Part II. V

V   MATHEMATICAL PROOF

22  In other cases, if I am looking for something, then even before it is found I can describe what finding it is; not so, if I am looking for the solution of a mathematical problem. Mathematical Expeditions and Polar Expeditions

‘How can there be conjectures in mathematics? Or better, what sort of thing is it that looks like a conjecture in mathematics? Such as making a conjecture about the distribution of the primes.

I might e.g. imagine that someone is writing primes in series in front of me without my knowing they are primes – I might for instance believe he is writing numbers just as they occur to him – and I now try to detect a law in them. I might now actually form an hypothesis about this number sequence, just as I could about any sequence yielded by an experiment in physics.

Now in what sense have I, by so doing, made an hypothesis about the distribution of the primes?’

‘Such as making a conjecture about the distribution of primes?’

i.e. is the distribution ordered or random?

and any theorem here will be an argument – open to question – open to doubt – uncertain

and any theorem adopted by practitioners will be an advance in prime theory –

and an extension of the prime-game

‘You might say that an hypothesis in mathematics has the value that it trains your thoughts on a particular object – I mean a particular region – and we might say “ we shall surely discover something interesting about these things”.’

an hypothesis in mathematics – is a proposal – open to question – open to doubt –

any such hypothesis is an exploration of uncertainty – and as such an exercise in propositional discovery

‘The trouble is that our language uses each of the words “question”, “problem”, “investigation”, “discovery”, to refer to such basically different things. It’s the same with “inference”, “proposition”, “proof”.’

any word we use is a proposal – open to question – open to doubt – uncertain

it’s not a ‘trouble’ – it’s propositional reality

‘The question arises, what kind of verification do I count as valid for my hypothesis? Or can I faute de mieux allow an empirical one to hold for the time being until I have a “strict proof”? No. Until there is such a proof, there is no connection at all between my hypothesis and the “concept” of a prime number.’

‘verification’? –

a verification – is a proposal – logically speaking – open to question – to doubt – uncertain –

a verification is an argument –

any propositional practise will develop verification proposals –

there is no logical end point to so called ‘verification’ – or for that matter ‘falsification’  –

theories of verification – confirmation etc – have a pragmatic function –

they facilitate movement – propositional movement –

and of course the idea is – that the movement is upward and onward –

that at any rate will be the press release

a ‘proof’ – is a deductive language-game

‘Only the so-called proof establishes any connection between the hypothesis and the primes as such. And that is shown by the fact that – as I said – until then the hypothesis can be construed as one belonging purely to physics. – On the other hand when we have supplied a proof, it doesn’t prove what was conjectured at all, since I can’t conjecture to infinity. I can only conjecture what can be confirmed, but experience can only confirm a finite number of conjectures, and you can’t conjecture the proof until you’ve got it, and not then either.’

with or without a so called ‘proof’ – the ‘hypothesis’ – is open to question – open to doubt – is uncertain

‘experience’ – is uncertain –

and ‘proof’ – the deductive language game – is of course – logically speaking – open to question – open to doubt – uncertain

the point of proof – is to bring argument to an end

proof is not logical – it is rhetorical

‘proof’ – is a rhetorical device –

‘proof’ is that point at which the proposal – is no longer put to the question – no longer a subject of doubt –

the ‘proof’ – just is the device agreed upon by the practitioners to bring question and doubt to an end –

what we have here is a decision to stop the inquiry –

there is no logical basis for this – but there is a psychological and indeed a practical imperative – to find a conclusion

the deductive language game – the argument that is –‘proof’ – fits the bill

mathematics as with all propositional practices is a marriage of logic and rhetoric

‘Suppose that someone, without having proved Pythagoras’ theorem, has been led by measuring the sides and hypotenuses of right angle triangles to “conjecture” it. And suppose the latter discovered the proof, and said that he had proved what he had earlier conjectured. At least one remarkable question arises: at what point of the proof does what he had earlier confirmed by earlier trials emerge? For the proof is essentially different from the earlier method. – Where do these methods make contact, if the proof and the tests are only different aspects of the same thing (the same generalizations) if, as alleged, there is some sense in which they gave the same result?

I have said: “from a single source only one stream flows”, and one might say that it would be odd if the same thing were to come from different sources. The thought that the same thing can come from different sources is familiar from physics, i.e. from hypotheses. In that area we are always concluding from symptoms to illnesses and we know that the most different symptoms can be different symptoms of the same thing.’

yes – the ‘conjecture’ here is a proposal – a description of the proposals already put forward in the actions taken

his discovery of the proof is the application of a language game to the proposal

the ‘contact’ – is the application of a language game to the proposal (conjecture) –

making contact is really just a matter of proximity – a propositional hand shake let’s say

yes the proof – the language game – is really a restatement of the proposal / conjecture

it is not ‘the same thing’ or ‘from different sources’ –

what we have is propositional actions – different proposals – different propositions –

in an open logical setting –

whether we are talking about the propositions of physics – or the propositions of mathematics

the context is contingency – is uncertainty –

it is about what happens (propositionally) –

where and when it happens

‘How could one guess from statistics the very thing the proof later showed?’

‘statistics’ –  is a probability game

a ‘proof’ – a deductive word game

whether a proposal is ‘based on’ statistics – or a so called ‘proof’ – the proposal is open to question – open to doubt – uncertain

as to ‘guess’ –

you might say the ‘guess’ is a proposal – with no immediate background – with no basis –

a proposal from the unknown – and one recognised as such –

a proposal without pretence?

that is to say – a purely ‘logical’ proposal –

yes – surprise – surprise –

and yes – you can be lucky

‘How can the proof produce the same generalisation as the earlier trials made probable?’

any generalization – is uncertain – regardless of how it is arrived at

‘I am assuming that I conjectured the generalization without conjecturing the proof. Does the proof now prove exactly the generalization that I conjectured?!’

                                                                                                                                  ‘proof’ is a logical deception –

it is a deception because it pretends certainty

the premises of a proof – the conclusion of the proof – proposals – open to question – open to doubt – uncertain

conjectures!

a proof – is a piece of poetry – a poetic form

mathematics – you might say – is the grand poem of signs – a great poetry of syntax

the so called ‘proof’– and the ‘generalization that I conjectured’ – are different proposals – different conjectures –

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

‘Suppose someone was investigating even numbers to see if they confirmed Goldbach’s conjecture. Suppose he expressed the conjecture – and it can be expressed – that if he continued with this investigation, he would never meet a counterexample as long as he lived. If a proof of the theorem is then discovered, will it also be a proof of the man’s conjecture? How is that possible?’

in this case the man’s conjecture – can be viewed as just being the conjecture that there will be a proof –

if as it turns out that a proof is constructed – 

a lucky guess – nothing more

‘Nothing is more fatal to philosophical understanding than the notion of proof and experience as two different but comparable methods of verification.’

yes – ‘proof’ – is an ‘experience’ – and ‘experience’ is ‘proof’ –

a proof is a language –game – and the argument from experience is a language-game

what is fatal to philosophical understanding – is not understanding that any proposal –

of proof – any proposal of experience – is open to question – open to doubt – is uncertain

verification – is a decision to use a proposal – it is a decision of utility –

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

‘What kind of discovery did Sheffer make when he found that p v q and ~ p

can be expressed by p | q? People had no method of looking for p | q, and if someone were to find one today, it wouldn’t make any difference.

What was it we didn’t know before the discovery? (It wasn’t anything we didn’t know, it is something with which we weren’t acquainted.)

You can see this very clearly if you imagine someone objecting that p | q isn’t at all the same as is said by ~ p. The only reply of course is that it’s only a question of the system p | q, etc. having the necessary multiplicity. Thus Sheffer found a symbolic system with the necessary multiplicity.

Does it count as looking for something, if I am unaware of Sheffer’s system and say I would like to construct a system with only one logical constant? No!

Systems are certainly not all in one space, so that I could say there are systems with 3 and with 2 logical constants and now I am trying to reduce the number of constants in the same way. There is no “same way” here.’

Scheffer put forward a new proposal – a new representation of the logical proposal –

p v q and ~ p

‘What was it that we didn’t know before the discovery?’

we didn’t know the proposal – the representation – p | q

‘Does it count as looking for something, if I am unaware of Sheffer’s system and say I would like to construct a system with only one logical constant? No!’

yes – looking for a new proposal – a new system – counts as ‘looking for something’

‘Systems are certainly not all in one space, so that I could say there are systems with 3 and with 2 logical constants and now I am trying to reduce the number of constants in the same way. There is no “same way” here.’ –

just what ‘same way’ amounts to – is open to question –

if someone is looking to reduce the number of constants in the same way – it would be useful to ask him what he means by ‘same way’ –

if the issue is significant to the participants – then there will be argument

‘Suppose prizes are offered for the solution – say – of Fermat’s problem. Someone might object to me:  How can you say that this problem doesn’t exist? If prizes are offered for the solution, the surely the problem must exist. I would have to say: Certainly, but the people who talk about it don’t understand the grammar of the expression “mathematical problem” or of the word “solution”. The prize is really offered for the solution of a scientific problem; for the exterior of the solution (hence also for instance we talk about a Riemannian hypothesis). The conditions of the problem are external conditions; and when the problem is solved, what happens corresponds to the setting of the problem in the way in which solutions correspond to problems in physics.’

a mathematical problem is a propositional game problem –

i.e. what rules – what system of mathematical rules – offer an answer to this propositional (mathematical) question?

an exterior of the solution – or in fact an exterior of the problem –

is a representation of the problem / solution in other (non-mathematical) terms –

i.e. in physical terms

                                                                                                                                  

you can have one without the other – and one may well be a springboard to the other

a proposal – a proposition – can be variously interpreted –

i.e. mathematically – physically – or in terms of some other descriptive model

‘If we set as a problem to find a construction for a regular pentagon, the way the construction is specified in the setting of the problem is by the physical attribute that is to yield a pentagon that is shown by measurement to be regular. For we don’t get the concept of constructive division into five (or of a constructive pentagon) until we get it from the construction.’

the constructive pentagon – is a proposal –

let us say e.g. – in stone

when it is constructed – that is when we have the proposal

the concept of this construction –  is a proposal –

of a different form –

you might describe it as ideational –

two different proposal –

one leading to the other

‘Similarly in Fermat’s theorem we have an empirical structure that we interpret as a hypothesis, and not – of course – as the product of a construction. So in a certain sense what the problem asks for is not what the solution gives.’

so if what the problem asks for is not what the solution gives –

you either have a discovery – or – it’s back to the drawing board

in Fermat’s theorem we have a proposal –

which as with any proposal – is – logically speaking – open to interpretation –

as indeed is any work done in respect to the theorem – and any ‘solution’ – proposed

‘Of course a proof of the contradictory of Fermat’s theorem (for instance) stands in the same relation to the problem as a proof of the proposition itself. (Proof of the impossibility of a construction).’

yes –

bare in mind though – a proof either way is deductive argument –

really just a piece of rhetoric

‘We can represent the impossibility of the trisection of an angle as a physical impossibility, by saying things like “don’t try to divide the angle into 3 equal parts, it is hopeless!” But in so far as we can do that, it is not this that the “proof of impossibility” proves. That it is hopeless to attempt the trisection is something connected with physical facts.’

as for the ‘hopeless!’ argument –

it is possible to trisect an arbitrary angle using tools other than straightedge and compass – i.e. neusis construction which involves the simultaneous sliding  and rotation of a straightedge –

this was a method used by ancient Greeks

other methods have been developed over time by mathematicians

the ‘proof of impossibility’ comes down to an algebraic argument –

if you accept the premises – the mathematics of this argument – then the conclusion follows –

i.e. it can be shown that a 60° cannot be trisected

the question this raises is just whether the mathematics employed here fits the task –

and indeed – whether there is a ‘real’ problem here at all –

or is it just that we have a language-game – a clever algebraic game – played in the wrong context?

                                                                                                                                  

‘Imagine someone set himself the following problem. He is to discover a game played on a chessboard, in which each player is to have 8 pieces; the two white ones which are in the outermost files at the beginning of the game (the “consuls”) are to be given some special status by the rules so that they have a greater freedom of movement than the other pieces; one of the black pieces (the “general”) is to have a special status; a white piece takes a black one by being put in its place (and visa versa); the whole game is to have a certain analogy with the Punic wars. Those are the conditions that the game is to satisfy. – There is no doubt that that is a problem, a problem not at all like the problem of finding out how under certain conditions white can win in chess. – That problem would be quite analogous to the problems of mathematics (other than problems of calculation).’

yes – indeed the problem of game construction –

‘What is hidden must be capable of being found. (Hidden contradictions.)’

putting it bluntly – nothing is hidden –

what is ‘in view’ – is just what is proposed –

what is proposed is what there is

a contradiction is proposed – or it is not proposed

what needs to be understood is that our reality is propositional –

that is – what is proposed

‘Also, what is hidden must be completely describable before it is found, no less than if it had already been found’

the idea that you describe something that is not described?

might I propose – a contradiction here?

if the proposal is described – it is proposed in the first place –

if it is not proposed – there is nothing to describe

as for ‘completely describable’ –

logically speaking the proposition – is never ‘complete’ – it is always – open –

open to question – open to doubt –

the proposal – the proposition – is uncertain –

uncertainty undercuts any pretence of ‘completion’ –

propositional logic – is the exploration of uncertainty

‘It makes good sense to say that an object is so well hidden that it is impossible to find it; but of course the impossibility here is not a logical one; I.e. it makes sense to speak of finding an object to describe the finding; we merely deny that it will happen.’

the logical reality is this –

if we are talking about an object – then the object has been proposed

if it can’t be described or is not described – then the initial proposal (of the object) – is of no use

proposing an object – and the proposing that it doesn’t exist (‘deny that it will happen’) –

is just pretentious sophistry

‘[We might put it like this: if I am looking for something, I mean the North Pole, or a house in London – I can completely describe what I am looking for before I have found it (or have found that it isn’t there) and either way this description will be logically acceptable. But when I am looking for something in mathematics, unless I am doing so within a system, what I am looking for cannot be described, or can only apparently be described; for if I could describe it in every particular, I would already actually have it; and before it is completely described I can’t be sure whether what I am looking for is logically acceptable, and therefore describable at all. So it is only an apparent description of what is being “looked for”]*’

* [Editor’s note] This paragraph  is crossed out in the typescript.

there is no ‘complete description’ – there are only descriptive proposals –

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

‘apparently described’ –

well ‘apparent’ – is all any description is

whatever ‘the something in mathematics is’ – there will be a proposal –

and any proposal put – any description put – will be logically speaking – open to question – open to doubt – uncertain

‘logically acceptable’?–

this amounts to whether or not what is proposed – can be shown to have function in the propositional context that is in play

‘Here we are easily misled by the legitimacy of an incomplete description when we are looking for a real object, and here again there is an unclarity about the concepts “description” and “object”. If someone says, I am going to the North Pole and I expect to find a flag there, and that would mean on Russell’s account, I expect to find something (an x) that is a flag – say of such and such a colour and size. In that case too it looks as if the expectation (the search) concerns only an indirect knowledge and not the object itself; as if that is something that I don’t really know (knowledge by acquaintance) until I have it in front of me (having previously been only indirectly acquainted with it), But that is nonsense. There whatever I can perceive – to the extent that it is a fulfilment of my expectation – I can also describe in advance. And here “describe” means not saying something or other about it, but rather expressing it. That is, if I am looking for something I must be able to describe it completely.’

‘a real object’ – is a proposal

‘an unclarity about the concepts ‘description’ and ‘object’’? –

a ‘concept’ is a proposal – open to question – open to doubt – uncertain –

as to Russell’s account –

‘acquaintance’ – is an occasion of uncertainty –

yes – I can describe in advance –

my description – my expression – is a proposal – logically speaking – open to question

if I am looking for something – I am making a proposal –

this notion of ‘completeness’ – has no place in epistemology –

it is rhetoric –

which is to say – it represents a stand against knowledge –

a stand for prejudice – a stand for ignorance

‘The question is: can one say that at present mathematics is as it were jagged – or frayed – and for that reason we shall be able to round it off? I think you can’t say that, any more than you can say that reality is untidy, because there are 4 primary colours, seven notes in an octave, three dimensions in visual space, etc.

You can’t round off mathematics any more than you can say “let’s round off the four primary colours to eight or ten” or “let’s round of the eight tones in an octave to ten”.’

‘jagged’? – where does that come from?

mathematics is  ruled governed propositional activity –

‘frayed’? – perhaps –

our reality is propositional –

that is to say – open to question – open to doubt – uncertain

‘The comparison between a mathematical expedition and a polar expedition. There is a point in drawing this comparison and it is a very useful one.

How strange would it be if a geographical expedition were uncertain whether it had a goal, and so whether it had any route whatsoever. We can’t imagine such a thing, it’s nonsense. But this is precisely what it is like in a mathematical expedition. And so perhaps it is a good idea to drop the comparison altogether.

Could one say that arithmetical or geographical problems can always look, or can falsely be conceived, as if they referred to objects in space whereas they refer to space itself?

By “space” I mean what one can be certain of while searching.’

a ‘mathematical ‘expedition’ – is the construction of a sign game – a language game –

arithmetical problems are problems of calculation –

calculation is a rule governed action –

‘an object in space’ – is a proposal –

‘space’ is a proposal – open to question – open to doubt – uncertain

23. Proof and the truth or falsehood of mathematical propositions

‘A mathematical proposition that has been proved has a bias towards truth in its grammar. In order to understand the sense of 25 x 25 = 625 I may ask: how is this proposition proved? But I can’t ask how its contradictory is or would be proved, because it makes no sense to speak of the contradictory of 25 x 25 = 625. So if I want to raise a question that won’t depend on the truth of the proposition, I have to speak of checking its truth, not of proving or disproving it. The method of checking corresponds to what one may call the sense of the proposition. The description of this method is a general one and brings in a system of propositions of the form a x b = c.

We can’t say “I will work out that it is so”, we have to say “whether it is so”, i.e., whether it is so or otherwise.

The method of checking the truth corresponds to the sense of a mathematical proposition. If it’s impossible to speak of such a check, then the analogy between “mathematical proposition” and the other things we call propositions collapses. Thus there is a check for propositions of the form “($k)n/m …” and “($k)m/n …” which brings in intervals.’

if by ‘proof’ you mean a procedure that renders a proposition beyond question – beyond doubt – as certain – there is no proof –

if by proof you mean a procedure which practitioners use to end question – doubt – uncertainty – in order to proceed – then yes – such decisions – pragmatic decisions are made – and represented in language games such as deduction

‘25 x 25 = 625’ –

here there is no question of proof

25 x 25 = 625 – is a sign-game – a mathematical game –

if you play it – you play it according to the sign-game rules of multiplication

‘checking the truth’?

‘If it’s impossible to speak of such a check, then the analogy between “mathematical proposition” and the other things we call propositions collapses’ –

a mathematical proposition like – ‘25 x 25 = 625’ – is a game – it is a propositional game

the very point of a game is that it is played –

which means the rules of the game – are not questioned – they are observed

for if they are questioned – there is no game – there is no play

yes – you can question any proposal – any procedure – any set of rules – any practice

but doing so is not game playing –

if you do question and doubt – you are dealing with and in uncertainty –

you are involved in propositional discovery –

and in such an activity – there are no rules

‘Now consider the question “does the equation x² + ax + b = o” have a solution in real numbers?” Here again there is a check and the check decides between ($ …) , and

~ ($ …), etc. But can I in the same sense also ask and check “whether the equation has a solution”? Not unless I include this case too in a system with others.’

the equation is a sign game –

if properly constructed i.e. in terms of the rules governing the game – the equation has a solution –

the only question is what values you give the variables –

the game’s play is rule governed –

there is nothing to check

‘(In reality “the proof of the fundamental theorem of algebra …” constructs a new kind of number.)

Equations are a kind of number. (That is, they can be treated similarly to numbers.)’

a proof as a language game (deduction) can function as a game within a game

when we speak of a number – what we are talking about is the number game

a particular number is a token – in the number game

so the question is – can a proof have the same function as a number?

can the game – the language game proof – function as a token in a number-game?

the answer is yes – if such a game is constructed – and played –

that is to say – if the players accept the proof as a token in the game

the general point is that any proposal can be included in any game as a token – if the players accept what is proposed as a token –

and of course the new token if accepted into the game functions in accordance with the rules of the game

‘A “proposition of mathematics” that is proved by an induction is not a “proposition”

in the same sense as the answer to a mathematical question unless one can look for the induction in a system of checks.

“Every equation G has a root”. And suppose it has no root? Could we describe that case as we describe its not having a rational solution? What is the criterion for an equation not having a solution? For this criterion must be given if the mathematical question is to have a sense and if the apparent existence proposition is to be a “proposition” in the sense of an answer to the question.

(What does the description of the contradictory consist of? What supports it? What are the examples that support it, and how are they related to particular cases of the proved contradictory? These questions are not side-issues, but absolutely essential.)

(The philosophy of mathematics consists in an exact scrutiny of mathematical proofs – not in surrounding mathematics with a vapour.)’

‘A “proposition of mathematics” that is proved by an induction is not a “proposition”

in the same sense as the answer to a mathematical question unless one can look for the induction in a system of checks.’

induction is not a language-game that functions in mathematical games – unless provision is made for it – and even then it will not have the same status as deduction

“Every equation G has a root”. And suppose it has no root?’ –

either an exception is made to the rule –  or the rule holds – and the ‘equation’ is not regarded as genuine – as functional –

or the equation is placed in ‘quarantine’ –  as it were – where its status is undecided – and where it becomes the subject of further study and consideration

‘Could we describe that case as we describe its not having a rational solution?’ –

yes – it could be described as not having a ‘rational solution’ –

‘What is the criterion for an equation not having a solution?’ –

yes there is the question –

and the reality is that various answers will be proposed – and argued for

and the answer – in a practical sense will just be that which is decided upon by those doing the work

‘For this criterion must be given if the mathematical question is to have a sense and if the apparent existence proposition is to be a “proposition” in the sense of an answer to the question.’ –

if the equation (without a rational solution) is to have currency then indeed its provenance must be ‘established’ – that is well argued for – and that means its ‘ground’ must be acceptable to the practitioners that work in this area

‘(What does the description of the contradictory consist of? What supports it? What are the examples that support it, and how are they related to particular cases of the proved contradictory? These questions are not side-issues, but absolutely essential.)’

the contradictory is no more than a method of asserting the rules

‘(The philosophy of mathematics consists in an exact scrutiny of mathematical proofs – not in surrounding mathematics with a vapour.)’ –

the philosophy of mathematics?

the philosophy of mathematics should make clear that the rules of the mathematics game – are open to question – open to doubt – are – as with any other set of propositions – uncertain

the philosophy of mathematics should be an argument against prejudice –

intellectual prejudice is the worst kind – as it can run deep

the argument against prejudice is not particular to the philosophy of mathematics – it is the task of any philosophy – any application of free and critical thinking

‘In discussions of the provability of mathematical propositions it is sometimes said that there are substantial propositions of mathematics whose truth or falsehood must remain undecided. What the people who say that don’t realise is that such propositions, if we can use them and want to call them “propositions”, are not at all the same as what we call “propositions” in other cases; because a proof alters the grammar of a proposition. You can certainly use one and the same piece of wood first as a weathervane and then as a signpost; but you can’t use it fixed as a weathervane and moving as a signpost. If some one wanted to say “There are also moving signposts” I would answer “You mean ‘There are also moving pieces of wood”. I don’t say that a moving piece of wood can’t possibly be used at all, but that only that it can’t be used as a signpost”.’

if there are ‘substantial mathematical propositions whose truth must remain undecided’ –

then what we have is proposals in mathematics – that are under consideration – by mathematicians – for their possible utility in mathematics – in mathematical games

a proof does not alter the grammar – the logic of a proposition –

a ‘proof’ – that is to say a ‘deductive language-game’ – may be used to fashion a use of a proposition

a piece of wood is a piece of wood – however it might be used –

a proposition is a proposal – open to question – open to doubt – open to question – uncertain – regardless of how it is used

‘The word ‘proposition’, if it is to have any meaning at all here, is equivalent to a calculus: to a calculus in which p v ~p is a tautology (in which the “law of the excluded middle” holds). When it is supposed not to hold, we have altered the concept of proposition. But that does not mean we have a discovery (found something that is a proposition and yet doesn’t obey such and such a law); it means we have a new stipulation, or set up a new game.’

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

the point a language-game is that it is rule-governed –

if you play a language-game properly – the propositions of the game as played –

are not open to question – as played they are moves of the game –

and the game as played is not questioned

game-playing is a mode of propositional use – of propositional activity

when we engage in propositional behaviour that is not rule-governed –

if our propositional activity is rational – our activity – is logical –

‘logical’ – if we regard our propositions – our proposals – as open to question – open to doubt – as uncertain

our propositional reality is uncertain – we operate in and with uncertainty – our lives are explorations of uncertainty –

yes you can pretend otherwise – deny the logical reality – and attempt to hide yourself – and the world – in some pretence of certainty –

it doesn’t work –  it is dead end – and you miss out on so much of life

human beings think and play –

play is a relief from thinking –

and thinking the relief from play

24 If you want to know what is proved, look at the proof

‘Mathematicians only go astray, when they want to talk about calculi in general; they do so because they forget the particular stipulations that are the foundations of each particular calculus.

The reason why all philosophers of mathematics miss their way is that in logic, unlike natural history, one cannot justify generalizations by examples. Each particular case has maximum significance, but once you have it the story is complete, and you can’t draw from it any general conclusion (or any conclusion at all).

There is no such thing as a logical fiction and hence you can’t work with logical fictions; you have to work out each example fully.

The philosopher only marks what the mathematician casually throws off about his activities.’

logic is the critical investigation of propositional forms – natural history is a propositional form –

‘unlike natural history, one cannot justify generalizations by examples’ –

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

justification – is rhetoric

as with natural history the concepts and practises of ‘logic’ – are open to question – open to doubt – are uncertain

‘once you have it the story is complete’ –

yes – the story is ‘complete’ – if it is not questioned –

propositional reality is open – logically speaking no story is ‘complete’

what counts as ‘fact’ and what counts as ‘fiction’ – in any propositional context -         

when all is said and done is best understood as a question of fashion

(some fashions have a longer half life than others)

the philosopher investigates forms of language – whether casually thrown off –

or not

‘The philosopher easily gets into the position of a ham-fisted director, who instead of doing his own work and merely supervising his employees to see they do their work well, takes over their jobs until one day he finds himself overburdened with other people’s work while his employees watch and criticize him. He is particularly inclined to saddle himself with the work of the mathematician.’

mathematicians don’t need ‘supervision’ by non-mathematicians –

should a mathematician question the basis of what he does – that is – ask a philosophical question –

i.e. what is the epistemological status of the propositions I work with?

then a philosopher – someone who focuses on such matters – and has some expertise in this area – is a good person to talk to –

as for philosophers –  their expertise is in critical thinking with respect to the ground of human practises

with or without this expertise human practises go on

mathematics would be mathematics whether or not a question was ever raised concerning its basis

‘If you want to know what the expression “continuity of a function” means, look at the proof of continuity; that will show you what it proves. Don’t look at the results as it is expressed in prose, or in the Russellian notation, which is simply a translation of the prose expression; but fix your attention on the calculation actually going on in the proof. The verbal expression of the allegedly proved proposition is in most cases misleading, because it conceals the real purport of the proof, which can be seen in full clarity in the proof itself.’

‘If you want to know what the expression “continuity of a function” means, look at the proof of continuity; that will show you what it proves.’ –

the ‘proof’ is a language game – if you play the game – what will you know?

the action of the game – as described in the steps of the proof

‘Don’t look at the results as it is expressed in prose, or in the Russellian notation, which is simply a translation of the prose expression’

how the action is ‘expressed’ – is logically open – open to question

a prose expression might be useful in certain contexts – just as indeed a Russellian notation might well be of use in certain contexts

‘but fix your attention on the calculation actually going on in the proof.’ –

the calculation is an action – which if not described is unknown

the action is described in the ‘proof’ – the action may be described in prose – the action may be described in Russellian notation –

mathematicians describe the action in the language-game ‘proof’ – and if you are doing mathematics – that should do the trick

‘The verbal expression of the allegedly proved proposition is in most cases misleading, because it conceals the real purport of the proof, which can be seen in full clarity in the proof itself.’

perhaps the verbal expression is for mathematicians – misleading –

any proposal – any proposition – any language-game – can be regarded as ‘clear’ – if it is not put to question – not made the subject of doubt –

if a proposition – a proposal – is not held open to question – open to doubt – not understood as uncertain –

then we are dealing not with logic – but with rhetoric –

the rhetoric – dare I say – of mathematics

‘ “Is the equation satisfied by any numbers?”; “It is satisfied by numbers”; “It is satisfied by all (no) numbers.” Does your calculus have proofs? And what proofs? It is only from them that we will be able to gather the sense of these propositions and questions.’

the ‘sense’ of these propositions and questions?

‘sense’ is a question of context – of use –

and here we are clearly in the context of mathematics – of rule governed propositional action

‘proofs’ – deductive language games – do have a place in this context –

however the proof – or knowing the proof – is not necessary to establish a context –

you can get the sense here – with or without proofs

‘Tell me how you seek and I will tell you what you are seeking.’

a methodology that is logical – will be open to question – open to doubt – will be uncertain –

as indeed will be the description of the object of the method – of the inquiry – of the endeavour

we operate with and in uncertainty

and furthermore what you are looking for may determine how you proceed

there are no rules as to how to proceed – or rules for what a procedure will result in –

if you proceed rationally – you keep an open mind

and whether you begin with ‘how’ – in your search for ‘what’ – or ‘what’ leads you to ‘how’

is no more than a question of circumstance

‘We must first ask ourselves: is the mathematical proposition proved? If so, how? For the proof is part of the grammar of the proposition! – The fact that this is so often not understood arises from our thinking once again along the lines of a misleading analogy. As usual in these cases, it is an analogy from our thinking in natural sciences. We say, for example, “this man died two hours ago” and if someone asks us “how can you tell that?” we can give a series of indications (symptoms). But we can also leave open the possibility that medicine may discover hitherto unknown methods of ascertaining the time of death. That means that we can already describe such possible methods; it isn’t their description that is discovered. What is ascertained experimentally is whether the description corresponds to the facts. For example, I

may say: one method consists in discovering the quantity of haemoglobin in the blood, because this diminishes according to such and such a law in proportion to the time after death. Of course that isn’t correct, but if it were correct, nothing in my imaginary description would change. If you call the medical discovery “the discovery of a proof that the man died two hours ago” you must go on to say that the discovery does not change anything in the grammar of the proposition “the man died two hours ago”. The discovery is the discovery that a particular hypothesis is true (or agrees with the facts). We are so accustomed to these ways of thinking, that we take the discovery of a proof in mathematics, sight unseen, as being the same or similar. We are wrong to do so because, to put it concisely, the mathematical proof couldn’t be described before it is discovered.

The “medical proof” didn’t incorporate the hypothesis it proved into any new calculus, so it didn’t give it any new sense; a mathematical proof incorporates the mathematical proposition into a new calculus, and alters its position in mathematics. The proposition with its proof doesn’t belong to the same category as the proposition without the proof. (Unproved mathematical propositions – signposts for mathematical investigation, stimuli to mathematical constructions.)’

as to natural science –

yes we have ‘empirical’ hypotheses – descriptions – and experiments – descriptions – that are tests – put against the ‘facts’ – which are descriptions –

our hypotheses – our tests and our ‘facts’ – our descriptions – are proposals – open to question – open to doubt – uncertain

mathematical proof –

is a deductive language-game – a form of propositional argument –

the premises and conclusion of such an argument are proposals – open to question – open to doubt – uncertain

‘Are all the variables in the following equations variables of the same kind?

x² + y² + 2xy = (x + y) ²

x² + 3x + 2 = 0

x² + ax + b = 0

x²  + xy + z = 0          ?

That depends on the use of the equations. – But the distinction between no. 1 and no. 2 (as they are ordinarily used) is not a matter of the extension of the values satisfying them. How do you prove the proposition “No. 1 holds for all values of x and y” and how do you prove the proposition “there are values of x that satisfy No. 2?” There is no more or no less similarity between the senses of the two propositions than there is between the proofs.’

the sense of a proposition? –

in general aren’t we talking here about how a proposition is used?

and that means its use – in a propositional context –

and if that is the case – then we can’t speak of the sense of a proposition – in isolation from a propositional context

and any assessment of propositional context – will be open to question – open to doubt – uncertain

as regards equations – these propositions are sign-games –

and so the question is – does context change the sense of a game?

I think not – the sense of an equation – is an internal property of the game –

which is to say the sense of the game is the rules of the game –

there is no ‘external’ sense – or contextual sense to equations – to mathematical games –

the equation has the same ‘sense’ – if you like – wherever it is played

as to proofs –

what you have there is a form of argument –

which if applied – does not depend on propositional context –

it is a formal representation

‘But can’t I say of an equation “I know it doesn’t hold for some substitutions – I’ve forgotten now which; but whether it doesn’t hold in general, I don’t know?” But what do you mean when you say you know that? How do you know? Behind the words “I know …” there isn’t a certain state of mind to be the sense of those words. What can you do with that knowledge? That’s what will show what the knowledge consists in. Do you know a method for ascertaining that the equation doesn’t hold in general? Do you remember that the equation doesn’t hold for some values of x between 0 and 1000? Or did someone just show you the equation and say he had found values of x that didn’t satisfy the equation, so that perhaps you don’t yourself know how to establish it for a given value? etc. etc.’

‘I know it doesn’t hold for some substitutions – I’ve forgotten now which; but whether it doesn’t hold in general, I don’t know’?

‘in general’ here can only mean ‘some’ –

so the original statement can be either – ‘I know it doesn’t hold for some substitutions’ or ‘I know it does hold for some substitutions’ –

these statement amount to the same thing –

and so the ‘I don’t know’ here  – is wrong – is out of place

‘But what do you mean when you say you know that? How do you know? Behind the words “I know …” there isn’t a certain state of mind to be the sense of those words’

‘knowing’ here – means playing the game – playing the equation game –

you play the game with different values – to see if the game can be played with the values you have chosen

if it can’t – then the values are not applicable to the game –

your knowledge here consists in play –

and the play is simply a function of enacting or following the rules of the game

if you don’t know the rules you can’t play –

if you don’t accept the rules – you can’t play

the whole point of the equation is to find out which values satisfy the equation –

in this respect it is a game of trial and error

‘ “I have worked out that there is no number that …” – In what system of calculation does that calculation occur? – That will show us to which proposition-system the worked-out proposition belongs. (One also asks: how does one work out something like that?)’

‘I have worked out that there is no number that …’ –

means you are not using a number-game – that has the result that you are looking for

‘In what system of calculation does that calculation occur?’ –

if you are not playing a calculation game that gets you the result you want – then there is no calculation

your conclusion – ‘I have worked out that there is no number that …’ – is –

‘there is no number-game that .…’ –

or perhaps  ‘I don’t know of a number-game that ..’

look the bald fact is  – you are not doing mathematics –

doing mathematics is playing the game –

you have no game to play here –

and indeed – you have not worked-out anything at all

imagining a result (‘there is no number that ..’) – to a non-existent game – or a game you are not playing – is not mathematics

‘I have worked out that there is no number that …’ – strikes me as ignorant – speculation

what you have here is not mathematics – but rather a pretence of mathematics

‘ “I have discovered that there is such a number.”

“I have worked out that there is no such number.”

In the first sentence I cannot substitute “no such” for “such a”. What if in the second  I put “such a” for “no such”? Let’s suppose the result of a calculation isn’t the proposition “~ ($n)” but “($n) etc.” Does it then make sense to say something like “Cheer up! Sooner or later you must come to such a number, if only you try long enough”? That would only make sense if the result of the proof has not been

“($n) etc.” but something that sets limits to testing, and therefore a quite different result. That is the contradictory of what we call an existence theorem, a theorem that tells us to look for a number, is not the proposition “(n) etc.” but a proposition that says that in such and such an interval there is no number which … What is the contradictory of what is proved? – For that you must look at the proof. We can say the contradictory of a proved proposition is what would have been proved instead of it if a particular miscalculation had been made in the proof. If now, for instance, the proof that ~ ($n) etc. is the case is an induction that shows that however far I go such a number cannot occur, the contradictory of this proof (using this expression for the sake of the argument) is not the existence of a proof in our sense. This case isn’t like a proof that one or none of the numbers a, b, c, d has the property e; and that is the case that one always has before one’s mind as a paradigm. In that case I could make a mistake by believing that c had the property and after I had seen the error I would know that none of the numbers had the property. But at this point the analogy just collapses.

(This is connected with the fact that I can’t ipso facto use the negations of equations in every calculus in which I use equations. For 2 x 3 ¹ 7 doesn’t mean that the equation 2x 3 =7 isn’t to occur, like the equation 2 x 3 = sine; the negation is an exclusion within a predetermined system. I can’t negate a definition as I can an equation derived by rules.)

If you say that in an existence proof the interval isn’t essential, because another interval might have done as well, of course that doesn’t mean that not specifying an interval would have done as well. – The relation of a proof of non-existence to a proof of existence is not the same as that of a proof of p to a proof of its contradictory.

One should suppose that in a proof of the contradictory of “($n)” it must be possible for a negation to slip in which would enable “~ ($n)” to be proved erroneously. Let’s for once start at the other end with the proofs, and suppose we were shown them first and then asked: what do these calculations prove? Look at the proofs and then decide what they prove.’

the first statement should be –

‘I have constructed a new game’ –

and if we are to be consistent – the second statement would be –

‘I have not constructed a game’ – and of course – rather a pointless statement

numbers are markers – operatives in a game – in a sign-game –

to speak of numbers outside of a sign-game is logically incoherent

an ‘existence’ theorem – ‘a theorem that tells us to look for a number’ – makes no logical sense

‘construct a sign-game’ – yes but would we call that an ‘existence theorem’? –

constructing a sign game is what creative mathematicians do – but would we call that an ‘existence theorem’?

as for the ‘contradictory of an existence theorem’ –

where is the value – the sense in proposing – what – quite simply –‘doesn’t exist’?

‘the contradictory of a proved proposition’?

yes – you could put up such a proposal – such an argument – but all it actually means is that you do not use the ‘proved proposition’ – you don’t work with it – you don’t use it –

beyond that such a proposal – such an argument – is just verbiage

or as Wittgenstein says in relation to equations –

(‘For 2 x 3 ¹ 7 doesn’t mean that the equation 2x 3 =7 isn’t to occur, like the equation 2 x 3 = sine; the negation is an exclusion within a predetermined system. I can’t negate a definition as I can an equation derived by rules.)’

‘within a predetermined system’ – means according to rules of practise that the  practitioners adhere to

if you play the sign-game in accordance with the accepted rules of practice – there will be no error – ‘error’ has no place in the game

as for – ‘Look at the proofs and then decide what they prove.’ –

‘proofs’ – are deductive language games – if they are constructed properly – they ‘prove’ – what has been proposed –

and it is just what is proposed – as distinct from what is not proposed – that we operate with – go forward with – that we use

‘I don’t need to assert that it must be possible to construct the n roots of equations of the n-th- degree; I merely say that the proposition “this equation has n roots” hasn’t the same meaning if I’ve proved it by enumerating the constructed roots as if I’ve proved it in a different way. If I find a formula for the roots of an equation, I’ve constructed a new calculus; I haven’t filled the gap in an old one.’

yes – a new calculus – a new game –

‘Hence it is nonsense to say that the proposition isn’t proved until such a construction is produced. For when we do that we construct something new, and what we now mean by the fundamental theorem of algebra is what the present ‘proof’ shows us.’

the proof of a proposition – is a language-game – a deductive argument –

as regards ‘a new construction’ – that is a sign-game that is being proposed for use –

using the new construction – the new game – is a separate matter to the proof of the proposition

the ‘present proof’ – gives the proposition a functional validity  –

that is to say the practitioners regard the proof as a sign of the validity of their practise

and as the sign to proceed with the proposition

what ‘the fundamental theorem of algebra’ then amounts to – is – nothing more than –

the functional validity of the practise – the practise of algebra – the playing of the game

‘ “Every existence proof must contain a construction of what it proves the existence of.” You can only say “I won’t call anything an “existence” proof unless it contains such a construction”. The mistake consists in pretending to posses a clear concept of existence.

We think we can prove a something, existence, in such a way that we are then convinced of it independently of the proof. (The idea of proofs independent of each other – and so presumably independent of what is proved.) Really existence is what is proved by the procedures we call “existence proof”. When the intuitionists and others talk about this they say: “This state of affairs, existence, can be proved only this and not thus.” And they don’t see that by saying that they have simply defined what they call existence. For it isn’t at all like saying “that a man is in the room can only be proved by looking inside, not by listening at the door”.’

what exists is what is proposed –

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

as for an ‘existence proof’ – no more than an argument regarding which proposal to use

‘the man is in the room’ – is a proposal – open to question – open to doubt –  uncertain

and any evidence for the proposal – or argument for the proposal – is logically speaking – no more than another set of proposals – open to question – open to doubt –

and uncertain

‘We have no concept of existence independent of our concept of an existence proof.’

this is a presumptuous statement – to say the least –

an ‘existence proof’ is an argument – a proposal or set of proposals

and there is nothing against taking a particular view on the ‘concept of existence’ – i.e.  adopting the so called ‘existence proof’  –

and if that is the practice – within the given practice – so be it –

however the fact remains –

this will be just one practice among many in the whole range of propositional practices – of human practices –

all of which – from a logical point of view – are open to question – open to doubt –

and are uncertain

‘Why do I say that we don’t discover a proposition like the fundamental theorem of algebra, and that we merely construct it? – Because in proving it we give it a new sense that it didn’t have before. Before the so called proof was only a rough pattern of that sense in the word-language.’

any so called ‘fundamental theorem of algebra’ is a proposal – open to question – open to doubt – uncertain

what we do is propose – that is the basic logical action –

you can dress it up and call it a ‘discovery’ –  or a ‘construction’ – the fact remains – it is – a proposal –

‘proving it’ is just putting up an argument for it – which – when all is said and done is another – proposal

as to ‘rough pattern of that sense’ – another description

the real question is – what is the point?

what is the point of a so called ‘fundamental theorem of algebra’?

what is shown by such a proposal that is not shown in any algebraic game?

what is shown in such a proposal that is not shown in the practice of algebra?

you don’t need to underpin your practice with ‘fundamentals’ –

all that counts is doing the work with the tools you have –

how you describe that – after – or before the fact – is – I would say – irrelevant – to the practice

it is just packaging – rhetoric

‘Suppose someone were to say: chess only had to be discovered, it was always there! Or: the pure game of chess was always there; we only made the material game alloyed with matter.’

we can say – in retrospect – chess was proposed –

as to – ‘it was always there’ –

it was only ‘there’ – when proposed –

the ‘pure game of chess’ – is a proposal

the making of the material game – was a proposal – or set of proposals – that presumably followed the initial proposal of the game

‘If a calculus in mathematics is altered by discoveries, can’t we preserve the old calculus? (That is, do we have to throw it away?) That is a very interesting way of looking at the matter. After the discovery of the North Pole we don’t have two earths, one with and one without the North pole. But after the discovery of the law of the distribution of the primes, we do have two kinds of primes.’

do we have a use for the old calculus?

after the discovery of the North Pole we have two descriptions of the earth

after the discovery of the law of the distribution of primes – we have two proposals regarding primes –

two ‘prime’ proposals

‘A mathematical question must be no less exact than a mathematical proposition. You can see the misleading way in which the mode of expression of word-language represents the sense of mathematical propositions if you call to mind the multiplicity of a mathematical proof and consider that the proof belongs to the sense of the proved proposition, i.e. determines that sense. It isn’t something that brings it about that we believe a particular proposition, but something that shows us what we believe – if we talk of believing here at all. In mathematics there are concept words: cardinal number, prime number, etc. That is why it seems to make sense straight off if we ask “how many prime numbers are there?” (Human beings believe if only they hear the words …) In reality this combination of words is so far nonsense; until it is given a special syntax. Look at the proof  “that there are infinitely many primes,” and then at the question that it appears to answer. The result of an intricate proof can have a simple verbal expression only if the system of expressions to which this expression belongs has a multiplicity corresponding to a system of such proofs. Confusions in these matters are entirely the result of treating mathematics as a kind of natural science. And this is connected with the fact that mathematics has detached itself from natural science; for so long as it is done in immediate connection with physics, it is clear that it isn’t a natural science (similarly, you can’t mistake a broom for part of the furnishing of a room as long as you use it to clean the furniture).’

‘A mathematical question must be no less exact than a mathematical proposition ‘ –

a ‘mathematical question’ – as with any question – and indeed any proposal – any proposition – ‘mathematical’ or otherwise – is logically speaking – open to question – open to doubt – is uncertain –

‘determines sense’? –

the ‘sense’ of a proposition – is always up for grabs –

‘determination’ – if it means anything – means – practice – use

if it is the practice (and it is) – to ‘determine’ – a mathematical proposition in terms of a proof – so be it –

that is the practice – that is how mathematics is done

‘It isn’t something that brings it about that we believe a particular proposition, but something that shows us what we believe – if we talk of believing here at all.’

look – this is an empirical question – i.e. perhaps language users report that ‘something brings it about that we believe a particular proposition’ – and perhaps they say also – in certain circumstances – the proposition can be seen as showing us what we believe

‘if we talk of believing here at all’?

                                                                                                                               

yes – what you ‘believe’ – about the ‘mathematical proposition’ – is actually entirely irrelevant to the doing of mathematics – the playing of the game

mathematics is a rule governed language-game –

what counts in the doing of mathematics is – playing the game –

and you can only do that – if you play by the rules –

in the end it is the propositional action that counts

‘That is why it seems to make sense straight off if we ask “how many prime numbers are there?” (Human beings believe if only they hear the words …) In reality this combination of words is so far nonsense; until it is given a special syntax. Look at the proof  “that there are infinitely many primes,” and then at the question that it appears to answer. The result of an intricate proof can have a simple verbal expression only if the system of expressions to which this expression belongs has a multiplicity corresponding to a system of such proofs.’

the question – in ordinary language invites you – requires you – to engage in the special syntax of the mathematical game –

and an answer in ordinary language requires a translation from that syntax

‘(Human beings believe if only they hear the words …)’

what counts for believing – is open to question –

what we can say is that a proposition put – has a contingent reality – a contingent life

and at the very least is recognized – if not entertained – by those who hear it or see it written

to say that anything proposed is believed by those who witness the proposal –

is plainly not the case – regardless of how you define ‘believing’ –

and it paints the picture of human beings as stupid – in all propositional contexts –

which again – is over doing it

‘Confusions in these matters are entirely the result of treating mathematics as a kind of natural science’

well exactly the same happens in natural science as in mathematics –

the engaging in a technical (non-natural) language –

and the ‘translation’ – usually rough – back to ordinary language

the general point is – we operate with any number of languages and their multiplicity of forms – i.e. ‘ordinary language’– the language of mathematics – the language of physics – etc. etc. –

and we translate from one to the other – and to the other – and so on –

and yes – any ‘translation’ – is open to question – to doubt –

just as any language is – and any proposal – any proposition – in any language is

as to mathematics and physics –

whatever language a physicist uses to deal with the problems of physics –

is the language of physics

and any language use – in physics – mathematics – whatever –

will be subject to question by those involved

‘(similarly, you can’t mistake a broom for part of the furnishing of a room as long as you use it to clean the furniture).’

anything in the room can be variously described – and can have various uses –

logically speaking the room is never stable

‘The main danger is surely that the prose expression of the result of a mathematical operation may give the illusion of a calculus that doesn’t exist, by bearing the outward appearance of belonging to a system that isn’t there at all.’

the prose expression – when understood in a mathematical context – points to the mathematical operation – and its result

it functions as a propositional sign-post

‘A proof is a proof of a particular proposition if it goes by a rule correlating the proposition to the proof. That is, the proposition must belong to a system of propositions, and the proof to a system of proofs. And every proposition in mathematics must belong to a calculus of mathematics. (It cannot sit in solitary glory and refuse to mix with other propositions.)’

yes

‘So even the proposition “every equation of nth degree has n roots” isn’t a proposition of mathematics unless it corresponds to a system of propositions and its proof corresponds to an appropriate system of proofs. For what good reason have I to correlate that chain of equations etc. (that we call the proof) to this prose sentence? Must it not be clear – according to a rule – from the proof itself which proposition it is a proof of?’

‘every equation of nth degree has n roots’ –

this prose proposition points to the mathematics –

to a system of propositions and a proof in an appropriate system of proofs

‘what good reason have I to correlate that chain of equations etc. (that we call the proof) to this prose sentence?’

the association of the prose statement with the mathematics – that chain of equations etc. –

is simply an association of use –

if you like – a pure contingency of practice –

and one that has been adopted because it is useful –

useful as a way into the mathematics

no system of propositions – prose or other – i.e. mathematics – exists in isolation from other propositional practices

any proposition of any system is open to interpretation –

and this will mean interpretation in terms of other propositional  practices

language functions as a dynamic system –

that depends on the interaction of its many and varied practices

this interaction is the grand expression of propositional logic

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

is uncertain

‘Now it is a part of the nature of what we call propositions that they must be capable of being negated. And the negation of what is proved also must be connected with a proof; we must, that is, be able to show in what different, contrasting, conditions it would have been the result.’

‘And the negation of what is proved also must be connected with a proof’

here we have a deductive language game – the proof – of what is not proposed –

I see no point in this – it strikes me as a useless game of syntax

‘that is, be able to show in what different, contrasting, conditions it would have been the result.’

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

what we deal with is what is proposed – not – with what is not proposed –

and what is not-proposed – is logically speaking – not there –

so to pretend that it is – is to  perpetrate a deception –

for what reason I can’t see –

it strikes me that it is really just a result of bad logic – a failure to understand the nature of the proposition – and also the fact that this failure has become entrenched historically in logical practice

you don’t need to engage in this ‘negation game’ to consider ‘what different, contrasting, conditions it would have been the result.’

you just need to understand that the proposition is open to question –

that the proposition is – the focus of possibility

25 Mathematical problems, Kinds of problems, Search, “Projects” in mathematics

‘Where you can ask you can look for an answer, and where you cannot look for an answer you cannot ask either. Nor can you find an answer.’

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

any propositional response put to a proposition – to a proposal – is open to question – open to doubt – is uncertain

‘Where there is no method of looking for an answer, there the question too cannot have any sense. – Only where there is a method of solution is there a question (of course that doesn’t mean: “only where the solution has been found is there a question). That is: where we can only expect the solution of the problem from some sort of revelation, there isn’t even a question. To a revelation no question corresponds.’

a method is an explanation –

it is a proposal to account for how and / or why a proposal has been put

a method for an answer – is the explanation of the answer – a propositional account of a proposed answer

a question can be asked without explanation –

not-knowing is the ground of questioning

a method of solution – is an explanation of solution – a proposal for the how and / or why of a solution

a solution can be given – without explanation –

an explanation is the back story of any proposal –

as to whether such a proposal will be accepted – that is another matter

a revelation – is a solution –

a revelation is an explanation –

as a matter of fact much of the world – explains the world by revelation –

and regards such explanation as solution

that others don’t accept revelation as explanation or solution –

is an argument to be had

and a proposed revelation can be the answer to a question –

that such a proposal is put to question – put to doubt – is logical

any proposal is open to interpretation – and any proposal can be variously described

i.e. – do we not at times regard nature – the physical world – as a revelation?

and indeed could it not be said that an observation is a revelation?

and do we not have the expectation that the truth will be revealed?

‘The supposition of undecidability presupposes that there is, so to speak, an underground connection between the two sides of an equation; that though the bridge cannot be built in symbols, it does exist, because otherwise the equation would lack sense. – But the connection only exists if we have made it by symbols; the transition isn’t produced by some dark speculation different in kind from what it connects (like a dark passage between two sunlit places).’

yes – either the equation is stated – or there is no equation

if the game cannot be formulated – that is – stated – there is no game to play – there is no mathematics

speculation here –

‘so to speak, an underground connection between the two sides of an equation; that though the bridge cannot be built in symbols it does exist’ –

is I think best seen as pre-mathematical

‘I cannot use the expression “the equation E yields the solution S” unambiguously until I have a method of solution; because “yields” refers to a structure that I cannot designate unless I am acquainted with it. For that would mean using the word “yields”

without knowing its grammar. But I might also say: When I use the word “yields” in such a way as to bring in a method of solution, it doesn’t have the same meaning as when this isn’t the case. Here the word “yields” is like the word “win” (or “lose”) when at one time the criterion for “winning” is a particular set of events in the game (in that case I must know the rules of the game in order to be able to say that someone has won) and at another by “winning” I mean something that I could express roughly by “must pay”.’

to say ‘a method of solution’ rather than ‘a meaning’ – indicates that already there is a type of context in mind – specifically mathematical –

so to call for a method of solution here is to place ‘yields’ in a mathematical context – which is to say use a mathematical concept to explain ‘yields’

as to using ‘yields’ without knowing its grammar?

grammar is explanation of use – and in fact terms and indeed propositions are used without – ‘knowing the grammar’ – this is common practice

yes – we get pulled up when the use is put to question –

and while we do operate with ‘some idea’ of the explanation of our terms and propositions – our idea here is more often than not – indeterminate – fuzzy

and it must also be recognized – that – different ‘grammars’ – explanations – of terms of propositions – are invariably in play – in any language exchange –

pining down an accepted and workable – ‘grammar’ – is hard work – and I would say rarely seriously attempted –

our language use – our grammars – are uncertain –

and this is not a result of inattention – carelessness – or dissipation –

it is of the nature of the beast –

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

language is the expression – the great ever changing canvas – of uncertainty

                                                                                                                                   

‘But I might also say: When I use the “yields” in such a way as to bring in a method of solution, it doesn’t have the same meaning as when this isn’t the case’

yes – but no big deal –

meaning is a function of context – of use – it is uncertain – even when a decision has been made as to ‘how to use ..’ – the matter is still open to question

nevertheless we do make decisions as to meaning – as to how to proceed – but there is nothing to appeal to (outside of use – and its indeterminacy) to validate those decisions – this is a hard fact to face

to understand this and to live at peace with it is to be rational – it is a difficult acceptance –

and we can understand the fall back to prejudice – as a surety –

natural as that may be – it is weak – and without any genuine satisfaction –

the problem is you don’t get anywhere with stupidity –

and really it puts you out of the game – the hard game of living –

which of course – is the idea for some – and is in its own way a valid choice –

my argument against such a move is – you are kidding yourself – deluding yourself –

if you think it will work –

however try telling that to someone who doesn’t see that questioning and doubt show the futility of any claim to certainty –

some prejudices – for all intents and purposes – are rock solid – in certain hearts and certain minds –

infertile ground

‘If we employ “yields” in the first meaning, then “the equation yields S” means: if I transform the equation in accordance with certain rules I get S. Just as the equation

25 x 25 = 620 says I get 620 if I apply the rules for multiplication to 25 x 25. But in this case these rules must already be given to me before the “yields” has a meaning, and before the question whether the equation yields S has a sense.’

yes the rules must already be given – if ‘yields’ is to have a rule governed sense

and the rules must be presupposed if the question whether the equation yields S – is to have a rule governed sense –

which means the question must be understood to be asking for a rule governed answer

this discussion of rules does raise the question – are we to say that any rule governed propositional action – that is to say any propositional game – is properly regarded as mathematical – that is even without what we recognize as mathematical symbols or operations?

or another way of putting the question is to ask – is what we call ‘mathematics’ really just a form of what is mathematics – in a more general sense?

mathematics as any rule governed propositional action?

‘It is not enough to say “p is provable”; we should say: provable according to a particular system.

And indeed the proposition doesn’t assert that p is provable according to a particular system S, but according to its own system, the system that p belongs to. That p belongs to the system S cannot be asserted (that has to show itself). – We can’t say, p belongs to system S; we can’t ask, to which system does p belong; we cannot search for p’s system. “To understand p” means, to know its system. If p appears to cross over from one system to another, it has in fact changed its sense.’

the proposition ‘p is provable’ – is – as with any proposition – open to question – open to doubt – is uncertain

of course we can say ‘p belongs to system S’ – if by ‘belongs’ we mean p functions in system S

and indeed we can ask to which system p belongs if what we are doing is systematic – requires the use of systems – and p has been put – put before us – for consideration

it is no big deal to ask the question

as for searching for p’s system –

this is no more than looking for where we can place p in a systematic scenario – if that is what we are considering

to understand p – is to recognize that p is open to question – open to doubt – is uncertain

perhaps p has a function in a system – perhaps it doesn’t

it is not that p ‘may appear to cross over from one system to another’ –

however it may be the case that p is used in one system – and then in another –

a different system may involve – and most likely will involve a different use of p –

in any case – the sense of p – whatever system it is placed in – or even if it is not placed in a system –

                                                                                                                                   

will be open to question – open to doubt – will be uncertain –

and any system under consideration or in use – will be – as with p – open to question – open to doubt – uncertain

and I seriously doubt that this would be news to any working mathematician

‘It is impossible to make discoveries of novel rules holding of a form already familiar to us (say the sign of an angle). If they are new rules, then it is not of the old form.’

making discoveries – propositional discoveries?

well a new proposal – is a contingent fact – if it happens – it happens – if it doesn’t – it doesn’t –

impossibility?

who is to say what can’t happen in the way of proposal?

how could anyone know?

and if you can’t know – how can you say? –

as to novel rules –

to say the discovery of novel proposals is impossible – simply defies the fact –

for every proposal is at some time – is in some way – novel –

how can you know what will be proposed?

and yes you are likely to find new rules – for a new proposal –

but you may also ‘discover’ that old rules can work too –

playing God is a dead end –

and the real sin here is – irrelevance

‘If I know the rules of elementary trigonometry I can check the proposition sin 2x = 2 sin x. cos x, but not the proposition sin x = x – x³   + x⁵  – … but that means the sine

                                                                         3!     5!

function of elementary trigonometry and that of higher trigonometry are different concepts.

The two propositions stand as it were on different planes. However far I travel on the first plane I will never come to the proposition on the higher plane.

A schoolboy, equipped with the armory of elementary trigonometry and asked to test the equation sin x = x - x³

                                      3!

simply wouldn’t find what he needs to  tackle the problem. He not merely couldn’t answer the question, he couldn’t even understand it. (It would be like the task the prince set the smith in the fairytale: fetch me a ‘Fiddle-de-dee’. Bausch, Volsmarchen).’

different concepts – yes – different games – different rules

the two concepts on different planes? –

we don’t need a geometrical image here – what we have plainly and simply is – different games

we are not dealing here with nonsense – which I think is the point of the reference to the prince setting the smith the task of fetching him a fiddle-de-dee – the schoolboy who is only equipped to do elementary trigonometry – is being asked to complete a task he is not equipped for – he is being asked to play a game he can’t play – perhaps someone should teach him the game

‘We call it a problem, when we are asked “how many are 25 x 16”, but also when we are asked: what is ò sin² x dx. We regard the first as much easier than the second, but we don’t see that they are problems in different senses. Of course the distinction is not a psychological one; it isn’t a question of whether the pupil can solve the problem, but whether the calculus can solve it, or which calculus can solve it.’

a ‘problem’ – perhaps – I don’t know – but certainly a question

and if a question is asked – an answer is looked for

‘how many are 25 x 16’? – and ‘what is ò sin² x dx?’ –

two different questions

‘we don’t see that they are problems in different senses’ –

different questions – different answers

‘Of course the distinction is not a psychological one; it isn’t a question of whether the pupil can solve the problem, but whether the calculus can solve it, or which calculus can solve it.’

yes – exactly

‘The distinctions to which I can draw attention are ones that are familiar to every schoolboy. Later on we look down on those distinction, as we do on the Russian abacus (and geometrical proofs using diagrams); we regard them as inessential, instead of seeing them as essential and fundamental’

this is just a point of view on intellectual fashion – if not prejudice

forget ‘fundamental’ – ‘essential’ – ‘inessential’ – these are no more than rhetorical terms

yes – we have – different propositional forms – for different propositional tasks –

what we should teach the schoolboy – is firstly – to keep an open and mind on how to approach a question –

and to be aware of what propositional techniques have been developed and used –

and most importantly – to understand that different games – are played differently –

in the end little more than commonsense

‘Whether a pupil knows a rule for ensuring a solution to òsin² x. dx is of no interest; what does interest us is whether the calculus we have before us (and that he happens to be using) contains such a rule.

What interests us is not whether the pupil can do it, but whether the calculus can do it, and how it does it.’

yes – that is true –

but at the same time it must be appreciated that the calculus – doesn’t exist in a vacuum –

it is a human proposal –

if it works – it works because it can be shown to work –

that is to say it can be demonstrated – and so – understood

a ‘calculus’ that can’t be shown to work – that cannot be demonstrated –

is just a string of undefined syntax

‘In the case of 25 x 16 = 370 the calculus we use prescribes every step for the checking of the equation.

“I succeeded in proving this” is a remarkable expression. (That is something no one would say in the case of 25 x 16 = 400).’

‘checking the equation’ – is nothing more than following the rules of the calculus

following the rules of the calculus will show that 25 x 16 = 400

25 x 16 = 370 – in terms of the rules of the calculus – is meaningless

‘I succeeded in proving this’ – simply means – ‘I followed the rules of the calculus with respect to this proposition-game

a so called ‘proof’ of ‘25 x 16 = 400’ – is no more than – can only be – a statement of the rules of the calculation game with respect to this proposition

‘I succeeded in proving this’ – is ‘I followed the rules’ – and if that’s what you’ve actually done – why would you bother stating it?

any such statement would be redundant – a statement of the obvious –

nevertheless – no harm done

‘One could lay down: whatever one can tackle is a problem. – Only where there can be a problem, can something be asserted.’

an assertion – that is a proposal – a proposition – is open to question – open to doubt – is uncertain

that is the logic of the matter

if you want to introduce this concept of ‘problem’ – which I see to be unnecessary –

then in terms of propositional logic – any assertion (open to question – open to doubt – uncertain) – is a problem

only where there is a proposal (an assertion) – can there be a (problem) question

‘Wouldn’t all this lead to the paradox that there are no difficult problems in mathematics, since if anything is difficult it isn’t a problem? What follows is, that the “difficult mathematical problems”, i.e. the problems for mathematical research aren’t in the same relationship to the problem “25 x 25 = ?” as a feat of acrobatics is to a somersault. They aren’t related, that is, just as very easy to very difficult; they are problems in different meanings of the word.’

firstly –

yes – playing the mathematical game – is not problematic –

it is simply a matter of following the rules of the game –

you play the game

and by ‘game’ – is meant here a propositional construction  – that is without question – without doubt – without ‘problem’ –

if you question – if you doubt – if you look for and / or find ‘problems’ –

you are not playing the game –

you are not doing mathematics

secondly –

any game that is played must first be proposed – must be constructed –

as to any such proposal – any such construction – we face questions – doubt – uncertainty –

and you can call this level of activity ‘pure mathematics’ – or in fact – just ordinary propositional logic –

for any proposal – be it a game proposal or not – is open to question – open to doubt – is uncertain

the pure mathematician proposes and constructs the games that the practicing mathematician utilizes – calls on – plays

the pure mathematician is in the business of propositional game construction

finally –

the game as devised – as constructed – is a result of question and doubt –

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

a feat of acrobatics or a somersault?

rule governed? – perhaps

if so the question is can the player follow the rules?

that is the challenge

‘ “You say ‘where there is a question, there is also a way to answer it’, but in mathematics there are questions that we do not see any way to answer.’ Quite right, and all that follows from that is that in this case we are not using the word ‘question’ in the same sense as above. And perhaps I should have said “here there are two different forms and I want to use the word ‘question’ only for the first”. But this latter point is a side issue. What is important is that we are concerned with two different forms. (And if you say they are just two different kinds of question you do not know your way about the grammar of the word “kind”.)’

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

if you understand this – you also understand that – a proposition – a proposal – as uncertain – raises questions – is a ‘logical space’ – for questions –

‘a way to answer questions’?

a way to answer questions – is a proposal –

now it doesn’t follow that because a question has been asked – there is necessarily a way to answer it

there may be a proposal here – or there may not be –

it is a contingent issue

however ‘any proposal to answer’ – will be open to question – open to doubt – will be uncertain

‘not seeing a way to answer’ – is not peculiar to mathematics –

many questions in many propositional contexts are asked – for which ‘a way to answer’ – is not seen –

we are not dealing here with two different forms of question –

a ‘question’ – in whatever context – invites exploration –

exploration of propositional uncertainty

‘”I know that there is a solution for this problem, although I don’t yet know what kind of solution” – In what symbolism do you know it?’

a ‘solution’ – will be a proposal – that purports to resolve whatever the issue is –

and that proposal – will be open to question – open to doubt – will be logically speaking – uncertain –

but it will be there – it will be proposed

if you have no proposal – you have no solution

you don’t know that there is a solution – or what kind of solution there is –

unless you have a proposal

as to ‘in what symbolism do you know it’?

well – we await the proposal –

for if we have a proposal – it’s symbolism will be clear

‘ “I know that here there must be a law.” Is this knowledge an amorphous feeling accompanying the utterance of the sentence?’

a law is a proposal – in a propositional context – that has been accepted as a direction for proceeding – by those engaged with the issues of that propositional context –

whether there is such a law – such a proposal – is a contingent issue –

either there is – or there isn’t –

there is no ‘must’ – no ‘necessity’ – in propositional logic –

‘must’ – is a term that has no logical function – it’s function is rhetorical

all knowledge – is propositional – that is to say – open to question – open to doubt – uncertain

‘Is this knowledge an amorphous feeling accompanying the utterance of the sentence?’

‘an amorphous feeling’? –

yes – if by this is meant – a feeling of uncertainty –

and if so – what we then have is the question – ‘is there a law here?’ – and that is fair enough –

but a feeling of uncertainty – is not consistent with the utterance of the sentence – ‘I know there must be a law’ – for such an utterance speaks of certainty –

and to have ‘a feeling of certainty’ – is to be epistemologically deluded – and is anything but ‘amorphous’

‘That doesn’t interest us. And if it is a symbolic process – well then the problem is to  represent it in a visible symbolism.’

if the question is whether ‘feelings’ are of interest – then the answer is no – what we deal with is proposals – propositions

logically speaking there is no feeling – if by feeling is meant – some kind of non-public reality

a so called ‘feeling’ – expressed in a proposal – a proposition – is the best you can do here – and that is enough

we are talking here about that which is expressed – made public

and therefore publicly – open to question – open to doubt

as to – ‘a visible symbolism’ –

yes – visible and thus public

‘What does it mean to believe Goldbach’s theorem? What does that belief consist in? In a feeling of certainty as we state or hear the theorem? That does not interest us. I don’t even know how far this feeling may be caused by the proposition itself. How does the belief connect with the proposition? Let us look and see what are the consequences of this belief, where it takes us. “It makes me search for the proof of  the proposition.” – very well; now let us see what your searching really consists in. Then we shall know what belief in the proposition amounts to.’

we have a proposition – belief in it amounts to use of it –

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

the use of it – is a proposal – open to question – open to doubt – uncertain

belief is uncertain

‘We may not overlook a difference between forms – as we may overlook a difference between suits, if it is very slight.

For us – that is, in grammar – there are in a certain sense no ‘fine distinctions’. And altogether the word distinction doesn’t mean at all the same as it does when it is a question of a distinction between two things.’

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

thus logically speaking there is no distinction between propositions –

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

a proposal to distinguish between two things – is open to question – open to doubt – is uncertain

‘A philosopher feels a change in the style of a derivation which a contemporary mathematician passes over calmly with a blank face. What will distinguish the mathematician of the future will be a greater sensitivity, and that will – as it were – prune mathematics; since people will then be more intent on absolute clarity than on the discovery of new games.’

a change in the style of derivation –

is not a change in derivation – just a difference in the way the derivation is approached or perhaps described – represented

I can well understand that a mathematician would not be all that interested in what amounts to a change of fashion –

and frankly I think it would be of limited interest to a philosopher

the best way to prune mathematics is to get philosophers out of it –

absolute clarity – you’ve got to be joking!

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

we operate in and with uncertainty –

the discovery of new games – that is rule governed propositional actions – does not defy propositional uncertainty – it is a relief from it –

mathematicians can understand logic – propositional uncertainty – or not – and still do what they do –

we are all in that boat

wisdom is not necessary to action

the primary role of the philosopher is to challenge prejudice – that is propositions advanced or held without question – without doubt – with certainty –

anyone who does this in any propositional context is acting philosophically

a critical challenge to prejudice may or may not result in a change of practice –

however it may lead to a different understanding –

a rational understanding

‘Philosophical clarity will have the same effect on the growth of mathematics as sunlight has the growth of potato shoots. (In a dark cellar they grow yards long.)’

what effects the growth of mathematics – is a subject for speculation and argument

who’s to say?

‘A mathematician is bound to be horrified by my mathematical comments, since he has always been trained to avoid in indulging in thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and, to use an analogy from psycho-analysis (this paragraph is reminiscent of Freud), he has acquired a revulsion from them as infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking, demand clarification.’

who is to know how any one will regard anything?

speculation is all very well – and it often reveals more about the speculator – than his speculation –

in any case – it’s an empirical issue how someone responds to another’s musings on how they should or should not do their work

I think there is a touch of pretension here from Wittgenstein –

however to the matter at hand – I think a philosophically inclined mathematician would find Wittgenstein’s views on mathematics to be of great interest – Wittgenstein is a brilliant thinker – and his work is of lasting value

the repression argument?

this depends really on just how you regard mathematics – which is of course a philosophical issue –

i.e. – if you don’t think there are these problems in the first place –

then you are not repressing anything are you?

and really the repression argument is not much more than a stand-over tactic – from someone with the opposite view –

and when you start introducing Freud – you may as well be introducing Greek gods  pixies or soothsaying –

all very well in the right context – but not here

I don’t think the repression argument would be given the time of day by practicing mathematicians –

clarification?

our propositions – are proposals –

open to question – open to doubt – uncertain –

question – doubt – uncertainty –

is our daily bread

26 Euler’s proof

‘From the inequality

1 + 1/2 + 1/3 + 1/4 + …  ≠ 

      (1 + 1/2 + 1/2²  + 1/2³ + …) . (1 + 1/3 + 1/3²  + … )

can we derive a number which is still missing from the combinations on the right hand side? Euler’s proof that there are infinitely many prime numbers is meant to be an existence proof, but how is such a proof possible without a construction?’

yes –  there is no proof without a construction –

there is no existence without a construction

in the absence of a construction – we are effectively left with a ‘speculation space’ – and the question is – how functional is this?

and does mathematics really have a place for such?

‘infinitely many prime numbers’ – is properly understood as ‘the prime numbers game’

we play the game – we generate numbers – and the idea is – if we keep playing the game – we keep generating prime numbers –

‘infinitely many prime numbers’ – is not a number – it is a game designed to repeat

it is a game to be played – in a speculation space

and really can such a ‘mathematical proposition’ function – with a speculation space

and with such a game played in such a space?

is this mathematics?

if mathematicians say – yes – we use such proposals – such propositions – and they function –

then that is the end of it –

but then we are dealing with – or playing games – that can only really be described as mathematically indeterminate

and you could ask – what then is the point of it – you introduce an indeterminacy – and you end up with indeterminacy?

where’s the result?

(actually the same question can be asked in respect to mathematical determinacy) –

what it comes down to – is what it has always come down to –

it’s not the result – it’s the play

mathematics is the play

mathematicians are the players

‘ ~ 1 + ½ + 1/3 + … = (1 + 1/2 + 1/2² + …).(1 + 1/3 = 1/3² + … )

The argument goes like this: the product on the right is a series of fractions1/n in whose denominators all multiples of the form 2^v 3^u occur; if there were no numbers besides theses, then the series would necessarily be the same as the series 1 + 1/2 + 1/3 + … and in that case the sums also would necessarily be the same. But the left hand side is ¥ and the right hand side only a finite number 2/1. 3/2 =3, so there are infinitely many fractions missing in the right-hand series, that is, there are on the left hand side fractions that do not occur on the right. And now the question is: is this argument correct? If it were a question of finite series, everything would be  perspicuous. For then the method of summation would enable us to find out which terms occurring in the left hand series were missing from the right hand side. Now we might ask: how does it come about that the left hand series gives ¥? What must it contain in addition to the terms on the right to make it infinite? Indeed the question arises: does an equation, like 1 + 1/2 +1/3 …= 3 above have any sense at all? I

certainly can’t find out from it which are the extra terms on the left? How do we know that all the terms on the right hand side also occur on the left? In the case of finite series I can’t say until I have ascertained it term by term; and if I do see at the same time which are the extra ones. – Here there is no connection between the result of the sum and the terms, and only such a connection would furnish a proof. Everything becomes clearest if we imagine the business done with a finite equation:

1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ≠ (1 +1/2) . (1 +1/3) = 1 + 1/2 + 1/3 + 1/6

Here again we have that remarkable phenomenon that we might call proof by circumstantial evidence in mathematics – something that is absolutely never permitted. I might also be called a proof by symptoms. The result of the summation is (or is regarded as) a symptom that there are terms on the left that are missing on the right. The connection between the symptom and what we would like to have proved is a loose connection. That is, no bridge has been built, but we rest content with seeing the other bank.

All the terms on the right hand side occur on the left, but the sum on the left hand side is ¥ and the sum of the right hand side is only a finite number, so there must … but in mathematics nothing must be except what is.

The bridge has to be built.

In mathematics there are no symptoms: it is only in a psychological sense that there can be symptoms for mathematicians.

We might also put it like this: nothing can be inferred unless it can be seen.’

I understand Wittgenstein’s frustration here –

it’s apples and oranges – and what is the point?

the fact remains that if the rule is there – adopted and practiced by mathematicians –

then we have a mathematical game –

but is a ‘speculation space’ – acceptable in an equation?

clearly it can’t be reasonably seen as a ‘term’ –

it is in fact the absence of a term –

but then this absence is represented – i.e. ‘…’?

and what can you do with it?

accept it?

the point of the equation – of any equation is the ‘=’ sign –

and as such there really is no question – of just whether the left hand side and the right hand side are equal –

that is the rule

and then the question is –

is the proposed equation – of use in mathematics – is it used by mathematicians – and perhaps others – and do they regard its use as fruitful?

if the answer is yes –

then we have – dare I say it – a ‘valid’ – equation – a working equation –

and this – even though at its heart – right at its heart – is –

propositional uncertainty

‘That reasoning with all its looseness no doubt rests on the confusion between a sum and the limiting value of a sum.

We do see clearly that however far we continue the right hand series we can always continue the left hand one far enough to contain all the terms of the right hand one. (And that leaves it open whether it contains other terms as well).’

could you say here that the equation functions as a working hypothesis?

or even – a game – the play of which – determines the game – where the play is undetermined – by the game?

if so you have a new kind of game – a new kind of mathematical game –

a game where subjectivity or indeterminacy – is a feature – a characteristic – of the game –

and with such a game you could no longer conceive it as independent – of its being played

and it is the game being played – that is the indeterminate element of the game

we have here an ‘uncertainty principle’ –  and a ‘quantum’ mathematics

‘Could I add further prime numbers to the left hand side in this proof? Certainly not, because I don’t know how to discover any, and that means that I have no concept of prime number; the proof hasn’t given me one. I could only add arbitrary numbers (or series).’

if only arbitrary numbers or series could be added – then could not what is added just be ‘further prime numbers’?

if all we have here is arbitrariness – then there is no question of ‘knowing how to discover’ or ‘having a concept of prime number’ –

or another way of looking at it is –

in an arbitrary context – ‘knowing how’ or ‘having a concept of’ – would just be instances of arbitrariness?

even so I can’t see the point of any such game –

so who would play it and why?

‘(Mathematics is dressed up in false interpretations).’

it is really a question of what interpretations mathematicians give their assent to –

what interpretations they find useful

‘(“Such a number has to turn up” has no meaning in mathematics. That is closely connected with the fact that “in logic nothing is more general or more particular than anything else”).’

if the mathematical game you are playing – as constructed – leaves open the question of what numbers will occur – then you are dealing with an indeterminacy

to say ‘such a number must turn up’ – in a game of indeterminacy – is the gambler’s delusion

‘If the numbers were all multiples of 2 and 3 then

would have to yield

but it does not … What follows from that? (The law excluded middle). Nothing follows, except that the limiting values of the sums are different; that is, nothing. But

then we might investigate how this comes about. And in so doing we might hit on numbers that are not representable as 2n 3m. Thus we shall hit on larger prime  numbers, but we will never see that no number of such original numbers will suffice for the formulation of all numbers.’

the limiting value of the sums is different – and this you might expect –

so yes – nothing follows –

investigating this?

‘hitting on numbers not representable by 2n 3m’ –

takes us out of the 2n 3m. game –

and in going there – out of that game – we don’t actually effect that game –

we have moved to a different space – with perhaps the makings of new game

‘we will never see that no number of such original numbers will suffice for the formulation of all numbers.’

and yes you can say we shall never know –

but that is not the point –

in mathematics – the issue is what we know –

and what we know – is the determination of the games we propose – construct and play

‘1 + 1/2 +1/3 + … ≠ 1 + 1/2 + 1/2² + 1/2³

However many terms of the form 1/2^v I take they never add up to more than 2, whereas the first four terms of the left-hand series already add up to more than 2 (So this must already contain the proof.) This also gives us at the same time the construction of a number that is not a power of 1, for the rule now says: find a segment of the series that adds up to more than 2: this must contain a number that is not a power of 2.’

yes – here we have as the ‘≠’ makes clear – a game of inequality

‘(1 + 1/2 + 1/2² +…) . (1 + 1/3 + 1/3³ + …) …(1 + 1/n + 1/n2 …) = n

If I extend the sum 1+1/2 + 1/3 +… until it is greater than n, this part must contain a term that doesn’t occur in the right hand series, for if the right hand series contained all those terms it would yield a larger and not a smaller sum.’

‘If I extend the sum 1+1/2 + 1/3 +… until it is greater than n’ –

then – (1 + 1/2 + 1/22 +…) . (1 + 1/3 + 1/32 + …) …(1 + 1/n + 1/n2 …)  ≠  n

and therefore the proposition is not an equation –

however this can be interpreted otherwise –

if you were to extend 1+1/2 + 1/3 +… until it is greater than n – then the right hand side at any point in the extension would be n +

here you have a proposal for an ‘on-going equation’ – a ‘rolling equation’ – if indeed you would still call this proposition an ‘equation’ –

for in such an interpretation the ‘=’ sign functions as an axis on which left and right sides of the proposition turn

this is a mathematical game where the initial proposition functions as a door to indeterminacy

and so a game that generates itself – its terms – in play

‘(1 + 1/2 + 1/22 +…) . (1 + 1/3 + 1/32 + …) …(1 + 1/n + 1/n2 …) = n’ –

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

and in mathematics interpretation is a question of what rules are applied to the proposition

27 The trisection of an angle, etc.

      

‘We might say: in Euclidean plane geometry we can’t look for the trisection of an angle, because there is no such thing, and we can’t look for the bisection of an angle, because there is no such thing.

In the world of Euclidean elements I can no more ask for the trisection of an angle than I can search for it. It just isn’t mentioned.

(I can locate the problem of the trisection of an angle within a larger system but can’t ask within the system of Euclidean geometry whether it’s soluble. In what language should I ask this? In the Euclidean? But neither can I ask in Euclidean language about

the possibility of bisecting an angle within the Euclidean system. For in that language that would boil down to a question about absolute possibility, which is always nonsense.)

Incidentally, here we must make a distinction between different sorts of question, a distinction which will shows once again that what we call a “question” in mathematics is not the same as what we call by that name in everyday life. We must distinguish between the question “how does one divide an angle into two different parts?” and the question “is this construction the bisection of an angle?” A question only makes sense in a calculus which gives us a method for its solution; and a calculus may well give us a method for answering the one question without giving us a method for answering the other. For instance, Euclid doesn’t shew us how to look for the solution to his problems; he gives them to us and then proves that they are solutions. And this isn’t a psychological or pedagogical matter, but a mathematical one. That is, the calculus (the one he gives us) doesn’t enable us to look for the construction. A calculus that doesn’t enable us to do that is a different one.(Compare methods of integration with methods of differentiation, etc.)’

‘In the world of Euclidean elements I can no more ask for the trisection of an angle than I can search for it. It just isn’t mentioned.’ –

yes – so if you ask the question – you are not in the world of Euclidean elements –

you are coming from a non-Euclidian context

‘A question only makes sense in a calculus which gives us a method for its solution’ –

a question – asks for a method of solution –

and indeed ‘the method of solution’ – is open to question – open to doubt – is uncertain –

the ‘question’ – is not the captive of any language form

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

the logical reality – is the reality of question – of doubt – of uncertainty –

your question – is open to question

‘In mathematics there are very different things that all get called proofs, and the differences between them are logical differences. The things called ‘proofs’ have no more internal connection with each other than the things called ‘numbers’.

a proof in mathematics irrespective of what form it takes – is the decision to suspend question – to suspend doubt –

mathematical proof is a rhetorical device

‘What kind of proposition is “It is impossible to trisect an angle with ruler and compass”? The same kind, no doubt, as “There is no F(3) in the series of angle- divisions F(n) just as there is no 4 in the series of combination-numbers n.(n-1)”.

                                                                                                                         2

But what kind of proposition is that? The same kind as “there is no ½ in the series of cardinal numbers”. That is obviously a superfluous rule of the game, something like: in draughts there is no piece called the “queen”. The question whether trisection is possible is then the question whether there is such a thing in the game as trisection, whether there is such a piece in draughts called “the queen” that has some kind of role like that of the queen in chess. Of course this question could be answered simply by stipulation; but it wouldn’t set any problem or task of calculation, so it wouldn’t have the same sense as a question whose answer was; I will work out whether there is such a thing. (Something like: I will work out whether any of the numbers 5, 7, 18, 25 is divisible by 3). Now is the question about the possibility of trisecting an angle that sort of question? It is if you have a general system in the calculus for calculating the possibility of division into n equal parts.

Now why does one call this proof the proof of this proposition? A proposition isn’t a name; as a proposition it belongs to a system of language. If I can say “there is no such thing as trisection” then it makes sense to say “there is no such thing as quadrisection”, etc., etc. And if this is a proof of the first proposition (a part of its syntax), then there must be corresponding proofs (or disproofs) for the other propositions of the proposition system, otherwise they don’t belong in the same system.’

‘What kind of proposition is “It is impossible to trisect an angle with ruler and compass”?

first up like any proposition – it is open to question – open to doubt – uncertain

‘kind of’? –

here we are talking about contexts of use

i.e. in a logical context – a rule?

in an empirical context – a statement of fact?

and –

‘The question whether trisection is possible is then the question whether there is such a thing in the game as trisection’

yes – it can be so regarded

‘Now why does one call this proof the proof of this proposition?’

because this proof – that is – this argument – is offered in relation to this proposition

a question of time and place

as for ‘corresponding disproofs’ – it will depend on how far you want to take the argument

‘otherwise they don’t belong in the same system.’ –

in general – no propositional ‘system’ will be given comprehensive explication –

and the logical reality is that any so called ‘system’ – in use – will not be fixed

the system – as with the propositions associated with it – is fluid

open to question – open to doubt – uncertain

what’s in and what’s out – at any point of the argument – will fashion the system –

and this is an on-going process – until – for whatever reason –

it stops

‘I can’t ask whether 4 occurs among the combination-numbers if that is my number system. And I can’t ask whether ½ occurs in the cardinal numbers, or show that it isn’t one of them, unless by “cardinal numbers” I mean part of a system that contains ½ as well. (Equally I can’t either say or prove that 3 is one of the cardinal numbers.) The question really means something like this: “If you divide ½ do you get whole numbers?, and that can only be asked in a system in which divisibility and indivisibility is familiar. (The working out must make sense.)

If we don’t mean by “cardinal numbers” a subset of the rational numbers, then we can’t work out whether 81/3 is a cardinal number, but only whether the division 81/3 comes out or not.’

yes – always here –

a question of which propositional game you are playing

‘Instead of the problem of trisecting an angle with straightedge and compass we might investigate a parallel, and much more pernicious problem. There is nothing to prevent us restricting the possibilities of construction with straightedge and compass still further. We might for instance lay down the condition that the angle of the compass may not be changed. And we might lay down that the only construction we know – or better: that our calculus knows – is the one to bisect a line AB, namely

(That might actually be the primitive geometry of a tribe. I said above that the numbers “1, 2, 3, 4, 5, many” has equal rights with the series of cardinal numbers and that would go for this geometry too. In general it is a good dodge K,Kin our investigations to imagine the arithmetic or geometry of a primitive people.)

I will call this geometry the system µ and ask: “in the system µ is it possible to trisect a line?”

What kind of trisection is meant in this question? That’s obviously what the sense of the question depends on. For instance, is what is meant is physical trisection – trisection, that is by trial and error and measurement? In that case the answer is perhaps yes. Or optical trisection – trisection that is, which yields three parts which look the same length? It is quite imaginable that the parts a, b, and c might look the same length if, for instance, we were looking through some distorting medium.

We might represent the results of division in the system µ by the numbers 2, 2^2,, 2³,

etc. in accordance with number of the segments produced; and the question whether trisection is possible might mean: does any of the numbers in this series = 3? Of course that question can only be asked if 2, 2^2,, 2³, etc are imbedded in another system (say cardinal number system); it can be asked if these numbers are themselves our number system for in that case we, or our system, are not acquainted with number 3.  But if our question is: is one of the numbers 2, 2^2, , etc. equal to 3. then there is nothing really said about the trisection of a line. Nonetheless, we might look in this manner at the question about the possibility of trisection – We get a different view, if we adjoin to the system µ a system in which lines are divided in the manner of this figure. It can then be asked: is a division into 180 sections a division of type µ? And this question might again

boil down to: is180 a power of 2? But it might also indicate a different decision procedure (have a different sense) if we connected the systems µ and b to a system of geometrical constructions in such a way that it could be proved in the system that the

two constructions “must yield” the same division points B, C, D.

Suppose that someone, having divided a line AB into 8 sections in the system µ,  groups these lines into the lines a, b, c, and asks: is that a trisection into 3 sections? (We could make the case more easily imaginable if we took a larger number of original sections, which would make it possible to form groups of sections which

looked the same length). The answer to that would be a proof that 2³ is not divisible by 3; or an indication the sections are in the ratio 1: 3: 4. And now you might ask: but surely I do have a concept of trisection in the system, a concept of division which yields parts a, b, c, in the ration 1 : 1: 1? Certainly I have now introduced a new concept of ‘trisection of a line’; we might well say that by dividing the line AB into eight parts we have divided the line CB into 3 equal parts, if that is just to mean we have produced a line that consists of 3 equal parts.

The perplexity with which we found ourselves in relation to the problem of trisection was roughly this: if the trisection of an angle is impossible – logically impossible – how can we ask questions about it at all? How can we describe what is logically impossible and significantly raise the question of its possibility? That is, how can one put together logically ill sorted concepts (in violation of grammar, and therefore nonsensically) and significantly ask about the possibility of the combination? – But the same paradox would arise if we asked “is 25 x 25 = 620”; for after all it’s logically impossible that the equation should be correct; I certainly can’t describe what it would be like if … - Well, a doubt whether 25 x 25 = 620 (or whether it = 625) has no more and no less sense than the method of checking gives it. It is quite

correct that we don’t here imagine, or describe, what it is like for 25 x 25 to be 620; what that means is that we are dealing with a type of question that is (logically) different from “is this street 620 or 625 meters long”?

(We talk about a “division of a circle into 7segments” and also of a division of a cake into 7 segments).’

if the question is trisection – then the is no point in considering systems that do not allow – or do not have a place for trisection

and you can say – in general that such a question would not arise – unless there was such a system –

in the event of a question being asked for which there is no system –

then either the question is regarded as meaningless –

or a new system is proposed –

or an argument is put that it can be shown that the question has a place in an existing system

that the question is asked – that there is a system – is really about showing that mathematics is an exploration of – mathematics –

mathematical systems are proposed – and we explore their possibilities

as to ‘logically impossible’ –

a logically impossible system – is simply a construction that doesn’t make sense – a construction that cannot be used – that is not used

a logically impossible  construction – is propositional rubbish –

it only attracts our attention because it is presented in a recognizable propositional form –

it is a sham presentation

‘grammar’ is proposition theory – or account –

25 x 25 = 625 – is a propositional practice –

if  25 x 25 = 625 is in use and makes sense to those who use it – then there can be –most likely will be – an account of that use and sense

if you put up an explanation – that doesn’t explain the use –

then you contribute nothing – and keep yourself in the explanatory dark

in any case – explanations come and go – grammars come and go –

even use is on shaky ground –

nevertheless – the propositions that we use –

make the world we live in –

what we are – is what we propose

28 Searching and trying

‘If you say to someone who has never tried “try and move your ears”, he will first move some part of his body near his ears that he has moved before, and either his ears will move at once or they won’t. You might say of this process: he is trying to move his ears. But if it can be called trying, it isn’t trying in at all the same sense as trying to move your ears (or your hands) in a case where you already “know how to do it” but someone is holding them so that you can move them only with difficulty or not at all. It is the first sense of trying that corresponds to trying “to solve a mathematical problem” when there is no method for its solution. One can always ponder on the apparent problem. If someone says to me “try by sheer will power to move that jug at the other end of the room” I will look at it and perhaps make some strange movements with my face muscles; so that even in that case there seems to be such a thing as trying.’

‘trying’?

if you know how to perform a task – you know how to do it –

and in that case you will not be trying to perform it – you will perform it

if you don’t know how to perform the task – you don’t know –

and in that case any attempt at the task – will be pretentious

if you know how to perform the task – but it is not easily performed – i.e. there are obstacles to performing it –

then you can be said to be trying to perform it –

in that you are attempting to overcome the obstacles

as to mathematics –

you either know the game – the rules of the game – or you don’t

if you learn the rules – you will be able to play the game –

where a ‘mathematical’ proposition is put – for which there are no rules – no method of solution known –

then the proposition is not mathematical – i.e. rule governed –

if a ‘mathematical’ proposition is put – and rules are proposed for it – a method is proposed – then you have a game proposal

it might also be that you begin with a proposal – and explore various systems to see where it might have a place –

and further – the development of a new system or game  – in response to a proposition – is most unlikely – but is indeed possible –

you have to begin somewhere

‘Think of what it means to search for something in one’s memory. Here there is certainly something like a search in the strict sense.’

if you are trying to remember a name – you may for example focus on the face you associate with the name – hoping that that focus will result in you remembering the name

this is hope by association –

‘Here there is certainly something like a search in the strict sense.’

we don’t have a definition of ‘search in the strict sense’ from Wittgenstein –

perhaps it would amount to something like ‘retracing my steps’?

in any case some kind of method

the question then is – would a method – such as an association with the memory that you don’t have – but want to have – result in the memory?

the question here comes down to – how would you know?

how would you know that the method resulted in the memory?

you may well have remembered the name anyway – at the time you did remember it – without any method at all

the idea here is that you either remember or you don’t –

and that any so called searching here is really best described as you wanting to remember –

which is just where you started

‘But trying to produce a phenomenon is not the same as searching for it.

Suppose I am feeling for the painful place with my hand. I am searching in touch-space not in pain-space. This means: what I find, if I find it, is really a place and not a pain. That means that even if experience shows that pressing produces a pain, pressing isn’t searching for a pain, any more than turning the handle of a generator is searching for a spark.’

‘But trying to produce a phenomenon is not the same as searching for it.’ –

it could be – could be described that way

‘Suppose I am feeling for the painful place with my hand. I am searching in touch-space not in pain-space.’

here it is really no more than a question of how you describe your action –

I understand the argument that the touch-space is not the pain-space – but it is just an argument

I see no problem with the (unscientific) description – that my hand touches the ‘pain-space’ –

our actions in the absence of description – in the absence of any description – are unknown

description makes known

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

we run with whatever description we find useful / functional at the time

‘Can one try to beat the wrong time to a melody? How does such an attempt compare with trying to lift a weight that is too heavy?’

‘Can one try to beat the wrong time to a melody?

it’s called jazz

‘trying to lift a weight that is too heavy?’

is a problem – how is it to be done?

how do they compare?

in both cases – a question of figuring out how to do it –

in the melody case – you have to think counter to the natural beat

in the lifting case – once you realize you can’t lift it unassisted – you have to find another method

perhaps there is a degree of difficulty in one – that is not in the other –

it really all depends on the circumstances –

and the people involved

‘It is highly significant that one can see the group IIIII in different ways (in different groups); but what is still more noteworthy is that one can do it at will. That is there is quite a definite process of producing a particular “view” at will; and correspondingly a quite definite process of unsuccessfully attempting to do so. Similarly, you can to order see the figure below in such a way that the first one and then the other vertical line is the nose, and first one and then the other line becomes the mouth; in certain circumstances you can try in vain to do the one or the other

The essential thing here is that this attempt is the same kind of thing as trying to lift a weight with the hand; is isn’t like the sort of trying where one does different things, tries out different means, in order (e.g.) to lift a weight. In the two cases the word “attempt” has quite different meanings. (An extremely significant grammatical fact.)’

the figure ‘IIIII’ and    – are proposals – propositions –

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

open to interpretation

‘that one can do it at will’ – or ‘in vain’ –

are accounts – possible accounts –  descriptions of the doing –

these proposal – like the proposal ‘IIIII’  and the proposal   – are open to question – to doubt – uncertain

what we are dealing with here – all that we have here – is proposals – propositions –

‘The essential thing here’ – is propositional logic

yes – you can argue that ‘the word “attempt” has quite different meanings’ –

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

and I would say indeed – this understanding is – ‘extremely significant’