Russell: introduction to mathematical philosophy:
limits and continuity
Russell says –
the notion of ‘limit’ is a purely ordinal notion – not involving quantity
what makes אₒ the limit of finite numbers is the fact that in the series it comes immediately after them – which is an ordinal fact – not a quantitative fact
an ordinal issue yes – but what is missing from Russell’s analysis here is the fact that any limit is a fact of action and decision
there is a sense in which this issue of limits gains some prominence in mathematics – once the argument for infinite classes / series and numbers goes through
the reality is an infinite series is not one that qua infinite can be dealt with
for an infinite series to be functional – its infinity has to be effectively denied in an operational sense
and so some account of limit must be advanced – just to make any operation feasible
limits are drawn in order for operations to take place – (or for them to be conceptually valid)
given that this is the day to day business of mathematics – you might ask the question – what value the idea of the infinite – in any of its manifestations?
and it is worth pointing out that its foundation is a rather bizarre notion – the idea of reflexivity
reflexivity is not an action that anyone actually performs
it is an attribute of a class
to understand this you need to think of a class as something other than an action of classification – you need to regard it as an ideal entity – and one that ‘reflexes’
yes reflexes – has the potency to reflex into ‘itself’ – endlessly or infinitely
this is the idea
quite a lovely notion from the point of view of imaginative fiction – worlds within worlds
but one that has no relevance for the action of mathematics
the idea that we can speak of a class – and the class has having a ‘self’ (itself) that it in some magical manner reflexes into – is a ridiculous notion – that has its origin in the reification of classification
that is in the idea that a class is a thing of some kind
a class – let’s not get hoodwinked by grammar – is an action
ordering is a kind of action
and the marking of any such ordering is a primitive set of actions
it is mathematics
understanding mathematics is essentially the same as understanding the markings and the symbols of a primitive tribe
that is understanding the use of ‘special’ syntax – in this case primitive syntax – that is logical syntax
my argument against infinity mathematics is that it is just verbosity – that has no actual – practical value
it is in fact a whole branch of mathematics based on a logical mistake – or series of mistakes
anyway back to limits – and Russell’s arguments here –
he says there are various forms of the notion of ‘limit’ of increasing complexity –
the definitions are as follows –
the ‘minima’ of a class a with respect to a relation P are those members of a and the field of P (if any) to which no member of a has the relation P
the ‘maxima’ with respect to P are the minima with respect to the converse of P
the sequents of a class a with respect to a relation P are the minima of the
‘successors’ of a – and the ‘successors’ of a are those members of the field of P to which every member of the common part of a and the field of P has the relation P
the minima maxima and the sequents are simply descriptions of the boundaries of a class – what is included in it and its range
the making of a class is an action of classification – we can as it were describe the class after the fact of its making in terms of its boundaries – in relation to a (greater) field
such a description is effectively a description of the action of the classification
the action of making the class in a given field
in terms of Russell’s view of things these descriptions (minima, maxima, sequent) are basically ‘logical underpinning’ to the idea of class –
they are there to give the appearance of some kind of basis to this conception of class – the idea that the concept has logical foundation
and this logical foundation is to be found in the theory of limits
you see Russell as with Cantor and Frege thinks of the class as an ideal entity
if you understand it as an act – then the act of classification itself defines the collection – the class
and in such a case there is no point to the discussion of limits
unless of course there is a question of relation - of one class to another –
and in such a case the limits of one and the limits of the other will be apparent – as in obvious
there is in such a case no need for ‘after the fact’ descriptions and analyses
it just strikes me that this theory of limits is really just non-operational baggage
any class will be a limiting of a greater class - or put it this way – it can be seen in this light
the point being that it is all quite relative – it all depends finally on the reason for the class – for the classification –
one description will fit one purpose – and another purpose will demand another description – or indeed descriptions - if there is any demand for description at all
it is the purpose that determines the description – and in that sense the limits
on such a view there is no definite description of limits
any mathematical action will presume a field of discourse to begin with
how relevant that field is to the action will depend on the problem being addressed – and where it leads to
what I am getting at is that there is no field independent description of any class
a classification is an action in context – always
and generally speaking for the action to be performed the context is understood – if not entirely – in part
if it is understood that the act of classifying is primitive and necessary – there is little to be gained by speaking of it in a non-contextual manner – i.e. – so called ‘objectively’
on continuity –
continuity in my view is not a ‘natural’ attribute of mathematical entities in the way that ancestry might be regarded in families
continuity – is really a serial attribute – an attribute or characteristic of the making of a series
an attribute that is of a kind of action
there are going to be in this connection questions of the point of the series – and questions of its form – whether in fact it is a well formed series – but the general assumption in any rational series is that there is continuity
and I say this regardless of whether there are what Russell calls gaps
gaps just may be defining characteristics of certain kinds of continuous series
Russell says that our ordinary intuition regarding continuity is that a series should have ‘compactness’
well yes – this might be where one would naturally start – but this can easily be shown to have holes in it – as indeed Russell points out
continuity is determined – not by compactness – placement in relation to – but rather – reason for –
that is the act of placing in a series creates the continuity – assumes it –
the making of a series and the making of a continuity are effectively one in the same – though continuity is a broader concept – more general than series
and of course there can be argument about just whether the continuity argument of a series actually stands up – but that’s really another matter
once this is understood we don’t need to resort to the fiction of Dedekind cuts
I guess my point is that continuity is a characteristic – and essential characteristic of the series
we presume continuity in order for a series to ‘operate’ – to be
Cantor defines a series as ‘closed’ when every progression or regression has a limit in the series –
and a series is ‘perfect’ – when it is condensed in itself and closed – i.e. when every term is the limit of a progression or regression – and every progression or regression contained in the series has a limit in the series
in seeking a definition of continuity what Cantor is after is a definition that will apply to the series of real numbers – and to any series similar to it
in other words after Cantor needs a way of ‘defining’ real numbers so that they can function in a rational series
to my mind – shutting the gate after the horse has bolted – or perhaps trying to breed a new horse
Cantor’s closed and perfect series – really come from the shock discovery that our number systems need to work in the physical world – quite independently of their other - worldly qualities –
following on from this -
Cantor argues we need to distinguish between two classes of real numbers – rational and irrational
and the idea is that though the number of irrationals is greater than the number of rationals – there are rationals between any two real numbers – however little the two may differ
Cantor’s argument is that the number of rationals is אₒ
(אₒ in my view is something that means nothing – Cantor really is a master at making it look like something that means everything – when the occasion requires it)
the argument is אₒ gives a further property which he thinks characterizes continuity completely – namely the property of containing a class of אₒ members in such a way that some of this class occur between any two terms of the series – however close together –
the idea is that this property – added to perfection defines a class of series which are all similar and are in fact a serial number
this class Cantor defines as a continuous series
none of this actually establishes continuity – all it does is establish and define a series – or indeed a class of series –
and yes there is continuity in the series – but it is only because it is presumed that with a sequence of rationals – you have continuity
I am not against this assumption – in fact I am sure it is all that continuity is
Russell ends off with a shot at the man in the street and the philosopher –
‘They conceive continuity as an absence of separateness, the general obliteration of distinctions which characterizes a thick fog. A fog gives the impression of vastness without definite multiplicity or division. It is the sort of thing a metaphysician means by ‘continuity’, declaring it, very truly, to be a characteristic of his metal life and of that of children and animals.’
I take it Russell is referring here to substance theories where at the cost of continuity – discreteness is sacrificed
what gets me though is that at the same time he can with a straight face suppose that reflexivity is a logically coherent notion – enough to base a whole mathematics on
the idea that a class can ‘reflex’ itself into itself infinitely
the point is once you accept such a notion – class in fact has no definition
and the reason being – it is never complete – it is never well formed
you have no class – at the end or even at the beginning of such a process
as with the fog theorists – there is no particularity – no discreteness – with reflexivity it is destroyed from the inside
the point being reflexivity it is not a process – logical or not
hard to say what it is – perhaps it has a theological origin
continuity is away of seeing things
it is the assumption that the objects chosen for view are connected in a continuous manner
to understand this – you need to know when – under what circumstances there is a need for such a view
my point being – continuity is a conception – the very same things regarded as continuous for one purpose – may indeed be regarded as discontinuous for another
neither numbers (serial marks) or material objects are continuous or discontinuous
strictly speaking the best you can say is that their ‘natural relation’ is unknown
there are tasks that require us to regard their relation as continuous (or discontinuous)
to understand continuity – you have to understand its reason - its task
I will return to Russell's view of mathematics at another time
© greg. t. charlton. 2008.
