andAs an example, what's the probability of X+1=4 given that X=3? Probability 1. — fdrake
Neither am I. But if probability=1 and true=1, then fdrake's conclusion follows.I've never seen probabilities assigned to mathematical facts like that. Not sure what it means. — fishfry
These are different uses of "1", in different contexts (language-games).1 is a probability and 1 is the number of stars in our solar system. — fishfry
P(X=1|X+1=2). Where X is a random variable. That'll give you probability 1. — fdrake
These are different uses of "1", in different contexts (language-games).
(Iadded this later, to try and clarify). Compare a traditional example:- "John came home in disgrace, a flood of tears and a wrecked car." "In" is ambiguous, because "disgrace", "flood of tears", and "wrecked car" are different kinds of thing, are pieces of different language-games and "in" is polymorphous and has different senses, or uses" in each of them. That's the theme of this whole argument. — Ludwig V
Applying numbers to objects in the solar system is one kind of language-game. Applying numbers to probabilities is quite another. — Ludwig V
Actually, there are (at least) two ways of using numbers in the context of probabilities. There are 6 probabilities (I prefer "possibilities" or "outcomes" as less confusing) when throwing a die, each of which can be assigned a probability of 1/6, and if the 6 comes up we can, I suppose, assign a probability of 1 to that outcome. — Ludwig V
So I would prefer to say that probability is not applicable to either 2+2=4 or (x=3)&(x+1=4). Why? Because there are no other possibilities. Probability of a specific outcome is only meaningful if there is a range of possible outcomes. — Ludwig V
1 is conventionally used as the range of the outcomes. — Ludwig V
Assigning a probability to one outcome and then to another without outside that context is meaningless. 1 isn't counting or measuring anything - it's just the basket (range) within which we measure the probabilities (in relation to the evidence and if there is no evidence, then equally to all). (fdrake is right to emphasize the role of evidence - especially in the context of Bayesian probability) We use 100 as a basket in other contexts when it suits us. In the case of the die, P(1v2v3v4v5v6)=1 is just reasserting the rules. — Ludwig V
In the case of truth, the language-game that provides the context is different. In a sense, when we assign 1 to truth, it is not a number at all. We can equally well use "T" or a tick if it suits us. This reflects the point that "true" is one of a binary pair. Probability isn't. I want to say that probability and truth are different language-games. — Ludwig V
But that would be too quick, because they are related. Probability is what we retreat to when we cannot achieve truth, one might say. There are others - "exaggerated", "inaccurate", "vague", "certain", "distorted", "certain". I would be quite happy to say that truth is not binary, but multi-faceted; the language game of truth has more than two pieces - probability is just one of them. Probability itself has more pieces than are usually recognized. In the context of empirical probability, we find ourselves confronted with "likelihood" and "confidence" and, sometimes, "certainty" and, of course, in the context of Bayesian probability, "credence" - "degree of belief" turns up from time to time, as well. — Ludwig V
@fishfry There's one other point I would like to make, in the context of our previous discussion about time in mathematics. — Ludwig V
Given that, probability is a bit of a problem, because it seems to me that it has time, or at least change, built in to it. (I have seen it said that probability is inherently about the future). — Ludwig V
We build the table around the outcome, in the context of a thought-experiment such as tossing coins or throwing dice or drawing cards lotteries or roulette wheels. (I expect you know that Pascal built the theory around a desire to help his gambing friends) We expect an outcome, when everything changes. Time isn't essential. The outcome could be unknown, for example. Even if it is known, we can pretend that we don't know it. But there is an expectation of change, without which probability makes no sense. So the timeless present does not describe what is going on here. — Ludwig V
One could regard probability theory as applied mathematics, but probability isn't a prediction. Probability statements are neither confirmed nor refuted by the actual outcome. (That's not quite black and white, because we do use deviations from probability predictions as evidence that something is wrong. But still...) — Ludwig V
I prefer to say, however, that the probability table does not change when the outcome is known. It describes a situation and that description is correct even after the outcome is known - it just doesn't apply any longer. So probability = 1 doesn't really apply. — Ludwig V
Ok. That will do. Maybe some of that is helpful. — Ludwig V
Full disclosure - I haven't formally studied probability either, any more than I've studied mathematics. But I have discussed both and thought about both a good deal, in various philosophical contexts. — Ludwig V
Yes. Is that a definition or an axiom? Whatever it is, it isn't just another assignment of a probability because it enables the actual assignments to the outcomes to be made. But I don't see that anything is wrong with representing them as percentages, in which case the probability of the entire event space is 100. Meteorologists seem to be very fond of this.That the total probability of the entire event space is 1. — fishfry
Timeless present? It looks like it. In which case it is what I'm looking for.A probability measure is a function from some event space to the set of real numbers between 0 and 1, inclusive, satisfying some additional rules. That's it. — fishfry
Yes. Most of the discussions I get involved in are at the applied level. But I have seen some posts that are completely abstract. So I think I understand what "event space" means. It is a metaphor to describe a formulation that doesn't identify actual outcomes, but only gives, for example, E(1), E(2)... - variables whose domain is events. In particular applications, that domain is limited by, for example, the rules of the game. That's not a complaint - just an observation.Now particular applications of probability often involve real life, temporal events, such as tomorrow's weather or the next card dealt from a deck. The underlying theory is abstracted from that. — fishfry
Yes. But the mathematical table you draw up doesn't change when it does happen. Assigning a probability to the outcome that happened isn't a change to the table, but just a misleading (to me, anyway) way of saying "this is the outcome that happened (and these are the outcomes that didn't happen)". The table doesn't apply any more.The probability that something, anything at all will happen, is 1. — fishfry
Yes. It's a rule, not an assignment of a probability.The probability that something, anything at all will happen, is 1. That's one of the rules of probability in the Wiki article. — fishfry
Yes. To be honest, the value, throughout our dialogue, is the opportunity for me to see how mathematics reacts to these questions. So the difference is the point. I'm very grateful to you for the opportunity.It's the philosophical contexts that I don't know much about. — fishfry
Neither do I. But given that intermediate probabilities don't apply, I would say that probability in this case doesn't apply. Probability theory has no traction. Perhaps that's too strong. So I'll settle for a philosopher's solution. Philosophers have (at least) two ways of describing statements like this - "trivial" or "empty".
Yes, and I once I realized that, I withdrew. Perhaps I wasn't clear enough.Earlier you said there was something off about using 1 as a probability and that .999... = 1. But that's two uses of the same number 1. — fishfry
Yeah no I ain't assigning random variables to generic mathematical expressions. — fdrake
Yes. Is that a definition or an axiom? Whatever it is, it isn't just another assignment of a probability because it enables the actual assignments to the outcomes to be made. But I don't see that anything is wrong with representing them as percentages, in which case the probability of the entire event space is 100. Meteorologists seem to be very fond of this. — Ludwig V
Timeless present? It looks like it. In which case it is what I'm looking for.[/quote[
I feel like that's a poetic phrase to describe the fact that there's no time in math; that when we say 1 + 1 = 2. Works for me.
— Ludwig V
Yes. Most of the discussions I get involved in are at the applied level. But I have seen some posts that are completely abstract. So I think I understand what "event space" means. It is a metaphor to describe a formulation that doesn't identify actual outcomes, but only gives, for example, E(1), E(2)... - variables whose domain is events. In particular applications, that domain is limited by, for example, the rules of the game. That's not a complaint - just an observation. — Ludwig V
Yes. But the mathematical table you draw up doesn't change when it does happen. Assigning a probability to the outcome that happened isn't a change to the table, but just a misleading (to me, anyway) way of saying "this is the outcome that happened (and these are the outcomes that didn't happen)". The table doesn't apply any more. — Ludwig V
Yes. It's a rule, not an assignment of a probability. — Ludwig V
Yes. To be honest, the value, throughout our dialogue, is the opportunity for me to see how mathematics reacts to these questions. So the difference is the point. I'm very grateful to you for the opportunity. — Ludwig V
To be honest, the use of "probability=1" is so widespread that it seems absurd to speak as if it should be banned. So far as I can see, it doesn't create any problems in mathematics. But in the rough-and-tumble of philosophy, it's a different matter. People asking what the probability is of God existing, — Ludwig V
Neither do I. But given that intermediate probabilities don't apply, I would say that probability in this case doesn't apply. Probability theory has no traction. Perhaps that's too strong. So I'll settle for a philosopher's solution. Philosophers have (at least) two ways of describing statements like this - "trivial" or "empty".
But now consider "There is one star in the solar system". Given that there is just one star in the solar system, intermediate probabilities don't apply. So assigning a probability of 1 is trivial or empty.
But, once I have won the lottery, intermediate probabilities don't apply. — Ludwig V
Most believe it's true. — fishfry
I'm still antsy about assigning a random variable to the truth of a theorem. How do you sample from mathematical theorems? What would it even mean for a mathematical theorem to be expected to be true 9 times out of 10? How do you put a sigma algebra on mathematics itself... — fdrake
Yes, I get that. In the sense that we've discussed, it is a speech act either way. However, axioms and definitions are not the same kinds of speech act. I expect there's a mathematical explanation of the difference. But they are both setting up the system (function?) - preparatory. So they are both different from the statements you make when you start exploring the system, whether proving theorems or applying it.This is one of those times a def is an ax and vice versa. You can say probability is the study of measurable spaces with total measure 1; or you can say that this property is one of the axioms of a probability space. It's the same thing, really. — fishfry
This is a different speech act, even though it may be the same sentence. The context is different.Posterior probability. Updating your probability with new information. Of course once something has happened, the probability is 1 that it happened. But then the probability is 1, so it makes sense to say that, right? — fishfry
So what does it mean to update the table? Are you correcting it, or changing it, or what? It seems like something that happens in time. You might be constructing a new table, I suppose.The point, or my point anyway is that the mathematical theory of probability is entirely abstracted from any meaning or interpretation or philosophy of "probability" that anyone has ever had. — fishfry
In a way, yes.But then the probability is 1, so it makes sense to say that, right? — fishfry
We are very close here. However, I take the fact that intermediate probabilities don't apply to mean that in a context where intermediate probabilities don't apply, "probability = 1" is empty.P(X=1|X+1=2). Where X is a random variable. That'll give you probability 1.
— fdrake
Yes ok, a true proposition has prob 1 and a false one 0. I don't see how intermediate probabilities could apply. — fishfry
.In that sense, my view of probability is not overloaded with philosophical interpretations. Whether that's good or bad I'm not sure. — fishfry
Hardly irrelevant. I think I understand your point about abstract systems and I am interested in interpreting or applying the abstract formal system; but that begins with the system.Perhaps the formalisms are irrelevant to your thoughts. — fishfry
Yes, I get that. There are even some beautiful arguments in philosophy. I'm sometimes tempted to think that the beauty is the meaning. I would, sometimes, even go so far as to agree with Keats' "‘Beauty is truth, truth beauty,—that is all/Ye know on earth, and all ye need to know." But only if all the philosophers are safely corralled elsewhere.Math doesn't do meaning. That is the beauty of abstraction. — fishfry
I've lost the context of this. I do hate the way that some people talk of chance and probability as if they were causes. Most philosophers (after their first year or two) will jump on that very firmly and, yes, the conventional doctrines about causation have little to recommend them. As for real world applications, they are derived from the mathematics, but heavily adapted. For one thing, they don't atually assign probabilities, but estimate them, and buffer them with likelihoods and confidence intervals. Almost a different concept, linked to the mathematics by the "frequentist" approach.All the real world usages of probability, from games of chance to the insurance industry. The way people think about all these correlations actually being causations, somehow. The way philosophers try to think about causality. — fishfry
You're welcome. I agree that there is something universal here. It is the faith that there is order to be found in the chaos we confront in our lives. Some people think that is a truth about the world, but I'm not at all sure it is that. The evidence points both ways. However, chaos is worse than anything. We will do anything, think anything, to achieve some way of organizing the world. Probability is not ideal, but it is better than nothing.When pressed, I believe there must be something universal in all this. If it's just random, that's too nihilistic for me to bear. That would be my philosophy of God, which I never thought of that way before. Thanks for the example! — fishfry
If you think about why you select specialists to ask, you will see that your are not escaping from the serious difficulties about achieving knowledge, in particular, the fact that conclusive proof of anything is very hard to achieve (not impossible, I would say, but still difficult). We have to weigh one argument against another, one piece of evidence against another, and there seem to be few guidelines about how to do that. Eliciting the consensus of those who are competent is one way of doing that - although far from certain. Asking 10,000 random people in the street what credence they have in the Riemann hypothesis won't help much, will it?Credence, or subjective degree of belief. You ask 10,000 specialists in analytic number theory whether they think the Riemann hypothesis is true. You take the percentage of yesses out of the total to be the credence of the group. — fishfry
Oh, I agree that there is a fact there. The question is what it's value is and that takes us back to the evidence.Better clarify that. Everyone's personal opinion is subjective, that's the beauty of the concept of credence. But the FACT that 75% of them think X and 25% think not-X, that's objective. So we can use the rules of probability without having to do metaphysics. — fishfry
But each of those people, if they are rational, will be assigning their credence on the basis of the evidence. But in this case, and many others, the issue is what counts as evidence and how much weight should be placed upon it.Even though we can't know probability of God; every single person in the world can assign that proposition a credence. That's why I'm big on credence. It takes the metaphysics out of probability. We aren't studying anything "out there," we are only studying our own subjective degrees of belief. — fishfry
Yes, I get that. In the sense that we've discussed, it is a speech act either way. However, axioms and definitions are not the same kinds of speech act. I expect there's a mathematical explanation of the difference. But they are both setting up the system (function?) - preparatory. So they are both different from the statements you make when you start exploring the system, whether proving theorems or applying it. — Ludwig V
This is a different speech act, even though it may be the same sentence. The context is different. — Ludwig V
So what does it mean to update the table? Are you correcting it, or changing it, or what? It seems like something that happens in time. You might be constructing a new table, I suppose. — Ludwig V
We are very close here. However, I take the fact that intermediate probabilities don't apply to mean that in a context where intermediate probabilities don't apply, "probability = 1" is empty. — Ludwig V
It depends whether you are a mathematician or a philosopher. — Ludwig V
Hardly irrelevant. I think I understand your point about abstract systems and I am interested in interpreting or applying the abstract formal system; but that begins with the system.
However, I can't help remembering that Pascal was interested in helping his gambling friends, so the application drove the construction of the theory. — Ludwig V
In the same way, counting and measuring drove the construction of the numbers - not that I would reduce either probability theory or numbers to their origins. — Ludwig V
But I do think that interpretations and applications are not an optional add-ons to an abstract system. — Ludwig V
Yes, I get that. There are even some beautiful arguments in philosophy. I'm sometimes tempted to think that the beauty is the meaning. I would, sometimes, even go so far as to agree with Keats' "‘Beauty is truth, truth beauty,—that is all/Ye know on earth, and all ye need to know." But only if all the philosophers are safely corralled elsewhere. — Ludwig V
I'm not a normal philosopher, with a fixed (dogmatic, finalized) doctrine. I'm exploring, with a view, if I'm successful (and I rarely am), I'll be able to understand how these concepts are related and maybe even construct some sort of map or model of them. (I'm heavily influenced by Wittgenstein, I'm afraid, though I'm incapable of imitating him. But that is why I don't do metaphysics.) — Ludwig V
I've lost the context of this. — Ludwig V
I do hate the way that some people talk of chance and probability as if they were causes. Most philosophers (after their first year or two) will jump on that very firmly and, yes, the conventional doctrines about causation have little to recommend them. — Ludwig V
As for real world applications, they are derived from the mathematics, but heavily adapted. For one thing, they don't atually assign probabilities, but estimate them, and buffer them with likelihoods and confidence intervals. Almost a different concept, linked to the mathematics by the "frequentist" approach. — Ludwig V
You're welcome. I agree that there is something universal here. It is the faith that there is order to be found in the chaos we confront in our lives. Some people think that is a truth about the world, but I'm not at all sure it is that. The evidence points both ways. However, chaos is worse than anything. We will do anything, think anything, to achieve some way of organizing the world. Probability is not ideal, but it is better than nothing. — Ludwig V
If you think about why you select specialists to ask, you will see that your are not escaping from the serious difficulties about achieving knowledge, in particular, the fact that conclusive proof of anything is very hard to achieve (not impossible, I would say, but still difficult). We have to weigh one argument against another, one piece of evidence against another, and there seem to be few guidelines about how to do that. Eliciting the consensus of those who are competent is one way of doing that - although far from certain. Asking 10,000 random people in the street what credence they have in the Riemann hypothesis won't help much, will it? — Ludwig V
Oh, I agree that there is a fact there. The question is what it's value is and that takes us back to the evidence. — Ludwig V
So - the great virtue of Bayesian probability is that it will give you a probability for a single case, which neither mathematical nor empirical probability can do. I still have a problem, because we normally express a probability in terms of the number of times it can be expected to show up in a sequence of trials. But that limitation, strictly speaking, means that its application to a single case, which we very often want to know, is extremely murky. Expressing it in terms of making bets helps. — Ludwig V
But each of those people, if they are rational, will be assigning their credence on the basis of the evidence. But in this case, and many others, the issue is what counts as evidence and how much weight should be placed upon it. — Ludwig V
We started off talking about "probability - 1" and in order to understand that, we've explored the construction and meaning of the probability table. — Ludwig V
I think that was all constructive, but we've got as far as we can with it. Now we are talking about Bayesian probability and what credence is. — Ludwig V
I know that I can be a bit relentless. If I'm boring or annoying you, please tell me and I'll shut up. — Ludwig V
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