That makes perfect sense. I understand you now! Yes I was thinking of the empty set. — fishfry

:smile:

A tautology that doesn't assert anything is kind of a dead end in the reality tree, if I may wax poetic. If we're processing it, it's an error. It doesn't have any meaning.
Is that about right? — fishfry

Yes.

But anyway, mathematicians are trained to get used to empty objects. There's the empty set, and the empty topological space, and so forth. You get used to accepting vacuous arguments. So I don't see empty ideas as a problem. An empty idea is still and idea. The empty set is a set. — fishfry

Empty sets, etc., are defined in a context, which assigns a use to them (though perhaps not a meaning!). So that's different. The criticism is directed against ideas or uses that are not in a context that gives a use to them.

But if that -- then I still don't get it! Probability 1 says that something is certain to happen. If I add 1 plus 1, I am certain to get 2. If we have a slow computer, we put in 1 plus 1 today, and we are certain, with probability 1, that the computer will output 2 tomorrow. What's wrong with that? — fishfry

Quite so. I'll overlook the intrusion of time. My point is different.
The use of "probability=1" is defined in the context of the table (function), that is, in context where a range of possible outcomes is given, one of which will turn out to be the outcome. Outside that context, it's use is not defined. Or rather, its use is defined as "= true". That is quite different from "probability (A v B vC..) = 1" meaning "the total of the probabilities of A v B v C... is 1", that is, its use in defining the range of the probabilities of the outcomes. So it serves no purpose, apart from confusing me.

When the knight is captured it doesn't feel good or bad. The player may feel good or bad. I'm back to the Chinese room. Searle says the room doesn't know what any of the Chinese sentences mean. So if you agree meaning is in the mind, that's what I believe also. — fishfry

Meaning is a slippery word. One might want to object that the meaning of the word "table" is an object in the world. But we make the words and we use them.

I say that it is NICE if my credence is based on some evidence. Maybe I put some work into forming my opinion. — fishfry

I would put it stronger, but it is true that credence is not necessarily based on conclusive evidence, and may be not be based on evidence at all.

I hope I'm making my point. We are all obliged to place high credences on many things that we can't possibly have the slightest idea about. The electric grid will be up tomorrow. How the hell do I know? Did I personally inspect every faulty transformer that's about to blow, and take down half the county with it? — fishfry

You're right. Most of what we know, we know at second hand. If we had to prove everything ourselves from scratch, we would be very limited. Standing on the shoulders of giants and even midgets is essential. Philosophers like to brush that aside and only pursue the gold standard. There shouldn't be any problem about assigning a credence to what we are told by others. I would count it as evidence. Why not?

Point being that I have a credence, which I found by simply thinking about it for a moment, about a situation in which I can't possibly know the first thing, and actually I haven't looked into it much. So I know nothing. But I have an opinion! — fishfry

Quite so. We react instantaneously and without conscious thought to most of what's going on around us. We would never keep up if we had to sit down and reason everything out.

But, if I've got any sense, I will give more credence to credences assigned by someone who knows what they're talking about over credences assigned by someone who doesn't. That's reasonable, surely?

I would say at the least, that many of the authoritarian types in our society took advantage of the situation, in a manner not supported by the science. And anyone who pointed that out, was cancelled, had their career ruined, their jobs or professional licenses taken away. — fishfry

Well, the opposition in the UK were certainly not silenced. Their voices were heard throughout. The problem is that without an estimate of what would have happened without lockdowns, we have no way of assessing their success. It's has always been regularly used with Ebola outbreaks, so it must have its uses. But those incidents have been relatively contained. I think the scope and duration of the COVID lockdowns was the problem.

I don't like the idea of giving up national sovereignty to such an undemocratic institution as the EP. "Brussels" has become a pejorative and not just the name of a city. — fishfry

It's not that simple. Every time you sign a treaty, you give up some sovereignty. It's a question of balance - quid pro quo.

That last bit I didn't know anything about, the Northern Irish. — fishfry

It's long and peculiar story. There'll be lots of stuff on the internet if you want to look it up. The problem was that it needed free access to both UK and Republic markets. While both were in the EU, it wasn't a problem. But when the UK left, it was not possible for them to continue free trade with both and yet could not give up either. It was obviously insoluble from the beginning, but nobody bothered until the reality hit.
They seem to be reasonably satisfied with the most recent arrangements, but they are a bit of a lash-up.

I went on a business trip to Cork once, it was so lovely. — fishfry

I'll bet. It's a very beautiful place. The whole island is - outside Belfast.

Is that right? Is there Churchill revisionism about? — fishfry

Well, there's always been a counter-narrative. The left wing have never liked him. There was the Sidney Street siege, Gallipoli, the famine in Assam in 1943, and pet research projects that wasted a lot of money and it took a lot of persuading to get him to accept the invasion of France. No financial scandal that I know of, which makes a nice change. I think most people accept he made a critical difference in WW2.

I think it does challenge foundationalism, which is why Lee Braver named his book on Wittgenstein and Heidegger ‘Groundless Grounds’. — Joshs

As it happens, I'm in the middle of reading this. However, I was hoping to find something that questioned the need for bedrock assumptions. I was also commenting on this metaphor. Wittgenstein seldom relies on just one metaphor. I like the river-bed much better. Nonetheless, if I ask myself what the river-bed is founded on, I find myself confronted with the planet earth. No bedrock there.

That's how I understand them: "hinges" are almost too mechanical for foundations: and in a way hinges can only be placed upon structures build on foundations... hrm. The up-down metaphor — Moliere

I think he presents the hinge metaphor in the context of analysing a debate - elaborating the idea that the debate turns on a fixed point. I would assume that this only applies to the context of the debate, and that what was a hinge may become a bone of contention in another context.

In his own metaphorical terms, I think when Wittgenstein says that his spade is turned when he hits the bedrock of "forms of life," many would simply suggest that he buy himself a shovel or a pick axe. — Count Timothy von Icarus

Well, even the expanded metaphor makes a good point. Foundations need to be in a different category from what is founded. More logic doesn't give you the foundations of logic and so forth. However, I do agree that the "bedrock" metaphor doesn't challenge foundationalism itself, and that's always puzzled me. The radical issue is whether foundations are always necessary. After all, it turned out that there are no foundations of the planet.

If our culture and language impact brain development in early childhood, there is not just an abstract difference between individuals of different cultures, but a physical one. — Lionino

But even that accepts that we are the product of our environment as well as our genes, and so undermines one form of essentialism. However, if I embed myself in second culture, that will affect what got embedded with my first culture and change that. The brain continues to change and develop throughout life.

Wittgensteinians often make claims that are the opposite of "common sense." For example, the claim that a man who washes ashore on desert Island loses his ability to make and follow rules, but then regains this capacity when a second person washes ashore later. Obviously, a great many Wittgensteinians (as well as people generally) find this to be somewhat absurd. — Count Timothy von Icarus

Does Wittgenstein appeal to common sense? He certainly relies on our intuitions, since we are expected to think things our for ourselves, but that's not the same thing, is it?
That claim is indeed absurd. Robinson Crusoe didn't forget everything he learnt when he washed ashore. On the contrary, he continued to embody the culture he came ashore with. No doubt, as he continues alone, he is very likely to adapt and develop what he has learnt and may move away from the starting-point without being aware of the fact. (In the fiction, Defoe ensures that doesn't happen.)

That's one way of framing it in the "Tarzan Versus Crusoe," discussion at least, but there is also the idea that Crusoe cannot make new rules so long as he is alone, and any continued rule following can only be judged by an absent community. — Count Timothy von Icarus

That's not wrong. But it is likely that after enough time, his rescuers will find deviations and adaptations in his way of life.

Unless some "tribe" (a favorite thought example of Wittgenstein) is in possession of the truth itself and the rest itself, we are dealing with opinions and beliefs held at that time and place to be true. The truth is, we are not in possession of the whole of the immutable truth. Throughout history human beings have held things to be true that turn out not to be. This is not something to be solves by attacks on the truth of relativism so understood. — Fooloso4

There's a lot of scary rhetoric about relativism. But it seems to me the greatest danger is precisely believing that one is in possession of the absolute truth and therefore does not need to compromise. That has serious real-world consequences.

all of Wittgenstein's complaints about "philosophers using language wrong," can be waved away by simply claiming that Wittgenstein is not privy to the language game used by these philosophers. Perhaps being a metaphysician, a Thomist, etc. are all discrete "forms of life?" — Count Timothy von Icarus

You're not wrong. I would rather say "discrete language-games" or, in the case of religion and science "discrete practices". But there's always the common ground of human life to appeal to. After all, if we can agree that we disagree, we must have something in common. Mapping that is always a useful first step.
That confidence - that there are rights and wrongs about how to use language - seems to me to be something of a hangover from the Tractatus. But his practice seems to have been different. Anscombe's story about why it looked as if the sun goes round the earth indicates a rather different practice.

Nevertheless when people use Wittgenstein's ideas, they have to interface with other arguments. Perhaps you can do that solely in his terms, but honestly trying to do it equitably makes it very difficult for Wittgenstein. Which is a weakness of his, rather than of philosophy. — fdrake

Quite so. In fact, if you want to engage in debate, you need to meet on common ground, and starting from a Wittgensteinian position is unlikely to do that.

Despite the theories about forms of life, I do not think it is vague unless one treats it as a theory. He has no theory about forms of life, he is simply pointing beyond language as something existing in and of itself to our being in the world and all that entails conceptually and practically. The boundaries between one way of life and another or one practice and another are not fixed and immutable. — Fooloso4

I agree with that. I prefer to think of those notions as ways of approaching problems, needing to be adapted to apply to specific situations, rather than doctrines or protocols.

We can agree to disagree, but I don't understand why you think probability 1 is "empty." — fishfry

This is a bit embarrassing. I was using a bit of philosophical jargon, which seems to be out of date. You must have been wondering how empty sets were relevant. The expression derived from the logical positivists who classified tautologies as empty or trivial because, although they are not false, they do not assert anything. For them, proper, non-empty, statements were those that could be verified or falsified. The idea is used in Peter Unger's book Empty Ideas

Unger’s argument is that thinkers used to put forward arguments whereby “if what they said was true then reality was one way. If it was untrue then it was another way .... They were sticking their necks out.”

See Review of Unger "Empty Ideas (I don't recommend the book. For all that the review talks about philosophy being fun, which I approve of, this book is hard going for rather small rewards.) I don't agree with this application of the argument, but the idea can be useful.

Some examples may help.
1) An obvious case is "This sentence is true".
2) If I assert that snow is white, it is empty for me to assert in addition that I believe that snow is white.
3) Tarski's redundancy theory of truth (which, in case you don't know, is popular among philosophers) says that "snow is white" is true iff snow is white.
4) The probability of p = 1 iff p is true iff p

Law of unintended consequences is a rule of general life too, right? — fishfry

Yes. But I thought that unintended consequences were events in the empirical world.

I don't see why. The importance to some people of the world chess championship is not inherent in the rules of chess. Symbolic systems have no meaning in them. It's the people who supply meaning. — fishfry

We're using "meaning" in slightly different ways. The paradigm case of a symbolic system for me is language, and that has meaning - if it didn't, it wouldn't be a symbolic system. A symbol is created by setting up rules for the use of an arbitrary character or object. So the rules of chess set up rules for the use of the various elements of the game. I'm inclined to say that establishes the meaning of the symbolic characters within the game, and I would agree that that meaning is "in the minds of" the players and spectators.
I also agree with you that the significance of the game (e.g. its interpretation as a war game, suggested by the names of some of the pieces, or the value attached to titles like "grandmaster") is not established by the rules of the game. So there are layers of meaning (or significance), depending on context.

Credence is not fantasy. — fishfry

Yes, I'm agreeing with you. But I want to distinguish between the two by saying that credence should be based on evidence or at least plausibility and that fantasy has neither of those. That's all. How else would one separate them?

The security incompetence is of a degree that invites suspicions of complicity. Just as in the JFK assassination, where the Secret Service was likewise grossly incompetent. Biden has other problems this week. Rumor has it he's dropping out of the race this weekend. But that might just be spin from his enemies (in his own party) leaking to the press to weaken him. — fishfry

Yes, I remember the JFK story. I was once, briefly, an auditor (annual accounts for companies and other institutions). They drummed into me that when something was wrong, cock-up was more likely than conspiracy. But that doesn't prevent suspicions.

Well if he's not free-market he's a collectivist! Generally speaking. — fishfry

The days of dogmatic nationalization of the means of production are long gone. Nowadays, at least in the UK, it's a pragmatic issue and we have a number of half-way houses and regulators for specific areas.
But isn't the free market a collective social institution? One of the basic functions of the state is to supervise and enforce contracts, and the companies and other collectives that operate in the market are themselves collective institutions - and they aren't accountable to voters.

I did hear that he wants "closer cooperation with Brussels," meaning that he'll be yet another British PM stabbing Brexit in the back. I think it might have had a chance to produce good results if the politicians had respected the will of the people. — fishfry

There was a lot of back-stabbing in the aftermath of the referendum. It was not pretty. But I don't think any of the Prime Ministers intended that. Brexiteers told everyone that the EU could be adjusted to suit what they wanted. The EU were reluctant to do so - and why should they? It's not as if public opinion in
the EU thought Brexit was a good idea. Brexiteers labelled any compromise as "stabbing Brexit in the back"; it seems they didn't grasp what negotiation is all about. The only people who were stabbed in the back were the Northern Irish who were thrown under a bus by Boris Johnson.
I voted remain, but had serious doubts about the ultimate EU project ("ever closer union"). Europhiles didn't pay enough attention to the longer-term history of the UK (since, say, 1700).

Miscalculation or malevolence, take your pick. — fishfry

Forgive me, but I can't think of anyone, malevolent or not, who actually benefited from the lockdowns apart from the vulnerable groups - older people, people with health issues. I plump for miscalculation, in spite of the UN warnings, so by British politicians.

Interesting that after the war, the British people showed Churchill their appreciation by voting him and his party out of office at the first opportunity. — fishfry

Yes. The conservatives thought they could go back to the way things were before the war. The voters wanted a fresh start. They got it - even the conservatives had to accept the new ways. It took them 50 years to unpick it and they're still not done.

I believe he (sc. Churchill) said that "History shall be kind to me, for I shall write it." — fishfry

Well, people were kind to him for quite a long time. But that's changing now.

Well I'm cycled out on this I think. At the end of most of the convos I'm in. I could let this go soon. — fishfry

I don't think I have any more to say about probability = 1. So let's agree to disagree. I think I understand at least where and why we disagree. I'm sorry I can't make myself clear to you.

That's a tall order (sc. the relationship between the purely mathematical abstractions in the context of what I'll call the everyday world). You mean differential geometry, the super-abstract geometry of Riemann, applied to general relativity? Or the math of quantum field theory? Or do you mean something far more prosaic? — fishfry

If I were qualified to tackle those areas, I would take them on. But I know better than to talk about them without a reasonably thorough understanding of them - which I don't have. I have to settle for the prosaic. Which matters too, I think.

Since the context of the use of the concept has changed, the meaning of the original concepts may or may not have changed, but may well be seen differently. Does that help?
— Ludwig V
No, I think you obfuscated the point. — fishfry

I can see why you think that. But I'm fascinated by the fact that we can posit some relatively simple rules and draw such startling and unexpected conclusions from them. How is that possible? For you, that's your home, but for me it is foreign - and confusing - territory.

I said there is no meaning in math. That when we manipulate symbols according to rules, there is no meaning that's part of the formal game. But of course "in the back of our minds," we do know what it all means. We have some every day experience in mind, even though that has no bearing on the symbology we write down. — fishfry

To me, that's paradoxical. But, from another perspective, very helpful.

You're the pro, so when I say metaphysics it just means, "What's really true about ultimate reality." Or something like that. — fishfry

That's good enough for this discussion.

The metaphysics is that when we say, "The probability of rain is 25%," we're making a statement about the REAL WORLD. When I say that "My credence it will rain is 25%," I am making a factual, verifiable statement about my subjective state of mind. I don't need to know anything about the real world, though I do base my credence on the available evidence. Clouds in the sky, for example. But in credence, I'm not making a claim about the world. I'm making a claim about my own subjective degree of belief. — fishfry

OK. It's just that a link to the real world (whatever that is) is what makes the difference between something interesting and useful and a fantasy.

Maybe that's all there is.
— Ludwig
Ok. Not disagreeing. — fishfry

For me, the formal representations in decision theory do have the prospect of articulating our decisions more precisely and enabling us to make more coherent and better balanced decisions.

I read Spiked Online (https://www.spiked-online.com/) as my main source of British politics. They're slightly right of center. I gather Starmer is a typical collectivist leftist, but that the so-called "conservatives" mucked up their own charter so badly they deserved to go. Maybe he's a better guy than I've heard. — fishfry

"Slightly right of centre" is about right. "typical collectivist leftist" sounds like slapping a conventional label on something without thinking about it very much. So it's very likely that he is better than you've heard. Most of the British media is right wing, so most of what was written was, essentially, political. (Perhaps the most significant thing about our election is that the normally right wing press abandoned the Conservative party. That's not happened since Blair got elected in 1997.) You have to realize that our right wing political people have no hesitation about government action when it suits them; but they often disguise it so they don't have to take responsibility for the outcome. Starmer's programme is very moderate and addresses areas where almost everybody agrees that existing, supposedly free market, structures have completely failed to deliver.

I'm not saying one thing or another, just that transparency and accountability are in short supply from the government this week. — fishfry

I'm not surprised. It's clear that there was a major screw-up on the security front. So the Government was bound to take some flak. So it went in to self-protection mode. All Governments do that. It doesn't usually work very well. It seems likely to reinforce Trump's lead in the election stakes. Biden must surely wish it had not happened.

Is Charlie someone's idea of a philosopher king? Poor guy, his entire role in life from the time he's a child is wait for his mum to die, then she turns out to have great genes and lives till 96. And a year later the poor guy gets a serious cancer. Feel bad for him. I always like Liz, she was a very great lady. — fishfry

I also feel sorry for Charlie. He's never been comfortable in his role. No, he's nobody's idea of a philosopher-king. He's there to be the unity that ties us all together, despite our disagreements and whatever happens in politics. Simply by existing. A philosopher-king would be completely unsuited to the role. It needs someone who doesn't think. He does, though not very well. That's one big reason why he's not suited to the role. But he will do his best, and I'm sure it will serve. In the US, that role was served by the Constitution. That seems to have become a political and legal football too, which really does not help.

Winston Churchill said that the greatest argument against democracy was a five minute conversation with the average voter. I believe that! — fishfry

He did, and he's right. But the full quote is:-

Many forms of Government have been tried, and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed it has been said that democracy is the worst form of Government except for all those other forms that have been tried from time to time. — House of Commons, 11 November 1947

He also said: -

My idea of it (sc. democracy) is that the plain, humble, common man, just the ordinary man who keeps a wife and family, who goes off to fight for his country when it is in trouble, goes to the poll at the appropriate time, and puts his cross on the ballot paper showing the candidate he wishes to be elected to Parliament—that he is the foundation of democracy.
And it is also essential to this foundation that this man or woman should do this without fear, and without any form of intimidation or victimization. He marks his ballot paper in strict secrecy, and then elected representatives together decide what government, or even in times of stress, what form of government they wish to have in their country. If that is democracy, I salute it. I espouse it. I would work for it. — House of Commons, 8 December 1944

Great man. But his record before WW2 was, let's say, mixed.

Covid lockdowns were scientism, not science. Science as a means of social control, not as a path to enlightenment. — fishfry

Covid wasn't dangerous enough. When people realized that it wasn't the plague or Ebola or HIV, they felt, not unreasonably that the risks and benefits were not sufficient. They were misapplied as a result of a political miscalculation. IMO.
The problem got serious in the two world wars 100 years ago. It was very successful in developing new weapons - arguably, it was a major factor in winning them. And, then, of course, "science" got taken up by institutions that were not capable of grasping what it was all about and misused in the service of other interests.

Ok, list of events and their associated probabilities. — fishfry

I wish I had thought of that. But I do think the layout is significant. But I think that's over.

Please do not hesitate to make several arguments at once. Thank you in advance for your insights! — LFranc

1 Singer doesn't ask himself what "needs" and he doesn't distinguish between relative and absolute poverty. (There may be stuff about this that I'm not aware of, but I very much doubt there's anything definitive.) Singer's "argument" is grossly exaggerated and seems more like a slogan than a proper argument. When you get in to the detail, it's obvious that things are not anything like as clear-cut as his rhetoric. Charity has its place in life, but this isn't it.

2 The argument, as we can see, and as every charity knows, is almost completely ineffective. Everybody (most people, many people) feel guilty for a while and then either works out a defence or just forgets it. This is not what motivates people.

3

So my conclusion for this topic is -- we don't have an answer. Nothing. Rien. Morality is a chore. — L'éléphant

A moral argument that presents morality as a duty and a chore has missed the point of morality - or at least the point of charity. It should not be about lecturing and bullying people. Not only is it counter-productive, but it leaves out love (prioritizing the welfare of some people over others) and compassion, which, if not patronizing, is the only proper motivation for charity. (There's a much better idea than Singer's in Indian philosophy - that the opportunity to give is a privilege and the we should thank the people that we give to rather than expect them to thank us. Whether it is more effective than Singer's is debatable, but still I prefer it to Singer's hectoring.)

However, there are other considerations here.

A Justice. The benefit of society is that we are stronger and better together. If resources are not shared, especially in times of trouble, there is no point to it and in extremis society does break down - so everybody suffers. That's not well defined, and you can always dismiss it as envy, but there are powerful motivators at work here. (Neither communism nor capitalism). (This comes under the heading of requirements for a society to function, which may or may not count as morality. But whatever its name, it matters).

B Enlightened self-interest. It isn't just those are homeless who benefit from help. We all do, because we don't have pictures of misery lining the streets we walk down, we don't have so much thieving and robbery, we don't have hungry mobs rioting and looting and so on. Helping an alcoholic get their life under control does not just benefit the alcoholic, but also the rest of us. It's not grand morality, but it is an effective motivator.

Probability = 1

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

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
I was building on his point and your reply. We have somewhat different opinions. I'm not sure that anything important hangs on it, so perhaps we should leave it at that.

MATHEMATICS

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

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

I'm interested in the relationship between the purely mathematical abstractions in the context of what I'll call the everyday world. I'm not trying to undermine the concept of mathematics in any way.

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? — fishfry

Yes. Not perfect, but better. I understand meaning to be the use of a symbol, in the context of related symbols. So I would say that pure mathematics does have a meaning, defined by the interacting concepts in play. When the interpretations and applications come into play, we have a new context. Since the context of the use of the concept has changed, the meaning of the original concepts may or may not have changed, but may well be seen differently. Does that help?

But I do think that interpretations and applications are not an optional add-ons to an abstract system.
— Ludwig V
They are not optional add-ons. So they are mandatory add-ons? Or not add-ons at all? Didn't understand that. — fishfry

That's a very good question. What I said was not quite right. I refer you to what I said about meaning and use above.

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. — fishfry

Well, I've explained what I mean by meaning. I hope that meets the case. But I'm not at all clear what you mean by metaphysics. I would hope that nothing that I say is metaphysical, but the word is so badly defined that I might have erred unwittingly.

POSTERIOR PROBABILITY

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

I read the Wikipedia article. The context seems to be Bayesian probability, which is a different kettle of fish. It's not, if I understand you right, about the basic mathematical function, but about the inputs to the function, so we're talking about an application, right?

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. — fishfry

The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition (such as a scientific hypothesis, or parameter values), given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating. — Wikipedia

OK. It's a small point, but wouldn't be clearer to say and more consistent with the timelessness of mathematical functions, to say that when new information becomes available, a new probability is established, which is substituted for the old one? I think that's compatible with what Wikipedia says.
Bayesian probability is a scenario, or posits a scenario. There's nothing wrong with that. Traditional probability does the same thing with its reliance on gambling scenarios. You're right that it is not a question of truth or falsity, but of enabling us to apply an existing concept in a new way - and one that is particularly interesting in view of the fact that we do ask about the probability of single cases.
I don't see the metaphysics in standard versions of probability. Can you explain?

BAYES

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. — fishfry

This way of articulating chance or probability depends on a "frequentist" concept of probability. One can then understand what the probability means as a phenomenon over a number of cases. But that makes it difficult to see how it applies to a single case. I guess a way of making it concrete is to see it is a question of the odds on a bet. That'll work for insurance and precautions in general, and in planning to take account of possible eventualities. But that only has application in the context of balancing risk and reward - decision theory. Maybe that's all there is.

MISCELLANEOUS

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

Yes. Public/political life - the "state of the world" - has all the ghastly fascination of watching a shipwreck. I expect you know that there's been a change of government in the UK. Suddenly I found myself unreasonably optimistic. Well, until I heard about the events in Pennsylvania.

... 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. — fishfry

Yes. If you expect the democratic vote to determine policy, you are going to come unstuck. Whether it was Socrates or Plato who rejected democracy is underdetermined and likely always will be. Small correction. The view in the Republic is that democracy will always turn into tyranny, because demagogues will take over and establish themselves. Say no more. The thing is, Plato blocks a proper discussion of the issues by positing someone who gets the answers right. But sometimes there is neither right nor wrong and sometimes actual people get things wrong. So his appeal to the philosopher-kings avoids the real issues. Popper says that the vital thing about democracy is that you can get rid of the ruler when they screw up.
Well, perhaps one can quote the old saying that those who do not understand history are doomed to repeat it.

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. — fishfry

Yes, that bit of the Tractatus is much misunderstood. There are suspicions that he was flat wrong, but that would be heresy. He is, perhaps, a rather specialist taste. Yes, his interpretation of Cantor and Godel is vigorously contested. I have the impression, however, that almost everything about those two is contested. I'm not taking sides yet.

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! — fishfry

H'm. Metaphysics again. Ants know what they need to know. There's a concept of the "lived world" that's quite useful in cases like this. Sure, whether you call it a metaphysics or a lived world, we have one too.
But there's a difference. We contemplate Euclid's geometry and start wondering whether the parallel postulate is really necessary. Next thing you know, whole new worlds have opened up. Or Mercator realizes that conventional maps are all wrong and works out how to project a spherical surface into two dimensions. So something new happens. We can do this in a generation or two, whereas evolution can take a very long time indeed.
We'll never know everything because we'll always find new things to know.
There are too many people around who think that science has the answer to everything or can discover the answer to anything. That view is overblown and we do need a more tempered attitude to it.

I haven't explained what I mean by a probability table. I meant something like this. (Forgive my primitive graphics)
Probability
{E(1) v E(2)} 1
Possible outcome E(1) 0.5
Possible outcome E(2) 0.5
not{E(1) v E(2)} 0

When the outcome is known, all that is required is a foot-note - "The outcome was <E(1) v E(2)>" - nothing more.

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

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? — fishfry

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. — 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.

But then the probability is 1, so it makes sense to say that, right? — fishfry

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. — 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.

In that sense, my view of probability is not overloaded with philosophical interpretations. Whether that's good or bad I'm not sure. — fishfry

.
It depends whether you are a mathematician or a philosopher.

Perhaps the formalisms are irrelevant to your thoughts. — 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.
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. — 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.

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. — 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.
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! — 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.

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

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. — 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.
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. — 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.

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.

Calculus or analysis is the perfect example of us getting the math right without any concrete foundational reasoning just why it is so. — ssu

Oh dear! That's a real can of worms, isn't it? Some philosophers would argue that the engineers have got it right. Perhaps it is best to start with the foundation of philosophy - a question. "What do you mean by a foundation?" But I do know that some mathematicians regard philosophers in much the same light as they regard engineers. Still, it's all great fun and often elegant and beautiful; I don't want t be a grinch.

All of these sets are of finished "actual infinity", not the potential infinity as the Greeks thought. — ssu

Yes. I remember. I don't think I ever replied properly. I can see why those definitions might seem reasonable. But it seems better to me to say that "potential", "actual" and "complete" have no application here. On the other hand, I can see that there are real problems here, so I'm not sure that these labels matter very much. Do they solve any problems?

To my reasoning it doesn't. And both Leibniz and Newton could simply discard them too with similar logic. — ssu

The trouble is that, like plastic, if you discard them, they just come back to haunt you. Perhaps Berkeley had a point. Perhaps the concept of incommensurability could help here?

Writing x^2 means x². A bit lazy to use this way of writing the equation. — ssu

Thanks. One has to do something when one doesn't have the keyboard for the symbolism. Handwriting is much more flexible.

Yet using the diagonalization method we get also many other very interesting theorems and proofs and also paradoxes, which in my opinion are no accident. — ssu

I didn't realize that argument was so powerful.

“They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

He was a great wit. I'm still trying to make up my mind whether he was a great philosopher or a complete charlatan - even possibly both. This comment is typical. It is very sharp, very pointed. But the calculus is embedded in our science and technology.

I just wanted to describe the seemingly paradoxical nature of the infinitesimals. — ssu

Yes, I see. You can remove an infinitesimal amount from a finite amount, and it doesn't make any difference - or does it?

Set theory gives us the actual infinity — ssu

What do you mean by "actual infinity"?

I'm afraid I don't know what "^" means. But the paradox in the concept of the infinitesimal - that it both is and is not equal to zero - Is not difficult to grasp - and I realize that that's what the concept of limits is about. (But the idea of a limited infinity is, let's say, a bit counter-intuitive.)

Zeno's least eating dog has to eat something, but then if let's say eats from Platons dog 1, then the food hasn't decreased! — ssu

I don't get this. There's enough food for all the dogs, so why does it have to take some from Plato's dog? If it does, then of course the amount of food for Plato's dog has decreased, but the food supply is infinite, so the amount of food available overall hasn't decreased. What's the problem?

Well, in my view mathematics is elegant and beautiful. And it should be logical and at least consistent. If you have paradoxes, then likely your starting premises or axioms are wrong. Now a perfect candidate just what is the mistake we do is that we start from counting numbers and assume that everything in the logical system derives from this.
And if someone says that everything has been done, that everything in ZFC works and it is perfect, I think we might have something more to know about the foundations of mathematics than we know today. — ssu

Right from the beginning, 2,500 years ago, people have been thinking that everything has been done and is perfect. But then they found the irrationality of sqrt(2) and pi. A paradox is not necessarily just a problem. Perhaps It's an opportunity. Oh dear, what a cliche!

This simply goes back to in the story of Plato's rejection of Zeno's most eating dog, just in a different form. — ssu

Believe it or not, I can see that.

This is why idea of infinitesimals is rejected in standard analysis. — ssu

I'm a bit confused about infinitesimals. Are they infinitely small? Does that mean that each one is equal to 0 i.e. is dimensionless? Is that why they can't be used in calculations? (I thought that Newton used them in calculus and Leibniz took exception.)

In fact you yourself brought up an old thread of four years ago, which is topic sometimes even banned in the net as it can permeate a nonsensical discussion. — ssu

Well, actually, someone else mentioned it. I misunderstood what it is about and off we go. Once I realized it was about the sum of an infinite sequence, I withdrew, with some embarrassment. But I've learnt some interesting snippets.

That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers.
— Ludwig V
Unfortunately... yes. — ssu

There is another way, mentioned in the video. Just relax and live with your paradox. It's like a swamp. You don't have to drain it. You can map it and avoid it. Perhaps I just lack the basic understanding of logic.

That the total probability of the entire event space is 1. — fishfry

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. — fishfry

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. — 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.

The probability that something, anything at all will happen, is 1. — 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. That's one of the rules of probability in the Wiki article. — fishfry

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

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

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. — 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".
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. — fishfry

Yes, and I once I realized that, I withdrew. Perhaps I wasn't clear enough.

And here's then the problem: not only Plato started from counting, but even today Set Theory starts from counting too with the Peano Arithmetic. — https://en.wikipedia.org/wiki/Class_(set_theory)ssu

I see your point.
I don't quite get that "fork" argument. The notation using lower case beta for a member of the set and upper case beta for the set is confusing, and I think there's a typo in the statement of the paradox. But I know better than to challenge an accepted mathematical result.
Wikipedia defines proper classes as "entities that are not members of another entity."
That's always a good solution to a difficulty - slap a name on it and keep moving forward. Sometimes mathematicians remind me of lawyers. That's what happened with sqrt2 etc. Also when defining the limits of infinite sequences.

Can you know or compute C, if you know both A and B? No, if A and B are as above, then only thing you know is that C can be a natural number 6 or 7 or 8 or larger. It might be six, but then it might be three googol also. — ssu

Yes. I always thought that was the point. Why should everything have a definite, computable result? Stating the range of a result is not pointless.

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. — fdrake

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. — fishfry

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. — fishfry

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.

This is why I argue that with infinite you cannot start counting. This also shows why 1+ ∞ = ∞ and ∞ + ∞ = ∞. — ssu

That's exactly what I have been trying to say all along! :smile:

I am the one who apologises for derailing the topic in an inconsistent scenario — javi2541997

I don't think you have derailed anything. If there's any derailing going on, it's me that's doing it.

Just think of a finite line you draw and put at the start zero and in the end — ssu

You can do that, but it's very misleading. It suggests that an infinite line is just a very long line. That's wrong. The best way I can think of is to draw your line and put your ∞ or ω at the end of it, but remember that those symbols mean that the line goes on forever - it has no end. That's why we always just write down the first few elements of the sequence and then ... or "and so on". That's not just an abbreviation or laziness or lack of time. It's telling you that the sequence has no end.

The whole story is about the problem of definition that math has. And for the Grand Order you refer to, there is the Well Ordering Theorem. — ssu

I don't know about all those theorems. I know I should, but I had a deprived education.
But what strikes me about your Grand Oder is that the only fixed point you have is Plato's dog. It is the only possible origin for the ordering of the dogs that eat more than Plato's dog, in which case we have to call it dog 0. But it is also the only possible origin for the ordering of the dogs that eat less than Plato's dog. We can call it Dog 0 or Dog 1, but either way, it won't look much like a single order from the dogs that eat less to the dogs that eat more.
The short version of this is that you have to start both sequences from a point in the middle of the line.

Thanks.

I think Athena never thought about it either. But since this mysterious dog showed up in this game yesterday, I started to think about his interference in the counting. Well, if we imagine there is actually a dog who doesn’t eat anything, it means that it should be represented with a zero (0) in the counting. As ssu pointed out, it took a while for Western mathematics to accept zero as a number. According to this issue, maybe Plato would never have taken the dog who doesn’t eat anything into account, but yet it is clear we should take the dog into account, and thus, the dog exists. Right? — javi2541997

That's a complicated thought process. This is a story. It was made up. Speculations about what Athena thought or didn't think beyond what we are told in the text can be plausible or implausible but there's no criterion for truth or falsity. The same applies to ideas about what Plato would or would not have done. For what it's worth, I don't think the real Plato would have done any of what the story attributes to him. But it doesn't matter. But there's no truth or falsity beyond what is stated in the the text - and what follows logically from that.
The mathematical "problem" is based on truth and the only question is what is consistent or not consistent with that structure.
A non-existing dog doesn't exist. The clue is in the description. That's all that needs to be said - unless you want to visit Meinong's jungle.
I'm sorry to be a bit abrupt, but if you don't keep your feet on the ground, you're bound to lose contact with reality.

It shows how we get the fraction representation of repeating decimals. — Lionino

OK. I wondered if it worked a bit more widely than that. I don't think that it would work for sqrt2, since Aristotle could prove that it was "incommensurable" without involving decimals. What about π? I was taught that it was 22/7 or 3.14....?

For me, this issue has a wider context.

This may be a step too far. But there are many people who turn up on this forum - and elsewhere - who deeply believe that nothing is true and everything is probable.

The usual basis for this is traditional (since Descartes) scepticism, and one usually tries to meet it by arguing about that.

But what if they have been introduced to probability theory and infinity? Suddenly, there is a mathematical proof.
Sometimes probability = 1 and 1 = 0.9999... So everything is probability,

I think this is a mistake, because it neglects context. But it is new angle on the mistake.

I'm basing this on an assumption that both theses are correct - in their context.

There is none. Why do you think there is one or should be one? That's why I think you're misunderstanding. There's no element of the a sequence immediately preceding the limit point. — fishfry

I think it follows that "0.999...." does not equal 1.

Sadly, my best time for philosophy is first thing in the morning...

There is an answer. The answer is that there is no number greater than all the terms of the sequence, and less than 1. — fishfry

Yes. I assume you mean all the terms of the infinite sequence?

I think I'm a little bit puzzled that you have this confusion after I've explained it in the other thread. — fishfry

And I'm puzzled why you think I'm disagreeing with you.

The limit is equal to 1, in exactly the same sense that 1 + 1 equals 2. — fishfry

So it is. But what is the element of the sequence immediately preceding 1?

I'm becoming increasingly astonished that this thread continues. — Banno

It's probably just that everyone who joins needs to be taken through it. Each person has to learn everything for themselves.

Now it is also true that 4/0=∞ and 9.7181=∞. And with a little more leg work I shall demonstrate that all numbers are actually equal to each other. Multiplicity is mere illusion, a result of the Fall and Adam's sin. — Count Timothy von Icarus

I look forward to mankind's return to the Garden of Eden.

Thank you very much for those.

If I've understood, your argument shows what the sum of the infinite series is.

Right?

Oh I see what happened. Ludwig brought up the old .999... = 1 chestnut in the staircase thread, and it apparently got moved over here to revivify this four year old thread. — fishfry

It probably saves time and energy. Actually, you mentioned it and I got curious. I'm afraid I innocently asked a question and set off a land-mine.

What number can possibly get between ALL the terms of that sequence, and the number 1? — fishfry

Well, if I've understood how this works, there is a number that gets between each element of the sequence - the next element in the sequence - and is there is no last element of the sequence. So there is no answer to your question.

However, it is also true that 1 is the sum of the infinite series 0.999... - and therefore the limit.
But an infinite series never reaches its limit. To put it another way, "=" in this context (an infinite series) does not mean what it usually means.

Ah, the so-called non-existing dog is the one who doesn’t anything at all. I get it now. But I assumed every dog ate at least a bit. — javi2541997

Is there a non-existing dog? If there is, it doesn't exist. If there isn't, it doesn't exist.

Yes, this is how I see the tricky game. If I'm not mistaken, the dog who eats less than the preceding dog would be represented by 0.00000000…, and so on. However, this dog does exist. It consumes something, even when it is infimum. — javi2541997

Exactly.

Sorry, I was foolish in trying to follow usual norms when infinity is involved. :sweat: — javi2541997

I've been bitten by that infinity more times than I can count. All common sense has to go out the window. It is possible to get used to it.

a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1 — Lionino

I do appreciate your help.

But this doesn't seem to work with other similar sequences, such as 0.333... or 0.444... or 0.1212....
What have I got wrong?

By accepting transcendental dogs and their transcendental food, I argue that you have already accepted (perhaps unintentionally) the existence of Zeno's least eating dog. — ssu

The transcendental food was a joke, playing on the absurdity of transcendental dogs. I must be more careful about jokes.

What is the criterion for Zeno's least eating dog?
Is there an infinite number of dogs?
What is the difference between transcendental dogs and ordinary dogs?

The one at the top (the dog who eats the most) and the one at the bottom (the dog who eats the least). — javi2541997

That is only possible if there is a finite number of dogs.
There cannot be a dog that eats the most - there's bound to be another one that eats more. Similarly for the dog that eats the least. Infinity doesn't follow the normal rules.

Honestly, I think those two are always ‘there’ but it is a mistake to try to identify them with numbers. — javi2541997

Well, strictly speaking they are identified by the amount of food they eat, which determines their position in the line.
The numbers identify their position in the line.
So, since they are identical in every way, apart from the amount of food they eat, there is no other way to identify them.
It is easy to think that they must exist, but if the line is infinite, any specified dog has another dog after it.

As I stated to Ludwig V, just having finite, but transcendental numbers like π or e that aren't Constructible numbers already gives the problem of Zeno's dogs, even if we would dismiss the two Zeno's dogs mentioned. — ssu

I don't know the math well enough to be sure, but I think it is possible to place numbers like π or sqrt2 in order among the natural numbers. So every dog will have a different place in the order, depending on how much they eat. So dogs numbered π etc. will be like every other dog in having a number assigned according to how much they eat. Each dog will be different from every other dog and each dog will be the same as every other dog. It depends how you look at it.

I though it might help to quote the rules again:-

1. The dogs are totally similar in every way except that every dog eats a different quantity of food. All the dogs eat the same food, which is divisible and there is enough of it for every dog. — ssu

Once these were put into the line, then came the dogs which ate quantities between these dogs. — ssu

As all dogs do eat something, we have a problem with the non-existent dog that doesn't eat anything, — ssu

I don't recall mentioning any non-existent dogs, nor any that don't eat anything.

Well, a dog eating ⅚ of Plato's dog's food amount isn't either a natural number, so would you deny it to be a dog?
I would not deny it to be a dog and I would be happy to assign a natural number to it depending on where it comes in the ordering.
— ssu

So, let’s say, there is a dog who eats 15 pieces of meat, and there is another dog who eats only 0.0001 pieces of that meat. — javi2541997

I didn't realize, though I should have done, that you are placing the dogs in a single continuous order. But you have defined two infinite sequences, with a common origin. So the start of your Grand Order is not defined, any more than the finish. Your ordering means you have to start from a dog that you cannot identify.

And what about transcendental dogs? They are finite, but the dog that eats π amount compared to Plato's dog? — ssu

You didn't mention them. In any case, they would naturally eat transcendental food - not being able to digest natural food. As for the dog that eats π amount of food, it will have its place in the order, so there's no problem.

Then again, one could reject that the equation for the sum applies. The equation of the infinite sum relies on the notion of limit, and it is the notion of limit that is at play on the 0.999... debate. — Lionino

I can see that point. I didn't look at the issue in the light of infinite series or take on board that it was a question of the sum of an infinite series. I apologize for the distraction.

There is
a=0.999...,
10a=9.999...,
10a-a=9,
9a=9,
a=1 therefore 0.999...=1 — Lionino

That's very neat.

Everybody agrees that mathematics applies to the physical world, but nominalists will broadly say that 2+2=4 is not about the world, so it is not true of it. — Lionino

Here's how I look at it. I think that everyone will agree that a formula is not about anything specific and, in itself is neither true nor false. x + y = z doesn't make any assertions, until you substitute values for the variables. So 2 +1 = 4 is false, but 2 + 3 = 5 is true. So there's a temptation to think it must be true of something. Hence realism. But 2 + 3 = 5 is itself like a formula in that once we specify what is being counted, it does make an assertion about the world - 2 apples + 2 apples = 4 apples. It is true of the world. Of course, 2 drops of water plus 2 drops of water doesn't make 4 drops of water, (until we learn to measure the volume of water). The domain of applicability and truth is limited.

For example if you randomly pick a real number in the unit interval, it will be irrational with probability 1, even though there are infinitely many rationals. — fishfry

If I said anything about that, I would be way out of my depth. So I'm afraid I shall have to ignore it - until another time, maybe.

1 is a perfectly sensible probability. — fishfry

.. in the context of probability theory, that may be so. But I'm interested in probability in the context of truth and falsity, which is a different context. So when you say that 1 is a perfectly sensible probability, are you saying that probability = 1 means that the relevant statement is true? (I don't want to disappear down the rabbit hole, so I just want to know what you think; I have no intention of arguing about it.

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze

Which one do you think should be rejected?

3×13=1 and 3×0.333...=0.999... — Michael

That makes 0.999999..... = 1 just an illusion created by the notation you have decided to use. It is not a proof. In my opinion. You might have a different idea of what a proof is.

On the other hand, it does show that looking at a problem another way might show that the problem is an illusion. But that would be philosophy.

I'll refer you to this: — TonesInDeepFreeze

I'm deeply flattered. But that is far too much for me to grasp in less than a month or two.

I saw an argument in a video that is much simpler, but I didn't get around to fully checking out whether it's rigorous. — TonesInDeepFreeze

Perhaps it would serve our purposes. I could probably get the point even if it isn't completely rigorous.

But let me explain why I need convincing.
In my book 0.9 + 0.1 = 1 and 1 - 0.1 = 0.9 and so 0.9 does not equal 1. There's a similar argument for 0.99 and 1 and so on. So for each element of 0.99999....., I have an argument that it does not equal 1. However, I see that your proof involves limits and I know that in that context words change their meanings. So I'm curious.

The argument shows that the premises entail a contradiction, so at least one of the premises must be rejected. — TonesInDeepFreeze

Well, it seems clear that at any specific time, it will be on or off depending on whether the button has been pushed an even number of times or an odd number of times since 11:00.

So at each of the times specified in the sequence, it will be on or off depending whether the number of times it has been pushed since 11:00 is odd or even.

rP6: At 11:00 the button is pushed to turn the lamp On, at 11:30 Off, at 11:45 On, and alternating in that way ad infinitum.* — TonesInDeepFreeze

The contradiction is created here - specifically in the last two words, which make it impossible to know whether it has been pushed an even or odd number of times since 11:00.

This issue was actually resolved a long time ago by Aristotle, — Metaphysician Undercover

I'm not deeply versed in Aristotle, but my impression is that he did indeed resolve the issue, as it was understood in his time (and what more than that could he possibly resolve?). In doing so, he invented or discovered or recognized the concept of categories, which was a titanic moment in philosophy. It's a pity that there seem to be so many people around who are completely unaware of it.

The unintelligibility is due to a thing's matter or potential. — Metaphysician Undercover

I think it would be more accurate to say "The apparent unintelligibility is due to a thing's matter or potential."

So in the example, when the lamp is neither on nor off, rather than think that there must be a third state which violates the excluded middle law, we can say that it is neither on nor off, being understood as potential. — Metaphysician Undercover

I don't think that's quite right. It is true that if the lamp is on, it has the potential to be off, and if the lamp is off, it has the potential to be on. But that's not the same as having the potential to be neither off nor on. A lamp, by definition, is something that is on or off, but not neither and not both. There are things that are neither off nor on, but they are not lamps and the point about them is that "off" and "on" are not defined for them. Tables, Trees, Rainbows etc.

As what may or may not be, "potential" violates the law of excluded middle. — Metaphysician Undercover

I don't think that's quite right. The LEM does not apply, or cannot be applied in the same way to possibilities and probabilities. "may" does not usually exclude "may not". On the contrary, it is essential to the meaning that both are (normally) possible - but not both at the same time.

Relax! I was talking about the traditional Aristotelian approach to infinity which was orthodox before Descartes but not since, so far as I know. Though I have since seen someone apparently still using the terms in Two Philosophers on a beach with Viking Dogs

Who says anything about probability when merely mentioning that .9... = 1. — TonesInDeepFreeze

Yes, I didn't think of the possible application of that idea to this discussion. I've only ever encountered it in the context of probability.

we prove that .9... = 1. — TonesInDeepFreeze

That's interesting. Can you refer me to a source?

No, it's not a matter of knowledge. Rather, at 12:00 the lamp is neither Off nor On, which contradicts that at all times the lamp is either Off or On. — TonesInDeepFreeze

I'm sorry. It's probably not worth pursuing, but I was struck by the point that "at all times the lamp is either Off or On" appears to be true while "the lamp is neither Off nor On" appears to be false, by reason of a failed referent. It's true by definition that a lamp is either off or on, so if some object is capable of being neither off nor on is not a lamp. The story is incoherent from the start. We cannot even imagine it.

And actual infinity is the completed infinity. — ssu

Forgive my stupidity, but I don't understand what a completed infinity is.

Notice in the story Athena, the goddess of wisdom, might very well know the answer as she did use the two philosophers for amusement for the other gods. — ssu

Well, it's your story. You are the only person who can provide an answer.

Even in the story Zeno is well aware of this. — ssu

But back to the story: Then doesn't that ω in the story relate to distinct dog? You even referred yourself of ω being a number. — ssu

A transfinite number isn't a natural number, so it doesn't get attached to (aligned with) a dog. Nor could it be.

First of all, notice that ω here refers to the largest Ordinal number. — ssu

I was careful to notice that - and. at least by implication, the cardinal numbers.

you put all the dogs that food amount is exactly divisible by dog 1's food (let's call them natural dogs) in a line from smaller to bigger — ssu

That will take you, and even the gods, an infinite time. But I guess Plato, Zeno and certainly the gods, have that amount of time available, and are bored.

start counting the dog line from their places on the line, from the first, second, third, fourth... and then get to infinity in the form of ω. — ssu

You can start, but you can't finish in less than infinite time. And even Plato, Zeno and the gods will be bored by the time they get to the end of a second infinite count.

Well, you already referred to completed infinity or actual infinity with the example of ω as that is Cantorian set theory. — ssu

If you choose to call ω completed or actual, that's your choice. I can't work out what you mean. I don't know enough to comment on Cantorian set theory.