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.
