Part II. On Logic and Mathematics. 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 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 they can 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 mathemathics 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 put here – what is proposed here – is that a chess problem is interpreted as an arithmetical problem – and that is fair enough

however the arithmetical interpretation is just one possible interpretation

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

and yes a strategy problem – in chess – could well be re-interpreted arithmetically –

characteristically in philosophy the issue is then seen to be – which approach – which description – is fundamental

and that usually – implicitly or explicitly – comes down to a question of authority

the only authority is authorship –

and authorship of a proposition – is logically irrelevant

beyond authorship – any claim to authority – is rhetorical –

which is to say logically – deceptive

the world runs on rhetoric – on deception –

persuasion is the art of deception – and rhetoric is its tool

in the absence of rhetoric – the logical situation is that different proposals – different descriptions – of the problem – whatever it is – are advanced – are proposed –

for whatever reason –

there is no one description of any problem –

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

you go with – what you go with – for whatever reason –

and you keep an open mind

‘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 –

games can be and are the basis of other games –

games within games

ultimately it is a question of description

how you describe what is going on –

and that is open to question – open to doubt –

however there is nothing against describing an activity – as a relation between games

a relation between rule govern activities –

yes you can go the reductionist route – but why?

what is gained by pretending – one description – is something else – do you have a ‘broader’ view as a result? –

or is it in fact a neurotic narrowing –

which whatever the end result – just does not accord with reality –

the reality of propositional behaviour

we can and do operate with different descriptions of the reality before us

this propositional diversity –

is both a source and the expression of our creativity –

it is knowledge

‘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 his description of why he plays the arithmetical game is different to another player

then there is an argument for saying that what he learns will be different to the other player

that is they play the same game – but interpret it differently

different reasons for playing it – different interpretations of the outcome

it is all a question of interpretation –

and any proposal – any proposition – any set of propositions – any game of propositions –

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 –

here the truth is a given

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 –

the truth  – is a given –

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 to play with a cat

‘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?

yes – you play a game for a result – a result that is already determined in the rules of the game –

you play the game to arrive at the result

is arriving at the result – a finding out?

yes it is ‘finding out’ – the result – and completing – ‘finding out’ – the game

what does he lack – an application for the game?

I wouldn’t call it a ‘lacking’ –

the game can be played for the pleasure of the game – the pleasure of the playing –

my view is that the point of mathematics is its applications –

clearly this is how mathematics is used –

but this view of mathematics – is not the only view possible

‘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.’

‘calculation’ –

calculation in a game with a competitor – is quite a different matter to when I calculate  21 x 8 = 148 by myself –

when I calculate 21 x 8 = 148 – my calculation is the game

when I calculate in a competition – yes I play according to rules – and hence play a game – but my ‘calculating’ – is really having a bet within the rules –

it is if you like an ‘intelligent speculation’

in any competition game – the rules are in the background – what is in the foreground

is the unknown –

what my competitor will do next is unknown – what I will do next is unknown – that is the mystery – that is a good part of the pleasure of the game

epistemologically speaking –

such a game is a domain definition of the unknown

‘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.’

yes – a billiard problem can be described as a physical problem –

but the physical actions of a billiard player occur in the context of any number of

variables – uncertainties

just as in a game of tennis

yes – a chess problem qua problem – that is – not in an actual game – could be read as a mathematical problem –

a game is a rule governed propositional exercise – 

be it a sign-game – as in arithmetic – a board game as in chess – a table game as in billiards – or a court game as in tennis

different contexts – different propositions – different games –

and we need to distinguish between solitary games and games between competitors

underpinning any rule governed propositional exercise is propositional logic –

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

yes you can play a game without question – in accordance with the rules –

however any game – and the rules of any game – logically speaking – are open to question – open to doubt – are uncertain

when you question – you are not playing

‘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?

it would be verified as any statement is verified – that is in terms of some agreed process and standard proposed as ‘verification’

clearly – an uncertain matter

and how the rule is interpreted – as with ‘verification’ – is open to question – open to doubt – is uncertain

if a teacher is trying to inculcate the rules of a game – it matter’s 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

‘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.

‘a war in the ordinary sense’ – is no different to any other state of affairs –

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

chess as a propositional – model of war – as indeed – any propositional model of human behaviour – may well be intellectually interesting –

but any interpretation of human affairs in terms of rule governed behaviour – and here I include the paradigm case of mathematics – is open to question to open to doubt is uncertain –

rule governed propositional action – only works when the actors – the players – act –

play according to the rules –

if you want to do mathematics – if you want to play mathematics – you play according to its rules –

the community of mathematicians decide the rules –

mathematics works because it’s practitioners in the main – stick to the rules –

you can always be one out

war never works – because in fact – despite the rhetorical rubbish of some – there are no rules –

and I think it is fair to say war is a consequence of the breakdown of rule governed behaviour –

war is an argument for rules – not an argument with rules

‘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

problem solving per se – is not a game –

a problem is a proposition put that is then subject to question – to doubt

the activity of problem solving is the activity of propositional uncertainty –

a thorough examination of a proposal – of a proposition – puts everything on notice

in a game played – there is no questioning of rules –

there is no ‘problem’

in any rule governed propositional activity – and this includes mathematics –

there can be questions of the reach of rules – the adequacy of rules – the compatibility of rules – or indeed in a critical moments the absence of rules –

in such circumstances the players – the practitioners – are not playing the game – the mathematician is not doing mathematics

the player – the mathematician is questioning the game

the player steps back from the game – from playing the game – and becomes a logician –

the mathematician stops doing mathematics – and does logic

in the sea of propositional uncertainty – the  good ship ‘game’ – is just a speck – always vulnerable – always in troubled waters

but who doesn’t like a cruise?

‘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 – and in that case ‘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 – and whatever that comes to – 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 matter of rhetoric

‘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 – 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 is a game – a sign game –

calculus is a game played with the ‘chess pieces of mathematics’

calculus is the action of mathematics –

the word ‘calculus’ – is a description of that action

the word will not be used in a calculus – in the sign game –

it can be used to describe the sign game –

a meta-calculus will be an interpretation of the sign game – called ‘calculus’ –

‘calculus’ – is a name of a sign game

‘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

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

if you take the view that logic is the inquiry which has for its object the principles

of correct reasoning – or the view that logic is chiefly the inquiry into deductive reasoning i.e. into inferences into which the conclusion follows necessarily from the premises – then mathematics as a prime example of correct reasoning or as a prime example of deductive reasoning may well be considered an expression of logic – a case study of logic –

and in that case – logic may well be proposed to in some way account for mathematics

‘(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 – of doubt – 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 –

foundation is description –

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

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

this just presumption

‘(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, an 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 –

but here we are talking about interpretation –

and interpretation is – post-proposal –

and is in fact a use of a proposition

there will be no interpretation of a proposition as dependant –

unless in there is an independent proposition in the first place

‘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

any such representation – will come with an interpretation –

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

the real pleasure of any game – I would suggest – is not in the interpretation –

but rather in the play

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 concept “solution” must occur in the proof’? –

or could we equally say – if we solve the problem the concept proof must occur somewhere in the solution?

solution in proof – proof in solution

solution or proof – we perform a propositional action or series of actions on a proposition –

we apply a propositional model to the proposition –

the propositional model – the word game – is called ‘proof’

logically speaking there is no question of proof – if by proof is meant – establishing a proposition beyond question – beyond doubt – establishing a proposition as certain

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

‘proof’ – as a language-game applied to a proposition – or as a language-game within language-games –

is just another play

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

this to say that the proof of the proposition is the proof of the proposition –

such a statement leaves the question of the proof a proposition with nowhere to go

proof or provability is without explanation – without any explanation – and the result is that this concept of proof is – in terms of this definition – is vacuous

proof and provability come off as empty rhetoric

‘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.’

the equation is a propositional game – the game is that one side of the ‘=’ sign –

equals the other side

‘equals’ here – just what it means – can be put to the question – as indeed can any term in any context –

but in this game we can say a standard – operating definition is ‘can be substituted for’

the actual game – the play of the game here is the ‘x’ sign – the multiplication sign

if you understand the rules of the game – if you understand how the game is played – then the game proposition – the equation (i.e. 17 x 38 = 456) – says it all –

there is no question of proof – the notion is an irrelevant – and unnecessary piece of

theatre

to you play the game – you must understand the rules

if you understand the rules – you see all there is to see – in the game- proposition – in the equation

‘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 ca actually use the sign ‘1, 2, 3, 4, and so on ad infin”. I ca define the word “number” too by pointing to various. 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 logically – definitions of the words “proof” and “number” – (as with any word – any proposal) – are open to question – open to doubt – are uncertain –

we create our own uses

pointing is best seen as proposing a logical space

and if you ask – in response to my pointing – ‘what is it you are pointing at? – or ‘what is that?’

any answer to those questions – is a proposal –

and if you think you know what I am proposing – that is you don’t ask the question – ‘what is it?’ – then you have a proposal yourself in response to the pointing –

and this proposal you may make public  – by word or gesture – or not

in any case – clearly whatever is proposed by me in pointing – or by another in response to the pointing – be it made public or not – is logically speaking – open to question – open to doubt – is uncertain

‘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”.’?

do we? – if we do there are different ways of regarding this –

perhaps we regard our use of ‘proof’ as a conjecture – i.e. – it might function in contexts we haven’t come across?

such an attitude is logical – in that it leaves open the question of just whether our current use will work in future contexts

or perhaps we regard our definition – our use of proof ‘proof’ – as fixed –

if so this attitude betrays a logically illiteracy

if ‘we want to speak absolutely precisely about proofs of relevance, of consistency etc.’?

precision is simply use of a word that is not questioned – or a use of a definition of precision – that is not questioned

again there are different ways of seeing this –

we suspend questioning – doubt – uncertainty – when we play a game – be it described as mathematical – or in some other terms

you could regard this as a pragmatic use of language – the point of which is to get on with it

even so – it is not as if logic disappears out the window – a question can always be asked – a doubt can always be raised –

however questioning – doubting – recognizing propositional uncertainty – is not playing the game – it is questioning the game – it is not a pragmatic use of language

it is a logical exploration of use

on the other hand – if you hold that your ‘absolutely precise proofs of relevance of consistency etc.’ – are to be regarded as ‘certain’ – then the view you are taking is illogical and is in fact a denial of logic – 

such a view is delusional

Wittgenstein says – ‘to the extent that the concept of proof is sharply circumscribed’

by italicizing ‘sharply’ – it looks as if he wants to believe that there can be sharp  circumscription – and yet ‘to the extent’  indicates reservations  –

a bet each way?

all we can say here is that ‘sharp’ – like any other word is open to question – open to doubt

we can make decisions in regard to the definitions of words – in regard to usage – and stick to them as best we can for whatever reason e.g. – to see where the lead – to finish a program etc. –

however no decision is – is beyond question – beyond doubt –

and that is about as ‘sharp’ as it gets

‘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 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 may 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

a calculus is a game – a game does not require justification

and lets be clear here – justification – in any context – is pure rhetoric

there is no question of ‘justifying’ writing down the equation 17 x 23 = 391 –

the proposition can be proposed in any form

mathematics if it is done cleanly – is non-rhetorical

as for ‘legitimacy’ – in general it can be regarded as another – authoritarian piece of rhetorical – rubbish

any proposal can be  put – and any proposal can be questioned –

whether a proposal is appropriate – and in this sense ‘legitimate’ – in a particular context – is a fair enough question at any time

mathematics is a propositional framework that has universal application –

its application and utility speaks for itself

we are best to just drop this notion of justification – and speak in terms of how a proposal is used

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 proof of the verifiability of 17 x 23 = 391 is not a “proof” in the same sense of the word as the proof of the equation itself.

‘the proof of the equation itself’– is understanding equations – understanding the equation-game –

a first order matter – relative to specific equations

“proof” here is understanding the game – the propositional game

yes – if you understand how the proposition-game 17 x 23 = 391 – is played – you grasp it – i.e. you understand it

there is no question of ‘verifiability’ here –

there is a proposition-game – if you play it according to its rules – then you play it –

if you don’t understand the rules – or don’t play according to the rules – you simply don’t play it

‘look at the proof , and you will see what is proved here and what is proved here, what get  called “the proposition proved”.’ –

look at the game – and you see how it is played – and seeing how it is played is seeing what the game is – the game-proposition played

‘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. Since the form in which the rule is applied is in  fact a new one at every step. But it is not a matter of an act of insight, but of an act of decision.

correct

‘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.’

different accounts of a mathematical game – are possible – different explanations

the playing of the game though does not depend on an interpretation of its basis – the basis of its rules –

the playing – the doing of mathematics – depends only on – following the rules –

and as a result – strangely enough – so called ‘proofs of relevance’ – are in fact –

irrelevant

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.’

a ‘proof’ here – is an account (a set of proposals) of the steps taken to reach a result –

and yes there is a sense in which a contraction in the proof can be seen as trivial –

it might also be that the result is seen to have merit – and the proof needs to be reworked to fit the result?

consistency is appearance – is presentation –

the logical reality is that any proposition of mathematics – is open to question – open to doubt – is uncertain

putting forward a ‘proof’ – doesn’t change this –

if the idea is that the proof  ‘settles the matter’ – then the proof – is nothing more than a rhetorical device –

which the next imaginative mind may well shoot down in flames

this is not to say that there cannot be a genuine argument as a result of the discovery of a contradiction

in practice  consistency is seen to be a mathematical touch stone – and this is really a comment on how mathematicians behave – how they proceed –

it’s the way the game is played –

and let’s also remember that the discovery of inconsistency may well prove to be instructive – and beneficial in the long run

perhaps it can be said that in mathematics what you have is something of a dialectical tension – between consistency and inconsistency –

and that it is that tension leads to mathematical advances or discoveries

some mathematicians may not be comfortable with this – but really – everything is up in the air –

we only bring it down to earth – on the page as it were – for a breather

‘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 –

it is irrelevant what equals what –

the point is that one designated sign can be substituted for another

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

at the end of such a rewriting of arithmetic ‘x’ would still ‘equal’ ‘y’ –

the game would have been rewritten – but in the end – it’s the same game

the point of ‘equals’ is substitution – not what is substituted

consistency is a condition of the game as played –

without a consistency in the rules – there is no game – to be played –

but here we are talking about game-playing –

the status of the propositions used in game playing is the issue of propositional logic –

propositional logic is the meta-underlay of the game – of mathematics –

the propositions – the proposals of mathematics – are open to question – open to doubt – are uncertain

‘meta-mathematics’ – if you like – is propositional uncertainty

the game as played – as played in accordance with its ‘rules’ – is not open to question

the mathematician as a calculator’ – is a game player

the mathematician as a thinker is a logician

‘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 – as to rules all we have here – logically speaking is proposals – propositions – open to question – open to doubt – uncertain

the idea of the ‘rule’ – as a proposition that stands – has not to do with the logic of the proposal – but rather the practice – the use – of the proposition –

and the use here is rhetorical

that is to say the ‘rule’ – or the proposal used as a rule – is a logical deception

the ‘rule’ – is a logical deception

that is to say that the rule is a deception – if it is held to be beyond question – beyond doubt – certain

I really don’t think anyone seriously holds this view – but the pretence does hold

and seems necessary to the practice of mathematics

yes – pretence – but can human beings operate without it?

‘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.’

the only foundation to the proposition – is propositional

agreement?

here I think we are talking about the great deception – ‘authority’ –

and the ground of this deception is rhetoric –

“this is how you do it – this is how it is” – “ok”

whatever is the case here – is as with any other issue – any other definition –

open to question – open to doubt - uncertain

but – yes we do have agreement – for whatever reason –

and indeed we  have disagreement – for whatever reason

logically speaking – the argument goes on

in practice – it stops –

or perhaps a better way to put is to say – there are stops –

and I think the reason is – we all need respite from logic –

but then as soon as we begin to think –

question – doubt – uncertainty –

once again take centre stage

‘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 rules.’

the rules here – are the practice – formalized –

the ‘rules’ simply reflect the practice – they are statements of it –

and any statement of practice – as with the practice itself – is open to question – open to doubt – uncertain –

it’s the practice that ‘determines’ the rules – not the rules that determine the practice –

rules come into their own – when someone has not engaged in the practice – and needs a start –up –

or someone who is not confident in the practice – and needs some guidance

they are also used as a quick check – that what you are doing – conforms with the given or accepted practice

‘“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.’

a contradiction?

is not a proposal – is not a proposition –

it has the form of proposition – however nothing is proposed –

it’s a dummy proposition –

if regarded as a proposition – then it is a pretence

so to rule out a contradiction is?

is to – rule in the proposal – the proposition –

and there is no sense in ‘ruling in’ a proposal – a proposition –

so there is no sense in ruling out a contradiction

the foundation of the rule – the proposal – the proposition?

there is no foundation

the rule – the proposal – the proposition –

is open to question – open to doubt – is uncertain

the ground of the proposition is –

uncertainty

‘In the indirect proof that a straight line can have only one continuation through a ‘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.’

 ok –

‘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 you say it can only have one continuation – you are with the use of ‘can only’ denying that it could have two

that is your assertion –

you are denying that possibility –

the ‘possibility of two’ – is not being entertained –

and the point of denying the possibility – of the ‘can only’ – is to give the assertion a logical status

I don’t see that there is an issue of proof here

the assertion is as it is –

supposing the opposite – that a straight line could have two continuations

and then putting that it is logically contradictory – is really just elaborating on ‘can only’ – which is if you read the proposition as written – is unambiguous – from the get go

so – the ‘proof’ here is just a nice little piece of rhetoric –

the point of which is I think to furnish the mathematician with the pretence of foundation – and perhaps even – the delusion of certainty

but really if you understand what is said – you understand – what is not said

no need for the windmills

‘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.’

no there is nothing absurd in the drawing –

but is it consistent with the assertion ‘that a straight line can have only one continuation through a certain point’

clearly not

and is a straight line bifurcating – a straight line?

no

if this is supposed to be part of the ‘proof’ – then the ‘proof’ is just a stupid word-game – that adds nothing to the original assertion – and takes nothing away from it

this ‘proof’ is just a diversion – a waste of space

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

this is one interpretation of the significance of a contradiction – but it is not the only one

the contradiction could well be read as an instruction to stop

how we proceed in mathematics – is at any point is open to question – open to doubt – is uncertain

mathematics is argument

‘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

what we are dealing with is proposals – propositions

there is nothing ‘hidden’ in propositional reality

what is proposed – is proposed – what is not proposed – is not there

a contradiction – can be proposed in argument

it’s not that it was hidden – it is proposed –

and you deal with what is proposed – not with what is not proposed

and yes a contradiction can be removed by organizing the grammar

but it’s not good enough to say – therefore it doesn’t matter

whether it matters of not – is open to question – open to debate

what significance you give a contradiction – has little to do with it’s grammar –

a contradiction may be seen as a simple grammatical issue – or it might well be viewed as demonstrating that the ‘system’ – doesn’t work – doesn’t work – as constructed –

it can be viewed as an indication that you are on the wrong track

whatever the case – any decision in respect of a proposition – or indeed a propositional system –

is open to question – open to doubt – is uncertain

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

best to forget about rules per se

what you have is established practices – and yes these have been described – codified – for convenience

in any so called conflict of rules – what you have is question – doubt –  uncertainty –

just exactly what you have – in the absence of rules –

before the rhetoric of rules

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 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; 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 to 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.’

the most general form of a propositional game?

the most general form of a game?

what these question presume is that what makes a game is the description that it applies to

first up – in any propositional game – the description it applies to – is irrelevant to the playing of the game

any so called ‘application’ is simply a context – a setting for the game –

if you like a reason for playing the game – a motivation to play the game

as for this so called ‘general form’ –

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

every proposal – every proposition – is of this form

‘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.)

any description – representation is possible –

arithmetic is a propositional game –

but if you think the point of arithmetic is what it is applied to – what context it is used in – then you miss the point – of the game

the arithmetic game – is played – for itself – just as any game is played – for itself –

the point of a game – any game – is play

and as to the point of play – I would say – pleasure

that a game is played in a particular context – may incidentally add to the pleasure of the game

and any propositional game –  that gives pleasure – I would say – is useful

people do think that in using arithmetic they are applying it to other problems –

hence the question – what is the most general application?

and the question is instructive – for once asked – its apparent significance dissolves –

for you see – the question of application – general or not – is actually irrelevant –

put it this way – the game is played – wherever it is played –

and once you see this – you see ‘wherever’ – is seen to be irrelevant

‘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.’

a game – any game is logically autonomous – in its play – not in its construction – or description – but in its play

it’s construction is a propositional exercise – open to question – open to doubt – uncertain –

and yes – any description of a propositional exercise – likewise is open to question – open to doubt – uncertain

however once a game – has been constructed – that is – decided on –

then it is – for the purpose of play – not open to question – open to doubt – or regarded as uncertain –

we play – as it were – in the absence of logic –

it is not that logic is ever actually absent – but when we play we have a rest –

a rest from logic – from question – from doubt – from uncertainty –

and that might just be the real source of a game’s pleasure

‘(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 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 –

we have a series of marks –

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

Wittgenstein here is suggesting that the above series of marks can be interpreted as an ‘arithmetical construction’ and a ‘geometrical one’ –

the point of his description here is to suggest a relationship between arithmetic and geometry –

earlier he put that arithmetic is a ‘more general kind of geometry’ –

presumably that means he is suggesting that geometry can be seen as a product of arithmetic – an outcome of it – or something along those lines

the truth about reductionism – of any kind – is that it is nothing more than a language game – a description game –

and Wittgenstein’s description above is a good example –

‘an arithmetical construction’ –

a clear case of shifting the goal posts – of playing a game

we just don’t normally describe an arithmetical proposition – as a ‘construction’ –

and yes we might quite naturally describe a geometrical proposal as a construction –

all we have here is an arithmetical proposal (rather awkwardly) described in what would be regarded as geometrical terms –

just a language game – a description game –

look the real question here is why?

it just seems to me to be a waste of thought to play such games – if the idea is to suggest some unified theory – of arithmetic and geometry – with arithmetic as the basis

this is just that old chestnut of the search for the fundamental – the basis – the essence

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

then it is clear that a so called ‘fundamental’ – is open to question – open to doubt – is uncertain

point being there is no proposition that is beyond question – beyond doubt – that is certain –

the search for a logical basis – is just wrongheaded –

the logical approach to the world is one of exploration – exploration of – propositional

uncertainty

‘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.’

if ‘3 strokes on the paper are the sign for the number 3’ –

is simply a definition –

then what we have here is a little word game –

all very well – but of no relevance to how language is actually practised

as a ‘rule’ of usage – the fact is we can’t say how 3 will be used –

so the idea of a ‘rule’ here is rather pointless

what we have here is a proposal –

and just whether or not this proposal – holds up in practise –

is open to question – open to doubt – is uncertain

‘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 that a count takes place –

how you described 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 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 –

at best these matters are side issues

as to foundations –

speculation on the basis of arithmetic – 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

so logically speaking we are best to drop the notion of ‘foundation’ altogether

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.)’

a proposition of arithmetic – can be variously described – can be given various descriptions

the proposition exists and functions independently of any description given to it

however putting a description on the arithmetical proposition – on an arithmetical game –

introduces it to descriptive reality

and thus enables its use in the world of description

and in the example given above – in the world of ‘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

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? What concept do we have of one?’

what it means is that someone has constructed a game with a 100-place relation –

if – indeed such a game is proposed

if it is not proposed – it is not a game mathematicians play –

who is to say what will be proposed – what games will be constructed?

the idea that you can ‘prepare the way’ – is just philosophical presumption –

there is enough to do in this world without playing God –

on the other hand logic and mathematics as with any propositional activity –

just is open – open to hypothesis – to invention

yes – arithmetic is a sign game played –

but behind any game played – are the game makers – game designers –

mathematics at it’s heart is a creative endeavour – a creative proposal

as to ‘our concept of application’ –

an application – is a description applied to a calculation –  a semantics put to the syntax

a calculation is a game – a sign game –

yes description can be applied to the calculation – to the game –

the description however adds nothing to the calculation –

nothing to the game –

it simply provides a setting for its play

‘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.’

is – ‘love is a 2-place relation’ – an example of a 2-place relation?

well it is a statement – a statement that – love is a 2 place relation –

‘love’ – is – as it were – hoisted on to the ‘2-place’ relation form –

you really need to know what a 2-place relation is – for the statement ‘love is a 2-place relation’ – to have a chance of making sense – of being of some use

so the idea of it being an example of a 2-place relation only works if you already know what a 2-place relation is –

and if you know that – what point any example?

a rule for the use of such a word?

if we actually use the word in this way –

what’s the point of a ‘rule’?

and if in fact we don’t use it in this way –

what’s the point of a ‘rule’?

do we need an example of how we us the word – if in fact we use it?

point being – our use is the example

if we don’t know the use of the word ‘love’ – ‘love is a 2-place relation’ – won’t get us too far at all

and really is – ‘love is a 2-place relation’ – an example ‘of using the word [‘love’] in such and such a way’?

‘such and such a way’?

well – “love is a 2-place relation”  is a use of ‘love’ – yes – but a pretty odd one –

I would suggest –

and what is it an example of?

itself  – an odd use of the word

‘love is a 2-place relation’ – is not only an odd statement in relation to how the word ‘love’ is used – but it is also a very vague statement –

if it was to be of any use it would require further elucidation

we must always remember that just because a proposal is put up – it doesn’t mean that it has any ‘inherent’ value – that it is of any use –

a lot of rubbish gets thrown up in propositional activity

‘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?’

do we put meaning into the husk of the calculus of relation – or rather load it up with irrelevance – and as a result obfuscation

let’s be clear ‘a 2-place relation’ – just is propositional form – if you like a grammatical explication – a structural proposal –

it is a proposal in the language game of grammar –

thought by those who take it seriously – as a way of ‘explaining’ language use –

and then there are those who think that the language of game of grammar – reveals the structure of thought – and is thus logic –

it is just this line of thinking that led to the development of formal languages – formal logic –

games within games – within games

‘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?’

the analytical symbols – the drawing – and indeed the cylinder –

are just different signs – different signs – that the game is played with –

and different signs means different contexts

same game – different signs – different settings

‘And does the word “love” in a declaration of love enter into my discussion of a 2-place relations?’

no – unless the declaration is given or received by a logician – in which case I would say – the point – however you might describe it – has been missed – badly –

and as a result – most likely – no love lost

‘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”.

the use of  ‘application’ – as with any word – is open to question – open to doubt – is uncertain –

the point being – it has no ‘inherent’ meaning – no ‘essential’ meaning –

its ‘meaning’ – in any context is open to question

as to the apples – the description ‘apples’ – is applied to the calculation –

‘apples’ – as it were – shadow the calculation –

that ‘apples’ refers to the fruit – is another matter –

the game is there to be played – in whatever descriptive context you wish –

however the descriptive context is irrelevant to the game – to its play –

the descriptive context – is nothing more than setting for the play

the ‘application’ of the game – if we are to continue with this notion of ‘application’ here –

just is the game played – the application is the play

‘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 when a context is applied to it

‘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.’

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 way I see it is that the calculus – is a language game – that we play –

there is no question of its reality – it’s real as a proposal – a proposition – and as an action – as a play –

that you might apply a semantics to it – a semantics to the syntax – is really neither here nor there –

such an application does not in any way touch the calculus –

any such application – is like a garment put on a skeleton

‘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 relation between a calculus and other propositional forms – is open to question – open to doubt – is uncertain

one puts up a proposal and argues for it

the ‘true’ position – or account – is the one you hold to for whatever reason –

yes – relative to one proposal – another will come off as ‘misleading’ –

but this is just rhetoric

‘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’ – will be simply an analysis of a proposal –

it will be an account of a language usage – in terms of a calculus

‘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

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.

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.’

‘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 –

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

‘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.’

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 it was not originally designed

‘hypothetical assumptions about the logical elements’ –

so called logical elements are no more than some 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 i.e. the history of pure mathematics –

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:

yes – I agree with Wittgenstein here –

and I don’t like Ramsey’s view – I think it takes liberties with logic – it’s vain –

but that really is just my prejudice

in the end – these views – Wittgenstein’s view and my view – as with Ramsey’s – are  no more than proposals –

open to question – open to doubt – uncertain

the rational game – is propositional exploration –

and when it comes to propositional play –

by all means – play whatever game you like – but play with an open mind –

and if you do – I think you will find – true delight

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 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 here – simply giving x – an emphasis – a focus –

as a rhetorical devise – yes ‘identity’ is used – and I would say – works –

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

x = x. by definition – a tautology

x = y by definition – a 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.’

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 leaves ‘ =’ – undefined –

his analysis – even if you accept his identity sign – (“je”) – and the argument that flows from it – fails – misses by a country mile –

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 definition, 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.’

extension – takes 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 extension argument has any value in logic – in mathematics

what you have with 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 –

extension – 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 as practiced – is a rule governed practice of language –

the manipulation of signs – of syntax –

mathematics is a genuine language game

mathematics as conceived is an art –

it is the art of making rules – within rules –

this is a creative endeavour –

and as with any creative endeavour – propositional –

that is the putting of proposals

the proposition is a proposal –

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

is uncertain

any interpretation of the signs – of the rules – of the game – of mathematics – is open to question – open to doubt – is uncertain – is speculative

what is not speculative – is the play – the playing of the game

and that is the practice of mathematicians – the rule governed practice –

what mathematicians  do

as to the question of applicability –

whatever context the game is played in –

it is the game that is played

of course mathematics is played in context –

I would suspect – every context

the point being –

we don’t live – in the absence of context –

we don’t live in a void –

any use – of any language form – is in context –

in a world – of context –

or strictly speaking – worlds of context –

and by ‘context’ – I mean descriptive context –

propositional context(s)

‘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 that 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 origin of the geometrical game – the reason for it – if you like – is open to question –

that is to say – it is a matter of propositional speculation –

and that speculation is really irrelevant to the game itself

i.e. you can come at it from a question of practice – or a as a matter of theory –

the game itself is a propositional game – a language game –

Wittgenstein calls it – ‘the grammar of statements about spatial objects’ –

the ‘about spatial objects’ – only occurs in a propositional form – is a proposal –

the point being the language game of geometry – is played in a propositional context

e.g. we have a proposal to survey a piece of land – in response to that proposal – a  propositional game – the geometrical game is played –

the geometrical game stands on its own terms –

it’s use is a proposal – and any use of the game will be open to question – open to doubt – uncertain

i.e. – just exactly where in fact – in actuality – is the line proposed in the geometry etc?

measurement – of course is always open to question – to doubt – is uncertain –

in practice – decisions get made – to proceed –

and proceed – in the face of uncertainty

the geometrical game – it’s rules – it’s play –

are not in any affected by such decision –

the game as played is not open to question –

any proposal of ‘application’ is

‘The relation between two colours isn’t like the relation between something fine and something course, but the relation between possibility and actuality. (Think of the notion of possibility as a shadow of actuality.)’

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

‘possibility as a shadow of actuality’

possibility is uncertainty – and that is the reality – the logical reality –

actuality – a decision –

open to question – open to doubt –

uncertain

‘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

proposing the same game in a different context –

is no more than the proposal that the playing of that game will not be useful –

it is as open – as indeterminate as that

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

(‘A proposition based on a wrong calculation (such as ‘he cut a 3 metre board into 4 one metre parts” 0 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 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

© greg t. charlton. 2015.