Russell: introduction to mathematical philosophy:
the definition of order
Russell begins –
the first thing to realise is that no set of terms has just one order to the exclusion of others – a set of terms has all the orders of which it is capable
the natural numbers occur to us most readily in order of magnitude – but they are capable of an infinite number of arrangements
when we say we ‘arrange’ numbers in various orders – that is an inaccurate description – what we do is turn our attention to certain relations between natural numbers – which themselves generate such and such an arrangement
we can no more arrange natural numbers than we can the starry heavens
one result of this view is that we should not look for the definition of order in the nature of the set of terms to be ordered – since one set of terms has many orders
the order lies not in the class of terms – but in a relation among the members of the class – in respect of which some appear earlier and some as latter
the fact that a class can have many orders is due to the fact that there can be many relations holding among the members of a single class
what properties must a relation have in order to give rise to order?
we must be able to say of any two terms in the class that one ‘precedes’ and the other ‘follows’
for these words to be used in this way we require that the ordering relation has three properties
(1) if x precedes y, y must not also precede x – a relation having this property is asymmetrical
(2) if x precedes y and y precedes z, x must precede z – a relation having this property is called transitive
(3) given any two terms of the class which is to be ordered – there must be one which precedes and the other which follows – a relation having this property is called connected
a relation is serial when it is asymmetrical transitive and connected
this is the definition of order or series
in response to this -
I would define order as a decision to regard individuals (of any kind) as being related
I see ordering as an essentially meta-geometrical activity – that is it is about where things are placed
the decision to place things in a common domain is the first act of ordering
the reason for this placement – for the setting up of a domain – is the reason or the purpose of the ordering
so the reason for the ordering is always outside of the ordering – outside of the domain
this is to make the point that ordering is an action –
it is a decision to relate individuals
against this you have Russell’s idea that there is something like a natural order – where e.g. the ordering of natural numbers can no more be an arrangement than the ordering of the starry heavens
this suggests that relations pertain between things quite independently of any purposes we may have for them or ‘designs’ we may have on them
my view here is that in a world without human consciousness there are no relations at all
relations – though not of blood and bone are very human constructions
constructions that are our basic method of ordering
it is true that we come into this world with a stock of categories and concepts that get us on our way –
which is to say that the making of order is a means of enabling us to function in whatever environment or domain we are engaged in – this is the idea of it
the ability to relate things is essential to our survival and happiness
my basic point here is that there are no inherent relations between things – we ‘make’ things relate
ok
Russell asks the question what properties must a relation have in order to give rise to order?
this question says it all
ordering as I have said is just the relating of things
you put any two things in relation to each other – which I argue is to make a meta-geometrical placement – then you have an ordering
ordering is not something different to relating
to relate is to order
what Russell calls the properties of a relation – are just descriptions of kinds of relations
now in this connection he mentions what he calls three ‘properties’ - these are asymmetry transitivity and connectedness
any relation that has these properties is a series or an order
an asymmetrical relation is if x precedes y – y must not precede x
what this amounts to is that from a meta-geometrical point of view x is placed before y
what is it to say x is ‘before’ y?
it is really just a decision to regard one term as having precedence spatially and / or temporarily
now if such a decision is made then clearly in terms of that decision the terms cannot be reversed
there would be no point in proposing that relation of terms in the first place if it was not to hold
so in terms of defining ‘precedes’ and ‘follows’ – all you have with this ‘property of asymmetry’ is the assertion that one will precede and the other will follow
no great step forward
the same point can be made with respect to the ‘property of transitivity’ – if x precedes y and y precedes z then x precedes z
this is just saying how things are placed – and that if they are placed in that manner then that is how they are placed – it is as simple as that
and as to the third ‘property’ – of connectedness
this is a good one
it is no more than to say that you have decided to place a number of individuals in relation to each other – that is what ‘connectedness’ comes down to – the decision ‘to connect’ things
'connectedness' seems like a rather clunky term to be used in logic – perhaps it’s a hang over from his Thomas the tank engine metaphor of the last chapter – anyway –
order presumes ontology
we order individuals or particulars be that numbers - stars - thoughts or whatever
a particular thing is what it is and not what it is not
which is to say ‘particularity’ presumes definition
what is included and what is not defines a particular
in common parlance we think of a particular thing as what it is – that is what is included
that is its positive definition
but a negative definition is just as essential
what follows from this is just that a thing cannot be ‘outside’ itself –
therefore it cannot be before itself – or after itself
‘before’ and ‘after’ – are relational terms – which means – they refer to particulars – and not a particular
and ‘relation’ here means – how things are placed in respect of each other
so ‘relation’ presumes multiplicity
there can be no relations unless there is a multiplicity
unless such an ontology is presumed
to make an order is to decide how to regard particulars – how to place them
that of course is determined by matters outside of the placement
an ordering is about how you want things to be in relation to each other
and why you want this relation depends on what you want to do with these things – and with these things in this arrangement
the series of natural numbers is really a language for progression – it is the argument that a repetitive act can be progressive - and if you go into negative numbers you have a language and a methodology of retrogression
in this way the series of natural numbers can be seen as a language of direction
the centre point of which is 0 – the place of rest relative to motion – the place you move from – or not
in conclusion – to order is to relate
to relate is to place particulars together
to place them in a context – in a domain
decisions are then made as to how the particulars are to be viewed
this is a matter of focus
i.e in xRy – we say the initial focus is x – the secondary focus y
in yRx the primary focus is y – the secondary x
where you begin is strictly speaking quite arbitrary
but in any ordering there must be a beginning - an initial focus
any relation is a series – in that any two terms related – form a series
asymmetry defines placement in a two term relation – if x precedes y – y does not precede x
transitivity is really no more than asymmetry with three terms -
order is the logic of placement
NB.
generation of series
Russell gives the example of the series of Kings of England
the series is generated by relations of each to his successor
here we pass from each term to the next – as long as there is a next – or back to the one before as long as there is one before
that is we generate a series by assuming that the term in question has an ancestry and has a posterity
my question is do we generate series?
or is it that we create a series by relating individuals – and then in terms of that series we can say the terms of the series have ancestry and posterity?
that is to say the properties of ancestry and of posterity are properties not of the terms of a series – but rather of the series
outside of the series the individual has no properties – i.e. ancestry or posterity
what I am putting here is really an argument against mathematical induction
my view is that properties such as ancestry and posterity are deductive of a series
that is they are properties we give to the terms of a series – given the series
and really what we are talking about here is description of the grounds of connection
the act of connection of the terms is just an ‘inductive' way of referring to the making of the series
in truth the terms are only connected given the series – it is the ordering that connects them – not the terms that ‘make’ the order
my sense is that mathematical induction is actually a false method if it is seen as a means of establishing order
mathematical induction only functions given that the order or series is presumed
and even so – what value does it have?
perhaps focusing on one term in a series and elucidating its properties as a member of the series might have some pedagogical value – that is it might be of use in the teaching and learning of mathematics
so it might have some value in elucidating the characteristics of a series
but the characteristics cannot be a product of mathematical induction
the characteristics of an ordering – of a series – are determined by the reason or the rationale of the series
also these characteristics are operational characteristics - they are directions for proceeding given the series - i.e with ancestry the direction is backward - with posterity forward
what I am saying basically is that a series if a series is given – it is not generated
generation given a series – is the elucidation of the principle of the series – this though is no more than to determine the series as an operation
a topical illustration of this is the ‘discovery’ of a new prime by Edson Smith of the University of California – the Guardian reports –
‘He installed software on the department’s computers from the Great Internet Prime Search (GIMPS) which uses downtime on volunteer’s PCs to hunt for ever larger numbers. Around 1000,000 computers add up to what is called a “grass roots super computer” that performs 29 trillion calculations a second.’
© greg. t. charlton. 2008.
