Russell: introduction to mathematical philosophy:
infinite series and ordinals
an infinite ordinal in Cantor’s and Russell’s sense is one which is reflexive
and a reflexive class we will remember from the discussion in the previous post is one which is similar to a proper part of itself
now my argument has been – and still is that there is no sense in this notion of a class being similar to itself
and this idea is the origin of the infinite class – and the infinite in mathematics
the reason it makes no sense is that – a class has no ‘self’ to be similar to
in the case where a class is ‘similar’– whatever this is supposed to mean – to another class – we are dealing with two classes – two classifications
a classification is just an operation of organization – of collecting
there is no entity as such that is a class – what is referred to as a class is in fact an action
granted we may represent the action diagrammatically – but this does make it something it is not
too much of the Cantor-Frege-Russell mathematics is buggered up by a substance theory of mathematical entities
numbers are not things – and classes and sets are not ideal entities
mathematics is simply a kind of action
as too the issue at hand – ordinals – as with cardinals it makes no sense at all to speak of infinite ordinals
in general we can say an ordinal number is defined as the order type of a well ordered set
and an order type is the set of all sets similar to a given set
sets are ordinally similar iff they can be put into a one to one correspondence that preserves their ordering
the question – is an order type a number – or rather a pattern?
the argument that it is a number comes from Russell’s argument that we can say one ordinal is ‘greater’ than another – if any series having the first number contains a part having the second number – but no series having the second number – contains a part having the first
the problem with this is that it really just identifies different collections
the fact that a sequence is common to two different collections – is irrelevant in terms of the character of the collections – that is the collections as whole collections
if you give a pattern a number – then you can give another pattern another number –
if one pattern is given the number 1 i.e. and another the number 2 – yes in terms of number theory – one is greater than the other –
this is all I think this idea of ordinal numbers as Russell puts it really comes down to – applying number theory to series and orderings –
and to my mind it is not a natural fit in the case of ordinals – that is patterns and pattern identification
(I will continue to use the terminology ‘ordinal number’ – with the understanding that what we are referring to is ordinal patterns)
a serial number is the name of a series – a mark for a series
a mark of the order of a series –
and yes we generalize this – to refer to any such ordering
this becomes the ordinal number –
it is important to realise that an ordinal number is only a number for an operation –that is the identification of such an ordered series
the ordinal number strictly speaking refers to a pattern
any number of patterns can be created – and named – thus given an identification
the question – is there a limit to the number of patterns (ordinal numbers) that can be made?
are we to say there are an infinite number of ordinal numbers?
what we can say is that there is an exhaustive number of ordinal numbers
that is to say the limit of ordinal numbers is a question of human endurance and purpose
this is not infinity
and the reality is that patterns will be identified for practical – that is real purposes
and in that sense then ordinal numbers are valid only for the purposes they serve
ordinal numbers that is must be seen as contingent –
that is as operations performed and identified for specific purposes
the fact that these patterns identified may in fact endure – is a fact of nature – in the fullest sense
that is how the world is
an enduring ordinal number is one that has high utility value
Russell says that cardinals are essentially simpler than ordinals – and on the face of it he has a point –
the cardinal identifies the number of a set – the number of its members
this would seem to be a simpler matter than identifying a pattern in an ordering
but once the identification is made – the result is the same –
separate classifications are given a common definition –
what we are dealing with here is different purposes – or different classes of purpose
cardinality is an identification of number of the membership
ordinality – can we say – the ‘character’ of the membership?
the idea of cardinal ‘number’ fits ok – but it is limited in its scope – the cardinal only identifies membership-number
ordinal numbers on the other hand open up the whole field of pattern mathematics
in this sense the ordinal is more significant
there is an argument too – that cardinals are in fact a subset of ordinals – in that the cardinal identifies a basic pattern in different classes
also it is worth asking the question – do ordinals put an end to number theory?
one gets the impression with Russell that the idea of the number must be maintained at any cost – any logical cost
the fact that he is even prepared to consider the notion of infinite numbers suggests a rather desperate hanging on – the full result of which is really the generation of a mathematics of irrelevance
after ordinals I see no reason to keep up the deception – ordinals are patterns – and we don’t need to continue to imagine they are numbers –
ordinals represent a post-number field of mathematics –
what is clear too is that we do not need to presume infinite classes to operate with ordinals
and in fact – the idea of infinite classes and infinite numbers – when you get the hang of ordinals – seems to be entirely irrelevant
the two subjects are best separated
the one ordinality - has a place in the real world of operating and defining –
the other - infinite classifications and numbers - has no utility value – and is best placed in the realm of imaginative fiction
© greg. t. charlton. 2008.
