Russell: introduction to mathematical philosophy:
similarity of relations
Russell begins –
the argument of chapter two was that two classes have the same number of terms when they are ‘similar’
that is there is a one-one relation whose domain is the one class and whose converse domain the other
in such a case we say that there is a ‘one-one correlation’ between the two classes
my view is
in the chapter on the definition of number the argument is –
‘the number of a class is the class of all those classes that are similar to it’
and from this to –
‘a number is anything which is the number of some class’
the concept of similarity is here employed to reach the idea or the definition of number
the thing is though this concept of similarity presumes number
the point is that in this context x is similar to y if x has the same number as y
the similarity of x and y is the number of x and y
or to put it another way – there is no similarity if is there is no number
number is the similarity
it is correct to use number as ‘the ground of similarity’
but not similarity as the ground of number
and I think from this it can be seen that similarity is a ruse
the intent of which is to make it appear that with the apparatus of class – number can be found and defined
again the basis of any class is its number
class as with similarity presumes number
Russell says – a number is anything that is the number of some class
clearly he wants to define number in terms of class
a class is a construction around or of number
and it should be noted that in terms of this view there is no such thing as a class with no number
for a class to exist – for a classification to be made – there must be particulars – that are the subject of the act of classification – or the act of making a class
these particulars can be marked as numbers
so in this context what are we to say of similarity?
two classes with the same number – are not similar
they are numerically identical
Russell can’t say this – because he wants to hold that the idea of class comes before that of number
and this is why you get such an unsatisfactory definition at the end of chapter 2 -
‘a number is anything which is the number of some class’
however you want to look at it this is no definition of number
‘anything’ does not qualify as definitive – of anything – excuse the pun
class does not ‘give birth’ to number
it is rather number in a sense that is the ground of class
classification is an act on numbers
I have argued that numbers themselves are acts of ordering
relative to this classification is a secondary act of ordering
we are dealing here then with primary and secondary acts
the secondary act is only possible – if you like – given the primary act
the primary act is primitive – it is the ‘marking out’ – as a means of ordering
its syntax are numbers
ok - to similarity of relations –
Russell gives the following definitions –
the relation-number of a given relation is the class of all those relations that are similar to the given relation
relation –numbers are the set of all those classes of relations that are the relation-numbers of various relations – or what comes to the same thing – a relation number is a class of relations consisting of all those relations that are similar to one member of the class
the class of all those relations that are similar to the given relation – is the relation number of a given relation
which is to say the number of all those relations –
a class of relations consisting of all those relations that are similar to one member of the class – is a relation number
again – the number of those relations
the point is isn’t that you have a relation – and a class with a number of instances of that relation
there is only one relation – however many instances there are of it
the instance of it are its relation-number
Russell begins this discussion with –
‘The structure of a map corresponds with that of the country of which it is a map. The space relations in the map have ‘likeness’ to the space relations in the country mapped. It is this kind of connection between relations that we wish to define.’
we are looking here at the relation between two sets of relations
one the actual geography of a country and the other a representation of that geography
we assume the map is an accurate representation of the country
what is the relation between the two?
Russell says they are ‘similar’
what we have here is not similarity
what we have is the one relation – however you describe this – in two expressions –
you might call it an identity of relations – but this is not strictly correct
there are not two identical relations
only one expressed differentially
Russell goes on to define Cardinal number as the number appropriate to classes –
and thus –
the ‘cardinal number’ of a given class is the set of all those classes that are similar to the given class
if classes are ‘similar’ – they have the same number of members
the number of those classes that have the same number (of members) is the cardinal number of such classes
Russell says two relations have the same ‘structure’ – when the same map will do for both – or when either can be a map for the other –
this is what he call ‘likeness’
and this is what he means by relation-number
and so relation-number is the same thing as structure
ok
from this he goes on to say –
‘…a great deal of speculation in traditional philosophy might have been avoided if the importance of structure, and the difficulty of getting behind it, had been realized. For example, it is often said that space and time are subjective, but they have objective counterparts; or that phenomena are subjective, but are caused by things in themselves, which must have differences inter se corresponding to differences in the phenomena to which they give rise. Where such hypotheses are made, it is generally supposed that we know very little about the objective counterparts. In actual fact, however, if the hypotheses as stated were correct, the objective counterparts would form a world having the same structure as the phenomenal world, and allowing us to infer from the phenomena the truth of all propositions that can be stated in abstract terms and are known to be true of phenomena. If the phenomenal world has three dimensions, so must the world behind phenomena; if the phenomenal world is Euclidean, so must the other be; and so on. In short, every proposition having a communicable significance must be true of both worlds or of neither: the only difference must lie in just that essence of individuality which always eludes and baffles description, but which, for that reason, is irrelevant to science.’
here is the theory of the correspondence of propositions to reality
the idea that the structure of a correctly formed proposition will correspond to the structure of the reality or piece of reality it is being put against
there are so many problems with such a proposal that it is hard to know where to start
the key thing to say is that such an idea presumes the possibility of a God’s eye view
a view that is outside of the reality and the proposition that is being put against it
with such an eye you could see if the two fitted up
this is the idea and it is really ridiculous
even on the assumption of a god’s eye view there is still the question – how would you know if one corresponded to the other?
the proposition and the piece of reality it is put against are two different things
still there would be the question - what is the connection – what is the relation?
presumably the only clear cut kind of ‘correspondence’ can be between two things of the same kind
and reality presumably does not have a double – and a proposition that is identical with itself – is just the same proposition
different things are different things
they relate only if made to relate – that is the relation is a construction
it is the bringing together of different things for a common purpose
it is in terms of the purpose that they relate
we order the world – we give it a structure – this is the very point of our actions
it is that structure that becomes the basis of our actions
we make the world – in order to operate in it
the structures that we give the world – are for all intents and purposes – literally – the structures that the world has
these structures indeed have objective reality – that is they become the actual practises of our living
but they are manufactured
they are structures imposed on the unknown – out of necessity
© greg. t. charlton. 2008.
