if instead of ‘infinite’ Russell used ‘unknown’ there would be more sense to his argument
he asks how are we to decide whether two collections belong to the same bundle?
well look there is no reason why anything belongs or does not belong to anything else unless you make it so
classification – the making of classes – is a contrivance - there are no ‘natural’ classes
to his question Russell says the answer that suggests itself is – find out how many members each has – and put them in the same bundle if they have the same number of members
but this he says - presupposes that we have defined numbers - and that we know how to discover how many terms a collection has
Russell’s view is that we cannot use counting here because numbers are used in counting
his argument here sounds cogent – on the assumption that numbers are something other than the operation of counting
of course you can ask – well ok counting – but what is being counted?
my view is that the act of counting is the act of numbering
the act of counting is the act of marking in some manner or another
the resultant markings are numbers
a number is a mark in a counting
and counting is the ordering of individuals in a series
Russell says it is simpler logically to find out whether two collections have the same number of terms – than it is to find out what a number is
this seems an odd statement to me – given what has preceded
it seems Russell thinks that the defining of number is the defining of some entity
when in fact all that ‘number’ is – is the term that we use to refer to the markings we make in counting
‘number’ to be fair comes up as a noun – as the name of something – and yes you can say the marks made in counting are something – but the real point is that ‘number’ refers to an action – so it is logically better understood as a verb
in any case Russell from the above statement seems to suggest you can understand a number without first knowing what ‘number’ is
this distinction doesn’t bother me or bear on my argument – but it seems to contradict what Russell just previously said regarding counting – that you need to know number first
he goes on to distinguish kinds of relations in this connection –
a relation is said to be ‘one-one’ when if x has the relation in question to y - no other term x’ has the same relation to y – and x does not have the same relation to any term y’ other than y
when only the first of these conditions is fulfilled – the relation is called ‘one-many’
when only the second is fulfilled – it is called ‘many-one’
Russell says it should be observed that the number 1 is not used in these definitions
it is true 1 is not used in this analysis
but the point is that for a relation of any kind to exist there must be at least two terms
that is a relation – is a relation between –
so it is clear that number is here presumed in any relation and any relational analysis
two classes are said to be ‘similar’ – when there is a one-one relation
he defines this more precisely –
one class is said to be ‘similar’ to another when there is a one-one relation of which one class is the domain while the other is the converse domain
it is obvious says Russell that two finite classes have the same number of terms if they are similar – but not otherwise
in what does this similarity consist?
granted you can have a one-one relation – why introduce similarity?
it seems like a weak word for what is very precise logical relation
and what is added by this notion of similarity?
the notion seems to me to be superfluous
Russell continues - the act of counting consists in establishing a one to one correlation between the sets of objects counted and the natural numbers (excluding 0) that are used in the process
the notion of similarity is logically presupposed in the operation of counting
the idea seems to be that you have a set of objects and a set of numbers – and then the act of co-relating the two
this presentation I think shows just how vacuous this idea of similarity really is
is Russell trying to suggest that the reason a number co-relates with an object is because of similarity?
he say the act of counting presupposes similarity
this is to suggest counting is like placing dominos on the ‘correct’ squares of a domino board
this seems incredibly naïve
numbers do not exist as objects – to be co-related or ‘imposed’ on other objects
numbering is simply the act itself of marking in a progressive manner the objects in a series
the numbers just are the marks of the numbering
no similarity exists or is required
he says we may thus use the notion of ‘similarity’ to decide when two collections belong to the same bundle
we want to make one bundle containing the class that has no members – one bundle of all classes that have one member – this will be for the number 1 etc.
given any collection we can define the bundle it is to belong to as being the class of all those collections that are similar to it
if a class has three members – the class of all those collections that are similar to it – will be the class of trios
whatever number of terms a collection may have – those collections that are ‘similar’
to it will have the same number of terms
and the number of a class is the class of those classes that are similar to it
and so to number – a number is anything that is the number of some class
Russell says at the end of this – such a definition has the verbal appearance of being circular – we define ‘the number of a given class’ – without using the notion of number
in general
therefore we define number in general in terms of ‘the number of a given class’ – without logical error
it is in this section that Russell reveals the point of ‘similarity’
it is a concept designed to establish the notion of number
the number of a class is the class of those classes that are similar to it
which is a very weak way of getting around saying that ‘the number of a class is the class of those classes which have the same number’
and Russell wants to avoid this statement for not only is it a circular definition -
and it brings down the whole edifice of classes –
for if a number is just a number (whatever that might mean) there really is no need to introduce classes at all
you also have the problem of classes that have the same number not being distinguishable
and there goes the neighbourhood
the idea of similarity is supposed to hold off these results
as I have said above – it just comes across as a very weak criterion in this context
but more than this it is at the very least – in this context an empty concept
we are it seems supposed to assume a similarity between classes with the same number – while not mentioning that they have the same number – which is of course the basis of their similarity
and if it doesn’t mean this it means nothing
the final point is that a number in general is any collection which is the number of one of its members
all its members are of course similar in that they have the same number
so the number of one of its members will be the number of the class
what else could it be?
the thing is Russell’s use of class here has not I think added to the issue
simply because in the end in order to identify class you need number
class does not elucidate number
now simply bundling things together that have the same number – and calling the greater bundle – the number –just doesn’t cut it for me
the greater bundle is just a greater bundle
Russell seems to think that we can in some way discover numbers in reality – and at this he has failed
reality as in the non-human reality has no numbers
numbering is an operation that human beings bring to reality – for their purposes
the human reality is one that demands at times an overlay of order
numbering is a basic operation to this end
Russell’s argument is like this – you use number to define class (even though you try to appear to not be doing this by using the phantom concept ‘similarity’) and then you use class to define number
it’s hard to credit really
and the result is that number is left undefined
as Russell says at the beginning of his discussion of the definition of number –
‘In seeking a definition of number, the first thing to be clear about is what we may call the grammar of our inquiry.’
number is not a noun – it is a verb
© greg. t. charlton. 2008.
