Russell: introduction to mathematical philosophy:
the series of natural numbers
Peano’ argument
Peano showed that the entire theory of natural numbers could be derived from three primitive ideas and five primitive propositions in addition to those of pure logic
the three primitive ideas in Peano’s arithmetic are:
0, number, successor
the five primitive propositions are:
(1) 0 is a number
(2) the successor of any number is a number
(3) no two numbers have the same successor
(4) 0 is not the successor of any number
(5) any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers
some preliminary thoughts -
by ‘number’ I mean ‘an operation or action in a series of actions’
the mark ‘1’ is the first in the series
‘the first in the series’ is defined by ‘series’ itself
the actual mark ‘1’ is a convention –
0 on this view is not a number – 0 is non-action – it the position prior to a series
it defines the series by marking the position of no series
the point being that a series ‘comes into being’ – there are thus no natural series
a series is a construction on action
succession is - relative to the series – a repetition of action
a mathematical series is thus a repetitive series
repetition is succession in time
in a repetitive series the successor of an action is an action
no two actions have the same successor because each action is unique in time
0 is not the successor of any number
any property of 0 is not shared by any number
on this view a number is just the mark of an action in a series of actions
not all actions are numbers of course –
but all numbers are actions in a series – in a conception
‘series’ thus is an ideal construction placed on action to create order
and order here is defined as repetition
repetition is the mast basic ordering – and it is on this basis that the series of natural numbers emerges
there are no numbers in the Pythagorean sense of ideal entities
the term ‘1’ or the mark ‘1’ has universality because any repetitive series has a first action
0 if you were to be metaphysical or poetical it is the ‘place’ of no action relative to any series
mathematics as a theory - or the mathematics of the series of natural numbers - is the formalization of the notion of a repetitive series
this is the point of Peano’s axiom 5
it is really a pragmatic theory in the sense that it provides us with terms that enable us to operate without actually performing every action that the terms name
numbers as marks refer to places in a series
so that whatever the series – we know that 501 – refers to a particular place in that series
we can therefore say numbers are ‘places in a series’ – relative of course to other places
and that these places are finally no more than actions performed in the series
Russell says of Peano’s three primitive ideas -
that they are capable of an infinite number of interpretations
he gives this example – let ‘0’ be taken to mean 100 – and let ‘number’ be taken to mean the numbers from 100 on in the series of natural numbers – then all our primitive propositions are satisfied – even the fourth – for though 100 is the successor of 99 – 99 is not a ‘number’ – in the sense we are now giving to number
he gives other examples – the point is that in Peano’s system there is nothing to enable us to distinguish between different interpretations of his primitive ideas –
that is it is assumed that we know what is meant by '0' – and we shall not suppose this symbol means 100 or Cleopatra’s needle
on the face of it - this seems like an ok argument –
however Peano’s clear intent is to distinguish ‘0’ – 0 is a number and it is not the successor of any number – unlike 100 – which like all numbers except 0 is a successor
so for Peano anyway there is no question that 100 can be 0
that possibility is excluded by definition –
0 is 0 – 100 is another number
so Russell’s argument is clever but it is not true to Peano’s definitions
the real problem I think – and one which Russell at least to this point does not address is the relation between axiom 2 and 4
the successor of any number is a number - 0 is not the successor of any number
this to me goes to the problem of defining 0 as a number
at the very least you end up with 0 as some special case number
and as a result your definition of number – whatever that might be – is problematic
for it is a definition that is not all inclusive – but one that has an exception
and so really – it fails
Russell goes on to say –
‘0’ ‘number’ and ‘successor’ cannot be defined by Peano’s definitions – and that they must be independently understood
he says it might be suggested that instead of setting up ‘0’ and ‘number’ and ‘successor’ as terms we know the meaning of although we cannot define them –
we might let them stand for any three terms that verify Peano’s axioms –
they will no longer have a meaning that is definite though undefined –
they will be variables – terms concerning which we make certain hypotheses – namely those stated in the five axioms – but which are otherwise undetermined
Russell says of this view – it does not enable us to know if there are any sets of terms verifying Peano’s axioms
and we want our numbers to be such as can be used for counting among common objects
and this requires that our numbers have a definite meaning
this is just to say if the three primitive ideas are regarded as variables then they will not have definite meanings
and of course you could then wonder what the actual value of any such terminology would be
his second point here that it does not enable us to know if there are any sets of terms verifying Peano’s axioms – is a strange argument
couldn’t you say just this of Peano’s argument as it stands?
it depends how you come at it – if you begin with the axioms – your question might be well – ok – we have these principles – but how are we to know what they refer to ?
what I am getting at is that you could be quite sceptical and ask – how are we to know that the terms ‘0’ ‘number’ and ‘successor’ verify the axioms
just because they are used is not verification
anyway
the basic problem with Peano’s approach as I see it is firstly that he wishes to define 0 and number – and he defines 0 as a number
the result of this is that elucidation of 0 is now dependent on the definition of number
that is a number – 0 – has already been singled out - without the ‘over riding’ definition of number being in place
so you could take the view that nothing has been accomplished by introducing 0 – it needs to wait until number is defined
you might then argue that Peano has failed to define 0
and another thing – 0 is not on the same level as number and successor
it is not primitive in this scheme – it is if anything derivative –
derivative that is from number
now I don’t know – but I suspect Peano did not envisage this implication
I think it probably undercuts his theory
‘successor’ is defined as ‘the next number in the natural order’
clearly then for the definition of ‘successor’ to proceed we need the definition of ‘number’
that is the integrity of the notion of ‘successor’ is dependent on that of number
so is ‘successor’ like 0 dependent on number?
and if so like 0 it is a derivative notion?
also defining ‘successor’ as ‘the next….’ is really not to give us anything at all –
it amounts to saying ‘a successor is a successor’
perhaps the point is that we should just focus on this notion as a key notion in the philosophy of mathematics
one thing we can say is that ‘successor’ is a relational term –
and this seems to me to be how Peano goes about defining number -
a number is that which is the successor to a number
is this to say – numbers are ‘points’ in a succession?
it is pretty clear that Peano’s three primitive ideas – 0 – number – successor
are not given clean - ‘stand alone’ – definitions
and I can’t see how you could say on the basis of Peano’s definitions that any of these notions are ‘primitive’
0 depends on number – number depends on successor - successor is a relation between numbers
what this is suggestive of – is that the idea that there are primitive notions in mathematics – that do no not depend on the meaning of the notions that derive from them – is not on
which is to say the quest for the foundations of mathematics is wrongheaded
that there are no foundations
what you have is an activity – perhaps a primitive human activity - and the description of that activity – the language of that activity – has given us working concepts – the meaning of which is not validated by analysis – but by the activity itself
this might suggest that the ‘foundations of mathematics’ – are not stable – that the activity of mathematics itself – whereever that is it goes will have repercussions on the concepts that are regarded as central or ‘foundational’
to be blunt – if you want to know what a ‘number’ is - look at what people do when they operate with numbers
you would have to say Peano leaves ‘number’ undefined – perhaps that is his sense of primitive
but really this is just a touch of the old Pythagoreans – what you might call a persistent mathematical malady
© greg. t. charlton. 2008.
