VII INFINITY
IN MATHEMATICS THE EXTENSIONAL VIEWPOINT
39 Generality
in arithmetic
‘ “What is the sense of such a proposition as ‘($n).
3 + n = 7’?” Here we are in an old difficulty: on the one hand we feel it to be
a problem that the proposition has the choice between infinitely many values of
n, and on the other hand the sense of a proposition seems guaranteed in itself
and only needing further research on our part, because after all we “know ‘what
‘($x)
jx’ means”. If someone said he
didn’t know what was the sense of ‘($n). 3 + n = 7’, he would be
answered “ but you do know what this proposition says: 3 + 0 = 7 .v. 3 + 1 = 7. v. 3 + 2 = 7 and so on!” But to that one can reply “Quite correct –
so the proposition isn’t a logical sum, because a logical sum doesn’t end with
‘and so on’. What I am not clear about is this propositional form ‘j(o)
v j(1) v j
(2) v and so on’ – and all you have
done is to substitute a second unintelligible proposition for the first one,
while pretending to give me something familiar, namely a disjunction.” ’
‘($n). 3 + n = 7’ – is a game
the question – what value to give n – if the result
is to be 7? –
so yes – there is a ‘choice’ –
and if the proposition is to be functional – if the
game is to be playable – a choice must be made – a calculation made
you could say ‘($n). 3 + n = 7’ is game that has a
fixed external form – and an indeterminate internality
or that with ‘($n). 3 + n = 7’ – there is a game
within the game – and that the game within is logically of a different type to
the game without –
i.e. the game within is a game of infinitely many
values
we are not in an ‘old difficulty’ –
it is simply a matter of seeing that there are
different propositional games –
and further that games can be and are played within games
‘That is, if we do believe that we do understand “($n)
etc.” in some absolute sense, we have in mind as a justification other uses of
the notation “($n
…) …”, or of the ordinary language expression “There is …” But to that one can
only say: So you are comparing the
proposition “($n)
…” with the proposition “There is a house in this city which …” or “There are
two foreign words on this page”. But the occurrence of the words “there is” in
those sentences doesn’t suffice to determine the grammar of the generalization,
all it does is indicate a certain analogy in the rules. And so we can still
investigate the grammar of the generalization “($n) etc.” with an open mind, that
is without letting the meaning of “($ …) …” in other cases get in our
way.’
there is no ‘absolute
sense’ of ‘($n) etc.’ – or for that matter – any proposition –
there is only the sense it is given in whatever
propositional game it is employed in
and any sense given to it – that is any use it is
put to – is open to question – open to doubt – is uncertain
‘And so we can still investigate the grammar of the
generalization “($n)
etc.” with an open mind, that is without letting the meaning of “($
…) …” in other cases get in our way.’
yes – of course
‘ “Perhaps all numbers have the property e”.
Again the question is: what is the grammar of this general proposition? Our
being acquainted with the use of the expression “all …” in other grammatical
systems is not enough. If we say “you do know what it means: it means e(0).
e(1).
e(2)
and so on”, again nothing is explained except that the proposition is not a logical product. In order to
understand the grammar of the proposition we ask: how is the proposition used?
What is regarded as the criterion of truth? What is its verification? – If
there is no method provided for deciding whether the proposition is true or
false, then it is pointless, and that means senseless. But then we delude
ourselves that there is indeed a method of verification, a method that cannot
be employed, but only because of human weakness. This verification consists in
checking all the (infinitely many) terms of the product
e(0).
e(1).
e(2)
… Here there is confusion between physical impossibility and what is called
“logical impossibility”. For we think we have given sense to the expression
“checking of the infinite product” because we take the expression “infinitely
many” for the designation of an enormously large number. And when we hear of
“the impossibility of checking the infinite number of propositions” there comes
before our mind the impossibility of checking a very large number of
propositions, say when we don’t have sufficient time.’
‘perhaps all numbers have the property e’?
yes ‘perhaps’ – and there is nothing out of place
with ‘perhaps’ – with such a proposal
– such a speculation – and yes you can run with it – or not –
it’s a proposal
what is the grammar of this general proposition?
the ‘grammar’ will be the theory of its use – whatever that theory is
‘Our being acquainted with the use of the expression
“all …” in other grammatical systems is not enough’?
of course – different systems – different uses =
different grammars
and remember just what the use /grammar of ‘e(0).
e(1).
e(2)
and so on’ is – will never – logically speaking – be set in stone –
any theory of use –
as with any theory – any proposal – is open to question – open to doubt
– is uncertain
this is the case even when there is in fact a
stable practice –
that is – a particular view of the usage is adopted
and regarded as uncontroversial
‘In order to understand the grammar of the
proposition we ask: how is the proposition used?’
exactly – and there are any number of answers to
this question
‘What is regarded as the criterion of truth? What
is its verification? – If there is no method provided for deciding whether the
proposition is true or false, then it is pointless, and that means senseless.’
the criterion of truth? –
the criterion of truth is whatever it is decided to
be – by those engaged in the
propositional action
its verification? – the same
if there is no method? –
if there is no method – there is no use –
effectively – no functional proposition –
‘But then we delude ourselves that there is indeed
a method of verification, a method that cannot be employed, but only because of
human weakness’
look – the ‘method’ – whatever that amounts to –
must enable use – if it doesn’t – then there is no use –
it is not a matter of ‘human weakness’ at all –
either the method of verification – whatever that
comes to – facilitates the propositional
action or it doesn’t
we don’t run with a method of verification that
brings the propositional action to a halt –
or if we do – what that means – is quite simply we
have no use for that propositional action
you either affirm the propositional action or you
don’t
verification – in whatever form it takes – is –
affirmation
‘checking of the infinite product’?
there is no ‘checking’ as such – there is simply
the propositional action –
and whatever account is given of it –
and to be frank – any such account is no more than
a restatement of the propositional
action
‘we don’t have sufficient time’? –
if we did have sufficient time we would be dealing
with a different logic – a different grammar – a different propositional action
the very point of the infinity game – is that it
doesn’t come to an end in time –
even though the playing – and indeed – the players
– do
‘Remember that in the sense in which it is
impossible to check an infinite number of propositions it is also impossible to
try to do so. – If we are using the words “But you do know what it ‘all’ means
to appeal to the cases in which this mode of speech is used, we cannot regard
it as a matter of indifference if we observe a distinction between these cases
and the case for which the use of the words is to be explained. – Of course we
know what it means by “checking a number of propositions for correctness”, and
it is this understanding that we are appealing to when we claim that one should
understand also the expression “ …infinitely many propositions”. But doesn’t
the sense of the first expression depend on the specific experiences that
correspond to it? And these experiences are lacking in the employment (the
calculus) of the second expression; if any experiences at all are correlated to
it they are fundamentally different ones.’
‘checking a number of propositions for
correctness’? –
the rules of the game determine the play of the
game – the action of the propositions –
if the game does not play – then there is no game –
so called ‘checking for correctness’ is quite
simply – playing the game
in the case of a game with ‘infinitely many
propositions’ – you play the game –
and this is of course to say – you understand that
the game is on-going –
even if you are not
experiences?
the experience of the game – is the play of the
game –
regardless of what kind of game it is –
different games – different plays –
‘different experiences’ –
if that’s how you want to put it
‘Ramsey once proposed to express the proposition
that infinitely many objects satisfied a function f (x)
by the denial of all propositions like
~($x)
. fx
($x)
. fx. ~($x,
y) . fx . fy
($x,
y) . fx . fy. ~($x,
y, z) . fx. fy. fz
and so on
But this denial would yield the series
($x)
. fx
($x,
y) . fx . fy.
($x,
y, z) …, etc., etc.
But this series is quite superfluous: for in the
first place the last proposition at any point surely contains all the previous
ones, and secondly even it is of no use to us, because it isn’t about an
infinite number of objects. So in reality the series boils down to the
proposition:
“($x, y, z) … ad infin.) . fx . fy .
fz … ad infin.”
and we can’t make anything of that sign unless we
know its grammar. But one thing is clear: what we are dealing with isn’t a sign
of the form “($x,
y, z) . fx . fy . fz” but a sign whose similarity to that looks deceptive.’
what we have from Ramsey is a sign-game proposal–
and what we have from Wittgenstein is a questioning
of just whether Ramsey’s game can legitimately be played with the signs that he
uses – given Wittgenstein’s interpretation of those signs –
this is good argument –
it would be very interesting to know what Ramsey
would have said here –
and perhaps he would take Wittgenstein’s point that
we are not dealing with a sign of the form ‘($x, y, z) . fx . fy . fz’ – but a
sign whose similarity to that is deceptive –
if so – we might get from Ramsey – a new
interpretation – a new grammar of
‘($x, y, z) … ad infin.) . fx . fy .
fz … ad infin.’ – ?
any innovation – or novel argument in mathematics –
has to jump the hurdle of the reinterpretation of signs –
point being you have to work with the signs that
are available – proposing new signs –
without showing how they relate to what is in use –
is likely to be met with blank stares
and of course you will always find someone who says
with conviction – “it doesn’t mean
that” –
paradigm shifts – and it is all the business of
argument –
finally – I do take Wittgenstein’s point in
relation to Ramsey’s proposal – ‘it isn’t about an infinite number of objects’
–
really Ramsey’s proposal is just a sign-game – a
language-game –
and the idea of ‘an infinite number of objects’ –
is no more than a propositional setting for the game – for the play
an imagined setting – that gives the sign-game a
pretend domain
it should also be noted that Wittgenstein argues
that with or without such a setting – the game goes nowhere –
‘But this series is quite superfluous: for in the
first place the last proposition at any point surely contains all the previous
ones’ –
so all we have in effect is – a restatement of –
and a continual restatement of – the initial proposition
quite apart from any argument regarding the
legitimacy of Ramsey’s proposal – the issue is – does it have a use?
that’s the question
‘I can certainly define “m >
n” as ($x):
m – n = x, but in doing so I haven’t in any way analysed it. You think, that by
using the symbolism “($
…) …” you establish a connection between “m > n” and other propositions of the
form “there is …”; what you forget is that that can’t do more than stress a
certain analogy, because the sign
“($ …) …” is used in countlessly
many different ‘games’. (Just as there is a ‘king’ in chess and draughts.) So
we have to know the rules governing its use here;
and as soon as we do that it immediately becomes clear that these rules are
connected with the rules for subtraction. For if we ask the usual question “how
do I know – i.e. where do I get it from – that there is a number x that
satisfies the condition m – n = x? it is the rules of subtraction that provide
the answer. And then we see that we haven’t gained very much by our definition.
Indeed we might just as well have given the explanation of ‘m >
n’ the rules for checking a proposition of that kind – e.g. ‘32 >
17’.’
‘I can certainly define “m >
n” as ($x):
m – n = x – but in doing so I haven’t in any way analysed it’ –
analysis is what? –
effectively – in the end – restatement
in different terms – likely – long-winded
as to m > n and ($x):
m – n = x –
we have two proposals – and if the idea is that
they are interchangeable –
this proposal – this relational proposal will be up
for argument –
i.e. – under what circumstances and why?
and it could just be that the second can only
function in place of the first – if added to the argument are a number of
qualifications
you could end up asking well why – why bother with
the second – where’s the advantage?
it’s not as if it wins in the simplicity or
elegance stakes
the answer will be that the second represents the
first (if indeed it does) in a different
context –
in a different
game
translation here – from one to the other – is –
logically speaking – open to question – open to doubt – is uncertain –
but that’s translation
at base the issue is agreement – and that can be as complex as you want to make it
however if the move is made – the interchange is agreed to – for whatever reason –
that is – in the end – the only argument –
for the moment
‘If I say: “given any n there is a d
for which the function is less than n”, I am ipso facto referring to a general arithmetical criterion that
indicates when F(d) >
n.’
yes – ‘F(d) > n’ represents a practice
‘If in the nature of the case I cannot write down a
number independently of a number system, that must be reflected in the general
treatment of number. A number system is not something inferior – like a Russian
abacus – that is only of interest to elementary schools while a more lofty
general discussion can afford to disregard it.’
a number system is – more generally speaking – a
propositional game –
if you like – a meta-game
writing down a ‘number’ – presupposes a meta
background –
if you don’t presuppose a meta background when you
write down a number – all you do is make a mark on a piece of paper –
why would you do that?
‘Again, I don’t lose anything of the generality of
my account if I give the rules that determine the correctness and incorrectness
(and thus sense) of ‘m > n’ for a particular system like the decimal system.
After all I need a system, and the generality is preserved by giving the rules
according to which one system can be translated into another.’
yes
‘A proof in mathematics is general if it is
generally applicable. You can’t demand some other kind of generality in the
name of rigour. Every proof rests on
particular signs, produced on a particular occasion. All that can happen is
that one type of generality may appear more elegant than another. ((Cf. the
employment of the decimal system in proofs concerning d
and h)’
‘A proof in mathematics is general if it is
generally applicable’
this tells us nothing –
we have rules and we have propositional games –
this is what we work with
the rules apply where they apply
the games are played where they are played –
the rules don’t apply where they don’t apply –
and the propositional games are not played where
they are not played –
if there is an issue here at all – it is application
can this game be played in this propositional
context?
and any
decision about application –
is open to question – open to doubt – is uncertain
‘generality’ here strikes me as a notion that has no functional value
‘All that can happen is that one game can appear
more elegant than another’
and elegance?
in the eye of the beholder
‘We may imagine
a mathematical proposition as a creature which itself knows whether it is true
or false (in contrast with propositions of experience).
A
mathematical proposition itself knows that it is true or that it is false. If
it is about all numbers, it must also survey all the numbers. “Its truth or
falsity must be contained in it as is its sense.” ’
a true
proposition is a proposition that is assented to –
affirmed –
for whatever reason
a false
proposition is a proposition dissented from –
denied – for
whatever reason
a true
proposition is proceeded with –
a false
proposition is not
a proposition
– a proposal – about all numbers – if it
is to have any sense – is a game
a game-proposition
and a game-proposition is a rule governed
propositional action
a properly constructed game – can be played – or
not
if played – it is affirmed –
if it is not affirmed – it is not played
as for this idea of a proposition being self-aware
– and aware of its truth or falsity –
psychologism – yes
Wittgenstein has no account of propositional truth
and to cover this failure he resorts to pretention
and fantasy
a proposition – is a proposal – a proposal of a
human being –
a sign of a human being –
so the appropriate question here is – does a human
being know that its signs are true or false?
the answer is that a proposal – a proposition a
sign – is open to question – open to doubt – is uncertain
propositions – of whatever kind – are uncertain
human knowledge is uncertain –
truth is assent – falsity – dissent
assent and dissent – are open to question – open to
doubt –
are uncertain
‘ “It’s as though the generality of a proposition
like ‘(n). e(n)’
were only a pointer to the genuine, actual , mathematical generality, and not
the generality itself. As if the proposition formed a sign in a purely external
way and you still needed to give to the sign a sense from within.” ’
generality is non-specific application –
it is a proposal of non-specific application
so the question is –
where does a proposal that has the form of
non-specific application apply?
and any answer to this will be open to question –
open to doubt –
will be uncertain
‘As if the proposition formed a sign in a purely
external way and you still needed to give to the sign a sense from within’
if the proposition has the form of generality
– then the question is – does this proposition have function in this
propositional context?
‘ “We feel the generality possessed by the
mathematical assertion to be different from the generality of the proposition
proved.” ’
the mathematical assertion is a propositional game
‘the proposition proved’ is the mathematical
assertion – restated
‘We could say: a mathematical proposition is an
allusion to a proof’
a mathematical proposition is a propositional game
–
a ‘proof’ – is a restatement –
a proof refers to the proposition –
without the proposition – there will be no
restatement –
nothing to restate
to ‘allude’ is to ‘refer indirectly’ –
a proof refers directly – directly to the mathematical
proposition –
there is no allusion
‘What would it be like if a proposition itself did
not quite grasp its sense? As if it were, so to speak, too grand for itself?
That is what logicians suppose.’
‘grasping’ a proposition – is interpreting it –
a proposition does not grasp itself – does not interpret itself –
to suggest that it does is ridiculous
a proposition is interpreted by an interpreter – an
actor – a human being
an interpretation is an action upon –
an action upon a proposition
the proposition – in the absence of interpretation
– is an unknown –
is an un-interpreted sign
the sense of a proposition is what is proposed –
in terms of its meaning – its function
and a proposition’s sense is open to question –
open to doubt – is uncertain –
do logicians know what they suppose?
whatever it is they suppose – their suppositions
are open to question – open to doubt – are uncertain
grandiosity is pretension –
the true task of the logician is to expose
pretension
‘A proposition that deals with all numbers cannot
be thought of as verified by an endless striding, for, if the striding is
endless, it does not lead to any goal.
Imagine an infinitely long row of trees, and, so
that we can inspect them, a path beside them. All right, the path must be
endless. But if it is endless, then that means precisely that you can’t walk to
the end of it. That is, it does not
put me in a position to survey the row. That is to say, the endless path does
not have an end ‘infinitely faraway’, it has no end.’
a proposition that deals with all numbers is a proposal – a game proposition–
a game is played
– it is not verified
the rules of the game determine the game –
determine the play –
if the striding is endless – the point of the game
is that it is on-going –
that is to say – there is no end point
as to the row of trees – as with ‘a proposition that deals with all
numbers’ –
yes – it is not a question of observation
“Nor can you say: “A proposition cannot deal with
all the numbers one by one, so it has to deal with them by means of the concept
of number” as if this were a pis aller:
“Because we can’t do it like this, we
have to do it another way.” But it is indeed possible to deal with numbers one
by one, only that doesn’t lead to the
totality. That doesn’t lie on the
path on which we go step by step, not even at the infinitely distant end of
that path. (This all only means that “e(0). e(1). e(2) and so
on” is not the sign for a logical product.)’
yes – exactly there is no logical product in such a
propositional game
and it is just this that defines any game that
‘deals with all numbers’
‘ “It cannot be a contingent matter that all
numbers possess a property; if they do so it must be essential to them.” – The
proposition “men who have red noses are good-natured” does not have the same
sense as the proposition “men who drink red wine are good natured” even if the
men who have red noses are the same as the men who drink red wine. On the other
hand, if the numbers m, n, o are the extension of a mathematical concept, so
that it is the case that fm. fn. fo, then the proposition that the numbers
satisfy f have the property e has the same sense as “e(m). e(n). e(o)”. This is because the proposition f(m).
f(n). f(o)” and e(m). e(n). e(o)” can be
transformed into each other without leaving the realm of grammar.’
we can forget
this talk of numbers possessing a property –
the issue is
the use of numbers – the games numbers are used in –
the sense of
a proposition is a matter – open to question – open to doubt – uncertain
transforming
f(m). f(n). f(o)” and e(m). e(n). e(o)” into
each other –
is playing a game – a grammatical game
‘Now consider the proposition: “all the n numbers
that satisfy the condition F(x)
happen by chance to have the property e”. Here what matters is
whether the condition
F(x) is a mathematical one. If it
is, then I can indeed derive e(x) from f(x), if only via
the disjunction of the n values of F(x). (For what we have in this case
is in fact a disjunction). So I won’t call this chance. – On the other hand if
the condition is a non-mathematical one, we can speak of chance. For example,
if I say: all numbers I saw today on buses happened to be prime numbers. (But
of course we can’t say “the numbers17, 3, 5, 31 happen to be prime numbers” any
more than “the number 3 happens to be a prime number”), “By chance” is indeed
the opposite of “in accordance with a general rule”, but however odd it sounds
one can say that the proposition “17, 3, 5, 31 are prime numbers is derivable
by a general rule just like the proposition 2 + 3 = 5.” ’
well is it not chance – that a general rule is applied? –
that I describe the numbers on the bus as prime
numbers – interpret them in terms of the prime rule – is that not a chance use
of the prime rule?
the description of any proposition – mathematical
or not – is open to question – open to doubt – is uncertain
propositional
interpretation is uncertain –
whether you
chance it with rules – or not
‘If we now
return to the first proposition, we may ask again: How is the proposition “all
numbers have the property e” supposed to be meant? How
is one supposed to be able to know? For to settle its sense you must settle
that too! The expression “by chance” indicates a verification by successive
tests, and that is contradicted by the fact that we are not speaking of a
finite series of numbers.’
how is the
proposition to be meant?
logically
speaking – there is no – ‘to be meant’ –
there are
propositional practices and traditions yes –
however any
proposition – any proposal – is open to question – open to doubt –
is uncertain
how a
proposition is meant – is how it is used –
and any use
is open to question
how is one supposed to be able to know?
knowledge – is proposal –
and any proposal – is uncertain –
our knowledge is uncertain
we operate with and in uncertainty –
we proceed in uncertainty
as to verification –
in mathematics – we are dealing with number games –
we don’t verify games – we play them –
whether they have a logical end –
or they are on-going
‘In mathematics description and object are
equivalent. “The fifth number of the number series has these properties” says the same as “ 5 has these properties”.
The properties of a house do not follow
from its position in a row of houses; but the properties of a number are the
properties of a position.’
well it is a question of description – how you
describe the object –
in the absence of description – the object – number or house – or whatever – is an unknown
the object does not have properties independent of
description –
a properties-description – is logically speaking –
as good as any other description –
which is to say – it is open to question – open to
doubt – is – as with any proposal –
uncertain –
and what follows from what – is the art of argument
‘You might say that the properties of a particular
number cannot be foreseen. You can only see them when you’ve got there.
What is general is the repetition of an operation.
Each stage of the repetition has its own individuality. But it isn’t as if I
use the operation to move from one individual to another so that the operation
would be the means for getting from one to the other –
like a vehicle stopping at every number which we
can then study: no, applying the operation +1 three times yields and is the number 3.
(In the calculus process and result are equivalent
to each other.)
But before deciding to speak of “all
individualities” or “the totality of these individualities” I had to consider
carefully what stipulations I wanted to make here for the use of the
expressions “all” and “totality”.’
‘You might say that the properties of a particular
number cannot be foreseen. You can only see them when you’ve got there’? –
really what this amounts to is – you can’t see how
a numbers game will be used –
and that’s fair enough
the repetition of the operation – is the game-play
you play the game – the action of the game is
repetition –
the game is a game of repetition
‘In the calculus process and result are equivalent
to each other’ –
this is the case if you regard the ‘result’ of a
calculus as a play in the game
as Wittgenstein notes here
– the use and function of ‘all’ and ‘totality’ – are of course – open to
question
and this business of
determining use – is the on-going logical issue in any propositional context
‘It is difficult to
extricate yourself completely from the extensional viewpoint: You keep thinking
‘Yes, but there must still be an internal relation between x3 + y3
and z3 since at least extensions of the expressions if I only knew
them would have to show the result of such a relation”. Or perhaps: “It must
surely be either essential to all numbers to have the property or not,
even if I can’t know it.’
it is not as if numbers – through some ‘internal’
action or dynamic – determine their relation to each other –
to suggest so – is to peddle some kind of Platonic
or Pythagorean myth
numbers are marks – signs – put into play in rule
governed propositional actions –
their relations are determined by the rules governing
the propositional action –
by the rules governing the game
“If I run the number series, I either eventually
come to a number with the property e or never do.” The
expression “to run though a number series” is nonsense; unless a sense is given
to it which removes the suggested analogy with “ running through the numbers
from 1 to 100”,’
a number
series – finite or infinite – will only have any significance in terms of a
game proposal
‘When Brower
attacks the application of the law of the excluded middle in mathematics, he is
right in so far as he is directing his attack against a process analogous to
the proof of empirical propositions. In mathematics you can never prove
something like this: I saw two apples
lying on the table, and now there is only one
there, so A has eaten an apple. That is, you can’t by excluding certain
possibilities prove a new one which isn’t already contained in the exclusion
because of the rules we have laid down. To that extent there are no genuine
alternatives in mathematics. If mathematics was the investigation of
empirically given aggregates, one could use the exclusion of a part to describe
what was not excluded, and in that case the non-excluded part would not be
equivalent to the exclusion of the others.’
an empirical proposition – is open to question –
open to doubt – is uncertain
mathematics is a rule governed propositional action
–
a rule governed propositional action – is a game
the rules – as with any proposition or set of
propositions – are logically speaking –
open to question –
if you question the rules – you involve yourself in
argument – you don’t play the game
there is no question of proof in play
and there is no question of excluding possibilities
in the play of a game – there is nothing to exclude
there is only the game and its play
‘The whole
approach that if a proposition is valid for one region of mathematics it need
not necessarily be valid for a second region as well, is quite out of place in
mathematics, completely contrary to its essence. Although many authors hold
just this approach to be particularly subtle and to combat prejudice.’
a proposition is valid if it functions in the
propositional context in which it is placed
validity = function
and whether or not a proposition functions in a
particular context – one way or another will be a decision for the
practitioners
so of course a proposition that functions in one
region of mathematics need not function in a second
from a logical point of view to suggest otherwise is what is out of place in mathematics
– or for that matter any other rational propositional activity
any proposition – in any context – is open to
question – open to doubt – is uncertain
if you reject this logical reality –
you go down the road of rhetoric and prejudice
‘It is only if you investigate the relevant
propositions and their proofs that you can recognize the nature of the
generality of the propositions of mathematics that treat not of “all cardinal
numbers” but e.g. of real numbers.’
‘investigation of the relevant propositions’ –
that is seeing them as open to interpretation –
understanding them as creative possibilities –
and this can leads to proposals –
proposals for propositional games –
i.e. the cardinal numbers game – the real numbers
game
‘How a proposition is verified is what it says.
Compare generality in arithmetic with the generality of non-arithmetical
propositions. It is differently verified and so is of a different kind. The
verification is not a mere token of the truth, but determines the sense of the
proposition. (Einstein: how a magnitude is measured is what it is.)’
a proposition is a proposal – open to question – open to doubt – uncertain
how and why you accept a proposition is the
question of verification
the
generality of any proposition – is the range of application of that proposition
the range of
application of one proposition or one type of proposition may well be different
to range of another proposition or another type of proposition –
generality –
is a form of proposal – open to question – to doubt – it is uncertain –
as is any
proposition of this form
verification is decision –
to accept a proposition is to decide how you will
understand it
verification is a decision on sense
we could speak of the proposition independently of verification
– but not for long
that is to say a proposition is put – it is open to
question – and the key question is –
acceptance or rejection –
the basis on which we accept or reject a
proposition – the argument we employ –
will determine how we understand the proposition
however any argument we put for acceptance or
rejection – will itself be open to
question
we never leave uncertainty –
we can proceed with a proposition – i.e. affirm it
– in spite of its uncertainty –
or we can further explore its uncertainty
finally –
might I put that the logical position in relation to the assessment of any proposition is the ‘un-excluded middle’ –
if – the
‘un-excluded middle’ – is understood as – ‘uncertain’ –
‘true’ and ‘false’ – are operational – pragmatic –
decisions –
uncertainty is at the centre of any operational
decision
40 On set
theory
‘A misleading
picture: “The rational points lie close together on the number-line.
Is a space
thinkable that contains all rational points, but not the irrational ones? Would
this structure be too coarse for our space, since it would mean that we could
only reach the irrational points approximately? Would it mean that our net was
not fine enough? No. What we would lack would be the laws, not extensions.
Is this space
thinkable that contains all rational points but not the irrational ones?
That only
means: don’t the rational numbers set a precedent for the irrational ones?
No more than
draughts set a precedent for chess.
There isn’t any
gap left open by the rational numbers that is filled up the irrationals’
‘a misleading
picture: “The rational points lie close together on the number-line’?
yes – a
misleading picture –
what we have
is a number game – not a number-line
–
the ‘number-line’
introduces – the idea of space – and numbers in space –
when all we
are really talking about is a rule governed propositional game
imagination –
‘spatial’ imagination – doesn’t help us here –
in fact it
throws us off the track
‘is a space
thinkable that contains all rational points, but not the irrational ones?’ –
this should
read –
‘is a game
thinkable that contains all rational numbers – but not the irrational ones?’
of course
is this game
too coarse? – I wouldn’t think so
a game that contains
only rational numbers – is relative to a game that contains rational and
irrational – a different game –
our game not
fine enough? –
no – this
game is a game of rational numbers –
there is no
other kind of number in this game –
it’s as
‘fine’ as it is
and as to
laws – the real question is what game are you playing?
if the game
is the game of rational numbers – the rules are in place –
if you are
proposing a game that deals with rational and irrational numbers –
then a rule
or rules that relates the two is required
‘that only
means: don’t the rational numbers set a precedent for the irrational ones?’
does a number
game set a precedent for a number
game?
perhaps – but
so what?
‘there isn’t
any gap left open by the rational numbers that is filled up the irrationals’
if the game
is rational numbers – then that’s it –
there are no
‘irrational’ numbers in it
‘We are
surprised to find that “between the everywhere dense rational points”, there is
still room for the irrationals. (What balderdash!) What does a construction
like that for Ö2
show? Does it show how there is yet room for this point in between all the
rational points? It shows that the point yielded
by this construction, is not rational – And what corresponds to
this construction in arithmetic? A sort of number which manages after all to
squeeze in between the rational numbers? A law that is not a law of the nature
of a rational number.’
by way of
analogy – you could ask here i.e. – is there room between physical objects –
for spirit entities?
yes we could
play such a game – but it is a different
game to the physical object game
there is
nothing against constructing new
games – combining games –
but if you do
that – recognize what you are doing –
and don’t get reluctantly stuck in the old faithful –
however if
the game you want to play is the old faithful – stick to it – play that game
it’s being
clear in your head what you are on about –
and the
getting on with it
‘The
explanation of the Dedekind cut pretends to be clear when it says: there are 3
cases: either the class R has a first member and L no last number, etc. In fact
two of these 3 cases cannot be imagined, unless the words “class”, “first
member”, “the last member”, altogether change the everyday meanings they are supposed
to have retained.
That is, if
someone is dumbfounded by our talk of a class of points that lie to the right
of a given point and have no beginning, and says: give us an example of such a
class – we trot out the class of rational numbers; but that isn’t a class of
points in the original sense.’
the Dedekind
cut – is a good example of an argument to a new game – a new construction –
it can be
seen as a response to – the number-line
image – and issue
that is the
apparent incompatibility or the difficulty of reconciling – placing –
irrational numbers in a rational line up
the Dedekind
cut is a clever argument –
and
Wittgenstein is right – in this argument the goal posts get shifted – meanings
get changed –
but that is
the point –
and without these
changes there is no conceptual shift –
there is no
new game
‘The point of
intersection of two curves isn’t the common member of two classes of points,
it’s the meeting of two laws. Unless, very misleadingly, we use the second form
of expression, to define the first.’
the proposal
– ‘The point of intersection of two curves’ – as with any proposal is open to account –
‘the common
member of two classes of points’? –
‘the meeting
of two laws’? –
different
accounts – and with them – different arguments –
different
propositional backgrounds
different
propositional baggage –
that’s how it
goes
‘After all I
have already said, it may sound trivial if I now say that the mistake in the
set-theoretic approach consists time and time again in treating laws and
enumerations (lists) as essentially the same kind of thing and arranging them
in parallel series so that one fills in the gaps left by another.’
the set
theoretic approach – can be seen as
treating laws and enumerations as essentially the same thing – and arranging
them in parallel series so that one fills in the gaps left by another –
yes you can
interpret the set-theoretic approach in this way –
and if you do
– then you have – you are dealing with – a new
relationship between laws and enumeration –
I can see the
argument too – that the set theoretic approach provides a context for laws and enumerations to function in a common
setting
it’s a
‘mistake’ only if you don’t accept the set-theoretic approach –
or if you
think a new relationship between laws and enumeration is not possible or
acceptable
the set
theoretic approach – is a new operational paradigm – a new game –
a different
proposal with different possibilities
if you don’t
see the value in this approach – in this game
–
then presumably
you will not utilize it – you will not play the game
it is as
simple as that
‘The symbol
for a class is a list.’
well – if you
say so –
I think the
point is that the concept of the class is not exhausted by a list –
this ‘the
symbol of a class is a list’ – is actually an argument to trash the idea of the
class
‘Here again,
the difficulty arises from the formation of mathematical pseudo-concepts. For
instance, when we say that we can arrange the cardinal numbers, but not the
rational numbers, in a series according to their size, we are unconsciously
presupposing that the concept of an ordering by size does have a sense for rational numbers, and that it turned out on an investigation that the
ordering was impossible (which presupposes that the attempt is unthinkable). – Thus one thinks that it is possible to
attempt to arrange the real numbers
(as if that were a concept of the same kind as ‘apple on this table’) in a
series, and now it turned out to be impracticable.’
this has to
do with game construction – what works – what doesn’t
as to
‘pseudo’ – that will depend on where you are coming from and where your
allegiances and prejudices lie –
much wiser to
keep an open mind – and not to bunker down
any proposal
– in or out of mathematics is open to question – to doubt – is uncertain –
the trick is
to find a way through the maze of argument to a functional product – a game
that that can be played – that finds acceptance –
and one that
enables you to do something that wasn’t done before – or not done in the same
way
any such
result – any proposed construction – will be open to question
‘For its form
of expression the calculus of sets relies as far as possible on the form of
expression of the calculus of cardinal numbers. In some ways that is
instructive, since it indicates certain formal similarities, but it is also
misleading, like calling something a knife that has neither blade nor handle
(Lichtenberg.)’
‘formal
similarities’ – a matter of description –
the real
issue here is whether set theory takes us further than the cardinal number
calculus if there is a question of which approach to take
you have to
bear in mind that with set theory – relative to earlier or other mathematical proposals you have a conceptual
paradigm shift –
now really –
as with any paradigm shift – you either get with it – or not –
there is no
necessity here – just the option
and yes –
there is always argument for and against –
but looking
at and considering different propositional models –
and being
open to making the jump to a different propositional framework
is – if
nothing else – logical behaviour
‘(The only
point there can be to elegance in a mathematical proof is to reveal certain
analogies in a particularly striking manner, when that is what is wanted;
otherwise it is a product of stupidity and its only effect is to obscure what
ought to be clear and manifest. The stupid pursuit of elegance is a principle
cause of the mathematicians’ failure to understand their own operations; or
perhaps the lack of understanding and the pursuit of elegance have a common
origin.)’
‘what ought
to be clear and manifest’ –
‘clarity’ –
is pretence – and as with all pretence
it has its uses
any proposal
– or any result of any propositional action or exercise – is – logically
speaking – open to question – open to doubt – is uncertain –
so what is
‘clear’ – comes out as what is not a subject of question or doubt –
in a word –
clarity – is not logical – it is rhetorical –
and again I
make the point – this is not to say that such pretence – such rhetoric doesn’t
have a place and function in
propositional realities –
in
mathematics i.e. it can have a prominent role
‘manifest’? –
is what is described and given descriptive prominence –
again – any
description – is open to question – open to doubt – is uncertain –
and this
‘giving prominence’ to a description – giving a description some authority – is
a rhetorical move
as for
elegance?
yes –
elegance for elegance’s sake – has no value
however the
real issue here is decision – and the criteria for decision –
and the
logical reality here is that there is no absolute criterion or standard –
we tend to
run with those proposals for criteria that are in play –
and that have
become entrenched in the practice
a careful
survey of the practice shows that in any decision there are options for how we
go about deciding
elegance – in
certain propositional traditions and endeavors – is one such option – one such
criteria
if it has a
use – it has a use –
but as with any
proposal – open to question – open to doubt
‘Human beings
are entangled all unknowing in the net of language’
so
beautifully put
our reality
is propositional –
our reality
is open to question – open to doubt –
our reality
is uncertain
“There is a
point where the two curves intersect.” How do you know that? If you tell me, I
will know what sort of sense the proposition “there is …” has.’
‘There is a
point where the two curves intersect’ – is a proposal
our knowledge
is what is proposed –
and what is
proposed is open to question – open to doubt – is uncertain
‘there is …’
is a proposal
–
what sense it
has – is open to question –
sense is
uncertain
‘If you know
what the expression “the maximum of a curve” means, ask yourself: how does one
find it? – If something is found in a different way it is a different thing. We
define the maximum as the point on the curve higher than all the others, and
from that we get the idea that it is only our human weakness that prevents us
from sifting through the points of the curve one by one and selecting the
highest of them. And this leads us to the idea that the highest point among a
finite number of points is essentially the same as the highest point of a
curve, and that we are simply finding out the same by two different methods,
just as we find out in two different ways that there is no one in the next
room; one way if the door is shut and we weren’t strong enough to open it, and
another if we can get inside. But, as I said, it isn’t human weakness that is
in question where the alleged description of the action “That we cannot
perform” is senseless. Of course it does no harm, indeed it is very
interesting, to see the analogy between the maximum of a curve and the maximum
(in another sense) of a class of points, provided that the analogy doesn’t
instill the prejudice that in each case we have fundamentally the same thing.’
the notion of ‘the maximum of a curve’ – is best
seen as a game-proposal –
the question is how would you play this game – what
rules would you put in place to enable a way of playing it – and where would
such a game have an application – a use?
‘It’s the same
defect in our syntax which presents the geometric proposition “a length may be
divided by a point into two parts” as a proposition of the same form as
“a length may
be divided for ever”; so that it looks as if in both cases we can say “Let’s
suppose the possible division to have been carried out”. “Divisible into two
parts” and “infinitely divisible” have quite different grammars. --We mistakenly treat the word “infinite” as
if it were a number word, because in everyday speech both are given as answers
to the question “how many?” ’
it is not a defect in our syntax –
there is no
defect in propositional
representation – but there is uncertainty –
as
Wittgenstein demonstrates here – the issue is interpretation –
he puts an
argument –
‘We mistakenly treat the word “infinite” as if it were a number word,
because in everyday speech both are given as answers to the question “how
many?” ’
fair enough
–
that just
is what it is about – dealing with propositional uncertainty –
putting a
case and arguing for it –
and I would
add –
recognizing
that any argument put –,any interpretation proposed –
is open to
question – open to doubt – is itself – uncertain
“But after
all the maximum is higher than other arbitrary points of the curve.” But the
curve is not composed of points, it is a law that points obey, or again, a law
according to which points can be constructed. If you now ask: “which points?” I
can only say “well, for instance, the points P, Q, R, etc.” On the one hand we
can’t give a number of points and say that they are all the points that lie on
the curve, and on the other hand we can’t speak of a totality of points as
something describable which although we humans cannot count them might be
called the totality of all the points on the curve – a totality too big for
human beings. On the one hand there is a law, and on the other points on a
curve; – but not “all the points of the curve”. The maximum is higher
than any point of the curve that happens to be constructed, but it isn’t higher
than the totality of points, unless the criterion for that, and thus the sense
of the assertion, is once again simply construction according to the law of the
curve.’
yes –
Wittgenstein’s final point here makes this notion of ‘the maximum of the curve’ – intelligible –
relative to an actual curve – a constructed curve
and if the
‘maximum of the curve’ is not meant
to have a relation to an actual curve –
a constructed
curve –
where is the
notion of ‘curve’ supposed to come from?
so what do we
have here?
a game that
begins with an image – the curve – an image we are all familiar with – but a
game that in fact has no relation to the image when all is said and done?
the same old
‘infinity game’ – wrapped up in a curve?
‘Of course
the web of errors in this region is a very complicated one. There is also e.g.
the confusion between two different meanings of the word “kind”. We admit, that
is, that the infinite numbers are a different kind of number from the finite ones, but then we misunderstand what
the difference between different kinds amounts to in this case. We don’t
realize, that is, that it’s not a matter of distinguishing between objects by
their properties in the way we distinguish between red and yellow apples, but a
matter of different logical forms. – Thus Dedekind tried to describe an infinite class by saying
that it is a class which is similar to a proper subclass of itself. Here it
looks as if he has given a property that a class must have in order to fall
under the concept “infinite class” (Frege). Now let us consider how this
definition is applied. I am to investigate in a particular case whether a class
is finite or not, whether a certain row of tress, say, is finite or infinite.
So, in accordance with the definition, I take a subclass of the row of trees
and investigate whether it is similar (i.e. can be coordinated one to one) to
the whole class! (Here already the whole thing has become laughable.) It hasn’t
any meaning; for, if I take “finite class” as a sub-class, the attempt to
coordinate it one to one with the whole class must eo epso fail; and if I make the attempt with an infinite class –
but already that is a piece of nonsense, for if it is infinite, I cannot make
an attempt to coordinate it. – What we call ‘correlation of all the members of
a class with others’ in the case of a finite class is something quite different
from what we, e.g., call a correlation of all cardinal numbers with all
rational numbers. The two correlations, or what one means by these words in the
two cases, belong to different logical types. An infinite class is not a class
which contains more members than a finite one, in the ordinary sense of the
word “more”. If we say that an infinite number is greater than a finite one,
that doesn’t make the two comparable, because in that statement the word
“greater” hasn’t the same meaning as it has say in the proposition 5 > 4!’
what we are
dealing with is different games – different propositional games
Dedekind
defines an infinite class as a class which is similar to a proper subclass of
itself
ok – if the subclass is infinite –
and what is
the point of such a ‘definition’?
defining x as x – is hardly clever – it is just a
statement of the obvious – and a waste of breath –
and saying x
is not y – is blindingly obvious as well
and let’s be
clear – this issue is games – propositional games –
games are
played – and their play is their
definition
furthermore –
the notion of
class is irrelevant and pointless
once it is understood that what we are dealing with is not different classes
but rather different games or different types of game –
the ‘finite
game’ and the ‘infinite game’
the idea or
rule of the finite game is that it has a definitive logical end –
the idea or
rule of the infinite game is that its logic is on-going
the terms or
numbers in a finite game reflect the rule of the game and are reflected in the
action of the game
and the terms
or numbers in an infinite game reflect the rule of the game and are reflected
in the action of that game
and yes
‘different’ means different – different games – different logics – different
practices
and
‘different’ does not mean ‘comparable’ – it means ‘incomparable’ –
yes we can
say that different games – are games – but that is really where any relevant or
significant comparison ends
and again –
to say this is only to state the obvious
‘That is to
say, the definition pretends that whether a class is finite or infinite follows
from the success or failure of the attempt to correlate a proper subclass with
the whole class; whereas there just isn’t any such decision procedure –
‘Infinite class’ and ‘finite class’ are different logical categories; what can
be significantly asserted of the one category cannot be significantly asserted
of the other.’
yes –
different logical categories – different logics – different games
‘With regard
to finite classes the proposition that a class is not similar to its subclasses
is not a truth but a tautology. It is the grammatical rules for the generality
of the general implication in the proposition “k is a subclass of K” that
contain what is said by the proposition that K is an infinite class.’
a finite
class will be similar to its sub-classes if the criterion of similarity is type
of class i.e. – finite class
where there
is a difference in type of class – i.e. – the class is infinite – and the
sub-class is finite – you have an internal contradiction –
a finite
subclass in an infinite class – is a dead zone
a logical
black hole – ready to implode –
in any case
either a dysfunctional – or at least a
very odd game –
a game of
contradicting logics
‘A
proposition like “there is no last cardinal number” is offensive to naive – and
correct – common sense. If I ask “Who was the last person in the procession?”
and I am told “There wasn’t a last person” I don’t know what to think: what
does “There wasn’t a last person” mean? Of course, if the question had been
“Who was the standard bearer?” I would have understood the answer “There wasn’t
a standard bearer”; and of course the bewildering answer is modeled on the
answer of that kind. That is, we feel, correctly, that where we can speak at
all of a last one, there can’t be “No last one”. But of course that means: The
proposition “There isn’t a last one” should rather be: it makes no sense to speak
of a “last cardinal number”, that expression is ill-formed.’
yes
if someone
says ‘there is a last cardinal number’ – they don’t understand the cardinal
number game
to say ‘There
wasn’t a last person in the procession’ – is to fail to understand the finite
numbers game
“Does the
procession have an end?” might also mean: is the procession a compact group?
And now someone might say: “There, you see, you can easily imagine a case of
something not having an end; so why can’t there be other such cases?” – But the
answer is: The “cases” in this sense of the word are grammatical cases, and it
is they that determine the sense of the question. The question “Why can’t there
be other such cases” is modeled on: “Why can’t there be other minerals that
shine in the dark”; but the latter is about cases where a statement is true,
the former about cases that determine sense.’
‘you can
easily imagine a case of something not having an end’ –
yes – you can
imagine this game – you can conceive
its logic –
‘other cases’?
– whatever ‘the case’ or ‘other cases’ are –
it is the
same game – the one game –
the game
without an end
‘The form of
expression “m = 2n correlates a class with one of its proper subclasses” uses a
misleading analogy to clothe a trivial sense in a paradoxical form. (And
instead of being ashamed of this paradoxical form as something ridiculous,
people plume themselves on a victory over all prejudices of the understanding).
It is exactly as if one changed the rules of chess and said it had been shown
that chess could also be played quite differently. Thus we first mistake the
word “number” for a concept word like “apple”, then we talk of a “number of
numbers” and we don’t see that in this expression we shouldn’t use the word
“number” twice; and finally we regard it as a discovery that the number of the
even numbers is equal to the number of the odd and even numbers.’
a game – is a
rule governed propositional action –
the rules of
a game are propositions – and as with any proposition – the rules of a game –
are open to question – open to doubt – and are as propositions – proposals –
uncertain
so rules can
be reinterpreted – can be changed
if you change
the rules of chess and say it can be shown that chess could also be played
quite differently –
the only
question then would be – is this different game of chess – still to be called
‘chess’?
because there
is no question that with different rules we have a different game
I think it is
quite unlikely that we would call both games ‘chess’ –
but this is
no argument against proposing a new game
and the same
is true with number games –
there is
nothing against proposing a new type of number game –
however any
such proposal will be the subject of argument
and whether
or not a new game is put to play–
will finally
be a decision for the players
‘It is less
misleading to say “m = 2n allows the possibility of correlating every time with
another” than to say “m = 2n correlates all numbers with others”. But here too
the grammar of the meaning of the expression “possibility of correlation” has
to be learnt.’
this has to
do with just how the game ‘m = 2n’ can be played –
in what
domains it has application
the grammar
of the meaning of the expression ‘possibility of correlation’ – is open to
question – is open to doubt – is uncertain –
for a game to
be – and to be played – rules – ‘rules of grammar’ – if you like – need to be
set
‘(It’s almost
unbelievable, the way in which a problem gets completely barricaded in by the
misleading expressions which generation upon generation throw up for miles
around it, so that it becomes virtually impossible to get at it.)’
there are ‘no
misleading expressions’ –
any
expression – any proposal – is open to question – to doubt – is uncertain
but if you
don’t see this – you may well get trapped with the generations upon generations
–
who didn’t
see it either
‘If two
arrows point in the same direction, isn’t it in such a case absurd to call
these directions equally long,
because whatever lies in the direction of one arrow, lies in that of the other?
– The generality of m = 2n is an arrow that points along the series generated
by the operation. And you can even say that the arrow points to infinity; but
does that mean there is something – infinity – at which it points, as at a
thing? – It’s as though the arrow designates the possibility of a position in
its direction. But the word “possibility” is misleading, since someone will
say: let what is possible now become actual. And in thinking this we always
think of a temporal process, and infer from the fact that mathematics has
nothing to do with time, that in its case possibility is already actuality.
The “infinite
series of cardinal numbers” or “the concept of cardinal number” is only such a
possibility – as emerges clearly from the symbol “| o,
x, x +
1 |
”. This symbol is itself an arrow with the “o” as its tail and the “x +
1”
as its tip. It is possible to speak of things which lie in the direction of the
arrow, but misleading or absurd to speak of all possible positions for things
lying in the direction of the arrow itself. If a search light sends out light
into infinite space it illuminates everything in its direction, but you can’t say it illuminates
infinity.’
we use the
term ‘long’ in connection with length – with measurement
the notion of
‘direction’ comes from geometry
if the arrow
points to infinity – then infinity is no
place
if
mathematics has nothing to do with time – we can’t say possibility is already actuality
the symbol “½o,
x, x +
1 | ”
is best understood as a game symbol
‘the
‘infinite series of cardinal numbers’ or ‘the concept of cardinal number’ –
are game
terms
the infinite
game is not one that actually goes on forever
rather it is
a game that whenever played – has no logical end point
possibility
in a game is rule governed indeterminacy
actuality is
play
‘It is always
right to be extremely suspicious when proofs in mathematics are taken with
greater generality than is warranted by the known application of the proof.
This is always a case of the mistake that sees general concepts and particular
cases in mathematics. In set theory we meet this suspect generality at every
step.
One always
feels like saying “let’s get down to brass tacks”.
These general
considerations only make sense when we have a particular region of application
in mind.
In
mathematics there isn’t any such thing as a generalization whose application to
particular cases is still unforeseeable. That’s why the general discussions of
set theory (if they aren’t viewed as calculi) always sound like empty chatter,
and why we are always astounded when we are shown an application for them. We
feel that what is going on isn’t properly connected with real things.’
‘It is always
right to be extremely suspicious when proofs in mathematics are taken with
greater generality than is warranted by the known application of the proof’? –
proofs in
mathematics – are arguments – are proposals –
any proposal
is open to question to doubt – is uncertain
and it is
this uncertainty that gives a proposal scope beyond a practiced application
Wittgenstein
here is not putting forward an open or logical perspective – rather he advances
a closed or dogmatic point of view
it is not
really a question of generality – or as he puts it – ‘suspect’ generality –
what it is
about is understanding that – whatever the use a proposition is put to –
whatever the practice – the proposition qua
proposition – logically speaking is open
or if you
want to put it in terms of ‘generality’ – generality is open
of course it
can be limited to ‘known applications
of the proof’
however if we
only think in terms of known applications
– there will be no prospect of any advance in mathematical thinking –
the real
importance of the ‘known application’ to mathematical thinking is that it
provides a starting point for speculation beyond what is accepted and practiced
taking a step beyond the known application – may
just result in a new way of understanding – and indeed a new theory – a new
game – a new calculus –
this is the realm of pure mathematics
‘This is
always a case of the mistake that sees general concepts and particular cases in
mathematics. In set theory we meet this suspect generality at every step.
there is no
‘mistake’ here – what we have is a different perspective – and set theory is –
a different perspective
‘One always
feels like saying “let’s get down to brass tacks”’ –
what I
suspect this means is – ‘let’s go down well trodden paths’ –
and of course
if that is how you want to proceed – why not?
‘These
general considerations only make sense when we have a particular region of
application in mind.’
having a
particular region of application in mind – does not exhaust the possibilities
of a ‘general consideration’
‘a general
consideration’ – is a proposal – open to question – open to doubt –
uncertain
‘In
mathematics there isn’t any such thing as a generalization whose application to
particular cases is still unforeseeable. That’s why the general discussions of
set theory (if they aren’t viewed as calculi) always sound like empty chatter,
and why we are always astounded when we are shown an application for them. We
feel that what is going on isn’t properly connected with real things.’
the
particular case will be represented as a result of the generalization – if the
generalization is applied to it
the same is
true in set theory
‘We feel that
what is going on isn’t properly connected with real things’? –
‘real things’
– is what is proposed –
and any
proposal – is open to question – open to doubt – is uncertain –
‘The
distinction between the general truth that one can know, and the particular one
doesn’t know, or between the known description of the object, and the object
itself that one hasn’t seen, is another example of something that has been
taken over into logic from the physical description of the world. And that too
is where we get the idea that our reason can recognize questions but not the
answers.’
what we
‘know’ is what is proposed – however you describe the proposal – i.e. ‘general’
– ‘particular’
in the
absence of description – the object of knowledge – is unknown –
description –
proposal – makes known –
a proposal –
be it in ‘logic’ – or ‘the physical sciences’ – is open to question – open to
doubt – is uncertain
‘Set theory
attempts to grasp the infinite at a more general level than the investigation
of the laws of real numbers. It says that you can’t grasp the actual infinite
by means of mathematical symbolism at all and therefore it can only be
described and not represented. The description would encompass it in something
like the way in which you carry a number of things that you can’t hold in your
hand by packing them in a box. They are then invisible but we still know we are
carrying them (so to speak, indirectly). One might say of this theory that it
buys a pig in a poke. Let the infinite accommodate itself in this box as best
it can.
With this
there goes too the idea that we can use language to describe logical forms. In a description of this sort the
structures are presented in a package and so it does look as if one could speak
of a structure without reproducing it in the proposition itself. Concepts which
are backed up like this may, to be sure, be used, but our signs derive their
meaning from definitions which package the concepts in this way; and if we
follow up these definitions, the structures are uncovered again.’
set theory is
a different game to the game of real numbers
the ‘actual infinite’?
what is proposed is what is actual –
and what is proposed – is open to question – to
doubt – is uncertain
what counts as ‘mathematical symbolism’ – will
depend on what mathematical game you play
described but
not represented? –
any
description is a representation –
any
description / representation – is a proposal
a theory that buys a pig in a poke? –
yes – a game – a true game – where the moves that
are made – are not known –
before they are made –
where the domain
of the game is an unknown
‘With this
there goes too the idea that we can use language to describe logical forms’
what is a ‘logical form’ – but a description – a
proposal?
‘In a description of this sort the structures are presented in a package
and so it does look as if one could speak of a structure without reproducing it
in the proposition itself’? –
this is a
clumsy way of putting it but – if there is a structure – we have a proposal –
we have a proposition
‘Concepts
which are backed up like this may, to be sure, be used, but our signs derive
their meaning from definitions which package the concepts in this way; and if
we follow up these definitions, the structures are uncovered again.’ –
yes – but
what this amounts to is proposals for – or regarding – proposals –
a sign – is a
proposal – but as with any proposal it will have a propositional background –
which may well be described in terms of concepts – definitions – structures –
or you can
run it any way you like –
a concept
resolved in terms of structures – which can be put as definitions and
represented as a sign
a definition
proposed as a structure – represented as a sign – accounted for conceptually –
etc. –
‘When ‘all
apples’ are spoken of, it isn’t, so to speak, any concern of logic how many
apples there are. With numbers it is different; logic is responsible for each
and every one of them.
Mathematics
consists entirely of calculations.
In
mathematics everything is algorithm
and nothing is meaning; even when it
doesn’t look like that because we seem to be using words to talk about
mathematical things. Even these words are used to construct an algorithm.
In set theory
what is calculus must be separated off from what attempts to be (and of course
cannot be) theory. The rules of the
game have to be separated off from inessential statements about the chessmen.’
how are you
to be responsible for all your children – if don’t know how many you have?
look – it is
true that ‘how many’ is irrelevant to ‘all’
the ‘how many
game’ is a different game altogether to the ‘all game’
‘responsibility’
is a misplaced notion here
calculation is
the play of the game –
there is more
to the game than its play – if there wasn’t there would be no play
the
‘algorithm’ – is a precisely described procedure that is applied and can be
systematically followed through to a conclusion –
so rather
than saying ‘nothing is meaning’ – I would put it that the algorithm – that is
the game-play – just is the meaning – and all it comes to –
as to
‘mathematical things’ – there is always a place for imagination – for
visualization – in language use
a calculus is
the rule of play –
the play only
occurs in a game –
theory – is
the game proposed
‘chessman’ –
are nothing – without chess
‘In Cantor’s
alleged definition of “greater”, “smaller”, “+”, “-” Frege replaced the signs
with new words to show the definition wasn’t really a definition. Similarly in
the whole of mathematics one might replace the usual words, especially the word
‘infinite’ and its cognates, with entirely new and hitherto meaningless
expressions so as to see what the calculus with these signs really achieves and
what it fails to achieve. If the idea was widespread that chess gave us
information about kings and castles, I would propose to give the pieces new
shapes and different names, so as to demonstrate that everything belonging to
chess has to be contained in its rules.’
proposing a
system – a new game – will inevitably begin with what is in practice –
and
presumably such a proposal will come out of a view that the standard practice
is deficient in some respect –
or possibly
be a consequence of an entirely new perspective – ‘out of left field’ – so to
speak
and for a new
game-proposal to gain support it must be argued
part of this
process may well be replacing terms in standard practice with new words
and with the
idea of seeing ‘what the calculus with the new terms achieves and fails to
achieve’ –
but here we
are not simply ‘replacing words’ for the sake of it –
new terms
will foreshadow – hint at – or introduce new ways of thinking –
all in all a
very messy – uncertain business
if there is a
result that gains acceptance among practitioners – then what you will have is a
new game
a new and different game
‘If the idea
was widespread that chess gave us information about kings and castles, I would
propose to give the pieces new shapes and different names, so as to demonstrate
that everything belonging to chess has to be contained in its rules.’
chess is
chess – yes
if a new and
different game emerges out of chess –
it is not
chess
‘What a
geometrical proposition means, what kind of generality it has, is something
that must show itself when we see how it is applied. For even if someone
succeeded in meaning something intangible by it it wouldn’t help him, because
he can only apply it in a way which is quite open and intelligible to every
one.
Similarly, if
someone imagined the chess king as something mystical it wouldn’t worry us
since he can only move him on the 8 x 8 squares of the chess board.’
the
proposition – the proposal – ‘geometrical’ – or not – is open – open to question
– open to doubt – is – uncertain –
how the
proposition is interpreted – is open
to question
‘something
intangible by it’? –
intangible
relative to what?
presumably a
given – or the given interpretation
‘open and
intelligible to everyone’
the idea here
is that there is one interpretation – one perspective – and every one has it
from a
logical point of view – this is no more than rhetoric – authoritarian
rhetoric –
I mean really
– how is anyone to know – what
intelligibility is – amounts to – for
everyone?
that there
may be a common understanding in a particular culture at a particular time – is
one thing –
that such is
the only understanding – or the only possible understanding – is quite
another
rhetoric may
be the final bastion of most arguments – but that is really where logic begins
yes – imagining
the chess king as something mystical – would make no difference in a game of
chess –
if on the
other hand – the ‘chess king’ piece – had been co-opted to another game – say a
mystical game –
we would say
the piece – formerly known as the ‘chess king’ had been re-interpreted
‘We have a
feeling “There can’t be possibility and actuality in mathematics. It’s all on one level. And is in a sense, actual. – And that is correct. For
mathematics is a calculus; and the calculus does not say of any sign that it is
merely possible, but is concerned
only with the signs with which it actually operates. (Compare the foundations
of set theory with the assumption of a possible calculus with infinite signs).’
it is
important to understand that when we play
the game of mathematics – we play in
terms of the signs of the game
as to the
business of game construction – yes
we come face to face with uncertainty –or – possibility – if you like
the
foundations of any theory – are open to question – open to doubt – are
uncertain
‘possible
calculus with infinite signs’ – if it is to be anything more than speculation –
will have to
be fashioned in a rule governed propositional system –
brought to
heel – so to speak
‘When set
theory appeals to the human impossibility of a direct symbolization of the
infinite it brings in the crudest imaginable misinterpretation of its own
calculus. It is of course this very misinterpretation that is responsible for
the invention of the calculus. But of course that doesn’t show the calculus in
itself to be something incorrect (it would be at worst uninteresting) and it is
odd to believe that this part of mathematics is imperiled by any kind of
philosophical (or mathematical) investigations. (As well say that chess might
be imperiled by the discovery that wars between two armies do not follow the
same course as battles on the chessboard.) What set theory has to lose is
rather the atmosphere of clouds of thought surrounding the bare calculus, the
suggestion of an underlying imaginary symbolism, a symbolism which isn’t
employed in its calculus, the apparent description of which is really nonsense.
(In mathematics anything can be imagined, except for a part of our calculus.)’
‘human impossibility’
–
yes – well
whatever that is supposed to be – it is open to question – open to doubt –
is indeed –
uncertain
‘direct
symbolization of the infinite’ –
all
symbolization is direct
symbolization
– of anything – is a proposal to make
known
in
mathematics what a symbol signifies is a propositional structure
where a
symbol gains operational credence – it functions as a game – or a game within a
game
‘the
infinite’ – is a recursive sign game
no
proposition is imperiled by any investigation
a calculus is
the game as played
it could well
be said that an ‘atmosphere of clouds of thought’ – envelopes – any proposition
–
‘imaginary
underlying symbolism’ – is speculation –
nothing to be
afraid of
mathematics
is an imagined propositional action
it will
continue to be imagined –
and indeed –
re-imagined
41 The extensional conception of the real numbers
‘Like the enigma of time for Augustine, the enigma
of the continuum arises because language misleads us into applying to it a
picture that doesn’t fit. Set theory preserves the inappropriate picture of
something discontinuous, but makes statements about it that contradict the
picture, under the impression that it is breaking with prejudices; whereas what
should really have been done is to point out that the picture doesn’t fit, that
it certainly can’t be stretched without being torn, and that instead of it one
can use a new picture in certain respects similar to the old one’
if you are going to put forward a new proposal – it
will only be taken seriously if it is placed in contrast to an accepted or
given outlook –
and the new perspective put in contrast to the
alternative view – will have an argumentative context and a base from which to
move on from –
and to move to – a new picture
‘The
confusion in the concept of the “actual infinite” arises from the unclear
concept of irrational number, that is from the fact that very different things
are called “irrational numbers” without any clear limits being given to the
concept. The illusion that we have a firm concept rests on our belief that in
signs of the form “o. abcd … ad infinitum” we have a pattern to which they (the
irrational numbers) have to conform whatever happens.’
‘the unclear
concept of irrational number’
any concept –
is open to question – open to doubt – is
uncertain –
‘very
different things are called irrational numbers’ –
ok – so then
we need new terminology –
or it might
be that ‘irrational numbers’ are used differently in different propositional
games –
and in that
case what needs to be distinguished is use
any ‘limits’
placed on the concept – will be a result of how it is used
placing limits
on a concept is simply defining it for use –
and of course
any such definition – will always be open to question
‘the illusion
that we have a firm concept’ –
is really
just the failure to understand the reality of propositional logic –
that any proposal
is open to question
‘The illusion
that we have a pattern to which they the irrational numbers conform to whatever
happens’
yes – this is
just dumb – dogmatic rhetoric –
and it
betrays a real lack of understanding of logic and mathematics –
we operate in
propositional uncertainty –
if we are to
avoid getting stuck in self-satisfied ignorance – we must reflect this
propositional reality in our thinking and action –
we need to be
open minded and flexible –
with all our
concepts – all our proposals
‘Suppose I
cut a length at a place where there is no rational point (no rational number).”
But can you do that? What sort of a length are you speaking of? “But if my
measuring instruments were fine enough, at least I could approximate without
limit to a certain point by continued bisection”! – No, for I could never tell
whether my point was a point of this kind. All I could tell would always be
that I hadn’t reached it. “But if I carry out the construction of Ö2
with absolutely exact drawing instruments, and then by bisection approximate to
the point I get, I know that this
process will never reach the constructed point.” But it would be odd if one
construction could as it were prescribe something to the others in this way!
And indeed that isn’t the way it is. It is very possible that the point I get
by means of the ‘exact’ construction of Ö2 is reached by the bisection
after say 100 steps; but in that case we could say: our space is not
Euclidean.’
‘Suppose I
cut a length at a place where there is no rational point (no rational number)’
then the
length cannot be measured in terms of rational points – end of story
if on the
other hand an irrational system of measurement is used – the result will be
irrational – indeterminate –
such a result
put against the rational standard – will be regarded as – inadequate – if not a
straight out failure
on the other
hand the ‘irrationals’ can say that the rational system is simplistic –
outdated and clunky –
can they say
it is not as precise?
this is a
tricky one – is an infinitesimal ‘precise’?
I think you
would have to say that the irrational system challenges the rational notion of
precision –
or as
Wittgenstein says –
‘but
in that case we could say: our space is not Euclidean’
‘The “cut at the rational point” is a picture, and
a misleading picture’
we are dealing with different mathematical systems here – different mathematical games
and it really is naïve to use the term ‘misleading’
‘cut at the rational point’ – may just be all you
need –
meaning the rational system and perspective – suits
your purposes –
if it doesn’t then you won’t use it
‘A cut is a
principle of division into greater and smaller’
that’s fair
enough – as far as it goes
‘Does a cut
through a length determine in advance the results of all bisections meant to
approach the point of the cut? No.’
the point of
the cut is indeterminate
any approach
to the point – is indeterminate –
and you are
left wondering why talk about a point
at all?
there is no
point
so what is
going on here?
clearly the
notion of the point as a determination
–
and
‘determination’ is a concept that comes from a rational framework
and it is this framework that makes space for the irrational game –
that enables
its extension
so how are we
to see this relation?
does the
irrational game undermine the rational game – and is it therefore illegitimate?
or is it that
in effect we have a combining of the two games – into a larger game?
a larger game
that really cannot be assessed in terms of either game?
it is easy to
understand the logical uneasiness here
we have a
practice – but does it have a secure foundation?
I think if
you are looking for secure foundation – the answer is – there is no such
thing – wherever you are looking
the
foundation of practice – just is practice –
and theorists
rush to cobble together a conceptual basis – for what happens – for what occurs
–
and there is
value in doing this –
in that it
throws up arguments – and can lead to valuable insights that enrich the
practice
nevertheless
the hard reality is just what occurs – the practice
and in terms
of explanation – logically speaking – we are in the realm of uncertainty
I can see the
view that the irrational game is a breakaway – that leads nowhere
against this
I would say that recognizing uncertainty – facing its reality – and exploring
it – is logical –
and is in
fact – the rational way to proceed in an uncertain reality
‘In the
previous example in which I threw dice to guide me in the successive reduction
of an interval by the bisection of a length I might just as well have thrown
dice to guide me in the writing of a decimal. Thus the description “endless
process of choosing between 1 and 0” does not determine a law in the writing of
a decimal. Perhaps you feel like saying: the prescription for the endless
choice between 0 and 1 in this case could be reproduced by a symbol like “o 000
… ad. infin.”. But if I
111
adumbrate a
law thus ‘0.00100100 … ad infin.”, what I want to show is not the finite
section of the series, but rather the kind of regularity to be perceived in it.
But in
“o 000 … ad.
infin.”. I don’t perceive any law, on the contrary, precisely that a law is
111
absent.’
yes – and so
the question is how to represent or present the game –
what
propositional construction suits one’s philosophical perspective or argument?
it is just
the choice of description
what bothers
me here is that the description ‘endless process of choosing between 1 and 0’
strikes me as a ‘game’ – for which it is hard to see how rules could be written
–
and if that’s
the case – are we in fact talking mathematics?
perhaps we
are here in the realm of mysticism –
mysticism
masked as mathematics?
and as for
mysticism – anything goes there
on the other hand –
within the
game – the framework of – ‘the endless process of choosing between 1 and 0’ –
‘anything
goes’ – just does fit with ‘logic’ of indeterminacy –
a different
logic – different game – different applications
‘(What
criterion is there for the irrational numbers being complete? Let us look at an
irrational number: it runs through a series of rational approximations. When
does it leave this series behind? Never. But then, the series also never comes
to an end.
Suppose we
had the totality of all irrational numbers with one exception. How would we
feel the lack of this one? And – if it were to be added – how would it fill the
gap? Suppose that it’s p. If an irrational number
is given through the totality of its approximations, then up to any point taken at random there is a
series coinciding with that of p. Admittedly for each such
series there is a point where they diverge. But this point can lie arbitrarily
far ‘out’, so that for any series agreeing with p I can find
one agreeing with it still further. And so if I have the totality of all
irrational numbers except p, and now I insert p I cannot cite a point at which p is now really needed. At every point it has
a companion agreeing with it from the beginning on.
To the
question “how would you feel the lack of p” our answer
must be “if p were an extension, we
would never feel the lack of it”. i.e. it would be impossible for us to observe
a gap that it filled. But if someone asked us ‘But have you then an infinite
decimal expansion with the figure m in the r-th place and in the n in the s-th
place, etc? we could always oblige him.)’
‘(What
criterion is there for the irrational numbers being complete? Let us look us
look at an irrational number: it runs through a series of rational
approximations. When does it leave this series behind? Never. But then, the
series also never comes to an end.’ –
yes – the
‘irrational number’ – just is an approximation
to a rational number
the
irrational number only has any sense – as an approximation to a rational number
imagine the
absence of rational numbers – a world without
rational numbers
a world
without rational numbers – is a world without irrational numbers
here is the
argument that irrational numbers are a subset
of rational numbers –
is there a
better way of putting it?
as for
irrational numbers being complete? –
the essence –
or rule – of the irrational number is that it is incomplete
and
‘incomplete’ here – means incomplete relative to a whole number – or a series
of whole numbers
can we say
that mathematics recognizes and deals with the complete and incomplete –
and in that
sense covers the full range of logical constructions?
that there is
no unifying theory must drive some logicians to distraction –
however the
real value of mathematics rests just in the fact that it comprehensively
represents the conceptual possibilities that we encounter in our reality – that
we make of our reality –
we design and
play games that are complete –
we design and
play games that are incomplete
‘Suppose we
had the totality of all irrational numbers with one exception’. How would we
feel the lack of this one? And – if it were to be added – how would it fill the
gap?’ –
in a game
with all irrational numbers with one exception?
in a board
game with an infinite numbers of squares – and there is one square you cannot
fall on – the challenge I presume would be to avoid that square –
or conversely
the challenge could be to find it
how would we
feel the lack? –
there would
be no ‘lacking’ – the exception – would be what defines the game –
the game is
complete
‘And – if it
were to be added – how would it fill the gap?’ –
if it were
added – it would change the game –
we would then
have the makings of another game
‘To the
question “how would you feel the lack of p” our answer
must be “if p were an extension, we
would never feel the lack of it”. i.e. it would be impossible for us to observe
a gap that it filled. But if someone asked us ‘But have you then an infinite
decimal expansion with the figure m in the r-th place and in the n in the s-th
place, etc? we could always oblige him.)’ –
what we have here is another game
‘ “The
decimal fractions developed in accordance with a law still need supplementing
by an infinite set of irregular infinite decimal fractions that would be
“brushed under the carpet” if we were to restrict
ourselves to those generated by a law.”
Where is there such an infinite decimal that is generated by no law? And how
would we notice that it was missing? Where is the gap it is needed to fill?’
is there a need
for supplementing – or is any such ‘supplementing’ – really just a ‘filling
out’ of a theoretical background?
there is no infinite decimal – generated by no law
if such is generated – we have a game proposal – a ‘law’
how would we notice that it is missing?
we would only notice that it was missing if it was
required for the game –
and if it was required for the game – it would be
there – in the game
the ‘game’ has no missing parts – no gaps –
(you can’t play
a game with missing parts)
you have to be clear in your head what game you are
playing – and what games you are not playing –
or what is relevant to the game you are playing –
and what is not
it’s a question of focus
‘What is it
like if someone so to speak checks the various laws for the construction of
binary fractions by means of the set of finite combinations of the numeral 0
and 1? –
The results
of a law run through the finite combinations and hence the laws are complete as
far as their extensions are concerned, once all the finite combinations have
been gone through.’
this is
simply to apply a rule – and in so doing – make a game
‘If one says:
two laws are identical in the case where they yield the same result at every
stage, this looks like quite a general rule. But in reality the proposition has
different senses depending on what is the criterion for their yielding the same
result at every stage. (For of course there is no such thing as the supposed generally
applicable method of infinite checking!) Thus under a mode of speaking derived
from an analogy we conceal the most various meanings, and then believe that we
have united the most various meanings into a single system.’
it is not that two laws / rules are identical in
the case where they yield the same result – the two rules are different – but they yield the same
result
if the laws are identical there is only one law
that two rules can yield the same result – just
points to the logical reality that a result can be arrived at – through
different methods – via different paths
it is not a matter of ‘concealing various meanings’
– any proposition / rule is open to
interpretation –
the various meanings are ‘united’ – if you want to
use that term – in the fact that a proposition – a proposal is open to question
– open to doubt – is uncertain
‘(The laws corresponding to the irrational numbers
all belong to the same type to the extent that they must ultimately be recipes
for the successive construction of decimal fractions. In a certain sense the
common decimal notation gives rise to a common type.)
We could also put it thus: every point in a length can be approximated to by repeated
bisection. There is no point that can only be approximated to by irrational
steps of a specified type. Of course, that is only a way of clothing in
different words the explanation that by irrational numbers we mean endless
decimal fractions; and that explanation in turn is only a rough explanation of
the decimal notation, plus perhaps an indication that we distinguish between
laws that yield recurring decimals and laws that don’t.’
I prefer the
idea – that the point is a mark –
the notion
that a point can only ever be approximated – strikes me as contradictory
or to but it
bluntly – it’s either there – or it’s not
what sense in
talking about approximating existence?
or are we
happy talking about approximating non-existence –
approximating
– nothing?
if a point –
is a mark in a game – a given – a construct – yes we can have approximation –
we have a
functional point of reference
the infinite
game – as the ‘on-going’ game is one thing –
‘points’ as
non-existent reference points – quite another
we don’t need
this ridiculous fiction to make sense of repeated bi-sections or endless
decimal fractions
repeated
bi-section or endless decimal fractions are language
games – propositional games –
that is to
say – rule or law governed propositional actions
‘The
incorrect idea of the word “infinite” and of the role of ‘infinite expansion”
in the arithmetic of the real numbers gives us the false notion that there is a
uniform notation for irrational numbers (the notation of the infinite
extension, e.g. of infinite decimal fractions).
The proof
that for every pair of cardinal numbers x and y (x)2 ¹ 2
does not correlate
y
Ö2
with a single type of number – called “the irrational numbers”. It is not as if
this type of number was constructed before I construct it; in other words, I
don’t know any more about this new type of number than I tell myself.’
the
representation of irrational numbers will be open to question –
and will be
determined context by context – game by game
‘this type of number’ –
any ‘number’ is a proposal open to question –
and we assess the value of any new construction –
in terms of its usefulness –
an on-going issue
42 Kinds of
irrational numbers (p¢ P, F)
‘p¢ is a
rule for the formation of decimal fractions: the expansion of p¢ is
the same as
the
expansion of p
except where the sequence 777 occurs in the expansion of p;
in
that case instead of the sequence 777
there occurs the sequence 000. There is no method
known in our calculus of discovering where we encounter such a sequence in
the expansion of p.
P is a rule for
the construction of binary fractions. At the nth place of the expansion there occurs
a 1 or a 0 according to whether n is prime or not.
F is a rule for
the construction of binary fractions. At the nth place there is a 0 unless
a triple x, y, z from the first 100 cardinal
numbers satisfies the equation
xn + yn
= zn.
I am tempted to say, the individual digits of the expansion
(of p
for example) are always only the results, the bark of the fully grown tree.
What counts, or what
something new can still grow from, is the inside of the
trunk, where the tree’s vital energy is. Altering the surface doesn’t change
the tree all. To change it, you have to penetrate the trunk which is still
living.’
p
is a proposal –
if it is to function – it will function in a rule governed
propositional context –
in a rule governed propositional action – p is a game –
and as such the individual digits of the expansion of p –
are the play of the game
we do have and play ‘irrational games’
Wittgenstein’s tree analogy is irrelevant to mathematics –
to the game
‘I call “pn” the expansion of p up to the nth place. Then
I can say: I understand what
p¢100
means, but not what p¢
means, since p
has no places, and I can’t substitute others for none. It would be different if
I e.g. defined the division 5®3 as a rule for the
a/b
formation of decimals by division and replacements of every
5 in the quotient by a 3. In this case I am acquainted, for instance, with the
number 5®3.
– And if our calculus
1/7
contains a method, a law, to calculate the position of 777
in the expansion of p,
then the law of p
includes a mention of 777 and the law can be altered by the substitution of 000
for 777. But in that case p¢ isn’t the same as what I defined above; it has a
different grammar to the one I supposed. In our calculus there is no question
of
=
p
> p¢ or not, no such equation
or inequality. p¢ is
not compatible with p.
And one can’t say “not yet compatible”, because if at some time I
construct something similar to p¢ that is compatible with p, then for that very
reason it will not be p¢. For p¢ like
p
is a way of denoting a game, and I cannot say that draughts is not yet played
with as many pieces as chess, on the grounds that it might develop into a game
with 16 pieces. In that case it will no longer be what we call “draughts”
(unless by this word I mean not a game, but a characteristic of several games
or something similar; and this rider can be applied to p and p¢
too). But since being comparable with other numbers is a fundamental
characteristic of a number, the question arises whether one is to call
p¢ a
number, and a real number; but whatever it is called the essential thing is that p¢ is not a number in the
same sense of p.
I can also call an interval a point and so on occasion it may even be practical
to do so; but does it become more like a point if I forget that I have used the
word “point” with two different meanings?’
yes – p¢100 as distinct
from p¢ is
given definition by a rule – and is by the rule – made functional –
p¢100
is a playable game
and yes – it
would be different if a rule for the formation of decimals is given –
and the
calculus contains a method – a law – a rule to calculate the position of 777 in
the expansion of p –
then the law / rule can be altered by the substitution of
000 for 777
and yes – p¢ – so defined – that is – rule governed in this manner –
is not the same as p¢ – as
Wittgenstein defined it above
p¢ is a
proposal – open to question – open to doubt – open to interpretation
different interpretations – different games
‘whether one is to call p¢ a number, and a real
number’?
the real question here is whether to call p¢ – a game – that is a rule governed
propositional action –
and the answer is straightforward – yes – if p¢ is
rule governed
and just in general here – we can talk of numbers – as we
can talk of chess pieces –
but the central focus is – or I would say – should be – the game –
for without the game – numbers – of whatever kind – mean
nothing
you can call
an ‘interval’ a ‘point’ –
as a proposal
– a proposition – the ‘interval’ is as with any proposition – open to
interpretation –
‘does
it become more like a point if I forget that I have used the word “point” with
two different meanings?
yes – but
really we are not interested in confusion here –
or in
mysticism
‘Here it is clear that the possibility of the decimal
expansion does not make p¢ a number in the same sense as p. Of course the rule for
this expansion is unambiguous, as unambiguous as that for p¢ or
√2; but that is no proof that p¢ is a real number, if one takes comparability with
rational numbers as an essential mark of real numbers. One can indeed abstract
from the distinction between rational and irrational numbers, but that does not
make the distinction disappear. Of course, the fact that p¢ is
an unambiguous rule for decimal fractions naturally signifies a similarity
between p¢ and p or
√2; but equally an interval has a similarity with a point etc. All errors that
have been made in this chapter of the philosophy of mathematics are based on
the confusion between internal properties of a form (a rule as one among a list
of rules)
and what we call “properties” in everyday life (red as a
property of this book). We might also say: the contradictions and unclarities
are brought about by people using a single word, e.g. “number”, to mean at one
time a definite set of rules, and at another time a variable set, like meaning
by “chess” on one occasion the definite game we play, and on another occasion the
substratum of a particular historical development.’
I think it is technically irrelevant whether p¢ is a
number – it is a game – or at least can be a game with relevant rules
‘All errors that have been made in this chapter of the
philosophy of mathematics are based on the confusion between internal
properties of a form (a rule as one among a list of rules) and what we call
“properties” in everyday life (red as a property of this book)’
there are no ‘errors’ here –
what you have is different
proposals – different descriptions –
‘We might also say: the contradictions and unclarities are
brought about by people using a single word, e.g. “number”, to mean at one time
a definite set of rules, and at another time a variable set, like meaning by
“chess” on one occasion the definite game we play, and on another occasion the
substratum of a particular historical development.’
yes – but this is not to do with ‘contradictions and
unclarities’ – rather differences –
different proposals
– different – descriptions
one advantage of understanding mathematical constructs and
mathematical action in terms of games
– is that in the end – different approaches – different descriptions – can be
resolved in the model – can be seen to fit the model of the game
‘ “How far
must I expand p in order to have some acquaintance with it?” – Of
course that is nonsense. We are already acquainted with it without expanding it
at all. And in the same sense I might say that I am not acquainted with p¢ at
all. Here it is quite clear that p¢ belongs to a different system from p;
that is something we recognize if we keep our eyes on the nature of the laws
instead of comparing “the expansions” of both.’
‘acquaintance’ – is a somewhat fuzzy notion –
you acquaint by
observing – comparing – working with – etc.
in the first instance p¢ is different from p –
syntactically –
if p¢ and p¢ did
not ‘belong to different systems’ –
there would be no syntactical differentiation –
and yes – you can deduce from this –
that they signify different laws – different rules –
different games
‘Two mathematical forms, of which one and not the other can
be compared in my calculus with every rational number, are not numbers in the
same sense of the word. The comparison of a number to a point on the number-line
is valid only if we can say for every two numbers a and b whether a is to the
right of b or b to the right of a.’
yes – the real issue here is that the number system or game
– is radically different to the point game
different games – there is no comparison
that the two are played together – combined – (if they are)
– is quite ridiculous – and the idea is not to be taken seriously
‘It is not
enough that someone should – supposedly – determine a point ever more closely
by narrowing down its whereabouts. We must be able to construct it. To be sure, continued throwing of a die indefinitely
restricts the possible whereabouts of a point, but it doesn’t determine a
point. After every throw (or every choice) the point is still infinitely
indeterminate – or, more correctly, after every throw it is infinitely
indeterminate. I think we are here misled by the absolute size of the objects
in our visual field; and on the other hand, by the ambiguity of the expression
“to approach a point”. We can say of a line in the visual field that by
shrinking it is approximating more and more to a point – that is, it is
becoming more and more similar to a point. On the other hand when a Euclidean
line shrinks it does not become any
more like a point; it is always totally dissimilar, since its length, so to
say, never gets anywhere near a point. If we say of a Euclidean line that it is
approximating to a point by shrinking, that only makes sense if there is
already a designated point which its ends are approaching; it cannot mean that
by shrinking it produces a point. To
approach a point has two meanings: in one case it means to come spatially near
to it, and in that case the point must already be there, because in this sense
I cannot approach a man who doesn’t exist; in the other case, it means “to
become more like a point”, as we say for instance that the apes as they
developed approach the stage of being human, their development produced human
beings.” ’
determining a
point by narrowing down it’s whereabouts?
well it has
to be there – for this to happen
and if it is
indeterminate – in any functional sense – it’s not there –
and therefore
there is no ‘narrowing down’ –
there is no
‘approximating’
‘to become
more like a point’ – is not to be a
point
in an
irrational game – there is no rational
point
this notion
of the point is duplicitous –
you can’t
have it both ways –
either there
is a definite end to the game
or the game
is on-going –
it makes no
sense – is contradictory – to speak of an indeterminate end to a game
an irrational
game per se – is an on-going game –
without end
should you
construct an irrational game – and propose a determinate end –
the ‘end’ –
will be no more than pragmatic –
simply a
‘breaking-off’ of the process – a stopping of the play
the ‘point’ –
as an indeterminate – is perhaps best seen as a characterization of the action
of an on-going game
the ‘point’
is a way of describing any such game
‘To say “two
real numbers are identical if their expansions coincide in all places” only has sense in the case in which, by producing a
method of establishing coincidence, I have given
a sense to the expression “to coincide in all places”. And the same
naturally holds for the proposition “they do not coincide if they disagree in
any one place”.’
‘two real
numbers are identical if their expansions coincide in all places’
if the
expression ‘to coincide in all places’ is ‘given sense’ –
what we have
is –
two games –
same play – same result –
think of two
chess games coincidentally played the same
‘they do not
coincide if they disagree in any one place’
two games –
different play – different results
‘But conversely couldn’t one treat p¢ as
the original, and therefore as the first assumed point, and then be in doubt
about the justification of p? As far as concerns their extension, they are naturally
on the same level; but what causes us to call p a point on the
number-line is its compatibility with the rational numbers.’
‘But conversely couldn’t one treat p¢ as
the original, and therefore as the first assumed point, and then be in doubt
about the justification of p?’ –
yes – you might well ask – what is the difference – what is
the point?
what is required here with p¢ – is – as was given above by Wittgenstein – a
statement of what distinguishes p¢ from p -
that is the rule of the p¢ game –
if there is no difference – then p¢ – is pointless
‘but what causes us to call p a point on the
number-line is its compatibility with the rational numbers.’
isn’t it rather that if ‘placed on a number
line’ – ‘its compatibility with rational numbers’ – is proposed?
or – is it rightly seen as a ring-in?
‘If I view p or let’s say Ö2
as a rule for the construction of decimals, I can naturally produce a
modification of this rule by saying that every 7 in the development of Ö2
is to be replaced by a 5; but this modification is of quite a different nature
from one which is produced by an alteration of the radiant or the exponent of
the radical sign or the like. For instance, in the modified law I am including
a reference to the number system of the expansion which wasn’t in the original
rule for Ö2.
The alteration of the law is of a much more fundamental kind than might at
first appear. Of course, if we have the incorrect picture of the infinite
expansion before our minds, it can appear as if appending the substitution rule
7 –> 5 to Ö2
alters it much less than altering Ö2 into
Ö2.1,
because the expansion of 7 –>5 are very similar to those of Ö2,
whereas the
Ö2
expansion of Ö2.1
deviates from that of Ö2
from the second place onwards’
true –
a ‘modified law’ – is a different game
‘Suppose I give a rule p for the formation of
extensions in such a way that my calculus knows no way of predicting what is
the maximum number of times an apparently recurring stretch of the extension
can be repeated. That differs from a real number because in certain cases I
can’t compare p – a with a rational
number, so that the expression p – a
= b becomes nonsensical. If for instance the expression of p so far known to me
is 3.14 followed by an open series of ones (3.1411 11 ..), it wouldn’t be possible to
say of the difference p –3.141 whether it was greater or less than 0; so in
this sense it can’t be compared with 0 or with a point on the number axis and
it and p can’t be called a number in
the same sense as one of these points.’
yes – different number systems – signify different games
‘|The extension of a concept of a number, or of the
concept ‘all’, etc. seems quite harmless to us; but it stops being harmless as
soon as we forget that we have in fact changed our concept. |’
yes – another reason for dropping the notion of
number from our central focus – and instead recognising the centrality of the
concept of the game
‘|So far as concerns the irrational numbers, my investigation
says only that it is incorrect (or misleading) to speak of irrational numbers
in such a way as to contrast them with cardinal numbers and rational numbers as
different kinds of number; because what are called “irrational numbers” are a
species of number that are really different – as different from each other as
the rational numbers are different from each other. |’
what we have is different games
‘“Can God know all the places of the expansion of p?”
would have been a good question for the schoolmen to ask.’
is ‘God’ the possibility of endless play?
perhaps that is all ‘God’ amounts to –
the infinite
irrational game –
p
‘In these discussions we are always meeting something that
could be called an “arithmetical experiment”. Admittedly the data determine the
result, but I can’t see in what way
they determine it. That is with the occurrences of the 7s in the expansion of
p;
the primes likewise are yielded as the result of an experiment. I can ascertain
31 is a prime number, but I do not see the connection between it (its position
in the series of cardinal numbers) and the condition it satisfies. – But this
perplexity is only the consequence of an incorrect expression. The connection
that I think I do not see does not exist. There is not an – as it were
irregular occurrence of 7s in the expansion of p, because there isn’t any
series that is called the expansion
of p.
There are expansions of p, namely those that have been worked out (perhaps 1000)
and in those the 7s don’t occur “irregularly” because the occurrence can be
described. (The same goes for the “distribution of the primes”. If you give us
a law for this distribution, you give us a new
number series, new numbers.) (A
law of the calculus that I do not know is not a law). (Only what I see is a law; not what I describe. This is the only thing
standing in the way of my expressing more in my signs that [than?] I can
understand.)’
the rules of
the game – determine the result –
the ‘data’ –
the tokens of play – are determined by the game –
by the design of the particular game
a law – or
rule that I do not know – is not a rule that I can use
what I see
(in the calculus) – is a law – is a rule
‘what I
describe’ – description – is speculation –
speculation –
is not rule governed
‘expressing
more in my signs than I can understand’ –
corrupts the
signs – renders them non-functional – useless
what my signs
express – if they are functional – is the play
of the game –
nothing more
it is the
game that determines – the signs
‘arithmetical
experimentation’ – is speculation –
it has a
place – is of interest –
but it is not
to be confused with calculation
‘Does it make no sense to say, even after Fermat’s last
theorem has proved, that
‘F = 0.11’? (If, say I were to read about it in the papers.)
I will indeed then say, “so now we can write ‘F = 0.11’.” That is, it is
tempting to adopt the sign “F” from the earlier calculus, in which it didn’t
denote a rational number, into the new one and now to denote 0.11 with it.
F was supposed to be a number of which we did not know
whether it was rational or irrational. Imagine a number, of which we do not
know whether it is a cardinal number or a rational number. A description in the
calculus is worth just as much as this particular set of words and it has
nothing to do with an object given by description which may someday be found.
What I mean could also be expressed in the words: one cannot
discover any connection between parts of mathematics or logic that was already
there without one knowing.’
Wittgenstein
proposes a history of F
what F meant
– and what F now means – is the argument of F
there is no
contradiction between what F was supposed to mean – and what F comes to mean
F – was and
is open to interpretation
yes – ‘a
description in the calculus is worth just as much as this particular set of
words’
and any
description in the calculus – as with any particular set of words – is open to
question – to doubt – is uncertain
‘one cannot discover any connection between parts of
mathematics or logic that was already there without one knowing.’?
knowledge is proposal
any connection between propositional systems – is a proposal
propositions are not ‘already there’ – they are proposed – or they are not there at all
knowledge is not
already there – it is proposed
and any proposal is open to question – to doubt – is
uncertain
‘In mathematics there is no “not yet” and no ‘until further
notice’ (except in the sense in which we can say that we haven’t further
multiplied two 1000 digit numbers together.)’
mathematics is not an experiment
a
mathematical action is a propositional game-proposal –
if the game
is properly constructed its play is determined by its rules
where there
is any questioning of the rules –
the game
stops
“Does the operation yield a rational number for instance?” –
How can that be asked, if we have no method for deciding the question? For it
is only in an established calculus that the operation yields results. I mean “yields” is essentially timeless. It doesn’t
mean “yields given time” – but: yields in accordance with the rules already
known and established.’
yes
‘The position of all primes must somehow be predetermined.
We work them out only successively, but they are already determined. God, as it
were knows them all. And yet for all that it seems possible that they were not
determined by a law.” – Always this picture of the meaning of a word is a full
box which is given us with its contents packed in it already to investigate. –
What do we know about the prime numbers? How is the concept of them given to us
at all? Don’t we ourselves make up the decisions about them? And how odd that
we assume that there must have been decisions taken about them that we haven’t
taken ourselves! But the mistake is understandable. For we use the expression
“prime number” and it sounds similar to “cardinal number”, “square number”,
“even number” etc. So we think it will be used in the same way, and we forget
that for the expression “prime number” we have given quite different rules – rules
different in kind – and we find
ourselves at odds with ourselves in a strange way. – But how is that possible?
After all the prime numbers are familiar cardinal numbers – how can we say that
the concept of prime number is not a number concept in the same sense as a
cardinal number? But here again we are tricked by the image of an “infinite
extension” as an analogue to the familiar “finite extension”. Of course the
concept ‘prime number’ is defined by means of the concept ‘cardinal number’,
but “the prime numbers” aren’t defined by means of “cardinal numbers”, and the
way we derived the concept ‘prime
number’ from the concept ‘cardinal number’ is essentially different from that
in which we derived, say, the concept ‘square number’. (So we cannot be surprised
if it behaves differently.) One might well imagine an arithmetic which – as it
were – didn’t stop at the concept ‘cardinal number’ but went straight on to
that of square numbers. (Of course that arithmetic couldn’t be applied in the
same way as ours.) But then the concept “square number” wouldn’t have the
characteristic it has in our arithmetic of being essentially a
part-concept, with the square numbers essentially a
sub-class of the cardinal numbers; in that case the square numbers would be a
complete series with a complete arithmetic. And now imagine the same done with
prime numbers! That will make it clear that they are not “numbers” in the same
sense as e.g. the square numbers or the cardinal numbers.’
don’t we ourselves make decisions about them?
yes – and as the concept has proven its utility in practise
– it is stable
decisions about them that we haven’t taken ourselves?
‘ourselves’ here – just is the totality of – or the history
of decisions made
what we are really talking about here is not numbers – of
whatever kind – but games –
i.e. – the ‘prime game’ – the ‘cardinal game’ etc –
and yes – game
‘similarities’ can always be proposed –
and who is going to be surprised – that one game ‘behaves
differently’ – is in fact different to
another?
‘mathematics’ as the game of games
‘Could the
calculations of an engineer yield the result that the strength of a machine
part in proportion to regularly increasing loads must increase in accordance
with the series of primes?’
a machine
part could have a design that involves a prime
calculation –
just how far
you would take it – would depend on the limits of the material –
and the
overall design of the machine –
and of course
there may well be other calculation games that could be used to reflect or
explain the same result
43 Irregular infinite decimals
‘ “Irregular
infinite decimals”. We always have the idea that we only have to bring together
the words of our everyday language to give the combinations a sense, and all we
then have to do is inquire into it – supposing it’s not quite clear right away.
–
It’s as if
words were ingredients of a chemical compound, and we shook them together to
make them combine with each other, and then had to investigate the properties
of the compound. If someone said he didn’t understand the expression “irregular
infinite decimals” he would be told “that’s not true, you understand it very
well; don’t you know what the words “irregular”, “infinite”, and “decimal”
mean? – well, then you understand their combination as well.” And what is meant
by “understanding” here is that he knows how to apply these words in certain
cases, and say connects an image with
them. In fact, someone who puts those words together and asks “what does it
mean” is behaving rather like small children who cover a paper with random
scribblings, show it to grown-ups, and ask “what is this?” ’
a word is a proposal – and any use of words – any use – is open to question – to doubt
is uncertain –
to ask the
question ‘what does it mean?’ – is to recognize the logic of language use –
it is to
behave logically
that we stop
asking that question and proceed – is a pragmatic move
the question
is still there
‘ “Infinitely
complicated law”, “infinitely complicated construction” (“Human beings believe,
if only they hear the words, there must be something that can be thought by
them”).’
‘human beings
believe, if only they hear the words, there must be something that can be
thought by them’ –
if a proposal
is put – it is worth considering
just what it
amounts to – is open to question – to doubt – is uncertain –
that it has a
definite meaning – a meaning that is beyond question – beyond doubt – or is
certain –
is rhetorical
– rubbish
‘infinitely
complicated law’ –
can a ‘law’
or a ‘rule’ be stated as infinitely complicated?
I think not
if it is
‘infinitely complicated’ – you would never come to a statement of it
can it be
shown to be more complicated than its statement suggests?
can it shown
to be so complicated that it has no direct or specific application –
that it is
effectively useless? – yes
‘an
infinitely complicated construction’?
one could
well say any construction – is infinitely complicated –
if you wish
to look at it that way
the point is
there is no end to what you see in a construction – if you have the wherewithal
– to keep looking
however that
is only possible if you have a defined construction to begin with –
to think
within
‘How does an
infinitely complicated law differ from the lack of any law.’
a law that
proposes an ongoing action – is quite straight forward – ‘simple’ – in fact
an
‘infinitely complicated law’ – as in one that cannot be given a statement –
is no
different to the lack of any law
‘(Let us not
forget: a mathematicians’ discussions of the infinite are clearly finite
discussions. By which I mean the come to an end.)’
yes – but any
proposal – is open to question – open to doubt – is uncertain –
whenever it
is proposed – whenever it is taken up
discussing
‘the infinite’ is no different to discussing anything else –
the question
remains – open
‘One can
imagine an irregular infinite decimal being constructed by endless dicing, with
the number of pips in each case being a decimal place.” But if the dicing goes
on forever, no final result ever comes out.’
yes – no
result to this game – rather an on-going action
so the
question is – in what context is such an exercise – or a section of it – of
use?
it’s a form
of calculation that could be used in measurement
and in such a
case the materials involved set the limit of the on-going calculation
‘ “It is only
the human intellect that is incapable of grasping it, a higher intellect could
do so!” Fine, then describe to me the grammar of the expression “higher
intellect”; what can such an intellect grasp and what can’t it grasp and in
what cases (in experience) do I say that an intellect grasps something? You
will then see that describing is itself grasping. (Compare: the solution of a
mathematical problem).’
higher or
lower intellect?
the proposition
put – by whatever level of intellect – is open to question – open to doubt – is
uncertain
that is the
logic of the proposition
what is
‘behind the proposition’ – where it comes from – if you like –
is logically
irrelevant
‘Suppose we
throw a coin heads or tails and divide an interval AB in accordance with the
following rule: “Heads” means: take the left half and divide it in a way the
next throw prescribes. “Tails” says “take the right half, etc.” By repeated
throws I then get dividing points that move in an ever smaller interval. Does
it amount to a description of the position of a point if I say that it is
infinitely approached by the cuts as prescribed by the repeated tossing of the
coin? Here one believes oneself to have determined a point corresponding to an
irregular infinite decimal. But the description doesn’t determine any point explicitly; unless one says that the words ‘point on this line’
also “determine a point”! Here we are confusing the recipe for throwing with a
mathematical rule like that of producing decimal places of Ö2.
Those mathematical rules are the
points. That is, you can find relations between those rules that resemble in
their grammar the relations “larger” and “smaller” between two lengths, and
that is why they are referred to by those words. The rule for working out
places of Ö2
is itself the numeral for the irrational number; and the reason I here speak of
a “number” is that I can calculate with these signs (certain rules for the
construction of rational numbers) just as I can with rational numbers
themselves. If I want to say similarly that the recipe for endless bisection
according to heads and tails determines a point, that would mean that this recipe could be used as a numeral, i.e.
in the same way as other numerals. But of course that is not the case. If the
recipe were to correspond to a numeral at all, it would at best correspond to
the indeterminate numeral “some”, for all it does is to leave a number open. In
a word, it corresponds to nothing except the original interval.’
the notion of
the point – the term ‘point’ – in the heads / tail game here – really refers to
– an on-going process – or action –
an on-going-game
and yes – an
on-going game played in the original
interval –
the original
interval is the ground of play
© greg t.
charlton. 2017.
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