Part II. III

III. FOUNDATIONS OF MATHEMATICS

11. The comparison between mathematics and a game

‘What are we taking away from mathematics when we say it is only a game (or it is a game)?’

mathematics as an action is a game – that is an action in accordance with rules –

the propositions used in the game – and the propositions that function as ‘rules’ of the game –

as with any proposal – any proposition – are open to question – open to doubt – are uncertain

we take nothing away from mathematics when we say it is a game – any rule governed activity is properly termed  a ‘game’ – this is the practise of mathematics

the theory of mathematics is the proposal – the propositions used in the game –

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

when we play a game – we suspend questioning – suspend doubt – in order to play

when we question and doubt – we do not play – we question and doubt –

there are no rules to uncertainty

‘A game in contrast to what? What are we awarding to mathematics if we say it isn’t a game, its propositions have sense?’

a game in contrast to what?

a game in contrast to non-rule governed propositional action – non-rule governed propositional behaviour

mathematics is a game – as mathematics is practised –

as to the sense of a mathematical proposition –

the sense of mathematical propositions – as with any proposal – any proposition –

is open to question – open to doubt – is uncertain

subjecting mathematical propositions to question – to doubt – recognizing their uncertainty –

is not doing mathematics – it is doing logic

                                                                                                                                  

‘The sense outside the proposition.

What concern is it of ours? Where does it manifest itself and what can we do with it? (To the question “what is the sense of this proposition?” the answer is a proposition.)

(“But a mathematical proposition does express a thought” – What thought? –.)’

our proposals – our propositions – make known – make sense –

there is no sense out side of propositional reality –

outside of propositional reality is the unknown

‘to the question ‘what is the sense of this proposition?’ the answer is a proposition’

yes – exactly –

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

sense is uncertain

when we say a mathematical proposition expresses a thought – we are putting forward a proposal – an explanatory proposal – of the mathematical proposition

it may prove useful to some to operate with such an explanatory proposal –

that is neither here nor there really –

for any such proposal is open to question – open to doubt – uncertain –

there are no rules for how we account for our propositions – there are no rules for how we account for reality – for the unknown

‘Can it be expressed by another proposition? Or only by this proposition? – Or not at all? In that case it is no concern of ours.’

I take it this means –

can the sense of a mathematical proposition be expressed by another proposition?

one could say that the ‘sense’ of a mathematical proposition is expressed in a painting for example – or some other work of art

it is all a matter of interpretation – but any such interpretation – is of course – open to question

I would say if you are doing mathematics – it would only be of peripheral interest  that some proposition of the mathematics is expressed – ‘illustrated’ in some other propositional form –

and the other thing is – just what is to be regarded as a mathematical proposition – might depend on who you are talking to

‘Do you simply want to distinguish mathematical propositions from other constructions, such as hypotheses? You are right to do so: there is no doubt there is a distinction.’

for practical purposes – yes we distinguish propositions – proposition types – proposition uses

the logic of the matter is entirely different –

there is no logical distinction between a proposal – be it described as mathematical – or whatever – and the proposal described as an hypothesis –

these descriptions ‘mathematical’ and ‘hypothesis’ – have to do with context – ways of practise

any proposal – however described – is open to question – open to doubt – is uncertain

to say there is no doubt – is rubbish –

it is in fact a denial of logic

‘If you want to say that mathematics is played like chess or patience and the point of it is like winning or coming out, that is manifestly incorrect.’

if you want to know how mathematics is played – start by asking those who play it –

and any proposal they put forward will be of interest

mathematics is not ‘chess’ or ‘patience’ – yes chess or patience could be described in mathematical terms – but they are different rule governed activities – different games

the point of anything – of any propositional action – is open to question – open to doubt –is uncertain

‘If you say the mental processes accompanying the use of mathematical symbols are different from those accompanying chess, I wouldn’t know what to say about that’

the proposal of mental processes and all that involves is a description of propositional action –

it is one proposal – one explanatory proposal – among any number that can – that have been advanced to account for propositional action –

it is the doing of mathematics that is important –

how you explain – account for mathematical propositions – mathematical propositional action –  is frankly a matter of philosophical prejudice

and it’s only real only function is to provide  speculative context for the propositional action

and any such speculation if it enables you to do mathematics – if it gets you going – is useful

once you understand that any proposition – any propositional action is open to question – open to doubt – is uncertain – the differences between propositional practises – are logically irrelevant –

you can leave such differences to sociologists and their progeny – modern French philosophers

‘In chess there are some positions that are impossible although each individual piece is in a permissible position. (E.g. if all the pawns are still in their initial position, but a bishop is already in play.) But one could imagine a game in which a record was kept of the number of moves from the beginning of the game and then there would be certain positions which could not occur after n moves and yet one could not read off from a position by itself whether or not it was a possible nth position.’

‘certain positions that could not occur after n moves’ – is a rule

‘and yet one could not read off from a position by itself whether or not it was a possible nth position’

there is no ‘position by itself’ –  any place on the board is only a ‘position’ in terms of the rules of the game

the moves in chess – are expressions of the rules of chess – the rules of the game

‘What we do in games must correspond to what we do in calculating. (I mean: it’s there that the correspondence must be, or again, that’s the way that the two must be correlated with each other.)’

calculating is a game –

calculating is a sign-game – a language-game –

it is a paradigm of game-playing

is all rule governed activity – calculating?

of course other games can be described in terms of calculating –

but can they also be described in other terms?

is the chess player’s move a calculation – a spontaneous action – a careless action etc.?

the point is that a rule governed activity can be described variously –

and any description is open to question – open to doubt – is logically speaking uncertain

‘Is mathematics about signs on paper? No more than chess is about wooden pieces.

When we talk about the sense of mathematical propositions, or what they are about, we are using a false picture. Here too I mean it looks as if there are inessential, arbitrary signs which have an essential element in common, namely sense.’

mathematics is about proposals – propositions –

signs on paper?

I don’t know that paper is essential – but what mathematics – what proposal – do you have without signs?

and as to how signs are expressed – that is ‘inessential’

as to ‘sense’ –

sense here really has to be a catch phrase for ‘significance’

yes signs have – can have – significance –

just what that significance is – what is proposed –

will be open to question – open to doubt –

the matter is – logically speaking – uncertain

in different propositional practises – there will be different accounts of significance

what signs have in common –

is that that they are proposed

‘Since mathematics is a calculus and hence isn’t really about anything, there isn’t any metamathematics.’

just what mathematics is – and what it is about – is open to question – open to doubt is uncertain

by all means have a view – but keep an open mind

can you operate a calculus without an interpretation?

you may think you do – but every sign in a calculus has a logical history – a history of interpretation

and any interpretation – any ‘metamathematics’ – if you wish to call it that – is logically speaking – open to question – open to doubt – is uncertain

‘What is the relation between a chess problem and a game of chess? – It is clear that chess problems correspond to arithmetical problems, indeed that they are arithmetical problems.’

what is proposed here – is that a chess problem is to be interpreted as an arithmetical problem – and that is fair enough

however such an interpretation is just one of any number of possible descriptions of the matter

e.g. what if the chess problem is seen as a problem of strategy?

the point is that the interpretation – the description – that we give a problem in chess – or for that matter a problem in any other context – is open to question

there is no definite description

any description is open to question – open to doubt – is uncertain

                                                                             

‘The following would be an example of an arithmetical game: We write down a four-figure number at random, e.g. 7368; we are to get to as near to this number as possible by multiplying the numbers 7, 3, 6, 8 with each other in any order. The players calculate with pencil and paper, and the person who comes nearest to the number 7368 in the smallest number of steps wins. (Many mathematical puzzles, incidentally, can be turned into games.)’

arithmetic is a game – a rule governed sign game

a game can be the basis of other games –

games within games

‘Suppose a human being had been taught arithmetic only for use in an arithmetical game: would he have learnt something different from a person who learns arithmetic for its ordinary use? If he multiplies 21 by 8 in the game and gets 168, does he do something different from a person who wanted to find out how many 21 x 8 is?’

would he have learnt something different?

if the game player’s description of what he does when he plays the arithmetical game is different to the description  of person who plays the arithmetical in its ordinary use

then there is an argument for saying that what the game player has learnt is different to the other player

it is all a question of  description –

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

‘It will be said: the one wanted to find out the truth, but the other did not want to do anything of the sort.’

yes – anything can be said –

as for the truth –

the truth is what you give your assent to – for whatever reason

when you play a game – you give your assent in playing it –

you give your assent to the rules – to the result –

when you do an arithmetical calculation – if you do the calculation in accordance with the rules of the game – you get the outcome that is determined by those rules –

again – the truth here – is what you assent to in the performing of the calculation –

if you don’t assent to the game – you don’t play it

yes you can question and doubt the rules of the ‘arithmetical game’ – or the rules for ‘ordinary use’ – that is not playing the game –

that is not doing mathematics –

                                                                                                                                subjecting the proposals – the rules of mathematics – to question and doubt – recognizing them as uncertain –

is not mathematics – it is logic

‘Well we might want to compare this with a game like tennis. In tennis the player makes a particular movement which causes the ball to travel in a particular way, and we can view his hitting the ball either as an experiment, leading to the discovery of a particular truth, or else as a stroke with the sole purpose of winning the game.

But this comparison wouldn’t fit, because we don’t regard a move in chess as an experiment (though that too we might do); we regard it as a step in a calculation.’

where you play a game – with a competitor – yes the rules of the game govern it’s activity –

however in any such game – you are constantly facing the unknown –

that is – while there are rules – even so – you don’t know what your competitor will do –

you don’t know what mastery you competitor has over the rules –

and your competitor is in the same position of not knowing in relation to you

here the rules of the game as it were define or map out the domain of not-knowing –

and yes – when you play chess or tennis – or any game – you do experiment with not-knowing –

in any experiment you face the unknown – fair and square – and place a bet

as for calculating –

calculating in chess is a game within a game – a game within the larger game

calculating without a competitor is simply a matter of following rules –

yes – you can question those rules – but questioning the rules of a calculation – is not calculating

questioning the rules of a game – is not playing the game

                                                                                                                                   

‘Someone might perhaps say: In the arithmetical game we do indeed do the multiplication 21 x 8.

                         168

but the equation 21 x 8 = 168 doesn’t occur in the game. But isn’t that a superficial distinction? And why shouldn’t we multiply (and of course divide) in such a way that the equations were written down as equations?’

the equation is just another formulation or statement of the calculation

the equation may not occur in the game – but to play the game we do the multiplication – which could be restated in the form of the equation –

the point is that any proposal – and proposition – in this case – a calculation – is open to question – open to doubt – in any number of ways – including how it is formulated – how it is stated

‘why shouldn’t we multiply (and divide) in such a way that the equations were written down as equations?’

no reason at all –

it is just that there is more than one way of performing the propositional action

‘So one can only object that in the game the equation is not a proposition. But what does that mean? How does it become a proposition? What must be added to it to make it a proposition? – Isn’t it a matter of the use of the equation (or of the multiplication)? – And it is certainly a piece of mathematics when it is used in the transition from one proposition to another. And thus the specific difference between mathematics and a game gets linked up with the concept of proposition (not ‘mathematical proposition’) and thereby loses its actuality for us.’

a proposition is a proposal –

open to question – open to doubt – uncertain

an equation is a proposal

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

a proposal is a proposition – you could say whether or how it is used –

isn’t it the case that a proposal put – is a proposal used – is a proposition in use?

a game is a proposal – is a proposition –

the game as a proposal – is open to question – open to doubt – is uncertain

a game as played is played without question

any proposition of mathematics is open to question – open to doubt – is uncertain

mathematics as a game – is played – without question

a proposal – is that which is put

and that which is put – is actual

‘(Here we may remind ourselves that in elementary school they never work with inequations. The children are only asked to carry out multiplications correctly and never – or hardly ever – asked to prove an inequation.)’

the ‘unequals game’ is just as rule governed as the equals game

‘When I work out 21 x 8 in our game the steps in the calculation, at least, are the same as when I do it in order to solve a practical problem (and we could make room in a game for inequations also). But my attitude to the sum in other respects differs in the two cases.

Now the question is: can we say of someone playing the game who reaches the position “21 x 8 = 168’ that he has found out that 21 x 8 = 168? What does he lack? I think the only thing missing is an application for the sum.’

‘can we say .. that he has found out that 21 x 8 =168?’ –

it is not the only way to describe what has happened  but there is nothing wrong with the description ‘found out’ –

you calculate – follow a rule to ‘find out’ – the result –

that is the game you play

the game is complete in itself – there is nothing lacking

yes – it can be played for an application –

it can also be played for the pure pleasure of the exercise –

the pleasure of the game

‘Calling arithmetic a game is no more and no less wrong than moving chessmen

according to chess-rules; for that might be a calculation too.’

yes – moving chess men according to chess rules – as with any action – can be described in any number of ways

‘So we should say: No, the word ‘arithmetic’ is not the name of a game. (That too of course is trivial) – But the meaning of the word “arithmetic” can be clarified by bringing out the relationship between arithmetic and an arithmetical game, or between a chess problem and the game of chess.

But in doing so it is essential to recognize that the relationship is not the same as that between a tennis problem and the game of tennis.

By ‘tennis problem” I mean something like the problem of returning a ball in a particular direction in given circumstances. (A billiard problem isn’t a mathematical problem (although its solution may be an application of mathematics). A billiard problem is a physical problem and therefore a “problem” in the sense of physics; a chess problem is a mathematical problem and so a “problem” in a different sense, a mathematical sense.’

a game is a rule governed propositional action

arithmetic is a game

‘tennis’ is a rule governed propositional action involving two or more players

i.e. the next shot is a proposal

you would quite naturally describe it as a physical game – however it – like any game can be variously described –

i.e. some commentators focus on the personalities of the players

a billiard problem can be described as physical –

there are of course other ways to describe the game-

i.e. you might describe a problem in billiards in terms of strategy –

or might describe the problem in historical terms – how it has been dealt with in past games

a chess problem – can be described  as mathematical –

but again this is just one possible description –

again – it could be described descried in terms of strategy – personality – history etc.

any description of propositional action is open to question – open to doubt – is logically speaking – uncertain

‘In the debate between “formalism” and “contentful mathematics” what does each side assert? This dispute is so like the one between realism and idealism in that it will soon have become obsolete, for example, and in that both parties make unjust assertions at variance with their day to day practice’

how a propositional activity is described is open to interpretation –

and any interpretation – any description – is open to question – open to doubt – is in so far as it is a proposal – uncertain

‘Arithmetic isn’t a game, it wouldn’t occur to anyone to include arithmetic in a list of games played by human beings’

wrong – it occurred to me

‘What constitutes winning and losing in a game (or success in patience)? It isn’t of

course, just the winning position. A special rule is needed to lay down who is the winner. (“Draughts” and “losing draughts” differ only in this rule.)’

in a competition game – the winning position – is the result of the game

a rule determining the winner is not I think a special rule – it is a rule of the game –

without such a rule – there is no game – so it is integral to the game

success – in patience is rule governed

in patience – though you are not playing with a competitor – you are playing against a rule governed indeterminacy – and with the revelation of each card – you face uncertainty

the rule in almost all variants of draughts is that the player without pieces remaining or who cannot move due to being blocked – loses the game

‘losing draughts’ or ‘giveaway checkers’ is another game –

in ‘losing draughts’ the rules are the same as draughts but the aim is to lose all your pieces

‘Now is the rule which says “The one who first has his pieces in the other half is the winner” a statement? How would it be verified? How do I know if someone has won? Because he is pleased, or something of the kind? Really what the rule says is: you must try to get some pieces as soon as possible, etc.

In this form the rule connects the game with life. And we could imagine that in an elementary school in which one of the subjects taught was chess the teacher would react to a pupil’s bad moves in exactly the same way as to a sum worked out wrongly.’

is the rule a statement – yes – the rule is a proposal – a proposition

how would it be verified?

the game is played – not verified

how do I know if someone has won?

I know if someone has won – if they have played so as to satisfy the rule that determines the winner

‘In this form the rule connects the game with life.’

if a teacher is trying to inculcate the rules of a game – it matters little what the game is – the issue is the same

however I would put that teaching obedience to rules is close to a waste of time

what we should be doing is showing the rules to pupils  – explaining why we have them –

and getting them to think about the value of rules – even to question them in a thoughtful manner

people learn games not just by mastering rules –

but rather by learning to think within rules –

and thinking here is the logical activity of question and doubt –

to think is to recognise – appreciate – and operate – with and in –

uncertainty –

this is the game of life

‘I would like to say: It is true that in the game there isn’t any “true” and “false” but then in arithmetic there isn’t any “winning” and “losing”.’

that is correct as far as it goes –

however Wittgenstein’s statement here draws a distinction between the ‘game’ and ‘arithmetic’ –

when in fact – arithmetic is a game –

that is to say a rule governed proposition activity

and in fact there isn’t any ‘true’ or ‘false’ in the arithmetic game –

you play that game in accordance with the its rules –

if you don’t play the arithmetic game in accordance with its rules –

it is not that you play a ‘false’ game  – or the result of your calculation is ‘false’ –

it is rather that you don’t play the game –

when you calculate – it is not that you can ‘miscalculate’ – or ‘make a mistake’ – or get it ‘wrong’ –

if you don’t follow the rules – you don’t calculate

a true proposal – a true proposition – is one you assent to – for whatever reason

a false proposal – a false proposition – is one you dissent from – for whatever reason

if you play the game – you assent to it –

if you don’t play it properly –

you don’t play it

‘I once said that it was imaginable that wars might be fought on a kind of chessboard according to the rules of chess. But if everything really went simply according to the rules of chess, then you wouldn’t need a battlefield for war, it could be played on an ordinary board; and then it wouldn’t be a war in the ordinary sense. But you really could imagine a battle conducted in accordance with the rules of chess – if say, the “bishop” could fight with the “queen” only when his position in relation to her was such that he could be allowed to ‘take’ her in chess.’.

yes – chess as a propositional model for a rule governed war

how likely though in reality – when war in fact just is the result of the breakdown of rule governed behaviour?

so called ‘rules of war’ I think you will find are irrelevant to actual war

and I would suspect – most likely proposed by those who don’t do any actual fighting

‘Could we imagine a game of chess being played (i.e. a complete set of chess moves being carried out) in such different surroundings that what happened wasn’t something we could call the playing of a game?

Certainly, it might be a case of the two practitioners collaborating to solve a problem. (And we could easily construct a case on these lines in which such a task would have a utility).’

if the activity is rule governed – and it is played in accordance with the rules – it is a game

collaborating to solve a problem within a game – might just be figuring out which rule does apply in the circumstance in question

and where the problem is not a chess problem per se – if the model that is being used to deal with the problem – is a rule governed propositional model – then the question will be – which rule applies?

‘The rule about winning and losing really just makes a distinction between two poles. It is not concerned with what happens to the winner (or the loser) – whether, for instance, the loser has to pay anything.

(And similarly, the thought occurs, with “right” or “wrong” in sums.)’

it is a question of description

if you describe how the game works – the ‘logic’ of the game – independently of the players – then ‘winning’ and ‘losing’ – will not be part of that description

if on the other hand you describe the play – then ‘winning’ and ‘losing’ will be part of the description

as to arithmetic –

you play that game – if you play it according to its rules –

‘right’ – is irrelevant

if you don’t play it according to the rules – you don’t play it – and in that case – ‘wrong’ – has no place

‘In logic the same thing keeps happening as happened in the dispute about the nature of definition. If someone says that a definition is concerned only with signs and does no more than substitute one sign for another, people resist and say that that isn’t all a definition does, or there are many different kinds of definition and the interesting and important ones aren’t the mere “verbal definitions”.

They think, that is, that if you make definition out to be a mere substitution rule for signs you take away its significance and importance. But the significance of definition lies in its application, and its importance for life. The same thing is happening to day in the dispute between formalism and intuitionism etc. People cannot separate the importance, the consequences, the application of a fact from the fact itself; they can’t separate the description of the thing from its importance.’

yes we define – to make known – and any definition – and  any propositional knowledge – is open to question –

and yes we use definitions –

and any use of a definition – any application – like the definition itself – is open to question – open to doubt – is uncertain

we operate with uncertainty – in uncertainty – that is our life – from a logical point of view

‘they can’t separate the description of the thing from its importance.’ –

the ‘thing’ in the absence of description is unknown

description makes known –

and any ‘description’ – is open to question – open to doubt – is uncertain

the importance of the thing – is the importance of the description

and ‘importance’ is a rhetorical issue

‘We are always being told that a mathematician works by instinct (or that he doesn’t proceed mechanically like a chess player or the like), but we aren’t told what that’s supposed to have to do with the nature of mathematics. If such a psychological phenomenon does play a part in mathematics we need to know how far we can speak about mathematics with complete exactitude, and how far we can only speak with the indeterminacy we must use in speaking of instincts.’

there are any number of possible descriptions of mathematical behaviour –

‘how far can we speak of mathematics with complete exactitude?’

how far can we speak of anything with complete exactitude?

any proposal – any proposition we put forward – be it ‘mathematical’ – or otherwise –

is logically speaking – open to question – open to doubt – is uncertain –

‘exactitude’ or even better – ‘complete exactitude’(?) – is rhetorical rubbish

as for ‘instincts’ –

to say ‘a mathematician works by instinct’ – is to propose an ‘explanation’ of his behaviour

now any explanation – as with any description – is open to question – open to doubt –

is uncertain

nevertheless – if in a certain context – it proves useful – then it has value –

the reality is that the mathematician – as with everyone else – does not have an  explanation of his behaviour – an explanation – that is beyond question – beyond doubt – that is certain –

still we use explanations – and in general we run with what is at hand – whatever is the fashion

intuition has proved an enduring fashion for the mathematician –

I frankly don’t think it matters an iota whether the mathematician can explain – can account for what he does –

what matters is that he does – what does –

‘explanation’ – is always – after the fact –

it’s just packaging –

and I suppose we all prefer the gift wrapped

‘Time and again I would like to say: What I check is the account books of mathematics; their mental process, joys, depressions and instincts as they go about their business may be important in other connections, but they are no concern of mine.’

nor of mine

12. There is no meta mathematics

‘No calculus can decide a philosophical problem.

A calculus cannot give us information about the foundations of mathematics.’

‘No calculus can decide a philosophical problem’?

a calculus is game – a sign game

a philosophical problem – is a proposal – subjected to question – to doubt –

a calculus – a sign game – in fact any ‘game’ – as played – is played – without question

if questioned – if doubted – then the ‘game’ – is not being played – it is being proposed

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

philosophical problems – that is – propositions questioned – are not – logically speaking ‘decided’ in any final sense

any decision taken – is open to question – open to doubt – uncertain

a game is not a propositional decision in response to question – to doubt – to uncertainty–

a game is a propositional application

‘A calculus cannot give us information about the foundations of mathematics’?

the foundations of mathematics?

the foundations of mathematical proposals – propositions –

a proposal – a proposition – however described – in whatever contexts it is used –

is open to question – open to doubt – is uncertain

the logical ‘foundation’ of the proposal – of the proposition – is uncertainty –

is ‘uncertainty’ a foundation?

I think it best to speak of the proposal – the proposition – as foundationless.

‘So there can’t be any “leading problems” of mathematical logic, if those are supposed to be problems whose solution would at long last give us the right to do arithmetic as we do.’

in the words of Bentham – talk of ‘rights’ is nonsense on stilts – rhetorical rubbish –

and that’s all we have here

arithmetic is a sign game – a calculation game – a propositional game –

a game developed over centuries – a game played because it is useful

‘We can’t wait for the lucky chance of the solution of a mathematical problem.’

what does he mean here?

‘a mathematical problem’ – is what?

a problem emerges when there are different interpretations of a rule – or a question about what rules apply –

a ‘problem’ if you like – is the question – ‘how to proceed?’ –

any such questioning – is logical –

in a mathematical context – such questioning is logical

so on this view there are no mathematical problems as such

any problem per se is logical

in a mathematical context – the logical question might be which mathematical game –

which sign game – which calculation – to use?

as for ‘lucky chance’ –

there are games – rules to games – games played in accordance with rules –

there is no chance – no luck

you play the game or you don’t

‘I said earlier “calculus is not a mathematical concept”; in other words, the word “calculus” is not a chess piece that belongs to mathematics.

There is no need for it to occur in mathematics. – If it is used in a calculus nonetheless, that doesn’t make the calculus into a meta calculus; in such a case the word is just a chessman like all the others.’

‘calculus’ – as with any word – any proposal – is open to question – open to interpretation

it occurs in mathematics

a meta calculus?

would be a calculus that explains the calculus in use –

a meta theory of calculus – is more likely

that is an account – a theory of – the calculus in use

and any such proposal – like the calculus – it purports to explain – will be open to question – open to doubt – is uncertain

‘Logic isn’t meta mathematics either; that is work within logical calculus can’t bring to light essential truths about mathematics. Cf. here the “decision problem” and similar problems in modern mathematical logic.’

the point is that meta mathematics is no more than a proposal to account for – to explain – to underwrite mathematics

a calculus is a propositional operation

there are no essential truths

any proposal – any decision –

is open to question – open to doubt – is uncertain

regarding mathematics in this light –

is to regard it logically

‘(Through Russell and Whitehead, especially Whitehead, there entered philosophy a false exactitude that is the worst enemy of real exactitude. At the bottom of this there lies the erroneous opinion that a calculus could be the foundation of mathematics.)’

there is no foundation to mathematics

any propositional activity – mathematics included – is open to question – open to doubt – is uncertain

quite apart from the logic of the matter – the history of mathematics demonstrates – that its concepts – its terms – its operations – its propositions – have been developed out of question – out of doubt – out of uncertainty

any concept of exactitude – is uncertain

as for a calculus as the foundation of mathematics –

this – or any other proposal of foundation – cannot be taken seriously

what you have in any such proposal – is the desire for foundation – nothing more –

and it is a desire based on a corruption of propositional logic

‘Number is not at all a “fundamental mathematical concept”.

There are so many calculations in which numbers aren’t mentioned.

So far as concerns arithmetic, what we are willing to call numbers is more or less arbitrary. For the rest, what we have to do is to describe the calculus – say of cardinal numbers – that is, we must give its rules and by doing so we lay foundations of arithmetic.’

if by ‘fundamental’ – is meant that which is beyond question – beyond doubt –

a certainty –

then logically speaking there is no fundamental in any propositional context

numbers are signs – in sign games

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

mathematics is the language of sign games

its foundation – is utility

‘Teach it to us and then you have laid its foundations’

yes – it ‘foundation’ – is use

        

‘(Hilbert sets up rules of a particular calculus as the rules of meta mathematics.)’

the rules of a calculus – as the rules (of an account) of mathematics?

the question is – how could you know that one set of rules – applies to all expressions of mathematics?

at best you have a conjecture

mathematics will be described – accounted for – in various ways – at various times – by various people – to various ends

yes we will have these descriptions – these meta mathematics proposals –

but this descriptive activity – is not mathematics

‘A system’s being based on first principles is not the same as its being developed from them. It makes a difference whether it is like a house resting on its lower walls or like a celestial body floating in free space which we have begun to build beneath although we might have built anywhere else.’

first principles – are just descriptions of starting points –

there will always be a question as to whether where you start is wise

this question is always live – at any stage of the activity –

any starting place – as with any action from that starting place – and indeed any assessment of the result of any action – is at any time – open to question – open to doubt – uncertain

yes – it will make a difference how you go about your enterprise – how you envisage the project – where you start – how you approach the construction

‘Logic and mathematics are not based on axioms, any more than a group is based on the elements and operations that define it. The idea that they are involves the error of treating the intuitiveness, the self-evidence, of the fundamental propositions as a criterion for correctness.

A foundation that stands on nothing is a bad foundation.’

logic and mathematics are propositional activities

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

any proposal put forward to ‘explain’ a propositional activity – is itself –

open to question – open to doubt – uncertain

axioms are proposals

intuitiveness – is a proposal to account for certain propositional behaviour

any proposal – is evidence of itself

the ‘fundamental propositions’ – at best are starting blocks

a criterion for correctness –  is whatever you decide it is – in whatever context requires it

any so called ‘foundation’ is a rhetorical device

rhetoric stands on deception

there is no foundation to the proposition

the proposition stands on nothing

‘(p.q) v (p. - q) v (- p.q) v ( - p. – q.) : That is my tautology, and then I go on to say that every “proposition of logic” can be brought into this form in accordance with specified rules. But that means the same as: can be derived from it. This would take us as far as the Russellian method of demonstration and all we add to it is that this initial form is not itself an independent proposition, and that like all “laws of logic” it has the property that p, p v Log = Log.’

the tautology is a game – a game-proposition –

a propositional game – played according to rules –

and yes – you can extend the game to every proposition of logic

the game can be played as a derivation game – then yes – it means the same as – ‘can be derived from it’

any proposal – any proposition as put – is independent

a proposition can be represented as dependent i.e. – ‘explained’ – accounted for – in terms of other proposals

proposals – propositions – open to question – open to doubt – uncertain

‘It is indeed the essence of a logical law that when it is conjoined with any proposition it yields that proposition. We might even begin with Russsell’s calculus with definitions like

p É p : q. = .q

p : p v q. = .p, etc.’

yes you can take what is proposed – what is presented – what is given – and reconfigure it into a word-game – a sign-game –

any such representation – will come with an interpretation –

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

13. Proofs of relevance

‘If we prove that a problem can be solved, the concept “solution” must occur somewhere in the proof. (There must be something in the mechanism corresponding to the concept.) But the concept cannot have an external description as its proxy; it must be genuinely spelt out.’

the solution to a problem – will be a proposal – open to question – open to doubt – uncertain –

the so called ‘proof’ – is the argument to the ‘solution’ – to the proposal

the argument to the ‘solution’ – the so called ‘proof’ – as with the solution – is open to question – open to doubt – is uncertain

‘The only proof of the provability of a proposition is a proof of the proposition itself.’

the ‘provability’ of a proposition – is the argument for the proposition

as for the proof of the provability – that would be – the argument for the argument for the proposition

does the argument for a proposition – need argument itself?

perhaps –

but all this would mean is that we have is a complex of arguments –

all of which are open to question – open to doubt – uncertain

‘But there is something we might call a proof of relevance: an example would be a proof convincing me that I can verify the equation 17 x 38 = 456 before I have actually done so. Well how is it that I know that I can check 17 x 34 = 456, whereas I perhaps wouldn’t know, merely by looking, whether I could check an expression in the integral calculus? Obviously it is because I know the equation is constructed in accordance with a definite rule and because I know the kind of connection between the rule for the solution of the sum and the way in which the sum is put together. In that case a proof of relevance would be something like a formulation of the general method of doing things like multiplication sums, enabling us to recognize the general form of the proposition it makes it possible to check. In that case I can say I recognize that this method will verify the equation without having actually carried out the verification.’

‘In that case a proof of relevance would be something like a formulation of the general method of doing things like multiplication sums, enabling us to recognize the general form of the proposition it makes it possible to check.’

this so called ‘proof of relevance’ then –

is nothing more than –

recognizing a propositional game –

and knowing how it is played

‘When we speak of proofs of relevance (and other similar mathematical entities) it always looks as if in addition to the particular series of operations called proofs of relevance, we had a quite definitive inclusive concept of such proofs or of mathematical proofs in general; but in fact the word is applied with many different, more or less related, meanings. (Like words such as “king”, “religion”, etc; cf Spengler.) Just think of the role of examples in the explanation of such words. If I want to explain what I mean by “proof”, I will have to point to examples of proofs, just as when explaining the word “apple” I point to apples. The definition of the word

“proof” is in the same case as the definition of the word “number”. I can define the expression “cardinal number” by pointing to examples of cardinal numbers: indeed instead of the expression I can actually use the sign “1, 2, 3, 4, and so on ad infin”. I can define the word number too by pointing to various kinds of number; but when I do so I am not circumscribing the concept of “number” as definitely as I previously circumscribed the concept cardinal number, unless I want to say it is only the things at present called numbers that constitute the concept “number”, in which case we can’t say of any new construction that it constructs a kind of number. But the way we want to use the word “proof” in is one in which it isn’t simply defined by a disjunction of proofs currently in use; we want to use it in cases of which at present we “can’t have any idea”. To the extent that the concept of proof is sharply circumscribed, it is only through particular proofs, or series of proofs (like the number series), and we must keep that in mind if we want to speak absolutely precisely about proofs of relevance, of consistency etc.’

yes – ‘proof’ as with any word – any concept – is open to question – open to doubt – is uncertain

the issue is use – where and how we use the term

to suggest that there is a use for the term ‘in cases of which at present we “can’t have any idea” –

is really to suggest that there is a meaning to proof regardless of actual use

what are we to call this?

a mystical proof?

‘We can say: a proof of relevance alters the calculus containing the proposition to which it refers. It cannot justify a calculus containing the proposition, in the sense in which carrying out the multiplication 17 x 23 justifies the writing down of the equation 17 x 23 = 391. Not, that is, unless we expressly give the word “justify” that meaning. But in that case we mustn’t believe that if mathematics lacks this justification, it is some sense more general and widely established sense illegitimate or suspicious. (That would be like someone wanting to say: “the use of the expression ‘pile of stones’ is fundamentally illegitimate, until we have laid down officially how many stones make a pile” but it wouldn’t “justify” it in any generally recognized sense; and if such an official definition were given, it wouldn’t mean that the use earlier made of the word would be stigmatized as incorrect.)’

17 x 23 = 391 – is a rule governed propositional game

a game is played – played according to its rules

a game is played – not justified

there is no ‘proof of relevance’ – there is just the game – its rules and its play

the expression ‘pile of stones’ – is not a rule governed propositional game –

it is a proposal

there is no question of ‘justifying’ a proposal or justifying a propositional use

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

‘The proof of the verifiability of 17x 23 = 391 is not a “proof” in the same sense of the word as the proof of the equation itself. (A cobbler heels, a doctor heals: both …) We grasp the verifiability of the equation from its proof somewhat as we grasp the verifiability of the proposition “the points A and B are not separated by a turn of the spiral” from the figure. And we see that the proposition stating verifiability isn’t a “proposition” in the same sense as the one whose verifiability is asserted. Here again, one can only say: look at the proof, and you will see what is proved here, what gets called “the proposition proved”.’

the so called ‘proof of the verifiability of 17 x 23 = 391’ – just is the arithmetical game – 17 x 23 = 391

the game and its rules of play

a propositional game is not verified – it is played

look at the game – and you’ll see the play –

what gets played is the game

‘Can one say that at each step of a proof we need a new insight? (The individuality of numbers.) Something of the following sort: if I am given a general (variable) rule, I must recognize each time afresh that this rule may be applied here too (that it holds for this case too). No act of foresight can absolve me from this act of insight. Since the form in which the rule is applied is in fact a new one each at every step. But it is not a matter of an act of insight, but of an act of decision.’

‘Can one say that at each step of a proof we need a new insight?

it is not a question of insight – insight is mysticism

here the issue is the application of the rule

and yes questions can be asked

and to proceed mathematically –

decisions have to be made

‘What I call a proof of relevance does not climb the ladder to its proposition – since that requires that you pass every rung – but only shows that the ladder leads in the direction of that proposition. (There are no surrogates in logic). Neither is an arrow that points the direction a surrogate for going through all the stages towards a particular goal.’

if you understand that mathematics is a network of propositional games –

rule governed propositional actions –

yes – you might in a ‘meta- moment’ – inquire into the history and so called provenance of a game – of the rules

but the mathematical issue is – the game and the playing of the game –

and the question is – are you a player?

if so – you follow the rules – you play the game

14. Consistency Proofs

‘Mathematicians nowadays make so much fuss about proofs of consistency of axioms. I have the feeling that if there were a contradiction in the axioms of a system it wouldn’t be such a great misfortune. Nothing easier than to remove it.’

yes –

but consistency does indicate order in your propositional action

and the value of the contradiction is that it brings undisciplined thinking to a stop

‘Suppose someone wanted to add to the usual axioms of arithmetic the equation 2 x 2 = 5. Of course that would mean that the sign of equality had changed its meaning, i.e. that there would now be different rules for the equals-sign.’

no – the sign of equality hasn’t changed its meaning –

what you have is a new equals game

2 x 2 = 5 – would not be accepted as an axiom of arithmetic unless it was decide that all other relevant axioms be made consistent with it –

this would be a very disruptive action – hard to see how there could be any advantage in such a change

‘If I inferred “I cannot use it as a substitution sign” that would mean that its grammar no loner fitted the grammar of the word “substitute” (“substitution sign”, etc.) For the word “can” in that proposition doesn’t indicate a physical (physiological, psychological possibility.’

yes

‘“The rules may not contradict each other” is like “negation, when doubled, may not yield a negation”. That is, it is a part of the grammar of the word “rule” that if “p” is a rule, “p. –p” is not a rule.’

That means we could also say: the rules may contradict each other, if the rules for the use of the word “rule” are different – if the word “rule” has different meanings.’

yes – it’s really a question of propositional practice –

and that always involves decision – and the acceptance or rejection of any decision by those engaged in the practice

‘Here too we cannot give any foundation (except a biological or historical one or something of the kind); all we can do is to establish agreement, or disagreement between rules for certain words, and say that those words are used with these rules.’

yes – exactly

‘It cannot be shown, proved that these rules can be used as the rules of this activity.

Except be showing that the grammar of the description of the activity fits the rules.’

again – the question of practice – what works – what doesn’t – what’s accepted – what is not accepted

a correlation of grammars – is a good start.

‘“In rules there mustn’t be a contradiction” looks like an instruction: “In a clock the hand mustn’t be loose on the shaft.” We expect a reason: because otherwise … But in the first case the reason would have to be: because otherwise it wouldn’t be a set of rules. Once again we have a grammatical structure that cannot be given as a logical

foundation.’

yes – we have grammatical practice

as to logical foundation –

any proposal – be a game proposal or not – logically speaking is open to question – open to doubt – is uncertain

the point of logic – is that there is no foundation to propositional action

‘In the indirect proof that a straight line can have only one continuation through a certain point we make the supposition that a straight line could have two continuations. – If we make that supposition, then the supposition must make sense. – But what does it mean to make that supposition? It isn’t making a supposition that goes against natural history, like the supposition that a lion has two tails. – It isn’t making a supposition that goes against an ascertained fact. What it means is supposing a rule; and there’s nothing against that except that it contradicts another rule, and for that reason I drop it.

Suppose that in the proof there occurs the following drawing

to represent a straight line bifurcating. There is nothing absurd (contradictory) in that unless we have some stipulation that it contradicts.’

‘that a straight line can have only one continuation’ –

you could well ask – really is there any need for such a definition?

if you have definition of ‘straight line’ – does the phrase ‘can only have one continuation’ – in anyway add to the definition?

i.e. isn’t ‘can only have one continuation’ – contained in the definition?

as for Wittgenstein’s drawing – yes he has made the point that a straight line – can a have more than one continuation

ok – but does the bifurcated line – fit the definition of straight line?

or if the focus is on the bifurcated line – don’t we have to change our definition of straight line?

no big deal really –

it is just a matter of being clear about what it is you are proposing –

‘Suppose that in the proof there occurs a drawing to represent a straight line bifurcating. There is nothing absurd (contradictory) in that unless we have some stipulation that it contradicts.’

yes

‘If a contradiction is found later on, that means that hitherto the rules have not been clear and unambiguous. So the contraction doesn’t matter, because we can now get rid of it by enunciating a rule.’

if a contradiction is found later on – you would say the propositional action was not properly conceived – sloppy work –

there is a rule against that

‘In a system with a clearly set out grammar there are no hidden contradictions, because such a systems must include the rule which makes the contradiction discernable. A contradiction can only be hidden in the sense that it is in the higgledy-piggledy zone of the rules, in an unorganized part of the grammar; and there it doesn’t matter since it can be removed by organizing the grammar.’

yes

‘Why may not the rules contradict one another? Because otherwise the wouldn’t be rules.’

it is not there wouldn’t be rules –

the point is there would be no game

15. Justifying arithmetic and preparing it for applications (Ramsey, Russell)

‘One always has an aversion to giving arithmetic a foundation by saying something about it’s application. It appears firmly grounded in itself. And that derives from the fact that arithmetic is its own application.’

arithmetic is a propositional game –

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

if a ‘foundation’ is that which is – not open to question – not open to doubt – is that which is certain –

there is no foundation – to the proposition

certainty is ignorance – is prejudice

the propositions of arithmetic – as with any proposition – however described or classified – are open to question – open to doubt

arithmetic is a propositional game –

a game is played – and played in accordance with its rules – 

any ‘rules’ here are simply agreed propositions – agreed propositional practices

the game as played – is played without question – without doubt

if you approach the propositions that make up a game – from a logical point of view –

that is if you put them to question – to doubt – recognize their uncertainty – you are not playing the game

if you put the propositions of arithmetic to question – you are not doing arithmetic –

your activity is logic

‘it appears firmly grounded in itself’ – is to say that as played – the arithmetic game – is played without question –

this is true of any game

‘arithmetic is its own application’?

a game – is its play

that the idea of a game – any game – is that it be played

the arithmetic game – is a play

‘You could say: why bother to limit the application of arithmetic, that takes care of itself. (I can make a knife without bothering about what kinds of materials I will have cut with it; that will show soon enough.)

What speaks against our demarcating a region of application is the feeling that we can understand arithmetic without having any such region in mind. Or put it like this: our instinct rebels against anything that isn’t restricted to an analysis of the thoughts already before us.’

as to the question of the limits of the application of arithmetic –

it is like asking – what are the limits of play?

you play when you play

and furthermore there are no limits to the making of games –

new games – or modifications or developments of older games can be proposed – at any time for any reason

their significance in any game playing arena will be decided by the game players

it is not a question of ‘the analysis of the thoughts already before us’ –

what is ‘before us’ – is the game – you play it – or you don’t –

you need to understand the rules – but the game itself is an explanation of its rules –

and how do we learn arithmetic? –

a player shows us how the game is played

‘You could say arithmetic is a kind of geometry; i.e. what in geometry are constructions on paper in arithmetic are calculations (on paper). You could say it is a more general kind of geometry.’

in geometry proposals are put – propositions are put –

and yes geometry can be seen as a game – a spatial game – a construction game

arithmetic ‘a more general kind of geometry’?

what does ‘more general’ mean here?

that when you do arithmetic – you are actually doing geometry – that geometry is just another form of arithmetic – that without arithmetic – there would be no geometry?

isn’t it rather the case that arithmetic and geometry are different games –

                                                                                                                                  different propositional games – but different propositional games that complement each other?

‘It is always a question of whether and how far it’s possible to represent the most general form of the application of arithmetic. And here the strange thing is that in a certain sense it doesn’t seem to be needed. And if in fact it isn’t needed, then it’s also impossible.’

arithmetic is a propositional game

the application of arithmetic – that is the play of the game

is the representation of the game –

in whatever context the game is played

‘The general form of its application seems to be represented by the fact that nothing is said about it. (And if that’s a possible representation, then it is also the right one.)

nothing is said about it – because there is no ‘general form of its application’ –

arithmetical propositions are used where they are used – arithmetical games are played where they are played

propositional application is not ‘general’ –

it is specific

‘The point of the remark that arithmetic is a kind of geometry is simply that arithmetical constructions are autonomous like geometrical ones and hence so to speak themselves guarantee their applicability.

For it must be possible to say of geometry too that it is its own application.’

arithmetic applies – where it is applied

geometry is applies – where it is applied

‘(In the sense in which we can speak of lines which are possible and lines which are actually drawn we can also speak of possible numbers.)’

as soon as you propose a ‘possible number’ – the number is proposed – it is there –

and then the issue is – just what is the point of this proposal – this possible number – why the proposal – what does it do – what is its use?

and of course the question is – how does this proposal stand in relation to practised mathematics – number theory – as we know it?

proposing ‘a possible number’ is really no different to any other proposal – in any other context –

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

it is propositional uncertainty – that just is the source of possibility

‘That is an arithmetical construction, and in a somewhat extended sense also a geometrical one.’

what ‘that’ is –

is open to question – open to doubt – uncertain

we have a series of marks –

‘a series of marks’ – you could say is a basic description – a starting point for further description

and as with any description – any proposal –

it is open to further description – and so on –

so the bottom line here is –

you run with whatever description suits your purpose

‘Suppose I wish to use this calculation to solve the following problem: if I have 11 apples and want to share them among some people in such a way that each is given 3 apples how many people can there be? The calculation supplies me with the answer 3. Now suppose I was to go through the whole process of sharing and at the end 4 people had 3 apples in their hands. Would I then say that the computation gave me the result? Of course not. And that of course means that the computation was not an experiment.

It might look as though the mathematical computation entitled us to make a prediction, say, that I could give three people their share and there will be two apples left over. But that isn’t so. What justifies us in making this prediction is an hypothesis of physics, which lies outside the calculation. The calculation is only a study of logical forms, of structures, and of itself cannot yield anything new.’

true – the computation is not an experiment

the computation is a propositional game – a game of signs

three people with two apples over?

however the units of the computation are described – apples – numbers – whatever –

the logical game is the same

‘If 3 strokes on the paper are the sign for the number 3, then you can say the number 3 is to be applied in our language in the way in which the three strokes can be applied.’

well yes – in those circumstances – those propositional contexts – where three strokes on paper – represent ‘3’ –

however in another propositional context – where we have an entirely different interpretation of three strokes on paper – i.e. – in an artistic proposal –

this proposed equivalence of signs will not be in play –

will for all intents and purposes – in this different context – be of no significance

equivalence of signs – will be context dependent

and yes you can mix contexts – but that is just an idle intellectual exercise – which might be an entertaining game for some  – but it will have no functional value for the contexts in question

the point is any ‘mark’ – any marking – is only a sign – when given propositional interpretation

in the absence of interpretation – a  mark – logically speaking – does not signify

it is an unknown –

which is to say it is a logical place for interpretation –

for propositional action

‘I said “One difficulty in the Fregean theory is the generality of the words “concept’ and ‘Object’. For, even if you can count tables, tones, vibrations and thoughts, it is difficult to bracket them all together.” But what does “you can count them” mean? What it means is that it makes sense to apply the cardinal numbers to them. But if we know that, if we know these grammatical rules, why do we need to rack our brains about the other grammatical rules when we are only concerned to justify the application of cardinal arithmetic? It isn’t difficult “to bracket them all together”; so far as is necessary for the present purpose they are already bracketed together.’

bracketing them all together? – finding a common description –

but what does that have to do with the action of counting?

it is not what is counted that is relevant here – it is that a count takes place –

how you describe what is counted – is irrelevant –

and in fact – what is counted (however described) – is irrelevant

counting is a rule governed action – the rule governed manipulation of numbers

a logical game –

numbers – cardinal numbers – are in the end – simply tokens – logical tokens – for the game – the game of counting –

tokens that enable the game to be played –

tokens for the action of the game

how you ‘cash in’ the logical tokens – is irrelevant to the playing of the game

there is no question of ‘justifying’ the use of numbers – of logical tokens

if you play the counting game – you use tokens –

that is the game

‘But (as we all know well) arithmetic isn’t at all concerned about this application.

It’s applicability takes care of itself.’

the point is we play the game – where we play the game

the world – is just setting for the game

the world’s regions – contexts – if you will – are just descriptive settings for the game

the game is played –  and the arithmetic game is powerful just because – as a logical game – it has no description – yet – just because it has no description – it is open to description –

and just where it is played can be the source of its description

yes I am counting – but right know – my context is apples –

so the description I use – is ‘counting apples’

‘Hence so far as the foundations of arithmetic are concerned all the anxious searching for distinctions between subject-predicate forms, and contrasting functions ‘in extension’ (Ramsey) is a waste of time.’

analysis of subject-predicate forms and contrasting functions in extension –

has no relevance at all to understanding the nature of arithmetic –

to understanding the nature of the game – to understanding the action of the game –

as to foundations –

speculation on the basis of arithmetic – on the ‘foundations’ of arithmetic – is all very well –

but mathematics gets on quite well without it

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

any proposal for ‘foundation’ – is open to question – open to doubt – is uncertain

any claim that a foundation is not open to question – not open to doubt – is certain –

is illogical –

such claims are baseless and only have rhetorical value –

that is if you think rhetoric has value

‘The equation 4 apples + 4 apples = 8 apples is a substitution rule which I use if instead of substituting the sign “8” for the sign “4 + 4”, I substitute “8 apples” for the sign “4 + 4 apples.

But we must be aware of thinking that “4 apples + 4 apples = 8 apples” is the concrete equation and 4 + 4 = 8 is the abstract proposition of which the former is only a special case, so that the arithmetic of apples, though much less general than the truly general arithmetic, is valid in its own restricted domain (for apples). There isn’t any “arithmetic of apples”, because the equation 4 apples + 4 apples = 8 apples is not a proposition about apples. We may say that in this equation the word “apples” has no reference. (And we can always say this about a sign in a rule which helps to determine its meaning.)’

yes – here we have an arithmetical game – applied to apples

‘How can we make preparation for something that may happen to exist – in the sense in which Russell and Ramsey always wanted to do this. We get logic ready for the existence of many placed relations, or for the existence of an infinite number of objects, or the like.

Well we can make preparations for the existence of a thing: e.g. I may make a casket for jewellery which may be made some time or another – But in this case I can say what the situation must be – what the situation is – for which I am preparing. It is no more difficult to describe the situation now than after it has already occurred; even, if

it never occurs at all. (Solution of mathematical problems). But what Russell and Ramsey are making preparations for is a possible grammar.’

grammar is an account of – a theory of – language use –

it is an explanatory proposal in relation to language use

any theory of language use – is after the fact – the fact of the usage –

you need a language use – a language practise before you have ‘an account of it’ –

a theory of it – a grammar of it

what we deal with in propositional logic – is what is proposed

as to what might be proposed –  unless it is proposed –

it doesn’t exist – it’s not there

preparations for a possible grammar?

just strikes me as moving into a realm of logical fiction.

‘On the one hand we think that the nature of the functions and of the arguments that are counted in mathematics is part of its business. But we don’t want to let ourselves be tied down to the functions now known to us, and we don’t know if people will ever discover a function with 100 argument places; and so we have to make preparations and construct a function to get everything ready for a 100 place-relation in case one turns up. – But what does “a 100-place relation turns up (or exists)” mean at all? What concept do we have of one? Or a 2-place relation for that matter? – As an example of a 2-place relation we give something like the relation between a father and son. But what is the significance of this example for the further logical treatment of 2-place relations? Instead of “aRb” are we now to imagine “a is the father of b”? – If not, is this example or any example essential? Doesn’t this example have the same role as an example in arithmetic, when I use 3 rows of 6 apples to explain 3 x 6 = 18 to somebody?

Here it is a matter of our concept of application. – we have an image of an engine which first runs idle, and then works a machine?

But what does application add to the calculation? Does it introduce a new calculus? In that case it isn’t any longer the same calculation. Or does it give substance in some sense which is essential to mathematics (logic)? If so, how can we abstract from the application at all, even temporarily?

No, calculation with apples is essentially the same calculation with lines or numbers. A machine is an extension of an engine, an application is not in the same sense an extension of a calculation.’

‘But what does “a 100-place relation turns up (or exists)” mean at all?

look – the idea of a 100 place relation can be proposed –

does it have an application?

well that needs to be proposed too

I see no problem with imaginative mathematics

what we are talking about here is game proposal and possibly construction

such might lead nowhere or it might be of use

the significance and value of any proposal – is always live

however the idea that we can complete our understanding of reality –

by covering all possibilities in some propositional format –

is quite simply ridiculous

those philosopher and logicians who have gone down this path –

have only demonstrated their ignorance –

of propositional logic – of propositional reality

they are clowns – and it is no laughing matter –

they have done a lot of damage

when I use 3 rows of 6 apples to explain  3 x 6 = 18 –

I propose an illustration – another propositional form

it’s a way of showing how the game is played – it’s an aide or a prop to showing how the game is played

application adds nothing to the calculation

the application is a use of the calculation –

in some descriptive context

‘Suppose that, in order to give an example, I say “love is a 2-place relation” – am I saying something about love?  Of course not.  I am giving a rule for the use of the word “love” and I mean perhaps that we use this word in such and such a way.’

am I saying something about love?

quite possibly –

if you think – as most would – that the word ‘love’ – has application in various contexts

to say ‘love is a 2-place relation’ –

could well be interpreted i.e. as saying that love is not a three a place relation – or a four place relation – or an n-place relation –

giving a rule for the use of the word – is to define the word –

is to say what it is to mean –

and therefore what love is –

and of course any such proposal –

is open to question – open to doubt –

is uncertain

‘Yet we do have feeling that when we allude to the 2-place relation ‘love’ we put meaning into the husk of the calculus of relation. – Imagine a geometrical demonstration carried out using the cylinder of a lamp instead of a drawing or analytical symbols. How far is this an application of geometry? Does the use of the glass cylinder in the lamp enter into the geometrical thought? And does the word “love” in a declaration of love enter into my discussion of a 2-place relations?’

a geometrical demonstration carried out using the cylinder of a lamp instead of a drawing or analytical symbols – is an illustration of the drawing or the analytical symbols

does the glass cylinder enter into the geometrical thought – or should we say proposal

no – it’s a representation of it

and does the word ‘love’ in a declaration of love enter into my discussion of 2-place relation?

no

‘We are concerned with different uses or meanings of the word “application”. “Division is an application of multiplication”. “The lamp is an application of the glass cylinder”. “The calculation is applied to these apples”.’

division can be represented as a form of multiplication

the lamp is a development of the glass cylinder

a calculation is made with these apples

‘At this point we can say: arithmetic is its own application. The calculus is its own application.

In arithmetic we cannot make preparations for a grammatical calculation. For if arithmetic is only a game, its application too is only a game, and either the same game (in which case it takes us no further) or a different game – and in that case we could play it in pure arithmetic also.’

yes –

the game’s application is its play

here ‘game’ is a noun – and then a verb –

pure arithmetic – is the game played in the absence of a descriptive context

it is the same game whatever descriptive context it is represented in

‘So if the logician says he has made preparations in arithmetic for the possible existence of 6-place relations, we may ask him: when what you have prepared finds its application, what will be added to it? A new calculus? – but that’s something you haven’t provided. Or something that doesn’t affect the calculus? – then it doesn’t interest us, and the calculus you have shown us is application enough.’

yes

if someone designs a new calculus – or a new game – all to the good –

its ‘application’ will be its action – its play

‘What is incorrect is the idea that application of a calculus in the grammar of real language correlates it to a reality or gives it a reality that it did not have before.’

the application of a calculus in the grammar of real language – is the making of a use of the grammar

the grammar of a real language is an account or theory of a real language

the reality of a real language – is whatever use it is put to

the application of a calculus in the grammar of real language

is putting the real language to a particular use

‘Here as so often in this area the mistake lies not in believing something false, but in looking in the direction of a misleading analogy.’

the ‘true’ position – or account – is the one you give your assent to for whatever reason

any use of analogy is open to question – open to doubt – is uncertain

‘So what happens when the 6-place relation is found? Is it like the discovery of a metal that has the desired (and previously described) properties (the right specific weight, strength, etc.)? No; what is discovered is a word that we in fact use in our language as we used, say the letter R. “ Yes, but this word has meaning, and ‘R’ has none. So now we see that something can correspond to ‘R’.” But the meaning of the word does not consist in something’s corresponding to it, except in a case like that of a name; but in our case the bearer of the name is merely an extension of the calculus,

of the language. And it is not like saying “this story really happened, it was not pure fiction”.’

the ‘discovery’ of a 6-place relation – is the proposal of a 6-place relation

‘This is all connected with the false concept of logical analysis that Russell, Ramsey and I used to have, according to which we are writing for an ultimate logical analysis of facts, like a chemical analysis of compounds – an analysis which will enable us really to discover a 7-place relation, like an element that really has the specific weight 7.’

what we deal with is what is before us – that is proposals – propositions –

any account of – any ‘analysis’ of – a proposal – a proposition – is itself – a proposal

and any proposal – or any account of a proposal – is open to question – open to doubt

is uncertain

‘writing for an ultimate logical analysis’ – is philosophical hubris –

predicated on a corruption of propositional logic

‘Grammar is for us a pure calculus (not the application of a calculus to reality).’

grammar is a theory of language use – a language-game –

the game is the reality – when the game is played

‘How can we make preparations for something which may or may not exist” means how can we hope to make an a priori construction to cope with all possible results while basing arithmetic upon a logic while we are still waiting for results of an analysis of our propositions in particular cases?

One wants to say: “we don’t know whether it may not turn out that there are no functions with 4 argument places, or that there are only 100 arguments that can significantly be inserted into functions of one variable. Suppose for example (the supposition does appear possible) that there is only one four place function F and 4 arguments a, b, b, c, d; does it make sense in that case to say ‘2 + 2 = 4’ since there aren’t any functions to accomplish the division into 2 and 2?” So now one says to oneself, we will make provision for all possible cases. But of course that has no meaning. On the one hand the calculus doesn’t make provision for possible existence; it constructs for itself all the existence that it needs. On the other hand what looks like hypothetical assumptions about the logical elements (the logical structure) of the world are merely specifications of elements in a calculus; and of course you can make these in such a way that the calculus dose not contain any 2 + 2.

‘an a priori construction to cope with all possible results’ –

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

‘preparations for what may or may not exist’ – are simply speculative proposals –

it is playing the a priori game – if you like –

and there is no reason why you can’t do this –

any proposal concerning what might happen – is a priori in a sense –

what you have to understand is this –

that a proposal – however you describe it – is open to question – open to doubt – is uncertain

that goes for so called ‘a priori’ proposals

‘while basing arithmetic upon a logic while we are still waiting for results of an analysis of our propositions in particular cases.’ –

arithmetic is a sign game –

any proposal to do with it’s basis – is open to question – open to doubt – is uncertain

its ‘basis’ – is irrelevant – to the game – to the playing of the game –

for all intents and purposes – it has no basis – or its basis is itself –

it is simply a language game – a language practice – that we engage in – for whatever reason –

if you want to be logical about it – its basis is no different to the basis of any proposal – any propositional practice –

its basis is uncertainty

yes – ‘the calculus does not make provision for possible existence’ –

the calculus is a sign game – it does not make provision – for existence – actual or possible

it is a sign-game construction –

yes – you can – as is done –  ascribe existence to the calculus – hypothesize existence – from the calculus –

but any such speculation has nothing to do with the calculus as such –

it is just a use of the calculus – in the same way as any proposal can be used – for purposes for which it was not originally designed

‘hypothetical assumptions about the logical elements’ –

so called logical elements are no more than an interpretation of the calculus –

again the calculus – gets on quite well regardless of such burdens –

and yes –you can construct any kind of calculus – any kind of sign game you like –

and underpin it with ‘logical elements’ – and if you want to then say the ‘the world is thus’ – fair enough –

by all means propose a novel and interesting metaphysics – in the end we will all be better for it

the one thing to remember though is that any such proposal – is just that – a proposal

open to question – open to doubt – uncertain

it is really quite irrelevant how a game comes about –

once it is up and running – that is the point

‘Suppose we make preparations for the existence of 100 objects by introducing a hundred names and a calculus to go with them… There isn’t any question here of a connection with reality which keeps grammar on the rails.  The “connection of language with reality”, by means of ostensive definitions and the like, doesn’t make the grammar inevitable or provide a justification for the grammar. The grammar remains a free-floating calculus which can only be extended and never supported. The “connection with reality” merely extends language, it doesn’t force anything on it. We speak of discovering a 27-place relation but on the one hand no discovery can force me to use the sign or the calculus for a 27-place relation, and on the other hand I can describe the operation of the calculus itself simply by using this notation.’

‘Suppose we make preparations for the existence of 100 objects by introducing a hundred names and a calculus to go with them.’ –

we don’t make preparations for existence – we propose in relation to that which is before us –  that is – the propositions we encounter

and the logic of it is – any proposal – any proposition – is open to question – open to doubt – is uncertain

our world is a propositional world – a world of uncertainty –

another way of putting it is to say – what exists – is uncertain – existence is uncertain

‘There isn’t any question here of a connection with reality which keeps grammar on the rails’

grammar is a proposal – in relation to language – to proposals –

it is open to question – open to doubt – it is uncertain

any grammar – any calculus – is ‘free floating’ – if by that you mean – open to question – open to doubt – uncertain

a calculus – such as arithmetic – is a propositional game –

that is a rule governed propositional activity

as played – it is not in question – not subject to doubt – the game is not uncertain –

games – propositional games – are played as a relief from question – from doubt – from uncertainty – from logic

of course the propositions that make up a game – are open to question –

but the game as played – is not –

our ‘reality’ – is a propositional reality –

a propositional game – is its own reality

the proposition – is reality

‘When it looks in logic as if we are discussing several different universes (as with Ramsey), in reality we are considering different games. The definition of a “universe” in a case like Ramsey’s would simply be a definition like:

’

a game could be used as a springboard for metaphysical speculation –

but this is a use of logical games –

it is not logic per se

the real logic of the matter is this –

any proposal – metaphysical or not – is open to question – open to doubt –

is uncertain

when you play logical games – you are involved in a rule governed propositional exercise

and if you play – you play in accordance with the rules – you play without question – without doubt

if you question – if you doubt – you don’t play the game

metaphysics is not a game

metaphysics is speculation

16. Ramsey’s theory of identity

‘Ramsey’s theory of identity makes the mistake that would be made by someone who said that you could use a painting as a mirror as well, even if only for a single posture. If we say this we overlook that what is essential to a mirror is precisely that you can infer from it the posture of a body in front of it, whereas in the case of the painting you have to know that the postures tally before you can construe the picture as a mirror image.’

saying ‘x = y’ – is to say you can substitute x for y – y for x –

this is not say x is identical to y –

x is not identical to y

x is x – y is y –

there is no identity in x = y

what you have is a proposal for substitution –

any by that I mean – the proposal that y can function in place of x – and visa versa –

that two things can substitute for each other – can behave in the same way –

does not mean they are the same thing

x is x – y is y –

is this to mean x is identical to x – y identical to y?

I would put that a relation only exists between different things

so ‘x is x’ – if that means – x is related to x – then such a statement is meaningless

x is not identical to x –

and the idea that x is a substitute for x – that identity might be explained in terms of substitution – is ridiculous

in x = y – x is not identical to y – y is not identical to x –

the concept of identity makes no logical sense –

in the example given above by Wittgenstein – using a painting as a mirror –

a painting is not a mirror –

and a mirror image is not the thing that is reflected –

it is an image of it –

there is no question of identity – even if the painting – is a good representation of a  particular posture of the subject

the subject and the painting are different things

just as a mirror image and the subject are different things

the concept of identity is a pseudo concept –

and it is hard to see what logical purpose this ‘identity relation’– could possibly serve in any context –

if I say “yes x is x” – I am proposing x – and at best by saying “x is x” here –  I am simply giving an emphasis – to x – to its proposal

yes the notion of ‘identity’ is used – and I would say – works –

as a rhetorical devise

but that is the best you can say for it

Wittgenstein considers Ramsey’s argument for the identity sign

and goes on to show that Ramsey’s explanation of the identity sign –  fails –

the point being the explanation is of no use

‘If Dirichlet’s conception of function has a strict sense, it must be expressed in a definition that uses the table to define the function-signs as equivalent.

Ramsey defines x = y as

(je). je X º je y

But according to the explanation he gives of his function-sign “je”

(je). je X º je X is the statement: “every sentence is equivalent to itself”

(je). je X º je y is the statement: “every sentence is equivalent to every sentence

so all he has achieved by his definition is laid down by the two definitions

           def

x = x . = . Tautology

          def

x = y . = . Contradiction’

Wittgenstein continues –

‘It goes without saying that an identity sign defined like that has no resemblance to the one we use to express a substitution rule.’

correct

                                                                                                                                 

my point here would be that any definition here is irrelevant – irrelevant to the use of the  ‘=’ sign

                                                                                                                                 

all you really have from Ramsey is a conception (“je”) –

that is ‘defined’ – that is expressed in other terms –

i.e. – x = x by definition – a tautology – x = y by definition – a contradiction –

and we are expected to believe that these terms – are ‘equivalent to’ – “je” –

when it is ‘equivalent to’ that is up for definition –

Ramsey’s analysis – even if you accept his identity sign – (“je”) – and the argument that flows from it – is ill-conceived

the point here is that ‘=’ – does not require definition –

it is a sign for the action of substitution –

call that a ‘definition’ – if you like – but calling it so – is neither here nor there –

the issue is not – how do you define the sign – the issue is what you do with it –

that is – how it is used –

and that is quite simply a matter of propositional practise –

which is quite straightforward –

the action of substitution

‘What is in question here is whether functions in extension are any use; because Ramsey’s explanation of the identity sign is just such a specification by extension. Now what exactly is the specification of a function by extension? Obviously it is a group of definitions, e.g.

fa = p  Def.

fb = q  Def.

fc = r   Def.

These definitions permit us to substitute for the known propositions “p”, “q”, “r” the signs “fa” “fb” “fc”. To say that these three definitions determine the function f(x) is either to say nothing, or to say the same as the three definitions say.’

yes – exactly – nothing –  nothing on top of nothing

‘Moreover, the purpose of the introduction of functions in extension was to analyse propositions about infinite extensions, and it fails of this purpose when a function in extension is introduced by a list of definitions.’

functions in extension – take us nowhere – but really the point is – that there is nowhere to go

it doesn’t matter what propositional context you are operating in – there is no final analysis of any kind – no final definition –

and the search for such is just woolly headed – and  should just be abandoned as a waste of time

any program of analysis is open to question – open to doubt – is uncertain –

the question here – is whether the function in extension argument has any value in logic – in mathematics

what you have with the function extension is effectively replication or if you like translation –

no great sin itself – but it does alter focus – and inevitably introduces considerations that are in fact irrelevant – and unnecessary –

unnecessary to the game as played – to the playing of the game –

and if that is so – then such argument in logic and mathematics – is not logical – but rhetorical –

‘rhetorical’ – that is – if we are to give it any status at all –

‘useless’ – is more to the point

another way of putting it is to say –

the function in extension – is a form of speculation –

and of course there is – and should be a speculative dimension to logic – to mathematics

but in this context – the context of the use of the ‘=’ sign in mathematical and logical games – such speculative argument is out of place –

it’s in the wrong place – it misses the point

‘There is a temptation to regard the form of an equation as the form of tautologies and contradictions, because it looks as if one can say x = x is self evidently true and x = y self evidently false. The comparison between x = x and a tautology is of course better than that between x = y and a contradiction, because all correct (and “significant”) equations of mathematics are actually of the form x = y. We might call x = x

a degenerate equation (Ramsey quite correctly called tautologies and contradictions degenerate propositions) and indeed a correct degenerate equation (the limiting case of an equation). For we use expressions of the form x = x like correct equations, and when we do so we are fully conscious that we are dealing with degenerate equations. In geometrical proofs there are propositions in the same case, such as  “the angle x is equal to the angle B, the angle y is equal to itself …”

 At this point the objection might be made that the correct equations of the form x = y must be tautologies, and incorrect ones contradictions, because it must be possible to prove a correct equation by transforming each side of it until an identity of the form

x = x is reached. But although the original equation is shown to be correct by this process and to that extent the identity x = x is the goal of the transformation, it is not its goal in the sense that the purpose of the transformation is to give the equation its correct form – like bending a crooked object straight; it is not that the equation at long

last achieves its perfect form of identity. So we can’t say: a correct equation is really an identity. It just isn’t an identity.’

a tautology is not a proposal – is not a proposition – it is a corruption – a propositional corruption –

the point of the tautology – is to bring the proposition process to an end – to bring argument to an end

that is to say to bring an end to question – to doubt – to uncertainty

the tautology we are told is ‘self-evidently true’ – which is just a neat turn of phrase for – no more questioning – no more doubt – the matter is certain

the contradiction – is a straight up denial of propositional reality – masked – in the form of a proposition –

the contradiction is simply a subversion of logical – that is – propositional reality

the contradiction brings the propositional process to an end by denying – or blocking  propositional action

both the tautology and the contradiction – are best seen as rhetorical devises – rhetorical devises – impersonating logical forms

to say that mathematical games are played with rhetorical devises – or that logical games are played with rhetorical devises – might strike some as quite an outrageous statement –

the fact is that if a game is to be – if it is to be played – logic – question – doubt uncertainty – must be circumvented – and this is where rhetorical devises come into play

it is in logic and mathematics that some of the most effective rhetorical devises have been devised – i.e. – the tautology – the contradiction –  the proof

as to the form of the equation –

what we can say of the equation is that it is a propositional game – a game of substitution

identity doesn’t figure in it

                                                                                                                               

17. The concept of the application of arithmetic (mathematics)

‘If I say “it must be essential to mathematics that it can be applied” we mean that its applicability isn’t the kind of thing I mean of a piece of wood when I say “I will be able to find many applications for it”.’

mathematics is not an application –

rather it is better understood as the theory of application

‘Geometry isn’t the science (the natural science) of geometric planes, lines and points, as opposed to some other science of gross physical lines, stripes and surfaces and their properties. The relation between geometry and propositions of practical life, about stripes, color boundaries, edges and corners, etc. isn’t the things geometry speaks of, though ideal edges and corners resemble those spoken of in practical propositions and their grammar. Applied geometry is the grammar of statements about spatial objects. The relation between what is called a geometrical line and a boundary between two colours isn’t like the relation between something fine and something course, but like the relation between possibility and actuality. (Think of the notion of

possibility as a shadow of actuality.)’

yes – geometry is a rule governed propositional game –

applied geometry – is the game played in context – the context of physical lines – stripes – surfaces and their properties –

‘The relation between what is called a geometrical line and a boundary between two colours isn’t like the relation between something fine and something course …’

no it’s the relation between two proposals –

a geometrical proposition and a physical proposition –

and this propositional relation – is open to account – to description

‘but like the relation between possibility and actuality. (Think of the notion of

possibility as a shadow of actuality.)’

here we have one such account – one such description

‘You can describe a circular surface divided diametrically into 8 congruent parts, but it is senseless to give such a description of an elliptical surface. And that contains all that geometry says in this connection about circular and elliptical surfaces.’

‘describing a circular surface’ –

is to propose that a particular geometrical game be played – in a particular context

‘a circular context’ –

is no more than the proposal that the playing of that game will be useful

the same game in a different context – may not work – may not be useful –

game utility – is open to question – open to doubt – is uncertain

‘(A proposition based on a wrong calculation (such as ‘he cut a 3 metre board into 4 one metre parts” is nonsensical, and that throws light on what is meant by “making sense” and “meaning something by a proposition”.)’

is ‘he cut a 3 meter board into 4 one metre parts’ – nonsensical?

what we are likely to say of the proposition – is that it represents – a failure to understand the rules of arithmetic – the rules of the arithmetic game –

as an illustration of that – it is not – nonsensical –

relative to the rules of arithmetic – it ‘makes sense’ – and ‘means something’ – as  an example of not properly applying those rules – of what not to do – if you want to play the game

in this respect such a proposal – such a proposition – is quite pertinent

the point is that this proposition – as with any proposition – is open –

open to question – open to doubt –

if it was not open – open to question – it would not be a proposal – it would not be a proposition –

it wouldn’t be here – and if it wasn’t here –

obviously we wouldn’t be discussing it

‘What about the proposition “the sum of all angles of a triangle is 180°”? At all events you can’t tell by looking at it that it is a proposition of syntax.

The proposition “corresponding angles are equal” means that if they don’t appear equal when they are measured I will treat the measurement as incorrect; and “the sum of the angles of a triangle is 180°” means that if it doesn’t appear to be 180 degrees when they are measured I will assume that there has been a mistake in the measurement. So the proposition is a postulate about the method of describing facts, and therefore a proposition of syntax.’

the proposition “the sum of all angles of a triangle is 180° is a game proposition

appearance here is irrelevant

the game as such – is not open to questions of truth or falsity – it is not correct or incorrect

it is not a postulate about the method of describing facts – it is a propositional game –

and yes – it can be described as a proposition of syntax