Russell: introduction to mathematical philosophy:
limits and continuity of functions
Russell is here concerned with the limit of a function (if any) as the argument approaches a given value
and also what is meant by continuous function
the reason for their consideration is that through the so-called infinitesimal calculus – wrong views have been advanced
it has been thought ever since Leibnitz that differential and integral calculus required infinitesimal quantities
Weierstrauss proved that this is an error
limits and continuity of functions are usually defined involving number
this is not essential as Whitehead has shown
consider the ordinary mathematical function fx – where x and fx are both real numbers – and fx is one-valued –i.e. – when x is given there is only one value that fx can have
we call x the argument – and fx the value for the argument of x
when a function is ‘continuous’ we are seeking a definition for when small differences in x – correspond to small differences in fx
and if we make the differences in x small enough – we can make the differences in fx fall below any assigned amount
the ordinary simple functions of mathematics have this property - it belongs to x², x³,…….log x, sin x, and so on
for discontinuous functions consider the example – ‘the place of birth of the youngest person living at time t’
this is a function of t - its value is constant from the time of one person’s birth to the time of the next birth
and then the value of t changes suddenly from one birthplace to another
a mathematical example would be ‘the integer next below x’ – where x is a real number
Russell’s argument is that there is nothing in the notions of the limit of a function or the continuity of a function that essentially involves number
both can be defined generally
and many propositions about them can be proved for any two series – one being the argument series – and the other the value series
the definitions do not involve infinitesimals
they involve infinite classes of intervals – growing shorter without any limit short of zero
but they do not involve any limits that are not finite
this is analogous to the fact that if a line an inch long is halved – then halved again – and so on indefinitely
we never reach infinitesimals this way
after n bisections – the length of our bit is ½n of an inch – and this is finite – whatever finite number n may be
infinitesimals are not to be found this way
ok – just a few thoughts -
infinite classes of intervals?
what you have is repetitive action that is progressive – in the case of a continuous function – the progression is continuous – in the case of a discontinuous function – it is discontinuous
hence – as Russell goes to quite a lot of effort to show – continuity (and discontinuity) are attributes or descriptions which are determined by the relations within a function
his definitions of continuity are really no more than second order descriptions of what occurs in various types of continuous function
the limit of such progressions is an issue of contingency – that is the possibility of performance
such a limit cannot be set in advance – or in concrete as it were –
the question of operation is an open question
it will depend on the state of the science of the day – in practise this means the theory of technology and its practise
so we cannot in advance assume that an operation is finite in the sense that it comes to a natural end of action –
you just have to see in practices what happens – and what in a predictive sense is possible
we can discount infinite operations as such - just in terms of the finite capabilities of human beings
infinity here – or the infinite performance of an operation – is really no more than keeping an open mind on contingent possibilities –
in general we can say the limit of a function and /or the continuity of a function is in any final sense unknown
practise determines these conceptions and the matter is finally undetermined
the point of contingency is just that it is undetermined – that its possibility is unknown
Russell is correct in dismissing infinitesimals
however his argument of infinite classes of intervals is wrong headed
first up the idea of infinite classes is based on a logical error
a class is a classification – it is an action – it is not an ideal entity – despite the fact that we characteristically speak of it in substantial terms –
this is no more really than a problem of grammar
the argument for infinity in this context is the argument of reflexivity –
the idea that a class can ‘reflex into’ itself
and such an idea presumes that a class has a dimension that is ‘itself’
what is to be meant by ‘self’ in this context?
clearly - a class x¹ within a class x – that is identical to x
this presumes the relation of identity
a relation exists if it exists between unique – that is distinct entities
there is no such thing as the relation of identity
an entity is not identical with itself – and not identical with an other thing
identity is a false relation
this is not to say that we can’t speak of equivalence in a mathematical sense
a classification of entities which has 10 members – can be regarded as equivalent to another classification that has 10 members – in terms that is of its membership number
but in such a case there is no question of the identity of entities
as I see it the great beauty of mathematics is that it enables a simple an elegant language of relations via number theory - that it completely dispenses with such questions as that of the substance of entities -
mathematics has really nothing at all to do with substance issues - it is the language of activity
and to my mind the theory of classes – and of infinite classes and numbers that Russell endorses and develops – brings the activity to a dead halt –
the reason being that such a theory of mathematics is really based on scholastic metaphysics – i.e. notions such as identity and self identity – which to my mind have no place in mathematics to begin with
a classification being an action – even if we were to hold with some metaphysical theory of identity – it is hard to see how it could be applied to classes
also let’s be clear about reflexivity –
reflexivity – if it is to mean anything is an action –
the idea that anything reflexes into itself presumes that the entity is active
that is that it performs actions
a class is an action – but it is as it must be an action performed
the result of such an action – i.e. the collecting of things together – does not go on to perform actions
which is just to say that an action – to be an action has a natural terminus
reflexivity is supposed to be the action that enables infinity – it presumes ‘self’ –and is apparently an action that no one actually performs
and further is not performable
not really a good bases for a theory of mathematics
so to get back to Russell – there are no infinite classes – and therefore no infinite classes of intervals
© greg. t. charlton. 2008.
