Russell: introduction to mathematical philosophy:
selections and the multiplicative axiom
Russell argues –
the problem of multiplication when the number of factors may be infinite arises in this way -
suppose we have a class k consisting of classes
suppose the number of terms in each of these classes is given
how shall we define the product of all these numbers?
if we frame the definition generally enough – it will be applicable whether k is finite or infinite
the problem is to deal with the case where k is infinite – not with case where its members are
it is the case where k is infinite even when its members may be finite that must be dealt with
to begin let us suppose that k is a class of classes – in which no two classes overlap
say e.g. electorates in a country where there is no plural voting
here each electorate is considered to be a class of voters
now we choose one term out of each class to be its representative – as i.e. – when a member of parliament is elected
in this case with the proviso that the representative is a member of the electorate
we arrive at a class of representatives who make up the parliament
how many possible ways are there to choose a parliament?
each electorate can select any one of its voters – and if there are u voters in an electorate - it can make u choices
the choices of the different electorates are independent
when the total number of electorates is finite – the number of possible parliaments is obtained by multiplying together the numbers of voters in the various electorates
when we do not know whether the number of electorates is finite or infinite –
we may take the number of possible parliaments as defining the product of the numbers of the separate electorates
this is the method by which infinite products are defined
my thoughts are –
if we don’t know whether the number of classes (electorates) is finite or infinite – then quite simply and straight up we don’t know
whether they are infinite or not is not the issue – the issue is that we don’t know the number
now in such a case we cannot know the number of possible parliaments –
for in terms of the above argument – the number of possible parliaments depends of the number of electorates
the fact is you cannot multiply the unknown and expect its product to be known
Russell introduces possibility here as a something like a ‘known unknown’
it’s a trick to get past the fact that there are no infinite classes
the fall back position appears to be possible classes – and the idea is that possibles have numbers
which is really no more than to say the unknown has a number
if Russell was to accept this argument he would have to accept that mathematics is right back to square one – where you start – with the unknown
Russell goes on –
let k be the class of classes – and no two members overlap
we shall call a class a ‘selection’ from k when it consists of just one term of each member of k
i.e. u is a ‘selection’ from k if every member of u belongs to some member of k and if 'a' be a member of k – u and k have exactly one member in common
the class of all ‘selections’ from k we call ‘the multiplicative class’ of k
the number of terms in the multiplicative class of k - i.e. the number of possible selections from k is defined as the product of the members of the members of k
the definition is equally applicable whether k is finite or infinite
in response –
the first point is that this notion of class of classes -
a classification of all classifications
is what?
it is nothing -
we can ask – is the class of classes – a member of itself?
as many have –
my point though is that there is no sense to the idea of being – a member of itself
a classification is an action – you can represent it as an enclosed entity – but this is logically speaking a misrepresentation
something of the picture theory of the proposition seems to operate here
anyway –
to this notion of ‘selection’
this is a purely arbitrary devise designed to give the impression that we can operate with infinite classes
that is that we can make a selection – and operate with it as if it is definitive
my general argument is that there is no such thing as an infinite classification
a classification is closed – infinity is not – the two concepts cannot go together – without contradiction
and really – the truth be known a ‘selection’ cannot be made – for what is there to distinguish in infinite classes?
and if there is no distinction – there is no ground for ‘selection’ -
‘the product of the members of the members of k’ – is the multiplicative class of k
what you have here is a statement of the multiplication principle in a context where it cannot make any sense
the statement of the principle is ok – but it has no application in the world of infinite classes
this is no argument against the principle
rather it is an argument against its misapplication
the point that comes out most clearly for me is that the attempt to apply the multiplication principle in the (imaginary) context of infinite classes – shows quite clearly just how useless the is whole idea of infinite mathematics is -
it doesn’t work – and using various devises to prop it up – only results in demonstrating its impotence – and showing that it is not worthy of genuine mathematical intelligence
© greg. t. charlton. 2008.
