NOTICE: Wittgenstein’s Philosophical Grammar

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

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

'Wittgenstein's Philosophical Grammar'

by Greg T. Charlton.

ISBN: 978-0-9586687-0-5

Killer Press

This work also published in the blog:

wittgensteinsphilosophicalgrammar.blogspot.com

(c) greg t. charlton. 2016

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

VII  INFINITY IN MATHEMATICS THE EXTENSIONAL VIEWPOINT

39 Generality in arithmetic

‘ “What is the sense of such a proposition as ‘($n). 3 + n = 7’?” Here we are in an old difficulty: on the one hand we feel it to be a problem that the proposition has the choice between infinitely many values of n, and on the other hand the sense of a proposition seems guaranteed in itself and only needing further research on our part, because after all we all “know ‘what ‘($x) jx’ means”. If someone said he didn’t know what was the sense of ‘($n). 3 + n = 7’, he would be answered “ but you do know what this proposition says: 3 + 0 = 7 .v. 3 + 1 = 7. v. 3 + 2 = 7 and so on!” But to that one can reply “Quite correct – so the proposition isn’t a logical sum, because a logical sum doesn’t end with ‘and so on’. What I am not clear about is this propositional form ‘j(o) v j(1) v j (2) v and so on’ – and all you have done is to substitute a second unintelligible proposition for the first one, while pretending to give me something familiar, namely a disjunction.” ’

‘($n). 3 + n = 7’ – is a game

the question – what value to give n – if the result is to be 7? –

so yes – there is a ‘choice’ –

and if the proposition is to be functional – if the game is to be playable – a choice must be made – a calculation made

you could say ‘($n). 3 + n = 7’ is game that has a fixed external form – and an indeterminate internality

or that with ‘($n). 3 + n = 7’ – there is a game within the game – and that the game within is logically of a different type to the game without –

i.e. the game within is a game of infinitely many values

we are not in an ‘old difficulty’ –

it is simply a matter of seeing that there are different propositional games –

and further that games can be and are played within games

‘That is, if we do believe that we do understand “($n) etc.” in some absolute sense, we have in mind as a justification other uses of the notation “($n …) …”, or of the ordinary language expression “There is …” But to that one can only say: So you are comparing the proposition “($n) …” with the proposition “There is a house in this city which …” or “There are two foreign words on this page”. But the occurrence of the words “there is” in those sentences doesn’t suffice to determine the grammar of the generalization, all it does is indicate a certain analogy in the rules. And so we can still investigate the grammar of the generalization “($n) etc.” with an open mind, that is without letting the meaning of “($ …) …” in other cases get in our way.’

yes – of course

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

different games –

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

‘ “Perhaps all numbers have the property e”. Again the question is: what is the grammar of this general proposition? Our being acquainted with the use of the expression “all …” in other grammatical systems is not enough. If we say “you do know what it means: it means e(0). e(1). e(2) and so on”, again nothing is explained except that the proposition is not a logical product. In order to understand the grammar of the proposition we ask: how is the proposition used? What is regarded as the criterion of truth? What is its verification? – If there is no method provided for deciding whether the proposition is true or false, then it is pointless, and that means senseless. But then we delude ourselves that there is indeed a method of verification, a method that cannot be employed, but only because of human weakness. This verification consists in checking all the (infinitely many) terms of the product

e(0). e(1). e(2) … Here there is confusion between physical impossibility and what is called “logical impossibility”. For we think we have given sense to the expression “checking of the infinite product” because we take the expression “infinitely many” for the designation of an enormously large number. And when we hear of “the impossibility of checking the infinite number of propositions” there comes before our mind the impossibility of checking a very large number of propositions, say when we don’t have sufficient time.’

‘perhaps all numbers have the property e’?

yes ‘perhaps’ – and there is nothing out of place with ‘perhaps’ – with such a proposal – such a speculation – and yes you can run with it – or not –

it’s a proposal

what is the grammar of this general proposition?

the ‘grammar’ will be the theory of its use – whatever that theory is

‘Our being acquainted with the use of the expression “all …” in other grammatical systems is not enough’?

of course – different systems – different uses = different grammars

and remember just what the use /grammar of ‘e(0). e(1). e(2) and so on’ is – will never – logically speaking – be set in stone –

any theory of use –  as with any theory – any proposal – is open to question – open to doubt – is uncertain

this is the case even when there is in fact a stable practice –

that is – a particular view of the usage is adopted and regarded as uncontroversial

‘In order to understand the grammar of the proposition we ask: how is the proposition used?

exactly – and there are any number of answers to this question

‘What is regarded as the criterion of truth? What is its verification? – If there is no method provided for deciding whether the proposition is true or false, then it is pointless, and that means senseless.’

the criterion of truth? –

whatever it is decided to be –  by those engaged in the propositional action

its verification? – the same

if there is no method? –

if there is no method – there is no use – effectively – no functional proposition –

‘But then we delude ourselves that there is indeed a method of verification, a method that cannot be employed, but only because of human weakness’

look – the ‘method’ – whatever that amounts to – must enable use – if it doesn’t – then there is no use –

it is not a matter of ‘human weakness’ at all –

either the method of verification – whatever that comes  to – facilitates the propositional action or it doesn’t

we don’t run with a method of verification that brings the propositional action to a halt –

or if we do – what that means – is quite simply we have no use for that propositional action

you either affirm the propositional action or you don’t

verification – in whatever form it takes – is – affirmation

‘checking of the infinite product’?

there is no ‘checking’ as such – there is simply the propositional action –

and whatever account is given of it –

and to be frank – any such account is no more than a restatement of the propositional action  

‘we don’t have sufficient time’? –

if we did have sufficient time we would be dealing with a different logic – a different grammar – a different propositional action

the very point of the infinity game – is that it doesn’t come to an end in time –

even though the playing – and indeed – the players – do  

‘Remember that in the sense in which it is impossible to check an infinite number of propositions it is also impossible to try to do so. – If we are using the words “But you do know what it ‘all’ means to appeal to the cases in which this mode of speech is used, we cannot regard it as a matter of indifference if we observe a distinction between these cases and the case for which the use of the words is to be explained. – Of course we know what it means by “checking a number of propositions for correctness”, and it is this understanding that we are appealing to when we claim that one should understand also the expression “ …infinitely many propositions”. But doesn’t the sense of the first expression depend on the specific experiences that correspond to it? And these experiences are lacking in the employment (the calculus) of the second expression; if any experiences at all are correlated to it they are fundamentally different ones.’

‘checking a number of propositions for correctness’? –

the rules of the game determine the play of the game – the action of the propositions –

if the game does not play – then there is no game –

so called ‘checking for correctness’ is quite simply – playing the game

in the case of a game with ‘infinitely many propositions’ –  you play the game –

and this is of course to say – you understand that the game is on-going –

even if you are not

experiences?

the experience of the game – is the play of the game –

regardless of what kind of game it is –

different games – different plays –

‘different experiences’ –

if that’s how you want to put it

‘Ramsey once proposed to express the proposition that infinitely many objects satisfied a function f (x) by the denial of all propositions like

~($x) . fx

($x) . fx. ~($x, y) . fx . fy

($x, y) . fx . fy. ~($x, y, z) . fx. fy. fz

and so on

But this denial would yield the series

($x) . fx

($x, y) . fx . fy.

($x, y, z) …, etc., etc.

But this series is quite superfluous: for in the first place the last proposition at any point surely contains all the previous ones, and secondly even it is of no use to us, because it isn’t about an infinite number of objects. So in reality the series boils down to the proposition:

“($x, y, z) … ad infin.) . fx . fy . fz … ad infin.”

and we can’t make anything of that sign unless we know its grammar. But one thing is clear: what we are dealing with isn’t a sign of the form “($x, y, z) . fx . fy . fz” but a sign whose similarity to that looks deceptive.’

what we have here is different propositional games –

and yes – one is not the other

‘I can certainly define “m > n” as ($x): m – n = x, but in doing so I haven’t in any way analysed it. You think, that by using the symbolism “($ …) …” you establish a connection between “m > n” and other propositions of the form “there is …”; what you forget is that that can’t do more than stress a certain analogy, because the sign

“($ …) …” is used in countlessly many different ‘games’. (Just as there is a ‘king’ in chess and draughts.) So we have to know the rules governing its use here; and as soon as we do that it immediately becomes clear that these rules are connected with the rules for subtraction. For if we ask the usual question “how do I know – i.e. where do I get it from – that there is a number x that satisfies the condition m – n = x? it is the rules of subtraction that provide the answer. And then we see that we haven’t gained very much by our definition. Indeed we might just as well have given the explanation of ‘m > n’ the rules for checking a proposition of that kind – e.g. ‘32 > 17’.’

‘I can certainly define “m > n” as ($x): m – n = x – but in doing so I haven’t in any way analysed it’ –

analysis is what? –

effectively – in the end restatement in different terms – likely long-winded

as to m > n and ($x): m – n = x –

we have two proposals – and if the idea is that they are interchangeable –

this proposal – this relational proposal will be up for argument –

i.e. – under what circumstances and why?

and it could just be that the second can only function in place of the first – if  added to the argument are a number of qualifications

you could end up asking well why – why bother with the second – where’s the advantage?

it’s not as it wins in the simplicity or elegance stakes

the answer will be that the second represents the first (if indeed it does) in a different context –

in a different game

translation here – from one to the other – is – logically speaking – open to question – open to doubt – is uncertain –

but that’s translation

at base the issue is agreement – and that can be as complex as you want to make it

however if the move is made – the interchange is agreed to – for whatever reason –

that is – in the end – the only argument

‘If I say: “given any n there is a d for which the function is less than n”, I am ipso facto referring to a general arithmetical criterion that indicates when F(d) > n.’

yes – ‘F(d) > n’ represents a practice

‘If in the nature of the case I cannot write down a number independently of a number system, that must be reflected in the general treatment of number. A number system is not something inferior – like a Russian abacus – that is only of interest to elementary schools while a more lofty general discussion can afford to disregard it.’

a number system is – more generally speaking – a propositional game –

if you like – a meta-game

writing down a ‘number’ – presupposes a meta background –

if you don’t presuppose a meta background when you write down a number – all you do is make a mark on a piece of paper –

why would you do that?

‘Again, I don’t lose anything of the generality of my account if I give the rules that determine the correctness and incorrectness (and thus sense) of ‘m > n’ for a particular system like the decimal system. After all I need a system, and the generality is preserved by giving the rules according to which one system can be translated into another.’

yes

‘A proof in mathematics is general if it is generally applicable. You can’t demand some other kind of generality in the name of rigour. Every proof rests on particular signs, produced on a particular occasion. All that can happen is that one type of generality may appear more elegant than another. ((Cf. the employment of the decimal system in proofs concerning d and h)’

‘A proof in mathematics is general if it is generally applicable’

this tells us nothing –

we have rules and we have propositional games –

this what we work with

the rules apply where they apply

the games are played where they are played –

the rules don’t apply where they don’t apply –

and the propositional games are not played where they are not played –

if there is an issue here at all – it is application

can this game be played in this propositional context?

and any decision about application –

is open to question – open to doubt – is uncertain

‘generality’ here strikes me as  a notion that has no functional value

‘All that can happen is that one game can appear more elegant than another’

and elegance?

in the eye of the beholder

‘We may imagine a mathematical proposition as a creature which itself knows whether it is true or false (in contrast with propositions of experience).

A mathematical proposition itself knows that it is true or that it is false. If it is about all numbers, it must also survey all the numbers. “Its truth or falsity must be contained in it as is its sense.” ’

a true proposition is a proposition that is assented to –

affirmed – for whatever reason

a false proposition is a proposition dissented from –

denied – for whatever reason

a true proposition is proceeded with –

a false proposition is not

a proposition – a proposal –  about all numbers – if it is to have any sense – is a game 

a game-proposition

and a game-proposition is a rule governed propositional action

a properly constructed game – can be played – or not

if played – it is affirmed –

if it is not affirmed – it is not played

as for this idea of a proposition being self-aware – and aware of its truth or falsity –

psychologism – yes

Wittgenstein has no account of propositional truth

and to cover this failure he resorts to pretention and fantasy

a proposition – is a proposal – a proposal of a human being –

a sign of a human being –

so the appropriate question here is – does a human being know that its signs are true or false?

the answer is that a proposal – a proposition a sign – is open to question – open to doubt – is uncertain

propositions – of whatever kind – are uncertain

human knowledge is uncertain –

truth is assent – falsity – dissent

assent and dissent – are open to question – open to doubt –

are uncertain

‘ “It’s as though the generality of a proposition like ‘(n). e(n)’ were only a pointer to the genuine, actual , mathematical generality, and not the generality itself. As if the proposition formed a sign in a purely external way and you still needed to give to the sign a sense from within.” ’

‘the generality itself’?

as I see it the issue is application –

where and how does a proposition apply?

and any answer to this will be open to question – open to doubt –

will be uncertain

‘As if the proposition formed a sign in a purely external way and you still needed to give to the sign a sense from within’ –

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

is open to interpretation

“We feel the generality possessed by the mathematical assertion to be different from the generality of the proposition proved.”

the mathematical assertion is a propositional game

‘the proposition proved’ is the mathematical assertion – restated

“We could say: a mathematical proposition is an allusion to a proof’

a mathematical proposition is a propositional game –

a ‘proof’ – is a restatement –

a proof refers to the proposition –

without the proposition – there will be no restatement –

nothing to restate

to ‘allude’ is to ‘refer indirectly’ –

a proof refers directly – directly to the mathematical proposition –

there is no allusion

‘What would it be like if a proposition itself did not quite grasp its sense? As if it were, so to speak, too grand for itself? That is what logicians suppose.’

‘grasping’ a proposition – is interpreting it –

a proposition does not grasp itself –  does not interpret itself –

to suggest that it does is ridiculous

a proposition is interpreted by an interpreter – an actor – a human being

an interpretation is an action upon –

an action upon a proposition

the proposition – in the absence of interpretation – is an unknown –

is an un-interpreted sign 

the sense of a proposition is what is proposed –

in terms of its meaning – its function

and a proposition’s sense is open to question – open to doubt – is uncertain –

do logicians know what they suppose?

whatever it is they suppose – their suppositions are open to question – open to doubt – are uncertain

grandiosity is pretension –

the true task of the logician is to expose pretension

‘A proposition that deals with all numbers cannot be thought of as verified by an endless striding, for, if the striding is endless, it does not lead to any goal.

Imagine an infinitely long row of trees, and, so that we can expect them, a path beside them. All right, the path must be endless. But if it is endless, then that means precisely that you can’t walk to the end of it. That is, it does not put me in a position to survey the row. That is to say, the endless path does not have an end ‘infinitely faraway’, it has no end.’

a proposition that deals with all numbers is a proposal – a game proposition–

a game is played – it is not verified

the rules of the game determine the game – determine the play –

if the striding is endless – the point of the game is that it is on-going –

that is to say – there is no end point

as to the row of trees –  as with ‘a proposition that deals with all numbers’ –

yes – it is not a question of observation

“Nor can you say: “A proposition cannot deal with all the numbers one by one, so it has to deal with them by means of the concept of number” as if this were a pis aller: “Because we can’t do it like this, we have to do it another way.” But it is indeed possible to deal with numbers one by one, only that doesn’t lead to the totality. That doesn’t lie on the path on which we go step by step, not even at the infinitely distant end of that path. (This all only means that “e(0). e(1). e(2) and so on” is not the sign for a logical product.)’

yes – exactly there is no logical product in such a propositional game

and it is just this that defines any game that ‘deals with all numbers’

‘ “It cannot be a contingent matter that all numbers possess a property; if they do so it must be essential to them.” – The proposition “men who have red noses are good-natured” does not have the same sense as the proposition “men who drink red wine are good natured” even if the men who have red noses are the same as the men who drink red wine. On the other hand, if the numbers m, n, o are the extension of a mathematical concept, so that it is the case that fm. fn. fo, then the proposition that the numbers satisfy f have the property e has the same sense as “e(m). e(n). e(o)”. This is because the proposition f(m). f(n). f(o)” and e(m). e(n). e(o)” can be transformed into each other without leaving the realm of grammar.’

we can forget this talk of numbers possessing a property –

the issue is the use of numbers – the games numbers are used in –

the sense of a proposition is a matter – open to question – open to doubt – uncertain

transforming f(m). f(n). f(o)” and e(m). e(n). e(o)” into each other –

is playing a game – a grammatical game

‘Now consider the proposition: “all the n numbers that satisfy the condition F(x) happen by chance to have the property e”. Here what matters is whether the condition

F(x) is a mathematical one. If it is, then I can indeed derive e(x) from f(x), if only via the disjunction of the n values of F(x). (For what we have in this case is in fact a disjunction). So I won’t call this chance. – On the other hand if the condition is a non-mathematical one, we can speak of chance. For example, if I say: all numbers I saw today on buses happened to be prime numbers. (But of course we can’t say “the numbers17, 3, 5, 31 happen to be prime numbers” any more than “the number 3 happens to be a prime number”), “By chance” is indeed the opposite of “in accordance with a general rule”, but however odd it sounds one can say that the proposition “17, 3, 5, 31 are prime numbers is derivable by a general rule just like the proposition 2 + 3 = 5.”

well is it not chance that a general rule is applied?

that I describe the numbers on the bus as prime numbers – interpret them in terms of the prime rule – is that not a chance use of the prime rule?

the description of any proposition – mathematical or not – is open to question – open to doubt – is uncertain

propositional interpretation is uncertain –

whether you chance it with rules – or not

‘If we now return to the first proposition, we may ask again: How is the proposition “all numbers have the property e” supposed to be meant? How is one supposed to be able to know? For to settle its sense you must settle that too! The expression “by chance” indicates a verification by successive tests, and that is contradicted by the fact that we are not speaking of a finite series of numbers.’

how is the proposition to be meant?

logically speaking – there is no – ‘to be meant’ –

there are propositional practices and traditions yes –

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

is uncertain

how a proposition is meant – is how it is used –

and any use is open to question

how is one supposed to be able to know?

knowledge – is proposal –

and any proposal – is uncertain –

our knowledge is uncertain

we operate with and in uncertainty –

we proceed in uncertainty

as to verification –

in mathematics – we are dealing with number games –

we don’t verify games – we play them

whether they have a logical or they are on-going

‘In mathematics description and object are equivalent. “The fifth number of the number series has these properties” says the same as “ 5 has these properties”. The properties of a house do not follow from its position in a row of houses; but the properties of a number are the properties of a position.’

well it is a question of description – how you describe the object –

in the absence of description – the object –  number or house – or whatever – is an unknown

the object does not have properties independent of description –

a properties-description – is logically speaking – as good as any other description –

which is to say – it is open to question – open to doubt – is – as with any proposal –

uncertain –

and what follows from what – is the art of argument

‘You might say that the properties of a particular number cannot be foreseen. You can only see them when you’ve got there.

What is general is the repetition of an operation. Each stage of the repetition has its own individuality. But it isn’t as if I use the operation to move from one individual to another so that the operation would be the means for getting from one to the other –

like a vehicle stopping at every number which we can then study: no, applying the operation +1 three times yields and is the number 3.

(In the calculus process and result are equivalent to each other.)

But before deciding to speak of “all individualities” or “the totality of these individualities” I had to consider carefully what stipulations I wanted to make here for the use of the expressions “all” and “totality”.’

‘You might say that the properties of a particular number cannot be foreseen. You can only see them when you’ve got there’? –

the point is that the use of a particular proposal – proposition – in this case a particular number – is open to question – to doubt – is uncertain

if you understand this logical reality – then you will be open to the possibilities of that use – at any time of its use

the repetition of the operation – is the game-play

you play the game – the action of the game is repetition –

the game is a game of repetition

so yes –

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

now the use and function of ‘all’ and ‘totality’ – are of course – open to question

and this business  of determining use – is the on-going logical issue in any propositional context

‘It is difficult to extricate yourself completely from the extensional viewpoint: You keep thinking ‘Yes, but there must still be an internal relation between x³ + y³ and z³ since at least extensions of the expressions if I only knew them would have to show the result of such a relation”. Or perhaps: “It must surely be either essential to all numbers to have the property or not, even if I can’t know it.’

it is not as if numbers – through some ‘internal’ action or dynamic – determine their relation to each other –

to suggest so – is to peddle some kind of Platonic or Pythagorean rubbish

numbers are marks – signs – put into play in rule governed propositional actions –

their relations are determined by the rules governing the propositional action –

by the rules governing the game

“If I run the number series, I either eventually come to a number with the property e or never do.” The expression “to run though a number series” is nonsense; unless a sense is given to it which removes the suggested analogy with “ running through the numbers from 1 to 100”,’

a number series – finite or infinite – will only have any significance in terms of a game proposal

‘When Brower attacks the application of the law of the excluded middle in mathematics, he is right in so far as he is directing his attack against a process analogous to the proof of empirical propositions.  In mathematics you can never prove something like this: I saw two apples lying on the table, and now there is only one there, so A has eaten an apple. That is, you can’t by excluding certain possibilities prove a new one which isn’t already contained in the exclusion because of the rules we have laid down. To that extent there are no genuine alternatives in mathematics. If mathematics was the investigation of empirically given aggregates, one could use the exclusion of a part to describe what was not excluded, and in that case the non-excluded part would not be equivalent to the exclusion of the others

an empirical proposition – is open to question – open to doubt – is uncertain

mathematics is a rule governed propositional action –

a rule governed propositional action – is a game

the rules – as with any proposition or set of propositions – are logically speaking –

open to question –

if you question the rules – you involve yourself in argument – you don’t play the game

there is no question of proof in play

and there is no question of excluding possibilities in the play of a game – there is nothing to exclude

there is only the game and its play

‘The whole approach that if a proposition is valid for one region of mathematics it need not necessarily be valid for a second region as well, is quite out of place in mathematics, completely contrary to its essence. Although many authors hold just this approach to be particularly subtle and to combat prejudice.’

a proposition is valid if it functions in the propositional context in which it is placed

validity = function

and whether or not a proposition functions in a particular context – one way or another will be a decision for the practitioners

so of course a proposition that functions in one region of mathematics need not function in a second

from a logical point of view to suggest otherwise is what is out of place in mathematics – or for that matter any other rational propositional activity

any proposition – in any context – is open to question – open to doubt – is uncertain

if you turn on this logical reality –

you go down the road of rhetoric and prejudice

‘It is only if you investigate the relevant propositions and their proofs that you can recognize the nature of the generality of the propositions of mathematics that treat not of “all cardinal numbers” but e.g. of real numbers.’

‘investigation of the relevant propositions’ –

that is seeing then as open to interpretation – understanding them as creative possibilities –

and this can leads to proposals –

proposals for propositional games –

i.e. the cardinal numbers game – the real numbers game

‘How a proposition is verified is what it says. Compare generality in arithmetic with the generality of non-arithmetical propositions. It is differently verified and so is of a different kind. The verification is not a mere token of the truth, but determines the sense of the proposition. (Einstein: how a magnitude is measured is what it is.)’

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

how and why you accept a proposition is the question of verification

the generality of any proposition – is the range of application of that proposition

the range of application of one proposition or one type of proposition may well be different to range of another proposition or another type of proposition

verification is decision –

to accept a proposition is to decide how you will understand it

verification is a decision on sense

we could speak of the proposition independently of verification – but not for long

that is to say a proposition is put – it is open to question – and the key question is –

acceptance or rejection –

the basis on which we accept or reject a proposition – the argument we employ –

will determine how we understand the proposition

however any argument we put for acceptance or rejection  – will itself be open to question

we never leave uncertainty –

we can proceed with a proposition – i.e. affirm it – in spite of its uncertainty –

or we can further explore its uncertainty

finally –

might I put that the logical position in relation to the assessment of any proposition is the ‘un-excluded middle’ –

if –  the ‘un-excluded middle’ – is understood as – ‘uncertain’ –

‘true’ and ‘false’ – are operational – pragmatic – decisions

40 On set theory

‘A misleading picture: “The rational points lie close together on the number-line.

Is a space thinkable that contains all rational points, but not the irrational ones? Would this structure be too coarse for our space, since it would mean that we could only reach the irrational points approximately? Would it mean that our net was not fine enough? No. What we would lack would be the laws, not extensions.

Is this space thinkable that contains all rational points but not the irrational ones?

That only means: don’t the rational numbers set a precedent for the irrational ones?

No more than draughts set a precedent for chess.

There isn’t any gap left open by the rational numbers that is filled up the irrationals’

‘a misleading picture: “The rational points lie close together on the number-line’?

yes – a misleading picture –

what we have is a number game – not a number-line –

the ‘number-line’ introduces – the idea of space – and numbers in space –

when all we are really talking about is a rule governed propositional game

imagination – ‘spacial’ imagination – doesn’t help us here –

in fact it throws us off the track

‘is a space thinkable that contains all rational points, but not the irrational ones?’ –

this should read –

‘is a game thinkable that contains all rational numbers – but not the irrational ones?’

of course

is this game too coarse? – I wouldn’t think so

a game that contains only rational numbers – is relative to a game that contains rational and irrational – a different game –

our game not fine enough? –

no – this game is a game of rational numbers –

there is no other kind of number in this game –

it’s as ‘fine’ as it is

and as to laws – the real question is what game are you playing?

if the game is the game of rational numbers – the rules are in place –

if you are proposing a game that deals with rational and irrational numbers –

then a rule or rules that relates the two is required

‘that only means: don’t the rational numbers set a precedent for the irrational ones?’

does a number game set a precedent for a number game?

perhaps – but so what?

‘there isn’t any gap left open by the rational numbers that is filled up the irrationals’

if the game is rational numbers – then that’s it –

there are no ‘irrational’ numbers in it

‘We are surprised to find that “between the everywhere dense rational points”, there is still room for the irrationals. (What balderdash!) What does a construction like that show for Ö2 show? Does it show how there is yet room for this point in between all the rational points? It shows that the point yielded by this construction, is not rational – And what corresponds to this construction in arithmetic? A sort of number which manages after all to squeeze in between the rational numbers? A law that is not a law of the nature of a rational number.’

by way of analogy – you could ask here i.e. – is there room between physical objects – for spirit entities?

yes we could play such a game – but it is a different game to the physical object game

there is nothing against constructing new games – combining games –

but if you do that – recognize what you are doing – and don’t get reluctantly stuck in the old faithful –

however if the game you want to play is the old faithful – stick to it – play that game

it’s only about being clear in your head what you are on about –

and the getting on with it

‘The explanation of the Dedekind cut pretends to be clear when it says: there are 3 cases: either the class R has a first member and L no last number, etc. In fact two of these 3 cases cannot be imagined, unless the words “class”, “first member”, “the last member”, altogether change the everyday meanings they are supposed to have retained.

That is, if someone is dumbfounded by our talk of a class of points that lie to the right of a given point and have no beginning, and says: give us an example of such a class – we trot out the class of rational numbers; but that isn’t a class of points in the original sense.’

the Dedekind cut – is a good example of an argument to a new game – a new construction –

it can be seen as a response to – the number-line image – and issue

that is the apparent incompatibility or the difficulty of reconciling – placing – irrational numbers in a rational line up

the Dedekind cut is a clever argument –

and Wittgenstein is right – in this argument the goal posts get shifted – meanings get changed –

but that is point –

and without these changes there is no conceptual shift –

there is no new game

‘The point of intersection of two curves isn’t the common member of two classes of points, it’s the meeting of two laws. Unless, very misleadingly, we use the second form of expression, to define the first.’

the proposal – ‘The point of intersection of two curves’ – as with any proposal is open to account –

‘the common member of two classes of points’? –

‘the meeting of two laws’? –

different accounts – and with them – different arguments –

different propositional backgrounds

different propositional baggage –

that’s how it goes

‘After all I have already said, it may sound trivial if I now say that the mistake in the set-theoretic approach consists time and time again in treating laws and enumerations (lists) as essentially the same kind of thing and arranging them in parallel series so that one fills in the gaps left by another.’

the set theoretic approach – can be seen as treating laws and enumerations as essentially the same thing – and arranging them in parallel series so that one fills in the gaps left by another –

yes you can interpret the set-theoretic approach in this way –

and if you do – then you have – you are dealing with – a new relationship between laws and enumeration –

I can see the argument too – that the set theoretic approach provides a context for  laws and enumerations to function in a common setting

it’s a ‘mistake’ only if you don’t accept the set-theoretic approach –

or if you think a new relationship between laws and enumeration is not possible or acceptable

the set theoretic approach – is a new operational paradigm – a new game –

a different proposal with different possibilities

if you don’t see the value in this approach – in this game  –

then presumably you will not utilize it – you will not play the game

it is as simple as that

‘The symbol for a class is a list.’

well –if you say so –

I think the point is that the concept of the class is not exhausted by a list –

this ‘the symbol of a class is a list ‘ – is actually an argument to trash the idea of the class

‘Here again, the difficulty arises from the formation of mathematical pseudo-concepts. For instance, when we say that we can arrange the cardinal numbers, but not the rational numbers, in a series according to their size, we are unconsciously presupposing that the concept of an ordering by size does have a sense for rational numbers, and that it turned out on an investigation that the ordering was impossible (which presupposes that the attempt is unthinkable). – Thus one thinks that it is possible to attempt to arrange the real numbers (as if that were a concept of the same kind as ‘apple on this table’) in a series, and now it turned out to be impracticable.’

this has to do with game construction – what works – what doesn’t

as to ‘pseudo’ – that will depend on where you are coming from and where your allegiances and prejudices lie –

much wiser to keep an open mind – and not to bunker down

any proposal – in or out of mathematics is open to question – to doubt – is uncertain –

the trick is to find a way through the maze of argument to a functional product – a game that that can be played – that finds acceptance –

and one that  enables you to do something that wasn’t done before – or not done in the same way

any such result – any proposed construction – will be open to question

‘For its form of expression the calculus of sets relies as far as possible on the form of expression of the calculus of cardinal numbers. In some ways that is instructive, since it indicates certain formal similarities, but it is also misleading, like calling something a knife that has neither blade nor handle (Lichtenberg.)’

‘formal similarities’ – a matter of description –

the real issue here is whether set theory takes us further than the cardinal number calculus if there is a question of which approach to take

you have to bear in mind that with set theory – relative to earlier or other  mathematical proposals you have a conceptual paradigm shift –

now really – as with any paradigm shift – you either get with it – or not –

there is no necessity here – just the option

and yes – there is always argument for and against –

but looking at and considering different propositional models –

and being open to making the jump to a different propositional framework

is – if nothing else – logical behaviour

‘(The only point there can be to elegance in a mathematical proof is to reveal certain analogies in a particularly striking manner, when that is what is wanted; otherwise it is a product of stupidity and its only effect is to obscure what ought to be clear and manifest. The stupid pursuit of elegance is a principle cause of the mathematicians’ failure to understand their own operations; or perhaps the lack of understanding and the pursuit of elegance have a common origin.)’

‘what ought to be clear and manifest’ –

‘clarity’ – is pretence –  and as with all pretence it has its uses

any proposal – or any result of any propositional action or exercise – is – logically speaking – open to question – open to doubt – is uncertain –

so what is ‘clear’ – comes out as what is not a subject of question or doubt –

in a word – clarity – is not logical – it is rhetorical –

and again I make the point – this is not to say that such pretence – such rhetoric doesn’t have a  place and function in propositional realities –

in mathematics i.e. it can have a prominent role

‘manifest’? –

is what is described and given descriptive prominence –

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

and this ‘giving prominence’ to a description – giving a description some authority –is a rhetorical move

as for elegance?

yes – elegance for elegance’s sake – has no value

however the real issue here is decision – and the criteria for decision –

and the logical reality here is that there is no absolute criterion or standard –

we tend to run with those proposals for criteria that are in play –

and that have become entrenched in the practice

a careful survey of the practice shows that in any decision there are options for how we go about deciding

elegance – in certain propositional traditions and endeavors – is one such option – one such criteria

if it has a use – it has a use –

but as with any proposal – open to question – open to doubt –

in the mix of uncertainty –

that is the decision process

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

so beautifully put

our reality is propositional –

our reality is open to question – open to doubt –

our reality is uncertain

“There is a point where the two curves intersect.” How do you know that? If you tell me, I will know what sort of sense the proposition “there is …” has.’

‘There is a point where the two curves intersect’ – is a proposal

our knowledge is what is proposed –

and what is proposed is open to question – open to doubt – is uncertain

‘there is …’

is a proposal –

what sense it has – is open to question –

sense is uncertain

‘If you know what the expression “the maximum of a curve” means, ask yourself: how does one find it? – If something is found in a different way it is a different thing. We define the maximum as the point on the curve higher than all the others, and from that we get the idea that it is only our human weakness that prevents us from sifting through the points of the curve one by one and selecting the highest of them. And this leads us to the idea that the highest point among a finite number of points is essentially the same as the highest point of a curve, and that we are simply finding out the same by two different methods, just as we find out in two different ways that there is no one in the next room; one way if the door is shut and we weren’t strong enough to open it, and another if we can get inside. But, as I said, it isn’t human weakness that is in question where the alleged description of the action “That we cannot perform” is senseless. Of course it does no harm, indeed it is very interesting, to see the analogy between the maximum of a curve and the maximum (in another sense) of a class of points, provided that the analogy doesn’t instill the prejudice that in each case we have fundamentally the same thing.’

how we define the maximum of a curve – is the method we employ –

different methods – different definitions

that there might be argument regarding which method to employ –

is no surprise

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

we proceed in this uncertainty – whatever approach we take –

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

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

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

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

one way or another – it is a pragmatic decision

‘It’s the same defect in our syntax which presents the geometric proposition “a length may be divided by a point into two parts” as a proposition of the same form as

“a length may be divided for ever”; so that it looks as if in both cases we can say “Let’s suppose the possible division to have been carried out”. “Divisible into two parts” and “infinitely divisible” have quite different grammars. We mistakenly treat the word “infinite” as if it were a number word, because in everyday speech both are given as answers to the question “how many?” ’

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

furthermore any term – or proposal – any proposition – even given its use histories – will be open – open to question – to interpretation –

language is functional uncertainty

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

“But after all the maximum is higher than other arbitrary points of the curve.” But the curve is not composed of points, it is a law that points obey, or again, a law according to which points can be constructed. If you now ask: “which points?” I can only say “well, for instance, the points P, Q, R, etc.” On the one hand we can’t give a number of points and say that they are all the points that lie on the curve, and on the other hand we can’t speak of a totality of points as something describable which although we humans cannot count them might be called the totality of all the points on the curve – a totality too big for human beings. On the one hand there is a law, and on the other points on a curve; – but not “all the points of the curve”. The maximum is higher than any point of the curve that happens to be constructed, but it isn’t higher than the totality of points, unless the criterion for that, and thus the sense of the assertion, is once again simply construction according to the law of the curve.’

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

and so – the ‘points of a curve’ – are irrelevant to the maximum of the curve

‘the points of a curve’ – is a description – an explanation of the curve – an account of the curve

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

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

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

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

‘Of course the web of errors in this region is a very complicated one. There is also e.g. the confusion between two different meanings of the word “kind”. We admit, that is, that the infinite numbers are a different kind of number from the finite ones, but then we misunderstand what the difference between different kinds amounts to in this case. We don’t realize, that is, that it’s not a matter of distinguishing between objects by their properties in the way we distinguish between red and yellow apples, but a matter of different logical forms. – Thus Dedekind tried to describe an infinite class by saying that it is a class which is similar to a proper subclass of itself. Here it looks as if he has given a property that a class must have in order to fall under the concept “infinite class” (Frege). Now let us consider how this definition is applied. I am to investigate in a particular case whether a class is finite or not, whether a certain row of tress, say, is finite or infinite. So, in accordance with the definition, I take a subclass of the row of trees and investigate whether it is similar (i.e. can be coordinated one to one) to the whole class! (Here already the whole thing has become laughable.) It hasn’t any meaning; for, if I take  “finite class” as a sub-class, the attempt to coordinate it one to one with the whole class must eo epso fail; and if I make the attempt with an infinite class – but already that is a piece of nonsense, for if it is infinite, I cannot, I cannot make an attempt to coordinate it. – What we call ‘correlation of all the members of a class with others’ in the case of a finite class is something quite different from what we, e.g., call a correlation of all cardinal numbers with all rational numbers. The two correlations, or what one means by these words in the two cases, belong to different logical types. An infinite class is not a class which contains more members than a finite one, in the ordinary sense of the word “more”. If we say that an infinite number is greater than a finite one, that doesn’t make the two comparable, because in that statement the word “greater” hasn’t the same meaning as it has say in the proposition 5 > 4!’

what we are dealing with is different games – different propositional games

Dedekind defines an infinite class as a class which is similar to a proper subclass of itself

ok – if the subclass is infinite –

and what is the point of such a ‘definition’?

defining  x as x – is hardly clever – it is just a statement of the obvious – and a waste of breath –

and saying x is not y – is blindingly obvious as well

and let’s be clear – this issue is games – propositional games –

games are played – and their play is their definition

furthermore –

the notion of class is irrelevant and pointless once it is understood that what we are dealing with is not different classes but rather different games or different types of game –

the ‘finite game’ and the ‘infinite game’

the idea or rule of the finite game is that it has a definitive logical end –

the idea or rule of the infinite game is that its logic is on-going

the terms or numbers in a finite game reflect the rule of the game and are reflected in the action of the game

and the terms or numbers in an infinite game reflect the rule of the game and are reflected in the action of that game

and yes ‘different’ means different – different games – different logics – different practices

and ‘different’ does not mean ‘comparable’ – it means ‘incomparable’ –

yes we can say that different games – are games – but that is really where any relevant or significant comparison ends

and again – to say this is only to state the obvious

‘That is to say, the definition pretends that whether a class is finite or infinite follows from the success or failure of the attempt to correlate a proper subclass with the whole class; whereas there just isn’t any such decision procedure – ‘Infinite class’ and ‘finite class’ are different logical categories; what can be significantly asserted of the one category cannot be significantly asserted of the other.’

yes – different logical categories – different logics – different games

‘With regard to finite classes the proposition that a class is not similar to its subclasses is not a truth but a tautology. It is the grammatical rules for the generality of the general implication in the proposition “k is a subclass of K” that contain what is said by the proposition that K is an infinite class.’

a finite class will be similar to its sub-classes if the criterion of similarity is type of class i.e. – finite class

where there is a difference in type of class – i.e. – the class is infinite – and the sub-class is finite – you have an internal contradiction –

a finite subclass in an infinite class – is a dead zone

a logical black hole – ready to implode –

in any case either a dysfunctional –  or at least a very odd game –

a game of contradicting logics

‘A proposition like “there is no last cardinal number” is offensive to naive – and correct – common sense. If I ask “Who was the last person in the procession?” and I am told “There wasn’t a last person” I don’t know what to think: what does “There wasn’t a last person” mean? Of course. if the question had been “Who was the standard bearer?” I would have understood the answer “There wasn’t a standard bearer”; and of course the bewildering answer is modeled on the answer of that kind. That is, we feel, correctly, that where we can speak at all of a last one, there can’t be “No last one”. But of course that means: The proposition “There isn’t a last one” should rather be: it makes no sense to speak of a “last cardinal number”, that expression is ill-formed.’

yes

if someone says ‘there is a last cardinal number’ – they don’t understand the cardinal number game

to say ‘There wasn’t a last person in the procession’ – is to fail to understand the finite numbers game

“Does the procession have an end?” might also mean: is the procession a compact group? And now someone might say: “There, you see, you can easily imagine a case of something not having an end; so why can’t there be other such cases?” – But the answer is: The “cases” in this sense of the word are grammatical cases, and it is they that determine the sense of the question. The question “Why can’t there be other such cases” is modeled on: “Why can’t there be other minerals that shine in the dark”; but the latter is about cases where a statement is true, the former about cases that determine sense.’

‘you can easily imagine a case of something not having an end’ –

yes – you can imagine this game – you can conceive its logic –

‘other cases’? – whatever the ‘the case’ or ‘other cases’ are –

it is the same game – the one game –

the game without an end

‘The form of expression “m =2n correlates a class with one of its proper subclasses” uses a misleading analogy to clothe a trivial sense in a paradoxical form. (And instead of being ashamed of this paradoxical form as something ridiculous, people plume themselves on a victory over all prejudices of the understanding). It is exactly as if one changed the rules of chess and said it had been shown that chess could also be played quite differently. Thus we first mistake the word “number” for a concept word like “apple”, then we talk of a “number of numbers” and we don’t see that in this expression we shouldn’t use the word “number” twice; and finally we regard it as a discovery that the number of the even numbers is equal to the number of the odd and even numbers.’

a game – is a rule governed propositional action –

the rules of a game are propositions – and as with any proposition – the rules of a game – are open to question – open to doubt – and are as propositions – proposals –

uncertain

so rules can be reinterpreted – can be changed

if you change the rules of chess and say it can be shown that chess could also be played quite differently –

the only question then would be – is this different game of chess – still to be called ‘chess’?

because there is no question that with different rules we have a different game

I think it is quite unlikely that we would call both games ‘chess’ –

but this is no argument against proposing a new game

and the same is true with number games –

there is nothing against proposing a new type of number game –

however any such proposal will be the subject of argument

and whether or not a new game is put to play–

will finally be a decision for the players

‘It is less misleading to say “m = 2n allows the possibility of correlating every time with another” than to say “m = 2n correlates all numbers with others”. But here too the grammar of the meaning of the expression “possibility of correlation” has to be learnt.’

this has to do with just how the game ‘m= 2n’ can be played –

in what domains it has application

the grammar of the meaning of the expression ‘possibility of correlation’ – is open to question – is open to doubt – is uncertain –

for a game to be – and to be played – rules – ‘rules of grammar’ – if you like – need to be set

‘(It’s almost unbelievable, the way in which a problem gets completely barricaded in by the misleading expressions which generation upon generation throw up for miles around it, so that it becomes virtually impossible to get at it.)’

there are ‘no misleading expressions’ –

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

but if you don’t see this – you may well get trapped with the generations upon generations –

who didn’t see it either

‘If two arrows point in the same direction, isn’t it in such a case absurd to call these directions equally long, because whatever lies in the direction of one arrow, lies in that of the other? – The generality of m = 2n is an arrow that points along the series generated by the operation. And you can even say that the arrow points to infinity; but does that mean there is something – infinity – at which it points, as at a thing? – It’s as though the arrow designates the possibility of a position in its direction. But the word “possibility” is misleading, since someone will say: let what is possible now become actual. And in thinking this we always think of a temporal process, and infer from the fact that mathematics has nothing to do with time, that in its case possibility is already actuality.

The “infinite series of cardinal numbers” or “the concept of cardinal number” is only such a possibility – as emerges clearly from the symbol “½o, x, x + 1 | ”. This symbol is itself an arrow with the “o” as its tail and the “x + 1” as its tip. It is possible to speak of things which lie in the direction of the arrow, but misleading or absurd to speak of all possible positions for things lying in the direction of the arrow itself. If a search light sends out light into infinite space it illuminates everything in its  direction, but you can’t say illuminates infinity.’

we use the term ‘long’ in connection with length – with measurement

the notion of ‘direction’ comes from geometry

if the arrow points to infinity – then infinity is no place

if mathematics has nothing to do with time – we can’t say possibility is already actuality

the symbol “½o, x, x + 1 | ” is best  understood as a game symbol

‘the ‘infinite series of cardinal numbers’ or ‘the concept of cardinal number’ –

are game terms

the  infinite game is not one that actually goes on forever

rather it is a game that whenever played – has no logical end point

possibility in a game is rule governed indeterminacy

actuality is play

‘It is always right to be extremely suspicious when proofs in mathematics are taken with greater generality than is warranted by the known application of the proof. This is always a case of the mistake that sees general concepts and particular cases in mathematics. In set theory we meet this suspect generality at every step.

One always feels like saying “let’s get down to brass tacks”.

These general considerations only make sense when we have a particular region of application in mind.

In mathematics there isn’t any such thing as a generalization whose application to particular cases is still unforeseeable. That’s why the general discussions of set theory (if they aren’t viewed as calculi) always sound like empty chatter, and why we are always astounded when we are shown an application for them. We feel that what is going on isn’t properly connected with real things.’

‘It is always right to be extremely suspicious when proofs in mathematics are taken with greater generality than is warranted by the known application of the proof’? –

proofs in mathematics – are arguments – are proposals –

any proposal is open to question to doubt – is uncertainty

and it is this uncertainty that gives a proposal scope beyond a practiced application

Wittgenstein here is not putting forward an open or logical perspective – rather he advances a closed or dogmatic point of view

it is not really a question of generality – or as he puts it – ‘suspect’ generality –

what it is about is understanding that – whatever the use a proposition is put to – whatever the practice – the proposition qua proposition – logically speaking is open

or if you want to put it in terms of ‘generality’ – generality is open

of course it can be limited to ‘known applications of the proof’

however if we only think in terms of known applications – there will be no prospect of any advance in mathematical thinking –

the real importance of the ‘known application’ to mathematical thinking is that it provides a starting point for speculation beyond what is accepted and practiced

taking a step beyond the known application – may just result in a new way of understanding – and indeed a new theory – a new game – a new calculus –

this is the realm of pure mathematics

‘This is always a case of the mistake that sees general concepts and particular cases in mathematics. In set theory we meet this suspect generality at every step.

there is no ‘mistake’ here – what we have is a different perspective – and set theory is – a different perspective

‘One always feels like saying “let’s get down to brass tacks”’ –

what I suspect this means is – ‘let’s go down well trodden paths’ –

and of course if that is how you want to proceed – why not?

‘These general considerations only make sense when we have a particular region of application in mind.’

having a particular region of application in mind – does not exhaust the possibilities of a ‘general consideration’

‘a general consideration’ – is a proposal – open to question – open to doubt –

 uncertain

‘In mathematics there isn’t any such thing as a generalization whose application to particular cases is still unforeseeable. That’s why the general discussions of set theory (if they aren’t viewed as calculi) always sound like empty chatter, and why we are always astounded when we are shown an application for them. We feel that what is going on isn’t properly connected with real things.’

the particular case will be represented as a result of the generalization – if the generalization is applied to it

the same is true in set theory

‘We feel that what is going on isn’t properly connected with real things’? –

‘real things’ – is what is proposed –

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

draw your own conclusion

‘The distinction between the general truth that one can know, and the particular one doesn’t know, or between the known description of the object, and the object itself that one hasn’t seen, is another example of something that has been taken over into logic from the physical description of the world. And that too is where we get the idea that our reason can recognize questions but not the answers.’

what we ‘know’ is what is proposed – however you describe the proposal – i.e. ‘general’ – ‘particular’

in the absence of description – the object of knowledge – is unknown –

description – proposal makes known –

a proposal – be it in ‘logic’ – or ‘the physical sciences’ – is open to question – open to doubt – is uncertain

‘Set theory attempts to grasp the infinite at a more general level than the investigation of the laws of real numbers. It says that you can’t grasp the actual infinite by means of mathematical symbolism at all and therefore it can only be described and not represented. The description would encompass it in something like the way in which you carry a number of things that you can’t hold in your hand by packing them in a box. They are then invisible but we still know we are carrying them (so to speak, indirectly). One might say of this theory that it buys a pig in a poke. Let the infinite accommodate itself in this box as best it can.

With this there goes too the idea that we can use language to describe logical forms. In a description of this sort the structures are presented in a package and so it does look as if one could speak of a structure without reproducing it in the proposition itself. Concepts which are backed up like this may, to be sure, be used, but our signs derive their meaning from definitions which package the concepts in this way; and if we follow up these definitions, the structures are uncovered again.’

set theory is a different game to the game of real numbers

the ‘actual infinite’?

what is proposed is what is actual –

and what is proposed – is open to question – to doubt – is uncertain

what counts as ‘mathematical symbolism’ – will depend on what mathematical game you play

described but not represented? –

any description is a representation –

any description / representation – is a proposal

a theory that buys a pig in a poke? –

yes – a game – a true game – where the moves that are made – are not known –

before they are made –

where the domain of the game is an unknown

‘With this there goes too the idea that we can use language to describe logical forms’

what is a ‘logical form’ – but a description – a proposal?

‘In a description of this sort the structures are presented in a package and so it does look as if one could speak of a structure without reproducing it in the proposition itself’? –

this is a clumsy way of putting it but – if there is a structure – we have a proposal – we have a proposition

‘Concepts which are backed up like this may, to be sure, be used, but our signs derive their meaning from definitions which package the concepts in this way; and if we follow up these definitions, the structures are uncovered again.’ –

yes – but what this amounts to is proposals for – or regarding – proposals –

a sign – is a proposal – but as with any proposal it will have a propositional background – which may well be described in terms of concepts – definitions – structures –

or you can run it any way you like –

a concept resolved in terms of structures – which can be put as definitions and represented as a sign

a definition proposed as a structure – represented as a sign – accounted for conceptually – etc. –

‘When ‘all apples’ are spoken of, it isn’t, so to speak, any concern of logic how many apples there are. With numbers it is different; logic is responsible for each and every one of them.

Mathematics consists entirely of calculations.

In mathematics everything is algorithm and nothing is meaning; even when it doesn’t look like that because we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm.

In set theory what is calculus must be separated off from what attempts to be (and of course cannot be) theory. The rules of the game have to be separated off from inessential statements about the chessmen.’

how are you to be responsible for all your children – if don’t know how many you have?

look – it is true that ‘how many’ is irrelevant to ‘all’

the ‘how many game’ is a different game altogether to the ‘all game’

‘responsibility’ is a misplaced notion here

calculation is the play of the game –

there is more to the game than its play – if there wasn’t there would be no play

the ‘algorithm’ – is a precisely described procedure that is applied and can be systematically followed through to a conclusion –

so rather than saying ‘nothing is meaning’ – I would put it that the algorithm – that is the game-play – just is the meaning – and all it comes to –

as to ‘mathematical things’ – there is always a place for imagination – for visualization – in language use

a calculus is the rule of play –

the play only occurs in a game –

theory – is the game proposed

‘chessman’ – are nothing – without chess

‘In Cantor’s alleged definition of “greater”, “smaller”, “+”, “-” Frege replaced the signs with new words to show the definition wasn’t really a definition. Similarly in the whole of mathematics one might replace the usual words, especially the word ‘infinite’ and its cognates, with entirely new and hitherto meaningless expressions so as to see what the calculus with these signs really achieves and what it fails to achieve. If the idea was widespread that chess gave us information about kings and castles, I would propose to give the pieces new shapes and different names, so as to demonstrate that everything belonging to chess has to be contained in its rules.’

proposing a system – a new game – will inevitably begin with what is in practice –

and presumably such a proposal will come out of a view that the standard practice is deficient in some respect –

or possibly be a consequence of an entirely new perspective – ‘out of left field’ – so to speak

and for a new game-proposal to gain support it must be argued

part of this process may well be replacing terms in standard practice with new words

and with the idea of seeing ‘what the calculus with the new terms achieves and fails to achieve’ –

but here we are not simply ‘replacing words’ for the sake of it –

new terms will foreshadow – hint at – or introduce new ways of thinking –

all in all a very messy – uncertain business

if there is a result that gains acceptance among practitioners – then what you will have is a new game

a new and different game

‘If the idea was widespread that chess gave us information about kings and castles, I would propose to give the pieces new shapes and different names, so as to demonstrate that everything belonging to chess has to be contained in its rules.’

chess is chess – yes

if a new and different game emerges out of chess –

it is not chess

‘What a geometrical proposition means, what kind of generality it has, is something that must show itself when we see how it is applied. For even if someone succeeded in meaning something intangible by it it wouldn’t help him, because he can only apply it in a way which is quite open and intelligible to every one.

Similarly, if someone imagined the chess king as something mystical it wouldn’t worry us since he can only move him on the 8 x 8 squares of the chess board.’

the proposition – the proposal – ‘geometrical’ – or not – is open – open to question – open to doubt – is – uncertain –

how the proposition is interpreted – is open to question

‘something intangible by it’? –

intangible relative to what?

presumably a given – or the given interpretation

‘open and intelligible to everyone’

the idea here is that there is one interpretation – one perspective – and every one has it

from a logical point of view – this is no more than rhetoric – authoritarian rhetoric –

I mean really – how is anyone to know – what intelligibility is – amounts to – for everyone?

that there may be a common understanding in a particular culture at a particular time – is one thing –

that such is the only understanding – or the only possible understanding – is quite another

rhetoric may be the final bastion of most arguments – but that is really where logic begins

yes –imagining the chess king as something mystical – would make no difference in a game of chess –

if on the other hand – the ‘chess king’ piece – had been co-opted to another game – say a mystical game –

we would say the piece –formerly known as the ‘chess king’ had been re-interpreted

‘We have a feeling “There can’t be possibility and actuality in mathematics. It’s all on one level. And is in a sense, actual. – And that is correct. For mathematics is a calculus; and the calculus does not say of any sign that it is merely possible, but is concerned only with the signs with which it actually operates. (Compare the foundations of set theory with the assumption of a possible calculus with infinite signs).’

it is important to understand that when we play the game of mathematics – we play in terms of the signs of the game

as to the business of game construction – yes we come face to face with uncertainty –or – possibility – if you like

the foundations of any theory – are open to question – open to doubt – are uncertain

‘possible calculus with infinite signs’ – if it is to be anything more than speculation –

will have to be fashioned in a rule governed propositional system –

brought to heel – so to speak

‘When set theory appeals to the human impossibility of a direct symbolization of the infinite it brings in the crudest imaginable misinterpretation of its own calculus. It is of course this very misinterpretation that is responsible for the invention of the calculus. But of course that doesn’t show the calculus in itself to be something incorrect (it would be at worst uninteresting) and it is odd to believe that this part of mathematics is imperiled by any kind of philosophical (or mathematical) investigations. (As well say that chess might be imperiled by the discovery that wars between two armies do not follow the same course as battles on the chessboard.) What set theory has to lose is rather the atmosphere of clouds of thought surrounding the bare calculus, the suggestion of an underlying imaginary symbolism, a symbolism which isn’t employed in its calculus, the apparent description of which is really nonsense. (In mathematics anything can be imagined, except for a part of our calculus.)’

‘human impossibility’ –

yes – well whatever that is supposed to be – it is open to question – open  to doubt –

is indeed – uncertain

‘direct symbolization of the infinite’ –

all symbolization is direct

symbolization – of anything – is a proposal to make known

in mathematics what a symbol signifies is a propositional structure

where a symbol gains operational credence – it functions as a game – or a game within a game

‘the infinite’ – is a recursive sign game

no proposition is imperiled by any investigation

a calculus is the game as played

it could well be said that an ‘atmosphere of clouds of thought’ – envelopes – any proposition –

‘imaginary underlying symbolism’ – is speculation –

nothing to be afraid of

mathematics is an imagined propositional action

it will continue to be imagined –

and indeed – re-imagined

41 The extensional conception of the real numbers

Like the enigma of time for Augustine, the enigma of the continuum arises because language misleads us into applying to it a picture that doesn’t fit. Set theory preserves the inappropriate picture of something discontinuous, but makes statements about it that contradict the picture, under the impression that it is breaking with prejudices; whereas what should really have been done is to point out that the picture doesn’t fit, that it certainly can’t be stretched without being torn, and that instead of it one can use a new picture in certain respects similar to the old one’

if you are going to put forward a new proposal – it will only be taken seriously if placed it in contrast to an accepted or given outlook –

and the new perspective put in contrast to the alternative view – will have an argumentative context and a base from which to move on from –

and to move to – a new picture

‘The confusion in the concept of the “actual infinite” arises from the unclear concept of irrational number, that is from the fact that very different things are called “irrational numbers” without any clear limits being given to the concept. The illusion that we have a firm concept rests on our belief that in signs of the form “o. abcd … ad infinitum” we have a pattern to which they (the irrational numbers) have to conform whatever happens.’

‘the unclear concept of irrational number’

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

‘very different things are called irrational numbers’ –

ok – so then we need new terminology –

or it might be that ‘irrational numbers’ are used differently in different propositional games –

and in that case what needs to be distinguished is use

any ‘limits’ placed on the concept – will be a result of how it is used

placing limits on a concept is simply defining it for use –

and of course any such definition – will always be open to question

‘the illusion that we have a firm concept’ –

is really just the failure to understand the reality of propositional logic –

that any proposal is open to question

‘The illusion that we have a pattern to which they the irrational numbers conform to whatever happens’

yes – this is just dumb – dogmatic rhetoric –

and it betrays a real lack of understanding of logic and mathematics –

we operate in propositional uncertainty –

if we are to avoid getting stuck in self-satisfied ignorance – we must reflect this propositional reality in our thinking and action –

we need to be open minded and flexible –

with all our concepts – all our proposals

‘Suppose I cut a length at a place where there is no rational point (no rational number).” But can you do that? What sort of a length are you speaking of? “But if my measuring instruments were fine enough, at least I could approximate without limit to a certain point by continued bisection”! – No, for I could never tell whether my point was a point of this kind. All I could tell would always be that I hadn’t reached it. “But if I carry out the construction of Ö2 with absolutely exact drawing instruments, and then by bisection approximate to the point I get, I know that this process will never reach the constructed point.” But it would be odd if one construction could as it were prescribe something to the others in this way! And indeed that isn’t the way it is. It is very possible that the point I get by means of the ‘exact’ construction of Ö2 is reached by the bisection after say 100 steps; but in that case we could say: our space is not Euclidean.

‘Suppose I cut a length at a place where there is no rational point (no rational number)’

then the length cannot be measured in terms of rational points – end of story

if on the other hand an irrational system of measurement is used – the result will be irrational – indeterminate –

such a result put against the rational standard – will be regarded as – inadequate – if not a straight out failure

on the other hand the ‘irrationals’ can say that the rational system is simplistic – outdated and clunky –

can they say it is not as precise?

this is a tricky one – is an infinitesimal ‘precise’?

I think you would have to say that the irrational system challenges the rational notion of precision –

or as Wittgenstein says –

‘but in that case we could say: our space is not Euclidean’

‘The “cut at the rational point” is a picture, and a misleading picture’

we are dealing with different mathematical systems here – different mathematical  games

and it really is naïve to use the term ‘misleading’

‘cut at the rational point’ – may just be all you need –

meaning the rational system and perspective – suits your purposes –

if it doesn’t then you won’t use it

‘A cut is a principle of division into greater and smaller’

that’s fair enough – as far as it goes

‘Does a cut through a length determine in advance the results of all bisections meant to approach the point of the cut? No.’

the point of the cut is indeterminate

any approach to the point – is indeterminate –

and you are left wondering why talk about a point at all?

there is no point

so what is going on here?

clearly the notion of the point as a determination –

is a concept that comes from a rational framework

and that it is this framework that makes space for the irrational game –

that enables its extension

so how are we to see this relation –

does the irrational game undermine the rational game – and is it therefore illegitimate?

or is it that in effect we have a combining of the two games – into a larger game ?

a larger game that really cannot be assessed in terms of either game?

it is easy to understand the logical uneasiness here

we have a practice – but does it have a secure foundation?

I think if you are looking for secure foundation – the answer is – no – wherever you are looking

the foundation of practice – just is practice –

and theorists rush to cobble together a conceptual basis – for what happens – for what occurs –

and there is value in doing this –

in that it throws up arguments – and can lead to valuable insights that enrich the practice

nevertheless the hard reality is just what occurs – the practice

and in terms of explanation – logically speaking – we are in the realm of uncertainty

I can see the view that the irrational game is a breakaway – that leads nowhere

against this I would say that recognizing uncertainty – facing its reality – and exploring it – is logical –

and is in fact – the rational way to proceed in an uncertain reality

‘In the previous example in which I threw dice to guide me in the successive reduction of an interval by the bisection of a length I might just as well have thrown dice to guide me in the writing of a decimal. Thus the description “endless process of choosing between 1 and 0” does not determine a law in the writing of a decimal. Perhaps you feel like saying: the prescription for the endless choice between 0 and 1 in this case could be reproduced by a symbol like “o 000 … ad. infin.”. But if I

                                                                                    111

adumbrate a law thus ‘0.00100100 … ad infin.”, what I want to show is not the finite section of the series, but rather the kind of regularity to be perceived in it. But in

“o 000 … ad. infin.”. I don’t perceive any law, on the contrary, precisely that a law is

     111

absent.’

yes – and so the question is how to represent or present the game –

what propositional construction suits one’s philosophical perspective or argument?

it is just the choice of description

‘(What criterion is there for the irrational numbers being complete? Let us look at an irrational number: it runs through a series of rational approximations. When does it leave this series behind? Never. But then, the series also never comes to an end.

Suppose we had the totality of all irrational numbers with one exception. How would we feel the lack of this one? And – if it were to be added – how would it fill the gap? Suppose that it’s p. If an irrational number is given through the totality of its approximations, then up to any point taken at random there is a series coinciding with that of p. Admittedly for each such series there is a point where they diverge. But this point can lie arbitrarily far ‘out’, so that for any series agreeing with p I can find one agreeing with it still further. And so if I have the totality of all irrational numbers except p, and now I insert p I cannot cite a point at which p is now really needed. At every point it has a companion agreeing with it from the beginning on.

To the question “how would you feel the lack of p” our answer must be “if p were an extension, we would never feel the lack of it”. i.e. it would be impossible for us to observe a gap that it filled. But if someone asked us ‘But have you then an infinite decimal expansion with the figure m in the r-th place and in the n in the s-th place, etc? we could always oblige him.)

‘(What criterion is there for the irrational numbers being complete? Let us look us look at an irrational number: it runs through a series of rational approximations. When does it leave this series behind? Never. But then, the series also never comes to an end.’ –

yes – the ‘irrational number’ – just is an approximation to a rational number

the irrational number only has any sense – as an approximation to a rational number

imagine the absence of rational numbers – a world without rational numbers

a world without rational numbers – is a world without irrational numbers

here is the argument that irrational numbers are a subset of rational numbers –

is there a better way of putting it?

as for irrational numbers being complete? –

the essence – or rule – of the irrational number is that it is incomplete

and ‘incomplete’ here – means incomplete relative to a whole number – or a series of whole numbers

can we say that mathematics recognizes and deals with the complete and incomplete –

and in that sense covers the full range of logical constructions?

that there is no unifying theory must drive some logicians to distraction –

however the real value of mathematics rests just in the fact that it comprehensively represents the conceptual possibilities that we encounter in our reality – that we make of our reality –

we design and play games that are complete –

we design and play games that are incomplete

‘Suppose we had the totality of all irrational numbers with one exception’. How would we feel the lack of this one? And – if it were to be added – how would it fill the gap?’ –

in a game with all irrational numbers with one exception?

in a board game with an infinite numbers of squares – and there is one square you cannot fall on – the challenge I presume would be to avoid that square –

or conversely the challenge could be to find it

how would we feel the lack? –

there would be no ‘lacking’ – the exception – would be what defines the game –

the game is complete

‘And – if it were to be added – how would it fill the gap?’ –

if it were added – it would change the game –

we would then have the makings of another game

‘To the question “how would you feel the lack of p” our answer must be “if p were an extension, we would never feel the lack of it”. i.e. it would be impossible for us to observe a gap that it filled. But if someone asked us ‘But have you then an infinite decimal expansion with the figure m in the r-th place and in the n in the s-th place, etc? we could always oblige him.)’ –

what we have here is another game

‘ “The decimal fractions developed in accordance with a law still need supplementing by an infinite set of irregular infinite decimal fractions that would be “brushed under the carpet” if we were to restrict ourselves to those generated by a law.” Where is there such an infinite decimal that is generated by no law? And how would we notice that it was missing? Where is the gap it is needed to fill?’

is there a need for supplementing – or is any such ‘supplementing’ – really just a ‘filling out’ of a theoretical background?

there is no infinite decimal –generated by no law

if such is generated – we have a game proposal – a ‘law’

how would we notice that it is missing?

we would only notice that it was missing if it was required for the game –

and if it was required for the game – it would be there – in the game

the ‘game’ has no missing parts – no gaps –

(you can’t play a game with missing parts)

you have to be clear in your head what game you are playing – and what games you are not playing –

or what is relevant to the game you are playing – and what is not

it’s a question of focus

‘What is it like if someone so to speak checks the various laws for the construction of binary fractions by means of the set of finite combinations of the numeral 0 and 1? –

The results of a law run through the finite combinations and hence the laws are complete as far as their extensions are concerned, once all the finite combinations have been gone through.’

this is simply to apply a rule – and in so doing – make a game

‘If one says: two laws are identical in the case where they yield the same result at every stage, this looks like quite a general rule. But in reality the proposition has different senses depending on what is the criterion for their yielding the same result at every stage. (For of course there is no such thing as the supposed generally applicable method of infinite checking!) Thus under a mode of speaking derived from an analogy we conceal the most various meanings, and then believe that we have united the most various meanings into a single system.’

it is not that two laws / rules are identical in the case where they yield the same result – the two rules are different – but they yield the same result

if the laws are identical there is only one law

that two rules can yield the same result – just points to the logical reality that a result can be arrived at – through different methods – via different paths

it is not a matter of ‘concealing various meanings’ – any proposition / rule is open to  interpretation –

the various meanings are ‘united’ –if you want to use that term – in the fact that a proposition – a proposal is open to question – open to doubt – is uncertain

‘(The laws corresponding to the irrational numbers all belong to the same type to the extent that they must ultimately be recipes for the successive construction of decimal fractions. In a certain sense the common decimal notation gives rise to a common type.)

We could also put it thus: every point in a length can be approximated to by repeated bisection. There is no point that can only be approximated to by irrational steps of a specified type. Of course, that is only away of clothing in different words the explanation that by irrational numbers we mean endless decimal fractions; and that explanation in turn is only a rough explanation of the decimal notation, plus perhaps an indication that we distinguish between laws that yield recurring decimals and laws that don’t.’

I prefer the idea – that the point is a mark –

the notion that a point can only ever be approximated – strikes me as contradictory

or to but it bluntly – it’s either there – or it’s not

what sense in talking about approximating existence?

or are we happy talking about approximating non-existence –

approximating – nothing?

if a point – is a mark in a game – a given – a construct – yes we can have approximation –

we have a functional point of reference

the infinite game – as the ‘on-going’ game is one thing –

‘points’ as non-existent reference points – quite another

we don’t need this ridiculous fiction to make sense of repeated bi-sections or endless decimal fractions

repeated bi-section or endless decimal fractions are language games – propositional games –

that is to say – rule or law governed propositional actions

‘The incorrect idea of the word “infinite” and of the role of ‘infinite expansion” in the arithmetic of the real numbers gives us the false notion that there is a uniform notation for irrational numbers (the notation of the infinite extension, e.g. of infinite decimal fractions).

The proof that for every pair of cardinal numbers x and y (x)² ¹ 2 does not correlate                                                                                                                      

                                                                                              y                                                                                         

Ö2 with a single type of number – called “the irrational numbers”. It is not as if this type of number was constructed before I construct it; in other words, I don’t know any more about this new type of number than I tell myself.’

the representation of irrational numbers will be open to question –

and will be determined context by context – game by game

‘this type of number’ –

any ‘number’ is a proposal open to question –

and we assess the value of any new construction – in terms of its usefulness –

and on-going issue

42 Kinds of irrational numbers (p¢ P, F)

‘p¢ is a rule for the formation of decimal fractions: the expansion of p¢ is the same as        

     the expansion of p except where the sequence 777 occurs in the expansion of p; in                                                      that case instead of the sequence 777 there occurs the sequence 000. There is no                                                                          method known in our calculus of discovering where we encounter such a sequence                 in the expansion of p.

P   is a rule for the construction of binary fractions. At the nth place of the expansion         there occurs a 1 or a 0 according to whether n is prime or not.

F   is a rule for the construction of binary fractions. At the nth place there is a 0 unless      a triple x, y, z from the first 100 cardinal numbers satisfies the equation

     xⁿ + yⁿ = zⁿ.

I am tempted to say, the individual digits of the expansion (of p for example) are always only the results, the bark of the fully grown tree. What counts, or what something new can still grow from, is the inside of the trunk, where the tree’s vital energy is. Altering the surface doesn’t change the tree all. To change it, you have to penetrate the trunk which is still living.

p is a proposal –

if it is to function – it will function in a rule governed propositional context –

in a rule governed propositional action – p is a game –

and as such the individual digits of the expansion of p – are the play of the game

we do have and play ‘irrational games’

‘I call “p_n” the expansion of p up to the nth place. Then I can say: I understand what

p¢₁₀₀ means, but not what p¢ means, since p has no places, and I can’t substitute others for none. It would be different if I e.g. defined the division 5®3 as a rule for the

                                                                                                a/b

formation of decimals by division and replacements of every 5 in the quotient by a 3. In this case I am acquainted, for instance, with the number 5®3. – And if our calculus

                                                                                                1/7

contains a method, a law, to calculate the position of 777 in the expansion of p, then the law of p includes a mention of 777 and the law can be altered by the substitution of 000 for 777. But in that case p¢ isn’t the same as what I defined above; it has a different grammar to the one I supposed. In our calculus there is no question of

    =

p > p¢ or not, no such equation or inequality. p¢ is not compatible with p. And one can’t say “not yet compatible”, because if at some time I construct something similar to p¢ that is compatible with p, then for that very reason it will not be p¢. For p¢ like

p is a way of denoting a game, and I cannot say that draughts is not yet played with as many pieces as chess, on the grounds that it might develop into a game with 16 pieces. In that case it will no longer be what we call “draughts” (unless by this word I mean not a game, but a characteristic of several games or something similar; and this rider can be applied to p and p¢ too). But since being comparable with other numbers is a fundamental characteristic of a number, the question arises whether one is to call

p¢ a number, and a real number; but whatever it is called the essential thing is that p¢ is not a number in the same sense of p. I can also call an interval a point and so on occasion it may even be practical to do so; but does it become more like a point if I forget that I have used the word “point” with two different meanings?’

yes -  p¢₁₀₀ as distinct from p¢ is given definition by a rule – and is by the rule – made functional –

p¢₁₀₀ is a playable game

and yes – it would be different if a rule for the formation of decimals is given –

and the calculus contains a method – a law – a rule to calculate the position of 777 in the expansion of p –

then the law / rule can be altered by the substitution of 000 for 777

and yes – p¢ – so defined – that is – rule governed in this manner – is not the same as p¢ – as Wittgenstein defined it above

p¢ is a proposal – open to question – open to doubt – open to interpretation

different interpretations – different games

‘whether one is to call p¢ a number, and a real number’?

the real question here is whether to call p¢ – a game – that is a rule governed propositional action –

and the answer is straightforward – yes – if p¢ is rule governed

and just in general here – we can talk of numbers – as we can talk of chess pieces –

but the central focus is – or I would say – should be – the game –

for without the game – numbers – of whatever kind – mean nothing

you can call an ‘interval’ a ‘point’ –

as a proposal – a proposition – the ‘interval’ is as with any proposition – open to interpretation –

‘does it become more like a point if I forget that I have used the word “point” with two different meanings?

yes – but really we are not interested in confusion here –

or in mysticism

‘Here it is clear that the possibility of the decimal expansion does not make p¢ a number in the same sense as p. Of course the rule for this expansion is unambiguous, as unambiguous as that for p¢ or √2; but that is no proof that p¢ is a real number, if one takes comparability with rational numbers as an essential mark of real numbers. One can indeed abstract from the distinction between rational and irrational numbers, but that does not make the distinction disappear. Of course, the fact that p¢ is an unambiguous rule for decimal fractions naturally signifies a similarity between p¢ and p or √2; but equally an interval has a similarity with a point etc. All errors that have been made in this chapter of the philosophy of mathematics are based on the confusion between internal properties of a form (a rule as one among a list of rules) and what we call “properties” in everyday life (red as a property of this book). We might also say: the contradictions and unclarities are brought about by people using a single word, e.g. “number”, to mean at one time a definite set of rules, and at another time a variable set, like meaning by “chess” on one occasion the definite game we play, and on another occasion the substratum of a particular historical development.’

I think it is technically irrelevant whether p¢ is a number – it is a game – or at least can be a game with relevant rules

‘All errors that have been made in this chapter of the philosophy of mathematics are based on the confusion between internal properties of a form (a rule as one among a list of rules) and what we call “properties” in everyday life (red as a property of this book)’

there are no ‘errors’ here –

what you have is different proposals – different descriptions –

‘We might also say: the contradictions and unclarities are brought about by people using a single word, e.g. “number”, to mean at one time a definite set of rules, and at another time a variable set, like meaning by “chess” on one occasion the definite game we play, and on another occasion the substratum of a particular historical development.’

yes – but this is not to do with ‘contradictions and unclarities’ – rather differences –

different proposals – different – descriptions

one advantage of understanding mathematical constructs and mathematical action in terms of games – is that in the end – different approaches – different descriptions – can be resolved in the model – can be seen to fit the model of the game

‘ “How far must I expand p in order to have some acquaintance with it?” – Of course that is nonsense. We are already acquainted with it without expanding it at all. And in the same sense I might say that I am not acquainted with p¢ at all. Here it is quite clear that p¢ belongs to a different system from p; that is something we recognize if we keep our eyes on the nature of the laws instead of comparing “the expansions” of both.’

‘acquaintance’ – is somewhat fuzzy notion –

you acquaint by observing – comparing – working with – etc.

in the first instance p¢ is different from p – syntactically –

if p¢ and p¢ did not ‘belong to different systems’ –

there would be no syntactical differentiation –

and yes – you can deduce from this –

that they signify different laws – different rules –

different games

‘Two mathematical forms, of which one and not the other can be compared in my calculus with every rational number, are not numbers in the same sense of the word. The comparison of a number to a point on the number-line is valid only if we can say for every two numbers a and b whether a is to the right of b or b to the right of a.’

yes – the real issue here is that the number system or game – is radically different to the point game

different games – there is no comparison

that the two are played together – combined – (if they are) – is quite ridiculous – and the idea is not to be taken seriously

‘It is not enough that someone should – supposedly – determine a point ever more closely by narrowing down its whereabouts. We must be able to construct it. To be sure, continued throwing of a die indefinitely restricts the possible whereabouts of a point, but it doesn’t determine a point. After every throw (or every choice) the point is still infinitely indeterminate – or, more correctly, after every throw it is infinitely indeterminate. I think we are here misled by the absolute size of the objects in our visual field; and on the other hand, by the ambiguity of the expression “to approach a point”. We can say of a line in the visual field that by shrinking it is approximating more and more to a point – that is, it is becoming more and more similar to a point. On the other hand when a Euclidean line shrinks it does not become any more like a point; it is always totally dissimilar, since its length, so to say, never gets anywhere near a point. If we say of a Euclidean line that it is approximating to a point by shrinking, that only makes sense if there is already a designated point which its ends are approaching; it cannot mean that by shrinking it produces a point. To approach a point has two meanings: in one case it means to come spatially near to it, and in that case the point must already be there, because in this sense I cannot approach a man who doesn’t exist; in the other case, it means “to become more like a point”, as we say for instance that the apes as they developed approach the stage of being human, their development produced human beings.” ’

determining a point by narrowing down it’s whereabouts?

well it has to be there – for this to happen

and if it is indeterminate – in any functional sense – it’s not there –

and therefore there is no ‘narrowing down’ –

there is no ‘approximating’

‘to become more like a point’ – is not to be a point

in an irrational game – there is no rational point

this notion of the point is duplicitous –

you can’t have it both ways –

either there is a definite end to the game

or the game is on-going –

it makes no sense – is contradictory – to speak of an indeterminate end to a game

an irrational game per se – is an on-going game – without end

should you construct an irrational game – and propose a determinate end –

the ‘end’ – will be no more than pragmatic –

simply a ‘breaking-off’ of the process – a stopping of the play

the ‘point’ – as an indeterminate – is perhaps best seen as a characterization of the action of an on-going game

the ‘point’ is a way of describing any such game

‘To say “two real numbers are identical if their expansions coincide in all places” only has sense in the case in which, by producing a method of establishing coincidence, I have given a sense to the expression “to coincide in all places”. And the same naturally holds for the proposition “they do not coincide if they disagree in any one place”.

yes

‘But conversely couldn’t one treat p¢ as the original, and therefore as the first assumed point, and then be in doubt about the justification of p? As far as concerns their extension, they are naturally on the same level; but what causes us to call p a point on the number-line is its compatibility with the rational numbers.’

yes – exactly

‘If I view p or let’s say Ö2 as a rule for the construction of decimals, I can naturally produce a modification of this rule by saying that every 7 in the development of Ö2 is to be replaced by a 5; but this modification is of quite a different nature from one which is produced by an alteration of the radiant or the exponent of the radical sign or the like. For instance, in the modified law I am including a reference to the number system of the expansion which wasn’t in the original rule for Ö2. The alteration of the law is of a much more fundamental kind than might at first appear. Of course, if we have the incorrect picture of the infinite expansion before our minds, it can appear as if appending the substitution rule 7 –> 5  to Ö2 alters it much less than altering Ö2 into

Ö2.1, because the expansion of 7 –>5 are very similar to those of Ö2, whereas the

                                                    Ö2

expansion of Ö2.1 deviates from that of Ö2 from the second place onwards’

true –

a ‘modified law’ – is a different game

‘Suppose I give a rule p for the formation of extensions in such a way that my calculus knows no way of predicting what is the maximum number of times an apparently recurring stretch of the extension can be repeated. That differs from a real number because in certain cases I can’t compare p – a with a rational number, so that the expression p – a = b becomes nonsensical. If for instance the expression of p so far known to me is 3.14 followed by an open series of ones (3.1411 11 ..), it wouldn’t

                                                                 .

be possible to say of the difference p –3.141 whether it was greater or less than 0; so in this sense it can’t be compared with 0 or with a point on the number axis and it and p can’t be called a number in the same sense as one of these points.’

yes – different number systems – signify different games

‘|The extension of a concept of a number, or of the concept ‘all’, etc. seems quite harmless to us; but it stops being harmless as soon as we forget that we have in fact changed our concept. |’

yes – another reason for dropping the notion of number from our central focus – and instead recognising the centrality of the concept of the game

‘|So far as concerns the irrational numbers, my investigation says only that it is incorrect (or misleading) to speak of irrational numbers in such a way as to contrast them with cardinal numbers and rational numbers as different kinds of number; because what are called “irrational numbers” are a species of number that are really different – as different from each other as the rational numbers are different from each other. |’

what we have is different games

‘“Can God know all the places of the expansion of p?” would have been a good question for the schoolmen to ask.’

is ‘God’ the possibility of endless play?

perhaps that is all ‘God’ amounts to –

the infinite irrational game –

p

‘In these discussions we are always meeting something that could be called an “arithmetical experiment”. Admittedly the data determine the result, but I can’t see in what way they determine it. That is with the occurrences of the 7s in the expansion of

p; the primes likewise are yielded as the result of an experiment. I can ascertain 31 is a prime number, but I do not see the connection between it (its position in the series of cardinal numbers) and the condition it satisfies. – But this perplexity is only the consequence of an incorrect expression. The connection that I think I do not see does not exist. There is not an – as it were irregular occurrence of 7s in the expansion of p, because there isn’t any series that is called the expansion of p. There are expansions of p, namely those that have been worked out (perhaps 1000) and in those the 7s don’t occur “irregularly” because the occurrence can be described. (The same goes for the “distribution of the primes”. If you give us a law for this distribution, you give us a new number series, new numbers.) (A law of the calculus that I do not know is not a law). (Only what I see is a law; not what I describe. This is the only thing standing in the way of my expressing more in my signs that [than?] I can understand.)’

the rules of the game – determine the result –

the ‘data’ – the tokens of play – are determined by the game –

by the design of the particular game

a law – or rule that I do not know – is not a rule that I can use

what I see (in the calculus) – is a law – is a rule

‘what I describe’ – description – is speculation –

speculation – is not rule governed

‘expressing more in my signs than I can understand’ –

corrupts the signs – renders them non-functional – useless

what my signs express – if they are functional – is the play of the game –

nothing more

it is the game that determines – the signs

‘arithmetical experimentation’ – is speculation –

it has a place – is of interest –

but it is not to be confused with calculation

‘Does it make no sense to say, even after Fermat’s last theorem has proved, that

‘F = 0.11’? (If, say I were to read about it in the papers.) I will indeed then say, “so now we can write ‘F = 0.11’.” That is, it is tempting to adopt the sign “F” from the earlier calculus, in which it didn’t denote a rational number, into the new one and now to denote 0.11 with it.

F was supposed to be a number of which we did not know whether it was rational or irrational. Imagine a number, of which we do not know whether it is a cardinal number or a rational number. A description in the calculus is worth just as much as this particular set of words and it has nothing to do with an object given by description which may someday be found.

What I mean could also be expressed in the words: one cannot discover any connection between parts of mathematics or logic that was already there without one knowing.’

as to –

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

nothing was already there –

what is proposed is what is there

‘without one knowing’?

knowledge here is what is proposed –

and what is proposed – is open to question – open to doubt –

is uncertain

‘In mathematics there is no “not yet” and no ‘until further notice’ (except in the sense in which we can say that we haven’t further multiplied two 1000 digit numbers together.)’

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

what can occur is what does occur

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

we work within existing practices – existing forms –

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

mathematical activity of any kind is here and now

“Does the operation yield a rational number for instance?” – How can that be asked, if we have method for deciding the question? For it is only in an established calculus that the operation yields results. I mean “yields” is essentially timeless. It doesn’t mean “yields given time” – but: yields in accordance with the rules already known and established.’

yes

‘The position of all primes must somehow be predetermined. We work them out only successively, but they are already determined. God, as it were knows them all. And yet for all that it seems possible that they were not determined by a law.” – Always this picture of the meaning of a word is a full box which is given us with its contents packed in it already to investigate. – What do we know about the prime numbers? How is the concept of them given to us at all? Don’t we ourselves make up the decisions about them? And how odd that we assume that there must have been decisions taken about them that we haven’t taken ourselves! But the mistake is understandable. For we use the expression “prime number” and it sounds similar to “cardinal number”, “square number”, “even number” etc. So we think it will be used in the same way, and we forget that for the expression “prime number” we have given quite different rules – rules different in kind – and we find ourselves at odds with ourselves in a strange way. – But how is that possible? After all the prime numbers are familiar cardinal numbers – how can we say that the concept of prime number is not a number concept in the same sense as a cardinal number? But here again we are tricked by the image of an “infinite extension” as an analogue to the familiar “finite extension”. Of course the concept ‘prime number’ is defined by means of the concept ‘cardinal number’, but “the prime numbers” aren’t defined by means of “cardinal numbers”, and the way we derived the concept ‘prime number’ from the concept ‘cardinal number’ is essentially different from that in which we derived, say, the concept ‘square number’. (So we cannot be surprised if it behaves differently.) One might well imagine an arithmetic which – as it were – didn’t stop at the concept ‘cardinal number’ but went straight on to that of square numbers. (Of course that arithmetic couldn’t be applied in the same way as ours.) But then the concept “square number” wouldn’t have the characteristic it has in our arithmetic of being essentially a part-concept, with the square numbers essentially a sub-class of the cardinal numbers; in that case the square numbers would be a complete series with a complete arithmetic. And now imagine the same done with prime numbers! That will make it clear that they are not “numbers” in the same sense as e.g. the square numbers or the cardinal numbers.’

don’t we ourselves make decisions about them?

yes – and as the concept has proven its utility in practise – it is stable

decisions about them that we haven’t taken ourselves?

‘ourselves’ here – just is the totality of – or the history of decisions made

what we are really talking about here is not numbers – of whatever kind – but games –

i.e. – the ‘prime game’ – the ‘cardinal game’ etc –

and yes –  game ‘similarities’ can always be proposed –

and who is going to be surprised – that one game ‘behaves differently’ – is in fact different to another?

‘mathematics’ as the game of games

‘Could the calculations of an engineer yield the result that the strength of a machine part in proportion to regularly increasing loads must increase in accordance with the series of primes?’

a machine part could have a design that involves a prime calculation –

just how far you would take it – would depend on the limits of the material –

and the overall design of the machine –

and of course there may well be other calculation games that could be used to reflect or explain the same result

43 Irregular infinite decimals

‘ “Irregular infinite decimals”. We always have the idea that we only have to bring together the words of our everyday language to give the combinations a sense, and all we then have to do is inquire into it – supposing it’s not quite clear right away. –

It’s as if words were ingredients of a chemical compound, and we shook them together to make them combine with each other, and then had to investigate the properties of the compound. If someone said he didn’t understand the expression “irregular infinite decimals” he would be told “that’s not true, you understand it very well; don’t you know what the words “irregular”, “infinite”, and “decimal” mean? – well, then you understand their combination as well.” And what is meant by “understanding” here is that he knows how to apply these words in certain cases, and say connects an image with them. In fact, someone who puts those words together and asks “what does it mean” is behaving rather like small children who cover a paper with random scribblings, show it to grown-ups, and ask “what is this?” ’

a word is a proposal – and any use of words – any use – is open to question – to doubt is uncertain –

to ask the question ‘what does it mean?’ – is to recognize the logic of language use –

it is to behave logically

that we stop asking that question and proceed – is a pragmatic move

the question is still there

‘ “Infinitely complicated law”, “infinitely complicated construction” (“Human beings believe, if only they hear the words, there must be something that can be thought by them”).’

‘human beings believe, if only they hear the words, there must be something that can be thought by them’ –

if a proposal is put – it is worth considering

just what it amounts to – is open to question – to doubt – is uncertain –

that it has a definite meaning – that it is beyond question – beyond doubt – or is certain –

is rhetorical – rubbish

‘infinitely complicated law’ –

can a ‘law’ or a ‘rule’ be stated as infinitely complicated?

I think not

if it is ‘infinitely complicated’ – you would never come to a statement of it

can it be shown to be more complicated than its statement suggests?

can it shown to be so complicated that it has no direct or specific application –

that it is effectively useless? – yes

‘an infinitely complicated construction’?

one could well say any construction – is infinitely complicated –

if you wish to look at it that way

the point is there is no end to what you see in a construction – if you have the wherewithal – to keep looking

however that is only possible if you have a defined construction to begin with –

to think within

‘How does an infinitely complicated law differ from the lack of any law.’

a law that proposes an ongoing action – is quite straight forward – ‘simple’ – in fact

an ‘infinitely complicated law’ – as in one that can not be given a statement –

is no different to the lack of any law

‘(Let us not forget: a mathematicians’ discussions of the infinite are clearly finite discussions. By which I mean the come to an end.)’

yes – but any proposal – is open to question – open to doubt – is uncertain –

whenever it is proposed – whenever it is taken up

discussing ‘the infinite’ is no different to discussing anything else –

the question remains – open

‘One can imagine an irregular infinite decimal being constructed by endless dicing, with number of pips in each case being a decimal place.” But if the dicing goes on forever, no final result ever comes out.’

yes – no result to this game – rather an on-going exploration

so the question is – in what context is such an exercise – or a section of it – of use?

it’s a form of calculation that could be used in measurement

and in such a case the materials involved set the limit of the on-going calculation

‘ “It is only the human intellect that is incapable of grasping it, a higher intellect could do so!” Fine, then describe to me the grammar of the expression “higher intellect”; what can such an intellect grasp and what can’t it grasp and in what cases (in experience) do I say that an intellect grasps something? You will then see that describing is itself grasping. (Compare: the solution of a mathematical problem).’

higher or lower intellect?

the proposition put – by whatever level of intellect – is open to question – open to doubt – is uncertain

that is the logic of the proposition

what is ‘behind the proposition’ – where it comes from – if you like –

is logically irrelevant

‘Suppose we throw a coin heads or tails and divide an interval AB in accordance with the following rule: “Heads” means: take the left half and divide it in a way the next throw prescribes. “Tails” says “take the right half, etc.” By repeated throws I then get dividing points that move in an ever smaller interval. Does it amount to a description of the position of a point if I say that it is infinitely approached by the cuts as prescribed by the repeated tossing of the coin? Here one believes oneself to have determined a point corresponding to an irregular infinite decimal. But the description doesn’t determine any point explicitly; unless one says that the words ‘point on this line’ also “determine a point”! Here we are confusing the recipe for throwing with a mathematical rule like that of producing decimal places of Ö2. Those mathematical rules are the points. That is, you can find relations between those rules that resemble in their grammar the relations “larger” and “smaller” between two lengths, and that is why they are referred to by those words. The rule for working out places of Ö2 is itself the numeral for the irrational number; and the reason I here speak of a “number” is that I can calculate with these signs (certain rules for the construction of rational numbers) just as I can with rational numbers themselves. If I want to say similarly that the recipe for endless bisection according to heads and tails determines a point, that would mean that that this recipe could be used as a numeral, i.e. in the same way as other numerals. But of course that is not the case. If the recipe were to correspond to a numeral at all, it would at best correspond to the indeterminate numeral “some”, for all it does is to leave a number open. In a word, it corresponds to nothing except the original interval.’

the notion of the point – the term ‘point’ – in the heads / tail game here – really refers to – an on-going process – or action – an on-going-game

and yes – an on-going game played in the original interval –

the original interval is the ground of play

the playground

(c) greg t. charlton. 2016.