Appendix 7: Probability

‘The probability of an hypothesis has its measure in how much evidence is needed to make it profitable to throw it out.’

It is only in this sense that we can say that repeated uniform experience in the past renders the continuation of this uniformity in the future profitable.

If, in this sense, I now say: I assume the sun will rise again tomorrow, because the opposite is so unlikely, I here mean by “likely” and “unlikely” something completely different from what I mean by these words in the proposition “It’s equally likely that I’ll throw heads or tails”. The two meanings of the word “likely” are, to be sure, connected in certain ways, but they aren’t identical.

We only give up an hypothesis for an ever higher gain.’

‘The probability of an hypothesis has its measure in how much evidence is needed to make it profitable to throw it out.’?

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

and any so called ‘evidence’ – will be a proposal or proposals – open to question – open to doubt – uncertain

likewise this claim of ‘profitability’ –

‘profitability’ can only be viewed as pragmatic and / or rhetorical argument

yes you can run with this probability / profitability explanation – but it has no logical advantage – over any other explanation

in the context or contexts in which you operate – it may be the preferred method for deciding which way to go –

that is a question of practice – of fashion

so be it

‘It is only in this sense that we can say that repeated uniform experience in the past renders the continuation of this uniformity in the future profitable.’?

the proposal of – repeated uniform experience in the past – makes repeated uniform experience – a profitable proposal for the future?

well this a stretch – again just what ‘profitability’ is to be – is anything but clear –

and could I suggest – that it can be said –

that we just do run with this idea – the idea of ‘the continuation of this uniformity in the  future’ –

and that there really is no explanation for why?

by all means put one up – or as many as you like –

might well be interesting – useful – instructive –

all to the better

but in the end – speculation –

and if you are not expecting ‘more’ – or something else – here –

the world can be a delight

‘If, in this sense, I now say: I assume the sun will rise again tomorrow, because the opposite is so unlikely, I here mean by “likely” and “unlikely” something completely different from what I mean by these words in the proposition “It’s equally likely that I’ll throw heads or tails”. The two meanings of the word “likely” are, to be sure, connected in certain ways, but they aren’t identical.’?

any proposal as to what is likely – is really putting a positive spin on uncertainty

whether you toss heads or tails –

as with the sun rising tomorrow –

you take a punt

‘We only give up an hypothesis for an ever higher gain.’?

any so called ’gain’ will be a proposal – open to question – uncertain

and just whether in fact people give up an hypothesis for a ‘higher gain’ is open to question too –

what if the reason I change my point of view – is that I want to try something different

explore different ways of understanding  the world –

different ways of operating in it – different ways of being in it –

and do so –

without any calculation of gain –

or loss?

‘Induction is a process based on a principle of economy.’

you might just as well say – ‘a principle of economy is based on induction’ –

or is the point that induction is a ‘more economical’ process?

more economical than what?

and what exactly does ‘economy’ amount to here?

‘induction is a process based on a principle of economy’ – gets us nowhere – on one reading – it’s just verbiage

is the point that induction has a basis – and that basis is ‘a principle of economy’?

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

and as with any proposal has no basis – but it’s assertion

any claim of a basis is not logical – it is rhetorical –

the point of which is to persuade –

and persuade of an authority –

in this case ‘a principle of economy’

the only authority – is authorship –

and authorship has no logical significance

any other claim of authority –

is logically false

NB

rhetoric indeed has a place in human affairs –

but it should not be confused with logic

a rational life is a logical life –

rhetoric is the barrier to a rational life –

if you live logically – you live free

if you live rhetorically –

you live in bondage

‘The question how simple a representation is yielded by assuming a particular hypothesis is directly connected, I believe with the question of probability

why a ‘simple’ representation?

 are we talking here of ‘taste’ – of fashion?

and what of a ‘complex’ representation – any advantages there – more possibilities perhaps?

look – any representation is a proposal –

frankly it is irrelevant if it is characterized as simple – complex – or pretty –

any characterization is logically irrelevant –

what is relevant is that the representation – the proposal – is open to question – open to doubt –

that is the logic of it

you can play the probability game with any proposal –

any hypothesis

‘We may compare a part of an hypothesis with the movement of a gear, a movement that can be stipulated without prejudicing the intended motion. But then of course you have to make appropriate adjustments to the rest of the gear if it is to produce the desired motion. I’m thinking of a differential gear. – Once I’ve decided that there is to be no derivation from a certain part of my hypothesis no matter what the experience to be described may be, I have stipulated a mode of representation and this part of my hypothesis is now a postulate.’

well yes – you can decide to do this – and proceed on that basis –

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

we operate in and with uncertainty

logic does not paralyse action –

logic is action’s context

‘A postulate must be such that no conceivable experience can refute it, even though it may be extremely inconvenient to cling to the hypothesis. To the extent that we can talk here of greater or slighter convenience, there is a greater or slighter probability of the postulate.’

any decision to make a proposition – immune from question – from doubt – is illogical

it is to leave logic and operate rhetorically

whether you are inconvenienced or not – is logically irrelevant

any proposal – any proposition – is from a logical point of view –

open to question – open to doubt

probability is a calculation game

play the game if you think it useful to do so –

understand though –

any claim of ‘greater of slighter probability’ –

as with any other claim – any other proposal –

is open to question – open to doubt –

is uncertain

‘It is senseless to talk of a measure for this probability at this juncture. The situation here is like that in the case of two kinds of numbers where we can with a certain justice say that one is more like the  (closer to it) than a third, but there isn’t any numerical measure of similarity. Of course you could imagine a measure being constructed in such cases, too by say counting the postulates or axioms common to the two systems, etc. etc.’

measurement is a game – language-game

and yes – a measure can be constructed here –

and a probability game played out –

when you play a game – any game – you play it in accordance with the rules –

the rules that define the game

if you don’t play according to the rules – you don’t play –

you are not in the game

the so called ‘rules’ of any game – from a logical point of view – are proposals –

open to question – open to doubt – uncertain

you can of course play a game –

while understanding the uncertainty of its assumptions

game-playing is a mode of human behaviour –

in mathematics as in other areas –

it is a source of delight

I give someone the following piece of information, and no more: at such and such a time you will see a point of light in the interval AB.

Does the question now make sense “Is it more likely that this point will appear in the AC than CB”? I believe, obviously not – I can of course decide that the probability of the event’s happening in CB is to be in the ration CB/AC to the probability of it’s happening in AC; however, that’s a decision I can have empirical grounds for making, but also about which there is nothing to said a priori. It is possible for the observed distribution of events not to lead to this assumption. The probability, where infinitely many possibilities come into consideration, must of course be treated as a limit. That is, if I divide the stretch AB into arbitrarily many parts of arbitrary lengths and regard it as equally likely that the event should occur in any one of these parts, we immediately have the simple case of dice before us. And now I can – arbitrarily – lay down a law for constructing parts of equal likelihood. For instance, the law that, if the lengths of the parts are equal, they are equally likely. But any other law is just as permissible.

probability is supposedly a calculation of likelihood –

however –

regardless of how this game is constructed –

uncertainty is not diminished

any probability proposal – will be like any other proposal – open to question – to doubt – uncertain –

probability is a game we play with uncertainty –

it is philosophically immature – or if not this – pretentious –

we pretend we can limit uncertainty – by our calculation – by our game – by our play

of course this is foolish –

but it can be fun

‘Couldn’t I, in the case of dice too, take, five faces together as one possibility, and oppose them to the sixth as the second possibility? And what, apart from experience, is there to prevent me from regarding these two possibilities as equally likely?’

there is nothing to prevent you from proposing this game – constructing this game –

playing this game

‘Let’s imagine throwing, say, a red ball with one very small green patch on it. Isn’t it much more likely in this case for the red are to strike ground than the green? – But how would we support this proposition? Presumably by showing that when we throw the ball, the red strikes the ground much more often than the green. But that is nothing to do with logic. – we may always project the red and green surfaces and what befalls them onto a surface in such a way that the projection of the green surface is greater than or equal to the red; so that the events, as seen in this projection, appear to have quite a different probability ration from the one they had on the original surface. If, e.g. I reflect the events in a suitably curved mirror and now imagine what I would have held to be the more probable event if I had only seen the image in the mirror.

The one thing the mirror can’t alter is the number of clearly demarcated possibilities. So that if I had n coloured patches on my ball, the mirror would also show n, and if I had decided that these are to be regarded as equally likely, then I can stick to this decision for the mirror image too.

To make myself even clearer: if I carry out the experiment with a concave mirror, i.e.

make the observations in a concave mirror, it will perhaps look as if the ball falls more often on the small surface than on the much larger one; and it’s clear that neither experiment – in the mirror or outside it – has a claim to precedence.’

isn’t it much more likely?

we can’t know – what will happen –

the matter is – uncertain

given this – the question – what is more likely?

really ‘likely’ here –  is a subterfuge – a denial of the logical reality – the logical reality of uncertainty

for implicit in ‘likely’ – is that even given uncertainty – you can know –

and ‘know’ here must be a claim to certainty –

even if it is – as it were – a ‘small’ certainty

for otherwise – the matter is uncertain – and we are back to where we started

our ‘knowledge’ is just what we propose –

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

our knowledge is uncertain –

any claim to certain knowledge – even within a context of uncertainty –

is logically false – is a logical deception

probability – is a ruse –

you pretend to recognize uncertainty –

but play a game that denies it –

(‘all done with mirrors’?)

there is nothing against  putting forward a proposal about what will happen –

and given that there are possibilities – you can make a choice

and by all means bulk up your choice with reasons – calculations etc. –

whatever’s the fashion –

however – logically speaking – whatever you decide –

is no more than a shot in dark

if we are to reconcile probability with logic –

we need to see probability as a game –

a game played as a challenge to uncertainty –

a challenge to logic –

and I would suggest that the point of such a game – and indeed of any game –

if it is understood for what it is – and what it is not –

is pleasure

as to the use that the probability game – or any other game – is put to –

that is a question for its players

‘We may apply our old principle to propositions expressing a probability and say, we shall discover their sense by considering what verifies them.

If I say “That will probably occur” is this proposition verified by the occurrence or falsified by it non-occurrence? In my opinion, obviously not. In that case it doesn’t say anything about either. For if a dispute were to arise as to whether it is probable or not, it would always be arguments from the past that would be adduced. And this would be so even when what actually happened was already known.’

yes

you can’t know what will occur –

by all means have a punt – play the game –

the game of probability

you don’t verify a game or falsify it

you play the game or you don’t play the game

if you have some luck – so be it –

but luck is not verification –

and bad luck is not falsification

as to verification – or falsification –

it is propositions – proposals – not games –

that are the subject of verification and falsification

and furthermore –

any claim of verification or falsification –

from a logical point of view –

is open to question – open to doubt –

is uncertain

‘Causality depends on an observed uniformity. This does not mean that a uniformity so far observed will always continue, but what cannot be altered is that the events so far have been uniform; that can’t be the uncertain result of an empirical series which in its turn isn’t something dependent on another uncertain one and so on ad infinitum.

causality is a proposal – that relates events –

a proposal of observed uniformity – as with any proposal –  any proposition –

is open to question – to doubt – is uncertain

‘but what cannot be altered is that the events so far have been uniform’?

the proposal that the events so far have been uniform – is open to question – open  to doubt – is uncertain

it can be altered and it can be replaced – it can be dropped altogether –

regardless of what does or does not happen in relation to this (or any) proposal –  

the logic of the matter is that a proposal – is open to question – to doubt – is uncertain

holding a proposition as certain – is an act of logical corruption –

it’s the deception game – the rhetoric game

‘When people say that the proposition “it is probable that p will occur” says something about the event p, they forget that the probability remains even when the event p does not occur.’

yes – because the probability game – has nothing to do with what does or does not occur

‘The proposition “p will probably occur” does indeed say something about the future, but not something “about the event p”, as the grammatical form of the statement makes us believe.”

does playing a game – the probability game – say – anything –

anything at all?

perhaps it says – you are in play –

that you are playing the game?

‘If I ask for the grounds of an assertion, the answer to the question holds not only for this person and this action (assertion), but quite generally.’

the ground of an assertion – or statement of – like the assertion itself – is open to question – to doubt – is uncertain

the ‘ground of an assertion’ – is either an amplification of the assertion – that is a rewriting of it – effectively a reassertion of it –

or it is a ‘justification’ of it – which is to say an argument for it –

and here we are in the realm of persuasion – rhetoric

a statement ‘holds’ for another statement – if it is applied to that statement

as to the claim that it applies ‘quite generally’ –

the question is – have you made that application?

can you make that application?

that application – to every person –  every action (assertion)? –

at any time and place?

if you think so –

you are deluded

‘If I say “the weather looks like rain” do I say anything about the future weather? No; I say something about the present weather, by means of a law connecting weather at any given time with weather at an earlier time. This law must already be in existence, and we are using it to construct certain statements about our experience. –

We might say the same of historical statements too. But I was too quick to say that the proposition “the weather looks like rain” says nothing about future weather. It all depends what is meant by “saying something about something”. The sentence says just what it says.

The sentence “p will probably occur” says something about the future only in a sense in which truth and falsehood are completely independent of what will happen in the future.’

‘a law connecting weather at any given time with weather of an earlier time’?

this so called ‘law’ – is just an account of – an explanation of the statement – of the proposal – ‘the weather looks like rain’

an ‘explanation’ – which itself – is no more than a proposal –

open to question – open to doubt – uncertain

‘saying something about something’?

‘the sentence says just what it says’ –

yes – but how the sentence is interpreted – how it is understood – is a question of context – of circumstance –

point being – the matter – the sentence – is uncertain

‘p will occur’? –

really just is the proposition that you can say something about the future –

you can’t know the future –

so on one reading – a serious reading – this proposition is delusional –

as a statement of wishful thinking –

yes we can all understand it –

we understand it as fanciful

‘truth and falsehood are completely independent of what will happen in the future’?

actually no –

a statement is true – if you assent to it – for whatever reason –

false – if you dissent from it –

for whatever reason

and any reason you have for assent – for dissent –

is open question – open to doubt –

is uncertain

‘If we say: “the gun is now aiming at the point p” we aren’t saying anything about where the shot will hit. Giving the point at which it is aiming is a geometrical means of assigning its direction. That this is a means we use is certainly connected with certain observations (projectile parabolas, etc) but these observations don’t enter into our present description of the direction.’

what we have here is a proposal for direction

it is uncertain because the geometry of it may well be altered if other factors are taken into account – contingencies – which cannot be predicted

nevertheless it is a useful proposal – or can be

we can put forward a proposal for direction – geometrical or other – but in terms of what actually occurs –

it is no more than a shot in the dark –

we play these games

‘A Galtonian photograph is the picture of probability.

The law of probability is the natural law you see when you screw up your eyes.’

a Galtonian photograph as an average of other photographs is a picture of probability – it is the picture of a game

uncertainty is what you see when you screw up your eyes

and strictly speaking – it is what you see –

when you don’t

‘“On average the points yielded by the experiment lie on a straight line”. “If I throw with a good die, then on average I throw one every six throws”. What does that mean?

Is the proposition compatible with any experience I may have? If so, it says nothing. Have I decided in advance which experiences are incompatible with it and what is the limit beyond which exceptions may not go without upsetting the rule? No. But couldn’t I have set sch a limit? Of course – Suppose that the limit has been set thus: if 4 out of 6 successive throws turn out the same, then it’s a bad die. Now someone says: “But if that happens only very seldom, mayn’t be a good one after all?” To that the answer is as follows. If I permit the turning up of 4 similar throws among 6 successive ones to occur within a certain number of throws, then I am replacing the first limit with a different one. But if I say “any number of similar successive throws is allowed, as long as it happens sufficiently rarely”, then strictly speaking I’ve defined the goodness of the dies in a way that makes it independent of the result of the throws; unless by the goodness of a die I do not mean a property of the die, but a property of a particular game with it. In that case I can certainly say: in any game I call the die good provided that among N throws of the game there occur not more than log? N similar

successive throws. However, that doesn’t give a test for the checking of dice, but a criterion for judging a particular game.’

 “If I throw with a good die, then on average I throw one every six throws”. What does that mean?

it is a way of seeing – what has happened – a means of organising events – organizing propositions

that you regard it as a guide to the future is really just to make it into a game

however you average out – whatever it is you average – even as you play the game – you don’t know what will happen

(if you did know what would happen – there would be no probability game)

‘is this proposition compatible with any experience I may have?’

is an average an experience?

if you interpret your experience this way – yes

‘if so it says nothing’?

well – it says what it says –

no ‘experience’ – is without interpretation –

and no interpretation is beyond question – beyond doubt –

any interpretation is uncertain –

that’s experience

‘Have I decided in advance which experiences are incompatible with it and what is the limit beyond which exceptions may not go without upsetting the rule?’

if you play the game – you play the game

the game can always be questioned –

but that is not playing the game

‘Suppose that the limit has been set thus: if 4 out of 6 successive throws turn out the same, then it’s a bad die. Now someone says: “But if that happens only very seldom, mayn’t be a good one after all?” To that the answer is as follows. If I permit the turning up of 4 similar throws among 6 successive ones to occur within a certain number of throws, then I am replacing the first limit with a different one …’

what this is all about is how you construct the game –

constructing the game is not playing the game

‘We say if the dies is quite regular and isn’t interfered with then the distribution of numbers 1, 2, 3, 4, 5, 6 among the throws must be uniform, since there is no reason why one number should occur more often than another.’

a ‘reason’ – we might say is what you know – or at least what you work with –

how the dice will play out – is what you don’t know

you can of course interpret the game in terms of your reasoning –

this is just wishfulness

the very point of the game is that you don’t know –

and that – even if your wish is fulfilled –

you are surprised

‘But now let’s represent the throws by the function (x – 3) ² for the arguments 1 to 6, i.e. by the numbers 0, 1, 4, 9 instead of by numbers 1 to 6. Is there any reason why one of these numbers should turn up in the new results more often than another? This shows us that the a priori law of probability, like the minimum-principles of mechanics etc., is a form that laws may take. If it had been discovered by experiment that the distribution of the throws 1 to 6 with a regular die was such that the distribution of the values of  (x – 3) ²  was uniform, it would have been this regularity that was defined as the a priori regularity.

We do the same thing in the kinetic theory of gasses: we represent the distribution of molecular movements in the form of some sort of uniform distribution; but we make the choice of what is uniformly distributed – and in the case of what is reduced to a  minimum – in such a way that our theory agrees with experience.’

yes – you can call the ‘principle’ on which you establish a game – the principle you decide to base it on – ‘a priori’ –

really all that amounts to is – ‘a game-proposal’–

and if you want to argue the toss here – as to what that amounts to – what it should be

all the better to get it sorted before you begin – the play

the point of the game – of any game I would suggest – is to find – or more correctly propose – for the play – a regularity – a uniformity –

otherwise – whence the game?

“The molecules move purely according to the laws of probability” is supposed to mean: physics gets out of the way, and now the molecules move as it were according to the laws of logic. This idea is similar to the idea that the law of inertia is an a priori proposition : there too one speaks of what a body does when it isn’t interfered with. But what is the criterion for it not being interfered with? is it ultimately that it moves uniformly in a straight line? Or is it something different? If the latter, then it’s a matter of experience whether the law of inertia holds; if the former, then it wasn’t a law after all but a definition. So too with the proposition, “if the particles aren’t interfered with, then the distribution of their motion is such and such”. What is the criterion for their not being interfered with? etc.

“The molecules move purely according to the laws of probability” is supposed to mean: physics gets out of the way, and now the molecules move as it were according to the laws of logic.’

there are no ‘laws’ of probability –

there is the probability game – and the proposals that define the game

‘physics’ is a set of proposals – open to question – open to doubt – uncertain

as too – the so called – ‘the laws of logic’ –

any proposition of logic – any proposal – as to how to proceed logically – is open to question – open to doubt – is uncertain –

and indeed this very proposition that – any proposition – is open to question – to doubt – is uncertain – is itself – open to question – to doubt – and is – as with – any proposal –any proposition – uncertain 

what goes for a priori – is – when  stripped of its rhetoric – a proposal –

the a priori argument – is simply that – an argument –

for how to regard the proposal – how it is to be held – how it is to be viewed – how it is to be prosecuted

our propositional reality is uncertain –  and there are always epistemological options –

as the following discussion / argument illustrates –

‘This idea is similar to the idea that the law of inertia is an a priori proposition: there too one speaks of what a body does when it isn’t interfered with. But what is the criterion for it not being interfered with? is it ultimately that it moves uniformly in a straight line? Or is it something different? If the latter, then it’s a matter of experience whether the law of inertia holds; if the former, then it wasn’t a law after all but a definition.’

philosophy is not a result –

philosophy is not a conclusion

it is the argument

propositional reality

(the world we live in)

is the argument

be that logical or

rhetorical

‘To say that the points yielded in this experiment lie roughly on this line, e.g. a straight line. means something like: “seen for this distance they seen to lie on a straight line.”

I may say that a stretch gives the general impression of a straight line;  but I cannot say: “This bit of line looks straight, for it could be a bit of a line that as a whole gives me the impression of being straight. (Mountains on the earth and moon. The earth a ball)

‘seen for’ – or ‘seen from’ ? – anyway –

yes – you make a call – but there is always a question –

no proposal is beyond doubt

‘An experiment with dice lasts a certain time, and our expectations about the future can only be based on tendencies we observe in what happens during this experiment. That is to say, the experiment can only give grounds for expecting that things will go in the way shown by the experiment; but we can’t expect that the experiment, if continued, will now yield results that tally better with a preconceived idea of its course than did those of the experiment we have actually performed. So if, for instance, I toss a coin and find no tendency in the result of the experiment itself for the number of heads and tails to approximate to each other more closely, then the experiment gives me no reason to suppose that if it were continued such an approximation would emerge. Indeed, the expectation of such an approximation must itself refer to a definite point in time, since we can’t say we’re expecting something to happen eventually, in the infinite future.’

‘our expectations about the future can only be based on tendencies we observe in what happens during this experiment’?

if I expect the outcome of double sixes –

yes – that expectation is defined by the game

but it is an expectation that precedes the playing of the game

the playing of the game – and anything I may observe during the play – is not what I base my expectation on

what do I base my expectation on?

nothing – but the logical possibilities of the game –

or you might say having a win – having some fun

to say ‘the experiment can only give grounds for expecting that things will go in the way shown by the experiment’ –

is to say throwing the dice – gives grounds for expecting – that things will go – as they go

if so – there is no expectation –

and without an expectation – what is the point of the game?

‘but we can’t expect that the experiment, if continued, will now yield results that tally better with a preconceived idea of its course than did those of the experiment we have actually performed’

a so called ‘preconceived idea of its course’ – can only be an expectation

continuing to play the game is really no different than the first play –

you have an expectation – you see what happens

So if, for instance, I toss a coin and find no tendency in the result of the experiment itself for the number of heads and tails to approximate to each other more closely, then the experiment gives me no reason to suppose that if it were continued such an approximation would emerge.

there is no tendency in the fact of the result

there is just the result –

any so called ‘tendency’ is no more than wishful thinking

and not seeing a tendency – is either facing the facts –

or failing to keep up the wishful thinking

‘Indeed, the expectation of such an approximation must itself refer to a definite point in time, since we can’t say we’re expecting something to happen eventually, in the infinite future.’

that you expect an outcome to a game – is to play the game –

presumably you will stop playing at some point – and that’ll be the end of it –

but I don’t see why expecting or wishing for a particular outcome at the first toss – is really any different from expecting it or wishing for it at the second – or if the game was to continue – eventually

NB

as to the ‘infinite future’ –

perhaps God does play games –

if so – the question would be –

what is his expectation?

perhaps

to keep playing

‘Any ‘reasonable expectation’ is an expectation that a rule we have observed up to now will continue to hold.

(But the rule must have been observed and can’t, for its part too, be merely expected.)’

a ‘reasonable expectation’?

will be to place –  an expectation within the definition of a game

a ‘rule’ – is a definition of the game

but let’s be clear here – we are game playing

if we play a game – we don’t question – or doubt – we play

the game itself – and any of it’s definitions – as with any proposal – any set of propositions – is open to question – to doubt – is uncertain

but questioning and doubting is not playing

rules are what you have – when you don’t think – or decide not to think

let’s also be clear – rules are not observed –

rules are proposed

to expect a proposal – a rule –  to hold – is to imagine that it will

we don’t know what will happen –

whether a proposal continues to be held – effectively – to be used –

is uncertain

‘The logic of probability is only concerned with the state of expectation in the sense in which logic in general is concerned with thinking.’

logic is the domain of the proposition

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

that is the logic of it

probability is a game –

the propositions that make up the game – like any other proposition – are open to question – open to doubt – uncertain

however the game as played is not open to question – open to doubt

if you question and doubt the game –  you are not playing the game –

the game as played – is not logical

as to expectation –

as a proposal – i.e. ‘I expect to throw two sixes at the next roll of the dice’ –

it is open to question – open to doubt –

it is clearly uncertain

in the form of a game played – i.e. actually expecting an outcome –

it is not logical

the probability game can have the appearance of being logical –

especially when calculations are involved –

here it needs to be understood that calculation is no more than game playing – formalized

the probability game as played –

despite the machinations that can be involved in it –

is properly seen – not as logical –

but rather as a relief – from logic

‘A ray is emitted from the light source S striking the surface AB to form a point of light there and then striking the surface AB¢. We have no reason to suppose that the point on AB lies to the left or to the right of M, and equally none for supposing that the point on AB¢ lies on one side or other of m. This yields therefore incompatible probabilities. But if I make an assumption about the probability of the point on AB lying in AM, how is this assumption verified? Surely, we think, by a frequency experiment. Supposing this confirms the view that the probabilities of AM and BM are equal (and so the probabilities of Am and B¢m differ), then it is recognized as the right one and thus shows itself to be an hypothesis belonging to physics. The geometrical construction merely shows that the fact that AM = MB was no ground for assuming equal likelihood.

this is a probability game –

on the face of it – it looks as if AB = MB

we have no reason for assuming that the ray strikes any particular point of AB or AB¢

and because AB and AB¢ are not equal – we start with incompatible probabilities

just an aside –

this incompatible probabilities argument – is really something of a red herring

we don’t know where the ray will hit – so any probability proposal– will be ‘incompatible’ – with any other proposal

that’s the first point

the second point is that the AB / AB¢ argument proves to be irrelevant to the outcome of this game – a game of equal likelihood

but we are dealing with a game here – so you can red regard the AB / AB¢ argument

as a diversion which if nothing else – spices up the game

that’s the best I think you can say for it

so the question is – how can we assume equal likelihood?

the answer proposed here is to make the assumption that the point on AB lies in AM

and then to test this assumption with a frequency experiment

and the next move in the game is –

‘Supposing this confirms the view that the probabilities of AM and BM are equal (and so the probabilities of Am and B¢m differ)’

now if these steps in the game are followed – what we get is – bingo –

‘an hypothesis belonging physics’

that’s the game played out

Wittgenstein says –

‘The geometrical construction merely shows that the fact that AM = MB was no ground for assuming equal likelihood.’

well as good a reason – I would say –

as making the assumption ‘about the probability of the point on AB lying in AM’

or performing the frequency experiment

point being an assumption is just that – a shot in the dark –

regardless of what arguments are found to support it –

and as to the experiment –

what if the frequency experiment doesn’t show that AM and BM are equal?

well of course that’s not part of the game

and let’s be clear about experiments in general –

an experiment at any point – is open to question – to doubt – is uncertain

we are playing a game here – the game of equal likelihood –

and there are different ways of playing it –

that’s all

‘Suppose that measurement shows the die to be accurate and regular, that the numbers on its sides don’t influence the throws, and that it is thrown by a hand whose movements follow no definite rules: does it follow that the distribution among the throws from 1 to 6 will be uniform on average? Where is the uniform distribution supposed to come from? The accuracy and regularity of the die can’t establish that the distribution of throws will be uniform on average. (It would be, as it were, a monochrome premise with a mottle conclusion.) And we haven’t made any suppositions about the movements while throwing. (Making the bundles of hay equal gives reason to believe that the donkey will starve to death between them; it doesn’t give reason to believe that he will eat from each with roughly the same frequency.) –

It is perfectly compatible with our assumptions for one hundred ones to be thrown in succession, if friction, hand-movements and air resistance coincide appropriately. The experimental fact that this never happens is a fact about those factors, and the throws will be uniformly distributed is an hypothesis about the operation of those factors.’

experiments –

show nothing beyond what happens –

in the experiment –

and ‘what happens’ –

is a matter for interpretation –

and any interpretation is –

open to question – open to doubt –

is uncertain

an hypothesis of uniform distribution –

is no more than wishful thinking –

and why not? –

have a bet –

play the game

‘Suppose someone says that a lever with arms of equal length must remain at rest under the influence of equal and opposite forces, since there is no cause to make it move to one side rather than to the other. That only means that if the lever moves to one side after we have ascertained the equality of arms and the equal and opposite nature of the forces, then we can’t explain this on the basis of the preconditions we know or have assumed. (The form that we call “explanation” must be asymmetrical: like the operation which makes “2a + 3b” out of “a + b”). But on the basis of our precondition we can indeed explain the lever’s continuance at rest. – Could we also explain a swing to the left and right with equal frequency? No, because once again the swing involves asymmetry; we could only explain the symmetry in this asymmetry. If the lever had rotated to the right with a uniform motion, one could similarly have said: given the symmetry of the conditions I can explain the uniformity of the motion, but not its direction.’

the equal and opposite forces – here is an explanation of rest –

not of any motion

‘Could we also explain a swing to the left and right with equal frequency?’

we might put forward a proposal – an account – an explanation – for  what has occurred – but not an explanation for what will occur

as to the symmetry / asymmetry issue

if you begin with the symmetry explanation of an event – then clearly that explanation will be at odds with an asymmetry explanation

asymmetry – or symmetry – is the explanation – not the event

the event – in the absence of proposal – of ‘explanation’ – is unknown

the point of explanation is to make known

and any ‘explanation’ – as with any proposal – is open to question – to doubt –

is uncertain

any proposal regarding what will occur – i.e. a proposal of equal frequency – is a game-proposal –

yes we can play the game of equal frequency –

but play is not explanation

‘A lack of uniformity in the distribution of the throws is not to be explained by the symmetry of the die. It is only to this extent that the symmetry explains the uniformity of the distribution. – For one can of course say: if the numbers on the side of the die have no effect, then the difference between them cannot explain an irregularity in the distribution; and of course similar circumstances cannot explain differences; and so to that extent one might infer a regularity. But in that case why is there any difference at all between the different throws? Whatever explains that must also explain their approximate regularity. It is just that the regularity of the die doesn’t interfere with that regularity.’

‘a lack of uniformity in the distribution of the throws is not to be explained by the symmetry of the die’ –

yes

‘It is only to this extent that the symmetry explains the uniformity of the distribution. – For one can of course say: if the numbers on the side of the die have no effect, then the difference between them cannot explain an irregularity in the distribution; and of course similar circumstances cannot explain differences; and so to that extent one might infer a regularity.’

is not this – the regularity of irregularity?

‘but in that case why is there any difference at all between the different throws?’

we don’t know why –

we can speculate – theorize – and any proposal put forward – will be interesting –

but like the throws themselves – uncertain

‘regularity’ here – and ‘uniformity’ – and their opposites too – are terms – the point of which is to get a fix – or pretend that you can get a fix – on uncertainty – 

really just descriptions of the game – the logic of the game –

a logic that is grounded in uncertainty

game playing – including gambling – can be seen as a way of directly experiencing the logical reality – the logical reality of uncertainty –

game-playing is rightly seen as philosophical practice –

to play is to throw yourself into a metaphysical state –

it is to act logically – authentically

perhaps people take to gambling – and others forms of game-playing in reaction to  pretence – the many faced pretences of claims to certainty

‘whatever explains that must also explain their approximate regularity’

approximate regularity – approximate irregularity – take your pick

the real point is that there will be no definitive explanation

at best we can – as it were – map out or point to the uncertainties involved –

but even here – any proposal will be open to question to doubt – will be uncertain

something like throwing die – lays bear – uncertainty

we begin with uncertainty – and the try to track it –

but any tracking will be with propositions –

open to question – open to doubt – uncertain

here is an excellent illustration of the perfect fit between reality and the proposition

‘It is just that the regularity of the die doesn’t interfere with that regularity.’

the ‘regularity of the die’ – is to do with the definition of the game – the structure of the game –

a game of chance

‘Suppose that a man throwing dice every day threw nothing but ones for a week, using dice that proved good by every other method of testing and that gave the usual results when thrown by others. Has he grounds, now, for supposing that there is a law of nature that he will throws ones? Has he grounds for supposing it will go on like this, or has he grounds for believing that this regularity can’t last much longer? Has he reason to abandon the game since it has become clear that he can only throw ones, or reason to play on since in these circumstances it is all the more probable that he will throw a higher number at the next throw? In actual fact, he will refuse to accept the regularity as a natural law: at least, it will have to go on for a long time before he will entertain the possibility. But why? I believe it is because so much of his previous experience in life speaks against there being a law of nature of such a sort, and we have – so to speak – to surmount all that experience, before embracing a totally new way of looking at things.’

‘Has he grounds for supposing that there is a law of nature that he will throws ones?’

we cannot say what will occur –

any ‘law of nature’ – that makes such a claim – is at best – a speculation

‘Has he grounds for supposing it will go on like this, or has he grounds for believing that this regularity can’t last much longer?’

any grounds we have for any proposal – any proposition – are open to question – open to doubt – are uncertain

‘Has he reason to abandon the game since it has become clear that he can only throw ones, or reason to play on since in these circumstances it is all the more probable that he will throw a higher number at the next throw?’

whatever course he takes – is uncertain

‘In actual fact, he will refuse to accept the regularity as a natural law: at least, it will have to go on for a long time before he will entertain the possibility.’

to accept the regularity as a ‘natural law’ – or to think it is all the more probable that he will throw a higher number – is to fall into delusion –

by all means have a punt – but don’t take it – or yourself – seriously

‘But why? I believe it is because so much of his previous experience in life speaks against there being a law of nature of such a sort, and we have – so to speak – to surmount all that experience, before embracing a totally new way of looking at things.’

as for why ‘a totally new way of looking at things’?

any number of answers can be given – any number of proposals – can be put forward

the weight of previous experience may tie him down – and make it hard for him to see things in a new light –

on the other hand he might decide to  adopt a new way of seeing things – just because he feels too bound up in the past –

who’s to say with any certainty why anyone does anything? 

yes we have a go at explaining – and it can be useful – to put up proposals

but any ‘explanation’ – is open to question – open to doubt – is uncertain

the trick is to understand this and run with it

as to the past – as with the future – and indeed – the present

best to keep an open mind

‘If we infer from the relative frequency of an event its relative frequency in the future,

we can of course only do that from the frequency which has in fact been observed. And not from one we have derived from observation by some process or other for calculating probabilities. For the probability we calculate is compatible with any frequency whatever that we actually observe, since it leaves the time open.’

if it leaves the time open – it’s empirically irrelevant

any ‘calculation of probabilities’  – is a claim on the future

what will happen – is unknown

probability is a game –

imagining that the past or the present is a guide to the future –

just is the game

‘calculating’ – is playing the game

the only good reason –

for playing this game –

is to have fun

NB

you won’t find ‘fun’ –

in a philosophical dictionary

but dare I say it?

fun should be taken –

seriously

‘When a gambler or insurance company is guided by probability, they aren’t guided by the probability calculus, since one can’t be guided by this on its own, because anything that happens can be reconciled with it: no, the insurance company is guided by a frequency actually observed. And that, of course, is an absolute frequency.’

if you believe that what you observe is an indication of what will occur – then whatever occurs is compatible with that belief –

what might happen is never defeated by what does happen

the gambler or the insurance company play the probability game

an ‘absolute frequency’ – is where the game starts

© greg t. charlton. 2015.