Russell on mathematics VIII

Russell: introduction to mathematical philosophy:
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…….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 substraction 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

it’s garbage

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 rubbish

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

if the map of the map is not actually in the map – its 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

(its 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 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 – at the first post

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 same number

repetition is the key concept here

not in any way as ‘sexy’ as they say these days – 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 – but 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 consectiveness

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 this garbage 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 uncountablity – 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 defunctionalization 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 the 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 your 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 rubbish 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


© greg. t. charlton. 2008.

Russell on Mathematics VII

Russell: introduction to mathematical philosophy:
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 any thing

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 to 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 that 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 bullshit

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 its 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



© greg. t. charlton. 2008.

Russell on mathematics VI

Russell: introduction to mathematical philosophy:
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 instance 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 call ‘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 –

‘…a great deal of speculation in traditional philosophy 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



© greg. t. charlton. 2008.

Russell on mathematics V

Russell: introduction to mathematical philosophy:
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



© greg. t. charlton. 2008.