# running with harlots drunk or naked

running with harlots drunk or naked is a killer press philosophy blog. all posts by greg t. charlton. (c) 2008. all rights reserved. killer press.

- Benedict de Spinoza. Political Treatise. 1677.

## Friday, December 01, 2017

### NOTICE: Publication of 'Feyerabend's Against Method'

Killer Press announces the publication of 'Feyerabend's
Against Method' by Greg T. Charlton. This book is a critical study of
Paul Feyerabend's 'Against Method'. It is a work in the philosophy of
science. Published Decemeber 2017. ISBN: 978-0-9806987-3-2. Also published in the blog: feyerabendsagainstmethod.blogspot.com

## Monday, November 27, 2017

### on parfit

what follows are some notes on derrick parfit’s moral theory

I have focused on and borrowed extensively from professor
peter singer’s introduction to his book ‘does anything really matter: essays on
parfit on objectivity’ – as the basis of my consideration –

singer’s introduction is superb

what follows is not a assessment or criticism of singer’s views –

it is parfit’s ideas – as introduced by singer – that I
consider here

the way I am proceeding here is I will quote from singer’s
introduction – and follow that with my response to the text

‘Kantianism,
contractualism, and rule consequentialism—are in fundamental agreement,
identifying the same acts as wrong. Underlying and supporting this original and
important argument, however, is another, more fundamental claim, also defended
at considerable length: that there are objective moral truths, and other
normative truths about what we have reasons to believe, and to want, and to do.’

that we have an
action (a ‘wrong action’) – that is accounted for in terms of different
theories is nothing remarkable

that underlying this
concurrence of determination are objective moral truths – is one explanation

it is not the only
one

i.e. it may be that
as a matter of custom certain acts are regarded as wrong –

and all that
kantianism – contractualism and rule consequentialism – amount to is different explanations of
customary behaviour

any explanation of
any aspect of moral behaviour is a

*proposal*–
and logically
speaking a proposal is open to question – open to doubt and uncertain

in the absence of
explanation – of any explanatory proposal – an act is without description – and
from an epistemological point of view – an unknown

we proposes to make
known

the objective
reality – the reality we propose in relation to – is the unknown

the unknown is
silent

moral truths?

a true moral
proposition – is one that is affirmed – for whatever reason

a false proposition
is one that is denied – and for whatever reason

our affirmations –
our denials – and our reasons – are from a logical point of view –

open to question – open to doubt and uncertain

open to question – open to doubt and uncertain

our reality – our
moral reality is the reality of propositional uncertainty

and it is this
propositional uncertainty – this logical uncertainty – that has given rise to
different moral theories and perspectives

and it is through
the many and different moral theories and perspectives that we explore moral
uncertainty –

the role of the
moral philosopher is to critically investigate the various theories and
perspectives that have been proposed –

and in so doing
perhaps even to propose new and different accounts – that are open to question
– open to doubt and are uncertain

‘Hume assumes, and
we commonly believe, that morality must be able to influence what we do. Otherwise,
we may wonder, what is its point? But Hume also held that reason alone cannot
move us to action. Our wants and desires determine our ultimate goals, and the
role of reason is limited to telling us how best to achieve these goals. Reason
applies to means, not ends. Hence, Hume famously held, it is not contrary to
reason to prefer the destruction of the whole world to the scratching of my
finger, and equally not contrary to reason to choose my own total ruin to
prevent a trivial harm to a stranger. Even acting contrary to one’s own
interests—preferring “my own acknowledged lesser good to my greater” is, on
Hume’s view, not contrary to reason. What it is rational for me to do depends
on what I want. If Hume is right both in his assumption about the relation
between morality and action, and about the role of reason in action, then there
is an obvious problem for those who think that moral judgments can be
objectively true. Moral judgments will only be able to influence our actions if
they somehow connect with our desires, and my desires may differ from yours
without either of us making a mistake. Wants and desires are neither true nor
false. An objectively true moral judgment would have to be true for everyone,
irrespective of what he or she most desires, but what reason for acting would
it offer to those whose desires are not furthered by acting on it?’

morality must be
able to influence what we do – otherwise what is the point?

the way I would put
it is that every action we take has a moral dimension to it –

we constantly
question and decide what is right – what is wrong – what is good and what is
bad

the moral question
and the moral decision are entirely natural to human beings

such question – such
decision is of our nature

what hume and others
call ‘morality’ – is moral theory – of one kind or another

the question then is
– do such theories of morality influence what we do?

what influences our
moral decisions – is open to question – open to doubt – and is uncertain

for those who are
exposed – in one way or another to moral theory –

it is likely that
those theories will influence –

however they may not

whether they
influence of not – is a purely contingent matter – a matter of circumstance –

moral theory – is
one among any number of possible influences –

it is not necessary
to have such an influence to make a moral decision

and whether or not –
and to what extent moral theories influence people’s ethical decisions – is in
fact an empirical matter

hume held that
reason alone cannot move us to action?

reason here means a

*proposal*as to how to act
any such proposal
may move us to act – or it may not

the reality is that
we will act – one way or another – even if action here may mean the decision
not to proceed with a course of action

what underlies any
action in a philosophical sense – will be open to question – open to doubt –
will be uncertain

our wants and
desires determine our ultimate goals?

what we want and
what we desire may well determine our goals

however it is quite
possible that we may decide a course of action – that is against our wants and
desires

that is we may i.e.
decide a course of action that is based on a moral principle that we hold to

i.e. I may want to
kill my enemy – desire his demise – but decide not to murder him – because I
hold to the proposal that murder is wrong

as for ‘ultimate
goals’ – they can change with the wind

‘ultimate goals’ –
are proposals designed to give us some sense of direction – for the moment – or
perhaps longer –

we play the ultimate goal game – or just the
goal game – for direction – order – and coherence

and we are quite
adept at making necessary adjustments – and even radical revision – depending on the
circumstances we have to deal with

reason applies to
means not ends?

reason is the logical
action of question and doubt – reason is the exploration of propositional
uncertainty

means and ends – are
open to question – open to doubt – are uncertain

what is rational to
me is to do what I want?

what I want is what
I want –

if what I want is
put to question – is put to doubt – is regarded as uncertain –

then what I want is
rational

if it is not – then
what I want – is not rational

if hume is right
then there is a problem for those who believe moral judgments to be objectively
true?

if by ‘objectively
true’ – you mean a proposition – that is beyond question – beyond doubt – that
is certain –

such a proposal
defies logical reality – and is better termed a prejudice –

if your morality is
based on prejudice – it is not rational – it is irrational and pretentious

moral judgments will
only be able to influence our actions if they somehow connect with our desires?

our moral judgments
are open to question – open to doubt – are uncertain

our desires are open
to question – open to doubt – and uncertain –

the ‘connection’ is
propositional uncertainty

wants and desires
are neither true nor false?

a true proposition
is a proposition affirmed – a false proposition – a proposition denied

if a want or a
desire is expressed – that is

*proposed*– that proposal – that proposition
can be affirmed –
can be denied –

an objectively moral
judgment would have to be true for everyone – irrespective of what he or she
most desires?

there is no ‘have to
be true’ – the truth or falsity of a proposal – of a judgment –

is a matter of
affirmation or denial

which is to say – it
is an empirical issue

if as a matter of
empirical fact it could be demonstrated that everyone affirmed a particular
moral judgment at a particular time – then you could say that in effect you
have an ‘objective moral judgment’ –

I would call it a
freak event – and one not beyond question – and doubt

crudely put –
‘objectivity’ is a question of numbers

but what reason for
acting would ‘an objectively moral judgment’ offer to those whose desires are
not furthered by acting on it?

this objectively
moral judgment is an authoritarian fiction –

nevertheless – if
confronted with this fiction – what reasons for acting would such a fiction
offer to those whose desires are not furthered by it?

the point is you
can’t give a so called objective answer here –

first up – you have
a proposal – fictional or not – to be considered

and yes this
proposal conflicts with my desires –

if I am to consider
it rationally – and to consider my desires rationally – that is subject both to
question – and to doubt

who knows – I may
end up deciding for the proposal – and
against what I desire?

one’s desires are
not in a logical void – beyond question – beyond doubt – and certain

desires can change –
can be changed – if you have reason to change them

and – one’s reasons
too – are not set in stone –

if you are to deal
rationally – your desires and your reasons – are open to question – open to
doubt – and uncertain

‘desires’ and
‘reasons’ – are really modes of propositional expression –

a ‘desire’ may be a
reason – a ‘reason’ – a desire

what we are dealing
with is proposals – propositions – regardless of the terms in which they are
expressed – and regardless of how they are conceptually organized

‘Parfit’s critique
of the forms of subjectivism that draw on Hume’s view of the limits of
practical reason begins with a discussion of the role of reason in a situation
relating to self-interest rather than morality. He asks us to imagine a man who
cares, as most of us do, about what pleasures or pains he will experience in
future, but with this difference: if they will happen on a future Tuesday, he
doesn’t care about them at all. If he is
contemplating what will happen to him on a Monday, a Wednesday, or any other
day, he would much rather experience slight discomfort now than agony on that
day; but if the agony will be on a future Tuesday, he doesn’t care about it,
and so will choose it over slight discomfort now. This man is not under any
illusion that pains on future Tuesdays are less painful than pains on other
days, for he knows that when that future Tuesday becomes the present day, the
agony will be as terrible as it is on a Monday or Wednesday. He also knows
that—since it will then not be a future Tuesday—he will not be at all
indifferent to the agony he then experiences. Nor does he believe in a strange
deity who will reward him for his indifference to what will happen to him on
future Tuesdays. He differs from us purely in what he desires.’

‘If he is contemplating
what will happen to him on a Monday, a Wednesday, or any other day, he would
much rather experience slight discomfort now than agony on that day…’

he prefers a slight
discomfort now – to an imagined future agony –

but isn’t this just
effectively to

*accept*the discomfort of now – supposing that things could be worse?
so all we have here
is a psychological strategy for dealing with the discomfort of now

as to indifference –

can you be
indifferent to the idea of a future pain?

parfit defines indifference
as ‘not caring’ –

if you care about
something – it must occupy thoughts

what this ‘not
caring’ means here – is putting the idea of a future agony out of your mind

but if not
considering it – is the argument for preferring – the discomfort of now –

then in preferring
the discomfort of now – the future pain is necessarily a consideration –

either you drop the
argument for

*preferring*the discomfort of now –
and if that goes –
so does the whole future tuesday scenario –

or you have to take
the proposed future agony into consideration –

and there goes
indifference

‘but if the agony
will be on a future Tuesday, he doesn’t care about it, and so will choose it
over slight discomfort now’

if he doesn’t care
about it – that is doesn’t consider it – puts it out of his mind –

he can’t choose it –

if he does consider
it – does care about it

he is not
indifferent

so I think

*this*notion of indifference is a pretence –
he

*pretends*not to care – as a strategy for dealing with discomfort
perhaps for this
future tuesday man – a useful pretence – but a pretence and self-deception
nevertheless –

all we have here is
a rather convoluted psychological strategy for dealing with discomfort –

much simpler and
more straightforward to say – ‘it looks like I will have to put up with this’ –

or to ask yourself –
‘what can I do

*now*to ease my discomfort?’
to be indifferent in
a logical sense is to suspend judgment

the future tuesday
man does not suspend judgment

he would

*prefer*the discomfort of now to a future agony
this to make a
judgment

‘he differs from us
purely in what he desires’ –

does he really?

does anyone

*desire*discomfort?
preferring
discomfort now – to an imagined future agony –

is not

*desiring*discomfort –
preferring a future
agony – to the discomfort of now

is not

*desiring*agony
preferring a future
agony to the discomfort of now –

is an attempt to
deny reality – deny the reality of the present discomfort

such an approach
will have no effect at all on the discomfort of the present

it is pointless and irrelevant

‘Surely, Parfit
claims, this man’s desires are irrational: “That some ordeal would be much more
painful is a strong reason not to prefer it. That this ordeal would be on a
future Tuesday is no reason to prefer it.” It is difficult to deny that such a
man would be irrational, and the only possible source of this irrationality is
his desires. But Hume’s approach leaves no room for desires to be rational or
irrational. Hume’s followers may say that this a very odd set of desires to
have, and that as far as we know no one has ever had this set of desires, but
it remains conceivable that someone could have them, and that is enough to pose
a problem for Hume’s view.’

parfit has confused
desire with preference

that I would prefer
to experience pain – now – or indeed in the future – is not to desire pain –

it is to say – if I
am to experience pain – I have a preference for when I experience it

when it comes to the
question of rationality however – desires and preferences are in the same boat

if you regard your
desires – or your preferences – as open to question – open to doubt and
uncertain – then you behave logically – you behave rationally

if on the other hand
– you regard your desires or your preferences as beyond question – beyond doubt
and certain – you behave illogically – and irrationally

in my view parfit’s
future tuesday argument completely misses the point on the question of
rationality

hume too fails on
the issue of rationality –

he fails to see that
desires – and reasons too – logically speaking – are open to question – open to
doubt and uncertain

we behave rationally
when we recognize and embrace the logic of moral uncertainty

where we operate
without question and without doubt – with the pretence of certainty – be it with regard to desires – preferences
– or reasons – our moral propositions and our actions are irrational

if the future
tuesday man holds his proposal – odd as it might be – open to question – open
to doubt – and regards it as uncertain – he holds the proposal rationally –

if on the other hand
he regards his proposal as beyond question – beyond doubt – and thus certain –
he holds the proposal irrationally

‘Moreover, many
people have attitudes that are somewhat like future Tuesday. Many people put
off going to the dentist, for instance, even though they are well aware that
doing so will mean more pain overall than if they were to go to the dentist
now. At least in extreme cases, these desires also seem to be irrational. But
subjectivists about reason cannot, it seems, say that they are. Similarly,
subjectivists about reason cannot say that the fact that putting my hand in a
flame will cause me agony is a reason not to put my hand in the flame. They
must say that whether I now have a reason not to put my hand in the flame will
depend on whether I now desire to avoid agony. Parfit thinks this is a mistake:
desires do not give us reasons for acting. I may desire to experience agony,
but that does not give me any reason to put my hand in the flame, since I have
no reason to have this desire, and strong reason not to have it.’

a desire can be held
rationally or irrationally – a reason can be held either rationally or
irrationally

and by a ‘reason’
here – I mean a proposal – the point of which is to account for a proposed
course of action –

if the ‘reason’ is
held open to question – open to doubt – and uncertain –

it is held
rationally

if it is not held
open to question – it is not held rationally

the same is true of
desires

as to the hand in the
flame –

if the proposal to
put your hand in a flame – is considered critically – put to question – put to
doubt – and regarded as an uncertain proposal –

then that proposal
is held rationally –

if it is not
considered critically – the proposal is held irrationally

do I have reasons
for my desires?

we can have accounts
– explanations – that is reasons – for our desires

i.e. I may see my
desire as a result of a current circumstance – I might explain it in terms of
my biology – I might even explain in it terms of my upbringing –

I may have no
account of why I desire a certain outcome

having reasons is
about explanations – proposals to account for desires

there is no
necessity here –

you can operate
without having an account of what you do – or you can have an explanation

desires do not give
us reasons for acting?

a desire may be

*proposed*as a reason for acting –
and here – as it
were – the desire is becomes a reason – the desire functions as a reason

‘Parfit grants that,
on his view, reasons may not motivate us. Whether something will motivate me to
act in a certain way is, he says, a psychological fact, and quite distinct from
the normative fact that I have a reason to act in that way. I may have a reason
to do something without being motivated to do it. Since subjectivists deny that
there are any objective, or object-given, reasons for acting, if Parfit is
right that having a present desire for something does not give one a reason for
acting, it would follow that on the subjectivist view we have no reasons for
doing anything, and hence, though some things may matter to us, in a larger
sense, nothing matters.’

a reason as an
account of an action – and an explanation of an action –

can the way that I
account for any proposed action – be a motivation for my action?

if I understand
a proposed action as a good action – as the right thing to do –

might not my account
of the proposed action – be a reason for doing it?

I would say it could
be – that it might be –

on the other hand I
don’t think it need be

I might act quite
instinctively – without any reflection – without any consideration of whether
the act is good – or any thought that it is the right thing to do

so I think we can be
motivated by reasons – but that we are not necessarily motivated by reasons

motivation as a
psychological fact?

yes – you can
describe motivation as psychological

but having a reason
– an account – an explanation for an act – is psychological

reasons do not exist
in a non-psychological reality or dimension

yes – I may have a
reason to do something – without being motivated to do it

but equally my
reason may motivate me

having a present
desire for acting does not give one reason for acting?

if I have a present
desire for something and I think that having it would make me happy – yes – I
may regard that desire as a reason to act

however it could
also be said that – even though I have this reason – my happiness –

that reason may not
cause me to act – that too is possible

the reality is that
we cannot give a definitive – answer here

whether a reason
does or does not cause me to act – will be a matter of circumstance –

it would follow that
on the subjectivist view we have no reason for doing anything?

as a matter of fact
people can and do give some account of their actions –

and so they have
reasons for doing what they do

and by the same
token – it is quite possible that someone may well say ‘I had no reason for
doing it – but I did it just the same’

and hence that
though some things may matter to us – in a larger sense nothing matters?

yes I suppose that
‘nothing’ by definition – does not matter

what matters to you
is what you are concerned about

can anyone live
without being concerned about something – about many things?

I think not

‘Hence Parfit
eschews any middle ground that would allow us to accept subjectivism but go on
as if nothing much had changed. For him, if there are no ethical truths,
nihilism awaits and his life has been wasted…’

as for nihilism –

human beings

*propose*ethical truths –
as long as there are
human beings – there will be ethical proposals that they affirm –

that one’s life has
been wasted – it’s a fair enough proposal
–

but again – if it is
to be dealt with logically and rationally – it will be held open to question –
open to doubt – and regarded as uncertain

‘Parfit rejects not
only ethical subjectivism, but also ethical naturalism. To say that we have
reason to reduce suffering, other things being equal, is to make a substantive
normative claim that Parfit believes to be true, but it is not something that
we can deduce from the meanings of moral terms like “good” or “ought.” Here
Parfit agrees with Hume that we cannot deduce an “ought” from an “is,” meaning
that no set of natural facts implies, on its own, any normative truths. We
cannot identify normative truths with facts about the natural world, whether
about our biological nature, about evolution, or about what we would approve of
under some set of specified conditions, or any other causal or psychological fact.’

I can put the
proposal – i.e. ‘I ought to do x’ – and I can also put the proposal ‘the grass
is green’

yes these are
different proposals –

and as hume has
showed – these different proposals can be analysed in different terms

nevertheless – different
as they are – they have the same logical properties – they are open to question
– open to doubt – and as such – uncertain

yes – there are
naturalistic and non-naturalistic accounts of moral propositions –

but any such account
of propositions – as with the propositions themselves – is open to question

how we account for
propositions – how we describe them – and how we explain them – has I think to do with metaphysical
preferences – if not prejudices

we develop different
ways of understanding our propositions – we develop different ways of
understanding the world –

and these different
proposals in thought and language –
enrich us – enrich our world with propositional diversity

this diversity I
would put is a

*natural*response to propositional uncertainty
‘How then do we come
to know normative truths? Like many of his objectivist predecessors—Richard
Price in the eighteenth century, Henry Sidgwick in the nineteenth, and W. D.
Ross in the early twentieth, Parfit is an intuitionist. “We have,” he writes,
“intuitive abilities to respond to reasons and to recognize some normative
truths.” But these intuitive abilities are not, for Parfit, some special
quasi-sensory faculty, nor do we use them to discover some mysterious new realm
of non-natural facts. Rather, we come to see that we have reasons for doing
some things, in something like the way in which we come to see that two plus
two equals four.’

how do we come to
know normative truths?

how we come to know
normative truths – is we

*propose*them –
knowledge is
proposal –

in the absence of
proposal – what we face is the unknown –

we propose –

*to make known*–
as to the ground or
basis of such proposals – this too is a matter of proposal – open to question –
open to doubt – and uncertain

parfit goes with a
version of intuitionism

which to my mind
amounts to saying that there is no basis to a normative proposal – but the
proposal itself –

if that is what
intuitionism comes down to – I have no argument

but if intuitionism
is meant as a ground to an ethical proposal – that is beyond question – beyond
doubt – that is certain – then intuitionism is a pretence –

‘we have reasons for
doing some things – in something like the way we come to see that 2 + 2 = 4’

2 = 2 = 4 – is a
rule governed sign-game – a language game

this is not good
enough – it trivializes morality –

yes we propose
reasons – and these reasons emerge out of our experience of moral uncertainty –

and are in fact
uncertain themselves –

morality is this
exploration of uncertainty –

it is the
exploration of our lived propositional experience

it is not trivial

‘This rubs against
the widely held metaphysical view that the world can be fully explained by
reference to the kind of facts that are open to investigation by the natural
sciences. Rejecting this view seems to open the way to believing in all kinds
of spooky entities, and hence many non-religious philosophers have accepted
metaphysical or ontological naturalism. Parfit does not defend non-natural
religious beliefs, but argues that without irreducibly normative truths,
nihilists would be right, for nothing would matter. It is, for example, an
irreducibly normative claim that if we establish that the premises of a valid
argument are true, then we have a decisive reason for believing the conclusion
of the argument.’

the widely held view
that the world can be fully explained by the natural sciences?

I presume that what
is meant here is that we can in principle have a complete explanation –

logically speaking –
any proposal – any explanation – is open –

open to question –
open to doubt – uncertain –

to hold that any
matter is fully explained – is pretentious

even when we have
decided for a proposal – or settled on a course of action –

if we behave
logically – we continue to question –
continue to doubt –

what we investigate
and what we explore is propositional uncertainty

to hold that any
proposition is beyond question – that is certain – is to hold a prejudice

there is no sin in
this – a good deal of human propositional action is the assertion of prejudice

my point is that
such behaviour – is not logical

the world can be
explained in any number of ways

and any proposal put
will be open to question – open to doubt – and will be –

uncertain

without
irreducibility of normative truth – the nihilists win the day?

irreducibility –
amounts to the end of question – the end of doubt – the claim of certainty

the argument for
irreducibility – is the argument for prejudice

and to suggest that
in the absence of prejudice – there is nothing – nothing to be concerned about –

is for mine – plain
ignorant – and quite preposterous

so called ‘normative
proposals’ are made – are put in various forms – and acted upon – and this is
as natural as sunshine –

this is an empirical
reality –

and the logical
reality is that any such proposal – or any proposal regarding its basis –

is uncertain

and it is this
uncertainty that is the ground of our morality – of our freedom

the valid argument –

if we establish that
the premises of a valid argument are true – then we have a decisive reason for
believing the conclusion of the argument?

a proposal – a
proposition – a premise is true – if it is affirmed –

and whatever
argument is given – if an argument is given – for the affirmation –

is open to question
– open to doubt – and uncertain

in a valid argument
the conclusion is contained in the first premise

i.e. – all men are
mortal

john is a man

___

john is mortal

the valid argument
is a language game –

really a piece of
poetry

the propositions of
such an argument are open to question

we have a ‘decisive
reason’ for believing the conclusion of the argument –

*if we decide*– to stop questioning and to suspend doubt

‘Thus Parfit
challenges metaphysical naturalists: if the position you defend were true, he
says, we could not have any reason to accept it, for there would be no such
reasons. It still might be true, but the only position we have any reason to
hold is that metaphysical naturalism is false.’

parfit’s argument –
as I take it – is that moral naturalism is a form of reductionism –

and he argues that
moral statements – are irreducible – therefore naturalism is false –

and so we have no
reason to accept it – only reason – not to accept it

to my mind – not
much of an ‘argument’ here – more in the line of an arrogant slap down

and what’s with –
‘it still might be true’?

there is no ‘might
be true’ –

it is true – if it
is affirmed – it is true – for whoever affirms it – when they affirm it – and
for whatever reason they give

and indeed – it is
false for those who reject it – when they reject it – and for whatever reason
they give

and any such
affirmation or rejection – and any reasons given for the affirmation or
rejection – are open to question – open to doubt – and uncertain

I think that
whatever human beings do is natural – is in accordance with their nature –

and I don’t think
there is any complete or final explanation of human nature –

the matter is open
to question – open to doubt and uncertain

as a matter of
practice – I don’t hold to a non-natural – or spiritual conception of
reality –

though I like to
think I keep an open mind on the matter –

my ‘naturalism’ – is
a vague materialism – for me it is more in the line of a working hypothesis –

I am sceptical of
any explanation of the world – even of the view that I hold –

and I have no final
– knock down explanation for why I hold this view

I have held
different metaphysical views at different times

I regard any
proposal – metaphysical or otherwise – to be from a logical point of view –
uncertain

I try not to be
philosophically prejudiced – but find from time to time that I am

as for what I know –
it is just and only what I propose – or what is put to me that I affirm

and I don’t believe
there is any basis to the propositions I entertain – but the uncertainty that I
face and deal with

outside of any
proposal – the world – my world – is unknown

‘Some will object
that even if we accept Parfit’s arguments, it would be a pyrrhic victory for
objectivism. He can overcome Hume’s objections only by rejecting the assumption
that morality must be capable of moving us to action. And what is the point of
an objective morality, if we are not motivated to act in accordance with the
moral truths it contains? Parfit could respond, like Kant, that insofar as we
are rational beings, we will respond to the reasons that morality offers. And
if we are not, well, the truths of morality would remain true even if no one
were to act on them.’

by ‘morality’ here –
what we are talking about is those proposals that are put in response to moral
issues

these proposals –
are open to question – open to doubt – and are uncertain

moral propositions
may move us to action – or they may not

‘the reasons that
morality offers’ – are just and only those reasons proposed – by those who
propose them – when they propose them

we behave rationally
if we put these reasons to question – to doubt – if we explore their
uncertainty

whether we act on
them or not – is not rationally relevant

‘the truths of
morality will remain true’?

all this can mean is
that moral propositions have been put – are put – and will continue to be put –

however whether or
not a moral proposition or set of propositions is affirmed – and by how many –
and under what circumstances – is an empirical matter

© greg t. charlton.
2017.

## Friday, August 19, 2016

### NOTICE: Wittgenstein’s Philosophical Grammar

The work previously published in this blog on Wittgenstein's 'Philosophical Grammar' is being reviewed and edited.

(c) greg t. charlton. 2016

The finished work will be published in book form by killer press in May 2017:

'Wittgenstein's Philosophical Grammar'

'Wittgenstein's Philosophical Grammar'

by Greg T. Charlton.

ISBN: 978-0-9586687-0-5

### Part II. On Logic and Mathematics: VII. INFINITY IN MATHEMATICS THE EXTENSIONAL VIEWPOINT

VII INFINITY IN MATHEMATICS THE EXTENSIONAL VIEWPOINT

39

*Generality in arithmetic*
‘ “What is the sense of such a
proposition as ‘($n).
3 + n = 7’?” Here we are in an old difficulty: on the one hand we feel it to be
a problem that the proposition has the choice between infinitely many values of
n, and on the other hand the sense of a proposition seems guaranteed in itself
and only needing further research on our part, because after all we all “know
‘what ‘($x)

**j**x’ 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.’
yes – of course

we are not dealing with the applications of one grammar here – rather
different grammars –

different games –

and the form of the game –
rule governed propositional action – will have many and varied
expressions

‘ “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? –

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 here is

*different propositional games*–
and yes – one is not the other

‘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 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
‘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 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.” ’

‘the generality itself’?

as I see it the issue is

*application*–
where and how does a proposition
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’ –

this is no more than to say that
the proposition the sign – the ‘external’ sign –

is open to interpretation

“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 expect 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 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’? –

the point is that the

*use*of a particular proposal – proposition – in this case a particular number – is open to question – to doubt – is uncertain
if you understand this logical
reality – then you will be open to the possibilities of that use – at any time
of its use

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

so yes –

‘In the calculus process and
result are equivalent to each other’

now 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 x

^{3}+ y^{3}and z^{3 }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 rubbish

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 turn on 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 then 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

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

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 – ‘spacial’ 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 show 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 only about 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 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 –

in the mix of uncertainty –

that is the decision process

‘Human beings are entangled all unknowing in the
net of unknowing’

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.’

how we define the maximum of a curve –

*is*the method we employ –
different methods – different definitions

that there might be argument regarding which method
to employ –

is no surprise

any method is a proposal – open to question – open
to doubt – uncertain –

we proceed in this uncertainty – whatever approach
we take –

the determining issue will be whether the method
adopted – that is the way we go about it – enables us to proceed in the
propositional context in which we are working

if it doesn’t – if there are perceived problems of
one sort or another – then a different approach will be looked for

and of course – it just may be that different
methods produce the same result –

and that the choice of method might ends up being a
matter of theoretic consistency – simplicity – or dare I say it – elegance

one way or another – it is a pragmatic decision

‘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?” ’

the word ‘infinite’ – and in fact any word in use –
will have a history – indeed

*histories*of use –
furthermore any term – or proposal – any
proposition – even given its use histories – will be

*open*– open to question – to interpretation –
language is functional uncertainty

Wittgenstein’s concern here is a result of him not
getting – or at least not accepting as everyday – in every context – the
uncertainty – that i

*s*propositional*reality*
“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.’

‘the maximum of the curve’ – is not determined by
the curve –

and so – the ‘points of a curve’ –
are

*irrelevant*to the maximum of the curve
‘the points of a curve’ – is a description – an

*explanation*of the curve – an account of the curve
an explanation of the curve – is irrelevant to the
question of the maximum of the curve

the maximum of a curve – will depend the place of the curve – the
position of the curve – in a context – a setting – in a propositional
construction

and any decision regarding the maximum of the curve
will be a pragmatic decision –

in order to proceed – where do we need the maximum
to be?

‘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, 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 ‘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 “½o,
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 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
uncertainty

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 –

draw your own conclusion

‘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 placed it 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*–
is a concept that comes from a

*rational*framework
and that 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 – no –

*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 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 away 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 –

and 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

x

^{n }+ y^{n}= z^{n}.
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’

‘I call “p

_{n}” 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
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 i

*ndeterminate**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”.
yes

‘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.’

yes – exactly

‘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.’

as to –

‘any connection
between parts of mathematics or logic that was already there’

*nothing*was

*already there*–

what is

*proposed*is*what is there*
‘without one
knowing’?

knowledge here is

*what is proposed*–
and what is proposed
– is open to question – open 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.)’

yes – in
mathematical action – mathematical games – there is no ‘not yet’ in terms of
what can occur –

what can occur is
what does occur

and likewise in
mathematical speculation – that is in the design phase of mathematical games –

we work within
existing practices – existing forms –

and if a new way of
seeing things is proposed – then it is to be argued for

mathematical
activity of any kind is here and now

“Does the operation
yield a rational number for instance?” – How can that be asked, if we have
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 – that it 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
can not 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 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
exploration

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 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

the playground

(c) greg t. charlton. 2016.