Russell: introduction to mathematical philosophy:
kinds of relations
it is clear that for Russell the ground of number – of mathematics is – relations
my argument is that relations presuppose a plurality – and number theory is the marking of plurality
relations on this view are an ordering of plurality
relations that is are actions on the plurality
‘relations between’ do not on this view – underpin numbering
the question of relations – in this context – only emerges – given numbers
this is to say that the making of relations – and the act of numbering
are two aspects – or can be two aspects of ordering –
numbers mark – relations define the possibility of the action of the numbers
how they can be ‘acted’
relations define action
in the context of mathematics we might say ‘manipulation’
so the theory of relations defines what you can do with numbers
numbers – are not – that is a consequence of relations
the theory of relations is the theory of activity
number theory is the theory of marking
in so far as much of mathematics is concerned with the ‘activity of numbers’ – the theory of relations underpins mathematic activity
so relations are the logic of mathematic activity
number theory on this view is prior to relations
mathematics is a primitive language of order
ordered activity depends on such a primitive language
the general point of this is that mathematics is a primitive marking
logic as a theory of relations is a theory of activity – that can be applied to numbers
to use a contemporary metaphor – mathematics is an essential hardware – logic or the theory of relations is basic software
for the program to work the two dimensions are required
and again what this is to say – is that neither are fundamental – both are actions – designed to create a platform – to act upon
in this chapter Russell discusses the relations asymmetry, transitivity and connexity – which he has previously defined as the properties of a serial relation –his idea being that when these properties are combined you have a series
we might ask in connection with asymmetry – why is that if x precedes y – y must not also precede x?
that is what is the basis of this claim - of this relation?
it is clear that with two particulars x and y there is no necessary relation
the existence of any relation is determined by what is done with x and y
that is how they are ‘made’ to relate
asymmetry is about placement that is all – the decision to regard x as preceding y
so it is simply a decision about how to order the world
clearly it depends on an idea of space and or time
so in my terms the placing of x before y is an action
and while it might be more than familiar to speak of an asymmetrical relation – in fact what you are talking about is an asymmetrical action - or placement
asymmetry defines a kind of action
the same is true of transitivity – transitivity is an action
connexity is - what?
there is no given connection between x and y
any connection is made –
it is the decision to place them ‘together’
to place them – that is in a common domain
and ‘domain’ here is really ‘the place of the relating’
which comes down to just the decision to relate
a series of natural numbers is thus a series that is constructed and the decision is made to hold the construction – hold the series
we decide that is that 1 follows 2 follows 3 etc
these are actions and held as repeatable
and the series is then held in principle – which means it is repeatable
any series can be held in such a manner – not all are
the reason that the series of natural numbers is so held is that it is so very useful
and it is its primitiveness – the marking out – that is the key to its utility
its utility is based on need – the need to order
as to the origin of this need – we can only say it exists as a necessity – given the existence of conscious entities in the world
beyond this fact there is no explanation
© greg. t. charlton. 2008.
