## 0.999... = 1

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I apologize for this post. I'm just flailing around. Actually, I'm still not sure where the best place to begin is.

This is about categories or conceptual families or language-games and the importance of context and use. I won't try to give a general characterization of this. I think it will help more if I focus on something specific.

As an example, what's the probability of X+1=4 given that X=3? Probability 1.
and
quote="fdrake;916313"]The probability that 2+2=4 doesn't make too much sense.[/quote]
What fdrake is saying (I think) is that probability is inapplicable without a context of argument and evidence and has much to be said for it.
I've never seen probabilities assigned to mathematical facts like that. Not sure what it means.
Neither am I. But if probability=1 and true=1, then fdrake's conclusion follows.

1 is a probability and 1 is the number of stars in our solar system.
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.
Applying numbers to objects in the solar system is one kind of language-game. Applying numbers to probabilities is quite another. 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.

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. 1 is conventionally used as the range of the outcomes. 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.

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.

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.

@fishfry There's one other point I would like to make, in the context of our previous discussion about time in mathematics. 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). 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.

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...)

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.

Ok. That will do. Maybe some of that is helpful.

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.
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P(X=1|X+1=2). Where X is a random variable. That'll give you probability 1.

Yes ok, a true proposition has prob 1 and a false one 0. I don't see how intermediate probabilities could apply. Unless, say, we could poll a bunch of mathematicians and ask them to assign a probability to the Riemann hypothesis being true. That would be one example I suppose. But I think that's credence (degree of belief) rather than probability (whatever exactly probability is).
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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.

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. So I don't see your point. Of course 1 has many different uses. Why is this nontrivial or interesting?

Applying numbers to objects in the solar system is one kind of language-game. Applying numbers to probabilities is quite another.

It's another. It's not "quite" another. You seem to be saying that it's not only a different usage; but a super-different usage, if I'm understanding you. And it's not. It's just different.

In fact let me tell you what a probability is. It's just a real number between 0 and 1, inclusive. So it's the same real number one in the context of probability or anything else.

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.

Yes ok. You don't have to suppose. Probabilities are additive. That is, if the events are independent (meaning that one is not dependent on the other) then you can add the probabilities. It's one of the axioms of probability. Or one of the consequences of the axioms, depending on how you state the axioms.

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.

I see the point you are making but it doesn't seem right. If we roll a die the probability that it's either 1, 2, 3, 4, 5, or 6 is 1. There is no other possibility. In fact that's another one of the probability axioms: That the total probability of the entire event space is 1.

1 is conventionally used as the range of the outcomes.

1 is a probability. 0 to 1, inclusive, is the range of probabilities.

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.

Well, "evidence" is a term in the philosophy of probability, I suppose. But it's not a word in the formal mathematical theory of probability. In any event, I don't think that's right. Evidence can change the credence of an event -- your subjective degree of belief. But it doesn't change the probability.

I'm in way over my head on the philosophy of probability actually.

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.

Well I certainly agree with you. I am not the one trying to apply probability theory to true/false propositions. @fdrake is doing that. I'm a bit baffled by the attempted connection.

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.

Are you perhaps referring to credance, or the degree of belief? I can't really debate these issues, I know nothing about them. Truth in mathematics is binary. In real life, not so much. Also in intuitionist logic, where we reject the law of the excluded middle. That's another complication.

@fishfry There's one other point I would like to make, in the context of our previous discussion about time in mathematics.

Uh-oh. Was all the preceding not for me? Probably wasn't since it's not about anything I can sensibly talk about. To me a proposition is true or false. That's the definition of a proposition.

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).

Philosophical probability, I suppose. Mathematical probability has no time element in it. 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. No time involved.

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.

The mathematics of probability is abstracted from all that. No time element.

https://en.wikipedia.org/wiki/Probability_axioms

You don't need to follow the symbology. The point is that time is not mentioned. Probability is a mathematical function that outputs a real number in the range [0, 1] and satisfies some rules.

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.

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...)

Probability theory is abstract. Applied probability is applied.

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.

I don't know why you have that hangup about probability 1. Probability 1 is just the probability of the entire event space. It's the claim that out of all the possible outcomes, one of them will occur. After all, in any situation, something must happen, even if we don't know what. The probability that something, anything at all will happen, is 1. That's one of the rules of probability in the Wiki article.

Ok. That will do. Maybe some of that is helpful.

You have thought a lot more deeply about the real-world meaning of probability than I have. The math is just math, as in the article I linked. It's very mathy as you can see.

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.

It's the philosophical contexts that I don't know much about.
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Unless, say, we could poll a bunch of mathematicians and ask them to assign a probability to the Riemann hypothesis being true.

Yeah no I ain't assigning random variables to generic mathematical expressions.
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That the total probability of the entire event space is 1.
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.

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.
Timeless present? It looks like it. In which case it is what I'm looking for.

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.
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.

The probability that something, anything at all will happen, is 1.
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. That's one of the rules of probability in the Wiki article.
Yes. It's a rule, not an assignment of a probability.

It's the philosophical contexts that I don't know much about.
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.

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,

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.
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.

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.
Yes, and I once I realized that, I withdrew. Perhaps I wasn't clear enough.
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Yeah no I ain't assigning random variables to generic mathematical expressions.

Ok. But you know in this case we can. We can interpret probability as credence, the subjective degree of belief, which is an epistemological claim rather than an ontological one. Pretty much any mathematician in that field would be glad to offer a number. Most believe it's true. I'd guess Riemann has better than a 90% credence among specialists.

With this interpretation, we free ourselves from having to give an account of what probability "is." We just talk about our own subjective degrees of belief. Sort of removes the mysticism from interpretations of probability.

This way we can reason mathematically about our beliefs, using the technical apparatus of abstract probability theory.

I'm just trying to interpret this question. About applying probabilities to predicates, I don't know anything about that. But I do think that people could "vote" on predicates, even in situations where you can never know the truth.
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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.

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.

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.

In math, a random variable is just a measurable function on a probability space; which defined as a measure space with total measure 1. It's all very technical and precise, and completely avoids all of the murky metaphysics of randomness. In that sense, my view of probability is not overloaded with philosophical interpretations. Whether that's good or bad I'm not sure. :-)

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.

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.

Yes. I confess I'm not sure what is the main thesis you and I are discussing. But clearly there are two meanings of probability:

* The formal, mathematical one; which doesn't even have a metaphysics. A probability is a mathematical gizmo that obeys some formal rules. Then we prove theorems about gizmos that obey those rules.

* 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.

But I do confess I don't remember what we are talking about :-)

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.

I'm not entirely sure I followed that. I think if probabilities of 0 and 1 make you uncomfortable philosophically, by all means you should reject them, for all the reasons you've been explaining to me.

I don't know what is gained philosophically by my injecting the mathematical formalisms; because really, they don't even have anything to do with the way people think about probability. Perhaps the formalisms are irrelevant to your thoughts.

Yes. It's a rule, not an assignment of a probability.

That was about the total probability being 1. Yes, that's a definition or an axiom, either one. It's a property that characterizes the universe of things that we're interested in when we use the word probability. On the one hand it's arbitrary. On the other hand, it provides tremendous logical clarity, an is amenable to mathematical techniques. Such is the power of mathematical abstraction. But, perhaps in the end, not relevant to your own thoughts on the subject. I can't tell.

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.

Math does not react to those questions. Math can't even see those questions. A "random variable" is "a measurable function in a measure space[/i]. In math the word "random" doesn't mean anything at all. Math doesn't do meaning. That is the beauty of abstraction.

That's good news and bad news. The good news is that as a philosopher, you are free to think about randomness and probability any way you like, because math takes no position at all. And the bad news is that math takes no position at all! The rest of us have to figure out what it means.

What do you think?

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,

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.

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!

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.

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?

https://en.wikipedia.org/wiki/Posterior_probability

Posterior probability

https://en.wikipedia.org/wiki/Posterior_probability
Yes, and I once I realized that, I withdrew. Perhaps I wasn't clear enough.

Oh ok.
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Most believe it's true.

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...
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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...

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.

OR you ask each mathematician what is their subjective belief that it's true; and you average all those individual credences.

If I'm understanding your objection, the idea is to replace the idea of probability, with that of credence.

With probability, we have no idea what it "means" to say that a theorem might be 75% true. But with credence, we do. Even though the theorem itself must be either true or false; still we can each have a fractional "subjective degree of belief" that it's 75% likely to be true.

In this way we can apply the mathematical techniques of probability theory, sigma algebras and such, but without having to figure out what we even mean by probability. We go from the objective to the subjective. From ontology to epistemology. X may be true or it may be false. and no other outcomes are possible. Yet, I can still have a subjective belief, based on what I know, that it's 75% likely. It's just a guess, but it's objective. We ask everyone what they think.

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.

https://en.wikipedia.org/wiki/Credence_(statistics)
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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.
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.

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?
This is a different speech act, even though it may be the same sentence. The context is different.

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.
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.

But then the probability is 1, so it makes sense to say that, right?
In a way, yes.
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.
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.

In that sense, my view of probability is not overloaded with philosophical interpretations. Whether that's good or bad I'm not sure.
.
It depends whether you are a mathematician or a philosopher.

Perhaps the formalisms are irrelevant to your thoughts.
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. 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.
But I do think that interpretations and applications are not an optional add-ons to an abstract system.

Math doesn't do meaning. That is the beauty of abstraction.
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.

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.)

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.
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.
Probability is the main way that we try to limit uncertainty, find some order in the chaos.
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!
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.

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.
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?

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.
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.
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.

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.
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.

We started off talking about "probability - 1" and in order to understand that, we've explored the construction and meaning of the probability table. 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.

I know that I can be a bit relentless. If I'm boring or annoying you, please tell me and I'll shut up.
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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.

Sometimes they are pretty much interchangeable and other times not. It depends on if it's an "if and only if" definition or not.

The axioms of group theory are the definition of group theory.

This is a different speech act, even though it may be the same sentence. The context is different.

You keep trying to frame this discussion in terms of speech acts. I'm not sure what point you are making.

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.

What table? Lost me on that.

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.

I don't know what you mean that a probability can be empty. A probability is a real number between 0 and 1 inclusive.

It depends whether you are a mathematician or a philosopher.

Ok. So please remind me of what point we are trying to discuss.

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.

Applications are always at the historical origin of every abstract theory. Not specific to probability.

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.

Yes of course, no issues there.

But I do think that interpretations and applications are not an optional add-ons to an abstract system.

They are not optional add ons. So they are mandatory add ons? Or not add ons at all? Didn't understand that.

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.

I'm not saying there's no meaning in math. I'm saying that the math itself doesn't refer to its meaning when we're doing the formalizations. The meaning is not to be found in the math, but rather in the minds of those who do or use the math. Is that better?

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.)

I know that whereof I cannot speak, thereof I must put a sock in it. That's as far as my knowledge of Wittgy goes. Also, that he thoroughly misunderstood Cantor's diagonal argument. I seem to recall that.

I've lost the context of this.

Me too, for sure.

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.

Right. Well that's the beauty (or the flaw I suppose) of mathematical abstraction. Mathematicians just think a probability distribution is a particular kind of function on a probability space. There is no meaning or metaphysics.

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.

Why are you telling me this? I don't know what we are talking about.

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.

I'm a new mysterian. I don't think we're going to know. We can't know any more than an ant on a leaf in on a tree in a forest can know about the world as we understand it. But the ant knows warm from cool, what to eat and what eats it. It has a metaphysics!

https://en.wikipedia.org/wiki/New_mysterianism

But I'm not sure why you mentioned this. The point was that the concept of credence lets us apply the mechanics of probability theory, without regard for the metaphysics. Because even though I don't know what's going on, I can have an opinion about it. And we can tally people's opinions to quantify their frequency.

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?

No, not at all. Instead we ask a hundred million people in the street to vote on how we should run our society! I believe it was Socrates who distrusted democracy. "In Plato's Republic, Socrates depicts democracy as nearly the worst form of rule: though superior to tyranny, it is inferior to every other political arrangement." So says Wiki. We can certainly see his point.

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.

Ah. No. Not the point I'm making. I'm saying we can substitute credence for probability, so that we can apply the techniques of probability without being burdened by metaphysics. I didn't say it was more true, only more workable. A pragmatic shift in view.

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.

Yes ok. If a baseball hitter has a batting average of .250, we would say he has a 1/4 chance of getting a hit on his next at bat. But of course this is absurd, the specifics of his next at bat are subject to all kinds of variables, how he's feeling, how the pitcher's feeling, the humidity and temperature of the air, etc.

But I don't follow your point in bringing this up. And betters use credences! The odds are based on the credences of the betters, and NOT on any metaphysics of what is really going to happen. That's a good point. Gambling odds are based on collective credence, along with an attempt to judge "objective" reality. It's a bit of both.

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.

Yes. That's why we aggregate everyone's subjective opinion and evaluation. These are situations whee nobody can know all the evidence. Like a murder mystery with only circumstantial evidence. We can't know for sure but we can use our best judgment and have a credence.

We started off talking about "probability - 1" and in order to understand that, we've explored the construction and meaning of the probability table.

I don't know what you mean by probability table.

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.

I've said nothing about Bayesian probability. I like credence because we can always have one, even when we can't know enough to assign a metaphysical probability.

I know that I can be a bit relentless. If I'm boring or annoying you, please tell me and I'll shut up.

Well I'm concurrently dabbling in the political threads in the Lounge, so this all seems like light recreation by comparison.

But your idea about the nonexistence or vacuity of probability 1, that I don't follow.
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