Russell on mathematics XVI

Russell: introduction to mathematical philosophy:
descriptions


Russell begins by saying there are basically two kinds of descriptions – definite and indefinite

a definite description is a phrase of the form ‘a so-and-so’

an indefinite description a phrase of the form ‘the so-and-so’ – in the singular

we begin with the former –

consider the description – ‘I met a man’

what do I really assert?

it is clear that what I assert is not – ‘I met Jones’?

Russell says that in such a case – not only Jones – but no actual man enters into the statement

and he says the statement would remain significant if there were no man at all – as in ‘I met a unicorn’

he argues – it is only the concept that enters into the proposition

in the case of unicorn – there is only the concept

this he says has led some logicians to believe in unreal objects

probably the same lot that bang on about infinite numbers and classes

Meinong argued that we can speak about ‘the golden mountain’ and ‘the round square’ – and hence that they must have some kind of logical being

Russell’s view is that to say unicorns must have existence in heraldry or literature or the imagination is to make a pitiful evasion of the issue

he says –

‘In obedience to the feeling of reality, we shall insist that, that in the analysis of proposition, nothing “unreal” shall be admitted. But after all, if there is nothing unreal, how it may be asked could we admit anything unreal?’

his answer is that with propositions we are dealing firstly with symbols – and if we attribute meaning to groups of symbols that have no meaning we will end up with unrealities – in the sense of objects described

the first point I would like to make here is that it is rather artificial and frankly a little stupid to think that a sentence or proposition can be taken in isolation from its context and use – and regarded as significant

we do not operate with individual sentences in some kind of metaphysical void

to understand ‘I met a man’ or ‘I met a unicorn’ – or for that matter ‘I met Jones’ – one needs a lot more information

or one needs to assume a lot more than what is contained in the proposition

one could go so far as to say that to understand Tammy when she says ‘I met Jones’ you would need a complete analysis of Tammy’s use of that statement at that time –

and such of course would be to call for a complete understanding of the inherent metaphysics or world view of the speaker – at that time

now that is not about to happen – it is not even theoretically feasible

unless you think you have some indubitable like principles as the basis of your analysis

and to claim such I would submit is to talk rot

nevertheless when propositions are uttered by a speaker and received by a hearer much is assumed

you could say to cut to the quick - reality is assumed and within that any number of other secondary assumptions come into play

now what this actually means is that what is assumed is technically unknown – in the sense of a definitive analysis

but as I argued in the previous post in relation to propositional functions – this is the actual reality we are in and that we deal with –

we operate in the unknown

when I speak I assume some degree of definitiveness

when you hear me speak you assume some degree of definitiveness

this is not because I have logical grounds for definitiveness – or because you do

it is rather that without the assumption of definitiveness – we would not be able to assert
anything at all

and therefore not be able to communicate in language

so what I am saying is that in order to act propositionally – the assumption of defintiveness (of some degree) is necessary

and this necessity is no more than a practical necessity – the necessity to act

on this view all propositions are technically indefinite – but their form in practise is definite

one could be cynical and say well this suggests all language is fraudulent – or in common slang parlance say – every one is talking shit

strictly speaking this is correct

but given that there actually is no alternative – it becomes the gold standard

what I am saying is that ‘I met a man’ or ‘I met Jones’ or ‘I met a unicorn’ are all indefinite propositions – even when understood in some wider context of the user and the usage

it is just that I assume you understand what I am saying – and you assume that what I say is understandable

it is an assumption based on ignorance – but a necessary one

Russell it seems slips in and out of what he calls ‘unreality’ quite seamlessly and indeed elegantly

he has no scruples it appears in basing his philosophy of mathematics on the unreality of infinite numbers and classes – but baulks at unicorns

unicorns I would suggest have a better chance of making it

we need to get the bottom of all this -

we describe in order – and only to deal with – to get a handle on – the unknown

and if you accept this you will see that any description is no more than a shot in the dark

but that is where we are – and that is what we have to do

so that’s Kansas Toto

Russell seems to think that because we have as the first cab off the rank – objective language – language that refers to objects

an object world is what we have – and there is no where else to go

and this for Russell is the object world of common sense and perhaps science

the object language is the starting point simply because it has proved to be so successful

and by this I mean humans have been enabled by such a platform

nevertheless though – it is only a platform – and it is not successful or useful in all circumstances

we ask ‘what do you mean by that? when the simplicity of object language seems not to be up to the job

and here – it is not the nature of things that is being questioned – rather the appropriateness of the description – in a particular context

the nature of things for human beings is a function of description – which of course is a function of need

clearly ‘I met a unicorn’ – is a statement that though it appears to be an assertion describing an object in the physical world – is a statement describing something else

and it is all very well for Russell to dismiss other ways of describing as ‘pitiful’ and ‘paltry’

but what is behind Russell’s view is that there is only one way to describe the world – and further that language can be taken out of its context and use and regarded as some kind of specimen under a microscope

the ‘pitiful and paltry evasion argument’ – is actually no argument at all

it’s the kind of comment that might be made when some one doesn’t like a point of view – but doesn’t want to address it – just in case it might turn out to be on the money

and there goes the neighborhood

Russell doesn’t actually address the possibility of the indefiniteness of all description

and he doesn’t seem to get that we have developed alternative ontologies simply because the starting point – is just that – a starting point

objective – as in physical object description is most useful – and clearly we couldn’t get on without it – but the actual reality of human behaviour shows it is not taken as universally applicable – never has been

because physical object language has been so useful – the fact is we often describe in its terms – when even a preliminary analysis shows it is not what is required

which is to say physical object language cast a long shadow – and most of the time we are quite happy to play in the shadow – knowing full well that other players understand this

I think the hidden truth of human beings is that they know that their humanity is based on not knowing

human beings have developed alternative ways of seeing the world and of describing it because they have needed to – and that’s the end of it

if you are going to operate ‘in obedience to the feeling of reality’ – then you ought to have a look at what’s going on – and has been since the beginning of recorded history

still I don’t want to be too hard on the old boy – at the time he was trying to ‘describe descriptions’ – he was doing a stretch at Brixton

prison can do that to you

Russell goes on to consider definite descriptions

‘We have two things to compare: (1) a name, which is a simple symbol, designating an individual which is its meaning, and having this meaning in its own right independently of the meaning of all other words; (2) a description, which consists of several words, whose meanings are already fixed, and from which results whatever is to be taken as the “meaning” of the description.’

a name does identify – it is an identification act

a description – makes known the identification – it is an act on the act of identification

we operate in description –

the world as known is the world described

our descriptions are the platform for our actions

descriptions are in that sense meta-actions

they are what enable us to proceed – to act on –

in propositions where names occur – as in ‘Scott is the author of Waverley’ – you have a neat example of the logic of descriptive behaviour

for essentially what you have here is an identification (‘Scott’) described (‘is the author of Waverley’)

Russell says the name designates an individual – which is its meaning – and that it is a simple symbol

a name does designate – true – but what does it designate?

in my view what it designates is a particular unknown

it is an act that is designed to focus attention – focus consciousness on a particular

or you could even say it is an act that particularizes

granted as a matter of course we are aware – conscious of particulars –

and in general we operate in a world already – and well described – so in most cases our particulars come with description

but to get to the bottom of this we need to look at the logic of the situation – and this requires that we make a step back from the obvious

I would suggest that the act of naming singles out a particular

and it singles it out for description –

in the proposition ‘Scott is the author of Waverley’ – we have the name described

that is to say a bare particular is singled out and then given some clothes

Russell argues the meaning of the name is the individual designated

the view I put is that the name is empty

and what I mean by that is that the name is a description place

that the name identifies an unknown – and is then the place for description –

it is the name that is then described – or if you like – made known

the act of description – gives the name meaning

which is really just to say – it makes the name functional – that is it makes it active

so when we talk about meaning – what we are talking about is not some inherent quality that some propositions have and others do not –

rather what it amounts to is making symbols functional

meaning is about ‘getting on with it’

you could then say well what you have is symbols (words) making symbols operational

yes – this is essentially it

and the symbolic platform so created - becomes a basis for physical / mental action

you might ask how is it exactly that symbols make symbols functional?

in the case of ‘Scott is the author of Waverley’ what you essentially have is a decision to make one set of symbols ‘is the author of Waverley’ function in place of ‘Scott’

in principle this substitution could go on indefinitely

the point of all such propositions is to make the original identification functional (known)

one might be tempted to argue that there is a logical relation between any such set of propositions – i.e. that the last proposition in the series ‘contains’ or entails all that came before

no doubt with a bit of patience it could be written up like that

but no –

the point is that each proposition serves its own purpose

and each purpose would or could itself be the subject of indefinite description

there is no doubt that we seek definite descriptions

however the reality is not that we find them

it is rather that we make constructions that appear to be definite

and the appearance is what we run with

for in non-reflective action we need the illusion of the definite

and we need non-reflective action to function and survive

language is a very functional platform – and the fact that it creates or enables the illusion of definitiveness is its principle function

‘Scott is the author of Waverley’ is a proposition which analysed correctly shows that a particular is identified and given a description

a particular is only made known through description – through some description

‘Scott’ identifies the particular – or to be more precise – it marks the particular for description

and the whole point of description is just to make the unknown – supposedly known

which means setting up a structure so that the particular named or described can be functional

just because the particular in itself – the original state of things is in itself unknown - (which is the reason for description) there cannot be a definite description of it –

there is no definite description of any feature of the world or our experience of it

nevertheless we must and do proceed as if there is

we operate in illusion – and this is necessary given the reality we face

it is the fact of consciousness in the world - consciousness facing the unknown



© greg. t. charlton. 2008.

Russell on mathematics XV

Russell: introduction to mathematical philosophy:
propositional functions


Russell begins here with a definition of ‘proposition’-

he says ‘proposition’ should be limited to symbols – and such symbols as give expression to truth and falsehood –

much would depend here on the definition of symbol – and one’s basic idea of truth and falsehood

by symbol – could we not mean any descriptive act?

of course such would include the propositions of ordinary language – but would it not by definition include other artistic creations – poetic expression – and any act of visual art i.e. painting – sculpture architecture etc.– and perhaps even acts of gesture? etc.

so it depends how much you want to let into ‘symbol’ –

and ‘truth’ – to cut quickly to the chase I see it as assent – and falsehood – as dissent –

really just a jump to the left or a jump to the right

and of course acts of assent and dissent can take on any number of forms – any number of expressions

I favour the idea that a proposition is a proposal – of whatever kind or form

and in the most general sense it is a proposal ‘of a state of affairs’

now any observer of such a proposal can give their assent to the proposal – or can dissent from it

that is they can affirm it – or deny it

so a proposition is a proposal that can be affirmed or denied –

is capable of being affirmed or denied

in normal parlance – it would seem to be of the nature of a proposition (proposal) that it can be affirmed or denied in some manner of speaking

a visitor to an art gallery whose response to a work of abstract art is broadly speaking one of approval – has affirmed the proposal

the same proposition in the shape of abstract expressionism can be ‘denied’ by the very next observer

perhaps if analysed such a response would mean something like ‘I don’t agree with how the world is portrayed in this painting’

anyway

‘propositional function’ is defined by Russell as an expression containing one or more undetermined constituents – such that when the values are assigned – the expression becomes a proposition

it is a function –whose values are propositions

or as he also describes it – ‘a mere schema, a mere shell, an empty receptacle for meaning, not something already significant.’

an example –

‘x is human’ is a propositional function

as long as x remains undetermined it is a propositional function – it is neither true nor false

but when a value is assigned to x it becomes a true or false proposition

I like propositional functions – but I think for reasons quite different to Russell

the beauty of a propositional function in my terms is just that it is a function with undetermined values

‘undetermined values’ here means unknown values

and the point of the propositional function is that it shows that function is not dependent on determination – on knowing

which is to suggest that function is quite independent of knowledge

I think it is even possible that Russell might agree to this view of things – in a limited way

I think that the propositional function really points to the basis of logic in scepticism – and much as Russell was known for his sceptical frame of mind – I doubt that he would have ever conceived of such a notion

the propositional function is a proposal – in the absence of determination – of knowledge –

nevertheless a proposal

Russell wants to distinguish sharply between a propositional function and a proposition

and this is where the definition of – or one’s understanding of - the nature of proposition is relevant

if as I have put – a proposition is any proposal that can be asserted or denied – what then of a propositional function?

Russell as I noted distinguishes proposition and propositional function – in terms of truth function

the proposition can be regarded as true or false – but not the propositional function?

is that so?

that is in the example above ‘x is human’ – while x is left undetermined – as an unknown – a proposal is put –

and it is the proposal that there is something that can be described as human –

and it is a proposal that can be regarded as true or false -

now you might wonder how could it be rationally denied?

under what conditions could such a statement be false?

this matter only depends on one’s definition of ‘human’

i.e. it is conceivable for instance that in the future with developments in genetic engineering and or bio-technology that the classification ‘human’ could be regarded as obsolete

in such a circumstance it could well make sense to regard the statement ‘x is human’ as no straightforward matter – and quite possibly false - either in general or in relation to certain classifications of ‘species’

so in such a case even though x is undefined – ‘human’ is up for grabs

this is not perhaps the best example to take of propositional functions

a more interesting case is one Russell goes on to consider ‘all A is B’

Russell says ‘A and B’ have to be determined as definite classes before such expressions becomes true or false

but is that so?

‘all A is B’ is a proposal for identity

such a principle or a version of such is required for arithmetic – calculation depends on the assumption that the left and right hand sides of the ‘=’ sign are equivalent

however in other contexts it is not so straightforward -

can you i.e. apply it in philosophy of mind?

i.e. are all sensations brain processes?

so the question is really about the appropriate application of such a propositional function –

it is clear that in some contexts such a propositional function – does function – has value

in other contexts – its status is uncertain

the point is – it is a proposal for relating one class to another in a certain manner

even that ‘certain manner’ can be a question – that is the ‘is’ in ‘all A is B’ is not uncontroversial – it can have a number of meanings

the propositional function even though its values are indeterminate – is not a statement without meaning or significance

one needs to accept it as a proposition – for the determined propositions to follow

so it can be regarded as true or false

it is quite extraordinary that in the twentieth century – and I suspect even in this century – logicians have seriously put that the propositions of their activity are not subject to truth conditions –

the absurdity of it is quite staggering

do they seriously suggest that the propositions with truth value are derived from propositions with no value?

a more cynical view might be to suggest that they find security in not subjecting their own propositions to the question of truth value

perhaps it is just that logic – main stream logic has never got past Plato

anyway such a view of logic of propositional functions is deluded nonsense


my overall point is that the propositional function is a proposal – is a proposition

the issue is really all about function

in my view a propositional function – asserts function

and the proposition (in Russell’s terms) – is a function asserted – meaning the values are declared – the ‘variables’ determined

now as I have just argued – the propositional function does not exist in metaphysical empty space – its validity depends on its epistemological context

so it is true or false – but to see this you need to be able to look to its use – and the context of its use

both the propositional function and the functioning proposition are proposals

and in an even more general sense they are propositional acts

to understand an act you need to understand its context – or at least make start in that direction – get an idea of it


so finally in relation to propositional functions -

the variable in a propositional function is an unknown value

the fact of the propositional function shows us quite clearly that we can and do function with unknowns

that is the fact of the unknown value does not prohibit function

the function in a propositional function – is the act proposed – and the value of the act is unknown

it is on this foundation – the unknown – that all ‘determined’ propositions rest – it is their ground and source

if to be is to be the value of a variable

and the variable qua variable is unknown

to be is to be the value of the unknown


© greg. t. charlton. 2008.

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

Russell on mathematics XIV

Russell: introduction to mathematical philosophy:
incompatibility and the theory of deduction


by ‘incompatibility’ Russell means that if one proposition is true the other is false

this is obviously a form of inference

it is the incompatibility of truth values

he notes that it is common to regard ‘implication’ as the primitive fundamental relation that must hold between p and q if we are to infer the truth of q from the truth of p – but says for technical reasons this is not the best primitive idea to choose

before coming to a view on the primitive idea behind inference he considers various functions of propositions

in this connection he mentions five: negation, disjunction, conjunction, incompatibility and implication

first he puts forward negation – ‘~p’

this is the function of p which is true when p is false and false when p is true

the truth of a proposition or its falsehood is its truth value

next he considers disjunction – ‘p or q’

this is a function whose truth value is true when p is true and when q is true – false when both p and q are false

conjunction – ‘p and q’ – its value is true when both propositions are true – otherwise it is false

incompatibility – i.e. when p and q are not both true – this is the negation of conjunction

it is also the disjunction of the negations of p and q i.e. ~p or ~q

its truth value is true when p is false and when q is false – it is false when p and q are true

implication i.e. ‘p implies q’ or ‘if p then q’ – that is we can infer the truth of q if we know the truth of p

all five have this in common – their truth value depends upon that of the propositions which are their arguments

a function that has this property is a truth function

he says it is clear that the above five truth functions are not independent – that we can define some in terms of others

Russell chooses incompatibility as the indefinable

incompatibility is denoted by p/q

the next step is to define negation as the incompatibility of a proposition with itself – i.e. ~p is defined as p/p

he then goes on to define disjunction implication and disjunction in this manner

but the first step needs to be looked at carefully

that is negation as p/p

now Russell has put the idea that negation is the incompatibility of a proposition with itself

clearly what this presumes is that incompatibility is a relation –

and clearly this is so

the point though is that a relation here holds between propositions – it is propositions that are incompatible

and this is what is put forward in connection with disjunction implication and conjunction

for clearly disjunction implication and conjunction – are relations between propositions

so the idea of incompatibility on the face of it can be applied to these relations – just because they are relations

but negation?

negation is not a relation between propositions

and more to the point – negation is not a relation

secondly propositions have relations with other propositions

that is the only way in which a relation can exist – between propositions

a proposition does not have a relation with itself

it is not possible for a proposition to ‘have a relation’ with ‘itself’

for there is no ‘itself’ to a proposition

a proposition does not have a self – that it can relate to

God knows what the ‘self’ of a proposition is supposed to be

this idea of a proposition having a relation with itself is just nonsense

a proposition in the broadest sense of the term is a proposal

and as to proposal – in the broadest sense of the term again – it is an action

to negate a proposition is to deny it

that it is to say ‘it is not the case that p’

it is to determine the proposition negatively

if you begin in an argument with p

and then assert ~p

the assertion of p is one action

the assertion of the negation of p is another

yes these two propositions can be related

but the second one – the negated proposition

does not have a relation with itself

it is in every sense a separate proposition

the assertion of a proposition and the negation of a proposition are two different logical acts

the upshot of this that Russel’s theory of incompatibility collapses

incompatibility cannot be applied in the manner he wishes to apply it

and for this reason his account of incompatibility as the primitive idea of inference cannot go forward

Russell says of incompatibility it will be denoted by p/q

negation is p/p – disjunction is the incompatibility of ~p and ~q i.e. (p/p) / (q/q)

implication is the incompatibility of p and ~q i.e. p / (q/q)

and conjunction the negation of incompatibility i.e. (p/q) / (p/q)

so in all but conjunction propositions are rendered incompatible with themselves

and in the case of conjunction what you effectively have is the incompatibility of incompatibility

i.e. – incompatibility is incompatible with itself

this rendering of the various types of inference in terms of incompatibility makes the notion of inference incomprehensible

it brings inference to a dead halt

why incompatibility?

Russell italicizes ‘truth’ in his statement ‘…..it seems natural to take “implication’ as the primitive fundamental relation, since this is the relation that must hold between p and q, if we are able to infer the truth of q from the truth of p.’

now he rejects implication as the primitive

is this because he thinks that implication only applies when the truth value is true?

that is he rejects it on the grounds that it does not apply when the value is false?

it does seems clear that he regards implication proper as only applying in the case of where the issue is truth

and yet at the same time he calls for the ‘widest sense’ of the term.

now the problem with this view is that it ties implication – it ties inference - to truth value

it says only given these truth conditions does this inference occur – or can occur

this to my mind confuses and conflates truth conditions and inference

or to put it another way an inference is a logical act – that is made or can be made regardless of the truth conditions of the propositions involved

and so I would put that we can use implication just as well when the subject is falsity as when it is veracity

there is not a problem with if p is false q is true or if q is true p is false

the general point is that inference – the logical act of inference – is independent of the question of truth value

Russell’s mistake with implication was to limit it to inferences where the only value is truth

to account for falsity in implication he came up with incompatibility

now as I have argued the idea of a proposition being incompatible with itself makes no sense

and furthermore it is not necessary to entertain this concept if truth value is not tied to inference

this is not to say the two cannot be formally related – for this is the issue of validity or invalidity

there is also a more general point to be made about Russell’s incompatibility thesis –

the idea is to find a primitive truth function in terms of which the other truth functions can be derived

the fact of it is though that incompatibility is not on the same logical level as conjunction disjunction and implication

it is clearly a derived truth function

the use of negation in its formulation indicates it is a secondary construction

now straight up – a secondary construction by definition will not serve as a primitive

that is it will always be shown to be reducible – and for that reason fail as a primitive

the question then - is implication the primitive that Russell was seeking?

now that we have removed incompatibility from the equation does implication do the job?

that is can we translate conjunction – disjunction and incompatibility into implication?

we can indeed –

conjunction - if p and q are true the inference is true – if either p or q is false the inference is false

disjunction – if p or q is true then the inference is true – but if p or q are false – the inference is false

incompatibility – if p or q is false the inference is true – if p and q are true – it is false

and the great advantage of the form of implication is just that it really does make clear the separation of inference and truth value

that is it quite literally leaves the question of truth value up in the air

and there is a real intellectual honesty built into implication – the issue of truth and falsity is in the inference left undecided

that is we can make the inference without necessarily knowing the values

it is beautiful in the sense that we can infer without hesitation in a state of uncertainty

in fact the state of uncertainty becomes and is the ground of inference

in logical terms this action demands that it is performed without prejudice

on this view – what is primitive to inference is uncertainty

that is once you make the move to implication as the general form of inference – uncertainty is revealed as the ground of inference

this I think injects health into logic – puts life into it

certainty is a corpse

however it must be remarked that such a view is at odds with standard or given view of deductive inference

Russell says:

‘In order to be able validly to infer the truth of a proposition, we must know that some other proposition is true, and that there is a between the two a relation of the sort called “implication”, i.e. that (we say) the premise “implies” the conclusion.’

my argument is that in implication the truth values of the propositions are conditional and are conditional in relation to each other

and the real point of this is that in implication per se nothing is decided in terms of truth value

when we imply we are effectively leaving open the question of truth

Russell’s argument above is that to infer the truth of a proposition we must know that some other proposition is true

but this I think is wrong

it is not that we must know – it is rather if p is true – then q is true

here the truth of p is an open question

now if you accept the view that deductive inference is implication – and that all forms of deductive inference can be seen as instances of implication

then deductive inference does not depend at all on the truth value of the propositions

rather it only depends on the possibility of truth value

now on such a view of deductive inference – it would seem that validity is never at issue

or to put it another way a conditional argument is neither valid or invalid

what I am getting at in general is that what logic does is not provide us with knowledge – what is does is spell out the conditions for knowledge

and the basis of conditional arguments is uncertainty


© greg.t. charlton. 2008.

Russell on mathematics XII

Russell: introduction to mathematical philosophy:
selections and the multiplicative axiom


Russell argues –

the problem of multiplication when the number of factors may be infinite arises in this way -

suppose we have a class k consisting of classes

suppose the number of terms in each of these classes is given

how shall we define the product of all these numbers?

if we frame the definition generally enough – it will be applicable whether k is finite or infinite

the problem is to deal with the case where k is infinite – not with case where its members are

it is the case where k is infinite even when its members may be finite that must be dealt with

to begin let us suppose that k is a class of classes – in which no two classes overlap

say e.g. electorates in a country where there is no plural voting

here each electorate is considered to be a class of voters

now we choose one term out of each class to be its representative – as i.e. – when a member of parliament is elected

in this case with the proviso that the representative is a member of the electorate

we arrive at a class of representatives who make up the parliament

how many possible ways are there to choose a parliament?

each electorate can select any one of its voters – and if there are u voters in an electorate - it can make u choices

the choices of the different electorates are independent

when the total number of electorates is finite – the number of possible parliaments is obtained by multiplying together the numbers of voters in the various electorates

when we do not know whether the number of electorates is finite or infinite –

we may take the number of possible parliaments as defining the product of the numbers of the separate electorates

this is the method by which infinite products are defined


my thoughts are –


if we don’t know whether the number of classes (electorates) is finite or infinite – then quite simply and straight up we don’t know

whether they are infinite or not is not the issue – the issue is that we don’t know the number

now in such a case we cannot know the number of possible parliaments –

for in terms of the above argument – the number of possible parliaments depends of the number of electorates


the fact is you cannot multiply the unknown and expect its product to be known


Russell introduces possibility here as a something like a ‘known unknown’

it’s a trick to get past the fact that there are no infinite classes

the fall back position appears to be possible classes – and the idea is that possibles have numbers

which is really no more than to say the unknown has a number

if Russell was to accept this argument he would have to accept that mathematics is right back to square one – where you start – with the unknown


Russell goes on –


let k be the class of classes – and no two members overlap

we shall call a class a ‘selection’ from k when it consists of just one term of each member of k

i.e. u is a ‘selection’ from k if every member of u belongs to some member of k and if 'a' be a member of k – u and k have exactly one member in common

the class of all ‘selections’ from k we call ‘the multiplicative class’ of k

the number of terms in the multiplicative class of k - i.e. the number of possible selections from k is defined as the product of the members of the members of k

the definition is equally applicable whether k is finite or infinite


in response –

the first point is that this notion of class of classes -

a classification of all classifications

is what?

it is nothing -

we can ask – is the class of classes – a member of itself?

as many have –

my point though is that there is no sense to the idea of being – a member of itself

a classification is an action – you can represent it as an enclosed entity – but this is logically speaking a misrepresentation

something of the picture theory of the proposition seems to operate here


anyway –

to this notion of ‘selection’

this is a purely arbitrary devise designed to give the impression that we can operate with infinite classes

that is that we can make a selection – and operate with it as if it is definitive

my general argument is that there is no such thing as an infinite classification

a classification is closed – infinity is not – the two concepts cannot go together – without contradiction

and really – the truth be known a ‘selection’ cannot be made – for what is there to distinguish in infinite classes?

and if there is no distinction – there is no ground for ‘selection’ -


‘the product of the members of the members of k’ – is the multiplicative class of k

what you have here is a statement of the multiplication principle in a context where it cannot make any sense

the statement of the principle is ok – but it has no application in the world of infinite classes

this is no argument against the principle

rather it is an argument against its misapplication


the point that comes out most clearly for me is that the attempt to apply the multiplication principle in the (imaginary) context of infinite classes – shows quite clearly just how useless the is whole idea of infinite mathematics is -

it doesn’t work – and using various devises to prop it up – only results in demonstrating its impotence – and showing that it is not worthy of genuine mathematical intelligence



© greg. t. charlton. 2008.

Russell on mathematics XI

Russell: introduction to mathematical philosophy:
limits and continuity of functions


Russell is here concerned with the limit of a function (if any) as the argument approaches a given value

and also what is meant by continuous function

the reason for their consideration is that through the so-called infinitesimal calculus – wrong views have been advanced

it has been thought ever since Leibnitz that differential and integral calculus required infinitesimal quantities

Weierstrauss proved that this is an error

limits and continuity of functions are usually defined involving number

this is not essential as Whitehead has shown

consider the ordinary mathematical function fx – where x and fx are both real numbers – and fx is one-valued –i.e. – when x is given there is only one value that fx can have

we call x the argument – and fx the value for the argument of x

when a function is ‘continuous’ we are seeking a definition for when small differences in x – correspond to small differences in fx

and if we make the differences in x small enough – we can make the differences in fx fall below any assigned amount

the ordinary simple functions of mathematics have this property - it belongs to x², x³,…….log x, sin x, and so on

for discontinuous functions consider the example – ‘the place of birth of the youngest person living at time t’

this is a function of t - its value is constant from the time of one person’s birth to the time of the next birth

and then the value of t changes suddenly from one birthplace to another

a mathematical example would be ‘the integer next below x’ – where x is a real number


Russell’s argument is that there is nothing in the notions of the limit of a function or the continuity of a function that essentially involves number

both can be defined generally

and many propositions about them can be proved for any two series – one being the argument series – and the other the value series

the definitions do not involve infinitesimals

they involve infinite classes of intervals – growing shorter without any limit short of zero

but they do not involve any limits that are not finite

this is analogous to the fact that if a line an inch long is halved – then halved again – and so on indefinitely

we never reach infinitesimals this way

after n bisections – the length of our bit is ½n of an inch – and this is finite – whatever finite number n may be

infinitesimals are not to be found this way


ok – just a few thoughts -


infinite classes of intervals?

what you have is repetitive action that is progressive – in the case of a continuous function – the progression is continuous – in the case of a discontinuous function – it is discontinuous

hence – as Russell goes to quite a lot of effort to show – continuity (and discontinuity) are attributes or descriptions which are determined by the relations within a function

his definitions of continuity are really no more than second order descriptions of what occurs in various types of continuous function

the limit of such progressions is an issue of contingency – that is the possibility of performance

such a limit cannot be set in advance – or in concrete as it were –

the question of operation is an open question

it will depend on the state of the science of the day – in practise this means the theory of technology and its practise

so we cannot in advance assume that an operation is finite in the sense that it comes to a natural end of action –

you just have to see in practices what happens – and what in a predictive sense is possible

we can discount infinite operations as such - just in terms of the finite capabilities of human beings

infinity here – or the infinite performance of an operation – is really no more than keeping an open mind on contingent possibilities –

in general we can say the limit of a function and /or the continuity of a function is in any final sense unknown

practise determines these conceptions and the matter is finally undetermined

the point of contingency is just that it is undetermined – that its possibility is unknown


Russell is correct in dismissing infinitesimals

however his argument of infinite classes of intervals is wrong headed

first up the idea of infinite classes is based on a logical error

a class is a classification – it is an action – it is not an ideal entity – despite the fact that we characteristically speak of it in substantial terms –

this is no more really than a problem of grammar

the argument for infinity in this context is the argument of reflexivity –

the idea that a class can ‘reflex into’ itself

and such an idea presumes that a class has a dimension that is ‘itself’

what is to be meant by ‘self’ in this context?

clearly - a class x¹ within a class x – that is identical to x

this presumes the relation of identity

a relation exists if it exists between unique – that is distinct entities

there is no such thing as the relation of identity

an entity is not identical with itself – and not identical with an other thing

identity is a false relation

this is not to say that we can’t speak of equivalence in a mathematical sense

a classification of entities which has 10 members – can be regarded as equivalent to another classification that has 10 members – in terms that is of its membership number

but in such a case there is no question of the identity of entities

as I see it the great beauty of mathematics is that it enables a simple an elegant language of relations via number theory - that it completely dispenses with such questions as that of the substance of entities -


mathematics has really nothing at all to do with substance issues - it is the language of activity

and to my mind the theory of classes – and of infinite classes and numbers that Russell endorses and develops – brings the activity to a dead halt –

the reason being that such a theory of mathematics is really based on scholastic metaphysics – i.e. notions such as identity and self identity – which to my mind have no place in mathematics to begin with


a classification being an action – even if we were to hold with some metaphysical theory of identity – it is hard to see how it could be applied to classes

also let’s be clear about reflexivity –

reflexivity – if it is to mean anything is an action –

the idea that anything reflexes into itself presumes that the entity is active

that is that it performs actions

a class is an action – but it is as it must be an action performed

the result of such an action – i.e. the collecting of things together – does not go on to perform actions

which is just to say that an action – to be an action has a natural terminus

reflexivity is supposed to be the action that enables infinity – it presumes ‘self’ –and is apparently an action that no one actually performs

and further is not performable

not really a good bases for a theory of mathematics


so to get back to Russell – there are no infinite classes – and therefore no infinite classes of intervals


© greg. t. charlton. 2008.

Russell on mathematics X

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.

Russell on mathematics IX

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.