Part II. VI

VI  INDUCTIVE PROOF AND PERODICITY

29 How far is a proof by induction a proof of a proposition?

‘If a proof of induction is a proof of a + (b + c) = (a + b) + c, we must be able to say: the calculation gives the result that a + (b + c) = (a + b) + c (and no other result).

In that case the general method of calculating it must be already known, and we must be able to work out a + (b + c) straight off in the way we work out 25 x 16. So first there is a general rule taught for working out all such problems, and later the particular cases are worked out. – But what is the general method of working out here? It must be based on general rules for signs (– say the associative law –)’

so induction is irrelevant to proof – yes

as for ‘inductive proof’ – it can only be a suggestion of proof – a speculation

or if inductive argument is regarded by practitioners – mathematicians –  as being a form of proof – so be it –

but any such move – is I would suggest unconventional and a departure from  standard mathematical thinking and practice –

and it’s hard to see what value there would in calling suggestion and any argument based on it ‘proof’

‘If I negate a + (b + c) = (a + b) + c it only makes sense if I mean to say something like: a + (b + c) isn’t (a + b) + c isn’t (a +b) + c, but (a + 2b) + c. For the question is:  In what space do I negate the proposition? If I mark it off and exclude it, what do I exclude it from?

To check 25 x 25 = 625 I work out 25 x 25 until I get the right hand result: - can I work out a + (b + c) = (a + b) + c, and get the result (a + b) + c? Whether it is provable or not depends on whether we treat it as calculable or not. For if the proposition is a rule, a paradigm, which every proposition has to follow, then it makes more sense to talk of working out the equation, than to talk of working out a definition.’

negation –

at best a proposal not to proceed with –

and you make such a proposal in whatever space your in –

it’s an argument not to accept what has been proposed –

which in an argumentative context may well have some value –

however in general – what we deal with in life and mathematics is what is proposed – what is put –

you can of course – put an alternative – and by implication ‘negate’ – what has been proposed –

but this is just a matter of deciding which way you will go –

and that is the issue – where you will go – what you propose

not where you won’t go – and not propose

‘can I work out a + (b + c) = (a + b) + c, and get the result (a + b) + c?’ –

yes – if it has a calculus – and a recognisable / usable system of rules –

which is to say – if it is a game – and not just a string of symbols that has the form of  a game

as a definition it can only make sense in a propositional context –

and as a rule it will only make sense in a game – a rule structured propositional context

‘What makes the calculation possible is the system to which the proposition belongs; and that also determines what miscalculation can be made in the working out. E.g.

(a + b) ^2, is a^2, + 2ab + b^2, and not a^2, + ab + b^2,; but (a + b) ^2, = - 4 is not a possible miscalculation in this system.’

a miscalculation – is – relative to the system – of calculation – void –

it is not a calculation

‘I might also say very roughly (see other remarks): “25 x 64 = 160, 64 x 25 = 160; that proves that a x b = b x a” (this way of speaking  need not be absurd or incorrect; you only have to interpret it correctly). The conclusion can be correctly drawn from that; so in one sense a.b = b.a can be proved.

And I want to say: it is only in the sense in which you can tell working out such an example a proof of the algebraic proposition that the proof by induction is a proof of the proposition. Only to that extent is it a check of the algebraic proposition. (It is a check of its structure, not its generality).’

a.b = b.a can be ‘proved’ – only in terms of a substitution rules

induction – may lead you to the rules –

may suggest the rules –  may suggest the game –

it is no proof

‘(Philosophy does not examine the calculi of mathematics, but only what mathematicians say about these calculi.)’

philosophers speculate on what makes mathematics – mathematics –

what mathematicians say about mathematics – is just one place for philosophers to start

30 Recursive proof and the concept of proposition. Is proof a proof that a proposition is true and it’s contradictory false?

‘Is the recursive proof of

 a + (b + c) = (a + b) + c …A

an answer to a question? If so, what question? Is it a proof that an assertion is true and its contradictory false?’

a + (b + c) = (a + b) + c …A

is an example of the recursive game –

that is a game where the  propositional action that recurs indefinitely or until a specified condition is met –

is it an answer to a question?

no

is it a proof that an assertion is true and its contradictory false?

recurrence of a propositional action – is just that – recurrence –

a form of repetition

proof is an argument –

if recurrence is presented as an argument – it is a rhetorical argument

is its contradictory false?

no –

its contradictory is another recursive game

‘What Skolem calls a recursive proof of A can be written thus;

 a + (b +1) = (a + b) +1

a + (b + (c = 1)) = a ((b + c) + 1) = (a + (b + c)) + 1 }   B     

(a + b) + (c + 1) = ((a + b) + c) + 1

If three equations of the form µ, b g are proved, we say “the equation D is proved for all cardinal numbers”. This is a definition of this latter form of expression in terms of the first. It shows that we aren’t using the word “prove” in the second case in the same way as in the first. In any case it is misleading to say that we have proved the equation

D or A. Perhaps it is better to say that we have proved its generality, though that too is misleading in other respects.’

what we have done here is propose it’s generality

‘Now has the proof B answered a question, or proved an assertion true? And which is the proof of B? Is it the group of three equations of the form µ, b, g or the class of proofs of these equations. These equations do assert something (they don’t prove anything in the sense in which they are proved). But the proofs of µ, b g answer the question whether these three equations are correct and prove true the assertion that they are correct. All I can do is explain: the question whether A holds for all cardinal numbers is to mean: “for the functions

j(x) = a + (b + x), y(x) = (a + b) +x

are the equations µ, b, g valid?” And then that question is answered by the recursive proof of A, if what that means is the proofs of µ, b, g (or laying down of µ and the use of it to prove b and g).

So I can say the recursive proof shows that the equation A satisfies a certain condition; but it isn’t the kind of condition that the equation (a + b) ²  + = a² +2b + b ²

has to fulfill in order to be called “correct”. If I call A “correct” because equations of the form µ, b, g can be proved for it, I am no longer using the word “correct” in the

same way as in the case of the equations µ, b, g or (a + b) ² = a² +2ab + b ²

What does “1/3 = 0.3” mean? Does it mean the same as “1/3 = 0.3? Or is that division the proof of the first equation?                                             1

That is, does it have the same relationship to it as a calculation has to what is proved?                                                                                           

         “1/3 = 0.3” is not the same thing as

         “1/2 = 0.5”;

what “1/2 = 0.5”; corresponds to is “1/3 = 0.3” not

            0

         “1/3 = 0.3”

            1

Instead of the notation “1/4 = 0.25” I will adopt for this occasion the following

1/4 = 0.25”.  So, for example, 3/8 = 0.375.

  0                                              0

(NB: The dash underneath emphasizes that the remainder is equal to the dividend. So the expression becomes the symbol for periodic division.)

                                                                                                           .

Then I can say, what corresponds to this proposition is not 1/3 = 0.3, but e.g.

“1/3 =  0.333”  .0.3 is not a result of division (quotient) in the same sense as 0.375.

  1

For we are acquainted with the numeral “0.375” before the division 3/8; but what does “0.3” mean when detached from the periodic division? – The assertion that the

                                   .

division a : b gives o. c as quotient is the same as the assertion that the first place of the quotient c and the first remainder is the same as the dividend.

The relation B to the assertion that A holds for all cardinal numbers is the same as that of 1/3 = 0.3 to 1/3 = 0.3 to 1/3 = 0.3

      1   

         

The contradictory of the assertion “A holds for all cardinal numbers”; is one of the equations µ, b, g is false. And the corresponding question isn’t asking for a decision between a (x). fx and a ($x), ~fx.

The construction of the induction is not a proof, but a certain arrangement of proofs (a pattern in the sense of an ornament). And one can’t exactly say either: if I prove three equations, then I prove one. Just as the movements of a suite don’t amount to a single movement

We can also say: we have a rule for constructing, in a certain game, decimal functions consisting only of 3’s; but if you regard this rule as a kind of number, it can’t be the result of a division; the only result would be what we may call periodic division which has the form a/d = c.’

Wittgenstein is right there are no proofs here – 

but you could well say that the advantage of the recursive theory here – in this  discussion of proof – is just that it illustrates to us that the standard notion of proof is actually irrelevant

and this is not to say we have identified some kind of an anomaly in mathematical theory –

it is rather to make the point that mathematical propositions and mathematical propositional constructions – are proposals

and that there is no ‘proof’ in propositional reality –

any proposal – mathematical or otherwise – is open to question – open to doubt – is uncertain

‘proof’ – is a rhetorical device – a pretence – and one that is entirely unnecessary to a clear understanding of the proposition – and one that is entirely unnecessary to the full function of mathematics –

logically speaking ‘proof’ is a myth – basically harmless – but a confusion nevertheless – and like any confusion – one we can do without it –

it strikes me as bad habit or at least a quaint habit

what we have in this ‘certain arrangement of proofs’ – is a propositional practice –

or if you like a propositional speak that mathematicians can be comfortable with – in playing out this recursive exercise –

and ‘proofs’ could be seen here as enabling or legitimizing deices

the real issue is not whether there is a proof here – or just how the process stacks up against standard proof practice –

but rather where the propositional exercise leads – and whether what it leads to is regarded as useful –

and just what that amounts to is a question of mathematical theory – mathematical context –

and here of course there is always question – doubt – uncertainty –

it is of the nature of human space – however it is configured – however it is drawn

and finally always a question of just what mathematicians do –

what they construct – what propositional reality they make and inhabit –

and what their colleagues will assent to

everyone who plays – is in the equation – on the left-hand side of the = sign –

and in general – everyone comes out – on the right-hand side –

and you could well say – the argument – from either side –

is always – recursive

what else do we have but the assertion of the proposition?

if this doesn’t get the job done – we are likely to reassert –

and most likely in a different form

the game is rhetoric

31 Induction. (x).j and ($x). jx. Does the induction prove the general proposition true and the existential proposition false?

                             ‘3 x 2 = 5 + 1

3 x (a + 1) = 3 + (3 x a) = (5 + b) + 3 = 5 + (b + 3)

Why do we call this induction the proof that (n):n > 2. É. 3 x n ≠ 5?! Well don’t you see that if the proposition holds for n = 2, it holds for n = 3, and then for n = 4, and that it goes on like that for ever? (What am I explaining when I explain the way a proof by induction works?) So you call it a proof of “f(2).f(3).f(4), etc.” But isn’t it rather the form of the proofs of “f(2)” and “f(3)” and “f(4)”, etc.? Or does that come to the same thing? Well if I call the induction the proof of one proposition, I can do so only if that is supposed to mean no more than it proves every proposition of a certain form. (And my expression relies on the analogy with the relationship between the proposition “all acids turn litmus paper red” and the proposition “sulphuric acid turns litmus paper red”).’

yes – the form of the proofs –

and yes – it comes to the same thing –

and ‘form’ here is no more than a rule of presentation –

which comes down to – syntax

there is in truth – no ‘induction’ here at all

to represent this action as ‘inductive’ –

is – as Wittgenstein suggests –

to propose an analogy –

that gets mathematical action dead wrong

in any case – ‘sulphuric acid turns litmus paper red’ –

is a deduction from –

‘all acids turn litmus paper red’

‘Suppose someone says “let us check whether f(n) holds for all n” and begins to write the series

                                                3 x 2 = 5 + 1

                  3 x (2 + 1) = (3 x 2) + 3 = (5 +1) +3 = 5 + (1 =3)

         3 x (2 + 2) = (3 x (2 + 1)) + 3 = (5 + (1 + 3)) + 3 = 5 + ((1 + 3) + 3)

and then breaks off and says “I see it holds for all n” – So he has seen an induction! But was he looking for an induction? He didn’t have any method for looking for one. And if he hadn’t discovered one, would he ipso facto have found a number which does not satisfy the condition? – The rule for checking can’t be: let’s see whether there is an induction or a case for the law which does not hold.– If the law of excluded middle doesn’t hold, that can only mean that our expression isn’t comparable to a proposition.’

‘let us check whether f(n) holds for all n … and then breaks off and says “I see it holds for all n’

here you can make a rule that ‘f(n) holds for all n’ –

but you can’t see that ‘f(n) holds for all n’ – or check that ‘f(n) holds for all n’

‘So he has seen an induction!’ –

an induction – does not end in a generality –

so ‘seeing an induction’ – is better described – is more accurately described as – not seeing that f(n) holds for all n’ –

but assuming nevertheless that it will –

the ‘seeing’ – the ‘checking’ here – only makes sense if the generality is already in place –

‘that f(n) holds for all n’ – can only be a rule – a rule in a game –

and if this is understood – the question of proof – does not arise –

and even if it did – if you were to still hold to this notion of ‘proof’ – induction does not fit the bill –

perhaps he stopped ‘checking’ – because he got sick of being involved in a pointless exercise

‘When we say that an induction proves the general proposition, we think: it proves that this proposition and not its contradictory is true. But what would be the contradictory of the proposition proved? Well, that ($n). ~ fn is the case. Here we combine two concepts: one derived from my current concept of the proof of (n). fn, and another taken from the analogy with ($x). jx . Of course we have to remember that “(n).fn” isn’t a proposition until I have a criterion for its truth; and then it only has the sense that the criterion gives it. Although, before getting the criterion, I could look out for something like an analogy to (x). (fx). What is the opposite of what the induction proves? The proof of (a + b) ² = a² + 2ab + b² works out this equation in contrast to something like (a + b)^2, = a² + 3ab + b². What does the inductive proof work out? The equations: 3 x 2 = 5 +1, 3 x (a + 1) = (3 x a) + 3, (5 + b) + 3 = 5 + (b + 3) as opposed to things like 3 x 2 = 5 + 6, 3 x (a + 1) = (4 x a) + 2, etc. But this opposite does not correspond to the proposition ($x). jx – Further, what does conflict with the induction is every proposition of the form ~f (n), i.e. the propositions “~f(2)”,

“~f(3)”, etc.; that is to say the induction is the common element in the working out of f(2), f(3), etc.; but it isn’t the working out of “all propositions of the form f(n)”, since of course no class of propositions occurs in the proof that I call “all propositions of the form f(n)”. Each one of the calculations is a checking of the proposition of the form f(n)”. I was able to investigate the correctness of this proposition and employ a method to check it; all the induction did was bring this into a simple form. But if I call the induction “the proof of a general proposition”, I can’t ask whether that proposition is correct (any more than whether the form of the cardinal number is correct). Because the things I call inductive proofs give me no method of checking whether the general proposition is correct or incorrect; instead, the method has to show me how to work out (check) whether or not an induction can be constructed for a particular case within a system of propositions. (If I may so put it, what it checked in this way is whether all n have this or that property; not whether all of them have it, or whether there are some  that don’t have it. For example, we work out that the equation x² + 3x + 1 = 0 has no rational roots (that there is no rational number that …), and the equation x² + 2x + 1/2 = 0 has none, but the equation x² + 2x + 1 = 0 does etc.’

‘When we say that an induction proves the general proposition, we think: it proves that this proposition and not its contradictory is true.

an induction proves nothing

at best you can say – an induction proposes a generality

‘But what would be the contradictory of the proposition proved?

the contradictory?

if an induction proposes a generality – its contradictory – would be to deny the generality –

effectively – an argument against induction – or the idea that a generality can result from particular ‘observations’

‘Of course we have to remember that “(n).fn” isn’t a proposition until I have a criterion for its truth; and then it only has the sense that the criterion gives it.’

anything put – is open to question – open to doubt – is a proposition

the question of the ‘criterion of truth’ – is the question of the acceptance of the proposal – of the proposition

particular language forms – i.e. mathematics – have rules – rules of procedures – for determining truth / acceptance

if you play the game – you play within the rules of that game

‘What is the opposite of what the induction proves?’

an inductive ‘argument’ proposes a generality –

the ‘opposite of a generality’ will be the particular – going – nowhere –

so is there a proposal?

you could say the particular is proposed – and that’s it –

but quite clearly – if this is the best you can say about what is proposed –

then ‘the opposite of the induction’ –

is no proposal

‘Hence we find it odd if we are told that the induction is a proof of a general proposition; for we feel rightly that in the language of induction we couldn’t have posed the general question at all. It wasn’t as if we began with an alternative between which we had to decide. (We only seemed to, so long as we had in mind a calculus with finite classes).

Prior to the proof asking about the general proposition made no sense at all, and so wasn’t even a question, because the question would only have made sense if a general method of decision had been known before the particular proof was discovered.

The proof by induction isn’t something that settles a disputed question.’

‘for we feel rightly that in the language of induction we couldn’t have posed the general question at all’

from an observation or an instance – one might propose a generality –

but it is no more than speculation –

speculation is not proof

and this is why it seems odd that anyone would propose that the induction is a proof of a general proposition

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

this logic applies to any and all propositions

so the form of the proposition is not logically relevant – i.e. – particular or general

any proposal – any proposition – is open to question –

the inductive argument to my mind is no more than suggestive –

but strictly speaking that can be said of any argument – or any form of argument – to a proposal – to a proposition

does induction have a place in mathematics?

does suggestion – propositional suggestion – have a place in mathematics?

yes – and logically speaking – does it matter where the suggestion comes from – how it is explained?

no

in mathematics – and in forms of mathematics – procedures and practices have been developed –

their only justification – in the end – is use – is the practice

‘The proof by induction isn’t something that settles a disputed question’

logically speaking there is no settling of a disputed question

questions do get ‘settled’ – but here we are talking about pragmatics

as for induction – it is a form of speculation

if understood correctly – what it does is raise questions –

not settle them

‘If you say: “the proposition ‘(n)fn’ follows from the induction” only means that every proposition of the form f(n) follows from the induction and “the proposition  ($n). ~ fn contradicts the induction” only means “every proposition of the form ~ f(n) is disproved by the induction”, then we may agree; but we shall ask: what is the correct way to use the expression “the proposition (n).f(n)”? What is its grammar? (For from the fact that I use it in certain contexts it doesn’t follow that I use it everywhere in the same way as the expression “the proposition (x). jx”).’

the proposition is put

and yes you can argue that it is the result of an induction – but this is neither here nor there

all you do in such an argument is put forward a proposal as to how the proposition came about –

might be of interest to do this – might not be –

it’s basically a background story

the issue is the proposal – the proposition itself –

which is to say – does it have legs – will it run?

now to then say – ‘(n)fn follows from the induction’ – is – when all is said and done – to propose (n)fn –

linking it to another proposition – the proposition that you say resulted from the ‘induction’ – is an argument –

and if you are to say it follows from it – I would imagine that means the argument is deductive –

and yes – in a systematic propositional activity – we like to propose links –

but again the issue is the proposition that is proposed –

an account of ‘how it came about’ – is not important – it’s an accessory – and this applies to deductive argument as much as it does to inductive argument

‘what is the correct way to use the expression “the proposition (n).f(n)”? What is its grammar? (For from the fact that I use it in certain contexts it doesn’t follow that I use it everywhere in the same way as the expression “the proposition (x). jx”)’ –

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

and it is just this that enables the use of propositions – in different contexts –

how one uses one’s propositions – yes – is a matter of context –

grammar – or the logic of use – is a matter of context –

we are best to see ‘grammar’ as propositional flexibility

‘Suppose that people argue whether the quotient of the division 1/3 must contain only threes, but had no method of deciding it. Suppose one of them noticed the inductive property of 1.0 /3 = 0.3 and said: now I know that there must be only threes in the

quotient. The others had not thought of that kind of decision. I suppose that they had vaguely imagined some kind of decision by checking each step, though of course they could never have reached a decision in this way. If they hold onto their extensional viewpoint, the induction does not produce a decision because in the case of each extension of the quotient it shows that it consists of nothing but threes. But if they drop their extensional viewpoint the induction decides nothing, or nothing that is not decided by working out 1.0 /3 = 0.3, namely that the remainder is the same as the

dividend. But nothing else. Certainly there is a valid question that may arise, namely, is the remainder left after this division the same as the dividend? The question now takes the place of the old extensional question, and of course I can keep the old wording, but it is now extremely misleading since it always makes it look as if having

the induction were only a vehicle – a vehicle that can take us into infinity. (This is also connected with the fact that the sign “etc.” refers to an internal property of the bit of the series that precedes it, and not to its extension.)’

‘Suppose that people argue whether the quotient of the division 1/3 must contain only threes, but had no method of deciding it.’ –

if so these people are speculating –

I see ‘method’ as basically a back story – a ‘justification’ if you like – in any case –

an argument for a propositional use –

now advancing such an argument – may well put any critics at bay – and indeed – if the argument / method advanced is regarded by other practitioners as status quo – or at least acceptable – then the proposal will be endorsed by the practitioners

however this has to with practice – and the rhetoric of practice

whether the methodological argument – whatever it might be – is accepted – is logically speaking – irrelevant

any argument – any method – from a logical point of view – is open to question – open to doubt – and regardless of how it is received and acted upon – uncertain –

a proposal – a proposition – is – that put –

and as such – from a logical point view – only ever in the realm of speculation

propositional practice or use is only ever speculation proceeded with

it is the proposition put – that is of significance – that is where we start

methodological argument – is about persuasion for acceptance                                                                                                                                 

‘Certainly there is a valid question that may arise, namely, is the remainder left after this division the same as the dividend?’

yes – a valid question – but no more valid than any question of interpretation

really the issue is not ‘numbers’ as such – but the game – the numbers game –

numbers as such are game steps – or markers of game action

the real question is – how is the game being used –

in what propositional / mathematical context is the game being played?

there are always options –

when a practice is determined – decided upon – the play begins

‘Of course the question “is there a rational number that is the root of x² x 3x +1 = 0?”

is decided by an induction; but in this case I have actually constructed a method of forming inductions; and the question is only so phrased because it is a matter of constructing inductions. That is, a question is settled by induction, if I can look for the induction in advance; if everything in its sign is settled in advance bar my acceptance or rejection of it in such a way that I can decide yes or no by calculating; as I can decide, for instance, in whether 5/7 the remainder is equal to the divided or not. (The employment in these cases of the expressions “all…” and “there is …” has a certain similarity with the employment of the word “infinite” in the sentence “today I bought a straightedge with an infinite radius of curvature”)’

yes – the methodology – is the proposal put forward – to settle the question

an inductive method – an inductive proposal – may well be the standard approach to this question – in practice

however this is not to say that a rule governed deductive approach could not be proposed –

it could well be argued that such an approach is in fact the ground of any induction –

and that any induction here is just a methodological short cut – in the shadow of the deductive argument

the more general point here is that whatever is proposed – inductive – deductive – or other – is valid if its argument is accepted by the mathematical  community – whatever that amounts to

and this is an issue – not of logic – but of persuasion

if you propose an answer to a question – the answer – if it is to be convincing to yourself and to others will involve argument

argument only ever ‘settles the matter’ – if the argument – for whatever reason – stops

and there is agreement

however logically speaking the argument can always go on

any proposal – is open to question

‘The periodicity of 1/3 = 0.3 decides nothing that has been left open. Suppose

                                 1

someone had been looking in vain, before the discovery of the periodicity, for a 4 in the development of 1/3, he still couldn’t have significantly put the question “is there a 4 in the development of 1/3?” That is, independently of the fact that he didn’t actually discover any 4s, we can convince him that he doesn’t have a method for deciding his question. Or we might say: quite apart from the result of his activity we could instruct him about the grammar of his question and the nature of his search (as we might instruct a contemporary mathematician about analogous problems). “But as a result of discovering the periodicity he does stop looking for a 4! No. The discovery of the periodicity will cure him if he makes the appropriate adjustment. We might ask him: “Well how about it, do you still want to look for a 4?” (Or has the periodicity so to say changed your mind?)

The discovery of the periodicity is really the construction of a new symbol and a new calculus. For it is misleading to say that it consists in our having realised that the first remainder is the same as the dividend. For if we had asked someone unacquainted with periodic division whether the first remainder in this division was the same as the dividend, of course he would have answered “yes”; and so he did realise. But that doesn’t mean he must have realised the periodicity: that is, it wouldn’t mean he had discovered the calculus with the sign a/b =c’

                                                              a

 

‘The periodicity of 1/3 = 0.3 decides nothing that has been left open’ –

                                  1

this is quite correct

‘Or we might say: quite apart from the result of his activity we could instruct him about the grammar of his question and the nature of his search (as we might instruct a contemporary mathematician about analogous problems’ –

what this amounts to is that he is not playing the game as it is played – and that he is looking for a 4 in the development of  1/3 – suggests he is far from understanding the game as played

‘The discovery of the periodicity will cure him if he makes the appropriate adjustment.’

and the appropriate adjustment is understanding this mathematical game – as it is constructed and played

discovering the periodicity – might well point him in the right direction

                                                                                                                                 

and yes –

‘The discovery of the periodicity is really the construction of a new symbol and a new calculus’

a new game – a game that can be played within a game –

and – indeed the periodicity game is played because it is useful

a/b = c is a formal statement of the game –

  a

a formal representation of periodicity

‘Isn’t what I am saying what Kant meant, by saying that 5 + 7 = 12 is not analytic but synthetic a priori?’

what we have is rule governed propositional actions

if you follow the rules of the propositional constructions – you play the game

the ‘game’ simply is playing according to the rules –

if you don’t know the rules – or if you don’t follow them – you don’t play –

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

a rule – is a proposal – a game – a set of proposals –

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

however – if you want to play – and this is the essence of mathematics – you regard the proposals in mathematical games – as rules

how these games come about – how they are discovered – is not mathematics

any account of such matters is propositional speculation –

what we can say is that the games that are played in mathematics – are played because they are regarded – by the players – for whatever reason – as  significant – as useful –

if by ‘analytic’ here – is meant that 5 + 7 = 12 – is a rule governed propositional – then ‘analytic’ tells a part of the story

if ‘a priori’ – is understood to mean ‘rule governed’ – and  ‘synthetic’ –  that any proposition is open to question – open to doubt – is uncertain –

then you have with  ‘synthetic a priori’ – a comprehensive description / picture of the logic and practise of mathematics

32. Is there a further step from the recursive proof to the generalization? Doesn’t the recursion schema already say all that is to be said?

‘Is there a further step from the recursive proof to the generalization?’

the recursive proof to the generalization – is of course an induction –

and I think it can be said that every induction has a missing step –

either that –

or you have to accept an infinity of steps where the only cut off point is fatigue

seriously though – the relation of the recursive schema to the generalization – is best described as suggestive

that mathematicians take the recursive proof as a proof of a generalization – is I would suggest best seen as no more than mathematical speculation –

and again such speculation can be useful – can function as a platform or a step in propositional exploration

mathematics is presented as strict rule governed activity – and this works in so far as those involved buy that story –

however the practices and the rules governing mathematical practice – are open to question – open to doubt are logically speaking – uncertain –

the real charm of mathematics is its pretence

‘Doesn’t the recursion schema already say all that is to be said?’

well – the question is – just what does it say?

Wittgenstein goes on to say –

‘But really the recursion shows nothing but itself, just as periodicity too shows nothing but itself’ –

a sanguine view – but on the money nevertheless

for mine though the issue comes down to the question – how is recursion used? 

and that I would say is really an empirical question – a question of mathematical practice –

that is what mathematicians in fact do with the recursion argument

-.-

                                                                                                                                   

the recursive proof is a propositional game –

‘the point is whether it has the same clearly defined meaning in all cases’ –

‘clearly defined meaning’ in any case – is open to question –

‘And isn’t it the case that the recursive proofs in fact show the same for all proved equations?’

yes – that is the idea of the game

‘And doesn’t that mean that between the recursive proof and the proposition it proves there is always the same (internal) relation?’

yes – that is the game

‘Anyway it is quite clear that there must be a recursive, or better “proof” of this kind (A proof conveying the insight that “that’s the way it must be with all numbers”)’

I.e. it seems clear to me; and it seems that by a process of iteration I could make the correctness of these theorems for the cardinal numbers intelligible to someone else.’

‘that’s the way it must be with all numbers’ – just is the game that is played

mathematicians look for proof – simply because they are at sixes and sevens regarding mathematics as a game –

the idea of the proof is to ground mathematics in certainty –

to give it a status beyond that of the game

now you can put this down to psychological insecurity or simply epistemological ignorance –

nevertheless – it is standard practise

‘proof’ is a propositional game played with propositional games – that is the game

‘I.e. it seems to me that by a process of iteration I could make the correctness of these theorems for cardinal numbers intelligible to someone else’

iteration – the idea here is that if you assert long and hard enough – the poor bastard at the end of your iteration – will just nod –

the cat is out of the bag here – all this is about is rhetoric – not logic – persuasion – not critical thought –

and in any case not everyone is going to fall for it –

so Wittgenstein’s ‘it seems to me ..’ – is wishful thinking –

that is of course if he actually believes what he has proposed here –

still so far I haven’t seen from Wittgenstein any recognition of the central place of rhetoric is mathematics –

so I take him at his word

‘But how do I know that 28 + (45 + 17) = (28 + 45) + 17 without having proved it? How can a general proof give me a particular proof? I might after all go through the particular proof, and how would the two proofs meet in it? What happens if they do not agree?’

‘But how do I know that 28 + (45 + 17) = (28 + 45) + 17 without having proved it?’

what this comes down to is rules – rules of practice – specifically the rules governing addition –

any so called proof – will at best – be nothing more than – a restatement of those rules

and any ‘proof’ – presented as some form of addition to the rules – will thus be  irrelevant and unnecessary

the ‘proof-game’ is not necessary on any level – if it is realized that the practise – the mathematical practise – is rule governed

‘How can a general proof give me a particular proof? I might after all go through the particular proof, and how would the two proofs meet in it? What happens if they do not agree?’

I would think that a general proof and a particular proof are different arguments –

and if the general proof applies to the particular proof – that will require argument 

if they don’t agree – the reason as for why – will be the subject of further argument

                                                                                                                                 

mathematics is a rule governed propositional action –

if you know the rules – accept the rules – you can play the game –

if you don’t know the rules or don’t accept them – you are not in the game

‘In other word: suppose I wanted to show someone that the associative law is really a part of the nature of number, and isn’t something that only accidentally holds in this particular case; wouldn’t I use a process of iteration to try to show that the law holds and must go on holding? Well – that shows us what we mean here by saying that a law must hold for all numbers.

And what is to prevent us calling this process a proof of the law?’

‘that a law must hold …’ –

this is no more than assertion

and that is iteration – assertion – and reassertion

‘And what is to prevent us calling this process a proof of the law?’

well yes –

all we have here is rhetoric – persuasion

does iteration – persuade?

I guess so – if it is held to persuade – by those who practise it – and those who are subject to it

and I suppose assertion and reassertion – is one sense honest and transparent –

and yes – logically speaking – this is all proof is – all it comes down to –

reassertion – in whatever form that takes

so in the beginning and in the end all we have is assertion

‘proof’ – is not a logical concept –

proof is a form of rhetoric

‘The concept of “making something comprehensible” is a boon in a case like this.

For we might say: the criterion of whether something is a proof of a proposition is whether it could be used for making it comprehensible. (Of course here again all that is involved is an extension of our grammatical investigation of the word “proof” and not any psychological interest in the process of making things comprehensible.)’

a proof will be – despite any pretensions of grandeur – no more than a restatement of the proposition

perhaps a restatement – as in a different formulation – will facilitate comprehension?

and in any case comprehension – with or without the ‘proof’ – will in practice –

come down to understanding the rules governing the use of the proposition in whatever context it is put to use

beyond this – comprehension is not something we can put limits on –

there are always questions of breadth and depth

‘ “The proposition is proved for all numbers by the recursive procedure.” That is the expression that is so misleading. It sounds as if here a proposition saying that such and such holds for all cardinal numbers is proved by a particular route, and as if this route was a route through a space of conceivable routes.

But really the recursion shows nothing but itself, just as periodicity too shows nothing but itself.’

yes –

what can we say of the proposal – ‘The proposition is proved for all numbers by the recursive procedure’?

it’s only value seems to me to be that of  protecting the proposition from question – protecting it from doubt

and of course – logically speaking there is no protection –

so?

we have to view it as a rhetorical device –

and look – an unnecessary one at that –

the proposition – if it has value – will be put to work –

claiming something like ‘eternal proof’ – is irrelevant –

it will function – in whatever context it is used –

or it won’t

‘We are not saying that when f(1)  holds and when f(c + 1) follows from f (c), the proposition f(x) is therefore true of all cardinal numbers; but: “the proposition f(x) holds for all cardinal numbers” means “it holds for x= 1, and f(c + 1) follows from f(c)”.

Here the connection with generality in finite domains is quite clear, for in a finite domain that would certainly be a proof that f(x) holds for all values of x, and that is the reason why we say in the arithmetical case that f(x) holds for all numbers.’

the connection with generality in finite domains is functional – is operational

the claim of generality to an infinite domain – is functionally and operationally irrelevant

the claim of generality to an infinite domain might suggest greater power – suggest a greater – indeed endless scope –

but any such suggestion has no effective baring on any particular mathematical action performed

such a suggestion only has rhetorical value

‘At least I have to say that any objection that holds against the proof of B holds also e.g. against the formula (a + b)ⁿ  = etc.

Here too, I would have to say, I am merely assuming an algebraic rule that agrees with the inductions of arithmetic.’

mathematics is a rule governed propositional action –

so called inductions – if they have a role in mathematics – are pointers to the rules of mathematics – or applications of the rules

here we have the issue of the proper way to understand mathematical action

induction in mathematics – is the poor man’s explanation –

yes certain actions can be represented as inductions – but any such representation  lacks overall perspective – and is therefore inadequate

and as for proof –

if you dispense with the rhetoric of proof – what you end up with is decision – the decision to proceed –

and yes – any decision – is open to question – open to doubt – and is – despite its action – uncertain

‘f(n) x (a + b) = f(n + 1)

f(1) = a + b)

therefore f(1) x (a + b) = (a + b)² = f(2)

therefore f(2) x (a + b) = (a + b)³ = f(3), etc.

So far all is clear. But then: “therefore (a + b)ⁿ = f(n)”

Is a further inference drawn here? Is there still something to be established?’

here we can ask is (a + b)ⁿ = f(n) a conclusion of an induction – or the first premise of a deduction?

the inductive argument always leaves us with the problem of the final step to the generalization – and the problem is that there is no final step –

so you get nowhere with induction –

(a + b)ⁿ = f(n) – as a rule (and all rules are rules of thumb) – clears the way –

and enables us to get on with it –

get on with the game

‘But if someone shows me the formula shows me the formula (a + b)² = f(n) I could ask: how have we got there? And the answer would be the group

f(n) x (a +b) = f(n +1)

f(1) = a + b

So isn’t it a proof of the algebraic proposition? – Or is it rather an answer to the question “what does the algebraic proposition mean?”’

‘how have we got there?’ –

is to ask for the path taken to the proposition

now the fact of it is that logically speaking –

f(n) x (a +b) = f(n +1)

f(1) = a + b

does not get us to – (a + b)² = f(n)

however –

f(n) x (a +b) = f(n +1)

f(1) = a + b

may just be the fact of how (a + b)² = f(n) is arrived at –

the point being that yes there is the problem of the inductive argument – but nevertheless that is the route taken to (a + b)² = f(n)

it’s the argument of practise

and you can ask – really does it matter where (a + b)² = f(n) came from – how you got there?

the idea being that the issue is rather – does it make sense – is it consistent with operating rules – can we work with it?

yes there is a place for explanation – but what we are talking about here is argument –

and usually argument after the fact

as for proof – what proof amounts to is decision to proceed –

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

‘I want to say: once you’ve got the induction, it’s all over.’

once you’ve got the induction – you’ve got nothing –

what you have to have is the rule – and once you have the rule – you will see the poverty of induction

‘The proposition that A holds for all cardinal numbers is really the complex B plus its proof, the proof of b and g. But that shows that this proposition is not a proposition in the same sense as an equation, and this proof is not in the same sense as a proof of a proposition.

Don’t forget that it isn’t that we first of all have the concept of proposition, and then come to know that equations are mathematical propositions, and later that there are also other kinds of mathematical propositions.’

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

‘The proposition that A holds for all cardinal numbers is really the complex B plus its proof, the proof of b and g.’ – is a proposal – open to question – open to doubt – uncertain

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

the point is that there is no logical distinction between any proposition – any proposal

propositions are distinguished – differentiated – in terms of use – in terms of practice and practice traditions

mathematical propositions function as game-propositions – that is as rule governed propositions

if you wish to play the game – you look for the rules of practise – and if you are to play the game – you follow the rules

the propositions – the rules – the games – are all open to question – to doubt – are  uncertain

investigating propositional uncertainty is not playing the game – is not doing mathematics

investigating uncertainty is doing logic proper

doing mathematics is following the rules without question – doubt – or uncertainty

that is how you play the game

33 How far does a recursive proof deserve the name “proof”? How far is a step in accordance with the paradigm A justified by the proof of B?

‘We cannot appoint a calculation to be a proof of a proposition’

I think Wittgenstein has hit the nail of the head here –

but it strikes me that it was perhaps a lucky shot –

but nevertheless well put

if appointing a proof – is quite simply ad hoc – what is the alternative?

that the proof is in the proposition – and can be teased out?

if it’s in the proposition – then it is the proposition

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

if you ‘proof’ a proposition – as in make it certain – you destroy it –

what you have then is not a proposition – rather a prejudice

so the point is just that there is no relation between a proposition and a so called proof

or the relation between the two is contradictory

the best we can do regarding establishing a proposition is to argue for its place and function in a working context

‘I would like to say: Do we have to call the recursive calculation the proof of proposition I? That is won’t another relationship do?’

yes – the ‘recursive calculation’ might better be described as a game – a propositional game – the recursive game

‘(What is infinitely difficult is to “see all around” the calculus.)’

whether you can see ‘all around’ the calculus or not – what are dealing with is the calculus –

and the point is to use it

‘In the one case “The step justified” means it can be carried out in accordance with definite forms that have been given. In the other case the justification might be that the step is taken in accordance with paradigms that themselves satisfy a certain condition.’

yes

and in any case ‘justification’ – will be just what it is decided that it is –

‘justification’ of any kind – is rhetoric

‘Suppose that for a certain board game rules are given containing words with no “r” in them, and that I call a rule justified, if it contains no “r”. Suppose someone then said, he had laid down one rule for a certain game, namely, that its moves must obey rules containing no “r”s. Is that a rule of the game (in the first sense)? Isn’t the game played in accordance with the class of rules all of which have only to satisfy the first rule?’

the one rule – or a class of rules – one game or two games?

the difference is in what is given –

in the first case rules (plural) are given – those rules are the rules of play – and no others

in the second case the one rule is given –

the second case effectively argues for an extension of the game – in that more rules than those given in the first case are allowed – and the argument is that they are consistent with those rules given in the first case

if the extension is allowed the first game is seen to be contained in the second – and you now have one game –

if not you have two different – but similar – games

it’s really just a question of how the game or games are constructed –

do you have a set of rules – a finite set – that is not to be messed with?

or do you have one rule that allows for – an unknown number of other rules that it is proposed are consistent with that one rule?

it depends just on what kind of game you want to play –

that’s all

‘Someone shows me the construction of B and then says that A has been proved. I ask “How? All I see is that you have used µ[r] to build a construction around A” Then he says “But when that is possible, I say A is proved”. To that I answer: “That only shows me the new sense you attach to ‘prove’.”

‘Someone shows me the construction of B and then says that A has been proved’ –

and then says –

this is really all it ever comes to – what is said – and if you like – the force in which it is delivered and received –

which is to say – one way or another – the matter is purely rhetorical –

a straight out assertion – or an involved argument – what we are dealing with is persuasion

‘In one sense it means that you have used µ[r] to construct a paradigm in such and such a way, in another, it means as before that an equation is in accordance with the paradigm.’

ok – but what has any of this got to do with proof?

‘If we ask “is that a proof or not?” we are keeping to the word-language.’

‘is that a proof or not?’ –  is to ask –

do I accept this proposition or not?

one’s reasons for acceptance or rejection can be – will be – many and varied – and will depend on one’s state of knowledge

that ‘authorities’ – whoever they are at the time –

may decide one way or the other – will probably determine what happens next –

the point is – a decision is required –

and yes – given that the matter is not frivolous – reasons are to be expected – and inspected –

and if we stick with what Wittgenstein here calls ‘keeping to the word language’ –

we follow and endorse a language-ritual

‘Of course there can be no objection if someone says: if the terms of a step in a construction are of such and such a kind, I say that the legitimacy of the step is proved.’

‘of such and such a kind’ – is open to question –

what you have when ‘the terms of a step in a construction are of such and such a kind’ – is a proposal – open to question – open to doubt – uncertain

at best ‘the legitimacy of the step’ is argued –

to say that it is proved – is to say – there is no argument –

and if that is the case then what we have is a proof of ignorance and stupidity –

and in a place where you would least expect it –

be that as it may – we have language rituals –which function –

and largely because – they continue to function

‘What is it in me that resist the idea of B as a proof of A? In the first place I observe that in my calculation I now here use the proposition about “all cardinal numbers”. I used r to construct the complex B and then I took the step to the equation A; in all that there was no mention of “all cardinal numbers”. (This proposition is a bit of word-language accompanying the calculation, and can only mislead me.) But it isn’t only that this general proposition drops out, it is that no other takes its place.’

‘What is it in me that resist the idea of B as a proof of A?’ –

perhaps at base something of an ontological uneasiness – B is not A – so how would B prove A?

and following on from this the idea would be that the proof of A is in A – or just simply is A

and if so – we may as well drop the idea of proof – as something outside of whatever is to be proved

and if you do that the notion of proof evaporates

what this amounts to is A is A – meaning A is well formed and functional –

or it is not –

and if it is not well-formed and functional – it is not only not A – it is not anything of significance

‘But it isn’t only that this general proposition drops out, it is that no other takes its place’ –

yes – the argument is changed – a different argument is put – A and B are different arguments

B is as it were the platform that is used to launch to A

you might say there is a trajectory – a propositional trajectory –

and the action of one to the other?

perhaps best explained as a quantum jump

and that I suggest is in fact a good model for all propositional action

I understand that people want to pin down (propositional) reality –

and the reason for that is insecurity – metaphysical insecurity –

and there is nothing wrong with that –

the issue is how do you understand it?

do you regard it as something that should be can be eliminated – with proofs – with explanations – etc. –

or do you – as I would suggest – see it as a the reality we must embrace as the source of our freedom and creativity?

as the source of true joy – yes really –

embrace it or not – it is the reality we face

‘So the proposition asserting the generalization drops out; “nothing is proved”, “nothing follows”.

“But the equation A follows, it is that which takes the place of the general proposition.” Well, to what extent does it follow? Obviously, I am using “follows”

in a sense quite different from the normal one, because what A follows from isn’t a proposition. And that is why we feel that the word “follows” isn’t being correctly applied.’

A is proposed

whether A takes the place of the general proposition or not

‘Well, to what extent does it follow?’

best to understand ‘follows’ here – in a straight out literal sense –

what follows – is what comes after

if A follows from an argument regarding the generalization –

then A is proposed in response to that argument

whatever lead to A is what A follows from

if A came about some other way – so be it –

whatever led to A is what it follows from

here of course we are just dealing with circumstance –

and however it is described –

that is to say – however it is proposed

‘If you say “it follows from the complex B that a = (b + c) = (a + b) + c” we feel giddy. We feel that somehow or other you’ve said something nonsensical although outwardly it sounds correct.’

when it is said that a = (b + c) = (a + b) + c follows from the complex B – it is only ‘nonsensical’ – if the steps to the equation – do not follow a standard practice

‘That an equation follows, already has a meaning (has its own definite grammar).’

yes – and by ‘grammar’ here we mean an account – or a theory of practice or procedure

‘If I am told “A follows from B”, I want to ask: “what follows?” That a + (b + c) is equal to (a + b) + c, is something postulated, it doesn’t follow in the normal way from an equation.’

‘If I am told “A follows from B”, I want to ask: “what follows?’

what ‘follows’ mean here – is open to question – open to doubt – is uncertain

whatever is proposed for ‘follows’ will be subject to argument

and any argument will only be resolved – by practise –

by what those involved in the practise decide ‘follows’ means

‘That a + (b + c) is equal to (a + b) + c, is something postulated, it doesn’t follow in the normal way from an equation.’

the point is that if it is proposed that a + (b + c) is equal to (a + b) + c – follows from an equation –

then again we need argument to that effect

and if the argument is accepted by those involved –

then indeed it follows

‘We can’t fit our concept of following from to A and B; it doesn’t fit’ –

‘following from A and B’ – is whatever step is taken from A and B

‘following from’ – is taking the step –

‘following from’ – is not really the issue – the issue is whether the step taken has an acceptable argument –

if the argument of the step doesn’t have support –

then the step will be regarded as a – misstep

‘ “I will prove to you that a +(b +n) = (a + b) + n”. No one then expects to see the complex B. You expect to hear another rule for a, b, and n permitting the passage from one side to another. If instead of that I am given B with the schema r

I can’t call it a proof, because I mean something else by “proof”.

I shall very likely say something like “oh, so that’s what you call a “proof”, I had imagined …”’

yes – different concepts of proof – different propositional games – which is to say different rules – and different rules of procedure –

different decision making processes

‘The proof of 17 + (18 +5) = (17 +18) + 5 is certainly carried out in accordance with the schema B, and this numerical proposition is of the form A. Or again: B is a proof of the numerical proposition; but for that very reason, it isn’t a proof of A.’

yes – A and B are different propositional games

‘ “I will derive A_1, A_II, A₁₁₁ from a single proposition – This of course makes one think of a derivation that makes use of these propositions – We think we shall be given smaller links of some kind to replace all these large ones in the chain.

Here we have a definite picture; and we are offered something quite different.

The inductive proof puts the equation together as it were crossways instead of lengthways.’

the inductive process here does ‘put together’ – it is a method of construction –

the inductive process of constructing an equation – is not a proof of the equation

look at it this way –

if you are building a building – your method of constructing – is not a guarantee of the integrity of the materials that you use

‘ “If we work out the derivation, we finally come to the point at which the construction of B is completed. But at this point we say “therefore this equation holds”! But these words now don’t mean the same thing as they do when we elsewhere deduce an equation from equations. The words “The equation follows from it” already have a meaning. And although an equation is constructed here, it is by a different principle.’

yes – different constructions – different methods – different games

and as for ‘holds’ –

‘holds’ here – is best understood as – proposed –

which is to say ‘standing’ – or ‘up and running’ –

ready to go

‘If I say “The equation follows from the complex”, then here an equation is ‘following’ from something that is not an equation.

yes –

and the complex functions as an argument for the step to the equation

it is the equation – the game that is to be played – that is significant – that is relevant –

the argument to it – the road to it – yes – really amounts to context –

the context out of which the equation emerges –

it’s propositional history – if in fact that is its history

‘We can’t say: if the equation follows from B, then it does follow from a proposition, namely µ. b. g; for what matters is how I get A from that proposition; whether I do so in accordance with a rule of inference; and what the relationship is between the equation and the proposition µ. b. g. (The rule leading to A in this case makes a kind of cross-section through µ. b. g; it doesn’t view the proposition in the same way as the rule of inference does.)’

yes we can say – ‘if the equation follows from B, then it does follow from a proposition, namely µ. b. g’ –

for B is a proposition – is a proposal

look how you get from one proposition to another – in any propositional context –

is up for question – a matter of doubt – is at all times – uncertain –

where this matter is held to be of importance – as it is in mathematics – then it is the subject of argument –

and yes we do feel more comfortable if we can say how we got from ‘a’ to ‘b’ – and it is always of interest to see what is proposed – and what is finally agreed upon –

whatever propositional jump we make – and however it is explained or accounted for the point is we never leave – uncertainty

‘If we have been promised a derivation of A from µ and now see the step from B to A, we feel like saying “oh that isn’t what was meant.” It is as if someone had promised to give me something and then says: see, I’m giving you my trust.’

yes – it is not the way expected – it is not how we play the game

‘The fact that the step from B to A is not an inference indicates also what I meant when I said that the logical product µ. b. g does not express the generalization.’

yes – a logical product will always fall short of the generalization

‘I say that A_I, A_II etc. are used in proving (a + b)² = etc. because the steps from (a + b)²

to a² + 2ab + b²  are all of the form A_I or A_II, etc. In this sense the step in III from

(b + 1) + a to (b + a) + 1 is also made in accordance with A_I, but the step from

a + n to n + a isn’t!’

again – a + n to n + a – just is a different game to (b + 1) + a to (b + a) + 1 – and so how can A_I be relevant?

‘The fact that we say “the correctness of the equation is proved” shows that not every construction of an equation is a proof.’

a construction of an equation – any construction – is just that – a construction

a construction is really all you have – that’s it –

different constructions – different ways to the equation –

which means different proposals – different arguments or steps to the equation

proof is just a rhetorical overwrite that has nothing to do with the construction – and nothing to do with the equation –

at best it is the decision to proceed –

a decision that for whatever reason – is open to challenge

‘Someone shows me the complexes B and I say “ they are not proofs of the equations A”. Then he says: “You haven’t seen the system on which the complexes are constructed”, and points it out to me. How can that make Bs into proofs?’

the arguments of the complexes are propositional games –

a game is not a proof –

and furthermore this notion of proof is not logical –

proof is a rhetorical notion

the integrity of an equation –  if that is what is really at issue here –

rests in the rules governing its formation and action

‘This insight makes me ascend to another, a higher level; whereas a proof would have to be carried out at a lower level.’

any proposal – complex or not – is open –

open to proposal

there is only one level in propositional reality –

and all proposals – all propositions are equal –

all propositions are open to question – open to doubt – uncertain –

we begin with a proposal – and we propose in relation to it

‘Nothing but a definite transition to an equation from other equations is a proof of that equation. Here there is no such thing, and nothing else can do anything to make B into a proof of A.’

a definite transition to an equation from other equation – is a step in a game –

you transition from one proposition to another – in any context – as of course we do constantly – where’s the proof?

what goes for ‘proof’ – is a back story enforced with rhetoric

‘But can’t I say that if I have proved this about A, I have proved A? Wherever did I get the illusion that by doing this I had proved it? There must surely be some deep reason for this’

let me put the proposal that the ‘deep reason for this’ – is fear

fear of uncertainty –

and really an intellectual – and hence emotional – and hence behavioral –

cowering

fear of uncertainty is fear of the logical reality – of reality – as it is proposed

fear of uncertainty comes out of a lack of courage

the courage required – to embrace logic

‘Well, if it is an illusion, at all events it arose from our expression in word-language “this proposition holds for all numbers”; for on this view the algebraic proposition is only another way of writing the proposition of word-language. And that form of expression caused us to confuse the case of all the numbers with the case of ‘all the people in this room’. (What we do to distinguish the cases is to ask: how does one verify the one and the other?)’

is the algebraic proposition only another way of writing the proposition of word-language?

propositions are translated from one form to another – and the difference is uncertainty –

this goes on all the time

however a proposition of one form – i.e. word language – is not a proposition of another form – i.e. mathematical / algebraic

different forms – different contexts – different functions –

all open to question – to doubt – all uncertain

‘this proposition holds for all numbers’ – is a proposal – open to question – to doubt – uncertain –

‘all the people in this room’ – strikes me as an incomplete sentence – ‘all the people in this room …’ – what? – i.e. what about them? –

so do we or don’t we have a proposal here? – I suppose we do

still it’s a bit of a floater

‘how does one verify the one and the other? –

is verification the issue?

‘this proposition holds for all numbers’ –

do we start checking all numbers against the proposition?

well that would be an endless task – and what then of verification?

isn’t the issue –  how is the proposition used – in what contexts does it function?

perhaps ‘this proposition holds for all numbers’ – is used as a rule? –

you have to know where it comes from – and how it is used – to have a go at  assessing it

and as for ‘all the people in this room’ –

I suppose you could do a count –

and in that case you play a propositional game – the addition game

games are played they are not verified

‘If I suppose the functions j, y, F exactly defined and then write the schema for the inductive proof:

                                    R

           µ     j(1) = y(1)                             A

B        b      j(c + 1) = F{j(c)}  }   … jn = yn

       

           g      y(c + 1) = F{y(c)

Even then I can’t say that the step from jr to yr is taken on the basis of r (if the step

µ, b g was made in accordance with r – in particular cases r = µ). It is still the equation A it is made in accordance with, and I can only say that it corresponds to the complex B if I regard that as another sign in place of the equation A.

For of course the schema for the step had to include µ, b g.

In fact R isn’t the schema for the inductive proof B_III; that is much more complicated, since it has to include the schema B_I.’

the inductive argument – ‘the schema for the inductive proof’ – is best seen as an accepted practice of procedure –

and that I am afraid is all any kind of ‘proof’ amounts to

steps ‘made in accordance with’ – amounts to ‘careful speculation’ within a given propositional framework

complex B as another sign in place of the equation A? –

yes you can propose alternative signs but really what is the point?

all that is relevant is the equation

‘The only time it is advisable to call something a ‘proof’ is when the ordinary grammar of the word ‘proof’ doesn’t accord with the grammar of that object under consideration.’

isn’t this to say that a proof is a proof when it can’t be a proof?

that is when its grammar can’t or doesn’t correspond with the grammar of that to be proved

so there’s no proof?

or there is proof but it doesn’t correspond with anything that needs to be proved?

strange notion

‘there is no proof ‘– is the best way to put it –

and is the fact of it

‘What causes the profound uneasiness is in the last analysis a tiny but obvious feature of the traditional expression.’

perhaps the point here is that we just automatically assume that mathematical propositions are subject to the question of proof

if you are profoundly uneasy about this – that is good –

it’s a step in the right direction

the mathematical proposition – as with any proposition – is open to question – open to doubt – is uncertain

the idea of ‘proof’ runs quite contrary to the nature of the proposition –

‘proof’ is a rhetorical notion

mathematical propositions have integrity – not in terms of any rhetorical deception – but rather in terms of the rules – the accepted propositional practice – that govern their construction and use

all we have in human affairs is proposals – and differing uses of proposals –

beyond that there is nothing

‘What does it mean, that R justifies a step of the form A? No doubt it means that I have decided to allow in my calculus only steps in accordance with a schema B in which the propositions µ, b, g, are derivable in accordance with r (And of course that would only mean that I allowed only the steps A_I, A_II etc., and that those had schemata B corresponding to them).

It would be better to write “and the schemata had the form R corresponding to them”. The sentence added in brackets was intended to say that the appearance of generality – I mean the generality of the concept of the inductive method – is unnecessary, for in the end it only amounts to the fact that the particular constructions B_I, B_II, etc. are constructed flanking the equations A_I, A_II, etc. Or that in that case it is superfluous to pick out the common feature of the constructions; all that is relevant are the constructions themselves, for there is nothing there except these proofs, and the concept under which these proofs fall is superfluous, because we never made any use of it. Just as if I only want to say – pointing to three objects – “put that and that and that in my room”, the concept chair is superfluous even though the three objects are chairs. (And if they aren’t suitable furniture for sitting on, that won’t be changed by someone’s drawing attention to a similarity between them). But that only means the individual proof needs our acceptance of it as such (if ‘proof’ is to mean what it means); and if it doesn’t have it no discovery of an analogy with other such constructions can give it to it. The reason why it looks like a proof is that

µ, b, g and A are equations, and that a general rule can be given, according to which we can construct (and in that sense derive) A from B.

After the event we may become aware of this general rule. (But does that make us aware that the Bs are really proofs of A?) What we may become aware of is a rule we might have started with and which in conjunction with µ would have enabled us to construct A_I, A_II, etc. But no one would have called it a proof in this game.’

‘all that is relevant is the constructions themselves’ –

yes

‘for there is nothing there except these proofs, and the concept under which these proofs fall is superfluous’ –

yes the concept is superfluous – except in a rhetorical sense

as for the proofs –

‘But that only means the individual proof needs our acceptance of it as such’

yes the key term here is ‘acceptance’ –

and this acceptance – is the acceptance of the proposition

the so called ‘proof’ – is no more than a vehicle for the acceptance

‘(if ‘proof’ is to mean what it means)’ –

yes well – what it means – in my opinion comes down to bad (foundational) epistemology – and corrupt logic –

or it means – just what mathematicians do – how they behave

‘and that a general rule can be given, according to which we can construct (and in that sense derive) A from B’

yes this is a rule governed propositional activity –

which in this case is a propositional game

‘After the event we may become aware of this general rule. (But does that make us aware that the Bs are really proofs of A?) What we may become aware of is a rule we might have started with and which in conjunction with µ would have enabled us to construct A_I, A_II, etc. But no one would have called it a proof in this game.’

yes – well put

‘Whence the conflict: “That isn’t a proof!” “That surely isn’t a proof.”?’

‘whence the conflict?’ – really? –

any proposal put – in any context – is open to question – open to doubt – is uncertain

that is the nature of (human) propositions

I say “today it was hot” – you say “no it wasn’t” –

no great mystery –

it is everyday propositional reality –

the idea of proof is to try and kill off question – doubt – uncertainty –

to kill off reality

it really all gets back to Plato’s delusion – and his failure to face up to and to deal with propositional reality –

and we have all paid a great price for his eloquence –

for his rhetoric

‘We might say that it is doubtless true, that in proving B by µ I use µ to trace the contours of the equation A, but not in the way I call “proving A by µ”.’

yes different games of acceptance

‘The difficulty that needs to be overcome in these discussions is the difficulty of looking at the proof by induction as something new, naively as it were.’

the ‘inductive proof’ as with any ‘proof’ facilitates acceptance –

perhaps the value of any induction just is that it always leaves a question – a doubt –

uncertainty

induction is a methodology that gives us action in the face of uncertainty –

and in that sense it can be seen to be a logical representation of propositional reality

I don’t think uncertainty is naïve – in fact just the opposite –

uncertainty as the end of naivety

‘So when we said above we could begin with R, this beginning with R is in a way a piece of humbug. It isn’t like beginning a calculation by working out 526 x 718. For in the latter case setting out the problem is the first step on the journey to the solution.

But in the former case I immediately drop the R and have to begin again somewhere else. And when it turns out that I construct a complex of the form R, it is again immaterial whether I explicitly set it out earlier, since setting it out hasn’t helped me at all mathematically, i.e. in the calculus. So what is left is just that I now have a complex of the form of R in front of me.’

yes the ritual of R

‘We might imagine we were acquainted only with the proof of B_I and could then say: all we have is this construction – no mention of analogy between this and other constructions, or of a general principle in carrying out the constructions. – If I then see B and A like this I am bound to ask: but why do you call that a proof of A precisely? – (I am not asking: why do you call it a proof of A)! Any reply will have to make me aware of the relation between A and B which is expressed in V.’

‘why do you call that a proof of A?’ –

yes – you might see in the so called ‘proof’ – a path to A – even some sort of an approximation of A – propositional packaging for A –

but proof?

a proof is either internal to a proposition – or external to it –

if the former – then there is really nothing of consequence to be said – you just get on with it

if the latter – how can a different proposal do anything here – but be used to endorse the subject proposition?

and if so proof comes down to – endorsement – acceptance

yes you can put up a definition of the relation between A and B – the idea of V – and go from there –

still in all – that amounts to – acceptance – affirmation – of A

and so the term ‘proof’ is just a signal of this – a signal of acceptance

in the end – regardless of whatever propositional constructions are impressed on the issue – regardless of what arguments are developed and used –

that is regardless of your conception of proof –

‘proof’ has no more logical status than a nod of the head –

and as to all the work done on ‘proofs’ –

interesting as that might be – brilliant as it might be –

just ritual

nevertheless – that is the way of it –

how the game is in fact played

‘Someone shoes us B_I and explains to us the relationship with A_I, that is, that the right side of A was obtained in such and such a manner etc. etc. We understand him; and he asks us: is that a proof of A? Certainly not!

Had we understood everything there was to understand about the proof? Had we seen the general form of the connection between A and B? Yes!

We might also infer from that that in this way we can construct a B from every A and therefore conversely an A from every B as well.’

someone explains the relationship – is – someone proposes a relationship

is that a proof? –

if that is to ask – is that proposal – a guarantee of the integrity of A?

the answer in general terms is no –

and the reason is that any proposal is open to question – open to doubt – is uncertain –

a ‘guarantee’ defies logic – it is a creature of pretence and rhetoric

on the other hand – if by proof – what you mean – is that a proposal of proof is a good reason to proceed with A –

well that may or may not be the case –

it all depends on who you are proposing to and how they regard your proposal –

here we are deep in contingency

‘we can construct a B from every A and therefore conversely an A from every B as well.’ –

ok – a game proposal – a propositional game – why not – if you have nothing better to do?

‘The proof is constructed on a definite plan (a plan used to construct other proofs as well). But this plan cannot make the proof a proof. For all we have here is one of the embodiments of the plan, and we can altogether disregard the plan as a general concept. The proof has to speak for itself and the plan is only embodied in it, it isn’t itself a constituent part of the proof. (That is what I have been wanting to say all the time). Hence it is no use to me if someone draws my attention to the similarity between proofs in order to convince me that they are proofs.’

if the proof is constructed on a definite plan – but the plan cannot make the proof a proof –

what’s the point of the definite plan? –

is it a definite plan not to make a proof?

‘The proof has to speak for itself and the plan is only embodied in it, it isn’t itself a constituent part of the proof’ –

the proof speaks for itself?

I would say the proposition (to be ‘proved’) – speaks for itself –

and the ‘proof’ – speaks for the proposition –

which if the proposition speaks for itself –

is hardly necessary

and the plan embodied in the proof – isn’t a constituent part of the proof?

the plan is in the proof – but not in the proof?

and wasn’t the original point that the proof was in the plan ‘constructed on a definite plan’?

so what’s it to be – the proof is in the plan – or the plan is in the proof?

I don’t think we have a plan here – or a proof

‘Isn’t our principle: not to use a concept-word where one isn’t necessary? – That means, in cases where the concept-word really stands for an enumeration, to say so.’

any action proposed will be defined conceptually –

so the concept-word – is not the action – but an understanding of it

enumeration you would say is the action –

its understanding – in a propositional context – is a conceptual matter

‘When I said earlier “that isn’t a proof” I meant ‘proof’ in an already established sense according to which it can be gathered from A and B by themselves. In this sense I can say: I understand perfectly well what B does and what relationship it has to A; all further information is superfluous and what is there isn’t a proof. In this sense I am concerned only with A and B; I don’t see anything beyond them and nothing else concerns me.’

‘according to which it can be gathered from A and B themselves’ –

‘gathered’ – amounts to anything and nothing

if you understand the relationship – the relationship is proposed –

if there is no proposal – there is no relationship

‘In this sense I am concerned only with A and B; I don’t see anything beyond them and nothing else concerns me.’

the point is that if you see ‘A and B’ – you see a relationship –

the proposal of the relationship is – ‘something beyond them’

I understand Wittgenstein here on proof – but just because you throw out ‘proof’ – or a particular version of it – doesn’t mean – that you throw out all and any propositions regarding A and B

the question for Wittgenstein is –‘what is it you see?’

‘If I do this, I can see clearly enough the relationship in accordance with the rule V, but it doesn’t enter into my head to use it as an expedient in construction. If someone told me while I was considering B and A that there is a rule according to which we could have constructed B from A (or conversely), I could only say to him “don’t bother me with irrelevant trivialities.” Because of course it’s something that’s obvious, and I see immediately that it doesn’t make B a proof of A. For the general rule couldn’t show that B is a proof of A and not of some other proposition, unless it were a proof in the first place. That means, that the fact that the connection between A and B is in accordance with a rule can’t show that B is a proof of A. Any and every connection could be used as a construction of B from A (and conversely).’

I agree with Wittgenstein that the idea of proof – an argument for epistemological foundation – fails –

not because I think it is obvious – but rather because such a notion flies in the face of propositional logic

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

if you are after certainty in any form – what you are dealing with – what you are looking for is prejudice

and the idea of the claim of the obvious – just is that you can’t argue against it –

and that of course is rubbish

proof is rhetoric – and really we need more than rhetoric – to make the point –

Wittgenstein hasn’t delivered

‘So when I said “R certainly isn’t used for the construction, so we have no concerns with it” I should have said: I am only concerned with A and B. It is enough if I confront A and B with each other and ask: “is B a proof of A?” So I don’t need to construct A from B according to a previously established rule; it is sufficient for me to place the particular As – however many there are – in confrontation with particular Bs. I don’t need a previously established construction rule (a rule needed to obtain the As).’

yes – the construction of A from B – a propositional game –

if that’s what you call ‘proof’ – why not?

and so you run with a ‘previously established construction rule’ – if that’s how you do it – where’s the problem?

‘confront A with B’ – ‘place the particular As in confrontation with particular Bs’ –

ok – but ‘confrontation’ means what?

if you just want to do – with no account – I say – fair enough –

but there is no explanation here –

and don’t pretend that there is

I am all for the sharp focus on A and B – and as for construction that train has left the station when it comes to dealing the relationship of A and B –

as to the relationship – it may be – at least initially – a matter of speculation – but as that matter is thrashed out – the question will be – what rule applies here – in this context?

so be clear – it is not as if the A and B on the page – are all there is to the focus –

the central focus is the relationship –

and that will come down to – the rule that governs their relationship

without this you have virtually nothing – just signs on a page – no action

and also –

if you are not prepared to state – articulate – propose – that relationship

how do you know that there is one?

how do you know what you are doing?

all very well to say you do –

but that is hardly enough if you are working in a rule governed propositional discipline –

where – as a matter of fact – as a matter of practice –

there are standards and issues of accountability

playing the ‘seer’ just doesn’t work

‘What I mean is: in Skolem’s calculus we don’t need any such concept, the list is sufficient.

Nothing is lost if instead of saying “we have proved the fundamental laws A in this fashion” we merely show that we can co-ordinate with them constructions that resemble them in certain respects.’

‘we have proved the fundamental laws A in this fashion’ – is just rhetoric –

and I would say that learning that you can do without such rhetoric – or at least minimizing its use  – is to be recommended –

and I would suggest leads to a clearer sharper understanding

if such rhetoric is understood for what it is – just ‘propositional packaging’ –

and is not taken seriously – perhaps seen as an instance of a propositional tradition and ritual – then it is quite harmless

the real game is the  propositional action taken –

how that is packaged up is by and large irrelevant to the game –

still it needs to be called out for what it is and checked with a critical eye –

reason being – it can distract from the main game – and it might just send some epistemological innocent – off on the wrong track –

bullshit can be harmless – but it is bullshit

as to –

‘we merely show that we can co-ordinate with them constructions that resemble them in certain respects.’

yes that’s about it

‘The concept of generality (and of recursion) used in these proofs has no greater generality than can be read immediately from the proofs’

these ‘proofs’ are language games

the concepts are employed in these games

and these concepts of generality and recursion may have application in other propositional actions and games

‘The bracket } in R, which unites µ, b and g can’t mean anymore than that we regard the step in A (or a step of the form A) as justified if the terms (sides) of the steps are related to each other in the ways characterized by the schema B. B then takes the place of A. And just as before we said: the step is permitted in my calculus if it corresponds to one of the As, so we now say: it is permitted if it corresponds to one of the Bs.

But that wouldn’t mean we had gained any simplification or reduction.’

B takes the place of A –

a move – a step – from – one language game to another –

a simplification or reduction?

I think it is rather that B and A are different propositional games –

and that a step (an argument) has been made from one to the other –

that’s the proposal

‘We are given the calculus of equations. In that calculus “proof” has a fixed meaning. If I now call the inductive calculation a proof, it isn’t a proof that saves me checking whether the steps in the chain of equations have been taken in accordance with these

particular rules (or paradigms). If they have been, I say that the last equation of the chain is proved, or that the chain of equations is correct.’

yes – exactly right –

‘proof’ – if it is to have any functional meaning just is – the action of a rule governed

procedure

‘Suppose that we were using the first method to check the calculation (a + b)³ = …

and at the first step someone said: “yes, that step was certainly taken in accordance with a (b + c) = a . b + a . c, but  is that right?” And then we showed him the inductive derivation of that equation. –‘

different methods – different arguments – different games

‘The question “Is the equation G right?” means in one meaning: can it be derived in accordance with the paradigms? – In the other case it means: can the equations

µ, b and g be derived in accordance with the paradigm (or paradigms?) – and here we have put two meanings of the question  (or of the “proof”) on the same level (expressed them in a single system) and can now compare them (and see that they are not the same).’

yes different paradigms – different methods – different practices – different understandings of ‘proof’

‘And indeed the new proof doesn’t give you what you might expect: it doesn’t base the calculus on a smaller foundation – as happens if we replace the p v q and ~p by p|q, or reduce the number of axioms, or something similar. For if we now say that all the basic equations A have been derived from r alone, the word “derived” here means something quite different. (After this promise we expect the big links in the chain to be replaced by smaller ones, not by two half links.) And in one sense these derivations leave everything as it was. For in the new calculus the links in the old one essentially continue to exist as links. The old structure is not taken to pieces. So we have to say the proof goes on in the same way as before. And in the old sense the irreducibility remains.’

what we are considering here is different constructions –  different calculi – different methods – different ‘proofs’ –

yes – you can argue that one calculus can be accounted for by another –

and that therefore –

‘in the new calculus the links in the old one essentially continue to exist as links. The old structure is not taken to pieces. So we have to say the proof goes on in the same way as before. And in the old sense the irreducibility remains.’

and what then does it come down to?

one might be tempted to say it’s a question of style –

or perhaps at a deeper level – philosophical orientation –

which amounts to what?

and the answer here could be anything –

anything you want to propose

what we can say – as an empirical fact – is that we have different practices –

different practices – whose validation just is practice –

and that the idea of the reduction or translation of one practice to another – is trivial –

best to embrace the multiplicity

Oakum would not favor this –

‘plurality ought never be posed without necessity’ –

I say here – the issue is not necessity – it is propositional practice – propositional reality

‘So we can’t say that Skolem has put the algebraic system on a smaller foundation, for he hasn’t given it foundations in the same sense as is used in algebra.’

well yes we are dealing with different conceptions –

of course comparisons will be made –

but is there much to this?

(the old chestnut – ‘does size matter?’)

different calculi – and different understandings –

different world views

the common ground is what we don’t know – not what we do

the common ground is what is not proposed –

not what is

‘In the inductive proof doesn’t µ show a connection between the As? And doesn’t this show we are here concerned with proofs? – The connection shown is not the one that breaking up the A steps into r steps would establish. And one connection between the As is already visible before any proof.’

firstly – this notion of proof –

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

and if so – how is it in any sense different to any other proposal – any other proposition?

secondly –

I think proof is best understood as satisfaction –

satisfaction that the  game is properly constructed and that it is played in accordance with its rules –

however that is then interpreted and represented –

and of course – again – satisfaction – as we all know – is a matter – open to question –

open to doubt – is uncertain

thirdly –

now if your notion of proof is deductive – then unless – you are open to a propositional – conceptual – philosophical – diversity –

an ‘inductive proof’ – will make no sense to you – (and visa versa)

and an inductive methodology – or paradigm – will be regarded as illegitimate

and there really is no argument that will sway someone who cannot see beyond their own conception and practice –

what you are up against here is prejudice – intellectual prejudice

the empirical reality is diversity –

and this empirical reality is a reflection of (or indeed – reflected in) – the logical – that is – propositional – reality –

the reality of the proposal – as the basis of our thought feeling and action –

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

prejudice flies in the face of uncertainty –

uncertainty flies in the face of prejudice

against prejudice all you can do is put that any proposal – any proposition – just simply is – open to question –

and make the point that what goes on in this world – the world we live in – is diverse

that should be a straightforward empirical matter –

a simple observation

‘I can write the rule R like this:

         a + (1 + 1) = (a + 1) + 1          ½

         a + (x + 1)    (a + x) + 1          ½       S

a + ((x + 1) + 1)    (a +(x + 1)) = 1  ½

or like this:

         a + (b +1) = (a + b) + 1

if I take R or S as a definition or substitute for that form.

If I then say that the steps in accordance with the rule R are justified thus:

µ            a + (b + !) = (a + b) + 1

b       a + (b +(c +1) = a + (b + c) = 1) = (a + (b + c)) + 1   }    B

g    (a + b) + (c + 1) = ((a +b) + c) + 1

you can reply: “If that’s what you call a justification, then you have justified the steps. But you haven’t told us any more than if you had just drawn our attention to the rule R and its formal relationship to µ (or to µ, b, and g).”

So I might also have said: I take the rule R in such and such a way as a paradigm for my steps.

Suppose now that Skolem, following his proof of the associative law, takes the step to:

                             a + 1 = 1 + a

                    a = (b + 1) = (a + b) = 1                                 }   C

(b + 1) + a = b + (1 +a) = b + (a +1) = (b + a) + 1

If he says the first and third steps in the third line are justified according to the already proved associative law, that tells us no more than if he said the steps were taken in accordance with the paradigm a + (b + c) = (a + b) + c (i.e. they correspond to the paradigm) and a schema µ, b, g was derived by steps according to the paradigm µ. –

But does B justify these steps, or not?” “What do you mean by the word ‘justify’? – “Well the step is justified if a theorem has been proved that holds for all numbers” – But in what case would that have happened? What do you call a proof that a theorem holds for all cardinal numbers? How do you know that a theorem is valid for all cardinal numbers, since you can’t test it? Your only criterion is the proof itself. So you can stipulate a form and call it the form of the proof that a proposition holds for all cardinal numbers. In that case we really gain nothing by being first shown the

general form of these proofs; for that doesn’t show that the individual proof really gives us what we want from it; because I mean, it doesn’t justify the proof or

demonstrate that it is a proof of a theorem for all cardinal numbers. Instead the recursive proof has to be its own justification. If we really want to justify our proof procedure as a proof of a generalization of this kind, we do something different: we give a series of examples and then we are satisfied by the examples and the law we recognize in them, and we say: yes our proof really gives us what we want. But we must remember that by giving this series of examples we have only translated the notations B and C into a different notation. (For the series of examples is not an incomplete application of the general form, but another expression of the law.) An explanation in word-language of the proof (of what it proves) only translates the proof into another form of expression: because of this we can drop the explanation altogether. And if we do so, the mathematical relationships become much clearer, no longer obscured by the equivocal expressions of word-language. For example, if I put B right beside A, without interposing any expression of word-language like

‘for all cardinal numbers, etc.” then the misleading appearance of a proof of A by B cannot arise. We then see quite soberly how far the relationships between B and A and a + b = b + a extend and where they stop. Only thus do we learn the real structure

and important features of that relationship, and escape the confusion caused by the form of word-language, which makes everything uniform.’

what do you mean by the word ‘justify’?

‘well the step is justified if a theorem has been proved that holds for all cardinal numbers’ –

and if your only criterion is the proof itself –

then justification is what?

asserting – or affirming the proof –

and the proof is really what – an affirmation of the proposition?

so why the need for the so called ‘proof’ – isn’t it just a reaffirmation of the proposition?

the idea of the proof is to give backing to the affirmation of the proposition –

but what backing can you give to affirmation?

you either affirm a proposition or you don’t – you get on with the job at hand – or you don’t

reaffirmation adds nothing here

so justification comes out as – affirmation of the proof – and proof as – reaffirmation of the proposition

reaffirmation (proof) can of course go on indefinitely –

and so you ask – what is the point of the proof?

can I suggest a grammatical analogy here?

what I have in mind is that proof is like – has the same function as –  a full stop in word-language grammar

yes we can go on forever –  affirming – reaffirming –

but we are not going to – and we don’t want to –

and the idea of doing so actually destroys the enterprise

we have to bring our argument to an end

and the ‘have to’ here – is a pragmatic ‘have to’

so – what we know as proof – the process of proof – is best seen as the ritual  developed to bring argument to a stop – to an end

and as with any ritual – it requires adherence –

without adherence – there is no game

even so – the point remains that the ritual of proof is nothing more than rhetoric –

the proposition is put – you can affirm it or deny it –

you can play the game or not play the game –

the need for proof – just indicates propositional insecurity –

and as to that – propositional insecurity – only exists if you believe in propositional security

now there is no such thing

it takes some courage to recognise and accept that any proposition – any proposal – is open to question – open to doubt – is uncertain

but if you do accept this –

what is there to be insecure about?

embrace the uncertainty – work in it and with it –

recognise it as the mark of intellectual freedom – indeed of freedom

understand that each proposal – each proposition – is not a dead truth – but rather a field of possibility

Wittgenstein goes on to say –

that if we really want to justify our proof procedure of a generalization of this kind –

we give a series of examples – and then we are satisfied by the examples and the law we recognize in them –

and then we say – yes our proof really gives us what we want

which is what? – proof –

our proof gives us proof

this tells us nothing

it is clear that what proof amounts to – is affirmation of the proposition

and the various props brought into play – i.e. examples – laws – explanations etc. –

are expressions of that affirmation

do we need proofs – their translations and explanations – to affirm propositions?

no – a proposition can be affirmed and proceeded with – without proofs – that is without reaffirmation –

proofs may be rhetorically useful – but they are not logically relevant –

they are not relevant to the action of mathematics 

they are just shadows

I am not suggesting that mathematicians no longer engage in the proof-game

I am only saying that it should be seen for what it is –

a shadow-play

as to propositional generalization in mathematics –

the generalization functions as a ground or space for speculation –

it opens up possibilities  –

i.e. the possibility of recursion – recursive games

and if understood correctly – generalization sets the scene for a mathematics – that literally flies in the face of the earthbound notion of proof

‘Here we see first and foremost that we are interested in the tree of the structures B, C etc., and that in it is visible on all sides, like a particular kind of branching, the following form

      j(1) = y(1)

j(n + 1) = F(jn)

y(n + 1) = F(yn)

These forms turn up in different arrangements and combinations but they are not elements of the construction in the same sense as the paradigms in the proof of

(a + (b + (c + 1))) = (a + (b +c)) + 1 or (a + b)² = a² + 2ab + b². The aim of the “recursive proofs” is of course to connect the algebraic calculus with the calculus of numbers. And the tree of the recursive proofs doesn’t “justify” the algebraic calculus unless that is supposed to mean that it connects it with the arithmetical one. It doesn’t justify it in the sense in which the list of paradigms justifies the algebraic calculus, i.e. the steps in it.’

no it doesn’t – the list of paradigms illustrates the algebraic calculus –

the tree of recursive proofs – makes use of the algebraic calculus and its connection with the arithmetical one –

it places the algebraic calculus in a new context

and there is no ‘justification’ – apart from use

‘So tabulating the paradigms for the steps makes sense in the cases where we are interested in showing that such and such transformations are all made by means of those transition forms, arbitrarily chosen as they are. But it doesn’t make sense where the calculation is to be justified in a another sense, where mere looking at calculation – independently of any comparison with a table of previously established norms – must shew us whether we are to allow it or not. Skolem did not have to promise us any proof of the associative and commutative laws; he could simply have said he would show us a connection between paradigms of algebra and the calculation rules of arithmetic. But isn’t this hair-splitting? Hasn’t he reduced the number of paradigms? Hasn’t he, for instance replaced [e]very pair of laws with a single one, namely, a + (b + 1) = (a + b) + 1? No. When we prove e.g. (a + b)⁴ = etc. (k) we can while doing so make use of the previously proved proposition (a = b)²  = etc. (1). But in that case the steps in k which are justified by 1 can also be justified by the rules used to prove 1. And the relation of 1 to those first rules is the same as that of a sign introduced to the primary signs used to define it; we can always eliminate the definitions and go back to the primary signs. But when we take a step in C that is justified by B, we can’t take the same step with a + (b + 1) = (a + b) + 1 alone. What is called proof here doesn’t break a step in to smaller steps but does something quite different.’

‘mere looking’ – is that supposed to be some kind of justification?

and ‘whether we are allowed it or not’?

the issue is function – and there is no certainty here –

at best – we use what is at hand – and recognizable in the practice – and here we are talking about the rules – the games of calculation

there can always be argument regarding the practice – how the rules are interpreted –

how the calculation games are interpreted –  and in whatever (mathematical) context – large or small

and context – itself – of course – is open to question – open to doubt – is uncertain

the question ultimately is that of use – that is the game – do you go with it or not?

yes or no –

questions remain

‘Hasn’t he reduced the number of paradigms? Hasn’t he, for instance replaced (e)very pair of laws with a single one, namely, a + (b + 1) = (a + b) + 1? No.’

look – I think the question of paradigms here – do we have more or less? – is rather pointless –

I don’t think it bears on the mathematical action –

it’s a background issue – a question of explanation –

which I think in the end is a matter of style

34 The recursive proof does not reduce the number of fundamental laws

‘So here we don’t have a case where a group of fundamental laws is proved by a smaller set while everything else in the proofs remains the same. (Similarly in a system of fundamental concepts nothing is altered in the later development if we use definitions to reduce the number of fundamental concepts.)

(Incidentally, how very dubious is the analogy between “fundamental laws” and “fundamental concepts”!)’

we are dealing with different laws – different proofs – nothing else

how dubious is the analogy between fundamental laws and fundamental concepts?

what is ‘dubious’ – to say the least – is this notion of ‘fundamental’

fundamental this or fundamental that is just rhetorical rubbish –

the point of the notion of ‘fundamental’ – is to ‘establish’ an authority –

the only ‘authority’ – is authorship –

and the authorship of a proposal – a proposition – is logically irrelevant –

yes – in any propositional enterprise – we start – start somewhere – start with a proposal –

and disciplined propositional practises such as mathematics –

as a matter of practice will have base propositions – or propositions that can be so regarded –

and the practice will be that such propositions are seen to be essential to the practice – or characteristic of the practice –

how this comes about is really a question for the history of the development of the practice –

and as Wittgenstein’s ‘argument’ here shows – the status of any such base proposal – is open to question – open to doubt – is uncertain

clearly there is no ‘fundamental’ proposition – if by that is meant – a proposition – a proposal – that is beyond question – beyond doubt – a proposition that is certain

what we can say is that different propositional practices – are different –

that is the empirical / logical reality

this point applies as much to Skolem’s argument as it does to Wittgenstein’s

Wittgenstein half gets this – but is reluctant or unable to grasp the broader logical implication of uncertainty

the question for the working mathematician – if there is a question for him or her here – is – which propositional arrangement – or propositional argument will – under circumstances – be useful?

i.e. what am I going to use to get to where I am looking to go?

now – again – there will be no certainty here –

we play games with established propositional models –

or we develop new models for new games –

and if we argue for a new proposal – a new game – we hope we can attract some players – it’s as basic as that

‘It is something like this: all that the proof of a ci-deviant fundamental proposition does is to continue the system of proofs backwards. But the recursive proofs don’t continue backwards the system of algebraic proofs (with the old fundamental laws); they are a new system, that seems only to run parallel with the first one.’

yes – exactly – a new – a different practice

‘It is a strange observation that in the inductive proofs the irreducibility (independence) of the fundamental rules must show itself after the proof no less than before. Suppose we said the same thing about the case of the normal proofs (or definitions), where fundamental rules are further reduced, and a new relationship between them is discovered (or constructed).’

yes – the ‘inductive proof’ here becomes what? – an illustration – an approximation – a representation – some kind of picture of the ‘fundamental’ rules –

however you represent – the so called proof – is effectively irrelevant – irrelevant if you hold to the rules

as to definitions – the same applies – you either accept – and work with them – or they are of no use –

playing around with them – ‘reducing’ – and constructing new relations – might be an interesting thing to do – might lead to new ways of seeing and doing – but it is not affirming and then working with the definitions – it’s not – as it were getting on with the job

‘If I am right that the independence remains intact after the recursive proof, that sums up everything I have to say against the recursive “proof”’

this is to say that the recursive proof is in fact irrelevant –

ok – but this is an argument not just in regard to ‘recursive proof’ – but to any proof

it applies to the whole ‘proof enterprise’ –

to all and any of its production lines

‘The inductive proof doesn’t break up the step in A. Isn’t it that that makes me baulk at calling it a proof? It’s that that tempts me to say that whatever it does – even if it is constructed by R and µ – it can’t do more than show something about the step.’

it really just states that the step is made –

and you can ask – do we really need any such statement? –

really what logical purpose does it serve?

isn’t it just a piece of rhetorical flourish?

‘If we imagine a mechanism constructed from cogwheels made simply out of uniform wedges held together by a ring, it is still the cogwheels that remain in a certain sense the units of the mechanism.’

yes – you can indeed describe the mechanism this way –

however it is not the only valid description –

‘the cogwheels as units’ – is a focus –

a focus that will serve certain purposes – it is not the only possible focus

the action of the mechanism – could well be the primary focus

the mechanism – the entity– in the absence of description – any description – logically speaking – is an unknown –

we make known in proposal –

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

the ground of proposal – is the unknown –

at best our proposals – suit our purposes – our uses

‘It is like this: if the barrel is made of hoops and wattles, it is these, combined as they are (as a complex) that hold the liquid and form new units as containers’

yes you can explain the barrel in these terms –

however it is quite clear that for certain uses the barrel may not be regarded as a complex – but rather as a simple –  as a unit

the barrel can be variously described

that is the fact of it – the logical / empirical reality

‘Imagine a chain consisting of links which each can be replaced by two smaller ones. Anything which is anchored by the chain can also be anchored by smaller links instead of by the large ones. But we might also imagine every link in the chain being made of two parts, each perhaps shaped like half a ring, which together formed a link, but could not individually be used as links.

Then it wouldn’t mean at all the same to say, on the one hand: the anchoring done by the large links can be done entirely by small links – and on the other hand; the anchoring can be done entirely by half the links. What is the difference?

One proof replaces a chain with large links by a chain with small links, the other shows how one can put together the old large links from several parts.

The similarity as well as the difference between the two cases is obvious.

Of course the comparison between the proof and the chain is a logical comparison and therefore a completely exact expression of what it illustrates.’

yes – what we have here is different descriptions – different valid descriptions

the proof is a description of the chain – or the chain a description of the proof

and ‘an expression of what it illustrates’ – is ‘an illustration of what is expressed’ –

what we have here without the mumbo jumbo – is a proposal –

and the proposal will be ‘exact’ – to the extent that it is not subjected to question – to doubt –

‘exactness’ – is really just a rhetorical cover – that has no logical significance –

in certain propositional rituals – the description ‘exact’ amounts to propositional satisfaction –

or the decision to proceed

35 Recurring decimals 1/3 =

‘We regard the periodicity of a fraction, e.g. of 1/3 as consisting in the fact that something called the extension of the infinite decimal contains only threes; we regard the fact that in this division the remainder is the same as the dividend as a mere symptom of this property of the infinite extension. Or else we correct this view by saying that it isn’t an infinite extension that has this property, but an infinite series of finite extensions; and it is of this that the property of the division is a symptom. We may then say: the extension taken to one term is 0.3, to two terms 0.33, to three terms

0.333, and so on. That is the rule and the “and so on” refers to the regularity; the rule might also be written “½0.3, 0.x, 0.x3 |”. But what is proved by the division

1/3 = 0.3 is this regularity in contrast to another, not regularity in contrast to

irregularity. The periodic division 1/3 = 0.3 (in contrast to 1/3 = 0.3) proves a

                                                         1                                     1

periodicity in the quotient, that it determines the rule (the repetend), it lays down; it isn’t a symptom that a regularity is “always there”. Where is it already? In things like the particular expansions that I have written on this page. But they aren’t “the expansions”. (Here we are misled by the idea of unwritten ideal extensions, which are a phantasm like those ideal, undrawn geometric straight lines of which the actual lines we draw are mere tracings.) When I said “the ‘and so on’ refers to the regularity” I was distinguishing it from the ‘and so on’ in “he read all the letters of the alphabet: a, b, c and so on”. When I say “the extensions of 1/3 are 0.3, 0.33 and so on” I give three three extensions and – a rule. That is the only thing that is infinite, and only in the same way as the division 1/3 = 0.3

                                          1

One can say of the sign  that it is not an abbreviation.

And the sign “| 0.3, 0. x, 0. x3 |” isn’t a substitute for an extension, but the undervalued sign itself; and the “” does just as well. It should give us food for thought, that a sign like “” is enough to do what we need. It isn’t a mere substitute in the calculus there are no substitutes.

If you think that the peculiar property of the division 1/3 = 0.3 is a symptom of the

                                                                                      1

periodicity of the infinite decimal fractions, or the decimal fractions of the expansion, it is indeed a sign that something is regular, but what? The extensions that I have constructed? But there aren’t any others. It would be a most absurd manner of speaking to say: the property of the division is an indication that the result has the form “½0.a, 0.x, 0.x a½”; that is like wanting to say that a division was an indication that the result was a number. The sign “” does not express its meaning from any greater distance than “0.33 …”, because this sign gives an extension of three terms and a rule; the extension 0.333 is inessential for our purposes and so there remains only the rule, which is given as well by “½0.3, 0.x, 0.x3 |”. The proposition “After the first place the division will yield the same number to infinity”, means “The first remainder is the same as the dividend”, just as the proposition “This straightedge has an infinite radius” means it is straight

We might say: the places of a quotient of 1/3 are necessarily all 3s, and all that could mean would be again that the first remainder is like the dividend and the first place of the quotient is 3. The negation of the first proposition is therefore equivalent to the negation of the second. So the opposite of “necessarily all” isn’t what one might call

‘accidentally all”; “necessarily all” is as it were one word. I only have to ask: what is the criterion of the necessary generalization, and what would be the criterion of the accidental generalization (the criterion for all numbers accidentally having the property e)?’

‘periodicity’ is a property of a propositional game

that is a description – of a feature – a regularity – in a mathematical game

what do we mean by ‘infinite extension’?

what it means is we are proposing – a repeatable action with no end point

e.g. the periodicity of the fraction 1/3 consists in the fact that the extension of the infinite decimal contains only threes –

that the remainder is the same as the dividend – is a property of the game –

a description of its constant – a characterization of its rule

‘and the so on’?

does it refer to the regularity of the periodicity?

the regularity just is the ‘periodicity’

the ‘and so on’ is a propositional directive –

it refers to the action of the game – which is to say – its ‘on-going-ness’

a propositional direction – and nothing more

for the game as such – by definition – can never be completed

(are we to then say it is not a game?

yes you could take this line – with all the implications that follow

however it is important to see that though we have rules to a game –

and that the rules give the game an operational structure –

at the same time a game is defined by its play – its actual play

so the fact is – the practice is –

the game begins with the start of play – and the game ends – with the end of play –

that is – when the play stops)

‘One can say of the sign  that it is not an abbreviation.’

 is a game directive

‘If you think that the peculiar property of the division 1/3 = 0.3 is a symptom of the

                                                                                        1

periodicity of the infinite decimal fractions, or the decimal fractions of the expansion, it is indeed a sign that something is regular, but what?’

is this ‘peculiar’ property (‘the remainder is the same as the dividend’) a symptom of the periodicity?

saying that ‘the remainder is the same as the dividend’ –

expresses the constant of this game

so a ‘symptom’ – I suppose yes –

if ‘symptom’ is understood as the game (logical) constant

‘a sign that something is regular but what?’

the periodicity is a regularity –

or ‘periodicity’ is a description of a regularity

you can play games the principle of which is regularity – or games where the principle is irregularity –

where the game is based on the principle of regularity – what is regular is the action of the game –

periodicity is a form or expression of a regularity

and as to ‘regularity’? –

an ordered (rule governed) succession –

in short – a game

‘We might say: the places of a quotient of 1/3 are necessarily all 3s,…’

that is the rule in play –

‘necessity’ comes out as nothing more than the game as constructed and played –

how the game is constructed is a contingent fact

how the game is played is a contingent fact

36 The recursive proof as a series of proofs

‘A recursive proof is the general term of a series of proofs. So it is a law for the construction of proofs. To the question how the general form can save me the proof of a particular proposition, e.g. 7 + (8 + 9) = (7 + 8) + 9, the answer is that it merely gets everything ready for the proof of the proposition, it doesn’t prove it (indeed the proposition doesn’t occur in it). The proof consists rather of the general form plus the proposition.’

yes – that is fair enough –

the way I would put it though is that a ‘recursive proof’ is the general term for a series of propositional games

‘Our normal mode of expression carries the seeds of confusion right into its foundations, because it uses the word “series” both in the sense of “extension” and in the sense of “law”. The relationship of the two can be illustrated by a machine for making coiled springs, in which a wire is pushed through a helically shaped passage to make as many coils as are desired. What is called an infinite helix need not be anything like a finite piece of wire, or something that that approaches the longer it becomes; it is the law of the helix, as it is embodied in the short passage. Hence the expression “infinite helix” or “infinite series” is misleading.’

yes – the law of the helix – is a constructive proposal – not an operational proposal

what is termed an ‘infinite helix’ – is not a constructive proposal – rather an operational proposal

as to ‘infinite series’ –

an ‘infinite series’ is a proposed extension –

a series is a propositional action

a series is not a law –

a series will be ‘governed’ by a law –

that is to say – ‘shaped by an over-riding proposal’ –

and in so far as the series reflects the law –

it can be said that the law is ‘in’ the series

‘So we can always write out the recursive proof as a limited series “and so on” without it losing any of its rigour. At the same time this notation shows more clearly its relation to the equation A. For the recursive proof no longer looks at all like a justification of A in the sense of an algebraic proof – like the proof of (a + b)^2. . That proof with algebraic calculation is quite like calculation with numbers.’

we can propose a limitless series –

but we can’t write out a limitless series –

so yes ‘we can always write out the recursive proof as a limited series’

and it is just here that you might ask – what is the point of proposing a limitless series?

I can only think that the value of such a proposal – is epistemological

it makes the point that there is no end game

in practice – we operate within limitations – and therefore we play to end games

the notion of a limitless series – keeps the game open – logically – if not practically –

really it ushers in a different mind-set –

the ‘and so on’ indicates this

Wittgenstein here in saying that we can ‘write out the recursive proof as a limited series and the “and so on” without losing any of its rigour’ –

is really just suggesting we can reconcile recursive mathematics with the algebraic form –

(and I can’t see that Wittgenstein’s view of ‘and so on’ here – has any significance at all – it’s just a dangler)

this ‘reconciliation’ – may indeed be just what in fact happens in mathematics – and so be it

my point is the that the recursive game and the algebraic game – are two different mathematical games

that one is not the other – is no cause for concern –

we are not here defending an empire

and of course ‘the recursive proof no longer looks at all like a justification of A in the sense of an algebraic proof’

we can forget justification – it is just a rhetorical device – to bring consideration to an end

and the point of the ‘recursive proof’ so called – is to indicate an on-goingness –

in so far as Skolem regarded the recursive proof as a justification – he misrepresented it and warped it

what we have with recursion is a game that in a sense plays itself

it does not require justification – and it justifies nothing

‘5 + (4 + 3) = 5 + (4 + (2 + 1)) = 5+ ((4 + 2) + 1) =

= (5 + (4 + 2)) = 1 = (5 + (4 + ( 1 + 1))) + 1 =

= (5 + ((4 + 1) + 1)) + 1 = ((5 + (4 +1)) = 1) + 1 =

= (((5 +4) + 1) + 1) + 1 = ((5 + 4) + 2) + 1 = (5 + 4) +3) …(L)

That is a proof of 5 + (4 + 3) = (5 + 4) + 3, but we can also let it count, i.e. use it, as a proof of 5(+ (4 +4) = (5 + 4) + 4, etc.

If I say that L is the proof of the propositions a + (b + c) = (a + b) = c, the oddness of the step from the proof to the proposition becomes much more obvious.’

if you know the relevant rules to this sign-game ‘5 + (4 + 3) = (5 + 4) + 3’ – that is all that counts

and if you know the relevant rules to the sign-game ‘5(+ (4 +4) = (5 + 4) + 4, etc. – that is all that counts

two games – two sets of rules

there is in fact – no question of a so called ‘proof’ – being in any sense relevant here

the question – just does not arise

yes – you can play other sign games that reflect the relevant rules here –

however doing so adds nothing to the ‘5 + (4 + 3) = (5 + 4) + 3’ game and its rules

proofs are affirmation games – that are entirely unnecessary – if you know the rules of the game

they don’t in fact serve any logical or mathematical end

perhaps they serve some psychological need?

or as I like to think they are just fun to do

in any case the fact remains they are part and parcel of the practice –

perhaps they are really the advertisement for the product?

                                                                                                                                  

if so we shouldn’t be confusing the advertisement with the product

this whole issue of proof is a good example of just how rhetoric can infiltrate and take hold in logic

this is no great surprise – but is something philosophers need to be aware of

‘Definitions merely introduce practical abbreviations; we could get along without them. But is that true of recursive definitions?’

‘practical abbreviations’ of? –

rules have to have some formulation – and directions have to be given – stated –

in so far as definitions do this – they have a place and function

recursive definitions give rules and direction –

however it is almost as if they have an added dimension of motion  –

and so a more operational sense or focus to them

‘Two different things might be called applications of the rule a + (b + 1) = (a + b)  +1:

in one sense 4 + (2 + 1) = (4 + 2) = (4 + 2) + 1 is an application, in another sense

4 + (2 + 1) = ((4 + 1) + 1) + 1 = (4 + 2) + 1 is.’

yes – and any rule – is open to different interpretations – different applications –

so just what application a rule is in fact given – is a matter of the context or circumstance in which it is used

‘The recursive definition is a rule for constructing replacement rules, or else the general term of a series of definitions. It is a signpost that shows the same way to all expressions of a certain form.’

the recursive definition defines the recursive game

and as for ‘a signpost that shows the way to all expressions of certain form’ – what else could it be?

any ‘definition’ is a ‘signpost that shows the way to all expressions of a certain form’

‘As we said, we might write the inductive proof without using letters at all (with no loss of rigour). Then the recursive definition a + (b + 1) = (a + b) + 1 would have to be written as a series of definitions. As things are, this series is concealed in the explanation of its use. Of course we can keep the letters in the definition for the sake of convenience, but in that case in the explanation we have to bring in a sign like

“1, (1) + 1, ((1) + 1) = 1 and so on”, or what boils down to the same thing,

“| 1, x, x + 1 |”. But here we mustn’t believe that this sign should really be

“(x). ½1, x, x + 1 |”!’

yes – exactly

‘The point of our formulation is of course that the concept “all numbers” is given only in a structure like “½1, x, x + 1 |”. The generality is set out in the symbolism by this structure and cannot be described by an (x). fx.

Of course the so-called “recursive definition” isn’t a definition in the customary sense of the word, because it isn’t an equation, since the equation a + (b + 1) = (a + b)  +1 is only a part of it. Nor is it a logical product of equations. Instead it is only a law for the construction of equations; just as ½1, x, x + 1 | isn’t a number but a law etc. (The bewildering thing about the proof of a + (b + c) = (a + b) + c is of course that it is supposed to come out of the definition alone. But µ isn’t a definition, but a general rule for addition).’

really is it a law for the construction of equations? –

isn’t it rather like this –

that an equation is given or proposed – and if the ‘recursive definition’ is brought into play – it sets out an approach to the equation – effectively – a never ending approach

it is a game played as were in response to the equation –

the equation here functions as the limit of the recursive game – a limit that logically cannot be reached – and so the game goes on – and that’s the idea

‘On the other hand the generality of this rule is no different from that of the period division 1 /3 = 0.3. That means, there isn’t anything that the rule leaves open or in

                1

need of completion or the like.’

‘the generality of this rule’ –

generality is non-restrictive – within a given propositional domain – i.e. the domain of numbers – e.g. ‘all numbers’

a general proposition expresses this non-restrictiveness

1 /3 = 0.3 – is an equation – a propositional game – it is a closed propositional system

   1

and yes it leaves nothing open –

however it is quite different to “½1, x, x + 1 |” – to the non-restrictive proposition

the equation is ‘general’ only in the sense that its domain is non-specific

‘Let us not forget: the sign “|1, x, x + 1|” … N interests us not as a striking expression for the general term of the series of cardinal numbers, but only in so far as it is contrasted with signs of similar construction. N as opposed to something like

|2, x, x +3|; in short, as a sign, or an instrument, in a calculus. And of course the same holds for 1 /3 = 0.3. (The only thing left in the rule is its application.)

                 1

1 + (1 + 1) = (1 + 1) + 1, 2 + (1 + 1) = (2 + 1) + 1, 3 + (1 + 1) = (3 + 1) + 1 …

and so on

1 + (2 + 1) = (1 + 2) + 1, 2 + (2 +1) = (2 + 2) + 1, 3 + (2 + 1) = (3 + 2) + 1 …

and so on

1 + (3 + 1) = (1 + 3) + 1, 2 + (3 + 1) = (2 + 3) + 1, 3 + (3 + 1) = (3 + 3) + 1 …

and so on

and so on.’

yes – the issue is application / use – in whatever way the rule is interpreted or signed

‘We might write the rule “a + (b + 1) = (a + b) + 1”, thus.

         a + (1 + 1) = (a + 1) + 1

                ¯                    ¯                    R

         a + (x + 1)     (a + x) + 1

 a + ((x + 1) + 1)   ((a +  x) + ) + 1 

In the application of the rule R (and the description of the application is of course an inherent part of the sign for the rule), a ranges over the series½1, x, x + 1 |; and of course that might be expressly stated by an additional sign, say “a ® N”. (We might call the second and third lines of the R taken together the operation, like the second and third term of the sign N.) Thus too the explanation of the use of the recursive definition “a + (b + 1) = (a + b) + 1” is a part of that rule itself; or if you like a repetition of the rule in another form; just as “1, 1 + 1, 1 + 1 + 1 and so on” means the same as (i.e. is translatable into) “|1, x, x + 1|”. The translation into word-language casts light on the calculus with new signs, because we have already mastered the calculus with the signs of word-language.

The sign of the rule, like any other sign, is a sign belonging to a calculus; its job isn’t

to hypnotize people into accepting an application, but to be used in the calculus in accordance with a system. Hence the exterior form is no more essential than that of an arrow ®; what is essential is the system in which the sign for the rule is employed.

The system of contraries – so to speak – from which the sign is distinguished etc.

What I am here calling the description of the application is itself of course something that contains an “and so on”, and so it can itself be no more a supplement to or substitute for the rule-sign.’

                            

yes the sign of the rule is a sign belonging to a calculus – a propositional system – a proposition tradition – a propositional practise

and in the event of different perspectives – different arguments – and in the development of different traditions –

different signs will be used  and developed with different descriptions – and different practises will come into play

‘and so on’ can well be seen as a supplement or substitute for the sign rule –

this is neither here nor there

what the game is played with – and how it is described at a theoretical level – is open to question – open to interpretation –

however when we get to doing mathematics – actually playing the game – at the least we begin with the propositional structures that are already in place

and yes – whether one set of symbols (and all that goes with it) – has some kind of advantage over another (i.e. simplicity – clarity – comprehensiveness etc.) – will always be an interesting and useful question

language is relevant – but action is the game –

and we should at any point in the propositional action understand that the ground of our symbolism is uncertainty

‘What is the contradictory of a general proposition like a + (b + (1 + 1)) = a + ((b + 1) + 1)? What is the system of propositions within which this proposition is negated? Or again, how, and in what form, can this proposition come into contradiction with others? What question does it answer? Certainly not the question whether (n).fn or

($n). ~ fn is the case, because it is the rule R that contributes to the generality of the proposition. The generality of a rule is eo ipso incapable of being brought into question.’

the negation of the generality of a rule – has the effect of just denying the rule –

and prime facie to do so is an operational dead end

it is not that the generality of the rule is eo ipso incapable of being brought into question – of course it can be brought into question –

it is of the nature of the proposition that it is open to question – open to doubt –

that it is uncertain

it is just that if it is brought into question – the game as it is – as it is practised –

will not proceed

and presumably – if it is denied or seriously questioned – any such denial or doubt would come out of an alternative point of view which proposes a better way to proceed –

and if it doesn’t come out of a positive alternative – then it will not be given any  consideration at all

‘Now imagine the general rule written as a series

P_11, P₁₂ P₁₃ …

P₂₁ P₂₂ P₂₃ …

P₃₁ P₃₂ P₃₃ …

……………….

and then negated. If we regard it as (x).fx, the we are treating it as logical product and its opposite is the logical sums of the denials of p_{11 ×}×   p₂₁ × p₂₂ ××× p_mn × Certainly if you compare the proposition with a logical product, it becomes infinitely significant and its opposite void of significance). (But remember that the “and so on” in the proposition comes after a comma, not after an “and” (“ . ”) The “and so on” is not a sign of incompleteness.)

Is the rule R infinitely significant? Like an enormously long logical product?

That one can run the number series through the rule is a form that is given; nothing is affirmed about it and nothing can be denied about it.

Running the stream of numbers through is not something which I can prove. I can only prove something about the form, or pattern, through which I run the numbers.

But we can’t say that the general number rule a + (b + c) = (a + b) + c …A) has the same generality as a + ( 1 + 1) = a + (a + 1) + 1 (in that the latter holds for every cardinal number and the former for every triple of cardinal numbers) and that the inductive proof of A justifies the rule A? Can we say that we can give the rule A, since the proof shows that it is always right? Does 1 /3 = 0.3 justify the rule

                                                                                  1

  1              2                  3

1/3 = 0.3, 1/3 = 0.33, 1/3 = 0.333 and so on” … P)

A is a completely intelligible rule; just like the replacement rule P. But I can’t give such a rule, for the reason that I can’t calculate the instances of A by another rule; just as I cannot give P as a rule if I have given a rule whereby I can calculate

  1              

1/3 = 0.3 etc.’

the ‘and so on’ – is not a sign of incompleteness – it is a sign of ‘on-goingness’

‘is the rule R infinitely significant?’

the rule R is a game sign – its action is on-goingness with no logical end point –

that is the game

‘like an enormous logical product’ –

R is not a logical product – though indeed – R is productive –

a way of looking at is to say –

R is a propositional game – the point of which just is that it has no logical product

‘That one can run the number series through the rule is a form that is given; nothing is affirmed about it and nothing can be denied about it.’

and –

‘Running the stream of numbers through is not something which I can prove. I can only prove something about the form, or pattern, through which I run the numbers.’

yes – what we have here is a game

the inductive proof of A – is no more than an account – an ‘explanation’ of A – effectively – a restatement of it

‘justification’ – if it comes to anything – just comes down to – use

it is not a question of whether the rule A is always right – it is rather does it have a function – does it have a use?

and it does not follow that because you can’t calculate the instances of A by another rule – that you can’t give the (‘completely intelligible’) rule A

the question is where and how is it to be used

‘But I can’t give such a rule, for the reason that I can’t calculate the instances of A by another rule; just as I cannot give P as a rule if I have given a rule whereby I can calculate

  1              

1/3 = 0.3 etc.’

well that you can’t calculate the instances of A by another rule – is only to say that  the instances of A will be calculated in terms of a rule and by a method appropriate to A

in any sophisticated mathematical environment there will be operative – different paradigms – different propositional systems – different rules and different methods of calculation –

in such a context we can have parallel and indeed conflicting propositional games – and of course argument regarding their value and utility

‘How would it be if someone wanted to lay down “25 x 25 = 625” as a rule in addition to the multiplication rules. (I don’t say “25 x 25 = 624”!) – 25 x 25 = 625 only makes sense if the kind of calculation to which the equation belongs is already known, and it only makes sense in connection with that calculation. A only makes sense in connection with A’s own kind of calculation. For the first question here would be: is that a stipulation, or a derived proposition? If 25 x 25 = 625 is a stipulation, then the multiplication sign does not mean the same as it does, e.g. in reality (that is we are dealing with a different kind of calculation). And if A is a stipulation, it doesn’t define addition in the same way as if it is a derived proposition. For in that case the stipulation is of course a definition of the addition sign, and the rules of the calculation that allow A to be worked out are a different definition of the same sign. Here I mustn’t forget that µ, b, g isn’t the proof of A, but only the form of the proof, or what is proved; so µ, b g is a definition of A.

Hence I can only say “25 x 25 = 625 is proved” if the method of proof is fixed independently of the specific proof. For it is this method that settles the meaning of

“x x h” and so settles what is proved. So to that extent the form a.b = c belongs to the

                                                                                                        a

method of proof that defines the sense of the proposition A.

Arithmetic is complete without a rule like A; without it it doesn’t lack anything. The proposition A is introduced into arithmetic with the discovery of periodicity, with the construction of a new calculus. Before this discovery or construction a question about the correctness of that proposition would have as little sense as a question about the correctness of “1/3 = 0.3, 1/3 = o.33 … ad inf.”

The stipulation of P is not the same thing as the proposition “1/3 = 0.3” and in that

                          .                     .

sense “a + (b + c) = (a + b) = c) is different from the rule (stipulation) such as A. The two belong to different calculi. The proof of µ, b, g is proof or justification of a rule like A only in so far as it is the general form of the proof of arithmetical propositions of the form A.’

laying down a rule – in addition to an arithmetical rule is either irrelevant –  or a restatement i.e. an abbreviation of the rule –

but Wittgenstein is correct ‘A only makes sense in connection with A’s  own kind of calculation –

or more generally you can say – calculation is paradigm dependent – game dependent

so yes –

‘If 25 x 25 – 625 is a stipulation, then the multiplication sign does not mean the same as it does, e.g. in reality (that is we are dealing with a different kind of calculation’

‘reality’ by the way is no more than an ‘accepted or dominant practice’

a stipulation – or – a calculation?

different forms – different approaches – different practices –

and any form or practice will have its argument – and its reasons –

what we have to understand is that there are different ways of proceeding – different methods – different practices – i.e. –

‘The stipulation of P is not the same thing as the proposition “1/3 = 0.3” and in that

sense “a + (b + c) = (a + b) = c) is different from the rule (stipulation) such as A. The two belong to different calculi’

and underlying this – is the logical reality – that any proposal – any proposition – any propositional system – is open –

open to question – open to doubt – is uncertain

understanding what fits where – what works with what – is finally a matter of observation – 

precise observation – of practice

proof as I have said is restatement – is rhetoric –

however in so far as this notion of proof – is standard practice – then Wittgenstein’s point that ‘Hence I can only say “25 x 25 = 625 is proved” if the method of proof is fixed independently of the specific proof’ – makes sense –

and yes – ‘The proof of µ, b, g is proof or justification of a rule like A only in so far as it is the general form of the proof of arithmetical propositions of the form A.’

this is all to do with the ritual of proof

the fact of it is though – arithmetic does not need a ‘rule’ like A –

and Wittgenstein is right – periodicity is the creation / construction of a new calculus

a new numerical game

the relation of periodicity to arithmetic – is to be described in terms of a family resemblance between numerical games

and if you were to go down that path –

you might say that periodicity launches from an arithmetical background – but once up and running is quite a different beast

‘Periodicity is not a sign (symptom) of a decimal’s recurring; the expression “it goes on like that forever” is only the translation of the sign for periodicity into another form of expression. (If there was something other than the periodic sign of which periodicity was only a symptom, that something would have to be a specific expression, which could be nothing less than the complete expression of that something.)’

periodicity –  is the periodic sign – in whatever way it is formulated

the periodic sign is a game-sign

‘it goes on forever’ – is a characterization of the logic of the recurrence-game – which is to say a characterization of the proposed action of the game

37 Seeing or viewing a sign in a particular manner. Discovering an aspect of a mathematical expression. “Seeing an expression in a particular way”. Marks of emphasis.

‘Earlier I spoke of the use of connection lines, underlining etc. to bring out the corresponding, homologous, parts of the equations of a recursive proof. In the proof

the one marked for µ for example corresponds not to b but to c in the next equation; and b corresponds not to d but to e; and g not to d but to c + d, etc.

Or in

i doesn’t correspond to x and e doesn’t correspond to l; it is b that i corresponds to; and b not correspond to x, but x corresponds to q and µ to d and b to g and g to m, not to q, and so on

What about a calculation like

(5 + 3)²  = (5 + 3).(5 + 3) = 5. (5 + 3) + 3.(5 + 3) =

    = 5.5 + 5.3 + 3.5 + 3.3 = 5² + 2.5.3 + 3² …R)

from which we can also read a general rule for the squaring of a binomial?

We can as we were look at this calculation arithmetically or algebraically.’

connection lines are proposals – relational conjectures –

and behind any such relational conjecture – argument

‘We can as it were look at this calculation arithmetically or algebraically’

yes – and the more general point that underlies this statement is that the proposition is open – open to interpretation 

that the ground of our reality – of propositional reality – is openness –

is uncertainty

and the point just is that our practice doesn’t diminish this openness – this possibility of propositional interpretation and innovation –

rather it feeds off it and reflects it

‘The difference between the two ways of looking at it would have been brought out e.g. if the example had been written

                        µb       b

(5 + 2)² = 5² + 2.2.5 + 2²

In the algebraic way of looking at it we would have to distinguish the 2 in the position marked µ from the 2s in the position marked b but in the arithmetical one they would not need to be distinguished. We are – I believe – using a different calculus in each case.’

yes – a different calculus in each case –

algebra and arithmetic – are related but different games –

that is different propositional constructs

                                                                                                                                  

so how you play them will be different

‘how you play’ – is the calculus

                                                                                                                                  

‘According to one and not the other way of looking at it the calculation above, for instance, would be a proof of (7 + 8)² = 8²  = 2 .7. 8 + 8².’

yes – you could play this propositional game

bare in mind – proof is a restatement –

and if you are operating within one propositional form – a restatement of a proposition in that form – should be possible –

and some restatements are quite ingenious

‘We might work out an example to make sure that (a + b)²  is equal to a² + b² + 2ab, not a² + b²  + 3ab – if we had forgotten it for instance; but we couldn’t check in that sense whether the formula holds generally. But of course there is that sort of check too, in the calculation

(5 + 3)² = … = 5²  + 2.5.3 + 3²

I might check whether the 2 in the second summand is a general feature of the equation or something that depends on the particular numbers occurring in the example.’

checking here is what?

proposal – and argument

the calculation is of course open – open to question – open to interpretation –

as is any proposal or argument in relation to it

‘I turn (5 + 2)²  = 5² + 2.2.5 into another sign, if I write

 µ    b      µ^-    -  bµ    b^-

(5 + 2)² = 5² + 2.2.5 + 2²

and thus “indicate which features of the right hand side originate from the particular numbers on the left” etc.

(Now I realize the importance of this process of coordination. It expresses a new way of looking at the calculation and therefore a way of looking at a new calculation.)’

yes – any proposition – any proposal – is open  -

open to propositional interpretation – open to propositional invention –

in mathematics we work with propositional structures –

the essential characteristic of these structure is uncertainty –

this the ground of creativity

‘ ‘In order to prove A’ – we could say – I first of all have to draw attention to quite definite features of B. (As in the division 1.0/3 = 0. ).’

this is just to say that ‘drawing attention’ – is proposing – 

and the only restrictions on this are those you impose yourself –

for whatever reason

‘(And µ had no suspicion, so to speak, of what I see if I do.)

Here the relationship between generality is like the relationship between existence and the proof of existence.’

the ‘proof’ of existence – is the assertion – of …

‘When µ, b g are proved, the general calculus has still to be discovered.’

when µ, b g are proposed and argued for –

the general calculus has still to be proposed – argued for –

and accepted (used)

‘Writing “ a + (b + c) = (a + b) + c” in the induction series seems to us a matter of course, because we don’t see that by doing so we are starting a totally new calculus. (A child just learning to do sums would see clearer than we do in this connection.)’

yes –

or it just puts pay to the induction as such

that there is this proposition in the inductive series – compromises the induction –

this doesn’t mean that the series with the proposition will not be used –

if it is – its description in logical terms becomes obviously questionable –

a moment of truth

‘Certain features are brought out by the schema R; they could be specifically marked thus:

Of course it would also have been enough (i.e. it would have been a symbol of the same multiplicity) if we had B and added

f₁ x = a + (b + x), f₂ x = (a + b) + x

(Here we must also remember that every symbol – however explicit – can be misunderstood.)’

the point is rather that regardless of how explicit a symbol is – it is open to interpretation

and what of ‘explicit’? –

apart from the fact that the symbol is made and is apparent – written –

‘explicit’ only has any meaning – sharp or not – in terms of the interpretation given it

‘The first person to draw attention to the fact that B can be seen in that way introduces a new sign whether or not he goes on to attach special marks to B or to write the schema R beside it. In the latter case R itself is the new sign, or if you prefer, B plus R. It is the way in which he draws attention to it that produces the new sign.’

that ‘B can be seen in that way’ –

if a new way of seeing is proposed – then the new proposal needs to be distinguished from the previous understanding if it is to be ‘real’ – functional and useful – it (the new proposal ) – must be signed – in some way or another – otherwise for all intents

and purposes – it doesn’t exist – it’s not there

how he draws attention to it – will most likely be an account of how he came up with the new sign –

but really this ‘how’– in terms of the signs utility –

is neither here nor there

‘We might perhaps say that here the lower equation is used as a + b = b + a; or similarly that here B is used as A, by being as it were read sideways. Or: B was used as A, but the new proposition was built up from µ. b . g, in such a way that though A is now read out of B, µ. b . g don’t appear in the sort of abbreviation in which the premises turn up in the conclusion.’

in some ways this is just like looking at a picture – i.e. a picture in a gallery

in this case a straightforward propositional picture –

but in logical terms – it is no more than to say that any proposal – any proposition – indeed – any propositional construct – is open to question – open to interpretation –

this is always on the go – in any propositional context –

a particular interpretation is adopted by a particular group – at a particular time – for a particular purpose

‘What does it mean to say: ”I am drawing your attention to the fact that the same sign occurs here in both function signs (perhaps you didn’t notice)”? Does that mean that he didn’t understand the proposition? – After all, what he didn’t notice was something which belonged essentially to the proposition; it wasn’t as if it was some external property of the proposition he hadn’t noticed (Here again we see what kind of thing is called “understanding a proposition”.)’

what does it mean?

well frankly it could mean anything –

it could be a statement of the obvious –

it could be a direction of focus that the other had not had –

it could be a lead into an argument –

who is to say?

the point is unless you are in that particular context – at that particular time you don’t really have a start here

of course we can speculate – but that as it turns out is all we can do –

even in a particular context at a particular time                                                                                                                                   

‘understanding a proposition’ –

just is recognizing that any proposition – that is – any proposal – is open to question – open to doubt – is – logically speaking – uncertain

drawing attention – is proposing –

it is simply – common and garden –

propositional activity

‘Of course the picture of reading a sign lengthways and sideways is once again a logical picture, and for that reason it is a perfectly exact expression of a grammatical relation. We mustn’t say of it “it’s a mere metaphor, who knows what the facts are really like?”’

yes we begin with a logical picture – but the truth is that any picture – any picture at all – is a logical picture –

‘logical’ – in that a ‘picture’ – any picture – is open to question – open to interpretation

in the absence of interpretation what you have is an unknown –

our ‘knowledge’ is our proposals – our responses to the unknown – our interpretations

reading a picture lengthways – is a different interpretation to reading it sideways

any expression of a grammatical relation – that is any proposal of a grammatical relation – is open to question – open to doubt – is uncertain

exactness is a rhetorical ploy

‘who knows what the facts are really like?’ –

the facts – are just what is proposed

‘When I said that the new sign with the marks of emphasis must have been derived from the old one without the marks, that was meaningless, because of course I can consider the sign with the marks without regard to its origin. In that case it presents itself to me as three equations (Frege), that is as the shape of three equations with certain underlings, etc.

It is certainly significant that this shape is quite similar to the three equations without the under linings; it is also significant that the cardinal number 1 and the rational number 1 are governed by similar rules; but that does not prevent what we have here from being a new sign. What I am now doing with this sign is something quite new.’

yes

‘Isn’t this like the supposition I once made that people might have operated the Frege-Russell calculus of truth functions with the signs “~” and “.” combined into

“~p. ~q” without anyone noticing, and that Sheffer, instead of giving a new definition, had merely drawn attention to a property of signs already in use.’

We might have gone on dividing without ever becoming aware of recurring decimals. When we have seen them, we have seen something new.’

we are not dealing here with the properties of signs –

or to put it bluntly signs as such have no properties –

signs are tools of use –

the question is only – how are they used? –

and yes you can background use with theory – i.e. talk of ‘properties’ –

and doing so you may well bolster (rhetorically) an argument for use –

but one has to be careful here about putting the cart before the  horse

Sheffer – in ‘drawing attention to a property of signs already in use’ –

was actually proposing a new and different use

and yes – unless something new is proposed – you will operate with the proposals in play – that is you will operate with the status quo

‘But couldn’t we extend that and say “I might have multiplied numbers together without ever noticing the special case in which I multiply a number by itself; and that means ‘x²’ the expression of our having become aware of that special case. Or, we might have gone on multiplying a by b and dividing it by c without noticing that we could write “a.b” as “a.(b.c)” or that the latter is similar to a.b. Or again, this is like a

                      c

savage who doesn’t yet see the analogy between ½½½½½ and  ½½½½½½, or between

½½and  ½½½½½.’

yes – what you are aware of – and what you might notice – depends in the first place on the propositional context or contexts that you can bring to bear –

in short the state of knowledge you have access to

which means – what has been proposed – and what – of what has been proposed – has a currency – as determined by yourself and the others who have a stake in the matter under consideration

as for the savage – the savage sees what he sees – and proposes what he proposes –

just like the mathematician

‘You might see the definition U, without knowing why I use that abbreviation.

You might see the definition without understanding its point. – But its point is something new, not something already contained in it as a specific replacement rule.’

any use of any proposal – definition or not – is open to question – to doubt – is uncertain – at any stage of its use

not knowing ‘why’ someone uses a proposition / definition – or understanding its point – opens its use up to question and invites explanation

and if a definition / proposition – is ‘new’ – what we will look for is a demonstration of its utility

‘Of course, “Á” isn’t an equals-sign in the same sense as the ones occurring in µ, b, g.

But we can easily show that that  “Á” has certain formal properties in common with =.’

the use of a sign – and an unusual sign in a particular context – begs explanation –

that is to say if we were to proceed with such a sign we would be looking for propositional elaboration

‘It would be incorrect – according to the postulated rules – to use the equals-sign like this:

D …½(a + b)²  = a.(a + b) + b.(a + b) = … =

= a² + 2ab + b²½. = .½(a + b)²  = a² + 2ab + b² ½

if that is supposed to mean that the left hand side is the proof of the right.

But mightn’t we imagine this equation regarded as a definition? For instance, if it had always been the custom to write out the whole chain instead of the right hand side, and we introduced the whole abbreviation.’

‘postulated rules’ – the rules in use –

and yes we can imagine – or re-imagine the ‘equation’ – the string of symbols –

in the end how the symbolism is used – is a matter of practice – that is to say – just how it is used –

if the symbolism is re-imagined in away that practitioners don’t understand – that is to say in a way that is not considered functional in the current propositional / mathematical environment –

then the new interpretation will not be used –

it will not get a guernsey

‘Of course D can be regarded as a definition! Because the sign on the left hand side is in fact used, and there is no reason why we shouldn’t abbreviate it according to the convention. Only in that case either the sign on the right or the sign on the left is used in a different way from the one now usual.

It can never be sufficiently emphasized that totally different kinds of sign-rules get written in the form of an equation.

The ‘definition’ x.x = x² might be regarded as merely allowing us to replace the sign “x.x” by the sign x²,” like the definition “1+ 1 = 2”; but it can also be regarded (and in fact is regarded) as allowing us to put a² instead of a.a, and (a + b)² instead of

(a + b).(a + b) and in such a way that any arbitrary number can be substituted for the x.

A person who discovers that a proposition p follows from one of the form q É p.q constructs a new sign, the sign for that rule. (I am assuming that a calculus with

p, q, É, has already been in use, and that this rule is now added to make it a new calculus.)’

yes

‘It is true that the notation “x²” takes away the possibility of replacing one of the factors x by another number. Indeed, we could imagine two stages in the discovery (or construction) of x^2.. At first, people might have written “x⁼” instead of “x²”, before it occurred to them that there was a system x.x, x.x.x, etc; later they might have hit upon that too. Similar things have occurred in mathematics countless times. (In Leibig’s sign for an oxide oxygen did not appear as an element in the same way as what was oxidized. Odd as it sounds, we might even today, with all the data available to us, give oxygen a similarly privileged position – only in the form of representation – by adopting an incredibly artificial interpretation, i.e. grammatical construction.)’

yes – one way or another it is a question of construction – propositional construction –

and indeed any proposition – any construction – is open to question – open to doubt –

the question is what will work in what propositional context?

‘The definitions x.x = x², x.x.x = x³ don’t bring anything into the world except the signs “x²” and “x³” (and thus so far it isn’t necessary to write numbers as exponents).’

it may not be ‘necessary’ – but it is a practise – an accepted practise – and I would say – obviously useful

‘½The process of generalization creates a new sign-system½’

ok – if that’s how it goes

‘Of course Scheffer’s discovery is not the discovery of the definition ~p. ~q = p½q. Russell might well have given that definition without being in possession of Scheffer’s system, and on the other hand Scheffer might have built up his system without the definition. His system is contained in the use of the signs “~p. ~q” for

“~p” and “~(~p. ~q). ~(~p. ~q)” for “p v q” and all “p½q” does is to  permit an abbreviation. Indeed, we can say that someone could well have been acquainted with the use of the sign “~(~p. ~q). ~(~p. ~q)” for “p v q” without recognizing the system

p½q. ½. p½q. in it.’

yes – an abbreviation

here we are talking about interpretation of symbolic proposals – symbolic propositions

and look an interpretation – variation – abbreviation – is valid if it functions –

and whether it functions or not is a matter of argument

the fact is logicians and mathematicians come at propositional representation from different points of view – different propositional paradigms –

so at the least – in any vibrant intellectual context – you might expect different formulations

and that a formal proposition can be written in a different manner just points to the underlying uncertainty of any propositional construction or use

a proposition – whatever form it takes – is a proposal – open to question – open to doubt – thus – uncertain

and this is to state the obvious –

logic and mathematics are explorations of uncertainty

‘It makes it clearer if we adopt Frege’s two primitive signs “~” and “.”. The discovery isn’t lost if the definitions are written ~p. ~p = ~p and ~(~p. ~p). ~(~q. ~q) = p.q. Here apparently nothing at all has been altered in the original signs.’

yes

and ‘primitive’ – is just the decision to suspend question – suspend doubt –

to make a play for certainty

‘primitive’ – is rhetoric

‘But we might also imagine someone’s having written the whole Fregian or Russellian logic in this system, and yet, like Frege, calling, “~” and  “.”. his primitive signs, because he did not see the other system in his propositions.’

yes – of course

‘It is clear that the discovery of Sheffer’s system in ~. p. ~p = ~p and ~(~p. ~p). ~(q. ~q) = p. q corresponds to the discovery that  x² + ax + a² is a specific instance of

                                                                                       4

 a² + 2ab + b².’

a fair enough argument

‘We don’t see that something can be looked at in a certain way until it is so looked at.

We don’t see that an aspect is possible until it is there.’

we don’t see what is there until it is proposed

‘That sounds as if Sheffer’s discovery wasn’t capable of being represented in signs at all. (Periodic division). But that is because we can’t smuggle the use of the sign into its introduction (the rule is and remains a sign, separated from its application).’

Sheffer put up a proposal – and in so doing – represented it in signs

yes – the rule is and remains conceptually separated from its application –

without the separation – no rule or application –

no game

‘Of course I can only apply the general rule for the induction proof when I discover the substitution that makes it applicable. So it would be possible for someone to see the equations

(a + 1) + 1 = (a + 1) + 1

1 + (a + 1) = (1 + a) + 1

without hitting on the substitution

’

first and foremost the equations are proposals – are propositions –

how they are interpreted – what paradigms they are used in – what overriding propositional constructions they are placed in – is open to question – open to doubt – is uncertain

the substitution here is a representation of an interpretation – of a use

‘Moreover if I say that I understand the equations as particular cases of the rule, my understanding has to be the understanding that shows itself in the explanations of the relation between the rule and the equations, i.e. what we express by the substitutions. If I don’t regard that as an expression of what I understand, then nothing is an expression of it; but in that case it makes no sense either to speak of understanding or to say that I understand something definite. For it only makes sense to speak of understanding in cases where we understand one thing as opposed to another. And it is this contrast that signs express.’

‘Moreover if I say that I understand the equations as particular cases of the rule, my understanding has to be the understanding that shows itself in the explanations of the relation between the rule and the equations, i.e. what we express by the substitutions.’

the equations express the rule –

the rule determines the propositional action of the equations

the equations reflect the rule

the rule is reflected in the equations –

if the rule is not consistent with the equation – or the equations are not consistent with the rule –

there is no functional relation –

no relation

any functional relation is first and foremost a proposal –

open to question – open to doubt – uncertain –

in short – a subject for argument

substitutions are proposals of equivalence

‘For it only makes sense to speak of understanding in cases where we understand one thing as opposed to another. And it is this contrast that signs express.’

a sign is a proposal

different proposals express different understandings

understanding any proposal – any proposition – logically speaking is never complete

and our proposals can be understood in a variety of ways

at different times – our understandings are different

‘Indeed, seeing the internal relation must in its turn be seeing that it can be described, something of which one can say: “I see that such and such is the case”; it has to be really something of the same kind as the correlation-signs (like connecting lines, brackets, substitutions, etc). Everything else has to be contained in the application of the sign of the general rule in a particular case.’

‘I see that such and such is the case’ – is accepting the proposal – it is affirmation of the proposal in play

and yes this acceptance will most likely be facilitated – if the ‘internal relation’ is expressed in terms that have ‘something of the same kind of correlation-signs’

‘everything else has to be contained in the application of the sign of the general rule in a particular case.’

well this is the idea – and the idea determines or guides the practice

still it must be remembered that whether in fact it is the case that ‘everything else is contained in the application of the sign of the general rule in a particular case’ – is a matter always open to question –

‘It is as if we had a number of material objects and discovered they had surfaces which enabled them to be placed in a continuous row. Or rather, as if we discovered that such and such surfaces, which we had seen before, enabled them to be placed in a continuous row. That is the way many games and puzzles are solved.’

yes – different applications – different uses – different games

‘The person who discovers periodicity invents a new calculus. The question is, how does the calculus with the periodic division differ from the calculus in which periodicity is unknown?’

look – and you will see

‘(We might have operated a calculus with cubes without having had the idea of putting them together to make prisms.)’

yes –

but you if have the idea of putting the cubes together to make a prism –

your idea will be a calculation

Appendix¹

^1. Remarks taken from the Manuscript volume. Wittgenstein omitted them. Even in the MS they are not set out together as they are here.

(On: The process of generalization creates a new sign-system)

‘It is a very important observation that the c in A is not the same variable as the c in

b and g. So the way I wrote out the proof was not quite correct in a respect which is very important for us. In A we could substitute n for c, whereas the cs in b and g are identical.

But another question arises: can I derive from A that i + (k + c) = (i + k) + c? If so why can’t I derive it in the same way from B? Does that mean that a and b in A are not identical with a and b in µ, b and g?

We see clearly that that the variable c in B isn’t identical with c in A if we put a number instead of it. Then B is something like

µ  4 + (5 + 1)         = (4 + 5) + 1

b  4 + (5 + (6 + 1)) = (4 + (5 + 6)) + 1      }  … W

g  (4 + 5) + (6 + 1) = ((4 + 5) + 6) + 1

but that doesn’t have corresponding to it an equation like Aw:

4 + (5 + 6) = (4 + 5) + 6!

What makes the induction proof different from a proof of A is expressed in the fact  that c in B is not identical with the one in A, so that we could use different letters in two places.

All that is meant by what I have written above is that the reason it looks like an algebraic proof of A is that we think we meet the same variables a, b, c in the equation A as in µ, b g and so we regard A as a result of a transformation of these equations.

(Whereas of course in reality I regard the signs µ, b g, in quite a different way, which means that c in b and g isn’t used as a variable in the same way as a and b. Hence one can express this new view of B, by saying that c does not occur in A)

What I said about the new way of regarding µ, b g might be put like this: µ is used to build up b  and g in exactly the same way as the fundamental algebraic equations are used to build up an equation like (a + b)² = a² + 2ab + b². But if that is the way they are derived, we are regarding the complex µ, b g in a new way when we give the variable c a function which differs from that of a and b (c becomes a hole through which the stream of numbers has to flow).’

so the proof of A is different to the proof of B –

the two arguments are not identical –

that I would have thought is obvious

what we have is different mathematical perspectives – different proofs – different propositional games

that they can be seen to be related – is no big news –

and yes – different letters in two places – might have saved – time –

and might have avoided a rather unnecessary discussion

38. Proof by induction, arithmetic and algebra

‘Why do we need the commutative law? Not so as to be able to write the equation

4 + 6 = 6 + 4, because that equation is justified by its own particular proof. Certainly the proof of the commutative law can certainly be used to prove it, but in that case it becomes just a particular arithmetical proof. So the reason that I need the law is to apply it when using letters.

And it is this justification that the inductive proof cannot give me.

However, one thing is clear: if the recursive proof gives us the right to calculate algebraically, then so does the arithmetical proof L.

Again: the recursive proof is – of course – essentially concerned with numbers. But what use are numbers to me when I want to operate purely algebraically? Or again, the recursion proof is only of use to me when I want to use it to justify a step in a number calculation.’

the ‘commutative law’ – is a sign game – a substitution game

and yes we play this came with letters

why do we need the commutative law?

it has its uses

a game is played – there is no question of justifying a game – you play it – or you don’t

as to induction –

induction is a form of suggestive proposal – it is not a game

a so called ‘inductive proof’ is no more than an exercise in speculation

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

it justifies nothing

and in any case justification is rhetoric – or if stripped of pretension –

it is the decision to stop – to stop questioning – and to proceed

yes – the recursive game – facilitates algebraic calculation

as does the arithmetical game L (see page 428)

‘But what use are numbers to me when I want to operate purely algebraically?’

of no use

‘Or Again, the recursion proof  is only of use to me when I want to justify a step in a number calculation.’

in so far as there is no ‘justification’ in a number game –

the recursion game – will be of no use to you

‘But someone might ask: do we need both the inductive proof and the associative law, since the latter cannot provide a foundation for calculation with numbers, and the former cannot provide one for the transformations in algebra?’

if they have a use – they have a use

‘Well, before Skolem’s proof was the associative law, for example, just accepted without anyone’s being able to work out the corresponding step in a numerical calculation? That is, were we previously unable to work out 5 + (4 +3) = (5 + 4) + 3, and did we treat it as an axiom?’

strictly speaking – if you understand the game – the equation-game – there is nothing as such to work out

Skolem’s proof – when it comes down to it is no more than a restatement of the equation –

and that is all a ‘working out’ will ever be – restatement 

we are dealing here with a propositional game

you play the game – and you play according to the rule –

there is no deeper explanation –

                                                                                                                               

and yes restating the game – even reconfiguring it – can be all part of playing it –

‘If I say that the periodic calculation proves the proposition that justifies me in those steps, what would the proposition have been like if it had been assumed as an axiom instead of being proved?’

it would be no different –

how you account for / ‘prove’ the game – is not the game –

any proof is simply an argument – open to question – open to doubt – uncertain –

of interest – yes – but it’s not the game – not playing the game

‘What would a proposition be like that permitted one to put 5 + (7 + 9) = (5 + 7) + 9) without being able to prove it? Is it obvious that there never has been such a proposition.’

well – Wittgenstein has proposed such a proposition here –

presumably to demonstrate that it is not consistent with relevant rules – that it has no utility

‘But couldn’t we also say that associative law isn’t used at all in arithmetic and that we work only with particular number calculations?’

yes – you could argue that

‘Even when algebra uses arithmetic, it is a totally different calculus, and cannot be derived from the arithmetical one.’

a different game – yes

‘To the question “is 5 x 4 = 20”? one might answer: “let’s check whether it is an accord with the basic rules of arithmetic” and similarly I might say: let’s check whether A is in accord with the basic rules. But with which rules? Presumably with µ.’

different constructions – different games – different rules –

and if the question arises – which rules apply? –

the matter will be open to discussion – open to question

‘But before we can bring µ and A together we need to stipulate what we want to call “agreement” here.

That means that µ and A are separated by the gulf between arithmetic and algebra,¹ and if B is to count as a proof of A, this gulf has to be bridged over by a stipulation.’

^1. to repeat, µ is: a + (b + 1) = (a + b) + 1

                  A is: a + (b + c)  = (a + b) + c.

the gulf between arithmetic and algebra –

bridged by a ‘stipulation’ – yes – that is bridged by a proposal –

and presumably such a proposal if it facilitates functionality will be entertained

‘It is clear that we do use an idea of this kind of agreement when, for instance, we quickly work out a numerical example to check the correctness of an algebraic proposition.

And in this sense I might e.g. calculate

25 x 16      16 x 25

25              32

150             80  

400            400

and say: “yes, it’s right, a.b is equal to b.a” – if I imagine that I have forgotten.’

so proposing a restatement of an algebraic proposal in arithmetical terms – shows what?

that a proposal can be translated –

the question is does doing so have any functional value –

or is it just effectively a word-game – a substitution game?

‘Considered as a rule for algebraic calculation, A cannot be proved recursively. We would see that especially clearly if we wrote down the “recursive proof” as a series of arithmetical expressions. Imagine them written out (i.e. a fragment of the series plus “so on”) without any intention of “proving” anything, and the suppose someone asks: does that prove a+(b + c) = (a + b) + c?”. We would ask in astonishment “How can it prove anything of the kind? The series contains only numbers, it doesn’t contain any letters”. – But no doubt we might say: if I introduce A as a rule for the calculation with letters, that brings this calculus in a certain sense into unison with the calculus of the cardinal numbers, the calculus I established by the law for the rules of addition (the recursive definition a + (b + 1) = (a + b) + 1).’

‘Considered as a rule for algebraic calculation, A cannot be proved recursively’ –

correct

‘if I introduce A as a rule for the calculation with letters, that brings this calculus in a certain sense into unison with the calculus of the cardinal numbers’

yes –

what we then have effectively – is a proposal for the translation of one propositional construct into another – or the use of one in place of the other –

however the algebraic argument and the recursive argument are different games

they come out of different logical perspectives –

so any ‘translation proposal’ one way or the other – strikes me as no more than a word-game – and a superficial one at that –

and in any case it’s really just a hijacking of one propositional construct by another –

and to what purpose?

I fail to see how this ‘translating’ one into another –

adds anything to either perspective –

to either game