Russell on mathematics XIII

Russell: introduction to mathematical philosophy:
the axiom of infinity and logical types


infinity is made axiomatic for there is no natural ground for it

an infinite operation is not performable

and it follows from this that there are no infinite entities –

for from a mathematical point of view – an ‘entity exists’ if it is countable

any other conception of infinity is of no interest to mathematics

a class or classification is an action of determination

the idea of infinite classes – that is classes that have the property of reflexivity – is not reconcilable with determination

I don’t think reflexivity makes any sense –

but if you were to entertain the idea – as it is put – for the argument’s sake –

you have the idea of a class reflexing into itself – infinitely

it becomes an endless action

or action with no terminus

so in that sense it is not a genuine action

but at the same time there is the idea that this reflexion – generates classes

as if in the one class – there is a constant generation of classes –

a kind of continual creation –

once you see this you see also its theological basis

a kind of equivalent in mathematical theory – to the current theological fashion of modern physics – namely the big bang theory – for which Stephen Hawkings was quite rightly given a papal medal –

quite apart from this though –

the idea that an act of classifying – in some sense has a self that it reflexes into – is quite absurd

even if you are to accept the argument that a class is some ideal entity with this endless potential to find itself in itself

you have to ask at what point are we talking about any kind of defined entity?

in Greek terms it is always in the state of becoming itself –

which is to say it is always in the state of not-being

and to get back to Kansas –

a thing either is or it ain’t

enough of my ramblings


Russell begins his discussion –

‘The axiom of infinity is an assumption which may be enunciated as follows: –

if n be any inductive cardinal number, there is at least one class of individuals having n terms’


the point here is that the above assumes the existence of an infinite cardinal number

if that assumption is accepted then it follows there will be a class of individuals having that number

so the axiom effectively just asserts the reality of infinite classes


Russell continues –

‘The axiom of infinity assures us (whether truly or falsely) that there are classes having n members – and thus enables us to assert that n is not equal to n + 1’


the essential issue here is with regard to the status of n

hate to break up the party but the real question is whether we can rationally speak of an infinite number at all –

what this comes down to is reflexive classes – for the infinite number per se is just a name or tag for such

the idea is that a reflexive class is based on the idea that a class is defined by its internal relations -

as distinct from i.e. its relation to other classes –

which would be to define a class in term of relations outside of itself –

that is in terms of external relations

so – the internal relations of a class –

this by the way is not to ask what is the relation between the members of a class

the class as class is a unity

the idea is that within an infinite class there are classes within classes and that this internal relation of ‘classes within classes’ has no terminus

such a class is defined by the fact that it does not have a logical end

now the issue here is internal relations – or internality

now in my view there are no grounds for asserting the internality of classes

internality is a dimension –

now this might upset some but I would say the internal / external relation only applies in relation to conscious entities

for on my definition internality is consciousness

but even putting this aside – you can legitimately ask in what sense can it be that a classification has an internal dimension –

and I mean here in the sense that Russell puts forward – of a class reflexing into itself

clearly on such a view we are not speaking of what is inside a classification – that is what is bound by the classification – its members –

we are talking about something else

we are talking about the class itself – independent of its membership

now I have a rather simple straightforward argument here – and it is that there is no sense in speaking of a class as in some sense independent of its membership

for in general terms it is its membership that defines a class

so what is in a class is its members

we can speak of the inside of a class – but this is not the same as the internality of a class

and it is internality that is required for reflexivity – for infinity

so it is obvious I think that Russell and those who are for infinite classes – confuse the fact of the inside of a class – its membership – with internality

the membership of a class – what it brings together – if the question should arise – is the external world –

and further there is no additional ghostly dimension to the act of making a classification

true – you can make a general classification – i.e. all Australians – and within that classification – create further endless classifications

but this is not setting up some infinite class –

in any such process what we are doing - to use a modern computer term is ‘drilling down’ –

that is what we are doing is offering further descriptions of the subject at hand

now the class of all Australians – is just where you start – or can start

any description ‘within’ this starting point is another description –

which in logical terms may or may not be seen as being connected to the original descriptions – ‘all Australians’

the descriptions can be related – but one is not internal to the other – they are quite logically independent

that you might relate them – as one being included in the other – is simply a decision to organize –

hate to upset the logical fraternity – but such is really an artistic issue

ummh – who would have thought?


so my point in essence is that we cannot establish n – as infinite number – and as a consequence there is no issue with n + 1

Russell says without this axiom we should be left with the possibility that n and n + 1 might both be the null class


well I have been arguing that there is no sense to n – so perhaps n is the null class

but this would be to say the null class is the class that makes no sense

and this is not what is usually understood or meant by null class


I cannot for the life of me understand how this concept of null class came about – by any mathematician or logician – with any sense

a classification – a class – is an action defined by its membership

a class with no membership – is no class

in such a case there is no act of classification

that is the idea of a null class is essentially a grammatical error – a misuse of terms

or to put it another way - an act of classification – presumes the existence of a world – and rightly so -

here we are in New South Wales

Russell goes on –

‘It would be natural to suppose – as I supposed myself in former days – that, by means of constructions, such as we have been considering, the axiom of infinity can be proved. It may be said: Let us assume that the number of individuals is n, where n may be 0 without spoiling our argument: then if we form the complete set of individuals, classes and classes of classes, etc., all taken together, the number of terms in our whole set will be

n + 2ⁿ + 2²ⁿ…….ad inf.,

which is אₒ . Thus taking all kinds of objects together, and not confining ourselves to objects of any one type, we shall certainly obtain an infinite class, and we shall not need the axiom of infinity. So it might be said.

Now, before going into this argument, the first thing to observe is that there is an air of hocus-pocus about it: something reminds one of the conjuror we who brings things out of a hat………So the reader if he has a robust sense of reality, will feel convinced that it is impossible to manufacture an infinite collection out of a finite collection of individuals, though he may not be unable to say where the flaw is in the above construction.’

the point is that a selection of individuals may be classified – in any number of ways

that is to say there is no definite description of anything

are we then to say there is an infinity of classes?

which is to say – an infinity of descriptions?

we might well be tempted to adopt such terminology –

therefore the question is – if there is no definite description is there an infinite number of descriptions - of any one thing – of any collection of things?

you see the trick here – and its crucial – its crucial to the whole of mathematics – to logic – to life itself – is to recognize what you don’t know

we cannot say in advance whether there is or there is not a limit to description

we just cannot say

the answer to such a question presumes a Spinozistic axiom – sub specie aeternitatis

that is the point of view of infinity – or as some have called it the ‘God’s eye view’

no amount of clever theoretical construction will get us to this height

but the result is not that we can therefore assume endlessness or infinity

it is that we cannot say

we cannot say because we do not have the vantage point required – and I would say such is logically impossible

the point being you cannot be inside reality and outside of it at the same time –

and there is no sense at all to the idea of being outside of reality

we may wish to know if there is a limit or not to things -

and for some the argument that we cannot is a source of woe

for me it is the ground of all wonder and creativity – I like it

be that as it may –

my argument is that the object of knowledge is the unknown

that the very reason for knowledge is the fact that reality is unknown

we make it known via our description and we do this in order to operate with it effectively

as there is no gold stand in human affairs – the issue is always alive

and for this reason we must continually describe and re-describe the world we live in

this does mean that the world is ‘infinite’ – or that it is finite – it is rather that it is undetermined


what is clear is that at the basis of this infinity argument of Russell’s is a confusion between the indeterminacy of description – which is the reality – and this fantasy of infinity – which is really just a misunderstanding of the unknown

and coming up behind this confusion is the mistaken belief that the class is some kind of ideal – real entity that reflexes infinitely into itself – therefore continually creates (infinite) reality

a class is an act –

any act is determinate – in the sense that its purpose is to determine –

the indeterminate

there is a natural end to this action – it’s called death

Russell goes on to introduce the issue of logical types

the necessity for some such theory he says results for example from the ‘contradiction of the greatest cardinal’

he argues that the number of classes contained in a given class is always greater than the number of members of a class

but if we could – as argued above – add together into one class the individuals, classes of classes of individuals etc

we should obtain a class of which its own sub-classes would be members

the class of all objects that can be counted – must if there be such a class – have a cardinal number which is the greatest possible –

since its subclasses will be members of it – there cannot be more of them – than there are members

hence we arrive at a contradiction


my view is there is no greatest cardinal – for there is no one classification that covers all possibilities – in which all possibilities are contained

the idea of such is really just the extension of the idea of order – to cover all possibilities

in real terms we only ever deal with parts of reality – sections – sequences

in a world with a greatest cardinal there would be no movement – no action – no mathematics

cardinals are class dependent – a cardinal is a description of a class –


there is no real sense to the idea of the class of all classes –

that is a classification of all classifications

such an idea is a misuse of class


another way of looking at it would be to say the class of all classes – is really a description of all descriptions

which is to say what? – that they describe

that is that a description is a kind of action

and in describing all descriptions – all you are doing is describing

or more technically – describing describing

which is simply – to describe

that is the description of all descriptions is an empty exercise


Russell goes on to say in considering this he came upon a new and simpler contradiction –

if the comprehensive class we are considering is to embrace everything – it must embrace itself as one of its members

if there is such a thing as ‘everything’ – then ‘everything’ is something and a member of the class of ‘everything’

but normally a class is not a member of itself

if we then consider the class of all classes that is not a member of itself –

is it a member of itself or not?

if it is – it is not a member of itself

if it is not – it is a member of itself


now in my opinion – this is the kind of mess you get into when you reify classes –

that is when you forget what you are doing is classifying – performing an action – the point of which is to bring things together – to create some order

even if you are to go with Russell’s metaphysics of classes –

the solution is obvious – a class is not a member of itself

which is to say the class is not one of the things classified – in the act of classification

this needn’t be put as another axiom of set theory – it is plainly obvious – if you understand – that is correctly describe what you are doing when you make a classification

the act is not that acted upon

to suggest so does result in incoherence


ok

Russell began this argument – in connection with the concept of ‘everything’

the class of everything is a member of itself

what is clear is that the description ‘everything’ – if is it is a description – is not closed

that is to use Russell’s terms – it does not ‘embrace’

it is of necessity an open concept

or if you like it is a non-definitive description

or in Russell’s terms it is an open class

which if it be so - is a different type of class – even a unique class

it is easy to see how some would say it is not a class – a classification at all –

just because it is a non-closed description

and the whole point of description as with class one would think is that it is closed

the other way to go is to say – well ‘everything’ – is not a class – is not a subject of description

that is everything but everything is –

the simplest point here is to say – well it’s a grammatical issue

we have a word here – which is as common as patty’s pigs – but it actually has no meaning

it refers basically to what cannot be classified or described

one thing is clear though – and this should be no shock to anyone – we do indeed need such a word

and if everyone who used it recognized that it is a word without meaning – then the world might be a better place

there’d be more dancing in the streets

what I am getting at really is that ‘everything’ and any other word of a similar logic – refers to the unknown

it is if you like a somewhat more interesting – perhaps more vital – description of the mystery

there really is no drama in including yourself as part of the greater unknown

the problem only occurs if you think you have a special place and a special status –
that you can in some sense stand apart –

and in response to this King Solomon was heard to say – all is vanity –

which if my analysis is right – is just to say the issue is open and never closed

everything is without bounds

after further discussion Russell has this to say –

‘If they are valid, it follows there is no empirical reason for believing the number of particulars in the world to be infinite, and that there never can be; also there is no empirical reason to believe the number to be finite, though it is theoretically conceivable that some day there might be evidence pointing, though not conclusively, in that direction.

From the fact that the infinite is not self-contradictory, but is also not demonstrable logically, we must conclude that nothing can be known a priori as whether the number of things in the world is finite or infinite. The conclusion is therefore to adopt a Leibnitzian phraseology, that some of the possible worlds are finite, some infinite, and we have no means of knowing to which of these two kinds of possible worlds our actual world belongs. The axiom of infinity will be true in some possible worlds and false in others; whether it is true or false in this world, we cannot tell.’


in essence this is the line of my argument – that we cannot know if the world is finite or infinite

after all the rubbish that has preceded in this book – I was more than surprised to come upon the above statement

as to the Leibnitzian argument of possible worlds – there is nothing to be gained by the attempt to give imaginative fiction the status of high logic

possibility in this context and its bastard children – possible worlds – are really no more than the attempt to dress up the unknown - and present it as something it is not –


© greg. t. charlton. 2008