Russell: introduction to mathematical philosophy:
finitude and mathematical induction
Russell –
in the case of an assigned number the proof that we can reach it is we define ‘1’ as ‘the successor of 0’ – then we define ‘2’ as ‘the successor of I’ and so on
the method is not available for showing all such numbers can be reached this way
is there any way this can be proved?
we might be tempted to say ‘and so on’ means that the process leading to the successor may be repeated any finite number of times
the problem we are engaged in is defining ‘finite’ – and therefore we can’t use this notion in the definition
our definition must not assume that we know what a finite number is
the key to this problem is mathematical induction
the idea is – any property that belongs to 0 – and to the successor of any number that has the property – belongs to all natural numbers
some thoughts –
they key to all this is the idea that numbers are entities – of some kind or another
and being so – they possess as all entities do – properties
Russell wants to enshrine ‘is the successor of’ as one such property
we have with Russell some confusion at the base of all this –
mathematical induction –
is it an action – the action of making one mark the successor of another and continuing this process?
or is it the property that a number has that enables one to perform such an action?
he wants it both ways – we can perform the action because the objects (in this case numbers) allow us to do so
which of course brings us fair and square back to numbers – the question of the nature of numbers
in terms of what Russell has said so far – if you were to accept that numbers are entities – you would also have to conclude they are unknown entities
which might not be such a problem – except that Russell wants to load them up with properties
now there is a logical issue here
properties if they are to have any reality presume the reality of the entities they are attached to
that is they are characteristics of something – something that is that has a reality apart form having properties
if your entities are unknowns – then the only properties they can have are – unknown
that is to say you can’t have perfectly intelligible properties – attached to ‘something’ that has know known – properties
Russell doesn’t want to be seen as a Pythagorean – holding the view that numbers have some kind of ideal – non-material – mysterious existence
but he does want to hang on to the ‘shell’ of this idea – and somehow run his analysis on it –
and this is what has lead to the talk of properties – but it doesn’t work
you could also say he falls back on to a kind of dispositional analysis –
the idea that we can get to the underlying entity (number) by looking at its ‘propensities’
in this case the ‘propensity to be a successor’
but this approach is just the properties argument again
the only way you get out of this dilemma is to recognise that with numbers you are not dealing with entities – but rather actions
so you just drop one side of the confusion I referred to above –
mathematics is about performing actions – and it is not actions in relation to entities
the so called entities of mathematics are just the actions – and their markings
again as I mentioned at the end of the last post – this point demands a wrenching of grammar – a realization that the grammar of the key term – number - must in light of the logic of the situation be – rewritten – it is best understood as a verb – not a noun
and the strange thing here is that you would have thought – if anyone was to see this straight up and understand it would have been Bertrand Russell –
his theory of description is just this point regarding logic and grammar
anyway
on the basis of this –
we can dispense with mathematical induction – it is an inference that could only apply if what was being discussed were entities of some kind
it is a concept designed to explain something that is not there
and the idea came about as a means of getting at an understanding and definition of ‘finite number’
Russell says – ‘Mathematical induction affords, more than anything else, the essential characteristic by which the finite is distinguished from the infinite. The principle of mathematical induction might be stated popularly in some form as ‘what can be inferred from next to next can be inferred from first to last’. This is true when the number of immediate steps between first and last is finite, not otherwise.’
following this quote comes Russell’s ‘Thomas the tank engine’ metaphor
again there is this confusion between action and object – with the problem of attributes
there is no such thing as a finite number
a number is an operation in a series
what Russell means by ‘finite number’ – is a defined or definite series
so if ‘finite’ is to be applied in this context it would have to be to a series – and
understood to mean definite – as in predefined
an ‘indefinite series’ is what Russell means by ‘infinite number’ – or what I am suggesting he should mean by it
‘infinite’ is to be properly understood as ‘indefinite’
and understanding this is a key to understanding the whole matter
indefinite applies to actions – not things
definite applies to actions - not things
when we are talking about a finite number we are talking about a definite number of actions – which is to say a definite (progressive) repetition
in the case of infinite number what is being proposed is that the act of repetition is in principle repeatable – this is the best you can say
the idea of a series that is defined as indefinite seems to me to be a mismatch of notions
the point being a series by definition is definite
or to put it another way – indefinite action has no coherence
perhaps the notion of ‘infinite’ only comes about as a result of the misapplication of the negation sign to finite
that is it is a logical mistake – and the term corresponds to no actual practise
© greg. t. charlton. 2008.
