RUSSELL ON MATHEMATICS

 1. The Series of Natural Numbers

 

Peano’ argument


Peano showed that the entire theory of natural numbers could be derived from three primitive ideas and five primitive propositions in addition to those of pure logic

the three primitive ideas in Peano’s arithmetic are:

0, number, successor

the five primitive propositions are:

(1) 0 is a number
(2) the successor of any number is a number
(3) no two numbers have the same successor
(4) 0 is not the successor of any number
(5) any property which belongs to 0, and also to the successor of every number which has the property, belongs to all numbers


some preliminary thoughts –

by ‘number’ I mean ‘an operation or action in a series of actions’

the mark ‘1’ is the first in the series

‘the first in the series’ is defined by ‘series’ itself

the actual mark ‘1’ is a convention –

0 on this view is not a number – 0 is non-action – it the position prior to a series

it defines the series by marking the position of no series

the point being that a series ‘comes into being’ – there are thus no natural series

a series is a construction on action

succession is – relative to the series – a repetition of action

a mathematical series is thus a repetitive series

repetition is succession in time

 

in a repetitive series the successor of an action is an action

 

no two actions have the same successor because each action is unique in time

0 is not the successor of any number

any property of 0 is not shared by any number


on this view a number is just the mark of an action in a series of actions

not all actions are numbers of course –

but all numbers are actions in a series – in a conception

‘series’ thus is an ideal construction placed on action to create order

and order here is defined as repetition

repetition is the most basic ordering – and it is on this basis that the series of natural numbers emerges

there are no numbers in the Pythagorean sense of ideal entities

the term ‘1’ or the mark ‘1’ has universality because any repetitive series has a first action

0 if you were to be metaphysical or poetical is the ‘place’ of no action relative to any series

mathematics as a theory – or the mathematics of the series of natural numbers – is the formalization of the notion of a repetitive series

this is the point of Peano’s axiom 5

it is really a pragmatic theory in the sense that it provides us with terms that enable us to operate without actually performing every action that the terms name

numbers as marks refer to places in a series

so that whatever the series – we know that 501 – refers to a particular place in that series

we can therefore say numbers are ‘places in a series’ – relative of course to other places

and that these places are finally no more than actions performed in the series


Russell says of Peano’s three primitive ideas –

that they are capable of an infinite number of interpretations

he gives this example – let ‘0’ be taken to mean 100 – and let ‘number’ be taken to mean the numbers from 100 on in the series of natural numbers – then all our primitive propositions are satisfied – even the fourth – for though 100 is the successor of 99 – 99 is not a ‘number’ – in the sense we are now giving to number

he gives other examples – the point is that in Peano’s system there is nothing to enable us to distinguish between different interpretations of his primitive ideas –

that is it is assumed that we know what is meant by '0' – and we shall not suppose this symbol means 100 or Cleopatra’s needle


on the face of it – this seems like an ok argument –

however Peano’s clear intent is to distinguish 0 – 0 is a number and it is not the successor of any number – unlike 100 – which like all numbers except 0 is a successor

so for Peano anyway there is no question that 100 can be 0

that possibility is excluded by definition –

0 is 0 – 100 is another number

so Russell’s argument is clever but it is not true to Peano’s definitions


the real problem I think – and one which Russell at least to this point does not address is the relation between axiom 2 and 4

the successor of any number is a number – 0 is not the successor of any number

this to me goes to the problem of defining 0 as a number

at the very least you end up with 0 as some special case number

and as a result your definition of number – whatever that might be – is problematic

for it is a definition that is not all inclusive – but one that has an exception

and so really – it fails


Russell goes on to say –

‘0’ ‘number’ and ‘successor’ cannot be defined by Peano’s definitions – and that they must be independently understood

he says it might be suggested that instead of setting up ‘0’ and ‘number’ and ‘successor’ as terms we know the meaning of although we cannot define them –

we might let them stand for any three terms that verify Peano’s axioms –

they will no longer have a meaning that is definite  –

they will be variables – terms concerning which we make certain hypotheses – namely those stated in the five axioms – but which are otherwise undetermined

Russell says of this view – it does not enable us to know if there are any sets of terms verifying Peano’s axioms

and we want our numbers to be such as can be used for counting among common objects

and this requires that our numbers have a definite meaning


this is just to say if the three primitive ideas are regarded as variables then they will not have definite meanings

and of course you could then wonder what the actual value of any such terminology would be

his second point here that it does not enable us to know if there are any sets of terms verifying Peano’s axioms – is a strange argument

couldn’t you say just this of Peano’s argument as it stands?

it depends how you come at it – if you begin with the axioms – your question might be well – ok – we have these principles – but how are we to know what they refer

to ?

what I am getting at is that you could be quite sceptical and ask – how are we to know that the terms ‘0’ ‘number’ and ‘successor’ verify the axioms

just because they are used is not verification

anyway


the basic problem with Peano’s approach as I see it is firstly that he wishes to define 0 and number – and he defines 0 as a number

the result of this is that elucidation of 0 is now dependent on the definition of number

that is a number – 0 – has already been singled out - without the ‘over riding’ definition of number being in place

so you could take the view that nothing has been accomplished by introducing 0 – it needs to wait until number is defined

you might then argue that Peano has failed to define 0

and another thing – 0 is not on the same level as number and successor

it is not primitive in this scheme – it is if anything derivative –

derivative that is from number

now I don’t know – but I suspect Peano did not envisage this implication

I think it probably undercuts his theory


‘successor’ is defined as ‘the next number in the natural order’

clearly then for the definition of ‘successor’ to proceed we need the definition of ‘number’

that is the integrity of the notion of ‘successor’ is dependent on that of number

so is ‘successor’ like 0 dependent on number?

and if so like 0 it is a derivative notion?


also defining ‘successor’ as ‘the next….’ is really not to give us anything at all –

it amounts to saying ‘a successor is a successor’

perhaps the point is that we should just focus on this notion as a key notion in the philosophy of mathematics


one thing we can say is that ‘successor’ is a relational term –

and this seems to me to be how Peano goes about defining number -

a number is that which is the successor to a number

is this to say – numbers are ‘points’ in a succession?


it is pretty clear that Peano’s three primitive ideas – 0 – number – successor

are not given clean – ‘stand-alone’ – definitions

and I can’t see how you could say on the basis of Peano’s definitions that any of these notions are ‘primitive’

0 depends on number – number depends on successor - successor is a relation between numbers

 

what this is suggestive of – is that the idea that there are primitive notions in mathematics – that do no not depend on the meaning of the notions that derive from them – is not on

which is to say the quest for the foundations of mathematics is wrongheaded

that there are no foundations

what you have is an activity – perhaps a primitive human activity – and the description of that activity – the language of that activity – has given us working concepts – the meaning of which is not validated by analysis – but by the activity itself

this might suggest that the ‘foundations of mathematics’ – are not stable – that the activity of mathematics itself – wherever that is – will have repercussions on the concepts that are regarded as central or ‘foundational’

to be blunt – if you want to know what a ‘number’ is – look at what people do when they operate with numbers


you would have to say Peano leaves ‘number’ undefined – perhaps that is his sense of primitive

but really this is just a touch of the old Pythagoreans – what you might call a persistent mathematical malady


2. Definition of Number


Russell says – defining number should not be confused with defining plurality

number is what is characteristic of numbers – as man is what is characteristic of men

a plurality is not an instance of a number – but of some particular number

a trio of men is an instance of the number 3 – and the number 3 is an instance of number

the trio is not an instance of number

my view is that numbering is the act of marking instances of a plurality

we follow conventions in doing this – that is established practises – and that means an established terminology

the act of counting is repetition with specific identification of instances – it is a progressive act                                                                                                                                   

                                                                                                                                    

each instant is marked as distinct i.e. by the marks ‘1’ ‘2’ ‘3’ etc.

 

 

these marks – are syntactical conventions - conventions not of the instances – but of the act

the act of counting is not particular to any circumstances – it is an act that has general application

as a matter of established practise we name these marks ‘numbers’

the term ‘number’ is thus a general term that refers to the marks made in the act of numbering


Russell goes on to say a number is something that characterizes certain collections – namely those that have that number

we can ask what is it to ‘have that number’?

how does a collection have a number?

firstly what is a collection?

clearly it is a conception

and I would say a conception is an ideal means of organizing individuals

anything can be ‘collected’ – anything can be a member of a collection

the reason for the organization – the collection – depends on other considerations

but essentially it is about what a collection is to be used for

when we speak abstractly about a collection – we refer to its members

an individual is a member of a collection – for the reason of that collection

that is the reason for the collection – what is it is designed for – is the reason for membership

on this view a number does not characterize a collection

the purpose of the collection is what characterizes it

the fact that one collection has 5 members and another has 5 actually tells us nothing of the character of the collections

that they are numerically the same is of no consequence

a numerical characterization of a collection simply gives us the number of its membership

 

it is really just a quantitative description of the collection

and yes trivially that does distinguish it from collections of another number

but it does not distinguish the collection in terms of its reason


Russell says a class or a collection can be defined in two ways – we may give an extensional definition – one that enumerates its members

and an intensional definition as when we mention a defining property


on the view that I have put above a class is not defined extensionally

that is to say enumerating the members of a class does not define the class

Fred and Jack and John – can be members of any number of collections

their membership is not what defines the class –

the class is defined by its reason for being – that is why it was constructed in the first place


now to intensional definition

can the class be defined by what the members have in common?

well they may have many things in common – but the reason for them being classified – made into a class – is not the fact they have something in common

it is the purpose that classification is to be put to

and that purpose – whatever it may be is outside of the classification – outside of the class – it is the reason for the class – and therefore is not internal to the class


Russell goes on to say when we come to consider infinite class we find that enumeration is not even theoretically possible for beings who live for a finite time

we cannot enumerate all natural numbers

our knowledge of such collections must be derived from intensional definitions

I don’t see this last point at all

 

intensional definition – finding what is common to the members of a group – tells us nothing of the number of the group

so intensional definition is quite irrelevant to the question of infinite classes

the infinity of natural numbers derives not from there being an actual infinity of things called numbers

but from the fact that we can understand progressive (as in continuing in time) repetition as being in principle without end

that is – the key to infinity is the concept of progression and the fact of repetition


Russell goes on to say –

firstly numbers themselves form an infinite collection – and cannot therefore be defined by enumeration

secondly – collections having a given number of terms themselves – presumably form an infinite collection – e.g. there are an infinite number of trios in the world

and thirdly we wish to define ‘number’ in such a way that infinite numbers are possible

thus we must be able to speak of the number of terms in an infinite collection

and such collections must be defined by intension


it is clear here that for Russell intensional definition is the key to his idea of infinite number

but as I have pointed out intension has nothing to do with infinity or number


the notion of infinity comes down to that of repetitive action

numbering is the marking of any such action

as such the idea of infinite number – has no sense to it


the point is this – infinity is not an attribute – it is an operation

on such a view there seems to be no sense at all in speaking of infinite collections

that is to say the description ‘infinite’ is not applicable to collection

we cannot speak of infinite classes

 

again – I say a collection – a class - is defined not by what is in it or its number but rather its reason – its purpose – its function

Russell says it is clear that number is a way of bringing together certain collections – those that have a given number of terms

we can suppose all couples in one bundle – all trios in another etc.

in this way we obtain various bundles of collections

each bundle consisting of all the collections that have a certain number of terms

each bundle is a class whose members are collections i.e. – classes

thus each is a class of classes

the bundle consisting of all couples e.g. is a class of classes –

each couple is a class with two members -

the whole bundle of couples is a class with an infinite number of members –

each of which is a class of two members


it is true that you can keep classifying – that you can classify within classification – there can be many good reasons for doing this

Russell says the whole bundle of couples is a class with an infinite number of members

‘infinite number of members’ as I have argued above makes no sense

what you can say here is that the whole bundle of couples is a class with an unknown number of members

if instead of ‘infinite’ Russell used ‘unknown’ there would be more sense to his argument

he asks how are we to decide whether two collections belong to the same bundle?

well look there is no reason why anything belongs or does not belong to anything else unless you make it so

classification – the making of classes – is a contrivance - there are no ‘natural’ classes

to his question Russell says the answer that suggests itself is – find out how many members each has – and put them in the same bundle if they have the same number of members

 

but this he says – presupposes that we have defined numbers – and that we know how to discover how many terms a collection has

Russell’s view is that we cannot use counting here because numbers are used in counting


his argument here sounds cogent – on the assumption that numbers are something other than the operation of counting

of course you can ask – well ok counting – but what is being counted?

my view is that the act of counting is the act of numbering

the act of counting is the act of marking in some manner or another

the resultant markings are numbers

a number is a mark in a counting

and counting is the ordering of individuals in a series


Russell says it is simpler logically to find out whether two collections have the same number of terms – than it is to find out what a number is


this seems an odd statement to me – given what has preceded

it seems Russell thinks that the defining of number is the defining of some entity

when in fact all that ‘number’ is – is the term that we use to refer to the markings we make in counting

‘number’ to be fair comes up as a noun – as the name of something – and yes you can say the marks made in counting are something – but the real point is that ‘number’ refers to an action – so it is logically better understood as a verb

in any case Russell from the above statement seems to suggest you can understand a number without first knowing what ‘number’ is

this distinction doesn’t bother me or bear on my argument – but it seems to contradict what Russell just previously said regarding counting – that you need to know number first


he goes on to distinguish kinds of relations in this connection –

 

a relation is said to be ‘one-one’ when if x has the relation in question to y - no other term x’ has the same relation to y – and x does not have the same relation to any term y’ other than y

 

when only the first of these conditions is fulfilled – the relation is called ‘one-many’

when only the second is fulfilled – it is called ‘many-one’

Russell says it should be observed that the number 1 is not used in these definitions


it is true 1 is not used in this analysis

but the point is that for a relation of any kind to exist there must be at least two terms

that is a relation – is a relation between –

so it is clear that number is here presumed in any relation and any relational analysis


two classes are said to be ‘similar’ – when there is a one-one relation

he defines this more precisely –

one class is said to be ‘similar’ to another when there is a one-one relation of which one class is the domain while the other is the converse domain

it is obvious says Russell that two finite classes have the same number of terms if they are similar – but not otherwise


in what does this similarity consist?

granted you can have a one-one relation – why introduce similarity?

it seems like a weak word for what is very precise logical relation

and what is added by this notion of similarity?

the notion seems to me to be superfluous


Russell continues – the act of counting consists in establishing a one to one correlation between the sets of objects counted and the natural numbers (excluding 0) that are used in the process

the notion of similarity is logically presupposed in the operation of counting

the idea seems to be that you have a set of objects and a set of numbers – and then the act of co-relating the two

this presentation I think shows just how vacuous this idea of similarity really is

is Russell trying to suggest that the reason a number co-relates with an object is because of similarity?

he say the act of counting presupposes similarity

this is to suggest counting is like placing dominos on the ‘correct’ squares of a domino board

this seems incredibly naïve

numbers do not exist as objects – to be co-related or ‘imposed’ on other objects

numbering is simply the act itself of marking in a progressive manner the objects in a series

the numbers just are the marks of the numbering

no similarity exists or is required


he says we may thus use the notion of ‘similarity’ to decide when two collections belong to the same bundle

we want to make one bundle containing the class that has no members – one bundle of all classes that have one member – this will be for the number 1 etc.

given any collection we can define the bundle it is to belong to as being the class of all those collections that are similar to it

if a class has three members – the class of all those collections that are similar to it – will be the class of trios

whatever number of terms a collection may have – those collections that are ‘similar’ to it will have the same number of terms

and the number of a class is the class of those classes that are similar to it

and so to number – a number is anything that is the number of some class

Russell says at the end of this – such a definition has the verbal appearance of being circular – we define ‘the number of a given class’ – without using the notion of number in general

therefore we define number in general in terms of ‘the number of a given class’ – without logical error

it is in this section that Russell reveals the point of ‘similarity’

it is a concept designed to establish the notion of number

the number of a class is the class of those classes that are similar to it

which is a very weak way of getting around saying that ‘the number of a class is the class of those classes which have the same number’

and Russell wants to avoid this statement for it a circular definition -

and it brings down the whole edifice of classes –

for if a number is just a number (whatever that might mean) there really is no need to introduce classes at all

you also have the problem of classes that have the same number not being distinguishable

and there goes the neighbourhood

the idea of similarity is supposed to hold off these results

as I have said above – it just comes across as a very weak criterion in this context

but more than this it is at the very least – in this context an empty concept

we are it seems supposed to assume a similarity between classes with the same number – while not mentioning that they have the same number – which is of course the basis of their supposed ‘similarity’

and if it doesn’t mean this it means nothing


the final point is that a number in general is any collection which is the number of one of its members

all its members are of course similar in that they have the same number

so the number of one of its members will be the number of the class

what else could it be?


the thing is Russell’s use of class here has not I think added to the issue

simply because in the end in order to identify class you need number

class does not elucidate number

 

now simply bundling things together that have the same number – and calling the greater bundle – the number – just doesn’t cut it for me

the greater bundle is just a greater bundle


Russell seems to think that we can in some way discover numbers in reality – and at this he has failed


reality as in the non-human reality has no numbers

numbering is an operation that human beings bring to reality – for their purposes

the human reality is one that demands at times an overlay of order

numbering is a basic operation to this end


Russell’s argument is like this – you use number to define class (even though you try to appear to not be doing this by using the phantom concept ‘similarity’) and then you use class to define number

it’s hard to credit really


and the result is that number is left undefined


as Russell says at the beginning of his discussion of the definition of number –

‘In seeking a definition of number, the first thing to be clear about is what we may call the grammar of our inquiry.’

number is not a noun – it is a verb

3. Finitude and Mathematical Induction


Russell –

in the case of an assigned number the proof that we can reach it is –

 

we define ‘1’ as ‘the successor of 0’ – then we define ‘2’ as ‘the successor of I’ and so on

the method is not available for showing all such numbers can be reached this way

is there any way this can be proved?

 

we might be tempted to say ‘and so on’ means that the process leading to the successor may be repeated any finite number of times

the problem we are engaged in is defining ‘finite’ – and therefore we can’t use this notion in the definition

our definition must not assume that we know what a finite number is


the key to this problem is mathematical induction

the idea is – any property that belongs to 0 – and to the successor of any number that has the property – belongs to all natural numbers


some thoughts –


they key to all this is the idea that numbers are entities – of some kind or another

and being so – they possess as all entities do – properties

Russell wants to enshrine ‘is the successor of’ as one such property


we have with Russell some confusion at the base of all this –

mathematical induction –

is it an action – the action of making one mark the successor of another and continuing this process?

or is it the property that a number has that enables one to perform such an action?

he wants it both ways – we can perform the action because the objects (in this case numbers) allow us to do so

 

which of course brings us fair and square back to numbers – the question of the nature of numbers

in terms of what Russell has said so far – if you were to accept that numbers are entities – you would also have to conclude they are unknown entities

which might not be such a problem – except that Russell wants to load them up with properties

now there is a logical issue here

properties if they are to have any reality presume the reality of the entities they are attached to

 

that is they are characteristics of something – something that is that has a reality apart from having properties

if your entities are unknowns – then the only properties they can have are – unknown

that is to say you can’t have perfectly intelligible properties – attached to ‘something’ that has no known – properties

Russell doesn’t want to be seen as a Pythagorean – holding the view that numbers have some kind of ideal – non-material – mysterious existence

but he does want to hang on to the ‘shell’ of this idea – and somehow run his analysis on it –

and this is what has led to the talk of properties – but it doesn’t work

you could also say he falls back on to a kind of dispositional analysis –

the idea that we can get to the underlying entity (number) by looking at its ‘propensities’

in this case the ‘propensity to be a successor’

but this approach is just the properties argument again

the only way you get out of this dilemma is to recognise that with numbers you are not dealing with entities – but rather actions

so you just drop one side of the confusion I referred to above –

mathematics is about performing actions – and it is not actions in relation to entities

the so called entities of mathematics are just the actions – and their markings

again – as I mentioned – this point demands a wrenching of grammar – a realization that the grammar of the key term – number – must in light of the logic of the situation be – rewritten – it is best understood as a verb – not a noun

and the strange thing here is that you would have thought – if anyone was to see this straight up and understand it would have been Bertrand Russell –

his theory of description is just this point regarding logic and grammar

anyway

on the basis of this –

we can dispense with mathematical induction – it is an inference that could only apply if what was being discussed were entities of some kind

 

it is a concept designed to explain something that is not there

 

and the idea came about as a means of getting at an understanding and definition of ‘finite number’

Russell says – ‘Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite. The principle of mathematical induction might be stated popularly in some form as ‘what can be inferred from next to next can be inferred from first to last’. This is true when the number of immediate steps between first and last is finite, not otherwise.’

following this quote comes Russell’s ‘Thomas the tank engine’ metaphor

again there is this confusion between action and object – with the problem of attributes

there is no such thing as a finite number

a number is an operation in a series

what Russell means by ‘finite number’ – is a defined or definite series

so if ‘finite’ is to be applied in this context it would have to be to a series – and
understood to mean definite – as in predefined

an ‘indefinite series’ is what Russell means by ‘infinite number’ – or what I am suggesting he should mean by it

‘infinite’ is to be properly understood as ‘indefinite’

and understanding this is a key to understanding the whole matter

indefinite applies to actions – not things

 

definite applies to actions - not things

when we are talking about a finite number we are talking about a definite number of actions – which is to say a definite (progressive) repetition

in the case of infinite number what is being proposed is that the act of repetition is in principle repeatable – this is the best you can say

the idea of a series that is defined as indefinite seems to me to be a mismatch of notions

 

the point being a series by definition is definite

or to put it another way – indefinite action has no coherence

 

perhaps the notion of ‘infinite’ only comes about as a result of the misapplication of the negation sign to finite

 

that is – it is a logical mistake – and the term corresponds to no actual practise



4. The definition of order


Russell begins –

the first thing to realise is that no set of terms has just one order to the exclusion of others – a set of terms has all the orders of which it is capable

the natural numbers occur to us most readily in order of magnitude – but they are capable of an infinite number of arrangements

when we say we ‘arrange’ numbers in various orders – that is an inaccurate description – what we do is turn our attention to certain relations between natural numbers – which themselves generate such and such an arrangement

we can no more arrange natural numbers than we can the starry heavens

one result of this view is that we should not look for the definition of order in the nature of the set of terms to be ordered – since one set of terms has many orders

the order lies not in the class of terms – but in a relation among the members of the class – in respect of which some appear earlier and some as latter

the fact that a class can have many orders is due to the fact that there can be many relations holding among the members of a single class

what properties must a relation have in order to give rise to order?

we must be able to say of any two terms in the class that one ‘precedes’ and the other ‘follows’

for these words to be used in this way we require that the ordering relation has three properties:

(1) if x precedes y, y must not also precede x – a relation having this property is asymmetrical

(2) if x precedes y and y precedes z, x must precede z – a relation having this property is called transitive

 

(3) given any two terms of the class which is to be ordered – there must be one which precedes and the other which follows – a relation having this property is called connected

a relation is serial when it is asymmetrical transitive and connected

this is the definition of order or series

in response to this –

I would define order as a decision to regard individuals (of any kind) as being related

I see ordering as an essentially meta-geometrical activity – that is it is about where things are placed

the decision to place things in a common domain is the first act of ordering

the reason for this placement – for the setting up of a domain – is the reason or the purpose of the ordering

so the reason for the ordering is always outside of the ordering – outside of the domain

this is to make the point that ordering is an action –

it is a decision to relate individuals

against this you have Russell’s idea that there is something like a natural order – where e.g. the ordering of natural numbers can no more be an arrangement than the ordering of the starry heavens

this suggests that relations pertain between things quite independently of any purposes we may have for them or ‘designs’ we may have on them

my view here is that in a world without human consciousness there are no relations at all

 

relations – are very human constructions

constructions that are our basic method of ordering

it is true that we come into this world with a stock of categories and concepts that get us on our way –

which is to say that the making of order is a means of enabling us to function in whatever environment or domain we are engaged in – this is the idea of it

the ability to relate things is essential to our survival and happiness

my basic point here is that there are no inherent relations between things – we ‘make’ things relate

 

ok

 

Russell asks the question what properties must a relation have in order to give rise to order?

this question says it all

ordering as I have said is just the relating of things

you put any two things in relation to each other – which I argue is to make a meta-geometrical placement – then you have an ordering

ordering is not something different to relating

to relate is to order

what Russell calls the properties of a relation – are just descriptions of kinds of relations

now in this connection he mentions what he calls three ‘properties’ – these are asymmetry transitivity and connectedness

any relation that has these properties is a series or an order

an asymmetrical relation is if x precedes y – y must not precede x

what this amounts to is that from a meta-geometrical point of view x is placed before y

what is it to say x is ‘before’ y?

it is really just a decision to regard one term as having precedence spatially and / or temporarily

now if such a decision is made then clearly in terms of that decision the terms cannot be reversed

there would be no point in proposing that relation of terms in the first place if it was not to hold

so in terms of defining ‘precedes’ and ‘follows’ – all you have with this ‘property of asymmetry’ is the assertion that one will precede and the other will follow

no great step forward

the same point can be made with respect to the ‘property of transitivity’ – if x precedes y and y precedes z then x precedes z

this is just saying how things are placed – and that if they are placed in that manner then that is how they are placed – it is as simple as that

and as to the third ‘property’ – of connectedness

 

this is a good one

it is no more than to say that you have decided to place a number of individuals in relation to each other – that is what ‘connectedness’ comes down to – the decision ‘to connect’ things

'connectedness' seems like a rather clunky term to be used in logic – perhaps it’s a hangover from his Thomas the tank engine metaphor of the last chapter – anyway –

order presumes ontology

we order individuals or particulars be that numbers – stars – thoughts or whatever

a particular thing is what it is and not what it is not

which is to say ‘particularity’ presumes definition

what is included and what is not defines a particular

in common parlance we think of a particular thing as what it is – that is what is included

that is its positive definition

but a negative definition is just as essential

what follows from this is just that a thing cannot be ‘outside’ itself –

therefore it cannot be before itself – or after itself

 

‘before’ and ‘after’ – are relational terms – which means – they refer to particulars – and not a particular

and ‘relation’ here means – how things are placed in respect of each other

so ‘relation’ presumes multiplicity

there can be no relations unless there is a multiplicity

unless such an ontology is presumed

to make an order is to decide how to regard particulars – how to place them

that of course is determined by matters outside of the placement

an ordering is about how you want things to be in relation to each other

 

and why you want this relation depends on what you want to do with these things – and with these things in this arrangement

 

the series of natural numbers is really a language for progression – it is the argument that a repetitive act can be progressive – and if you go into negative numbers you have a language and a methodology of retrogression

in this way the series of natural numbers can be seen as a language of direction

the centre point of which is 0 – the place of rest relative to motion – the place you move from – or not


in conclusion – to order is to relate

to relate is to place particulars together

to place them in a context – in a domain

decisions are then made as to how the particulars are to be viewed

this is a matter of focus

i.e. – in xRy – we say the initial focus is x – the secondary focus y

in yRx the primary focus is y – the secondary x

where you begin is strictly speaking quite arbitrary

but in any ordering there must be a beginning – an initial focus

any relation is a series – in that any two terms related – form a series

asymmetry defines placement in a two term relation – if x precedes y – y does not precede x

transitivity is really no more than asymmetry with three terms –

order is the logic of placement


NB.


generation of series


Russell gives the example of the series of Kings of England

the series is generated by relations of each to his successor

here we pass from each term to the next – as long as there is a next – or back to the one before as long as there is one before

that is we generate a series by assuming that the term in question has an ancestry and has a posterity

my question is do we generate series?

or is it that we create a series by relating individuals – and then in terms of that series we can say the terms of the series have ancestry and posterity?

that is to say the properties of ancestry and of posterity are properties not of the terms of a series – but rather of the series

outside of the series the individual has no properties – i.e. ancestry or posterity

what I am putting here is really an argument against mathematical induction

my view is that properties such as ancestry and posterity are deductive of a series

that is they are properties we give to the terms of a series – given the series

and really what we are talking about here is description of the grounds of connection

the act of connection of the terms is just an ‘inductive' way of referring to the making of the series

in truth the terms are only connected given the series – it is the ordering that connects them – not the terms that ‘make’ the order

my sense is that mathematical induction is actually a false method if it is seen as a means of establishing order

 

mathematical induction only functions given that the order or series is presumed

and even so – what value does it have?

perhaps focusing on one term in a series and elucidating its properties as a member of the series might have some pedagogical value – that is it might be of use in the teaching and learning of mathematics

so it might have some value in elucidating the characteristics of a series

but the characteristics cannot be a product of mathematical induction

the characteristics of an ordering – of a series – are determined by the reason or the rationale of the series

 

also – these characteristics are operational characteristics – they are directions for proceeding given the series – i.e. with ancestry the direction is backward – with posterity forward

what I am saying basically is that a series if a series is given – it is not generated

 

generation given a series – is the elucidation of the principle of the series – this though is no more than to determine the series as an operation

a topical illustration of this is the ‘discovery’ of a new prime by Edson Smith of the University of California – the Guardian reports –

‘He installed software on the department’s computers from the Great Internet Prime Search (GIMPS) which uses downtime on volunteer’s PCs to hunt for ever larger numbers. Around 1000,000 computers add up to what is called a “grass roots super computer” that performs 29 trillion calculations a second.’



5. Kinds of Relations


it is clear that for Russell the ground of number – of mathematics is – relations

my argument is that relations presuppose a plurality – and number theory is the marking of plurality

relations on this view are an ordering of plurality

relations that is are actions on the plurality

‘relations between’ do not on this view – underpin numbering

the question of relations – in this context – only emerges – given numbers

this is to say that the making of relations – and the act of numbering

are two aspects – or can be two aspects of ordering –

numbers mark – relations define the possibility of the action of the numbers

how they can be ‘acted’

relations define action

in the context of mathematics we might say ‘manipulation’

so the theory of relations defines what you can do with numbers

numbers – are not – that is a consequence of relations

the theory of relations is the theory of activity

number theory is the theory of marking

in so far as much of mathematics is concerned with the ‘activity of numbers’ – the theory of relations underpins mathematic activity

so relations are the logic of mathematic activity

number theory on this view is prior to relations

mathematics is a primitive language of order

ordered activity depends on such a primitive language

the general point of this is that mathematics is a primitive marking

logic as a theory of relations is a theory of activity – that can be applied to numbers

to use a contemporary metaphor – mathematics is an essential hardware – logic or the theory of relations is basic software

for the program to work the two dimensions are required

and again what this is to say – is that neither are fundamental – both are actions – designed to create a platform – to act upon

in this chapter Russell discusses the relations asymmetry – transitivity and connexity – which he has previously defined as the properties of a serial relation – his idea being that when these properties are combined you have a series

 

we might ask in connection with asymmetry – why is that if x precedes y – y must not also precede x?

that is what is the basis of this claim – of this relation?

it is clear that with two particulars x and y there is no necessary relation

the existence of any relation is determined by what is done with x and y

that is how they are ‘made’ to relate

asymmetry is about placement that is all – the decision to regard x as preceding y

so it is simply a decision about how to order the world

clearly it depends on an idea of space and or time
                                                                                                                                   

so in my terms the placing of x before y is an action

 

and while it might be more than familiar to speak of an asymmetrical relation – in fact what you are talking about is an asymmetrical action – or placement

 

asymmetry defines a kind of action

the same is true of transitivity – transitivity is an action

connexity is – what?

there is no given connection between x and y

any connection is made –

it is the decision to place them ‘together’

to place them – that is in a common domain

and ‘domain’ here is really ‘the place of the relating’

which comes down to just the decision to relate

a series of natural numbers is thus a series that is constructed and the decision is made to hold the construction – hold the series

we decide that is – that 1 follows 2 follows 3 etc

these are actions and held as repeatable

and the series is then held in principle – which means it is repeatable

any series can be held in such a manner – not all are

the reason that the series of natural numbers is so held is that it is so very useful

and it is its primitiveness – the marking out – that is the key to its utility

its utility is based on need – the need to order

as to the origin of this need – we can only say it exists as a necessity – given the existence of conscious entities in the world

beyond this fact there is no explanation

 

 

6. Similarity of Relations

Russell begins –

 

the argument of chapter two was that two classes have the same number of terms when they are ‘similar’

that is there is a one-one relation whose domain is the one class and whose converse domain the other

in such a case we say that there is a ‘one-one correlation’ between the two classes

my view is –

in the chapter on the definition of number the argument is –

‘the number of a class is the class of all those classes that are similar to it’

and from this to –

‘a number is anything which is the number of some class’

the concept of similarity is here employed to reach the idea or the definition of number

the thing is though – this concept of similarity presumes number

the point is that in this context x is similar to y if x has the same number as y

the similarity of x and y is the number of x and y

or to put it another way – there is no similarity if is there is no number

number is the similarity

it is correct to use number as ‘the ground of similarity’

but not similarity as the ground of number

and I think from this it can be seen that similarity is a ruse

the intent of which is to make it appear that with the apparatus of class – number can be found and defined

 

again – the basis of any class is its number

class as with similarity presumes number

Russell says – a number is anything that is the number of some class

 

clearly – he wants to define number in terms of class

a class is a construction around or of number

 

and it should be noted that in terms of this view there is no such thing as a class with no number

for a class to exist – for a classification to be made – there must be particulars – that are the subject of the act of classification – or the act of making a class

these particulars can be marked as numbers

so in this context what are we to say of similarity?

two classes with the same number – are not similar

they are numerically identical

Russell can’t say this – because he wants to hold that the idea of class comes before that of number

and this is why you get such an unsatisfactory definition at the end of chapter 2 –

‘a number is anything which is the number of some class’

however you want to look at it this is no definition of number

‘anything’ does not qualify as definitive – of anything – excuse the pun

class does not ‘give birth’ to number

it is rather number in a sense that is the ground of class

classification is an act on numbers

I have argued that numbers themselves are acts of ordering

relative to this classification is a secondary act of ordering

we are dealing here then with primary and secondary acts

the secondary act is only possible – if you like – given the primary act

the primary act is primitive – it is the ‘marking out’ – as a means of ordering

its syntax are numbers

ok – to similarity of relations –

Russell gives the following definitions –

 

the relation-number of a given relation is the class of all those relations that are similar to the given relation

relation – numbers are the set of all those classes of relations that are the relation-numbers of various relations – or what comes to the same thing – a relation number is a class of relations consisting of all those relations that are similar to one member of the class

the class of all those relations that are similar to the given relation – is the relation number of a given relation

which is to say the number of all those relations –

a class of relations consisting of all those relations that are similar to one member of the class – is a relation number

again – the number of those relations

the point is isn’t that you have a relation – and a class with a number of instances of that relation

there is only one relation – however many instances there are of it

the instances of it are its relation-number

Russell begins this discussion with –

 

‘The structure of a map corresponds with that of the country of which it is a map. The space relations in the map have ‘likeness’ to the space relations in the country mapped. It is this kind of connection between relations that we wish to define.’

we are looking here at the relation between two sets of relations

one the actual geography of a country and the other a representation of that geography

we assume the map is an accurate representation of the country

what is the relation between the two?

Russell says they are ‘similar’

what we have here is not similarity

 

what we have is the one relation – however you describe this – in two expressions –

 

you might call it an identity of relations – but this is not strictly correct

there are not two identical relations

only one expressed differentially

 

Russell goes on to define Cardinal number as the number appropriate to classes –

 

and thus –

the ‘cardinal number’ of a given class is the set of all those classes that are similar to the given class

if classes are ‘similar’ – they have the same number of members

the number of those classes that have the same number (of members) is the cardinal number of such classes

Russell says two relations have the same ‘structure’ – when the same map will do for both – or when either can be a map for the other –

this is what he calls ‘likeness’

and this is what he means by relation-number

and so relation-number is the same thing as structure

ok

from this he goes on to say –

‘There is a great deal of speculation in traditional philosophy which might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realized. For example, it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding to differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world, and allowing us to infer from the phenomena the truth of all propositions that can be stated in abstract terms and are known to be true of phenomena. If the phenomenal world has three dimensions, so must the world behind phenomena; if the phenomenal world is Euclidean, so must the other be; and so on. In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes and baffles description, but which, for that reason, is irrelevant to science.’

here is the theory of the correspondence of propositions to reality

the idea that the structure of a correctly formed proposition will correspond to the structure of the reality or piece of reality it is being put against

there are so many problems with such a proposal that it is hard to know where to start

 

the key thing to say is that such an idea presumes the possibility of a God’s eye view

a view that is outside of the reality and the proposition that is being put against it

with such an eye you could see if the two fitted up

this is the idea and it is really ridiculous

even on the assumption of a God’s eye view there is still the question – how would you know if one corresponded to the other?

the proposition and the piece of reality it is put against are two different things

still there would be the question – what is the connection – what is the relation?

presumably the only clear-cut kind of ‘correspondence’ can be between two things of the same kind

and reality presumably does not have a double – and a proposition that is identical with itself – is just the same proposition

different things are different things

they relate only if made to relate – that is the relation is a construction

 

it is the bringing together of different things for a common purpose

it is in terms of the purpose that they relate

we order the world – we give it a structure – this is the very point of our actions

it is that structure that becomes the basis of our actions

we make the world – in order to operate in it

the structures that we give the world – are for all intents and purposes – literally – the structures that the world has

these structures indeed have objective reality – that is they become the actual practises of our living – but they are manufactured

they are structures imposed on the unknown – out of necessity
                                                                                                                                    

7. Rational Real and Complex Numbers


Russell begins here –

 

arguing that he has defined cardinal numbers and relation numbers – of which ordinal numbers are a particular species – and each of these kinds of numbers may be infinite as well as finite –

he will now go on to define the familiar extensions – negative – fractional – irrational and complex numbers


my argument –

the series of natural numbers is an ordering –

it is not an ordering of anything in particular

it is just the basic ordering of repetitive acts in space and time

a series is a conception of ordering –

the most basic ordering is the action of marking and repetition of marking

marks are differentiated i.e. ‘1’ ‘2’ ‘3’ etc – for the reason that a series requires such differentiation – of the one operation – of the one act of ordering

because such a series is not tied to any particular state of affairs – we say numbers have universal application

that is the series – the ordering - can be applied in whatever circumstance

the language of ordering is not special it is just a matter of convention

that is the marks used – i.e. ‘1’ ‘2’ ‘3’ or e.g. ‘I’ ‘II” ‘III” etc.

numbering is the act of essential or basic ordering

‘numbers’ are the marks of this ordering

numbers – that is are acts – actions recorded in a basic terminology – or language

the ‘necessity’ of mathematics simply comes from the very contingent fact that human beings need and seek basic ordering

that is the need for order – for ordered systems – is unavoidable – for human beings

ok

 

now to different kinds of numbers –

the cardinal number Russell defined as –

 

‘the cardinal number of a given class is the set of all those classes that are similar to the given class’

 

the cardinal number is thus a classification of classes

the cardinal number is the name of a set

the number of a set – which is just the number of a grouping

when you get into class and set you have strictly speaking moved one step from pure mathematics

the purity of mathematics is its primitiveness

classifications – class and set – are really the proposing of domains for number

in a way objects for the numbering action

we speak of classes and sets as if they have some independent existence

in fact they are just actions of classification – which then can become the objects of mathematical explication

 

that is to say we go on to order these classification – in terms of numbers

if you have classified things in terms of relations

then I suppose you can talk as Russell does of the ‘relations number’

but this number like the cardinal is not as it sounds – a special kind of number – it is just an action (of mathematical /numerical) ordering applied to a particular ‘object’

as you can see I take the view that all of mathematics is applied mathematics

that is it is the application of the primitive ordering of numbers – on whatever

the point of so called ‘pure mathematics’ as Russell would understand is from my point of view – finding simpler or more general operations that enable us to do the work required more efficiently

in relation to ‘relation numbers’ and ‘cardinal numbers’ – again what we are really talking about is – relational actions and cardinal actions

Russell at every turn it seems to me commits the fallacy of mistaking action for entity

 

it infects his whole theory of mathematics

it is why he cannot give a satisfactory account of number

where number for him has to be – as he puts it ‘anything’ – anything which is the number of some class

as I have said before - mathematical markings (language) refer not to objects (that could be ‘anything’) – but to actions – to actions of ordering

mathematics is primitive action

anyway

Russell says of positive and negative integers that both must be relations

the definition is –

+1 is the relation of n+1 to n – and -1 is the relation of n to n+1

the relation of n+1 to n – is +1

so +1 is a relation to 1

+1 on this view cannot be identified with 1 – for it is a relation to 1

and 1 according to Russell is a class of classes – an inductive cardinal number

so +1 is a relation

1 – a class of classes

the argument – a relation is not a class of classes

therefore the two cannot be identified

+1 is not 1


1 as a class of classes all who have 1 member – is no definition of 1

such is a definition of a particular class – not of number

the class may be defined as that which has 1 member

number cannot be defined by class

for it is the ‘object’ – of the class -

 

that is the classification is brought to the ‘object’

the object exists prior to the classification

what is the object?

what is a number?


you might say here – well any number is essentially a classification

therefore it is a class

if so your definition of number is –

the class of all classes that have a class as a member

this results of course in defining class in terms of class -

a classification is a classification

yes

we are none the wiser

such a definition is verbal – and does not elucidate in any constructive fashion the nature of class

but the real point is here that ‘number’ disappears into class

 

class therefore cannot be used in any definition of number

you may still have your ‘class of classes’

but if you are to introduce number – any number – it must be from outside such an argument

and if you want number to be the basis of class it must have a separate rationale

to fail to do this – to argue as Russell does is to confuse the object of a classification
with the classification

it is to confuse an organizing principle – with that which is to be organized

this confusion is of the same type as confusing subject with object – or object with subject

we may all wish to find a unifying essence – yes

 

but this cannot be done by a process of logical implosion

unless of course your idea is mysticism

a number is the mark of an action in a series

 

it is thus an action of ordering

such a mark is characterized by its primitiveness

it is a mark only of order

it is not a mark of anything

things may then be ordered in terms of a series of numbers

a number system establishes a serial order

that can then be applied in whatever circumstance


Russell says of +1 that it is a relation to 1

therefore it cannot be 1

if a relation – then there are at least two terms in the relation

so 1 – and its relation to what?

what is +1 on this view?

if you are going to say +1 is a relation to 1 – are you thereby saying +1 is any relation to 1?

it seems on the face of it that Russell has no choice here

short of any more specific characterization – any relation to 1 – is +1

the problem is of course that on such a view –1 may well be a relation to 1

if so – +1 is -1

either that or +1 and -1 cannot be distinguished

either result renders Russell’s argument impotent

the idea of ‘any relation’ is way too vague for the purposes of mathematical definition

and the idea that +1 is something other than 1 is just a touch Platonic

as there are – in Russell’s terms two things 1 and +1

for if +1 is in a relation with 1 – for there to be a relation – there must be two terms

1 – we already know about – and +1 must be the other term

but how can it be – how can it be another kind or form of 1?


to straighten this mess out you need to understand that a series of positive and negative integers – i.e. -3, -2, -1, 0, +1, +2, +3

is a different series to 1, 2, 3, etc

the negative-positive series is a different ordering

yes it is a numerical ordering just as the series of natural numbers is a numerical ordering

the point is that the signs ‘-‘ and ‘+’ indicate the ordering has a specific function

it is designed for another purpose

that is – it is a different operation

the ‘-‘ and ‘+’ signs are directional signs

they indicate retrogression and progression – from a central point

such an ordering is useful in any operation that requires retrogression and progression

so on this view you can’t speak of +1 and -1 outside of the series they are marks in

there is no such thing as a +1 or a -1

there is only a series in which such terms are marked out

in such a series we can say that there is a symmetry between +1 and -1 –

but this is only so because such is the point of the series

it is a series designed to establish a symmetrical order of progression and retrogression

it is to give us an order for any operation that requires these progressions

the ordering itself – the syntax – is a representation of the acts performed in any such operation

 

Russell goes on to define fractions –

the fraction m/n as the relation which holds between two inductive numbers x,y when xn = ym.

 

this definition he says proves that m/n is a one-one relation – provided neither m or n is zero

and n/m is the converse relation to m/n

it is clear that the fraction m/1 is the relation between integers x and y which consists in the fact that x = my

this relation – like the relation +m is by no means capable of being identified with the cardinal number m because a relation and a class of classes are objects of an utterly different kind

Russell makes it clear here that his definition of fractions is based on the same principle as his definition of positive and negative integers

the points made in relation to the definition of positive / negative integers therefore apply here

something I didn’t address above is the issue of class of classes and relations being of an ‘utterly different kind’

my question to Russell is what is a class of classes – if not a relation?

the point being a ‘class of classes’ is a description of a classification of classes

if classes can be classified as a class – then clearly there is a relation between the original classes and the class they then become a members of –

or to put it another way a class is a classification – a way of bringing things together

 

a relation is ‘what exists between things’ when they are brought together

a classification sets up the ground of any relation

a relation is a representation of the classification

for all intents and purposes the difference is only one of description

and it is different tasks that determine the use of different descriptions

the act of relating and the act of classifying are one in the same

that is you cannot do one without doing the other

 

what this leads to in my opinion is the view that there is no final or absolute description of any such act

and by ‘any such act’ I just mean what it is you do when you describe your action in whatever manner

the point is – the description is the act defined

outside of the action of description – the act itself is unknown

description gives the act an epistemological status

and this means it has a tag – is identified –

and identified within a larger often presumed network of description

the act is real – its identification is indeterminate

when it is so determined it is determined in relation to some task or goal

and the meaning of this is something that is held within the network of descriptions that any such task presumes or entails

there is nothing solid about all this

description is necessary for effective rational action

strictly speaking any description can do the job

it just depends what the job is

 

and how it has been previously described

that is the epistemological background of the job is where you start

 

but any starting point is uncertain

the action of description is the action of setting up a platform that has the appearance of stability or even certainty – just so you can get on with the job

action determines epistemology


fractions are the marks of specific actions that are operations within a given ordering

these actions are determined by practical tasks that demand a particular ordering if they are to be successfully accomplished

any series of fractions – or any ‘making’ of fractions presumes the order the of natural numbers

 

the manipulation of the terms of this ordering reveal possibilities of calculation

 

these possibilities enable particular actions

fractions are – relative to natural numbers – functions of natural numbers

fractions are essentially marks of function


on irrational numbers Russell says –

‘Thus no fraction will express the length of a diagonal of a square whose side is one inch long. This seems like a challenge thrown out by nature to arithmetic………

Russell goes on to discuss the Dedekind cut and real numbers

the idea behind the Dedekind cut is to include the square root of two and other irrationals in mathematics – to somehow make these numbers real

that is we have to take the convergent sequences of rationals – which don’t have rational limits – and make them into numbers –‘real’ numbers

this is the idea –

to have a number theory that includes both rational and irrational numbers – a unified theory

and this is what the Dedekind cut presumes

the idea is – arrange all rationals in a row increasing from negative to positive as you go from left to right –

the ‘cut’ is the separation of this row into two segments – one on the left – one on the right

all rational appear in one of the two sets

the row can be cut in infinitely many places

all the rationals in L are less than the rationals in R

we have cut the line in two and the cut becomes the real number

Dedekind shows how to add – subtract multiply or divide any two cuts – not dividing by zero

he also defines ‘less than’ for cuts and the limit of a sequence of cuts

once these rules of calculation are set up – the cuts are established as a number system

 

for this number system to be a real number system it must be shown that the Dedekind cuts include the rationals and irrationals

so to the square root of 2 –

to show that this irrational is included we must identify a left half line and right half line associated with the square root of 2

what rationals are less than the square root of two?

certainly all the negative ones – and also all those whose squares are less than 2

all numbers x such that either x < 0 or x²

that specifies the left piece of the cut – the left half line associated with the square root of 2

its compliment is the corresponding right half-line

when this cut is multiplied by itself – it produces the cut identified with the rational number 2

among Dedekind cuts 2 does have a square root

so what are we to make of the Dedekind cut?

firstly it is a device to bring unity to number theory – to bring rationals and irrational together

 

and it does this is by assuming that irrational and rational numbers will be members of the set of real numbers

a real number is a Dedekind cut –

if you accept the Dedekind cut then yes by definition the square root of 2 is a real number – for it is a Dedekind cut

this may well be a useful devise for giving the appearance of unity and thus simplicity to number theory –

but is it no more than just a classification of kinds of numbers?

simply a category created that includes both rational and irrational?

so the question is – in what sense are these real numbers real?

 

Russell says –

‘Thus a rational real number consists of all ratios less than a certain ratio – and it is the rational real number corresponding to that number. The real number 1, for instance is the class of proper fractions. In the cases where we supposed an irrational must be the limit of a set of ratios the truth is that it is the limit of the corresponding set of rational numbers in the series of segments ordered by whole and part. For example the square root of 2 is the upper limit of all those segments of the series of ratios that correspond to ratios whose square is less than 2. More simply still the square root of 2 is the segment consisting of all those ratios whose square is less than 2.’

in the case of rational real numbers – 1 comes off as a name for the class of proper fractions

so it is a class and a name of a class

as a mark for an operation I have no real issue with this – but I don’t see the point of giving such an action a separate numerical category – ‘rational real’ number

in the case of the square root of 2 – as the upper limit of all those segments of the series of ratios whose square is less than 2 – I find this to be no advance on irrational

you can say the upper limit – define the square root as such – but the truth is there just isn’t any upper limit

on this real number analysis the square root of 2 comes off not as a number – but as the name of a non-existent limit –

so how real is that?

Russell says –

‘It is easy to prove that the series of segments of any series is Dedekindian. For given any set of segments, their boundary will be their logical sum, i.e. the class of all those terms that belong to at least one segment of the set.’

again numbers – in this case real numbers are defined in terms of class -

it seems to me that if you want to go with a class definition of numbers – and so far that is all that we have from Russell – as well as the very real logical problem of having number presumed in the construction of any class – how can class thereby be an explication of number? –

let’s say you just forget about that – as Russell seems to –

what you end up with is nothing more than a name theory of numbers

 

that is a number – of whatever kind – is just the name of a class

 

(a class that presumes number)

it seems like a real mess to me

and the only logic in it seems to me to be ‘a class’ of logical errors

 

the Dedekind cut in relation to irrationals strikes me as a con –

not a real number – but a real con

for it is an argument that presumes what it is trying to show

it presumes that the square root of 2 exists

when this is just what has to be shown

the argument is that we can segment less than the square root of 2 and greater than the square root of 2 – and thereby find the square root in the centre – in the cut

the logic of it is that if you multiply the cut by itself – you get 2

you must get 2

this is just a sleight of hand trick

what you actually have in the cut in this case – in the case of irrationals – is a proposal for the square root of 2 as a ‘real’ number

a proposals that exists because of the cut – the line arrangement of the rationals – and the cut made

the number as such does not exist – it is made to exist – in the Dedekind argument

 

and as such it exists as an unknown

an unknown which multiplied by itself – gives 2?

 

this is not mathematics – this is magic


complex numbers


there are no numbers that yield ~1 when squared

for that reason it might be said that the square root of ~1 does not exist

however –.                                                                                                                                                     

if i is regarded as a symbol so that by definition:

i² = ~1

real multiples of i – like 2i or 3i are called imaginary

numbers of the form z = x + iy – where x and y are real numbers are to be called complex numbers

x is the real component of z – and y the imaginary

either x or y or can be 0

so imaginary numbers and real numbers are complex numbers

we can ask since no real number satisfies x² = ~1

is it justifiable to simply introduce the square root of ~1

the problem only real exists if you think you are dealing with a real entity of some kind

if it’s not there and you want it to be there – well yes you can do as imaginary fiction writers to – create an imagined reality

and who is to say that will not work?

the basic point is that from an epistemological point of view – in a fundamental sense what we are dealing with is the unknown

any representation of the unknown is a construction

what you have here – in a Russellian view of number theory – is the assumption that numbers of whatever kind – have some kind of real – as in non-imaginary existence

to run with such a theory and then to have to ‘imagine’ numbers when in terms of your own theory – they don’t exist – is nothing less than failure

complex numbers are ‘real’ to the extent that they mark a class of numerical operations required for ‘complex’ orderings

the actions of mathematicians are not just part imaginative – they are in fact entirely so

the history of mathematics is a history of imagining the possibilities of order

the language of mathematics is the syntax of this imagining

 

 

8. Infinite Cardinal Numbers

 

 

the cardinal number as constructed is not a member of any series

therefore it is not ‘inductive’ in Russell’s sense of this term

the notion of series I would argue is by definition definitive

that is the idea of a series that doesn’t begin or end is senseless

the series of natural numbers – just simply is – ‘that series counted’

the point being the action of counting defines the series

or when the counting stops – for whatever reason –

the series is complete for that operation

the cardinal number of a given class is the set of all those sets that are similar to the given class

as I have argued this idea of ‘similarity’ depends on number – and therefore is not an explanation of number

the set of those classes that are similar to the given class – just is the number of those classes

i.e. if all the classes contain 10 members (and this is something we discover in the action of counting) then the cardinal number of the collections – is 10

Russell says –

 

‘This most noteworthy and astonishing difference between an inductive number and this new number is that this new number is unchanged by adding 1, or subtracting 1…….The fact of not being altered by the addition of 1 is used by Cantor for the definition of what he calls ‘transitive’ cardinal numbers, but for various reasons………. it is better to define an infinite cardinal number …….as one which is not an inductive number.’

on the face of it this is quite a bizarre definition

a cardinal number is not a series number

that is it is not a number in a series

the purpose of a cardinal number is not serial

the function of the cardinal number is to identify the number common to a set of classes

 

common that is to the collections in considerations

so – there just is – or there just would be no point at all in adding 1 or subtracting 1 – to or from this number

it is trivially true that it can’t be done – but the point is there is no reason to – there is nothing to add or subtract to or from a cardinal number

it is not a member of a series – on which such operations are to be performed

addition and subtraction only make sense in terms of a series – of numbers in a series

the cardinal number is not such a number

now to go from this to the argument – therefore it is infinite

therefore it is an infinite number is absurd

and the point is this and it is crucial

there are no finite or infinite numbers

finity and infinity are not attributes of numbers

numbers are simply the markings of operations

in a repetitive series such as that of natural numbers you have a progressive operation and marks that identify such

we call such a series finite – because the action of marking cannot go on for ever

the idea that it might go on forever – as I have argued above makes no sense – for a

series must if it is to be a series – be defined

what you have with a cardinal number is a non-serial number

it gets its sense from the fact that it refers to a class of series (plural)

it is an essential or ‘identifying number’ – that is its function

at the basis of Cantor and Russell’s argument is the Platonic like notion of the reality of numbers –

and as if this is not bad enough – then comes the epidemic of classes and then the pandemic of sets – that have been imagined to somehow – and not at all in a successful manner – to give reality to number

the class idea as I have argued depends on number – it doesn’t establish it –

but these fantasies of class set and number are adopted ‘in re’ as you might say –

 

and so – it might seem that there are different kinds of these things – numbers – just as there are different kinds of objects in the real world – the unimagined world –

the point I would also wish to make is that the properties of a number are determined by its function – what it is designed for – or determined to do – what function it is to fulfil

seen this way the notion of ‘infinite number’ – Cardinal or whatever – makes no sense

that is what sense an infinite operation?

as any ‘properties of numbers’ are in fact properties of use

so on such a view the issue of aligning the so called properties of natural numbers – with i.e. cardinal numbers – does not arise

an operation by its very nature is a defined action

and mathematics the primitive marking of such action

at this level – for all intents and purposes – there is no difference between action and its marking

the action of numbering is the making of numbers

we have if you like descriptions of ‘natural usage’ and descriptions of cardinal usage

it is thus clear that where there are different usages there will be different numbers

 

to understand the difference – you need to see what different operations are being performed

i.e. – on this view Peano’s axioms do not define ‘natural number’ in the sense Peano intended – which is that there are these things ‘numbers’ that ‘have’ these properties

I argued above that ‘0’ is not a number – that Peano does not actually define number – rather he assumes it – really as an unknown and that ‘successor’ depends for its coherence on the presumption of number

 

so I have an argument with Peano

but yes in the series of natural numbers we do have succession

my point here is that ‘the successor of’ is not a characterization of a number –

it is a characterization of the operation – or action with numbers

 

it is a characterization of a certain usage

 

a characterization that is not present in – not required by cardinal usage

different task – different number

you can say any mathematical act is an act of ordering

and in this lies the unity of mathematics

but clearly there are different possibilities in the action of ordering – different ways to order

these different ways are responses to different needs – different objectives

you can define ordering – mathematics – in terms of different kinds of order

e.g. – you can say – to order is to relate

my view is that ordering and the act of mathematics is primitive

that is to say it has no explanation

we know what we do when we order – when we act mathematically

we ‘see’ it in the marks made – and the operations they represent

these acts are the basis of mathematics

any so called ‘meta’ descriptions of such activity have the epistemological status of metaphor

that is – poetry

the best example of this is Russell’s argument that class defines number when number is used to define class

poetry – though Milton it’s not

any way back to Russell –

a reflective class is one which is similar to a proper part of itself

this notion is based on the idea that x can have a relation to itself

so it is the view that a relation need not be between different entities –

a relation can exist ‘between’ an entity…….

naturally we want to say here ‘…….and itself’

 

for even in the argument that there is such a thing as a relation ‘to itself’ –

we can’t avoid referring to the entity as something else

the reason being of course that a relation is ‘between’ and for there to be a relation ‘between’ you must have at least two entities – that are distinguished – as particulars – as individuals

so what I am saying here is that the idea of a relation between x and itself – makes no sense

x if it is to be placed in a relation – is placed in a relation to ~x – whatever that might be

so as I have put before this ‘similarity’ argument is – really just a con – and not a particularly clever one either – that comes from bad thinking

and its origin is in taking the idea of class – way too seriously – giving it an importance and status in logic – it just doesn’t have –

and as a result misunderstanding it – as a logical entity – when in fact all it is – is an action of collection – a form of ordering

when understood for what it is – it is clear that there is no sense in saying that an action (of classifying) is similar to itself

you might argue it is similar to something else – i.e. – some other action of ordering – but to itself – that is just gibberish

there is no relation between a thing and itself

 

so – I argue right from the get go that the notion of a reflexive class –

as that which is ‘similar to a proper part of itself’ – is just bad thinking

a collection of things cannot be similar to a member ‘of itself’ –

 

for the entity (the member) is only a member in virtue of the fact that it ‘has been collected’

outside of the action of the collection – the collecting

there is no class –

the action of collecting – of classifying – ontologically is in an entirely different category – to the subjects of the act

a class is not an entity – it is an action

 

an action on entities

 

for this reason the idea of a reflexive class – has no coherence at all

and with the end of the reflexive class – comes too – the end of the idea of reflexive cardinal number – as the cardinal number of such a class

Russell refers to Royce’s illustration of the map in this connection –

consider e.g. – a map of England upon a part of the surface of England –

the map contains a map of the map

which in turn contains a map of the map – of the map

ad infinitum

this is a delightful little argument – but it is incoherent

the map is only as good as its markings – as its syntax

if the map of the map is not actually in the map – it’s not there

that is the first point

and the thing is the map is not a representation of itself

in this case it is a representation of England

and further – the idea of a map of a map –

true this is an example of reflexivity

of apparently ‘creating a relation’ of an entity with itself

it is a version of the idea that x is included in x

(it’s amazing how much verbosity there is in a subject like pure logic)

and as such a misuse of the concept of inclusion

the real clear point is that there is no reason for a map of a map

what is the purpose?

and further – what would such a thing look like?

it would be a duplication of the map – verbosity

 

it could not be anything else

in that case you have – not a map of the map in Royce’s sense – rather a copy

 

Russell goes on to say -

‘Whenever we can ‘reflect’ a class into a part of itself, the same relation will necessarily reflect that part into a smaller part, and so on ad infinitum. For example, we can reflect, as we have just seen, all the inductive numbers into the even numbers; and we can, by the same relation (that of n to 2n) reflect the even numbers into the multiples of 4, these into the multiples of 8, and so on. This is an abstract analogue of Royce’s problem of the map. The even numbers are a ‘map’ of all the inductive numbers; the multiples of 4 are a map of the map; the multiples of 8 are a map of the map of the map; and so on.’

first up we cannot reflect a class into part of itself – a class may be included in another class – and this is the proper use of inclusion – but a class is not a member of itself

and my general point is that nothing is included in itself –

for there to be inclusion – there must be distinction and difference –

inclusion is a relation between things

this idea of a class and ‘itself’ – has no place in logic

a class – a classification is just that – an operation of ordering

it has no ‘self’

there is no entity residing in it

 

and this is obvious even if you do not accept my operational analysis of class

to suggest that there is ‘a self’ to class is to confuse it with consciousness

all we are talking about here is operations performed

and at some point – it is worthwhile to ask – is this operation performable?

 

that is does it make any functional sense to think of an operation as an operation within itself?

the idea – the notion is absurd – an operation has no self – an operation is an action directed to – or out – not in

there is no ‘in’

 

this theory of class that Russell runs with is some kind of hangover from his Hegelian days I think

so – a reflexive class as I have argued above is not a legitimate concept

 

a class that is ‘similar’ to part of itself

look all you can say here is that you have two classes – two classifications – and they have the same number of members

this is not one class ‘similar’ to itself

this is two classifications with the same number

the fact you can make any number of such classifications – that have the same number does not mean in any way – that that number is infinite –

it is to imply the same numerical classification – repeated in different orderings

this is all Russell’s ‘abstract analogue of Royce’s problem of the map’ can amount to – different classifications with the same number

repetition is the key concept here

not in any way as ‘sexy’ as infinite – but that is the end of it

again – mathematics – just is about operations

and to cut a long story short – there are no infinite operations

there are only genuine operations – and failed operations

the so called infinite operation – is a non-operation

so perhaps 0 is the only ‘infinite’ number?

for the idea of infinite numbers – or infinite reflexivity to go forward

given the fact that there are no infinite operations

you need to give a ‘theoretical’ account of infinite operation

that is something that can go on in some sense without actually being performed

and to the service of this issue the idea of progression is brought to bear

the infinite ‘operation’ that no one performs – that is without end

what number do you give such a progression?

 

the unknown number –

the number that is not a number

that is not a member of any genuine series

for if it was a mark in a real series – it would be a number – and known

my view is that progression is a linear serial action of repetition in time – and the marking of such an action

such an action can be progressive or retrogressive

that is you move from 0 in either a positive or negative direction

positive is defined as right of 0 – negative left of 0

markings to the right – positive numbers – and each number – its syntax – must be distinct

markings to the left – negative numbers

progression and retrogression are just basic linear (special ) orderings of repetitive action in time

a series is defined by its action

so a progressive series ends when its operation is complete – that is when the action stops

and the same of course is true of a retrogressive series

Russell I think imagines that mathematical induction somehow enables automatic infinite generation of numbers

 

this just is what happens when you de-operationalize mathematics and place it in some theoretical no man’s land

where actually nothing happens – but the imagination can run wild – based as it happens on the operational model – but not in any real world

this is mathematics adrift from nature – nature as action

 

but the essential point is this – any progression is an action in a series of actions in space and time

in the world we live in – the world we know

we do not need to imagine an alternative reality – to do mathematics

 

the permanence and universality of mathematics comes from the syntax – the markings – the fact that they have a reality beyond their thought – and more to the point – the culture that ‘holds’ such knowledge as stable

 

and this amounts to – cultural repetition

and perhaps all this is backed up too by myth

the mythology of mathematics – which really originates from Pythagoras – the ideality – the transcendence of numbers

perhaps too to really understand the origin of this kind of thinking you would need to have a good look at the stability of the culture and society out of which it came

that is ask what social and cultural purpose did such thinking serve?

and most importantly what were the political and economic circumstances it was a response to?

in my view the psychological source of any transcendent argument is anxiety

anyway – back to Russell – who on the face of it seems to be anything but a victim of anxiety

Russell goes on to consider the definition of the number which is that of the cardinals

the first step he says is to define the series exemplified by the inductive cardinals in order of magnitude –

the kind of series which is called a progression

it is a series that can be generated by a relation of connectiveness

every number is to have a successor – but there is to be one with no predecessor – and every member of the series is to be the posterity of this term – with respect to the relation ‘immediate predecessor’

these characteristics may be summed up in the following definition: -

‘a progression is a one-one relation such that there is just one term belonging to the domain but not to the converse domain, and the domain is identical with the posterity of this term’

Russell’s concern here is with cardinal numbers –

since two progressions are similar relations

it follows their domains are similar classes

the domains of progressions form a cardinal number –

 

since every class which is similar to the domain of a progression –

 

is easily shown to be itself the domain of a progression

this cardinal number is the smallest of the infinite cardinal numbers – אₒ

to say that a class has אₒ terms is the same thing as to say that it is a member of אₒ -

and this is the same as to say -

that the members of the class can be arranged in a progression


my view here is that אₒ simply ‘defines’ the ‘fact’ of infinite progression

but can it really be called a number?

when all it is – is a symbol of the infinite progression – any infinite progression

it identifies the ‘idea’ of infinite progression

yes – you could say – therefore the number of infinite progressions - is a number

this is the idea

but really what you are talking about here is an infinite operation (though this is not what Russell or Cantor would say) –

and some kind of tag for it – ‘אₒ’

it’s an operation that cannot be performed –

 

a progression that in fact never progresses

still they want to describe it – as a number

as אₒ - aleph null –

actually the name is spot on

mathematicians spruiking the reality of reflexive classes infinite cardinal numbers and the like need to be reminded of the first commandment

 

and also perhaps to consider that they are in the wrong department –

perhaps there are some places left in the creative arts course 101 – imaginative fiction and abstract art

 

as far as infinite numbers – infinite classes go –

the simple truth is

 

the members of an infinite class cannot be counted

so they are by definition – uncountable numbers

infinity – the introduction of it into number theory results in the paradox – that infinite numbers are not countable

so they are numbers – that are not numbers

as far as the cardinal number אₒ goes – first up it is not a number – let alone the smallest of infinite cardinals

not only does infinity destroy numbers –

it makes class impossible

a classification for it to be valid must be closed

otherwise there is no class

the point being you cannot have an open class – an ‘infinite’ class

the idea of infinity in number theory results in non-classes – whose members are non-members

it’s a lot of nothing – an infinity of it – as it happens


Russell goes on to say –

‘It is obvious that any progression remains a progression if we omit a finite number of terms from it……These methods of thinning out a progression do not make it cease to be a progression, and therefore do not diminish the number of its terms, which remain אₒ……Conversely we can add terms to the inductive numbers without increasing their number.’

the brutal fact is the reason that the number of terms remains אₒ – is because there are no terms – you add or subtract to nothing – there is no change – nothing is nothing

אₒ represents – nothing –

there is no progression here – there are no numbers – there is no class –

there is just a collection of logical mistakes –

the first is that there are such things as numbers that are not countable

that there is a series (of numbers) – the members of which – though not countable – have a number

 

that we can call this – undermining of number theory – the making of – infinite numbers

there is no infinite number – for there is no infinite operation

if you persist with this talk of the infinite – of infinite numbers – it is an easy step to theology

in fact this is really where all these notions belongs

and could it not be asked – well is not God the infinite number – the infinite operation?

or in a related manner – in terms of Spinoza’s idea of substance – could it not be asked – is not reality itself – infinite – and its operations rightly given a number?

as you can see – in basics – no different really to the line of Cantor and Russell

but the answer to this question is that beyond what actually happens – we don’t know

and the thing is that any talk of God as the infinite or as substance as the infinite – is no more than human vanity writ large – or just the refusal to accept that beyond our knowledge is the unknown – and the unknown is just that – without characterization – description – or number

the concept of infinity is really just the attempt to defy the reality of human limitation

Russell goes on to say –

it is not the case that all infinite collections have אₒ terms

the number of real numbers for example is greater than אₒ – it is in fact 2 to the power of אₒ

the domains of progressions from the cardinal אₒ

where אₒ represents the domains of progressions of inductive numbers

then yes the number of real numbers (any number represented as a non-terminating decimal) is relative to the progressions of inductive numbers – greater

 

this is really no more than to say that the number of real numbers is greater than that of inductive numbers

so really what is being argued here is that if you were to place real numbers as the domains of progressions – that is as an infinite cardinal – against inductive numbers as domains of progressions – as an infinite cardinal – then the infinite cardinal of the

real numbers – is necessarily greater than the infinite cardinal of the inductive numbers

 

that is to say one group is greater than the other – therefore one cardinal is greater than the other

the argument here is that ‘greater than’ is a relation between classes – in this case inductive and real numbers

that is the class of real numbers is greater than the class of inductive numbers

the fact that these classes are infinite – is on this view – not relevant – to the issue of ‘greater than’

infinity is not relevant because it is not a discriminating factor – or a discriminating property – because both classes possess this property

therefore it is not what distinguishes them –

the distinction is between type of number – (real or inductive) – not to do with cardinality

cardinality here – it seems is not really – as might be thought – a matter of magnitude (greater than) – it is rather to do with the characteristic of reflexivity

my point is this – that if you hold with Russell and Cantor’s argument here – then infinity is not numerical – and infinite progression is best seen as something like an internal property – that real or natural numbers can have

it is like an internal repetition – but one that has no number

if that is the case you can say yes – the real cardinal is greater than the inductive cardinal – just simply because cardinality has nothing to do with it –

but if we want to go down this track – the cost is that there are no cardinal numbers

and certainly no relation of one cardinal being greater than another

if on the other hand you want to say an infinite progression or progressions can be given a number –

then you need to see that counting won’t do the trick

and then what is left?

to straight up argue that an infinite number – is not like any other number – countable – it is in fact uncountable – and this property of uncountability – or would Russell say – non-inductiveness – is its essential property –

this I think would be an improvement on the argument Russell is offering

but the result is – still you cannot say any one instance of such – of the infinite number – is greater than another

 

for on the view I am putting there is only one infinite number

and if so – there can be no comparison of infinite numbers

so the idea of a mark that marks infinity – and we call this a number?

starting to get mystical in my old age –


Russell goes on to say –

'In fact, we shall see later, 2 to the power of אₒ, is a very important number, namely the number of terms in a series that has “continuity” in the sense in which this word is used by Cantor. Assuming space and time to be continuous in this sense (as we commonly do in analytical geometry and kinematics), this will be the number of points in space or of instants in time; it will also be the number of points in any finite portion of space, whether line area or volume. After אₒ 2 to the power of אₒ is the most important and interesting of infinite cardinal numbers.'

 

it just strikes me that infinity and the attempt to attach it to numbers – i.e. cardinal numbers results in the complete defunctionalisation of mathematics

it really is all about pretending mathematics has a substance – and in this sense it is very similar to Spinoza’s idea of substance as the foundation of everything

Spinoza’s substance is – without substance – it is really just a term that refers to the unknown – but has the appearance of ‘substance’ – it’s an intellectual devise designed to give a foundation where there is none

and this idea that 2 to the power of אₒ has value in relation to the calculation of points in space is quite the sham

as soon as you introduce the notion of infinity – of infinite points – you forgo any possibility of calculation

as if this is not bad enough – the result is to make space into something it is not – that

is something that we cannot – by definition define

and here I mean define in an operational sense

 

the infinite cardinal number is a dead number – it has no action and it refers to nothing

all you have with this Cantor / Russell view here is mysticism

it surprises me that Russell’s thinking in mathematics – is without any critical dimension

 

it’s as if his theory of mathematics is just a composition – with a tweak here and a tweak there – so that everything hangs together – reasonably well

  

there seems to no genuine questioning of the content of mathematical theory

very disappointing

Russell goes onto say –

‘Although addition and multiplication are always possible with infinite cardinals, subtraction and division no longer give definite results, and therefore cannot be employed as they are in elementary arithmetic’

contrary to what Russell asserts here the operations of addition and multiplication on infinite collections does not increase their sum

it is only with finite collections that there is any genuine increase as a result of the operations of addition and multiplication

an infinite collection if you believe that such exists is without limit – addition and multiplication can only be performed – with any genuine outcome – if there are distinct finite collections

this does raise the question whether it is valid to speak of infinite collections – plural

the identity of indiscernibles is crystal clear here –

if there is nothing to distinguish two collections – there is only one collection

and of course at this point in the argument it is realized there is nothing actually being proposed

for in a case where there is only one infinite collection – it is clearly of no operational use – unless you’re are a Trappist monk

anyway Russell goes on to mention subtraction and division

subtracting 0 from 0 – leaves you with 0

and the same with division of 0 and 0

the point being there is nothing to subtract from – or nothing to divide in a infinite collection

that is there is nothing that you can take from an infinite collection – that leaves it wanting

and here if nowhere else the utter absurdity of this mathematics of infinity is patently obvious

there is no mathematics – no operations can be performed if you give up any sense of definition

and it is just that which is discarded with this fantasy of infinite numbers


reflexivity is based on a logical howler –

the idea that a something can be a member of itself

something – can only be a member of something else

Russell of all people should have known better

 

 

 

9. 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 section 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

 

10. 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 is the minima with respect to the converse of P

the sequents of a class a with respect to a relation P is 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 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 mental 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 a way 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


11. 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 had 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 another 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

12. 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 fallback 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 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 – as many have – is the class of classes – a member of itself?

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 application

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

 

 

13. 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 –

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 standard 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 not 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

‘everything’ is without bounds

 

it refers basically to what cannot be classified or described

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 what 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

 

14. 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 – and if p or q are false – the inference is false – if both p and q are false in p v q – 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

 

 

15. 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?

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 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

 

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

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

 

16. 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 section 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 definitiveness (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 that a good deal of language is logically fraudulent –

and 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 nowhere 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 someone 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 neighbourhood

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 casts 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

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

17. Classes


as to the nature of classes –

Russell’s goal here is to define class in such a way that eliminates mention of class

so that the symbols for classes are mere conveniences – not representing objects called classes – but are rather logical fictions – incomplete symbols


it needs to be said from the outset that a class is an action – an action of classification

any reference to class in an a substantive or non-active sense is merely a reference to the representation of the class (of the act)

so i.e. when logicians are talking about classes in the manner that Russell does – what they are actually – or should I say logically referring to is a representation of the act of classification

that might be in whatever form – i.e. diagrammatical or symbolic – in the sense of logical symbols

 

now the point of this is that actions are not fictions – not incomplete symbols

their meaning may be ultimately unknown – or in practice indeterminate –

nevertheless they are as real as any natural event

the guts of the problem for Russell is that he never considered that classes are actions

he has from the beginning been hoodwinked into this idea of ideal entities – even though his own analysis shows they don’t make sense – they don’t function – but he soldiers on – I think because of an entrenched metaphysics and epistemology – which never really comes up for question in his discussion of pure logic or the logic of mathematics

it’s as if there is a template and mathematics has just got to fit – and that the metaphysics justifies the making of any devise or argument that serves the unstated purpose of a correct fit

anyway

 

Russell’s view is that classes cannot be regarded as part of the ultimate furniture of the world

this notion of ‘ultimate furniture’ really needs to be given some scrutiny

yes we have in Western philosophy various theories of the ultimate furniture – from Thales onwards

the problem with ‘ultimate furniture’ as the idea is usually understood – is an epistemological problem

who is to know what is ultimate – if indeed the notion of ‘ultimate’ makes any real sense?

the truth is any proposition can function as an ultimate proposition – if so constructed 

in short if given that status

what is clear is that systems of thought – radically different systems will produce very different accounts of the ‘ultimate furniture’

so from a straight out epistemological point of view – one would have to conclude there just is no ultimate account

unless you want to go down the path of epistemological fascism – and claim there is one true account and the rest are false

most Western philosophers have indeed taken this course –

some have even managed to fit freedom pluralism and tolerance into their absolutism – nice job that

in general I would have to say it’s just been a parade of charlatans and hucksters – all with the same mask – truth

or perhaps it’s all just a misunderstanding

and yes the notion of ultimate and ultimate furniture – does play a role in how we think act and construct our world

and that therefore such notions even though they do not have the epistemological credentials that have been claimed for them – are in fact useful and in that sense necessary

all very well –

I guess my point is that when it comes to logic and the logic of mathematics – we are better off with an open mind

 

the activity has indeed produced theory and technique – these are tools to begin with – to work with

and in that sense it is the activity itself that is basic – and indeed you may come up with any number of theories as to why this is so – but that is just structuring the unknown

killing the beauty really –

back to Russell –

his argument is –

‘If we had a complete symbolic language, with a definition for everything definable, and an undefined symbol for everything indefinable, the undefined symbols in this language would represent symbolically what I mean by “the ultimate furniture of the world”. I am maintaining that no symbols either for “class” in general or for particular classes would be included in this apparatus of undefined symbols.’

as to definition –

it is the act of definition which defines – there is no other basis to it – but the act of singling out and describing –

any act of definition presumes that a term can be translated into other terms

the point of definition is utility –

a term is made active – in terms of a definition

and a term may be defined in innumerable ways

that is there in no essential definition

 

so there is no limit on definition – which is to say – any term or any symbol that requires definition can be defined

Russell clearly has a notion of the indefinable –

I can’t really imagine what he thinks it is

except to say that if you are talking in terms of an essential or ultimate definition – then indeed – any symbol is indefinable

this is not how it works in practise however –

if you get yourself tangled up as Russell has with bizarre notions like his idea of class – and his theory of infinite numbers

then you do find yourself stuck – or as he so politely puts it ‘indefinable’

but the key point of the above is just that he thinks that class is a definable

another way of putting is to say that in his view ‘class’ is not some kind of ultimate category

which might be to say – even though it does define – it too is definable

all of which is to get us ready for the move where class will be defined as ‘something else’

and Russell will be able to say – yes I have defined class without using ‘class’

and with this idea that some or all of the problems associated with his idea of class will melt away

my bet though is that in fact it will be the notion of class as Russell understands it that will dissolve

he’s an old fox

but the question will be – what’s left and what was all this for?

no sick sparrows flew into his cell – so we got this run around?

what I will point out again is that if you understand class as the act of classification and the markings of any such act – none of these issues emerge

we are not dealing with entities – we are dealing with actions

back to Russell –

he argues classes cannot be regarded as a species of individuals – on account of the contradiction about classes that are not members of themselves and because he thinks we can prove that the number of classes is greater than the number of individuals

as I have argued a class can only be a ‘member’ of another classification – another class – and in such a case what you have is an act upon an act

to ascertain the number of individuals – you have to first determine them – this is an act of classification

a classification is not an individual – therefore there is no question of one being greater than the other –

 

to suggest such is to compare apples and oranges – or to commit what Ryle called a category mistake

also Russell says we cannot take classes in the pure extensional way as simply heaps and conglomerations – he says – if we did this we could not account for a null class

again as I have argued previously – there is no null class – that is there is no ‘classification of nothing’

I would argue there is no ‘nothing’ to classify

and further the act of classification is at the very minimum an act about something

an act on something

what all this is leading to is –

‘We shall come nearer to a satisfactory theory if we try to identify classes with propositional functions.’

not exactly a grand finale – would you say?

nevertheless this is the argument – classes as propositional functions

it’s a risky course – for if it is successful you might just end up with – propositional functions – and classes as a nostalgic memory –

prime facie though – what is the situation here?

that is intuitively what are we to say of the relation of classes and propositional functions?

firstly the propositional function is a structure for propositions –

Russell uses the statement ‘all men are mortal’ and says it involves the functions ‘x is human’ and ‘x is mortal’

 

in these functions the subject of the predicate is unknown –

the subject is left undefined –

now Russell’s argument is that every class is defined by some propositional function – which is true of the members of the class – and false of other things

so propositional functions define classes

it does seem to me that the x’s in the above functions may well function as places for classes – i.e. ‘x’ is the class of those things that satisfies the function –

this works on the level of pure verbalism –

that is – we can refer to the unknown represented by x – as the place of classes

but to do this with any logical significance we need to have the notion of class to begin with

that is functions provide ‘places’ for classes – if classes exist

so what I am getting at is that the two notions ‘class’ and ‘propositional function’ are independent concepts

and furthermore quite different

a classification per se is not a propositional function

and even though you may choose to determine your propositional function in the language of classes – there is no necessity here

x is x – is unknown

if you have as a part of your theory of propositional functions the axiom that all propositional functions are determined by classes – then yes class terminology fits

but the other side of the coin is not so intuitive

a classification may be written up in terms of a propositional function – in propositional logic

that is the idea of class may be applied in the context of propositional logic

even so this would not be the only valid use of the notion of class

so in general the point is – a classification is one thing

and a propositional function is not a classification –

though it may be how a class is used – that is it may be a context for the use of the idea of class

propositional functions are logical apparatus – tools to enable logical process

classification is not an action dependent on propositional function

Russell goes on to say –

‘But if a class can be defined by one propositional function, it can be equally defined by any other which is true whenever the first is true and false whenever the first is false. For this reason classes cannot be identified with any one such propositional function.’

yes – propositional function is a definition tool

a particular propositional function will define a class in a particular manner

another propositional function will define it in another manner

if what you are after is a theory of the nature of class – the theory of propositional function is not relevant

the propositional function is a tool that can be used to define particular classes

it is not a meta theory of the nature of class

at this point of the story Russell has pretty much written off his own argument –

“For this reason the class cannot be identified with any one such propositional function rather than any other…..’

he goes on –

‘When we have decided that classes cannot be things of the same sort as their members, and that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. This is an example of Occam’s razor, namely “entities are not to be multiplied without necessity”. But when we refuse to assert that there are classes, we must not be supposed to be asserting that there are none. We are merely agnostic with regard to them: like Laplace, we can say “je n’ai pas besoin de cette hypotheses.’

it strikes me as an early instance of the British axiom ‘don’t mention the war’

 

Russell bites the bullet and gets pragmatic – in the attempt to resurrect the idea of the propositional function as the definition of class

but this is really the argument you have when you haven’t got an argument

he wants to set forth the conditions that a symbol must fulfil if it is to serve as a class

he lists five –

(1) every class is rendered determinate by a propositional function
(2) two formally equivalent propositions must determine the same class
(3) we must find some way of defining not only classes – but classes of classes
(4) it must be meaningless not false to suppose a class a member of itself
(5) it must be possible to make propositions about all classes that are composed of individuals or about all classes composed of objects of one logical type

yes we can accept that a propositional function defines a class

and that two formally equivalent propositions determine the same class

these two ‘conditions’ are no more than just applying the apparatus of propositional logic to classes – and that has never been in question

and asserting these conditions does not address the issue of the logic of classes

classes of classes – is no more than classification of classification – action on action

no problem if you understand that a class is an action

there is no sense in the idea that an act of classification is performed on itself

if we speak of ‘all classes’ we are really only referring to the function of classes – that is what a classification does – classify – so no more than a trivial and unnecessary statement

of course we can speak of our classifications – in the same way as we can refer to and make propositions concerning any of our actions

 

 

18. Mathematics and Logic


according to Russell mathematics and logic are one –

logic is the youth of mathematics and mathematics the manhood of logic

after a survey of all that has come before in his book Russell asks the question

‘What is this subject, which may be called indifferently either mathematics or logic? Is there any way we can define it?’

to begin Russell says

in this subject we do not deal with particular things or properties –

we deal formally with what can be said to be anything or any property

logic does not deal with individuals – because they are not relevant or formal

in the syllogism the actual truth of the premises is irrelevant – all that is important is that the premises imply the conclusion

a syllogism is valid in terms of its form – not in virtue of the particular terms occurring in it

and we are therefore faced with the question – what are the constituents of a logical proposition?

if we take a relation between two terms we may represent the general form of such propositions as xRy – i.e. x has the relation R to y

in the assertion ‘xRy is sometimes true’ i.e. there are cases where dual relations hold – there is no mention of particular things or relations

we are left with pure forms as the only possible constituents of logical propositions

the form of a proposition is that which remains unchanged when every constituent of the proposition is replaced by another

logic is concerned only with forms – and stating that they are always or sometimes true

in the proposition ‘Socrates is human’ – the word ‘is’ is not a constituent of the proposition – but merely indicates the subject predicate form

in the proposition ‘Socrates is earlier than Aristotle’ ‘is’ and ‘than ‘ merely indicate form

 

however form can be the concern of a general proposition even when no symbol or word in that proposition designates form

Russell argues we can arrive at a language in which every form belonged to syntax and not vocabulary

in such a language we could express all the propositions of mathematics even though we did not know one word of the language

we should have symbols for variables such as ‘x’ an ‘R’ and ‘y’ arranged in various ways – and the way of arrangement would indicate something was being said of all or some of the values of the variables

 

there are symbols with constant formal meanings – these are ‘logical constants’

 

‘logical constants’ will always be derivable from each other – by term for term substitution

and that which is in common is ‘form’

all constants that occur in pure mathematics are logical constants

logical propositions are those that can be known a priori – that is without study of the actual world

logical propositions have the characteristic of being tautologous – as well as being expressed in terms of variables and constants

this gives us the definition of logic and pure mathematics

Russell says he does not know how to define tautology

and in a note to this matter says –

‘The importance of “tautology” for a definition of mathematics was pointed out to me by my former pupil Ludwig Wittgenstein, who was working on the problem. I do not know whether he has solved it, or whether he is alive or dead.’

yes – one gets the impression here that Russell was not all that keen on the tautology

or at the time of his writing the above all that keen on the student who introduced the ‘importance’ of it to him

for he doesn’t even bother to give a definition of tautology

we are left wondering – perhaps indeed it is just a bad smell

also one would imagine a query here a word there could have settled the question of whether Wittgenstein was alive or dead –

perhaps though the first world war was the reason for placing Wittgenstein in a disjunction


my view on all this is –

all propositions are actions – we can say propositional actions

their basis is necessity – practical necessity

that is we propose descriptions of the world – so as to be able to operate effectively in the world –

we can therefore say any proposal is a proposal for order

we need order so as to operate effectively

 

this is a premise for any propositional behaviour

logic is a description of the possibilities of propositional order

that is logic displays the order inherent in propositional behaviour

so my first point is that logic is a descriptive action

the propositions of logic describe what is possible with propositions – that is how they can be ordered – how they can be related

i.e. a proposition can be put – and its opposite can be put

 

the second proposition is a negation of the first – the relation here is negation

two propositions can be conjoined – and in such a case their relation is conjunction

propositions can be disjoined – and their relation is disjunction

implication is a relation where one proposition is said to imply another

describing the relations between propositions (negation conjunction disjunction implication) tells us not only how propositions are used – but also if the question arises – how they can be used – in relation to each other

in this sense logic is the study of propositional relations

and it is an account or description of propositional behaviour that applies to any propositional usage – mathematical or empirical

mathematics is primarily concerned with calculation

 

that is to say it is a particular or specialized propositional usage –

this is not to suggest that mathematics is in any way limited –

for it is clear that any kind of experience can be made the subject of calculation

logic though is not an activity of calculation – even though there is calculation in logic

it is a description of the possibilities of propositional behaviour – one form of which is mathematical action

and it is in that sense a description or a proposal about what actually occurs

 

for this reason – it makes no sense to speak of it as being a priori

 

logic as a descriptive activity only exists because propositional behaviour exists –

because that is how the world is in terms of human beings and their actions

the propositions of logic are descriptions of what occurs or can occur when people use propositions

Russell mentions the law of self-contradiction as a logical proposition – and somewhat reluctantly the tautology

‘it is raining and it is not raining’ is a self-contradiction – it is a proposition that contradicts itself – it is a logically false statement

which to my mind means quite simply it has no use

‘it is raining and it is raining’ – is tautologous – it is a proposition which takes the value true for all assignments of truth values to its atomic expressions

again it like the self-contradiction is a propositional form that has no utility – no use

now I make this point to raise the question whether it makes any sense to speak of ‘propositions of logic’

if as Wittgenstein argues and Russell comes along for the ride – the propositions of logic are all tautologous –

then as a set of propositions they are useless

but they are only useless in this sense because they are being treated in an artificial manner

they are being taken out of any context – even out of the world

and then the question is asked – well what is their significance or their meaning?

well the answer of course is that they have none – they’ve been placed in a void –
and the very point – theoretical point of a void is that it has no significance

this bizarre result is a consequence – firstly of regarding propositions as in some sense special entities – when in fact all they are is the expression of the human need to make known – which is I would suggest the most basic of human needs

and they are therefore actions in the unknown – actions of defiance if you like

now to describe these actions – the propositional actions – to get an idea of how they do and can work is just another propositional action designed to shine some light into the darkness

logical activity is just the same action as any other propositional action – it has no specials status

it is a descriptive activity

its subject is propositional behaviour

so it is a propositional account of propositional behaviour

it's an ‘in house’ activity – or action within the action

it’s ground if you like is all propositional behaviour

and the ground of all propositional behaviour is simply the unknown

for it is the unknown that is the object of all propositional behaviour

through our propositions we make platforms for action

it is on such platforms that we get about the business of living

logic is simply a way of seeing how we do this



© greg. t. charlton. 2025.