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.
