Russell: introduction to mathematical philosophy:
classes
as to the nature of classes –
Russell’s goal here is to define class in such a way that eliminates mention of class
so that the symbols for classes are mere conveniences – not representing objects called classes – but are rather logical fictions – incomplete symbols
it needs to be said from the outset that a class is an action – an action of classification
any reference to class in an a substantive or non-active sense is merely a reference to the representation of the class (of the act)
so i.e. when logicians are talking about classes in the manner that Russell does – what they are actually – or should I say logically referring to is a representation of the act of classification
that might be in whatever form – i.e. diagrammatical or symbolic – in the sense of logical symbols
now the point of this is that actions are not fictions – not incomplete symbols
their meaning may be ultimately unknown – or in practice indeterminate - nevertheless they are as real as any natural event
the guts of the problem for Russell is that he never considered that classes are actions
he has from the beginning been hoodwinked into this idea of ideal entities – even though his own analysis shows they don’t make sense – they don’t function – but he soldiers on – I think because of an entrenched metaphysics and epistemology – which never really comes up for question in his discussion of pure logic or the logic of mathematics
it’s as if there is a template and mathematics has just got to fit – and that the metaphysics justifies the making of any devise or argument that serves the unstated purpose of a correct fit
anyway
Russell’s view is that classes cannot be regarded as part of the ultimate furniture of the world
this notion of ‘ultimate furniture’ really needs to be given some scrutiny
yes we have in Western philosophy various theories of the ultimate furniture – from Thales onwards
the problem with ‘ultimate furniture’ as the idea is usually understood – is an epistemological problem
who is to know what is ultimate – if indeed the notion of ‘ultimate’ makes any real sense?
the truth is any proposition can function as an ultimate proposition – if so constructed –
in short if given that status
what is clear is that systems of thought – radically different systems will produce very different accounts of the ‘ultimate furniture’
so from a straight out epistemological point of view – one would have to conclude there just is no ultimate account
unless you want to go down the path of epistemological facism – and claim there is one true account and the rest are false
most Western philosophers have indeed taken this course –
some have even managed to fit freedom pluralism and tolerance into their absolutism –
nice job that
in general I would have to say it’s just been a parade of charlatans and hucksters – all with the same mask – truth
or perhaps it’s all just a misunderstanding
and yes the notion of ultimate and ultimate furniture – does play a role in how we think act and construct our world
and that therefore such notions even though they do not have the epistemological credentials that have been claimed for them – are in fact useful and in that sense necessary
all very well –
I guess my point is that when it comes to logic and the logic of mathematics – we are better off with an open mind
the activity has indeed produced theory and technique – these are tools to begin with – to work with
and in that sense it is the activity itself that is basic – and indeed you may come up with any number of theories as to why this is so – but that is just structuring the unknown
killing the beauty really –
back to Russell –
his argument is –
‘If we had a complete symbolic language, with a definition for everything definable, and an undefined symbol for everything indefinable, the undefined symbols in this language would represent symbolically what I mean by “the ultimate furniture of the world”. I am maintaining that no symbols either for “class” in general or for particular classes would be included in this apparatus of undefined symbols.’
as to definition –
it is the act of definition which defines – there is no other basis to it – but the act of singling out and describing –
any act of definition presumes that a term can be translated into other terms
the point of definition is utility –
a term is made active – in terms of a definition
and a term may be defined in innumerable ways
that is there in no essential definition
so there is no limit on definition – which is to say – any term or any symbol that requires definition can be defined
Russell clearly has a notion of the indefinable –
I can’t really imagine what he thinks it is
except to say that if you are talking in terms of an essential or ultimate definition – then indeed – any symbol is indefinable
this is not how it works in practise however –
if you get yourself tangled up as Russell has with bizarre notions like his idea of class – and his theory of infinite numbers
then you do find yourself stuck – or as he so politely puts it ‘indefinable’
but the key point of the above is just that he thinks that class is a definable
another way of putting is to say that in his view ‘class’ is not some kind of ultimate category
which might be to say – even though it does define – it too is definable
all of which is to get us ready for the move where class will be defined as ‘something else’
and Russell will be able to say – yes I have defined class without using ‘class’
and with this idea that some or all of the problems associated with his idea of class will melt away
my bet though is that in fact it will be the notion of class as Russell understands it that will dissolve
he’s an old fox
but the question will be – what’s left and what was all this for?
no sick sparrows flew into his cell – so we got this run around?
what I will point out again is that if you understand class as the act of classification and the markings of any such act – none of these issues emerge
we are not dealing with entities – we are dealing with actions
back to Russell –
he argues classes cannot be regarded as a species of individuals – on account of the contradiction about classes that are not members of themselves and because he thinks we can prove that the number of classes is greater than the number of individuals
as I have argued a class can only be a ‘member’ of another classification – another class - and in such a case what you have is an act upon an act
to ascertain the number of individuals – you have to first determine them – this is an act of classification
a classification is not an individual – therefore there is no question of one being greater than the other –
to suggest such is to compare apples and oranges – or to commit what Ryle called a category mistake
also Russell says we cannot take classes in the pure extensional way as simply heaps and conglomerations – he says – if we did this we could not account for a null class
again as I have argued in other posts in this sequence – there is no null class – that is there is no ‘classification of nothing’
I would argue there is no ‘nothing’ to classify
and further the act of classification is at the very minimum an act about something
an act on something
what all this is leading to is –
‘We shall come nearer to a satisfactory theory if we try to identify classes with propositional functions.’
not exactly a grand finale – would you say?
nevertheless this is the argument – classes as propositional functions
it’s a risky course – for if it is successful you might just end up with – propositional functions - and classes as a nostalgic memory –
prime facie though – what is the situation here?
that is intuitively what are we to say of the relation of classes and propositional function?
firstly the propositional function is a structure for propositions –
Russell uses the statement ‘all men are mortal’ and says it involves the functions ‘x is human’ and ‘x is mortal’
in these functions the subject of the predicate is unknown –
the subject is left undefined –
now Russell’s argument is that every class is defined by some propositional function – which is true of the members of the class – and false of other things
so propositional functions define classes
it does seem to me that the x’s in the above functions may well function as places for classes – i.e. ‘x’ is the class of those things that satisfies the function –
this works on the level of pure verbalism –
that is we can refer to the unknown represented by x – as the place of classes
but to do this with any logical significance we need to have the notion of class to begin with
that is functions provide ‘places’ for classes – if classes exist
so what I am getting at is that the two notions ‘class’ and ‘propositional function’ are independent concepts
and furthermore quite different
a classification per se is not a propositional function
and even though you may choose to determine your propositional function in the language of classes – there is no necessity here
x is x – is unknown
if you have as a part of your theory of propositional functions the axiom that all propositional functions are determined by classes – then yes class terminology fits
but the other side of the coin is not so intuitive
a classification may be written up in terms of a propositional function – in propositional logic
that is the idea of class may be applied in the context of propositional logic
even so this would not be the only valid use of the notion of class
so in general the point is – a classification is one thing
and a propositional function is not a classification –
though it may be how a class is used – that is it may be a context for the use of the idea of class
propositional functions are logical apparatus – tools to enable logical process
classification is not an action dependent on propositional function
Russell goes on to say –
‘But if a class can be defined by one propositional function, it can be equally defined by any other which is true whenever the first is true and false whenever the first is false. For this reason classes cannot be identified with any one such propositional function..’
yes – propositional function is a definition tool
a particular propositional function will define a class in a particular manner
another propositional function will define it in another manner
if what you are after is a theory of the nature of class – the theory of propositional function is not relevant
the propositional function is a tool that can be used to define particular classes
it is not a meta theory of the nature of class
at this point of the story Russell has pretty much written off his own argument –
“For this reason the class cannot be identified with any one such propositional function rather than any other…..’
he goes on –
‘When we have decided that classes cannot be things of the same sort as their members, and that they cannot be just heaps or aggregates, and also that they cannot be identified with propositional functions, it becomes very difficult to see what they can be, if they are to be more than symbolic fictions. And if we can find any way of dealing with them as symbolic fictions, we increase the logical security of our position, since we avoid the need of assuming there are classes without being compelled to make the opposite assumption that there are no classes. We merely abstain from both assumptions. This is an example of Occam’s razor, namely “entities are not to be multiplied without necessity”. But when we refuse to assert that there are classes, we must not be supposed to be asserting that there are none. We are merely agnostic with regard to them: like Laplace, we can say “je n’ai pas besoin de cette hypotheses.’
it strikes me as an early instance of the British axiom ‘don’t mention the war’
Russell bites the bullet and gets pragmatic – in the attempt to resurrect the idea of the propositional function as the definition of class
but this is really the argument you have when you haven’t got an argument
he wants to set forth the conditions that a symbol must fulfil if it is to serve as a class
he lists five –
(1) every class is rendered determinate by a propositional function
(2) two formally equivalent propositions must determine the same class
(3) we must find some way of defining not only classes – but classes of classes
(4) it must be meaningless not false to suppose a class a member of itself
(5) it must be possible to make propositions about all classes that are composed of individuals or about all classes composed of objects of one logical type
yes we can accept that a propositional function defines a class
and that two formally equivalent propositions determine the same class
these two ‘conditions’ are no more than just applying the apparatus of propositional logic to classes - and that has never been in question
and asserting these conditions does not address the issue of the logic of classes
classes of classes – is no more than classification of classification – action on action
no problem if you understand that a class is an action
there is no sense in the idea that an act of classification is performed on itself
if we speak of ‘all classes’ we are really only referring to the function of classes – that is what a classification does – classify – so no more than a trivial and unnecessary statement
of course we can speak of our classifications - in the same way as we can refer to and make propositions concerning any of our actions
© greg. t. charlton. 2009.
