philosophy of probability

Forums: Philosophy, Philosophy Of Science
Email this Topic • Print this Page

Consider the following.

You meet a stranger in the street. She introduces herself and her son jake. You ask her how many children she has. "Two she replies". What is the probability that the other child is a son too?

A quantum state of an atom is unstable such that it releases either two up particles and one down particle, or it releases two down particles and one up. Both scenarios are equally possible, and equal probability for up or down from any particle emitted. You know that the release has occurred and you know that one of the particles is a down. There is a possibility that the two other states are still entangled and have not collapsed yet. Whatever. What is the probability that the other two particles will manifest as both up when you measure them?

A switch is set up with a random generator. A box seals the generator from the environment. The generator gives either a one or a zero each time it is switched on. If the generator gives a one, then a light comes on the top of a box. However, this light only comes on after the generator has been switched on twice. So if one is generated twice the light comes on the same and at the same time as when a one and a zero have been generated. The generator just keeps being turned on by a battery powered switch. What is the probability that two ones have just been generated if you see the light on? (NB there is no recording device of the first or second number generated. After any number is generated ie one or zero, the generator quickly reverts to a blank display.)

A cube has two faces coloured red, two green and two blue. What is the probability of red showing face up on this randomly thrown cube before you throw it, during the throwing of it, during the looking for it........ and after losing it down a drain by accidentally kicking it before seeing it?

Consider thought experiments for each of the above real situations, what are the probabilities then?

The probability answer of course is 1/3 for each of the above. But they are very different situations. Future, present, past, knowable, unknowable, accidental and imaginary. Are there different philosophical implications for the application of probability theory to these different situations? If not then what is the common factor? Eg is the unknown the same as the unknowable? Is prediction of the future different to prediction of the past?

Topic Stats
Top Replies
Link to this Topic

Type: Discussion • Score: 1 • Views: 2,678 • Replies: 37

No top replies

@pagan,

pagan;138004 wrote:

Consider the following.

You meet a stranger in the street. She introduces herself and her son jake. You ask her how many children she has. "Two she replies". What is the probability that the other child is a son too?

[...]

Consider thought experiments for each of the above real situations, what are the probabilities then?

The probability answer of course is 1/3 for each of the above. [...]

Oh dear! I don't think much about this sort of thing, and even experienced professional mathematicians get these sorts of questions wrong (witness the infamous Marilyn Mach vos Savant affair, where she was right about the Monty Hall problem, and several mathematicians who poured scorn on her were wrong - I got it right!), so there's no shame in it if I'm mistaken.

Having made my excuses (and having skipped the two more complex cases, because I already have problems with the first one), I'll say this is wrong, and the probability of a boy in the first example is just what your instinct (probably!) says it is, i.e. 1/2.

The obvious argument is that she has two children, call them Kid A and Kid B, you have met Kid A, whom you know to be male, you know nothing about Kid B, and the a priori probability of Kid B being male is 1/2, so that's the answer, there's nothing to it.

Being alert for hidden traps, you decide to check your conclusion by means of a different argument. Now let her elder child be Kid 1, and her younger child be Kid 2. (Age is just an arbitrary ordering attribute, so the possibility of twins doesn't matter.) You don't know whether the child you have met is Kid 1 or Kid 2. You apply Bayes' theorem. A priori, there are 4 equally likely cases (for the sexes of Kid 1 and Kid 2, in that order): FF, FM, MF, MM, probability 1/4 each. The respective conditional probabilities of your meeting a male child in each case are 0, 1/2, 1/2, 1. Therefore, the a posteriori probability of the other child being a boy, i.e. of event MM, is:

(1 x (1/4)) / ((1 x (1/4)) + ((1/2) x (1/4)) + ((1/2) x (1/4))) = 1/2

as expected. (You can draw a Venn diagram to make this clearer. I did! I don't trust myself.)

1 Reply

@Twirlip,

Great subject!

When you ask whether or not there are philosophical implications for the application of probability theory (in terms of the situations you given and in general), what comes to mind is the false belief that probability calculus somehow shows what may come to pass in an accurate way. This is not the case. Though I am no expert of probability calculus, I can say that much like truth tables in propositional calculus, truth values (perhaps of complex statements) are determined from individual truth values of atomic components and therein used to determine validity or invalidity. But as far as I know, it's not exactly possible calculate the probability of some x (or even the inductive probability of some x) based on the probabilities of their atomic components (simply, the given variables of the proof). However, you can make adequate generalizations about probabilistic interpretations (thus probability calculus).

As to common factors, as far as probability calculus goes, there are a few to speak of (namely four). Of course, it depends on the modus of probability theory you use(note: P(A) stands for the probability of A). There is subjective interpretation [ the most philosophically abstract of the bunch (i.e. P(A) = degree of belief a person has in proposition a at a given time)], logical interpretation [logical set theory (i.e. P(A) = a-priori probability of A)], relative frequency interpretation [very familiar to those who like mathematics and number theory (i.e. P(A) were A Is a given event and P(A) is the frequency of A relative to some specified event)]. The fourth, and most applicable (in my opinion) is classical interpretation, which takes A to be an event (dependant on finite non-zero possibilities? like finding the probability of having a pair of dice read a certain number ceterus parabus.

But I bring up the different methods of probability calculus also to show the basic principles of probability. Commonly known as Kolmogorov's axioms, its important to note that these have deductive consequences, like propositional/predicate/etc. have inferences/replacement rules, though not the latter. Also, I'm sparring the equations in favor of a literal interpretation unless someone needs to know them. Axiom 1 essentially says that zero is the lowest bound. Axiom 2 says that 1 is the highest bound (equating to tautology). Axiom 3 says that of the probability of disjunction, it is the sum of probabilities of its disjuncts (as long as they are mutually exclusive). This all is just to show that there is an underline framework (axioms/inferences) within an inherently static system.

As far as whether or not the unknown is the same as the unknowable, its seems to me that something is unknown yet knowable if anything because we have yet to know it. It is not impossible to know the unknown (baring some impossibility). If something be unknowable, it has an inherent impossibility factor that "unknown" does not have attached to it.

As to prediction of the future being different than the prediction of the past, Napoleon Bonaparte said it best, "Forethought we have, but never foresight." But Napoleon is an excellent example of someone who should have utilized probability theory. Modern armies have never been able to successfully invade Russia, from Charles of Sweden, etc. onwards. Heck, even Hitler found that out and he should have known at least a little about Napoleonic blunders. Bernard Montgomery even said, "first rule of war on the first page of the book of war? don't invade Russia." LOL!

On a side note though, probability theory is what card counters use in blackjack. If anyone wants the probability readout, it's a fun little proof to see and say to yourself "there are actually people in the world who do this in their heads." LOL!

1 Reply

@VideCorSpoon,

hi twirlip

yes the first one is the dodgy one Smile

Trying the following reasoning to counter your objection .... You know that a woman has two children, and its equal probability of a girl or boy. You find out that at least one child is a boy. What is the probability that the other is a boy?

How is this different to seeing one is a boy, as compared to someone telling you?

The point i am trying to raise here philosophically is with regards to the nature of knowledge and probability.

I share your honesty for the need for excuses lolol re

Quote:

twirlip
Oh dear! I don't think much about this sort of thing, and even experienced professional mathematicians get these sorts of questions wrong (witness the infamous Marilyn Mach vos Savant affair, where she was right about the Monty Hall problem, and several mathematicians who poured scorn on her were wrong - I got it right!), so there's no shame in it if I'm mistaken.

Having made my excuses (and having skipped the two more complex cases, because I already have problems with the first one), I'll say this is wrong, and the probability of a boy in the first example is just what your instinct (probably!) says it is, i.e. 1/2.

i think discussing where this comparison is also wrong (or right but different) gets us to the nub of the problem, and especially with regards to incompleteness of knowledge.

hi videcorspoon

Quote:

As far as whether or not the unknown is the same as the unknowable, its seems to me that something is unknown yet knowable if anything because we have yet to know it. It is not impossible to know the unknown (baring some impossibility). If something be unknowable, it has an inherent impossibility factor that "unknown" does not have attached to it.

yes thats a good analysis of the word unknowable. But we could be more specific and mean 'unknowable' in the sense of say 'the future' or 'hidden variables' or just plain 'a secret' with respect to a particular time or place, yet which will yield knowable phenomena in a different time or place. This potentially distinquishes the past from the future re the meaning of probability theory ..... but not necessarily so maybe.

eg is it different to know the probability that the dice will be thrown such that a red face is up, compared to it has been thrown but you haven't recieved the information yet? You could argue in the latter case that the probability is 1 for whatever colour because it has happened! And again what is going on for the thought experiment versions of these two scenarios? If this is a different use of probability, then what validity does it have in the real world? If it isn't different in the mind, then what is the same between reality and the mind re probability?

There is something very odd about probability theory. Its sort of brain whirling. The bleeding obvious can turn out to be completely wrong. Why is that?

1 Reply

@pagan,

pagan;138266 wrote:

There is something very odd about probability theory. Its sort of brain whirling. The bleeding obvious can turn out to be completely wrong. Why is that?

Have you encountered the two envelopes paradox? and Bertrand's paradox?

1 Reply

@ughaibu,

hi ughaibu Smile

just looking but have already come across an interesting quote from wikepedia page ...

Quote:

The paradox itself is a proof that lack of knowledge of facts is not equivalent to random facts.

1 Reply

@pagan,

pagan;138266 wrote:

hi videcorspoon
yes thats a good analysis of the word unknowable. But we could be more specific and mean 'unknowable' in the sense of say 'the future' or 'hidden variables' or just plain 'a secret' with respect to a particular time or place, yet which will yield knowable phenomena in a different time or place. This potentially distinquishes the past from the future re the meaning of probability theory ..... but not necessarily so maybe.

"Unknown" and "unknowable" necessarily entail their own particular consequences and differ from one another. It's like saying "measurable" and "immeasurable." If something is unknown, there is a chance that it can be known later. To be unknowable is completely remove the possibility of it ever being known (it becomes a stable generalized set variable). If you attribute "unknowable" to the future, then you may possibly get away for it unless you dismiss induction. However, "hidden variables" are not unknowable, but unknown hidden variables yet to be known. We know they exist if we explicitly say they are hidden.

One other thing I would say though is that probability theory is not a crystal ball by any means. Probability theory is a closed system, like propositional calculus, predicate calculus, indicative calculus, modal calculus, etc. What is specified within the terms of a given proof is law. With this in mind, probability calculus itself is a subset of induction, and induction is never accurate because there exists the possibility that something else could occur which had not previously happened. You could have a 99.9% nearly identical set of events over a given amount of time and have within that frame some outlier which throws off the statistical probability by a potentially wide margin.

pagan;138266 wrote:

eg is it different to know the probability that the dice will be thrown such that a red face is up, compared to it has been thrown but you haven't recieved the information yet? You could argue in the latter case that the probability is 1 for whatever colour because it has happened! And again what is going on for the thought experiment versions of these two scenarios? If this is a different use of probability, then what validity does it have in the real world? If it isn't different in the mind, then what is the same between reality and the mind re probability?

There is something very odd about probability theory. Its sort of brain whirling. The bleeding obvious can turn out to be completely wrong. Why is that?

As an example of probability calculus and to answer your question, consider this extremely simple proof (not proof proof but calculus proof). A single die is tossed once, however, we need to the probability of an even number landing.

First, there are a few things to consider, such as what modus of probability we will use and what specific rules we choose to attribute to the proof. Modus wise, we use the classical interpretation. What specific rules we use are those of a conditional probability, which is essentially equated to P (A|B), or the probability of A, given B. P (A|B) is the volume of possible outcomes in which A occurs among the possible outcomes of B. The formula is simple;

In a worded explanation, the proof is such that it takes the number of possible outcomes in which A and B occur divided by total number of possible outcomes, which is divided by the number of possible outcomes of B and the number of possible outcomes. This in turn yields the number of correct outcomes in which A and B occur over the number of possible outcomes in which B occurs.

The proof works out to this;

E means the result of the die landing even and L means the possible number of E is less (L) than 5 (assuming the die is 6 sided). In the case of the number of possibilities in which E and L occur, the probability exists such that E and L is true in two of six outcomes, and landing on an even (less than 5) is four in six.

So take this example into context with your questions. You ask, "is it different to know the probability that the dice will be thrown such that a red face is up, compared to it has been thrown but you haven't received the information yet?" In some respects, the point sounds like a hypothetical-temporal issue that bears little relevance to probability in the calculable sense. In the case of the proof, the die is thrown. The nature of probability is such that it is a closed system where the die is thrown? the whole universe within the terms of this proof depends on it. However, the fact that a die will be thrown most likely means that if you wanted to find that out, you have a higher class probability proof to determine the possibility of the die being thrown in which case the die is thrown and then suppose that the die will land even. Actually, this seems like a probabilistic ontological issue.

As to your point that, "You could argue in the latter case that the probability is 1 for whatever color because it has happened," you are right. Based on Kolmogorov's axioms (specifically axiom II), the maximum probability is 1. Axiom 2 states that the minimum probability is zero. Thus, everything in between determines the probability of x, which is pretty much infinite.

As to "And again what is going on for the thought experiment versions of these two scenarios? If this is a different use of probability, then what validity does it have in the real world", probability is part of a closed system dependant on the truth functional molecular components. We could determine the probability of unicorns menacing the earth feeding on the blood of bunny rabbits and come up with a number. I think a mistake comes when probability theory is taken to be universally applicable and cross systematized. Suffice to say, my proof of unicorns not existing would not mesh well with the proof of unicorns menacing bunny rabbits, or even the ironic proof I have where the bunny rabbits explode once eaten by unicorns and only unicorns.

As to your question, "If it isn't different in the mind, then what is the same between reality and the mind re probability?" You best ask Rene Descartes about that... but even he could not come to ultimate conclusions, only those which he knew most clearly and distinctly.

As to your question, "There is something very odd about probability theory. Its sort of brain whirling. The bleeding obvious can turn out to be completely wrong. Why is that?" I like to think of probability theory as a different world existent in a universe full of all possible worlds? Like Leibniz's monadology. All we are trying to then do under that framework is determine the dominant monad, the best of all possible worlds which reflects the universe most accurately.

1 Reply

@VideCorSpoon,

Quote:

VideCorSpoon
"Unknown" and "unknowable" necessarily entail their own particular consequences and differ from one another. It's like saying "measurable" and "immeasurable." If something is unknown, there is a chance that it can be known later. To be unknowable is completely remove the possibility of it ever being known (it becomes a stable generalized set variable).

while i agree that your definition of unknowable makes sense, i don't think it is the only useful definition of unknowable. I think 'unknowable' can have a contextual meaning. and thus it isn't necessarily removed from the possibility of it ever being known. eg In relativity theory it is postulated that if sirius has been observed to explode by some frames of reference much closer to sirius than us, this fact would be 'unknowable' to us for a certain length of time.

What i want to draw a distinction between is ......

a) not knowing something within a particular context that makes it unknowable, as compared to other contexts where conversely it is possible to know it.

b) not happening to know something within a particular context whereby it is nevertheless possible to know it.

c) Not knowing something because there is no possible context at all for being able to know it.

The possibility of 'unknown' can apply within all of the above, but they are of very different quality. I agree that unknown and unknowable are different in quality, depending upon the context. Unknown can read as knowable within a context, but that context doesn't have to be all of time. Similarly unknowable doesn't have to be restricted to within the context of all time. ie ever being able to know. Time isn't the only context for the possibility or impossibility of knowledge.

1 Reply

@pagan,

pagan;138327 wrote:

while i agree that your definition of unknowable makes sense, i don't think it is the only useful definition of unknowable. I think 'unknowable' can have a contextual meaning. and thus it isn't necessarily removed from the possibility of it ever being known. eg In relativity theory it is postulated that if sirius has been observed to explode by some frames of reference much closer to sirius than us, this fact would be 'unknowable' to us for a certain length of time.

What i want to draw a distinction between is ......

a) not knowing something within a particular context that makes it unknowable, as compared to other contexts where conversely it is possible to know it.

b) not happening to know something within a particular context whereby it is nevertheless possible to know it.

c) Not knowing something because there is no possible context at all for being able to know it.

The possibility of 'unknown' can apply within all of the above, but they are of very different quality. I agree that unknown and unknowable are different in quality, depending upon the context. Unknown can read as knowable within a context, but that context doesn't have to be all of time. Similarly unknowable doesn't have to be restricted to within the context of all time. ie ever being able to know. Time isn't the only context for the possibility or impossibility of knowledge.

I'm sure there are many different applications that could be made for the term "unknowable," although that is where the line is (at least in my opinion). It almost seems as though you want to apply a word with a specific inherent notion (namely unknowable) to a notion which can be defined otherwise. This is not necessarily a bad thing (its interesting because it is potentially very abstract), although I have to say that the term "unknowable" is more of an absolutist term removed from penumbral interpretation. It is on this note that a pause when you say unknowable, "?isn't necessarily removed from the possibility of it ever being known." I tend to side with the belief that within our own formal (and informal) systems of logic, whatever they might be whether they are deduction, induction, etc, categorical statements carry a lot of weight (such as predicates like universal and existential quantifiers) whose applications lend to and embody absolute terms? like what it is to be unknowable. This I feel is a term of universal quantification. Specifically, for everything that is x, If x is A, then x is B. Perhaps a fundamental question is whether or not a universal can be taken as a particular? Maybe this would reveal some fundamental underpinning which allows the absolute term to be taken transcendentally and "unknowable" possibly revealing shades of the unknown, knowable, etc.

But it is interesting that you bring in special relativity theory and the Sirius example. In general, special relativity theory dictates that temporal relations are not absolute, that is, events happening simultaneously at different places in one frame of reference will not be simultaneous in all frames of reference. In your example of Sirius exploding, the relative perception of a stars' explosion witnessed by one frame of reference compared to those of a frame of reference farther away from the star would possibly yield that the explosion was "unknowable" to those farther away (for a certain length of time). However, would the fact that the stars explosion would inevitable be known to the frame of reference farther away discredit the term "unknowable." I suppose this is my central arguing point, which is that (taking into account special relativity theory), the simultaneous event of Sirius exploding entails that the knowledge of its explosion is not "unknowable," but merely "unknown" because it has happened yet bound to reveal itself to the latter's frame of reference. My beef is simply with unknowable as widely applicable, which I do not think it is (at least in this sense).

As far as your three distinctions, it seems like these are things more easily revealed in epistemological philosophies. Although I can see where you want to go with it, I don't think it is possible given the previous issues. It's like a kink in an ontological chain... you can go further, although that little kink is problematic and magnifies as we go on. But this is just my own opinion though.

As to the various uses of unknowable, like I said in the beginning, I'm sure there are a wide variety of applications. But a primary issue is as far as I am concerned the categorical factor which limit the usage significantly. But this is far from saying that what you want to convey is not possible.

1 Reply

@VideCorSpoon,

yeh

well i suppose the ultimate context for unknowable is for an individual life. Even if something is knowable in a region of space eventually, since an individual life is very finite in time, then it is a powerfully limiting context for what is and is not knowable.

And of course there are physical limitations. The physical ability to detect information that then yields knowledge. And further the physical/mental ability to make sense of that information to build further knowledge. These might not be restricted by time constraints at all. eg even an infinitely living goldfish will never understand quantum mechanics.

2 Replies

@pagan,

pagan;138382 wrote:

well i suppose the ultimate context for unknowable is for an individual life. Even if something is knowable in a region of space eventually, since an individual life is very finite in time, then it is a powerfully limiting context for what is and is not knowable.

But corporeal limitations do not really negate the possibility of something being known though, do they? I came across this wonderful example of such an issue in Vedanta Indian metaphysics. Suppose I have never been to India (honestly, never had). I do not know it exists, since I have never seen it. Is it then unknowable or simply unknown?

pagan;138382 wrote:

And of course there are physical limitations. The physical ability to detect information that then yields knowledge. And further the physical/mental ability to make sense of that information to build further knowledge. These might not be restricted by time constraints at all. eg even an infinitely living goldfish will never understand quantum mechanics.

A few things here. Do the physical limitations such as the ability to detect information which in turn yields knowledge dismiss a-priori knowledge (knowledge gained before the senses)? And to make sense and assimilate knowledge is relative to most everyone on the planet (and beyond). If I suffered a brain injury and went uttely dumb, and you sat next to me and a given event occur, was it then unknowable, or unknown? Much like adults and children see the same things and assimilate and interpret it in different ways, the given event is relative? but not unknowable. Perhaps known under different terms. Heck, philosophers and scientists have been doing this for as long as can be remembered.

As to the goldfish, if a 100 monkey's with typewriters were given all of time and put in one room together, they could write all of Shakespeare's works. Perhaps this is where the factors of probability come into play. However, I would not be keen to bring capabilities rather than probabilities into this. Some possess better capabilities than others, and some goldfish may be brilliant in their own right. It's all relative (especially when it comes to categorical predicates).

1 Reply

@VideCorSpoon,

no i was more trying to focus on understanding the philosophical implications of probability theory, and developing a consistency of terms to explore it.

taking example from above

Quote:

You meet a stranger in the street. She introduces herself and her son jake. You ask her how many children she has. "Two she replies". What is the probability that the other child is a son too?

i think twirlip is correct in saying that it is actually 1/2. But trying to explain why it is so in common language is tricky. Its a minefield in fact.

Here is my attempt ...

The key seems to be that knowing there are two children and one is a boy, is different to knowing anything about either boy that distinguishes them from the other.

It comes down to betting.
If I know that there is one boy and two children and that is all, I can bet on the chance that there is a girl. 2/3
If I know any identifier of a boy, then the bet changes immediately to what is the chance that the other with respect to this identifier is a girl. ?

the other is the crucial thing. In the first scenario the other cannot be defined for the bet. As soon as it is then it becomes ?. Eg I bet the other is a girl. Either the information source says "what do you mean by other?" or it accepts/refuses the bet.

eg suppose you get your partial information from a third distant person on a telephone who decides on the following strategy. She looks at the first child and if it is a boy then she says immediately there is a one boy in the family and doesn't look at the other child. So 'the other' is defined as the second child not looked at. If the first is a girl and the second is a boy, then 'the other' is the first child looked at (girl) when the boy was identified. The reciever of this information thus bets that it is 1/3 probability of two boys. The giver of the information however knows that it is either certain that two boys is not possible (having seen a girl and a boy) or that it is 1/2 having not looked at the second child. If such a person were asked 'what is the chance of the other being a boy', would be able to give a meaningful answer because 'the other' is definable. ie zero or 1/2.

Another information source might look at both children and know for certain before pronouncing there is at least one boy in the family. However, if asked to bet on 'the other' being a boy would have no definition of what 'the other' is.

One could say that the person doing the bet (the probability) cannot know 'the other' even if he knows that 'the other' is known outside his context (ie by the person on the telephone). So he is in the context that he doesn't know if the information 'there is a boy' is released after looking at the first or second child. With that piece of information either it is a certainty that the other was not a boy or it is ?. Thus even if 'the other' is defined in one information context, as long as the person making the bet is not in that context and therefore without 'the other' as defined information, then the odds are 1/3 for two boys.

In the example where you yourself see one child and it is a boy, you are now in the context of defining 'the other' through that knowledge. Therefore the odds of two boys is now 1/2.

You meet a stranger in the street. She introduces herself, you chat and she mentions she has 2 children. You ask her if she has two daughters like yourself. "No" she replies. What is the probability that she has two sons? 1/3. You cannot define 'the other' because a child has not been identified.

Now in the above scenario's there would seem to be no reason why the person making the bet couldn't get some extra information and change the odds. ie the information is obviously knowable in an existing context, and therefore is knowable in a universal context.

But does this necessarily follow? It may be that there are physical contexts where knowledge is known, but are unknowable in other physical contexts. eg the physical context itself destroys knowledge passed on to it. Like the infinite goldfish. You can give it all the information you want about QM, but the eye and brain of the goldfish will mash it up into fish fuzz. The context of being a goldfish renders knowledge of QM unknowable. It has a probability of zero. In other contexts it has a probability of 1 (it is known) and in others contexts it is a probability between zero and one.

2 Replies

@pagan,

the two envelope problem is an interesting mirror image in a sense to the two child problem. In the latter, information that is important is not realised as important (the knowledge that the mother has a boy has come from distinguishing him from the other child, rather than through a general impersonal description of the situation). In the former, information that appears important is in fact irrelevant.(the amount of money in the envelope)

ie sometimes particular information within a context affects probability, and sometimes it doesn't. We often get confused.

One thing that interests me is the contexts of past and future. Are they different in terms of probability, and can they be distinguished? We seem to make predictions both forward and backwards in time. Further, science in its drive for universality states that the laws of the universe are symmetric forward or backwards in time.

In the light of QM indeterminism, does this make the past indeterminate as much as the future?........ And would that clash with the theory of the conservation of information?

0 Replies

@pagan,

pagan;138428 wrote:

no i was more trying to focus on understanding the philosophical implications of probability theory, and developing a consistency of terms to explore it.

I think the philosophical implications of probability theory have been discussed within the context of this thread for a while though. Points such as finite logical inferences in a static system, the notion of closed system, absolute terms with categorical predicates, etc. have been brought up more than a few times in different posts.

pagan;138428 wrote:

taking example from above: You meet a stranger in the street. She introduces herself and her son jake. You ask her how many children she has. "Two she replies". What is the probability that the other child is a son too? i think twirlip is correct in saying that it is actually 1/2. But trying to explain why it is so in common language is tricky. Its a minefield in fact.

A problem with this assumption is the simple fact of statistical outliers. When you concretely say that the chance of the second child being either a boy or a girl, you negate the possible third alternative? in this case a hermaphrodite. Is it probable that the third child be a hermaphrodite? Sure. Though not a common occurrence, there exists the possibility that there as at least one hermaphrodite in the world, and subject to introduction to the probability. Simply put, there is no universal quantification here, but a wide degree of existential categorization. You had mentioned earlier that you wished to focus on the philosophical implications of probability theory. What about genderization issues? Would that factor in to the identification of a child as either a boy or a girl in terms of a probability proof? Would that then bring into the picture the concept of sex rather than gender affiliation? Is sex assumed in gender identification? Indeed, explaining in common language is tricky, which is why logical subsystems come into play (which I mentioned in my first post i.e. probability being a closed system) to resolve issues like these with bare inferences and axioms. Were we to take the fact that a boy is a boy and a girl is a girl, we narrow the truth-functional logical framework we are working with, removing it from reality and into the realm of deduction, induction, etc. I think the minefield you speak of is in fact littered with unconsidered possibilities not taken into account.

pagan;138428 wrote:

Here is my attempt ... The key seems to be that knowing there are two children and one is a boy, is different to knowing anything about either boy that distinguishes them from the other. It comes down to betting. If I know that there is one boy and two children and that is all, I can bet on the chance that there is a girl. 2/3 If I know any identifier of a boy, then the bet changes immediately to what is the chance that the other with respect to this identifier is a girl. ?

I am a little confused by this explanation. What is an identifier? Was is being bet upon? Also, one major point to factor in is that because there are two children, one being a boy, the fact that one is a boy does not affect the probability of the sibling being either a boy or a girl. That is determinate on other factors. The probability is mutually exclusive.

pagan;138428 wrote:

the other is the crucial thing. In the first scenario the other cannot be defined for the bet. As soon as it is then it becomes ?. Eg I bet the other is a girl. Either the information source says "what do you mean by other?" or it accepts/refuses the bet.

Can't it? Isn't that what probability is all about? From what I am assuming the first scenario is "If I know that there is one boy and two children and that is all, I can bet on the chance that there is a girl. 2/3" The other can be defined in terms of probability, probable components, etc. And when you bring in the "information source" you introduce a sub probability proof which in terms of the base issue seems to be irrelevant. Also, what is an (or the) information source?

pagan;138428 wrote:

eg suppose you get your partial information from a third distant person on a telephone who decides on the following strategy. She looks at the first child and if it is a boy then she says immediately there is a one boy in the family and doesn't look at the other child. So 'the other' is defined as the second child not looked at. If the first is a girl and the second is a boy, then 'the other' is the first child looked at (girl) when the boy was identified. The reciever of this information thus bets that it is 1/3 probability of two boys. The giver of the information however knows that it is either certain that two boys is not possible (having seen a girl and a boy) or that it is 1/2 having not looked at the second child. If such a person were asked 'what is the chance of the other being a boy', would be able to give a meaningful answer because 'the other' is definable. ie zero or 1/2.

This seems very confused. I follow you up until you say that "the other is defined as the second child not looked at." You then proceed to introduce a switch, saying that the first child (presumably the one identified over the telephone as the first role is switched) is a girl and the second a boy. Ok, this is understandable. But then you state that the "other" is the girl? the first child to be looked at. Isn't this a misuse of the word "other" or at least a relative attribute? This seems immaterial. When you then introduce your probability, hasn't it already been established? Honestly, I think a good deal of the confusion may come from the fact that you are focusing on the catalysts rather than the subjects of the probability. But then, it's not fair on my part to push standard probability theory (in the actual calculable sense) into this, since what you may want to do is rationalize it by other means.

pagan;138428 wrote:

Another information source (P1)? [omit]? context (P5).

Again, it wouldn't be fair on my part to push standard probability theory.

pagan;138428 wrote:

But does this necessarily follow? It may be that there are physical contexts where knowledge is known, but are unknowable in other physical contexts. eg the physical context itself destroys knowledge passed on to it. Like the infinite goldfish. You can give it all the information you want about QM, but the eye and brain of the goldfish will mash it up into fish fuzz. The context of being a goldfish renders knowledge of QM unknowable. It has a probability of zero. In other contexts it has a probability of 1 (it is known) and in others contexts it is a probability between zero and one.

This is definitely an epistemological issue (and a few others). I'm not quite sure where this came from exactly. It seems as though there are cognitive and capacitive arguments in a thread about probability here and probability is being given token usage.

1 Reply

@VideCorSpoon,

Quote:

videcorspoon
I am a little confused by this explanation. What is an identifier? Was is being bet upon? Also, one major point to factor in is that because there are two children, one being a boy, the fact that one is a boy does not affect the probability of the sibling being either a boy or a girl. That is determinate on other factors. The probability is mutually exclusive.

the identifier is anything that distinguishes something from another.

What is being bet upon is whether there are two boys, or reworded as 'the other is a boy' IFF there is an identifier. The two bets have different probabilities.

With regard to probability theory being a closed system, would you say that science necessarily views the universe as a closed system if it makes probability theory fundamental to its universal theories? Further it must reduce the universe to inferences and axioms to make it work. Is this a criticism of science?

1 Reply

@pagan,

pagan;138631 wrote:

the identifier is anything that distinguishes something from another.

So would this mean that the "another" which is being distinguished has a likewise capability of being an identifier? Is this not relative?

pagan;138631 wrote:

What is being bet upon is whether there are two boys, or reworded as 'the other is a boy' IFF there is an identifier. The two bets have different probabilities.

So you are saying that the probability of the child being a boy (or a girl) is dependent on the identifier? Doesn't that seem more like a choice rather than a probability?

pagan;138631 wrote:

With regard to probability theory being a closed system, would you say that science necessarily views the universe as a closed system if it makes probability theory fundamental to its universal theories? Further it must reduce the universe to inferences and axioms to make it work. Is this a criticism of science?

First thing to say is that science does not say anything itself, it is relative to those that follow a particular scientific methodology... so this encompasses a wide variety of approaches. Scientific relativism? But as to your question that, "?With regard to probability theory being a closed system, would you say that science necessarily views the universe as a closed system if it makes probability theory fundamental to its universal theories?

Now if probability theory is a closed system, then that entails that generalizations within that system are taken at truth-functional value for a determinate conclusion. In fact, let the framework of science supply its own relevant evidence of the nature of a closed system? namely the laws of thermodynamics. The first law of thermodynamics state that energy cannot be created nor destroyed, only undergo conversion from one form to another. The second law of thermodynamics states that disorder (entropy) increases. But look at what has been said, we have two axioms on which much (if not all) of our conceived system relies on. These are truth functional. Without this truth functionality, we could not calculate enthalpy (G=H-TS) (which funny enough contains an absolute temperature (in kelvin) K=C+*273? yet another axiom), endergonic and exergonic reactions, etc. Also look at the implications of probability, especially in the case of the second law of thermodynamics. If entropy increases, it necessarily means that energy (and other things as the law (or theory) is widely applicable) seeks equilibrium. Probability is inherent in the law. The product of a reactant seeks a lower level as energy is released fulfilling the terms of entropic reaction. Probability.

Whether or not there is an implicit or explicit assertion that science views the universe as a closed system, it is inherent in its fundamental axioms. And incorporated into those fundamental axioms are essential notions of probability. So in many respects inferences and axioms comprise our system of science. That being said, this does not consider the issues of induction (interestingly enough addressed in the Bayesian theorem and probability theory). But this brings to mind instrumentalism though.

1 Reply

@VideCorSpoon,

Quote:

videcorspoon
So would this mean that the "another" which is being distinguished has a likewise capability of being an identifier? Is this not relative?

whatever, but it makes a difference to the probability Smile

Quote:

So you are saying that the probability of the child being a boy (or a girl) is dependent on the identifier? Doesn't that seem more like a choice rather than a probability?

its only a choice if you are free to choose it. Knowing that there are not two girls doesn't identify either child. No choice.

Quote:

Now if probability theory is a closed system, then that entails that generalizations within that system are taken at truth-functional value for a determinate conclusion. In fact, let the framework of science supply its own relevant evidence of the nature of a closed system? namely the laws of thermodynamics. The first law of thermodynamics state that energy cannot be created nor destroyed, only undergo conversion from one form to another. The second law of thermodynamics states that disorder (entropy) increases. But look at what has been said, we have two axioms on which much (if not all) of our conceived system relies on. These are truth functional. Without this truth functionality, we could not calculate enthalpy (G=H-TS) (which funny enough contains an absolute temperature (in kelvin) K=C+*273? yet another axiom), endergonic and exergonic reactions, etc. Also look at the implications of probability, especially in the case of the second law of thermodynamics. If entropy increases, it necessarily means that energy (and other things as the law (or theory) is widely applicable) seeks equilibrium. Probability is inherent in the law. The product of a reactant seeks a lower level as energy is released fulfilling the terms of entropic reaction. Probability.

Whether or not there is an implicit or explicit assertion that science views the universe as a closed system, it is inherent in its fundamental axioms. And incorporated into those fundamental axioms are essential notions of probability. So in many respects inferences and axioms comprise our system of science.

yes interesting. But thermodynamics is not necessarily QM theory. Therefore thermodynamics is conceivably true even in a deterministic universe. Probability theory then becomes a matter of distribution and approximation..... rather than measurement itself.

1 Reply

@pagan,

pagan;138797 wrote:

whatever, but it makes a difference to the probability

But I would think more the probabilities rather than a general probability (because of the relative nature of the point).

pagan;138797 wrote:

its only a choice if you are free to choose it. Knowing that there are not two girls doesn't identify either child. No choice.

Not necessarily. You can have choice within an a-priori framework. Gottfried Leibniz comes to mind.

pagan;138797 wrote:

yes interesting. But thermodynamics is not necessarily QM theory. Therefore thermodynamics is conceivably true even in a deterministic universe. Probability theory then becomes a matter of distribution and approximation..... rather than measurement itself.

Actually, thermodynamics is inextricably connected to quantum mechanics. The atom is essentially comprised of a dense, positively charged core surrounded by negatively charged electrons which fill the remaining volume of a given atom. Electrons move around in this volume (resisting the electrostatic charge of the nucleus). The vibrating charge of relative electrons create these ripples (electromagnetic waves) which in turn cause the electron to lose energy, seeking lower amplitude (2nd law of thermodynamics). The electron is thrown into a spiral towards the nucleus. However, the death spiral does not continue into the nucleus? what the hell? (what is known as the collapsing atom paradox). Thus quantum mechanics is introduced. I feel I may be getting too technical, so I might as well just plainly say that the fundamentals of thermodynamics apply to quantum mechanics? indeed, quantum mechanics could not do much without it. Thinking about it, quantum mechanics gets a good deal of its public image as being mysterious and particularly deep when the abstract notions of duality come into play. In an interesting turn (out of necessity since physics could not precisely answer the question), it is established (hypothetically and as a result of experimentation) an electron has both a property of a particle, but the characteristics of a wave (of which thermodynamics applies). Interesting? Yes. Too much information? Probably. LOL!

But the point is to illustrate the relationship of quantum mechanics and thermodynamics and that they are inextricably connected. Of course they are not the same, but interdependent? like a substrate and its predicate. Also, some of the more abstract points raised about quantum mechanics, particularly the issue of duality, show a conflict in the truth functional framework (of at least physics). Thermodynamics is oddly tossed in this instance.

However, with all this in mind, thermodynamics is not true in a grand sense, no matter what universe it's in spared the hypothetical. It is a theory (granted a more firmly established one) nestled tight within a normative framework which much of the physical sciences depend on. Scientists never (or at least should never) establish definites and universals, since next month Dr. Tom introduces a new theory just as good with wider applicability and probability.

As to probability theory then becoming a matter of distribution and approximation rather than measurement, was probability theory ever definite to be used as a form of measurement?

0 Replies

@pagan,

the problem with probability is that it is by its very Nature inaccurate

the problem with exact calculations is that it is extremely complex , and therefore closer to the truth of the way things are ,( beyond us really at this point )

hence a connundrum

probability or exact calculations ?

2 Replies

@north,

north;138839 wrote:

the problem with exact calculations is that it is extremely complex , and therefore closer to the truth of the way things are ,( beyond us really at this point )

If you mean exact calculations of outcomes, ie predictions, Solomonoff has proved that they're impossible, regardless of whether the model is deterministic or probabilistic. What does that imply about "the truth of the way things are"?

1 Reply

philosophy of probability

Related Topics

Quick Links

My Account

able2know