Part II. IV

IV ON CARDINAL NUMBERS

18 Kinds of cardinal numbers

‘What are numbers? – What numerals signify; an investigation of what they signify is an investigation of the grammar of numerals.’

numbers are signs in a sign game – the sign game of calculation

‘what numerals signify’? –

numerals are a standard representation of numbers –

but what you have to understand is that ‘number’ – is no more than a term for ‘that which is counted’ – whatever that is

‘number’ is a standard or formal term for ‘that’ –

 and ‘that’ of course in a final sense – is without definition –

and we don’t concern ourselves with any definition – proposed definitions – for definitions are irrelevant to the action of counting – to the counting game

and so – simply because it is convenient and useful to call ‘that which is counted’ – something –

we give it the name – ‘number’ – knowing full well – it could be anything – and that it is nothing

nothing but a tag in an action – in a game –

a record of a step in an action – a record of a play in a game –

‘an investigation of what they (numerals) signify’?

there is nothing to investigate –

unless you are a linguist or an anthropologist – or a game designer

what do numerals signify?

to the mathematician – it doesn’t matter what they signify –

the point is their use in the calculation game

what do the draughtsmen in a game of draughts signify?

‘the grammar of numerals’?

the grammar of numerals – is the use of numerals –

which amounts to the rules that govern that use –

that govern the practice

‘What we are looking for is not a definition of the concept of number, but an exposition of the grammar of the word “number” and of the numerals.’

the definition of the concept of number – is whatever description is used to account for the use of numbers

in mathematics that amounts to an account of the rules that govern number practice by

mathematicians

‘an exposition of the grammar of the word “number” and of numerals –

is an account of just how ‘number’ is used – and an account of just how numerals are used

definition and exposition – description of use –

and any definition – any exposition – will of course be open to question – open to doubt – uncertain

‘The reason why there are infinitely many cardinal numbers, is that we construct this infinite system and call it the stem of cardinal numbers. There is also a number system “1, 2, 3, 4, 5, many” and even a system “1, 2, 3, 4, 5 ”. Why shouldn’t I call that too a system of cardinal numbers (a finite one)?’

the reason why there are infinitely many cardinal numbers –

is simply because the sign operation – the game – that is cardinality – is repeatable

where a decision is made that an operation is not repeatable –

the operation is finite – the game is finite

‘It is clear that the axiom of infinity is not what Russell took it for; it is neither a proposition of logic, nor as it stands a proposition of physics. Perhaps the calculus to which it belongs, transplanted into quite different surroundings (with a quite different “interpretation”), might somewhere find an application; I do not know.

One might say of logical concepts (e.g. of the, or a, concept of infinity) that their essence proves their existence’

yes – the axiom of infinity might find an application

it is not that the essence of a logical concept proves its existence

when you strip away all the obfuscation and pretension  (‘essence’) – what you have is a proposal – and yes – when put it exists

‘(Frege would have said: “perhaps there are people who have not got beyond the first five in their acquaintance with the series of cardinal numbers (and see the rest of the series only in an indeterminate form or something of the kind), but this series exists independently of us”. Does chess exist independently of us, or not? –)’

all very well to propose that something we do – or if you like something that we make or create – exists independently of us – but a tough one to argue

actually it’s just rubbish

‘Here is a very interesting question about the position of the concept of number in logic: what happens to the concept of number if a society has no numerals, but for counting, calculating, etc. uses exclusively an abacus like an Russian abacus?

(Nothing would be more interesting than to investigate the arithmetic of such people; it would make one really understand that there is no distinction between 20 and 21.)’

what is relevant is the action of counting –

what you call what you count with – numbers – beads – colours or whatever – is logically speaking – irrelevant

in different cultures you have different descriptions – different protocols –

                                                                                                                               

20 and 21 – are steps in the action in a game of counting –

they are actions – in the game

actions that have no logical distinction –

except as determined by the rules of the game

‘Could we also imagine, in contrast with the cardinal numbers, a kind of number consisting of a series like the cardinal numbers without the 5? Certainly; but this kind of number couldn’t be used for any of the things for which we use the cardinal numbers. The way in which these numbers are missing a five is not like the way in which an apple may have been taken out of a box of apples and can be put back again; it is of their essence to lack a 5; they do not know the 5 (in the way that the cardinal numbers do not know the number ½.). So these numbers (if you want to call them that) would be used in cases where the cardinal numbers (with the 5) couldn’t meaningfully be used.

(Doesn’t the nonsensicality of the talk of the “basic intuition” show itself here?)’

different ways of calculating – different games

‘basic intuition’ – a piece of rhetoric –

just a way of big-noting – a starting point –

and in any case how you describe where you start – is irrelevant –

irrelevant to the action of calculating –

to the play of the game – whatever the game

‘When intuitionists speak of the “basic intuition” – is this a psychological process? If

so, how does it come into mathematics? Isn’t what they mean only a primitive sign (in Frege’s sense); an element of a calculus?’

it depends on how they define basic intuition –

to my mind it’s a throw back notion – to the idea of certain knowledge –

a notion which has no place in propositional logic

it doesn’t ‘come into’ mathematics – mathematics gets on with its business – with or without such epistemological clouds

and as for ‘primitive sign’ – again – just rhetoric –

a primitive sign – is a sign you stop questioning – or it’s just simply whatever you decide to work with

there is no sign – no proposal – that is beyond question – beyond doubt

propositional logic – is the logic of uncertainty

the hard reality is – we propose signs – and we use them to suit our purposes –

they are a means to an end

and whatever the means – whatever the end –

a question – a doubt – an uncertainty

‘Strange as it sounds, it possible to know the prime numbers – let’s say – only up to 7 and thus to have a finite system of prime numbers. And what we call the discovery that there are infinitely many primes is in truth the discovery of a new system with no greater rights than the other.’

yes – and it is to say that we can continue to play the prime-game

‘If you close your eyes and see countless glimmering spots of light coming and going, as we might say, it doesn’t make sense to speak of a ‘number’ of simultaneously seen dots. And you can’t say “there is always a definite number of spots there, we just don’t know what it is”; that would correspond to a rule where you can speak of checking the number’

‘there is always a definite number of spots there we just don’t know what it is’ –

is to say we can’t count here –

and if we can’t count – we can’t say there is a number

and if we could count we could say

if this is so – the focus then is properly – not on the symbols (the numbers) used to count– but on the action of counting

if you can’t perform the action – the symbol – that would be the result of the action – just isn’t there –

there is no count – there is no number

‘(It makes sense to say: I divide many among many. But the proposition “I couldn’t divide the many nuts among the many people” can’t mean that it was logically impossible. Also you can’t say in some cases it is impossible to divide many among many and in others not”; for in that case I ask: in which cases is this possible and in which impossible? And to that no further answer can be given in the many-system.)’

                                                                                                                                 

it depends what the relation between the many and the many is –

if there is a relation of cardinality the division game can be played – with the result – a whole number

if there is no cardinality – the results can be either whole numbers – real numbers – or fractions

it depends too – on what result you are looking for –

if you are only after a whole number result – then clearly there will be contexts in which the division game will not function

‘To say of a part of my visual field that it has no colour is nonsense; and of course it is equally nonsense to say that it has colour (or a colour). On the other hand it makes sense to say it has only one colour (is monochrome, or uniform in colour) or that it has at least two colours, only two colours, etc.

So in the sentence “this square in my visual field has at least two colours” I cannot  substitute “one” for “two”. Or again: “the square has only one colour” does not mean – on the analogy of ($x).fx. ~ ($x,y). ).fx. fy – “the square has one colour but not two colours”.’

colour – or no colour – is really just a question of description – the use of a description – and whatever that description is taken to imply –

the visual field – in the absence of description – as with any other epistemological focus – is an unknown

and any so called ‘natural description’ –  is logically speaking no different to what you might call a non-natural description i.e. a theoretical description –

natural or non-natural – any description is open to question – open to doubt – is uncertain

what we count is description – not a particular description – but any description –

and ‘any description’ – in the mathematics game is given the formal name of ‘number’ –

numerals – colours – beads – whatever – descriptions –

all covered by ‘number’

and to understand ‘number’ you have to understand the number game

we play a game with description – and that game is arithmetic

‘one’ is not ‘two’ – therefore ‘one’ cannot be substituted for ‘two’ –

however the proposition – ‘this square has at least one colour’ – though a different proposition to – ‘this square in my visual field has at least two colours’ – is consistent with it –

and yes – the proposition – ‘the square has one colour but not two colours’ – does not mean the same as – ‘this square has only one colour’ – but the two propositions are consistent

‘I am speaking here of the case in which it is senseless to say “that part of space has no colour”. If I am counting the uniform (monochrome) patches in the square, it does incidentally make sense to say there aren’t any there at all, if the colour of the square is continually changing. In that case of course it does make sense to say that there are one or more uniformly coloured patches in the square and also that the square has one colour and not two – But for the moment I am disregarding that use of the sentence “the square has no colour” and am speaking of a system in which it would be called a matter of course that an area of a surface had a colour, a system, therefore in which strictly speaking there is no such proposition. If you call the proposition self-evident you really mean something that is expressed by a grammatical rule giving the form of a proposition about visual space, for instance. If you now begin the series of statements giving the number of colours in the square with the proposition “there is one colour in the square”, then of course that mustn’t be the proposition of grammar about the “colouredness” of space.’

the proposition – ‘that part of space has no colour’ – is not senseless

it points to the fact that the theory of colour in use has no account of that part of space – or that that the colour game in play – cannot be played in that part of space

and if you hold that all parts of space are coloured –

then relative to your theory of colour – you would regard the statement as false

there is no ‘self-evident proposition’ –

any proposition – any proposal – is open to question – open to doubt – is uncertain

‘self-evidence’ – is illogical –

it is pretentious – and the notion functions as a rhetorical device –

it’s point is to shut down  question and doubt – and pretend – certainty

‘the proposition of grammar about the colouredness of space’?

firstly ‘grammar’ – is an account of usage –

a proposition of grammar about the colouredness of space

is a theory of the usage of propositions about the colouredness of space –

it’s a back story –

and one that as with any proposition – is open to question – open to doubt – is uncertain

‘What do you mean if you say “space is coloured”? (And, a very interesting question: what kind of question is this?) Well, perhaps you look around for confirmation and look at different colours around you and feel the inclination to say: “wherever I look there is a colour”, or “it’s all coloured, all as it were painted.” Here you are imagining colours in contrast to a colourlessness, which on closer inspection turns into a colour itself. Incidentally, when you look around for confirmation you look first and foremost at static monochromatic parts of space, rather than at unstable unclearly coloured parts (flowing water, shadows, etc.). If you then have to admit that you call just everything that you see colour, what you want to say is that being coloured is a property of space in itself, not of parts of space. But that comes to the same as saying

of chess that it is chess; and at best it can’t amount to more than a description of the game. So what we must do is describe spatial propositions; but we can’t justify them, as if we had to bring them into agreement with an independent reality.’

I describe what I see as ‘coloured’ – I point out colours – I refer to colours –

perhaps too – I use the term ‘colourless’ – in certain contexts –

and if you ask me what these colour propositions mean – I will have a go at accounting for them – I will put forward proposals

and you might keep at it with questions – with doubts –

and I will try to answer your question – and address your doubts

this process of proposal and question can go on as long as there is breath to breathe

all we do in such analysis is explore propositional possibilities –

we explore uncertainty

and if we stop doing this and just proceed with whatever proposal we find useful at the time – that is what we do –

we proceed in uncertainty

yes – you describe i.e. spatial propositions – in whatever way you do describe them –

and any so called ‘justification’ – is no more than your argument for your account –

open to question – open to doubt – uncertain

independent reality – is reality independent of description – of proposal – is the unknown

you can only propose that a proposition is in agreement – with another proposition –

another proposal

‘In order to confirm the proposition “the visual field is coloured ‘one looks around and says ‘that there is black, and black is a colour; that is white, and white is a colour”, etc. And one regards “black is a colour” as like “iron is a metal” (or perhaps

better, “gypsum is a sulphur compound”).’

yes – and as simple as that –

but confirmation is what?

putting the to and fro of argument aside here for the moment–

confirmation – if it amounts to anything is – use

                                                                                                                                 

‘If I make it senseless to say that a part of my visual field has a colour, then asking for the analysis of a statement assigning the number of colours in a part of the visual space becomes very like asking for the analysis of a statement of the number of parts of a rectangle that I divide up into parts by lines.’

yes –

if the proposal is that a part of my visual field has no colour – then assigning a number of colours in a part of my visual field – is just to assign numbers –

dividing a rectangle into parts – just is a mathematical action

and is the point here that assigning numbers is a game that can be played however the context is described –

and is therefore an action that can be performed – is  performed – regardless of so called empirical consideration?

we can play the mathematical game however the world is –

and mathematics tells us nothing about the world – except that we play these games –

we play games – and we question – is that the sum of it?

‘Here too I regard it as senseless to say that the rectangle “consists of no parts”. Hence, one cannot say that it consists of one or more parts, or that it has at least one part. Imagine the special case of a rectangle divided by parallel lines. It doesn’t matter that this is a very special case, since we don’t regard a game as less remarkable just because it has only a limited application. Here I can if I want count the parts in the usual manner, and then it is meaningless to say there are 0 parts. But I could also imagine a way of counting which so to say regards the first part as a matter of course and doesn’t count it as 0, and counts only the parts that are added to this division. Again one could imagine a custom according to which, say, soldiers in rank and file were always counted by giving the number of soldiers in a line over and above the first soldier (perhaps because we wanted the number of possible combinations of fuglemen with another soldier of the rank). But a custom could also exist of always giving the number of soldiers as 1 greater than the real one. Perhaps this happened originally in order to deceive a particular officer about the real number, and latter came into general use as a way of counting soldiers. (The academic quarter). The number of different colours on a surface might also be given by the number of their possible combinations in pairs and in that case the only numbers that would count would be numbers of the form n/2(n-1); it would be as senseless then to talk of the 2 or 4 colours of a surface as it is now to talk of the Ö2 or i colours. I want to say that it is not the case that the cardinal numbers are essentially primary and what we might call the combination numbers – 1, 2, 6, 10 etc. are secondary. We might construct an arithmetic of the combination numbers and it would be as self-contained as the arithmetic of the cardinal numbers. But equally of course there might be an arithmetic of the even numbers or of the numbers 1, 3, 4, 5, 6, 7 …Of course the system is ill adapted for the writing of these kinds of number.’

it is not that it is senseless to say that the rectangle ‘consists of no parts’ – it is just that if you play the ‘game of parts’ – then a ‘game of no parts’ – has no place –

in another context – a ‘non-part context’ – it will have legs

‘But I could also imagine a way of counting which so to say regards the first part as a matter of course and doesn’t count it as 0, and counts only the parts that are added to this division.’ –

yes – another game of calculation

‘But a custom could also exist of always giving the number of soldiers as 1 greater than the real one. Perhaps this happened originally in order to deceive a particular officer about the real number, and latter came into general use as a way of counting soldiers. (The academic quarter).’ –

and again – a different calculation game –

look whatever mathematical game you are playing – will determine – the games you are not playing – it is that straightforward

to play a game – and then refer to another game as ‘senseless’ – is really just over- doing it –

the point of any game is its utility –

and just when a game is useful is any player’s guess – is any player’s decision

cardinality is a number game –

as to cardinal numbers being essentially ‘primary’ –

which game is primary – will be a matter of focus

the ‘primary game’ will be determined by what is at the time – in play

‘Imagine a calculating machine that calculates not with beads but with colours on a strip of paper. Just as we now use our fingers, or the beads on an abacus, to count the colours on a strip so then we would use the colours on the strip to count the beads on a bar or the fingers on our hand. But how would this colour-calculating machine have to be made in order to work? We would need a sign for there being no bead on that bar. We must imagine the abacus as a practical tool and as an instrument of language. Just as we now represent a number like 5 by five fingers of a hand (imagine a gesture language) so we would then represent it as a strip with five colours. But I need the sign for the 0, otherwise I do not have the necessary multiplicity. Well I can either stipulate that a black surface is to denote the 0 (this of course is arbitrary and a monochromatic red surface would do just as well); or that any one coloured surface is to denote zero, a two coloured surface 1, etc. It is immaterial which method of denotation I choose. Here we see that the multiplicity of the beads is projected onto the multiplicity of the colours on a surface’

yes it is immaterial which method of denotation I use –

the denotation – the game – must be defined – must be determined

and in the mode of game design – as distinct from the mode of game play –

we are in the realm of uncertainty –

any proposal put forward – i.e. method of denotation – rules to use etc. –

is open to question – open to doubt – is uncertain

when decisions are made – we have a game –

we have a play

‘It makes no sense to speak of a black two sided figure in a white circle; this is analogous to its being senseless to say that the rectangle consists of 0 parts (no part). Here we have something like a lower limit of counting before we reach the number one.’

a black two sided figure in a white circle – senseless?

it depends how you define ‘figure’ –

consider two sides of an equilateral triangle drawn in black– which is to say two lines – where of course the figure is not solid – nevertheless a black two sided figure?

as for a rectangle having no parts –

this is not senseless – it is just a possible description –

another possible description is that the rectangle has parts

it really depends on what  you want to do with the rectangle

it’s a question – once again of defining your game

the rectangle – has no properties – outside of the properties ascribed  to it –

how the rectangle is to be described – is open to question – open to doubt – is uncertain

how the rectangle will be described – will be a function of the propositional context in which it is to figure

 its description will be a function of the argument of that context

‘Is counting parts in I the same as counting points in IV? What makes the difference? We may regard counting the parts in I as counting rectangles; but in that case one can also say: “in this row there is no rectangle”; and the one isn’t counting parts. We are disturbed both by the analogy between counting the points and counting the parts, and by the breakdown of the analogy.’

look – you have to decide what it is you’re counting –

so it is a question of which description you are using – i.e. parts or rectangles

once that issue is decide – the analogy – if you want there to be an analogy here – holds –

as to the counting – in the end it makes no difference to the count – how you describe what you count –

the counting game – as such – really has nothing to do with what is counted –

however the counting game will have different descriptions i.e. ‘counting parts’ – ‘counting rectangles’ – ‘counting points’ –

you are not going to have a game –

if you don’t know what you’re counting

                                                                                                                                  

‘There is something odd in counting the undivided surface as “one”; on the other hand we find no difficulty in seeing the surface after a single division as a picture of 2. Here we would much prefer to count “ 0, 2, 3”, etc. And this corresponds to the series of “the rectangle is undivided”, “the rectangle is divided into 2 parts”, etc.’

it doesn’t matter whether its odd or not –

the issue is defining the counting game

yes ‘0’ – can be  correlated with ‘the rectangle is undivided’ – but so what?

we could well decide that yes the undivided surface is ‘one’ – and quite naturally speak of the divided surface as 2

and so 1 and 2 are the only numbers in this game –

and therefore 0 – is not in the game –

and so for the purpose of the game – I’m afraid to say – 0 doesn’t exist –

shock horror

‘If it’s a question of different colours, you can imagine a way of thinking in which you don’t say that here we have two colours, but that here we have a distinction between colours; a style of thought which does not see 3 at all in red, green and yellow; which does indeed recognize as a series a series like: red: blue, green: yellow, black, white: etc., but doesn’t connect it with the series |; ||; |||; etc., or in such a way as to correlate | with the term red.’

yes – of course

‘From the point of view from which it is ‘odd’ to count the undivided surface as one,  it is also natural to count the singly divided one as two. That is what one does if one regards it as two rectangles, and that would mean looking at it from the standpoint from which the undivided one might well be counted as one rectangle. But if one regards the first rectangle in I as the undivided surface, then the second appears as a whole with one division (one distinction) and division here does not necessarily mean dividing line. What I am paying attention to is the distinctions, and here there is a series of an increasing number of distinctions. In that case I will count the rectangles in I “0, 1, 2, etc.”

This is all right where the colours on a strip border on each other, as in the schema

But it is different if the arrangement is

or  

Of course I might also correlate each of these two schemata with the schema

and correlate schemata like

 

with

 ,

etc. And that way of thinking though perfectly unnatural is perfectly correct.

The most natural thing is conceive the series as

etc. And here we may denote the first schema by ‘0’, the second by ‘1’, but the third say with ‘3’, if we think of all possible distinctions, and the fourth by ‘6’. Or we may call the third schema ‘2’ (if we are concerned simply with an arrangement) and the fourth ‘3’.’

here we have games of correlation –

and as with any game you have to decide its rule –

then the question is how far can you take the game –

i.e. what proposed correlations work and what proposed correlations don’t work –

relative to the rule decided

and of course you can always experiment with different rules –

different rules different correlations – different games – different plays

‘We can describe the way a triangle is divided by saying: it is divided into five parts, or: 4 parts have been cut off it, or: its division-schema is ABCDE, or: you can reach every part by crossing four boundaries; or the rectangle is divided (i.e. into 2 parts), one part is divided again, and both parts of this part divided, etc. I want to show that there isn’t only one method of describing the way it is divided.’

yes – there isn’t any one method of describing the way the triangle is divided –

and in so far as we explore different possible descriptions – the game as such is an exploration of possible descriptions

a rule to play yes – but the rule  – its range – its application – is open to question –

the rule is only as it were fixed – when the questioning stops – when the exploration comes to an end

such is not a logical end – rather a pragmatic beginning

‘But perhaps we might refrain altogether from using a number to denote distinction and keep solely to the schemata A, AB, ABC, etc. or we might describe it like this: 1, 12, 123 etc., or, what comes to the same, 0, 01, 012 etc.

We may very well call these too numerals.’

any description – any denotation – any game – works – if works –

that is to say – if it functions in a propositional context –

and for a propositional context to function –

its terms and argument must be explainable or recognizable to those who participate in that context –

even though shadowing the proceeding – will be question – doubt –and uncertainty

that is just the way of it

.

‘The schemata A, AB, ABC etc, ½, ½½, ½½½, etc;

; etc.; 0, 1, 2, 3, etc; 1, 12, 121323, etc., etc., are all equally fundamental.’

yes ‘fundamental’ –

not a term that we need any more – certainly not in logic –

however it is quite functional in rhetoric

we operate with our proposals – that is our propositions – and our proposals – whatever they are – are open to question – open to doubt –

are uncertain

‘We are surprised that the number-schema by which we count soldiers in a barracks isn’t supposed to hold for the parts of a rectangle. But the schema for the soldiers in the barracks is       

 

etc, the one for the parts of the rectangle is

etc. neither is primary in comparison with the other.

I can compare the series of division-schemata with the series 1, 2, 3, etc. as well as with the series 0, 1, 2, 3, etc.

If I count the parts, then there is no 0 in any number series because the series

                                                                                                                                 

etc. begins with one letter whereas the series   etc. does not begin with one dot. On the other hand, I can represent any fact about the division by this series too, only in that case “I’m not counting the parts”.’

the proposal that counting soldiers in a barracks isn’t supposed to hold for the parts of a rectangle – is – as with any proposal – open to question

Wittgenstein in the above argument puts forward a representation that illustrates a correspondence

and yes – if the aim of the game is to show a correspondence – then the game must be defined in such a way that facilitates this goal.

and as Wittgenstein shows above – even though in the lettered series – the series begins with one letter – whereas in the dot series – it doesn’t begin with one dot –

any fact about the division by this series can be represented too– so long as the game is not defined as ‘counting parts’

and so we have a correspondence – if the game is not defined as counting parts’

the logical point here is that definition of the game – is – and must be – open to question –

otherwise you run the risk of missing all the fun

‘A way of expressing the problem which, though incorrect, is natural is: why can one say “there are 2 colours on this surface” but not “there is one colour on this surface?”

Or how can I express the grammatical rule so that it is obvious and so that I’m not tempted to talk nonsense? Where is the false thought, the false analogy by which I am misled into misusing language? How must I get the grammar so that the temptation ceases? I think that setting it out by means of the series

removes the unclarity.

What matters is whether in order to count I use the number series that begins with 0 or one that begins with 1.

It is the same if I am counting the lengths of sticks or the size of hats.

If I counted with strokes the lengths of I might write them thus

   in order to show that what matters is the distinction between the directions and that a simple stroke corresponds to 0 (i.e. is the beginning).’

all that is at issue here is understanding that a proposition is a proposal – and as such is open to question – open to interpretation –

and that in mathematics – there is in fact as much propositional uncertainty – or flexibility – as there is any propositional activity

and as for  – ‘nonsense’ –

nonsense is the proposal – that is held to be beyond question – beyond doubt

there is no misuse of language – there is only different uses –

‘false thought’?

a so called ‘false thought’ – will be a proposal that it is argued doesn’t fit in the broader propositional context under consideration –

note I said ‘argued’ –  for any proposal is logically speaking open to question

yes we make decisions as to what proposals we will use – but that is just pragmatics

a ‘false thought’ – will be a proposal – you decide not to use

and let’s get off the God truck – and recognize that there will be many uses of  language – of signs – that we will be confronted with – that initially – we don’t understand

and yes we have to work to get an understanding –

and the truth of the matter is – we can never be sure – we’ve got it – or indeed – ‘what getting it’ amounts to –

and further this uncertainty – has stopped no one –

and is in fact the very engine of exploration and creativity

‘Here incidentally there is a certain difficulty about the numerals (1), ((1) + 1), etc:

beyond a certain length we cannot distinguish them any further without counting the strokes, and so without translating the signs into different ones. “||||||||||” and “|||||||||||”

cannot be distinguished in the same sense as 10 and 11, and so they aren’t in the same sense distinct signs. The same thing could also happen incidentally in the decimal system (think of the numbers 1111111111 and 11111111111), and that is not without significance.’

yes – if and when this becomes a difficulty that inhibits propositional action – we make arrangements –

i.e. we modify the sign game we are playing – or we devise a new sign system – a new game

‘Imagine someone giving us a sum to do in a stroke-notation, say |||||||||| +|||||||||||, and, while we are calculating, amusing himself by removing and adding strokes without our noticing. He would keep saying: “ but the sum isn’t right”, and we would keep going through it again, fooled every time. – Indeed, strictly speaking, we wouldn’t have any concept of a criterion for the correctness of the calculation’

as presented – this game is being sabotaged

hence – no game – no calculation

either that – or we in fact have another ‘game’ in the guise of the expected or standard game –

and without rules for it – no game –

if there are rules and the player doesn’t know them – or comes to realize he doesn’t know them –

he should wake up to himself

‘Here one might raise questions like: is it only very probable that 464 + 272 = 736? And in that case isn’t 2 + 3 = 5 also very probable? And where is the objective truth which this probability approaches? That is, how do we get a concept of 2 + 3’s really being a certain number, apart from what it seems to us to be?’

the issue here is what game are you playing?

2 + 3 = 5 is not a probability game

2 + 3 = 5

how do we get this objective truth?

this ‘objective truth’ is nothing more than a sign game – a rule governed game –

that is in use

2 + 3 = 5 – is an example of – or representation of – a propositional practice –

that has had and continues to have endorsement – primarily because of its utility

in any case we can say arithmetic is a cultural practise – and its history is ancient

the same can be said of probability games – but arithmetic’s is not a probability game

here we have different games –

and there is no value is confusing the two

‘For if we were asked: what is the criterion in the stroke-notation for our having the same numeral in front of us twice? – the answer might be: “if it looks the same both times”. Or should it be: if a one-one correlation etc. is possible?’

yes there are different ways of answering the question –

the reality is that in the end what we have is a propositional practise –

and as with any propositional practice – how you account for it – how you explain it –

is open to question – open to doubt – is uncertain

such is propositional reality

‘How can I know that |||||||||| and |||||||||| are the same sign? After all it is not enough that they look alike. For having roughly the same gestalt can’t be what is to constitute the identity of the signs, but just their being the same in number.’

what is to constitute the identity of signs?

‘identity’ as with any other proposal – is open to question – open to doubt – is uncertain

so could we run with the gestalt?

I think yes –

would such a view of identity work in a mathematical context

I think not

in the mathematical context – you do the count

‘(The problem of the distinction between 1 +1 + 1 + 1 + 1 + 1 + 1 and 1 +1 + 1 + 1 + 1 + 1 + 1 + 1 is much more fundamental than appears at first sight. It is a matter of the distinction between physical and visual number.)’

physical and visual?

is there any distinction here?

19. 2 +2 = 4

‘A cardinal number is an internal property of a list.’

a cardinal number is a use of the number game –

it is an application of numbers on numbers –

a number game played with numbers

a ‘list’ – is a description of propositions – of proposals –

a ‘list’ is a description –

‘an internal property’ –  of a list’?

is a description – of a description –

and what of such a description?

it is to say that a list has an ‘internality’ –

a ‘dimension’ – distinguishable apparently from its – ‘external’ – dimension

I think this loads up the ‘list’ with unnecessary – totally unnecessary – baggage

we have a proposal –‘the list’ –

and then a further proposal – the cardinal number – in relation to that proposal

one proposal in relation to another

that’s all there is to it

‘internal’ / ‘external’ here – descriptions reminiscent of outdated epistemology and metaphysics

yes you can go there – by all means – look – you can go wherever you like –

but to my mind – doing so – makes what is essentially a straightforward matter of propositional action – into a rather woolly metaphysical concern –

and the fact is you can avoid going down this path –

so why not?

‘Are numbers essentially concerned with concepts?’

numbers are signs in a counting game – if they are anything at all –

the issue (if there is an issue) is not numbers – the issue is the action of counting – is the counting game

it is the counting game we describe – by the use of terms such as ‘numbers’

it is entirely irrelevant – how you describe – what you count – ‘numbers’ – ‘concepts’ – ‘objects’ – whatever –

it is the action of counting – the game of counting that is relevant –

it is the doing of it that is the point of it –

description – is up for grabs

‘I believe this amounts to asking whether it makes sense to ascribe a number to objects that haven’t been brought under a concept?’

whether you bring objects under a concept or not – and that is just a way of describing objects – of dealing with objects – is irrelevant to whether you ‘ascribe a number’ or not –

the number game is played – can be played – regardless of context –

that is regardless of descriptive context –

if your descriptive context is ‘objects under concept’ – you can of course play the number game in this context

if your descriptive context is something else – the same applies –

‘The concept is only a method for determining an extension, but the extension is autonomous and, in its essence, independent of the concept; for its quite immaterial which concept we have used to determine the extension. This is the argument for the extensional viewpoint. The immediate argument against it is: if a concept is really only an expedient for aiming at an extension, then there is no place for concepts in arithmetic; in that case we must simply divorce a class completely from the concept which happens to be associated with it. But if it isn’t like that then an extension independent of a concept is just a chimaera, and in that case it’s better not to speak of it at all, but only of the concept.’

the ‘extension’ is a set of proposals – that it is proposed are related in a particular manner to an initial proposal

all we have here is individual proposals – and a proposal relating the individual proposals –

the term ‘extension’ may be used to describe this language-game –

but that is all it is – a description of a kind of propositional game –

as for ‘concept –

yes a concept is an organising principle – a description of a proposal for organising – propositions –

all very well – but not the main game

Wittgenstein goes on to say –

‘The sign for the extension of a concept is a list. We might say as an approximation that a number is an external property of its extension (the list of objects that fall under it). A number is a schema for the extension of a concept. That is, as Frege said, a statement of a number is a statement about a concept (a predicate). It is not about the extension of a concept, i.e. a list that might be something like the extension of a concept. But a number-statement about a concept has a similarity to a proposition saying that a determinate list is the extension of the concept. I use such a list when I say “a, b, c, d, fall under the concept F(x)”: “a, b, c, d,” is the list. Of course this proposition says the same as Fa.Fb.Fc.Fd; but the use of the list in writing the proposition shows its relationship to “($x, y, z, u). Fx. Fy. Fz. Fu” which we can abbreviate as “($½½½½x).F(x).” ’

‘The sign for the extension of a concept is a list’ –

a ‘list’ is a description of propositions

a number – as an external property of its extension?

a number is a point in a number-game – a mark of action – if indeed there is any need to mark the action

to speak of numbers – outside of number games – is where you go wrong

the central focus is not number – but the number-game

if you wish to describe the game – yes – you can describe it in terms of the action of numbers

‘a number as a schema for the extension of a concept’

the number game is played in various contexts – settings –

if you use the description ‘concept’ – you can run the game in that descriptive context – but this is to say nothing about the number-game as such – it is just a reference to a context of use

‘That is, as Frege said, a statement of a number is a statement about a concept (predicate). It is not about the extension of a concept. But a number-statement about a concept has a similarity to a proposition saying that a determinate list is the extension of the concept.’

this kind of rigmarole is just what results from failing to understand mathematics

yes – you play with chess pieces – but they have no value as chess pieces – outside of the chess game

by all means come up with different proposals to describe what you are working with and different descriptions for how you have organized what you are working with  i.e. – ‘concepts’ –‘extensions’ –

but these descriptions have nothing to do with the number-game

a number-statement by the way is not mathematics – is not the numbers game – and is essentially irrelevant to it

it is a propositional use of the numbers-game – a description that uses the number-game

where the focus is the number – you have well and truly left mathematics – taken yourself out of the game

Frege wallows in descriptive metaphysics –

I have no objection to him proposing a view of reality – and stitching it up nicely –

with whatever descriptions and arguments fit his view

but I would argue that in so doing he misunderstands and misuses mathematics

any proposition – any proposal is open to question – open to doubt

however when we play propositional games –

the point of the game – be mathematics or some other game – is that it is a rule governed propositional activity

when you play – if you play – you play – without question – without doubt – in accordance with the rules – whatever they might be

this is what human beings do – propose – and – play

we propose and we play – logically different activities –

best not to confuse the two

‘What arithmetic is concerned with is the schema ½½½½. – But does arithmetic talk about the lines I draw with pencil on paper? – Arithmetic doesn’t talk about the lines, it operates with them.’

yes – exactly

‘If you want to know what 2 + 2 = 4 means, you have to ask how we work it out. That means that we consider the process of calculation as the essential thing; and that’s how we look at the matter in ordinary life, at least as far as concerns the numbers that we have to work out. We mustn’t feel ashamed of regarding numbers and sums in the same way as the everyday arithmetic of every trader. In everyday life we don’t work out 2 + 2 = 4 or any of the rules of the multiplication table; we take them for granted like axioms and use them to calculate. But of course we could work out 2 + 2 = 4 and children in fact do so by counting off. Given the sequence of numbers 12345 the calculation is 1 2 1 2

                      1 2 3 4’

‘If you want to know what 2 + 2 = 4 means, you have to ask how we work it out.’ –

yes – you have to be introduced into the game – and you have to be shown – how to play it – by whatever means

what you are being shown is the game – and game playing

and does anyone really ask what the game playing means?

‘what’s it mean to play a game?’ – ‘here – watch’

and this may sound like no answer at all – but the fact is we play it – because play is a basic human activity – it is what we do

it is what we do – in language – we play language-games

and mathematics is a central game

‘That means that we consider the process of calculation as the essential thing; and that’s how we look at the matter in ordinary life, at least as far as concerns the numbers that we have to work out.’

yes the process of calculation is the essential thing

‘the process of calculation’ – is the game play

I can show you how the game is played – or different ways of playing it –  and that is the ‘explanation’ –

the play

‘The difference between my point of view and that of contemporary writers on the foundations of arithmetic is that I am not obliged to despise particular calculi like the decimal system. For me one calculus is as good as another. To look down on a particular calculus is like wanting to play chess without real pieces, because playing with pieces is too particularized and not abstract enough. If the pieces really don’t matter then one lot is as good, i.e. as interesting, as another. And if the games are really distinct from each other, then one game is as good, i.e. as interesting, as another. None of them is more sublime than the other.’

yes – exactly –

there is no reason not to have an open mind – and to be flexible –

you can play with whatever ‘pieces’ suit you under the circumstances –

and play is play –

whatever the game – it is the play that is sublime

‘Which proof of ∊ ½½. ∊ ½½½. É . ½½½½½ expresses our knowledge that this is a correct logical proposition?

Obviously, one that makes use of the fact that one can treat ($x) … as a logical sum. We may translate … [this proposition] “if there is a star in each square, then there are two in the whole rectangle” into the Russellian one. And it isn’t as if the tautologies in that notation expressed an idea that is confirmed by the proof after first of all appearing plausible; what appears plausible to us is that this expression is a tautology (a law of logic).’

a ‘correct logical proposition’ –

that will depend on what logic – what language-game is being played – and how it is played – according to its rules – or not

a proof is a language game – principally an argument whereby p is derived from premises

a tautology is a repetition – if that is a ‘law of logic’ – so be it

plausibility?

what is plausible is a proposal – a proposition – open to question – open to doubt –

uncertain

the form of the tautology – or at least how it is traditionally represented – is quite the opposite to this –

and yet you wouldn’t grace it with description ‘implausible’ –

the tautology has use as a word game – a game of redundancy –

but as a proposition – as a proposal it is malformed – like an animal born with two heads

Wittgenstein goes on to consider the adequacy of Russellian notation – Russellian proposals – relative to ordinary language proposals – and it is clear from Wittgenstein’s argument that Russellian notation – as with any other propositional form is open to question – open to doubt – is uncertain

‘ “($3 x, y). F(x, y)” would perhaps correspond to the proposition in word-language “F(x, y) is satisfied by 3 things”; and that the proposition too would need an explanation if it was not to be ambiguous.

Am I now to say that in these different cases the sign “3” has different meanings? Isn’t it rather that the sign “3” expresses what is common to the different interpretations? Why else would I have chosen it? Certainly in each of these contexts,

the same rules hold for the sign “3”. It is replaceable by 2 + 1 as usual and so on. But at all events a proposition on the pattern of  ∊ ½½. ∊ ½½½. É . ∊ ½½½½½ is no longer a tautology. Two men who live at peace with each other do not make five men who live at peace with each other. But that does not mean 2 + 3 are no longer 5; it is just that addition cannot be applied in that way. For one might say: 2 men who … and 3 men who  …, each of whom lives at peace with each other of the first group, = 5 men who …

In other words. the signs of the form ($1x, y) . F(x, y), ($2x,y) . F (x, y) etc. have the same multiplicity as the cardinal numbers, like the signs ($1x) . jx, ($2x) . jx etc. and also like the signs (∊ 1x) . jx, (∊ 2 x) . jx, etc.

‘ “($3 x, y). F(x, y)” would perhaps correspond to the proposition in word-language “F(x, y) is satisfied by 3 things”; and that the proposition too would need an explanation if it was not to be ambiguous’. –

the point here is that the proposal – the proposition ‘($3 x, y). F(x, y) – as with the proposal – the proposition – ‘F(x, y) is satisfied by 3 things’ – is open to question – open to doubt – is uncertain

‘Am I now to say that in these different cases the sign “3” has different meanings?’

if it’s the same game played across different proposals – different contexts – then it’s the same game –

and that’s all very well – but any proposal – any proposition – and the terms of any proposition – are open to question – to doubt – are – logically speaking – uncertain

you may decide to regard a term as ‘the same’ in different proposals – (this is what you do when you play number games) – but playing a language-game with regard to propositions – is not ‘the same’ as regarding them logically – that is – as – open to question – open to doubt – as uncertain

cardinality is a numbers game – and yes – this game can be played across different propositions –

and that you might say – is the point of it

‘ “There are only 4 red things, but they don’t consist of 2 and 2, as there is no function under which they fall in pairs”. That would mean regarding the proposition 2 + 2 = 4 thus: if you can see 4 circles on a surface, every two of them always have a particular property in common; say a sign inside the circle. (In that case of course every three of the circles too will have to have a sign in common etc.) If I am to make any assumption at all about reality, why not that? The ‘axiom of reducibility’ is essentially the same kind of thing. In this sense one might say that 2 and 2 always make 4. (It isn’t only because of the utter vagueness and generality of the axiom of reducibility that we are seduced into believing that – if it is a significant sentence at all – it is more than an arbitrary assumption for which there is no ground. For this reason, in this and in all similar cases, it is very illuminating to drop this generality, which doesn’t make the matter any more mathematical, and in its place to make very specific assumptions.)’

‘If I am to make any assumption at all about reality, why not that?’ –

yes if it suits your purpose

reality is propositional – that is what is proposed – and any proposal – any proposition – is open to question – open to doubt – is uncertain

that 2 + 2 = 4 – is a mathematical game – that can be played in whatever context it is played in

if regarded as ‘a sentence’ – a proposal – a proposition – it is open to question – to doubt – is uncertain –

the different logical uses or modes of language – are the proposal (proposition) –

and the game

language as a game – that is a rule governed activity – and the use of language as proposal – language – open to question – open to doubt – uncertain – language if you like – in the absence of rules

Wittgenstein does not understand the difference – the difference between the game – and the proposition – and as a consequence – confuses the two

this ‘axiom of reducibility’ – is a proposal – open to question – open to doubt – uncertain –

this is clear from Wittgenstein’s own work in the Tractatus (6.1233) -  Zermelo (1908) – Wiener (1914) and indeed Russell himself in his 19217 Introduction to the second edition of ‘Principia Mathematica’ –

the point is that if this proposal is useful – for whatever philosophical agenda – it will have a run

as for ‘an arbitrary assumption for which there is no ground’ –

the ground of any assumption is its use –

and some will ask – what ground is that?

all you can put here is – the argument or arguments for use –

and any such argument –  whatever it might be –

is a back story – is rhetoric –

the world we operate in – is what is – proposed –

and what is proposed – reality – if you like –

is groundless

‘We feel like saying: 4 does not always have to consist of 2 and 2, but if it does consist of groups it can consist of 2 and 2 or of 3 and 1 two; but not of 2 and 1 or 3 and 2, etc. In that way we get everything prepared in case 4 is actually divisible into groups. But in that case arithmetic doesn’t have anything to do with the actual 268.

division, but only with the possibility of division. The assertion might just as well be the assertion that any two of a group of 4 dots on paper are always joined by a line.

Or that around every 2 such groups of 2 dots in the real world there is always a circle drawn.’

games can be played within games –

and yes the promise of any well constructed game – is possibility –

asking what does the game allow – is not questioning the game – it is exploring the game

a game – as such is not to be confused with a proposal – a proposition – what Wittgenstein refers to here – as an ‘assertion’ –

a proposal is not a game –

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

yes – you can propose that – ‘around every 2 such groups of 2 dots in the real world there is always a circle drawn.’

good luck with that

‘Add to this that a statement like “you can see two black circles in a white rectangle” doesn’t have the form “($x, y), etc. For, if I give the circles names, the names refer to the precise location of the circles, and I can’t say of them that they are either in this rectangle or in the other. I can indeed say “there are 4 circles in both rectangles taken together” but that doesn’t mean that I can say of an individual circle that it is in one rectangle or the other. For in the case supposed “this circle is in this triangle” is senseless.’

‘you can see two black circles in a white rectangle’ – doesn’t have the form ‘($x, y)’ –

and I guess you take from this that translation from one propositional form to another – is not without contention –

you go from one proposition – to another –

and any such move is best seen not as ‘translation’ – rather as propositional exploration

some call it the indeterminacy of translation – and that is not a bad start – but really you are dealing here with different propositions –

at any time with any proposition – any proposal – the logical reality – is the reality of uncertainty –

so called ‘translation’ – is no special case –

we move from proposition to proposition – in the context of uncertainty – in any propositional activity

‘For, if I give the circles names, the names refer to the precise location of the circles, and I can’t say of them that they are either in this rectangle or in the other.’ –

actually – it doesn’t follow that naming the circles – gives precise location of the circles –

and you could say which rectangle they are in – with further description –

you could nominate one square as coloured red – the other green

and I can’t see why you couldn’t do the same with the circles –

in so doing – of course you have modified the original description (‘two black circles in a white rectangle’ etc) –

but if the question is to do with distinction – this is what you need to do –

otherwise don’t carry on about distinguishing them

the general point is that any proposal – any proposition – is uncertain – is open to question – is open to doubt –

and therefore the pursuit of precision – as certainty – is logical nonsense –

we operate with uncertain propositions – and we explore this uncertainty –

‘this circle is in this triangle’ – is not senseless –

the question is – does it function in this context?

‘But what does the proposition “there are 4 circles in the 2 rectangles taken together” mean? How do I establish that? By adding the numbers in each? In that case the number of circles in two rectangles means the result of the addition of the two numbers. – Or is it something like the result of taking a count through both triangles? Or the number of lines I get if I correlate a line to a circle no matter whether it is in this rectangle or in the other? If  “this circle” is individuated by its position, we can say “every line is correlated either to a circle in this rectangle or to a circle in the other rectangle” but not “this circle is either in this rectangle or in the other” This can only be here if  “this” and “here” do not mean the same. By contrast this line can be correlated to a circle in this rectangle because it remains this line, even if it is correlated to a circle in the other rectangle.’

‘But what does the proposition “there are 4 circles in the 2 rectangles taken together” mean? How do I establish that? –

what it means – is how it is used –

and any use is open to question – open to doubt – is uncertain

how do I ‘establish’ that?

establish? – you use it in some propositional context –

use – is all ‘establish’ can come to

the proposition – ‘there are 4 circles in the 2 rectangles taken together’ – as with any proposal – is open to question – open to doubt – is uncertain –

which is to say – open to interpretation –

and what you get from Wittgenstein in what follows above (‘By adding … triangle’) –

is just that – interpretation – argument

‘In these two circles together there are 9 dots or 7? As one normally understands the question, 7. But must I understand it so? Why shouldn’t I count twice the points that are common to both circles?’

the point is there is an argument –

the proposal – the pictorial proposition – is open to question – open to doubt – is uncertain –

and obviously so – its logic – is not hidden – it is revealed –

there is no rhetorical packaging – it is plain – for all to see

here we have a fine display of propositional reality

a good illustration

                                                                                                                                                                                                                                                                    

‘It is a different matter if we ask “how many dots are within the black lines?” For here I can say: in the sense in which there are 5 and 4 in the circles, there are 7.’

yes – as the form of the question is specific – i.e. the black lines

‘Now we might say: by the sum of 4 and 5 I mean the number of the objects which fall under the concept jx v yx, if it is the case that (E 4x) . jx.( E 5x) . yx . Ind. That doesn’t mean that the sum of 4 and 5 may only be used in the context of propositions like ($ 4x) . jx; it means: if you want to construct the sum of n and m, insert the numbers on the left hand side of “É” in the form ($nx) . jx . ($mx) . yx, etc. and the sum of m and n will be the number which has to go on the right hand side in order to make the whole proposition a tautology. So that is a method of addition – a very long-winded one.’

yes – again – there is no ambiguity –

and as for the ‘construction’ – the ‘tautology’ – a language-game – as explanation of a language-game –

however – if you understand the game – its logic  – to begin with – what need any further ‘explanation’?

it’s really just a restatement in another form –

as if such has any ‘deeper’ meaning – any logical significance – beyond ‘4 +5 = 9’!

‘Compare: Hydrogen and oxygen yield water”, “2 dots and 3 dots yield 5 dots”.’

here we have two proposals – open to question – open to doubt – and uncertain –

i.e. just what are we to make of ‘yield’ – in both proposals?

‘So do e.g. 2 dots in my visual field, that I “see as 4” and not “as 2 and 2” consist of 2 and 2? Well what does that mean? Is it asking whether in some way they are divided into groups of 2 dots each? Of course not (for in that case they would presumably have had to be divided in all other conceivable ways as well). Does it mean that they can be divided into groups of 2 and 2, i.e. that it makes sense to speak of such groups in the four? – At any rate it does correspond to the sentence 2 + 2 = 4 that I can’t say that the group of 4 dots I saw consisted of separate groups of 2 and 3. Everyone will say: That’s “impossible”, because 3 + 2 = 5. (And impossible here means “nonsensical”.)’

2 dots in my visual field that I see as 4 –

are 4 dots in my visual field – not 2

if I see them as 2 and 2 – then I have applied a mathematical game to what I see –

and this I did too – when I described what I saw as ‘4’ –

‘is it asking whether in some way they are divided into groups of 2 dots each?’ –

if you apply the division game here – yes ‘in some way’ they are divided into groups of 2 dots each –

dividing them into groups of 2 dots each – is a way of describing what you see – a way of dealing with what you see –

and presumably playing this game is of some use to you –

‘of course not (for in that case they would presumably have had to be divided in all other conceivable ways as well)’ –

no – they don’t have to be divided in all conceivable ways –

what has been done – the game that has been played – is the division into groups of 2 dots each

‘does it mean that they can be divided into groups of 2 and 2, i.e. that it makes sense to speak of such groups in the four?’

why not?

‘at any rate it does correspond to the sentence 2 + 2 = 4 that I can’t say that the group of 4 dots I saw consisted of separate groups of 2 and 3’ –

correct – the numbers game here – has rules –

if you want to change the game – you have to change the rules –

it would be an entirely different game

‘everyone will say: That’s “impossible”, because 3 + 2 =5. (And impossible here means “nonsensical”.)’ –

‘3 + 2 = 5’ – has the form of a recognizable arithmetic game – but it is not a game we understand given the present propositional context

as a proposal – it is open to question  –

if you are to put that proposal – forward explain what you are on about –

otherwise it is no more than an illustration – of not following the rules of the game – 

of not understanding accepted practise in a given context

“Do 4 dots consist of 2 and 2?” may be a question about a physical or visual fact; for it isn’t a question in arithmetic. The arithmetical question, however, certainly could be put in the form: “Can a group of 4 dots consist of separate groups of 2?”

‘do 4 dots consist of 2 and 2 may be a question about a physical or visual fact’? –

‘the physical or visual fact’ – is the descriptive context – in which the arithmetical question is asked

‘can a group of 4 consist of separate groups of 2?’ – is an arithmetical question – asked in the absence of descriptive context

let’s be clear here – asking an arithmetical question – is not doing arithmetic –

such questions are game questions – questions about the game – the logic of the game the rules of the game –

if you do arithmetic – you play the game

questioning the game – is not playing the game

‘ “Suppose that I used to believe that there wasn’t anything at all except one function and four objects that satisfy it. Latter I realise that it is satisfied by a fifth thing too: does that make the sign ‘4’ become senseless?” – well if there is no 4 in the calculus then ‘4’ is senseless.’

what you believe – whatever that is – is open to question – open to doubt –

what you believe is uncertain

realizing latter that there is a fifth thing too – just makes the point –

beliefs are proposals

‘does that make the sign ‘4’ become senseless?” – well if there is no 4 in the calculus then ‘4’ is senseless’

what it means is that if you are going to continue to play the arithmetical game – in the new context – then presumably the terms of that game  – will change to fit the context

in any case ‘senseless’ is way too harsh – and somewhat stupid –

if a sign – a term – has a use – it has sense –

when it is not used – it is not it is senseless –

it is simply not used

in what follows Wittgenstein considers the use of the tautology when adding – and he

explains how it can be used

this is all very well but the point is that the tautology game as applied to arithmetic – is no more than a restatement of the arithmetic game – a restatement – that in no way adds to the arithmetic game – or takes away from it – if the tautology game is constructed successfully

symbolic logic – or some variant of it might  provide an interesting structure – and lead to interesting insights – but it is a language game – played on a language game

its basis is no different to the language it describes –

and its ‘basis’ – as with the basis to any language use – is open to question – open to doubt – is uncertain

Wittgenstein goes on to look at proofs of arithmetical game-propositions –

his ‘proofs’ simply amount to rewriting the game-proposition – in some other form – as if this provides proof

let’s be clear we can completely dispense with this notion of proof –

a game-proposition – is a proposition that is to be understood in terms of the rules of the game

these rules of the game are accepted for the purpose of playing the game –

if you are to play the game – play it –

if what you are on about is questioning the game – question it – but don’t think that in so doing you are playing it

any proposal – any proposition is open to question – open to doubt –

no game is –

that is the point of the game –

it is the point of play

‘It may sound odd, but it is good advice at this point; don’t do philosophy here, do mathematics’

any mathematical game – begins with proposals and involves the construction and the development of a language-game –

this is a speculative enterprise –

any such proposal – the utility of any such proposal or set of proposals – is – open to question – open to doubt – is uncertain –

mathematicians have the final word here – it is a question of what they regard as  useful to their practise

doing mathematics – is doing it – playing the game –  playing the games

the business of proposal and argument – is philosophical –

and this is the case – in whatever context – whatever context –

you find proposal and argument

Wittgenstein goes on to consider calculus construction –

this amounts to a discussion of language-game construction –

and the business of fitting one language-game to another –

this you might say is a game in itself – where rules are resurrected or indeed new rules proposed –

and there is indeed art to all of this

‘A question that suggests itself is this: must we introduce the cardinal numbers in connection with the notation ($x, y, …) . fx. fy … ? Is the calculus of the cardinal numbers somehow bound up with the calculus of the signs ($x, y, …) . fx. fy … ?

Is that kind of calculus perhaps in the nature of things the only application of cardinal numbers? So far as concerns the “application of the cardinal numbers in the grammar”, we can refer to what we said about the concept of the application of a calculus. We might put our question in this way too: in the propositions of our language – if we imagine them translated into Russell’s notation – do the cardinal numbers always occur after the sign “$”? This question is closely connected with another: Is a numeral always used in language as a characterization of a concept – of a function? The answer to that is that our language does always use the numerals as attributes of concept-words – but that these concept-words belong to different grammatical systems that are so totally distinct from each other (as you see from the fact that some of them have meaning in contexts in which others are senseless), that a norm making them all concept-words is an uninteresting one. But the notation

“($x, y, …)  etc.” is just such a norm. It is a straight translation of a norm of our word-language, the expression “there is …”, which is a form of expression into which countless grammatical forms are squeezed.’

‘A question that suggests itself is this: must we introduce the cardinal numbers in connection with the notation ($x, y, …) . fx. fy … ? –

there is no ‘must’ involved here –

it depends on what you have in mind and why

‘Is the calculus of the cardinal numbers somehow bound up with the calculus of the signs ($x, y, …) . fx. fy … ?’ –

is firstly to ask – can cardinality – the sign-game that is cardinality – be played in this calculus?

yes – of course

is it ‘bound up’ with the calculus of the signs – ($x, y, …) . fx. fy …?

cardinality –  is not ‘bound by’ any notation

we play the game cardinality –– in whatever context we find suitable – with whatever notation we are using

‘Is that kind of calculus perhaps in the nature of things the only application of cardinal numbers?’ –

it’s ‘application’ – as with any game – is the setting – in which it is played –

therefore – you can say cardinality has no application – or that it has whatever application it has

‘We might put our question in this way too: in the propositions of our language – if we imagine them translated into Russell’s notation – do the cardinal numbers always occur after the sign “$”? This question is closely connected with another: Is a numeral always used in language as a characterization of a concept – of a function? The answer to that is that our language does always use the numerals as attributes of concept-words – but that these concept-words belong to different grammatical systems that are so totally distinct from each other (as you see from the fact that some of them have meaning in contexts in which others are senseless), that a norm making them all concept-words is an uninteresting one. But the notation “($x, y, …)  etc.” is just such a norm. It is a straight translation of a norm of our word-language, the expression “there is …”, which is a form of expression into which countless grammatical forms are squeezed.’ –

yes – you can see it this way –

but you could also argue that ‘there is’ – is in fact a logical space – the carving out of a logical space – that is empty – and thus allows for any description –

and yes – the description – ‘concept-words’ – would fit there as indeed would any other characterization – any other description of the terms or signs used – in any language-game or any language use

the notation “($x, y, …)  etc.” – is simply a description – of a starting point in language use –

it is a formal characterization – not of any actual language use – but rather of the ground of propositional action –

it is a characterization of the domain of the proposal – of the proposition –

otherwise it is just an instance of an artificial language – set up to serve a philosophical cause – or objective –

Russell’s logical analysis is inventive and of interest – but the overall objective of providing a foundation to language – to thought – is fool’s gold

the proposition is a proposal – open to question – open to doubt – uncertain

all we do in any so called ‘analysis’ – is explore uncertainty

‘If we disregard functions containing “=” (x = a . v . x = b) etc. ), then on Russell’s theory 5 =1 if there are no functions satisfied by only one argument, or by only 5 arguments. Of course at first this proposition seems nonsensical; for in that case how can one sensibly say that there are no such functions? Russell would have to say that the statement that there are five functions and the statement that these are one-functions can only be separated if we have in our symbolism a five class and a one class. Perhaps he could say that his view is correct because without the paradigm of the class 5 in the symbolism, I can’t say at all that a function is satisfied by five arguments. That is to say, from the existence of the sentence “($j) : (E1 x) . jx” its truth already follows. – So you seem to be able to say: look at this sentence, and you will see that it is true.’ And in a sense irrelevant for our purposes that is indeed possible: think of the wall of a room on which is written in red “in this room there is something red”. – ’

the point is we can use whatever symbolism suits our purpose –

and as Wittgenstein points out in relation to Russell’s logic – there is – there will be questions whatever language-game is being played

as for – ‘look at this sentence, and you see that it is true’ –

look at that sentence and you see a game

‘This problem is concerned with the fact that in an ostensive definition I do not state anything about the paradigm (sample); I only use it to make a statement. It belongs to the symbolism and is not one of the objects to which I apply the symbolism.

For instance, suppose that “1 foot” were defined as the length of a particular rod in my room, so that instead of saying “this door is 6 ft high” I would say “this door is six times as high as this length” (pointing to the unit rod). In that case we wouldn’t say things like “the proposition ‘there is an object whose length is 1 ft’ proves itself,

because I couldn’t express the proposition at all if there were no object of that length”. (That is, if I introduced the sign “this length” instead of “1 foot”, then the statement that the unit rod is 1 foot long would mean “this rod has this length” (where I point both times to the same rod). Similarly one cannot say of a group of strokes serving as a paradigm of 3, that it consists of 3 strokes.

“If the proposition isn’t true, then the proposition doesn’t exist” means: “if the proposition doesn’t exist, then it doesn’t exist”. And one proposition can never describe the paradigm in another, unless it ceases to be a paradigm. If the length of the unit rod can be described by assigning it the length “1 foot”, then it isn’t a paradigm of the unit length; if it were, every statement of length would have to be made by means of it.’

an object to which I apply the symbolism is – by that symbolism – made known

in the absence of any application of any symbolism – ‘the object’ – if you can even call it that – is unknown

the object does not belong to the symbolism – the object is made – made known – by the symbolism

‘In that case we wouldn’t say things like “the proposition ‘there is an object whose length is 1 ft’ proves itself, because I couldn’t express the proposition at all if there were no object of that length”.’ –

let’s be clear here the proposition – is a proposal –

there is no ‘proving itself’ –

as a proposal – it is open to question – open to doubt – it is uncertain

it is not that I couldn’t express the proposition at all if there were no object of that length –

the proposition could well be advanced –

and if it was – in the absence of an object of that length – the proposal – would be open to question – open to doubt –

as indeed it would be – in the presence of an object of that length

‘Similarly one cannot say of a group of strokes serving as a paradigm of 3, that it consists of 3 strokes.’

you can say this – and as with any proposal – any proposition – it is open to question

a proposal – a proposition – is logically speaking – an argument place

“If the proposition isn’t true, then the proposition doesn’t exist” means: “if the proposition doesn’t exist, then it doesn’t exist”. And one proposition can never describe the paradigm in another, unless it ceases to be a paradigm. If the length of the unit rod can be described by assigning it the length “1 foot”, then it isn’t a paradigm of the unit length; if it were, every statement of length would have to be made by means of it.’

‘if the proposition isn’t true’ –

is only to say – if the proposal is not affirmed –

if the proposal is put – the proposition exists

this so called ‘paradigm’ – is a proposal – is a description – open to question – open to doubt – uncertain

and of course one proposition can describe another proposition –

if we are to elucidate a proposal (‘paradigm’) – we propose in relation to it

if the suggestion is that this ‘paradigm’ is in some way beyond description – or hidden in a proposition

that is just essentialist and obscurantist nonsense

‘1 foot’ is a description of length –

open to question – open to doubt –

uncertain

‘If we give any sense at all to a proposition of the form “~ ($j) : (E x) . jx” it must be a proposition like: “there is no circle on this surface containing only one black speck” (I mean: it must have that sort of determined sense, and not remain vague as it did in Russellian logic and in my own Tractatus).

If it follows from the propositions

            r) ~ ($j) : (E x) . jx

and      s) ~ ($j) : (E x, y) . jx . jy

that 1 =2, then here “1” and “2” don’t mean what we commonly mean by them, because in word-language the propositions r and s would be  ‘there is no function that is satisfied by only one thing” and ‘there is no function that is satisfied by only two things.” And according to the rules of our language these are propositions with different senses.

One is tempted to say: “In order to express ‘($x, y) . jx . jy’ we need 2 signs ‘x’ and ‘y’.” But that has no meaning. What we need for it, it is, perhaps, pen and paper; and the proposition means no more than “to express ‘p’ we need ‘p’.” ’

‘it must be a proposition like: “there is no circle on this surface containing only one black speck” (I mean: it must have that sort of determined sense, and not remain vague as it did in Russellian logic and in my own Tractatus).’ –

the proposition “~ ($j) : (E x) . jx” – is open to question – open to doubt – is uncertain

what Wittgenstein mistakes for vagueness – is propositional uncertainty –

no proposal – however constructed – is beyond question – beyond doubt –

if you are looking for ‘certainty’ – go for rhetoric – prejudice and ignorance –

‘And according to the rules of our language these are propositions with different senses.’ –

look the sense of any proposition is never settled –

our propositions are uncertain – and move in uncertainty –

we explore uncertainty – and we create forms with it and in it

‘to express ‘p’ we need ‘p’ ’ –

the proposition is a proposal – if it’s put – it’s put – if it’s not – it’s not

‘to express ‘p’ we need ‘p’ ’ –

is just another proposal – and relative to any proposition actually advanced –

of no significance or relevance whatsoever –

perhaps that is Wittgenstein’s point?

‘If we ask: but what then does “5 + 7 = 12” mean – what kind of significance or point is left for this expression after the elimination of the tautologies, etc. from the arithmetical calculus? The answer is: this equation is a replacement rule which is based on certain general replacement rules, the rules of addition. The content of 5 + 7 = 12 (supposing someone did know it) is precisely what children find difficult when they are learning the proposition in arithmetic lessons.’

‘supposing someone did know it’ – a joke from Dr. Wittgenstein?

what we have here with 5 + 7 = 12 – is a sign-game –

its ‘content’ – is the symbolism –

its content is just what you see

and as Wittgenstein says –

‘this equation is a replacement rule which is based on certain general replacement rules, the rules of addition’ –

this practise has developed – and developed – because human beings find it useful –

why? – because they do

‘No investigation of concepts, only insight into the number-calculus can tell us that

3 + 2 = 5. That is what makes us rebel against the idea that

“(E 3x) . jx . (E 2 x) . yx . Ind.: É . ( E 5x) . jx v yx”

could be the proposition 3 + 2 = 5. For what enables us to tell that this expression is a tautology cannot itself be the result of an examination of concepts, but must be recognisable from the calculus. For the grammar is a calculus. That is, nothing of what the tautology calculus contains apart from the number calculus serves to justify it and if it is number we are interested in the rest is mere decoration.

Children learn in school that 2 x 2 = 4, but not that 2 = 2’

3 + 2 = 5 –

yes you can rewrite it as –

“(E 3x) . jx . (E 2 x) . yx . Ind.: É . ( E 5x) . jx v yx”

and you describe the statement as a ‘tautology’ –

which however you then explain – comes back  to –

3 + 2 = 5

so –

‘nothing of what the tautology calculus contains apart from the number calculus serves to justify it and if it is number we are interested in the rest is mere decoration’

I would say – rather than ‘decoration’ – rhetoric – which if you don’t take a hard-arse line – amounts to the same thing

what this is about – and what Wittgenstein doesn’t actually get to is that understanding 3 + 2 = 5 is recognizing a game –

and that is seeing that a ‘game’ – is a rule governed proposition – a rule governed propositional activity

we recognize this in all manner of language uses – and we do it quite naturally

and as for 2 = 2 –

2 = 2 – is no profound and final logical analysis –

it is rather a perfect example of (logical) rhetoric –

a stupid attempt to provide foundation were there is none – and in fact – where there is no need for there to be any

it is rather a statement of the blinding obvious – a statement entirely unnecessary to make

and a statement any child would see as pointless

20. Statements of number within mathematics

‘What distinguishes a statement of number about a concept from one about a variable? The first is a proposition about the concept, the second a grammatical rule concerning the variable.

But can’t I specify a variable by saying that its values are to be objects satisfying a certain function? In that way I do not indeed specify the variable unless I know which objects satisfy the function, that is, if these objects are given me in another way (say by a list); and then giving the function becomes superfluous. If we do not know whether an object satisfies the function, then we do not know whether it is to be a value of the variable, and the grammar of the variable is in that case simply not expressed in this respect.”

‘What distinguishes a statement of number about a concept from one about a variable? The first is a proposition about the concept, the second a grammatical rule concerning the variable.’ –

a proposition about a concept – is a proposition about a proposition –

a proposal in relation to a proposal

a grammatical rule concerning the variable –

is logically speaking a proposal regarding a proposal –

the variable is formally undefined –

the concept – not so –

though any definition of a concept –

is open to question – open to doubt – is uncertain

the logic of the concept and the variable – is at base – the same –

the difference has to do with propositional space or location

a concept is located – even though this ‘location’ is –  and always will be open to question

the variable on the other hand – is open –

it’s location in propositional space in respect to any formal action – is irrelevant –

in fact strictly speaking – it has no location

here we are talking about different propositional functions –

or different types of propositional functions

different uses that have come about in propositional practice

‘But can’t I specify a variable by saying that its values are to be objects satisfying a certain function? In that way I do not indeed specify the variable unless I know which objects satisfy the function, that is, if these objects are given me in another way (say by a list); and then giving the function becomes superfluous.’

the ‘object’ – is a description of a point in propositional space – a starting point –

the function – a description of action in propositional space –

effectively different modes of the proposed variable –

different descriptions – different perspectives –

different uses

‘If we do not know whether an object satisfies the function, then we do not know whether it is to be a value of the variable, and the grammar of the variable is in that case simply not expressed in this respect.’ –

‘if we do not know’? –

what we do is propose values for a variable –

and any proposal is just that – a proposal – open to question – open to doubt – uncertain –

that is our ‘knowledge’ –

and as for the grammar –

grammar is an analysis of the proposal – an interpretation –

and as with any interpretation of any kind –

uncertain in every respect –

nevertheless – whatever the interpretation – we follow or invent –

that is what we proceed with

‘Statements of number in mathematics (e.g. “The equation x² = 1 has two roots”) are therefore quite a different kind of thing from statements of number outside of mathematics (“There are two apples on the table”).’

the statement ‘The equation x² = 1 has two roots’ – refers to a rule governed propositional action (equation) – a  mathematical game

the statement ‘There are two apples on the table’ refers to rule governed propositional action (addition) – a mathematical game

there is no ‘in or out’ of mathematics – if the propositional focus – in whatever context – is the mathematical game

‘If we say AB admits of 2 permutations, it sounds as if we had made a general assertion, analogous to “There are 2 men in the room” in which nothing further is said or need to be known about the men. But this isn’t so in the AB case, I cannot give a more general description of AB, BA and so the proposition that no permutations are possible cannot say less than that the permutations AB, BA are possible. To say that 6 permutations of 3 elements are possible cannot say less, i.e. anything more general, than is shown by the schema:

A B C

A C B

B A C

B C A

C A B

C B A

For it is impossible to know the number of possible permutations without knowing which they are. And if this weren’t so, the theory of combinations wouldn’t be capable of arriving at its general formulae. The law which we see in the formulation of the permutations is represented by the equation p = n! In the same sense, I believe as that in which the circle is given by its equation. – Of course I can correlate the number 2 with permutations AB, BA just as I can 6 with the complete permutations of A, B, C, but that does not give me the theorem of combination theory. – What I see in AB, BA is an internal relation which therefore cannot be described. That is, what cannot be described is that which makes this class of permutations complete. I can only count what is actually there, not possibilities. But I can e.g. work out how many rows a man must write if in each row he puts permutations of 3 elements and goes on until he cannot go further without repetition. And this means, he needs 6 rows to write down the permutations ABC, ACB, etc., since these just are “the permutations of A, B, C”. But it makes no sense to say that these are all permutations of A B C.’

‘For it is impossible to know the number of possible permutations without knowing which they are’ –

yes you have to play the permutations / calculation game – to see – to know – what the number of permutations is

‘What I see in AB, BA is an internal relation which therefore cannot be described. That is, what cannot be described is that which makes this class of permutations complete. I can only count what is actually there, not possibilities.’ –

what I see in AB, BA is an internal relation which therefore cannot be described? –

‘internal relation’ if it’s something you ‘see’ – is something that can be described –

‘internal relations’ – is a description

and i.e.  – to say ‘AB, BA are the 2 permutations admitted by AB’ – is another description

AB, BA – can be described – in any number of ways

to say ‘x cannot be described’ – is to describe with – ‘x’ –

what is – is what is described –

or you could say here – to be – is to be described

if you are in the business of not describing – don’t describe – be silent –

‘what cannot be described is that which makes this class of permutations complete. I can only count what is actually there, not possibilities.’ –

I can play the permutations game – the calculation game – in whatever context given e.g. AB or ABC –

that’s all this is about

‘We could imagine a combination computer exactly like the Russian abacus’

yes

‘It is clear that there is a mathematical question: “How many permutations of –

say 4 elements are there?”, a question of precisely the same kind as “What is 25 x 18?”. For in both cases there is a general method of solution.

But still it is only with respect to this method that this question exists.’

both questions are game questions –

and the game is calculation –

the questions will likely make no sense to you – unless you see them as game questions – and you know how to play these games –

that is to say –

you understand rule governed propositional play

‘The proposition that there are 6 permutations of 3 elements is identical with the permutations schema and thus there isn’t here a proposition “There are 7 permutations of 3 elements”, for no such schema corresponds to it.’

a permutations schema is a description of game moves –

there are 7 permutations of 3 elements – is not a description of a game –

a game that is played

‘You could also conceive the number 6 in this case as another kind of number, the permutations-number of A B C. Permutation is another kind of counting.’

6 is a number in a number game – in whatever context you play the counting game

permutations – is not ‘another kind of counting’ –

‘permutations’ is a propositional context – in which the counting game is played

‘permutations’ – is a game description

‘If you want to know what a proposition means, you can always ask “How do I know that?” Do I know that there are 6 permutations of 3 elements in the same way in which I know that there are 6 people in this room? No. Therefore the first proposition is of a different kind from second.’

the right question is – do I know how to play the calculation game – in relation to the question of permutations? –

and can I play the calculation in the context of the people in this room?

there is no difference in kind between the first question and the second –

both questions are game-propositions –

and the game is calculation –

the only difference is context – is setting – where you do the calculation –

that is to say – in what descriptive setting –

you play the game

‘You may also say that the proposition “There are 6 permutations of 3 elements” is related to the proposition “There are 6 people in this room” in precisely the same way as “3 + 3 = 6”, which you could also cast in the form “There are 6 units in 3 + 3”. And just as in the one case I can count the rows in the permutation schema, so in the other I can count the strokes in           

|    |    |

|    |    |

Just as I can prove that 4 x 3 = 12 by means of the schema

o    o    o

o    o    o

o    o    o

I can also prove 3! = 6 by means of the permutation schema.’

‘There are 6 permutations of 3 elements’ and ‘There are 6 people in this room’ are game propositions –

‘elements’ and ‘people in this room’ – are calculation settings –

that is descriptive settings for the action of  calculation

a permutations schema – is not ‘proof’ of anything –

a permutations schema is an illustration – a model for the permutations / calculation game

there is no proof in game playing – there is just the game – and its play

‘The proposition “the relation R links two objects”, if it is the same as “R is a two place relation, is a proposition of grammar.

the proposition – the proposal – ‘the relation R links two objects’ – is – a description of – a propositional form –

if applied to a proposition – it is an analysis of the proposition –

it tells you what ‘R’ is – and how to use it

‘R is a two place relation’ –

a description of how ‘R’ – is used in a particular (formal logic) context of use –

again a direction even – as to how to interpret ‘R’

Wittgenstein hypothesizes here –

all very well – to propose a view of these propositions – of their relationship

and yes – the propositions can be understood in Wittgenstein’s sense of ‘a proposition of grammar’ –

this being said however – all you have here from Wittgenstein – is a proposal – open to question – open to doubt

logically speaking – uncertain

21. Sameness of number and sameness of length

‘How should we regard the propositions “these hats are of the same size”, or “these rods have the same length” or “these patches have the same colour”? Should we write them in the form “($L).La .Lb”? But if that is intended in the usual way, and so is used with the usual rules, it would mean that it made sense to write “($L).La”, i.e. “the patch has a colour”, “the rod has a length”. Of course I can write “($L).La .Lb” for “a and b have the same length” provided that I know and bear in mind that “($L).La” is senseless; but then the notation becomes misleading and confusing (“to have a length”, “to have a father”). – What we have here is something that we often express in ordinary language as follows: “If a has the length L, so does b”; but here the sentence “a has the length L” has no sense, or at least not as a statement about a: the proposition should be reworded “if we call the length of a ‘L’, then the length of b is L” and ‘L’ is essentially a variable.’

what is clear here is that the formal representation does not deal with ‘same’ –

formal representation does not provide successful analysis

and analysis here – when you strip away the philosophical pretension of the logicists –

is no more than – duplication

and once analysis is seen for just what it is – you have to ask – successful or not –

what is the point – why bother?

‘The proposition incidentally has the form of an example, of a proposition that could serve as an example for the general sentence: we might go on: “for example, if the length of a is 5 metres, then the length of b is 5 metres, etc.” – “Saying “the rods a and b have the same length” say nothing about the length of each rod; for it doesn’t even say “that each of the two has a length”. So it is quite unlike “A and B have the same father” and “the name of the father of A and B is ‘N’”, where I simply substitute the proper name for the general description. It is not that there is a certain length of which we are at first only told that a and b both possess it, and of which ‘5m’ is the name. If the lengths are lengths in the visual field we can say the two lengths are the same, without in general being able to “name” them with number. –The written form of the proposition “if L is the length of a, the length of b too is L” is derived from the form of an example. And we might express the general proposition by actually enumerating examples and adding “etc.”. And if I say “a and b are the same length; if the length of a is L, then the length of b is L; if a is 5m long then b is 5 m long, if a is 7 m long, then b is 7 m long, etc.”, I am repeating the same proposition. The third formulation shows that the “and” in the proposition doesn’t stand between the two forms, as it does in “($x). jx. yx”, where one can also write “($x). jx” and “($x). yx”.’

the proposition has the form of an example?

well prime facie – it doesn’t

yes you can reinterpret it that way –

but if it is to be ‘an example’ – it is in effect a different proposition

still the question is – does such a reinterpretation account for ‘same’?

enumerating examples – as such – doesn’t do it 

enumeration of examples might well function as a proposal in a game of syntactical correspondence between two or more proposals –

but in that case enumeration is a means to the end –

not the end itself

‘Let us take as an example the proposition “there are the same number of apples in each of the two boxes”. If we write this proposition in the form “there is a number that is the number of the apples in each of the boxes” here too we cannot construct the form “there is a number that is the number of apples in this box” or “the apples in this box have a number”. If I write: ($x). jx. ~ ($x, y). jx . jy. =. $n Ix). j I,  etc. then we might write the proposition “the number of apples in both boxes is the same as “($n). jn. yn”. But “($n). jn” would not be a proposition.’

yes – once again the formal representation is not up to it

‘there is a number that is the number of apples in each box’

clearly is not an adequate representation of ‘there are the same number of apples in each of the two boxes’

the clearest way of putting it is this –

we have two propositions – (1) ‘there are 5 apples in this box’ – (2) ‘there are 5 apples in that box’

the proposal ‘there is the same number of apples in each box’ – relates the two proposals – and is in effect shorthand for the two proposals

the ‘number’ – whatever it is – is the result of an arithmetical game of addition

the real point of  ‘there is the same number of apples in each of the two boxes’ – is that the arithmetical game has been applied to the two proposals

‘The expressions “same number”, “same length”, “same colour”, etc. have grammars which are similar but not the same. In each case it is tempting to regard the proposition as an endless logical sum whose terms have the form jn.yn. Moreover, each of these words has several different meanings, i.e. can itself be replaced by several words with different grammars. For “same number” does not mean the same when applied to lines simultaneously present in the visual field as in connection with the apples in two boxes; and “same length” applied in visual space is different from “same length” in Euclidean space; and the meaning of “same colour” depends on the criterion we adopt for sameness of colour.’

any ‘expression’ – in any propositional form – is open to question – open to doubt – is logically uncertain –

and yes – in each case (‘same number’ –‘same length’ – ‘same colour’) – you could regard the proposition as an endless logical sum –

and as pointed out each of these words can have different meanings – different grammars

it all depends on where you are propositionally – and what you are doing propositionally

formal representation is just one propositional option – and one that I think has a very narrow application

to expect one propositional form to function in all contexts of use is really a failure to understand propositional reality

this ‘logical analysis’ enterprise – is wrongheaded –

at best it’s a harmless language game –

at worst – a pretentious waste of time

‘If we are talking about patches in the visual field seen simultaneously, the expression “same length” varies in meaning depending on whether the lines are immediately adjacent or at a distance from each other. In word-language we often get out of the difficulty by using the expression “it looks”.’

it’s not a ‘difficulty’ – the real world of propositional reality – just is that a proposal – in any language use – is open to question – open to doubt – is uncertain –

we live in uncertainty – we deal with uncertainty – language is the expression of uncertainty

the idea – put most stridently by Russell – that you can via some formal construction – some formal language – eliminate propositional uncertainty – should be seen for what it is – the attempt to defy and scuttle propositional reality

to present such an idea as ‘logical’ – is to my mind – the gravest form of deception –

any such endeavour is an undermining of logical reality – and nothing more than rhetoric – to the service of ignorance and base prejudice

‘Sameness of number, when it a matter of a number of lines “that one can take in at a glance” is a different sameness from that which can only be established by counting the lines.’

here we are talking about a hunch – a guess –

as distinct from a propositional game based on calculation

‘We want to say that the equality of length in Euclidean space consists in both lines measuring the same cm, both 5 cm, both 10 cm etc; but where it is a case of two lines in visual space being equally long; there is no L that both lines have.

One wants to say: two rods must always have the same length or different lengths. But what does that mean? What it is, of course, is a rule about modes of expression. “There must either be the same number or a different number of apples in the two boxes.” The method whereby I discover whether two lines are of the same length is supposed to be the laying of a ruler against each line: but do they have the same length when the rulers are not applied? In that case we would say we don’t know whether during that time the two lines have the same or different lengths. But we might also say that during that time they have no length, or perhaps no numerical length.’

if there is no L that both lines have – 

how can it be also said that they are ‘equally long’?

                                                                                                                                  

the real point here is – if there is no L that both lines have –

all that means is –  there has been no measurement taken –

or no measurement can be taken

‘do they have the same length when the rulers are not applied? –

we don’t know

‘we might also say that during that time they have no length, or perhaps no numerical length’ –

yes – the point being that length is a function of measurement – which is a calculation

                                                                                                                                   

calculation is a propositional game –

if you don’t play the game –

obviously – there can be no result

‘Something similar, if not exactly the same holds, holds of sameness between numbers.’

yes – there are no numbers outside of a counting-game

‘When we cannot immediately see the number of dots in a group, we can sometimes keep the group in view as a whole while we count, so that it makes sense to say it hasn’t altered during the counting. It is different when we have a group of bodies or patches that we cannot keep in a single view while we count them, so that we don’t have the same criterion for the group’s not changing while it is counted.’

we do ‘see’ the dots in a group – but we haven’t counted them

we assume that the group hasn’t altered during the counting –

and as with any assumption – this assumption is – open to question – open to doubt – is uncertain

you might say ‘there may have been a change that was not observed’ –

nevertheless in the absence of an observable change – we assume no change – and play the counting-game

what we have with a ‘single view’ – is a context –

where we have bodies or patches that we cannot keep in a single view – we have a different propositional context –

I think the criterion for not changing is the same – if we don’t see or recognize a change – then we assume that there is no change

the assumption that there is no change underlies the counting-game –

if there is a change – we have to start again –

if everything is changing – how can you count?

so – ‘that there is no change’ – is a methodological assumption –

it is if you like a meta-game assumption

‘Russell’s definition of sameness of number is unsatisfactory for various reasons. The truth is that in mathematics we don’t need any such definition of sameness of number. He puts the cart before the horse.

What seduces us into accepting the Russellian or Fregean explanation is the thought that two classes of objects (apples in two boxes) have the same number if they can be correlated 1 to 1. We imagine correlation as a check of sameness of number. And here we do distinguish in thought between being correlated and being connected by a relation; and correlation becomes something that is related to connection as the “geometrical straight line” is related to a real line, namely a kind of ideal connection that is as it were sketched in advance by Logic so that reality only has to trace it. It is possibility conceived as a shadowy actuality. This in turn is connected with the idea of ($x). jx as an expression of the possibility of jx.’

it is not that Russell puts the cart before the horse –

there is no cart

first up –

it is not two classes of objects (apples in two boxes)

what we deal with is two proposals –

‘x number of apples in this box’ and ‘x number of apples in that box’ –

two proposals – two propositions

and these propositions are different

correlation?

different propositions –

there is no 1 to 1 correlation here

as to the same number of apples –

what you have –

is an arithmetical game (addition) applied to the two propositions –

and with the statement the ‘same number of apples in each box’ –

you have a representation of that application

‘Logic’ – is not ‘an ideal connection sketched out in advance’ –

                                                                                                                                

‘Logic’ – is propositional reality

and that reality is that the proposition – the proposal – is open to question – open to doubt – is uncertain

‘Logic’ – propositional activity – is the exploration of propositional uncertainty

our propositional reality – is uncertain –

game playing as I see it – is relief from propositional reality

the game is rule-governed propositional behaviour

                                                                                                                                  

the very point of the ‘rule’ is that it is not questioned – not doubted –

any rule – as with any proposition – any proposal – can be questioned of course –

but the very point of the game is that it is not

if you don’t want to play – don’t play –

if you would prefer to question – knock yourself out –

we think and we play – that’s the sum of it

‘We can regard the concept of sameness of number in such a way that it makes no sense to attribute sameness of number or its opposite to two groups of points except in the case of two series of which one is correlated 1 to 1 to at least a part of the other. Between such series all we can talk about is unilateral or mutual inclination. This has really no more connection with particular numbers than equality or inequality of length in the visual field has with numerical measurement. We can, but need not, connect it with numbers. If we connect it with the number series, then the relation of mutual inclusion or equality of length between the rows becomes a relation of sameness of numbers. But then it isn’t only that y5 follows from II. j5. We also have II following from j5. y5. That means that here S = II.’

as to sameness of number –

really ‘sameness of number’ – is a propositional game–

which is to say a game where syntactical correspondence is proposed –

that is a correspondence between symbols – in a mathematical game – or in two or more mathematical games

there is no mystery to this