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. — Ludwig V

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

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. — Ludwig V

Well, the prosaic applications of math to everyday life are not really what mathematicians do.

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. — Ludwig V

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

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

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.

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. — Ludwig V

Credence is not fantasy.

"Slightly right of centre" is about right. "typical collectivist leftist" sounds like slapping a conventional label on something without thinking about it very much. — Ludwig V

That was my three-word summary of everything I know about Starmer. I agree I haven't thought about him 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. — Ludwig V

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

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.

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. — Ludwig V

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.

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. — Ludwig V

LOL. My impression too.

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. — Ludwig V

Don't think I've heard that before, that the Constitution is the US analog of a hereditary monarch. Maybe there are some parallels.

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. — Ludwig V

I believe he said that "History shall be kind to me, for I shall write it."

I believe he said that in the context of WWII. Most of Chamberlain's bad reputation is due to Churchill. Chamberlain was actually a pretty good guy, and his appeasement of Hitler was both rational and very popular at the time. So I've read from some alternative views of history.

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.

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. — Ludwig V

Miscalculation or malevolence, take your pick.

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. — Ludwig V

Yes.

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

Ok.

BUDDY IS A GONER — Lionino

They couldn't kill Trump so they're going to kill Biden. Bring in Andrew Cuomo and give him the old ventilator treatment.

Today's New York Times frontpage is basically BIDEN MUST GO, with terrible polling and yet more senior democrats calling for his retrenchment. His situation is plainly untenable, you can't go into the National Convention with the Party split and the press all over you, up against the Trump juggernaut, which really is a civilization-ending threat. — Wayfarer

Remind me why your party (technically still mine) chose not to have competitive primaries, where they could have solved their Biden problem in an open and democratic manner?

Isn't it ironic that the Dems are the victims of their own failed machinations?

And when they swap in Kam in a back room deal to avoid an open convention, will the rank and file just fall into line like they always do? Probably.

I have to admit I've been on record saying they won't be able to move Joe out. But he's looking pretty shaky right now. Lot of party heavyweights are against him.

I never spent any time thinking about what I was doing. I did it, and still do it because it is a fascinating realm of exploration. As was rock climbing when I was a lot younger. I never puzzled over the fundamental nature of mathematics. And I doubt my colleagues did either. — jgill

Interesting.

No. Gravity simply is. Some aspects could be said to be true. Word babble IMO. — jgill

You don't believe in the word truth, or that anything in the world is true, even outside of math?

I'm more of a free trader.
— fishfry

Well the Trump cult does love their delusions. — Mikie

That's a curious statement.

Trump is not a free trader. You asked what Trump did that I disagreed with, and I mentioned tariffs. He's big on tariffs. So he's the opposite of a free trader.

So when I said I'm a free trader (or more of one; it's a matter of degree, of course) you could have said, "You're delusional to be a free trader." We could have that argument.

But to say that the "Trump cult" is delusional, in the context of free trade, makes no sense. I PART with Trump on the issue of free trade.

Was that too subtle for you? You ask what Trump did that I disagreed with; then I tell you free trade; and you say I'm in the Trump cult because I DISAGREE with him on free trade.

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

You keep saying that. You have not yet articulated it in a way that makes me believe you are saying anything sensible.

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. — Ludwig V

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.

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. — Ludwig V

That's a tall order. 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?

The math of the everyday world is to be found in the grocery check-out lane and the baseball scores.

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? — Ludwig V

No, I think you obfuscated the point.

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.

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. — Ludwig V

Ok.

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. — Ludwig V

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

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? — Ludwig V

Well there's abstract and applied probability. I knew a grad student who got a Ph.D. in abstract probability and got a job as an actuary at an insurance company. The insurance companies know more about probability than anyone, it's their business. And bookies. Sports book operators know the theory and the practice.

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? — Ludwig V

I'm talking about credence, not Bayesian probability.

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.

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. — Ludwig V

Ok. Not disagreeing.

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. — Ludwig V

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.

As more news continues to come out, the Pennsylvania deal looks like an op. An operation. It is not as we are being told, and we will likely never be told. If it was some weirdo 20 year old kid with access to his father's gun, so be it. But the numerous incomprehensible malfeasances of the Secret Service raise many questions; and the Biden administration is actually stonewalling and slow-walking the case, raising even more suspicions. I'm not saying one thing or another, just that transparency and accountability are in short supply from the government this week.

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. — Ludwig V

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.

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

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. — Ludwig V

My sense is that he just didn't get it. That he was wrong, not just having a side. But I could be wrong too.

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. — Ludwig V

Yes agree. Science versus scientism. Science is using experiment and rationality to understand the world. Scientism is the belief that science is infallible, or that "trust the science" was ever anything other than an authoritarian political slogan. Covid lockdowns were scientism, not science. Science as a means of social control, not as a path to enlightenment.

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 — Ludwig V

Ok, list of events and their associated probabilities.

I don't see what this all has to do with your claim that a concept like a number, 5, could have a physical instantiation . Fingers are fingers, and are therefore physical instantiations of fingers, not of numbers, not matter how many of them you have. — Metaphysician Undercover

5 is an attribute of the fingers on your hand, would you grant me at least that?

I think of fingers as a physical instantiation of the concept of 5. But if you disagree, then we must be using the word differently. I'm ok with that. How about representation, in the same sense that the first cave man to kill five mastodons and make five marks in the ground to keep track.

Wittgenstein took up this issue in the Philosophical Investigations, showing why there is a lot more involved with learning a language than simple ostensive definition. Abstraction is very complex, and with complex concepts like number, an explanation of what it is about the thing which is being shown, which is being referred to with the word, is a requirement. — Metaphysician Undercover

As far as I know, Wittgy utterly failed to understand Cantor's diagonal argument; therefore his mathematical judgment is deficient in my opinion.

Perhaps abstraction is difficult to define or pin down with words. But we all know it when we see it, and with practice we become good at using it.

A person cannot simply look at the fingers on a hand and apprehend the concept 5. — Metaphysician Undercover

This is manifestly false. Not a matter of opinion or interpretation or language. Flat out false. On the contrary, it is exactly through the experience of looking at one's hand that one at first does apprehend the number 5; and only later, by analogy and induction, all the other natural numbers. Most of the others are far too big to have any such convenient physical representation. Our introduction to the natural numbers, our first concept of them, is by counting the things around us when we are babies. I"m no expert on child development, but there must come a time that a small human looks at his or her hand, and says, "Five." I just did it myself as an experiment. I looked at my hand and saw five. I believe you when you say you don't. You just lack the abstraction and math genes.

An explanation about quantity, or counting is required. The concept 5 is learned from the explanation, not from the ostensive hand, therefore the hand is not a physical instantiation of the number. — Metaphysician Undercover

Oh no. 5 is learned by bijection with the fingers, not with counting. Counting is a higher function. Bijection is more primitive or intuitive. If you've seen a mother cat missing a kitten from her litter, she is not going "One, two, three ..." She's comprehending the total number instinctively and knowing when she's one short.

It can be judged by anyone. The issue though, is that many, like yourself refuse to make such a judgement. — Metaphysician Undercover

There is a modern trend of misspelling judgment, and I can't let it go by. No middle 'e' in judgment.

You say that there is no truth or falsity to mathematical axioms, they are simply tools which cannot be judged for truth. Since mathematicians tend to think this way, they are not well suited for judging truth or falsity of their axioms. But I've shown how axioms can be judged for truth. If an axiom defines a word or symbol in a way which is inconsistent with the way that the symbol is used, then it is a false axiom. — Metaphysician Undercover

There's none in the theory. When you're thinking ABOUT the theory, you can have truth if you like. There is no "truth" in the axioms of group theory, but they are true about the symmetries of a triangle.

So for example, if a mathematical axiom defines "=" as meaning "the same as", — Metaphysician Undercover

There is no such axioms. You make stuff up then tilt your lance at strawmen.

yet in applied mathematics the mathematicians use "=" to mean "has the same value as", then the axiom makes a false definition. — Metaphysician Undercover

You are the only one making up these definitions so that you can disagree with them. Your characterizations of what mathematicians do is in your imagination.

This axiom will be misleading to any "pure mathematician" who uses it to produce a further conceptual structure with that axiom at the base, just like if anyone else working in speculative theories in other fields of science starts from a false premise. False propositions are fascinating, sometimes leading to theories which are extremely useful, because they are designed for the purpose at hand. — Metaphysician Undercover

You're off on your own thing here.

Sorry, I don't understand what you mean by "meta-false". I am talking about "literally false". — Metaphysician Undercover

You said that the axioms are false.

But you misunderstand what that means. Take the powerset axiom. Suppose as you claim, it's false. Then we would be studying the class of set theories that lack powersets. It would be interesting math. It's actually done. The powerset axiom is one that is often assumed to be false, so that we can work out the set theory that doesn't depend on it.

But you think that by an axiom being false, it's committing some kind of metaphysical no-no or faux pas. That's what I mean by meta-false. You think that axioms aren't properly defined or conceptualized or something. NOT that they are literally propositions that are to be taken as false. That's an entirely other thing.

False to me, means not corresponding with reality. — Metaphysician Undercover

If 2 + 2 is 5, then I am the Pope.

That is a true statement that does not correspond with reality.

Ahab is captain of the Pequod. That is a true statement that does not correspond with reality, since both Ahab and the Pequod are fictional entities.

So you are wrong. But isn't the knight move a truth of chess that does not correspond to reality? Axioms in math are like that. Statements assumed true in a fictional context so as to work out the consequences.

For example, if someone says that in the use of mathematics, "=" indicates "the same as", but in reality, when mathematicians use equations, "=" means "has the same value as", then the person who said that "=" indicates "the same as" has spoken a falsity. Do you agree that this would be an instance of "literally false"? — Metaphysician Undercover

I can't sort it out. That paragraph broke my parser. An instance of literally false? I have no idea how to approach this question; nor would I be inclined to do so even if I did.

That doesn't help. Numbers form discrete units, and discrete units cannot model an idealized continuum. — Metaphysician Undercover

Not sure how the discrete/continuous thing got into this convo. Do you mean the set of real numbers doesn't contain all the real numbers? What's a discrete unit? Starting from the natural numbers we can logically construct the real numbers, even there you're wrong.

There is an inconsistency between these two, demonstrated by those philosophers who argue that no matter how many non-dimensional points you put together, you'll never get a line. The real numbers mark non-dimensional points, the continuum is a line. The two are incompatible. — Metaphysician Undercover

If you didn't get the number line in high school, there is not much I can say. The modern mathematical theory of the real numbers is logically unimpeachable. We can indeed start from the empty set and the axioms of ZF, and construct a model of a continuum; that is, an infinite totally-ordered Archimedean set. And we can show that all models of such a thing are isomorphic. So we can indeed construct the real numbers out of discrete units.

It's call the arithmetization of analysis. It's a thing in late 19th century math. Basically founding math, including calculus and continuous processes, on set theory.

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

That doesn't make sense automatically because formalism is a program for foundations, platonism is an ontological claim. And idk what post of MU it is. — Lionino

I was only relating what @Metaphysician Undercover said. Of course it doesn't make sense :-)

I'm not really too expert on formalism versus Platonism, I have a very lay-person understanding of those philosophical terms.

Nor do I lean back and say, Wow, that's true! I simply don't use the words "true" or "truth" when doing math. I don't even think the words. But that's me, not other math people. — jgill

Did you think your work was "about" anything? Or pure symbol-pushing?

I'm pressing you on this point because I don't believe you did not believe in the things you were studying!

I don't think of myself doing anything. I only do. Or did. I'm pretty old and not in such great shape to do much of anything. — jgill

I'm sorry to hear that.

Doesn't surprise me. I am (was) a humble classical analysis drone, far from more modern and more abstract topics. Maybe young math profs these days use the word "truth" frequently. — jgill

I believe you are making too much of what someone on the forum might have said about truth. You are the only professional mathematician on this site and you are more authoritative on what mathematicians do than anyone.

(On the other hand I did point out what I considered the truth of a form of rock climbing many years ago by demonstrating and encouraging a more athletic, gymnastic perception of the sport. Even then I didn't use the word "truth".) — jgill

You know, I would think there is much truth to rock climbing. A famous theoretical physicist, Lisa Randall, famously had a rock climbing accident. She is a specialist in the most advanced theories of gravity. I always thought that was ironic, a world-class expert on gravitation being injured by that very force.

Gravity is true, wouldn't you say? You probably have a visceral sense of gravity, more so than most physicists.

I believe that the soul is non-algorithmic. — Tarskian

Ok! You're an anti-computationalist like me. I don't believe we're ever going to "upload our minds," I don't think we live in a computer simulation, I don't think our minds or our universe are Turing machines.

Concerning "human consciousness", I don't know how much of it is just mechanical. The term is too vague for that purpose. A good part of the brain can only be deemed to be a machine, i.e. a biotechnological device, albeit a complex one, of which we do not understand the technology, if only, because we did not design it by ourselves. — Tarskian

What we know of the human brain does not work like a digital computer. Some people say that neural nets work because they mimic the neural structure of brain. I don't believe that, but I have to admit that some of their recent achievements are impressive. Who knows.

But then again, even if the brain were entirely mechanical, its theory is undoubtedly incomplete, which ensures that most of its truth is unpredictable. — Tarskian

Something deterministic can be unpredictable, so that doesn't solve the problem.

Even things without a soul can have an incomplete theory and therefore be fundamentally unpredictable. — Tarskian

You're confusing determinism with predictability, but I thought we'd already covered this.

Wow. — Mikie

I did invite the person who asked to remind me of incidents or policies I'd forgotten. It's possible. I give you the same invitation. I didn't like Trump's tariffs, that's one that just came to mind. I'm more of a free trader. I didn't approve of the massive tax cut combined with a massive spending bill, that primed the pump for Biden's inflation to follow. Trump's no fiscal conservative.

The whole country was upset by the zero-tolerance immigration policy (the action behind the words), which is why the policy ended just weeks after people found out. Discussing this with you is pointless. You're beclowning yourself now. — RogueAI

The country got upset because the MSM hysterically broadcast Trump's border crisis. I personally watched the same MSM be silent during Obama's identical border crisis in 2014.

What you think you are measuring is not national outrage, but media coverage. Protecting Obama in 2014 and attacking Trump in 2018. Pretty much the same humanitarian crisis.

You're beclowning yourself now. — RogueAI

You're quite ignorant about the southern border and the striking difference in media coverage of Democratic versus Republican humanitarian crises on the border. You're ignorant of decades of Democratic militarization of the border leading to so much human misery both on the border and inside Mexico. All you care about is your little talking point. In the end you have no knowledge and no argument so you sling insults.

Nice chatting with you. All the best.

A provocative question: Why do you support the "cause of peace"? Not why were the wars in Iraq or Afghanistan wrong, but why is peace, generally, the most important consideration? — Echarmion

Most important, all things being equal. There might have been a good war among all the bad ones. WWII gets credited with being a war that needed to be fought. Maybe the last one. Most of the others have been gravy train operations for the military-industrial complex, which has had Washington by the throat since WWII. JFK turned toward peace, but the MIC didn't take kindly to that. Or else there was a lone nut and the MIC just got lucky, as they so often do.

Why peace? Because people living together harmoniously despite their differences, engaging in free trade and the free exchange of ideas, is better than tearing each other limb from limb every time they have a difference of opinion or a difference of interest. As JFK said at American University on June 10, 1963, five and a half months before his assassination:

So, let us not be blind to our differences--but let us also direct attention to our common interests and to the means by which those differences can be resolved. And if we cannot end now our differences, at least we can help make the world safe for diversity. For, in the final analysis, our most basic common link is that we all inhabit this small planet. We all breathe the same air. We all cherish our children's future. And we are all mortal.

That's why.

That's not how justice works. You have to actively work for it but I guess if you vote in the US you have to pretend justice is released like manna from the heavens because your political system is currently incapable of protecting it. — Benkei

We shall see. As Martin Luther King said, The arc of the moral universe is long, but it bends toward justice. Of course it can take a long time. Stalin sent a lot of people to the Gulag before the Soviet Union finally collapsed. There was justice, but not in time to help many individuals.

Some cages were built by Obama and some children were separated under exceptional circumstances. Trump wrote a blanket policy to prosecute all illegal border crossers which resulted in a blanket separation of kids from their families. A lot more cages were build. There's your "action" to judge. — Benkei

I'll see your Trump cages and raise you Clinton and Obama's militarization of the souther border, which has caused untold human misery. The Dems have a terrible human rights record re Mexico.

What was the worst thing Trump did while he was president, in your opinion? — RogueAI

Choosing Pence. Choosing all the disloyal people he did. Being so pathetically ignorant of how the government works and what a pile of snakes DC was that he just got rolled by the bureaucrats and deep staters. Giving in to the worst of the covid authoritarianism, though by then I don't think he had much choice. These are political or tactical mistakes. Off the top of my head I can't think of any particular policy I didn't like. Toss me out some egregious examples, I'm sure he must have done something I didn't like at the time.

Firing Comey, which by then everyone in Washington favored, but then the next day inartfully saying it was because of his own legal issues. That sent the left into a feeding frenzy. If he's just shut up, nobody would have complained. They all hated Comey for sinking Hillary at the end. So in general, being his own worst enemy. Saying things they could use against him.

But like I say, remind me of what I might have forgotten. Every time they called him Hitler, I'd think, "I'll reserve judgment to see if he invades Poland." He never did. Nor did he invade anywhere else. That overrides almost anything bad he might have done. If you'll just remind me what you think I should object to. You might remember something I didn't.

“Yeah, I guess so,” Trump said in the fall of 2002 when asked by Stern if he supported an invasion. “You know, I wish the first time it was done correctly.”

He's singing a different tune now, of course. But if you're going to call out the NYTimes for what they were saying back then, what's good for the goose... — RogueAI

The difference is that the NYT always supports the warmongers. Trump's off that train.

"Attorney General Jeff Sessions told Fox News' Laura Ingraham on Monday that he hopes the administration's new policy that separates children from their parents will serve as a deterrent to other immigrants considering crossing the border illegally." — RogueAI

Again, you are upset about words, something Sessions said on a right wing talk show. You can't prove he wasn't just saying that to (a) deter immigrants, or (b) suck up to the FOX news base. You are upset about what he SAID and you have once again presented no evidence about what he DID.

You just assume that Obama's heart is pure and Trump's is impure, and that causes you to not even realize that you are not making a substantive argument.

In the 1990's the Dems wanted to look tough on immigration, so they hardened and militarized the border (Hillary was front and center on that), leading to desperate immigrants dying of thirst in the desert. If you seek to paint Trump as lacking in human compassion for immigrants, you will have to come to terms with the Democrats' own cruelty. The Dems forced Mexico to run a war on the cartels in order to receive American economic aid, leading to many tens of thousands of deaths of Mexican citizens. The Democrats have an appalling human rights record regarding Mexico.

To recap: the Trump Admin started a new zero-tolerance policy to separate all families as a deterrent. This had never been done before and was ended just a few weeks after the public found out. Do you see how that is different than what Biden and Obama were doing? — RogueAI

Your proof consists only of verbal statements, no proof as to what was actually done.

The Democrats have an absolutely appalling human rights record in Mexico. And you want to give Joe Biden as an example of human decency in Mexico? Motherf*cker is a child trafficker, I have repeatedly pointed this out. He doesn't waste time separating families, he just turns the kids over to their traffickers at the border.

You are just being a partisan shill, your argument is devoid of logic. "Orange Man say bad thing on television." That's the extent of your argument. While being willfully ignorant of Democratic human rights abuses on the border and within Mexico for decades.

Less talked about was when Trump rid the GOP of their anti-gay stance, their homophobia, which for me was a pivotal realignment of that party’s social conservatism and neocon agenda into the more libertarian spirit we see today. — NOS4A2

Yes, good point. And he's softened on the abortion issue. He says to leave it up to the states. That's upsetting some hard core pro-lifers, but where are they going to go?

Trump's a New York city builder. He's been dealing with people from every walk of life and every socioeconomic level for decades. He didn't become a "racist" till he ran against Hillary. Before that, the Clintons came to his (third) wedding. Maureen Dowd wrote about When Hillary and Donald Were Friends.

I think that having "free will" versus having a "soul" are not the same thing. — Tarskian

They're closely related. A self-awareness and the ability to have preferences and desires, and to be able to act to bring them about.

As I see it, the soul is an object in religion while free will is an object in mathematics. — Tarskian

I'm using soul in a secular sense. And free will does not appear in any math text that I've ever seen. Free will is not an object of study of math at all.

I see free will and incompleteness as equivalent. I don't see why they wouldn't be. — Tarskian

I believe Penrose makes that argument.

https://en.wikipedia.org/wiki/The_Emperor%27s_New_Mind

"Penrose argues that human consciousness is non-algorithmic, and thus is not capable of being modeled by a conventional Turing machine, which includes a digital computer. Penrose hypothesizes that quantum mechanics plays an essential role in the understanding of human consciousness. The collapse of the quantum wavefunction is seen as playing an important role in brain function."

Yes, I get that. In the sense that we've discussed, it is a speech act either way. However, axioms and definitions are not the same kinds of speech act. I expect there's a mathematical explanation of the difference. But they are both setting up the system (function?) - preparatory. So they are both different from the statements you make when you start exploring the system, whether proving theorems or applying it. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

Yes of course, no issues there.

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.

Yes, I get that. There are even some beautiful arguments in philosophy. I'm sometimes tempted to think that the beauty is the meaning. I would, sometimes, even go so far as to agree with Keats' "‘Beauty is truth, truth beauty,—that is all/Ye know on earth, and all ye need to know." But only if all the philosophers are safely corralled elsewhere. — Ludwig V

I'm not 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.) — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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? — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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.

But who said I'm not a Platonist? I am? When it suits my argument. I'm a formalist as well at times.
— fishfry

Are the two really mutually exclusive? — Lionino

That's funny. @Metaphysician Undercover just told me that formalism is just a deeper kind of Platonism. Which actually makes sense in this context. It explains how one can be both.

I may have messed up the order of some of the paragraphs. I don't think I can hold up my any of this any more. I will bow out now. Thank you for the chat.

I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. — Metaphysician Undercover

Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics.

You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity. — Metaphysician Undercover

Actually some numbers happen to have physical instantiations and some don't. 5 does.

But it's a distinction without a difference. It's something that consumes you but nobody else. Least of all me.

If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads. — Metaphysician Undercover

Nah. Not buying any of it.

Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false. — Metaphysician Undercover

Why do you think I take a position on any of this?

Like I explained earlier, formulism is just a specific type of Platonism. It takes Platonist principles much deeper in an attempt to realize the ideal within the work of human beings, while other Platonists allow the ideal to be separate from human beings. — Metaphysician Undercover

The swine.

Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead. — Metaphysician Undercover

Judged by who? Politicians? Academic administrators? Philosophers? How about by their fellow mathematicians? That's the standard of what counts as math.

As I explained to jgill above, theory building is a form of problem solving, it just involves a different type of problem. There are many different types of problems which can be categorized in different ways. — Metaphysician Undercover

Just change the meanings of words to suit your argument. Pigs fly, if I redefine pigs as birds.

Yes, this is the problem, axioms of set theory are false, in the way described above. — Metaphysician Undercover

False. Not just "not true," or "lacking a truth value," but literally false. If they can be false then they can also be true.

I said that 5 is not an instance of a real number. Also, I would say that the fingers on my hand are not an instance of the number 5, they are an instance of a quantity of five. You see, this is the problem of mixing up the ideal with the physical. "The natural number 5" is an ideal, a type of Platonic object called "a number". There is no physical instantiation of numbers, they are by definition ideal. So we need to refer to the use of "5" to see its meaning, and then we can find a physical representation for its meaning. In the context of usage of the natural numbers my understanding is that 5 represents a specific quantity, and the fingers on my hand provide an example of this specific quantity. — Metaphysician Undercover

Meaningless word games. The fingers on your hand are a physical instantiation of the number 5. Positive integers have the property that the smaller among them may be physically instantiated. 12 as in a dozen eggs, 9 as in the planets unless an astronomical bureaucracy demotes Pluto. That's one for the philosophers, don't you agree? The number of planets turns out to be a matter of politics, not math or astrophysics.

If we say that the numeral 5 represents a number, which goes by that name, 5, we have no meaning indicated to assist us in finding a physical example of the number five. All we have is that there is a type of thing called a number, and one of them is named 5. In order for numbers such as 5 to be used in practise, we need to provide something more, otherwise we're stuck with the interaction problem of idealism, these ideal things have no bearing on the real world. — Metaphysician Undercover

I can establish a bijection between the fingers on one hand, and the elements of the set {0, 1, 2, 3, 4}. Done.

But if we give the number 5 further meaning, such as "a specific quantity", to allow it to be useful in the world, then the ideal, the number 5 becomes redundant, and completely useless. Why not just say that the numeral "5" means a specific quantity, and be done with it. Well I'll tell you why not. The numeral "5" is assumed to represent a number, 5, which is an abstract, Platonic object, for another purpose. The other purpose is mathematical philosophy, building structures and frameworks to be used as tools for understanding the development of math. However, as explained above, rather than assisting understanding, it misleads. — Metaphysician Undercover

Because sometimes it's useful to study things in the abstract; just as one guy goes fishing, and another studies ichthyology

You are free to abandon me anytime you want. — Metaphysician Undercover

Thank you. I should avail myself of that option at the end of this post. I think I will.

If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum". — Metaphysician Undercover

It depends on how you look at them.

Well, "the real numbers", and "5" being an instance of a real number, was your example. I agree that by some accepted principles of mathematics, the axioms of set theory, etc., 5 is an instance of a real number. This I believe to be the influence of Platonism which assumes that a number is an object. I disagree with this, and think that a number is a concept, and conceptions are quite different from objects. The way that one concept relates to another for example is completely different from the way that one object relates to another.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false. — Metaphysician Undercover

Suppose math is entirely fraudulent. Would it then be any less useful in the world? Would you fire all the professors? What is your suggested remedy for all these academic crimes?

Do you not look at mathematics, and mathematicians as real human beings, carrying out activities in the real world? If so, then don't you think that there is such a thing as true and false propositions about what those mathematicians are doing? If you follow, and agree so far, then why wouldn't you also agree that mathematical philosophies, as tools, or models, ought to be judged for truth and falsity? If a mathematical philosophy provides false propositions about what mathematicians are doing, offering this philosophy as a tool for understanding the structure and development of math, it is likely to mislead. — Metaphysician Undercover

You are free to judge things as you wish.

You might think that it doesn't matter whether a number is an object or not. You might think that within the confines of the logical system of "the real numbers", a number can be whatever the mathematician who states the axiom wants it to be. My argument is that numbers are used billions of times a day by human beings, and according to that usage there is some truth and falsity about what a number is. Therefore when an axiom makes a statement about what a number is, and it's not consistent with how numbers are actually used, the axiom can be judged as false. — Metaphysician Undercover

They're meta-false, as I understand you. They're not literally false. If the powerset axiom is false, you get set theory without powersets. You don't get some kind of philosophical contradiction. You are equivocating levels.

If you truly believe this, then how would you validate your claim that the number 5 is an instance of a real number. Do you see that when you talk about "a real number", and "the real numbers", you validate the claim that "the real numbers" refers to a collection of individual objects? And that is contrary to what you say here. And do you see that in set theory, "numbers" also must refer to individual things, and this is contrary to being a description of "the idealized continuum". — Metaphysician Undercover

A model, not a description. Is that better?

I guess so.

As you have probably noticed, Lionino does not talk about metaphysics or about mathematics but about me. That is apparently his obsession. He incessantly talks about me, very much like I incessantly talk about Godel. I don't know if I should feel flattered. — Tarskian

I've noticed that some posters have personal obsessions with others. For me, when I find it unpleasant to interact with someone, I just don't interact. Don't disagree with them, don't bait them, don't troll them, don't interact with them directly or directly.

But then again, the metaphysical implications of the foundational crisis in mathematics, are truly fascinating. — Tarskian

Well, you are saying that historically contingent opinions about math, have some bearing on the ultimate nature of math. But I imagine that if there is such a thing as an ultimate nature of math, it incorporates and transcends all such opinions. Math is more than the sum of all philosophies about it.

How can something that "isn't about anything at all" suddenly become about the fundamental nature of everything? — Tarskian

Well now, that is a great question. Wigner asked about the Unreasonable Effectiveness of Mathematics in the Physical Sciences. How can math be so fictional, so idealized, so much about nothing at all; and yet so relevant and useful. I think one answer is that math is useful to humans in the same way that echoes are useful to bats. Our brains are wired to makes sense of the world through math. Our approach isn't better or worse than any other creature's. We flatter ourselves to imagine that the world is "like" math; when in fact, we're just wired that way.

Of course there is nothing wrong with using the word "true" in math. But in the papers I have written (around thirty publications and over sixty more as recreation) I doubt that I ever used the word - but I could be wrong. On the other hand, "therefore" is ubiquitous. — jgill

That was not my point. Mathematicians don't use the word true in their formal work.

But when you make a discovery, don't you feel that you are discovering something that is true, or factual, about whatever it is you're studying? Surely you don't lean back and say, "That's a cool formal derivation that means nothing." On the contrary, I imaging that you say, "I learned something about nonabelian widgets" or whatever. Am I wrong? I would be surprised if I'm wrong?

"True but verify" might be my motto. I suppose I would consider myself a Platonist were I to care, but this type of philosophical categorization - although relevant to this forum - matters very little to me. — jgill

In your work, do you think of yourself as discovering formal derivations? Or learning about nonabelian widgits?

"concept of truth in first order arithmetic statements" — jgill

Yes but it's so unlike you to have an interest in Tarski's undefinability theorem.

If there are any practicing or retired mathematicians reading these threads I wish you would speak up. I would ask my old colleagues what they think of these philosophical discussions, but they are pretty much all gone to greener pastures. — jgill

The "philosophical" discussions in this forum do not reflect the actual work of philosophically inclined mathematicians. For example the early category theorists like Mac Lane were very philosophically oriented. "He was vice president of the National Academy of Sciences and the American Philosophical Society, and president of the American Mathematical Society ..." Impressive resume.

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

It's a bit like physicists. Most of them are of the "shut up and calculate" school, while others -- a minority -- are interested in what it all means, what it can tell us about the ultimate nature of reality. They call the latter foundations of physics. So math is to math foundations as physics foundations. Most don't care, some do.

Meanwhile, as Biden and politicians around the word pile on the platitudes about 'unification' and 'coming together', and the abhorrence of violence in politics, guess which side is using the episode as ammunition in the culture war? — Wayfarer

Can't you dial it back for five minutes?

No, I mean how the Capitol police would respond to black or brown people. I doubt very much they’d be “letting them in,” to the extent that that even happened (you know, apart from breaking windows and ramming down doors). — Mikie

So your best argument against me is a hypothetical scenario you just made up. If you had actual facts, or an actual argument, you'd make it.

Well I'm taking the day off from partisanship so thanks for the chat.

Adderral: — Lionino

Uh-oh, insert Biden joke here.

If you think Iraq was such a bad idea (which it was), why are you voting for the political party that got us into it? A majority of House Dems voted against the Iraq war authorization. — RogueAI

I was a full-on liberal Democrat then, and I was fervently against that awful, misbegotten war. And I still resent Obame for not holding the Bush administration accountable (though I concede he did so for sound political reasons).

But if you recall, in the 2016 GOP debates, Trump ripped Jeb over his brother's war. Trump came out four square against the Iraq war and against Bush. And when the audience cheered, you knew there was a sea change in the Republican party. Trump in, Bush(es) out. Peace in, and war out.

That's exactly why I support Trump. He called out Jeb on his brother's war and the GOP audience cheered. Then as president he started no new wars.

I oppose Bush-ism and support most of Trumpism. I oppose the bipartisan neocon wars. I support the cause of peace. Anyone for peace should take a look at the track record of Trump versus the Hillerys and Pelosis and Schumers and Bidens of this world. DiFi, my own Senator for so many years. Voted for the wars while her construction business husband profited from them. That's the system Trump is fighting.

Ask yourself how the Capitol police would respond to black or brown people. I doubt very much they’d be “letting them in,” to the extent that that even happened (you know, apart from breaking windows and ramming down doors). — Mikie

You mean when a black cop shot an unarmed young white woman to death? If the races were reversed we'd still be having riots.

But nevermind— just go on pretending it the insurrection was nothing. Years from now I’m sure it’ll be remembered as a tour — in conservative media anyway. — Mikie

Time will tell. If there's justice in the universe, there will be justice for J6.

https://www.axios.com/2018/06/19/sessions-says-he-hopes-child-separation-policy-will-serve-as-a-deterrent — RogueAI

As usual you confuse what Trump SAYS from what he DOES.

He's a negotiator. He puts up skyscrapers in Manhattan. How do you know he wasn't expecting his statement to go out to prospective migrants, and make them decide to stay home/

You can't prove otherwise. You can't actually cite statistics on what was in Obama or Trump's heart for each of the families separated and kids caged.

Instead you choose to judge Trump on his words, and not on his actions, which were in fact no different than Obamas. He used Obama's cages for God's sake, you can't say with a straight face that they were Lightbringer cages when Obama stuck kids in them and Orange Hitler cages when Trump did it.

You're operating from emotion, choosing to overreact to Trump's words, because you can't cite facts in his actual actions.

You have no credibility when you do this, because the left has been doing this for eight years. Just think taco bowl tweet. The left went ballistic. I took one look at that, cracked up laughing, and said, "Trump is a master troll and a brilliant performance artist!"

That's the deal. I get Trump. If he does something wrong, he did something wrong. I'm just not emotionally trigged by the guy, and many on the left are. And that clouds their, and your, judgment.

I'm still antsy about assigning a random variable to the truth of a theorem. How do you sample from mathematical theorems? What would it even mean for a mathematical theorem to be expected to be true 9 times out of 10? How do you put a sigma algebra on mathematics itself... — fdrake

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)

If you desire to avoid the long posts, I think, by the end of my reply here, that I have isolated the primary point of disagreement between us. It is exposed in how you and I each relate to what is referred to by "the real numbers", and what is referred to with "5" in the context of "the real numbers". And further, how this relates to the extension/intension distinction. — Metaphysician Undercover

Looking ahead, you wrote a long post. To which, in my own verbose style, I will reply to at length para by para, increasing the overall length of the thread.

I see an out. In this para you have stated your aim about the real numbers and the number 5. I don't think I have any interest in this topic. I know it's important and meaningful to you, but it isn't to me. Perhaps I'm to dim to grasp all these philosophical subtleties such as you raise. If so, so be it.

But secondly, and I'd be remiss if I didn't add, that I have formally studied the real numbers and the number 5. That doesn't make me right and you wrong, by any means. What it does mean is that I'm not likely to ever defer to your opinions about the real numbers or the number 5.

And if, as you say, that's all you want me to know, then now I know it. We disagree on the real numbers and the number 5. Ok. I am a pluralist. It doesn't bother me when people have different opinions than I do. I don't have to convert you nor you me. We can let the matter rest. I'm for that.

Therefore, I think you might just read through my post and reply to the aspects which are related to this issue. However, the issue of what mathematics is, how you and I would each describe "what mathematicians do", might also be important and relevant. — Metaphysician Undercover

I have no strong or absolute opinion about "what math is," nor do I feel any need to argue for or against any particular interpretation of that question. The history of math is the evolution of the answer to that question! "What math is," is always changing. It literally is what mathematicians do.

The issue though, is that even supposedly "pure" mathematicians work toward resolving problems, and problems always have a real world source or else they are really not problems, but more like amusements. — Metaphysician Undercover

If your claim is that by definition, what pure mathematicians do amounts to amusements, in the sense that they solve problems that were only inspired by their own meaningless work; and not by anything that we currently know about in the world.

If so, it's perfectly and trivially true, if that's your definition. What of it? Doesn't mean anything.

Read Hardy's A Mathematician's Apology. He'd have been insulted if you told him his work was useful. The irony is that he did number theory, which had been a beautiful but utterly useless branch of math for over 2000 years. And then finally in the 1980s, people invented public key cryptography, and Hardy's work was at the heart of grubby world commerce.

So you just never know.

But still. Amusement? Ok. Whatever. Like Picasso, he made amusements too.

A mathematician working in pure abstractions works with abstractions already produced, and may not even know how real world problems have shaped the already exist abstract structure. Even if we attempt to step aside from existing conceptions, and 'start from scratch' as philosophers often do, we are guided by our intuitions which have been shaped and formed by life in the world. And intuition comes from the subconscious into the mind, so we cannot get our minds beneath it, to free ourselves from that real world base. And since it is from the subconscious, we have no idea of how the real world effects it. — Metaphysician Undercover

Such a triviality. Everyone knows that math started when some caveman put a mark in the ground when he killed a wooly mammoth, and then put another mark next to it when he killed another one. That was the first mathematical abstraction, and it is the paradigm for all others.

Everybody knows this. You think you just discovered it. Just like someone could say that abstract art is an evolution of representational art, or is influence by it or is a reaction to it.

It's just the process of abstraction. And abstraction is always based on our own experience of the world. Nobody is denying that. It's your strawman.

I agree, but the description of what mathematicians do, is very difficult to get an agreement on. — Metaphysician Undercover

Correct. Mathematics is a historically contingent human activity that changes every day as new papers are published. And every few decades new ideas come out that change our very conception of what math is.

Even in contemporary practice, there is professional disagreement about what is mathematics. I refer to the amazing dispute over the work of Shinichi Mochizuki, about which mathematicians have been arguing for over a dozen years, and illustrating the fact that mathematical truth is subject to social agreement.

In view of the history of math, we see that this has always been so.

It's not a circular definition, but a proposal of how to produce a definition. So to actually provide the definition of mathematics, we need that description. It will be very difficult for you and I to agree on such description. You will probably place as the primary defining feature, (the essential aspect), of what mathematicians do, as working with abstractions. I will say, that description is problematic because then we need some understanding of what an abstraction is, and what it means to "work" with this type of thing. This almost certainly will lead to Platonism because we've already assumed as a premise, the existence of things called "abstractions". — Metaphysician Undercover

Trying to nail down a definition of mathematics is like a cat chasing its own tail. Not as cute though. Why do you persist? Why do you even think it matters? It changes throughout history, and it's not even agreed on by all professional practitioners today!

But who said I'm not a Platonist? I am? When it suits my argument. I'm a formalist as well at times. Mathematical philosophies are tools, nothing more. Conceptual tools, frameworks for thinking about the development and structure of math. They aren't "true" or "false," they're just models, if you will.

Therefore I look at what mathematicians are doing as "solving problems". That's what they do, and there is a specific type of problem which they deal with. You are most likely not going to like this proposal for a description of what mathematicians are doing, because it eliminates the distinction between "pure" mathematics and "applied" mathematics. — Metaphysician Undercover

I don't like it because it's factually false. There's a famous essay on that, about the kind of mathematicians who solve problems, and the kind of mathematicians who build theories. To the Internet! Yes here it is, The Two Cultures of Mathematics by Timothy Gowers.

Problem solvers and theory builders. The theory builders don't solve problems at all. They create conceptual frameworks in which others can solve problems.

In the way described above, there is no such thing as "pure" mathematics. However, my starting point has the advantage of applying equally to all mathematicians, by applying the initial assumption of pragmaticism. Instead of saying "mathematicians are working with abstractions", we say "mathematicians are working with symbols (language), to solve problems. This way we avoid the messy ontological problem of "abstractions" It is only when we start sorting out the different types of problems which mathematicians work on, do we get the divisions within mathematics. — Metaphysician Undercover

Ok, there's no such thing as pure mathematics. So what? Why do you care? Why should I? I've already explained that not only is math historically contingent, there's not even universal agreement today about what math is.

You are arguing against a strawman of your own imagination.

This is not the point at all, and you are not paying respect to the difference between the two distinct fields, mathematics, and mathematical logic, so your analogy is not well formed. If the field of mathematics is represented by the sculptor, then the field of mathematical logic is represented by the critic. Whenever the critic mistakenly represents what the sculptor is doing, then the critic is wrong. When mathematical logic represents mathematicians as using = to symbolize identity, the logic is wrong. — Metaphysician Undercover

Ok you didn't like my sculptor analogy.

Fishfry, wake up! Was it getting late there or something? There is no physical object involved! There is no star! I think we've been through this before. The intensional/extensional distinction is completely irrelevant in this case because everything referred to is meaning (intensional). There is nothing extensional, no objects referred to by "1+1", or "2". That is the heart of the sophistic ruse. This intensional/extensional rhetoric falsely persuades mathematicians. It wrongly misleads them due to their tendency to be Platonist, and to think of mathematical abstractions as objects. As soon as meaning is replaced by objects, then "extensional" is validated, the sophist has succeeded in misleading you, and down the misguided route you go. In reality, there is only meaning referred to by "1+1", and by "2", everything here is intensional, and there is nothing extensional. — Metaphysician Undercover

LOL. 1 + 1 and 2 are each representations of the same set in ZF, with "1" and "2" interpreted as defined symbols in the inductive set given by the axiom of infinity; and likewise "+" is formally defined.

But we've been having this conversation for years, and I don't think today is the day for any more.

This is why I was very steadfast on the previous issue, to explain that "5" is not "an instance of a real number". It is that type of nomenclature, that type of understanding, which leads one into allowing that there is a place for extensional definitions in mathematics. Really, "5" in that example is just a part of that conception called "the real numbers". It receives it's meaning as part of that conception. there are no extensional objects referred to by "the real numbers", and "5" is just an intensional aspect of that conception. When you apprehend "the real numbers" as referring to a collection of things, instead of as referring to a conception, then you understand "5" as referring to an instance of a real number, instead of understanding it as a specific part of that conception. Then you may be misled into the "extensionality" of real numbers, instead of understanding "the real numbers" as completely intensional. — Metaphysician Undercover

Well.

I will stipulate that the real number 5 has a relatively shake ontological status. The mathematical real numbers are a very strange gadget, and I genuinely doubt that they are instantiated or exemplified by anything in the physical world. That is, the real world is not a continuum in the sense of being isomorphic to the mathematical real numbers. The real numbers are far too weird to be real.

BUT! Are you telling me that you don't believe in the physical instantiation of the natural number 5? Just look at the fingers on your hand. I rest my case.

Again, you are not distinguishing between "mathematics", and the "mathematical logic" which the head sophist preaches. One is the artist, the other the critic. My beef is not with mathematics (the art), it is with mathematical logic (the critic). I see mathematical logic as sophistry intended to deceive. And I will explain the reason why i say there is an intent to deceive. — Metaphysician Undercover

Mathematical logic. Anyone in particular? Do you go back to Aristotle, or are you annoyed with Russell, or Godel, or what? What is the specific nature of your beef, as you put it.

Mathematics has a long history of exposing us to problems which we just cannot seem to solve. These are issues such as Zeno's paradoxes, and other apparent paradoxes discussed at TPF, which generally amount to problems with the conception of infinity, the continuity of space and time, etc.. — Metaphysician Undercover

Your level of mathematical knowledge is so naive, that you actually think that Zeno's paradox is a mathematical problem; and that the extremely naive conceptions of mathematics exhibited by innumerate philosophers that drive these inane discussions of supertasks, has any relationship whatsoever to the professional activities of mathematicians. You just have no idea. In argument, you wield your ignorance like a club. You have no idea what you are talking about.

What mathematical logic does, is create the illusion that such problems have been solved. So, the intent to deceive is inherent within the conceptual structure, which makes these problems solvable. It deceives mathematicians into thinking that they have solved various problems, by allowing them to work within a structure which makes them solvable. The problem though is that the basic axioms (extensionality for example) are blatantly wrong, and designed specifically so as to make a bunch of problems solvable, regardless of the fact that incorrect axioms are required to make the problems solvable. — Metaphysician Undercover

Instead of addressing these bitter complaints to me, have you thought about going down and picketing the math department at your local university? "Lying corrupt sophists all! Should not drink and derive!" Get some press for sure.

Why me?

Future application is not the issue here. The issue is that mathematicians work toward problem solving, by the very nature of what mathematics is. The problems are preexistent. Therefore mathematics by its very nature is fundamentally "applied". If you remove problem solving from the essence of mathematics, then it would be random fictions. But mathematics is not random fictions, the mathematicians always follow at least some principles of "number", already produced. — Metaphysician Undercover

Yeah ok, all math is applied. I have no strong opinion. It's all an abstraction of the first caveman who made a bijection between marks on the ground and wooly mammoths he killed. I have no problem with that. But actually you have zero idea what pure mathematicians do, and why they are so regarded by their peers. You're just making stuff up that you don't know anything about. You keep saying mathematicians do this and mathematicians do that, and you have repeatedly demonstrated to me that you have no understanding of mathematics nor mathematicians.

What I think, is that there is really no such things as sameness in math, and this is better described as a misleading subject. Mathematics actual deals with difference, and ways of making difference intelligible through number. Similarity is not sameness, but difference which can be quantified. To me, "essential the same" just means similar, which is different.[/quotem

You keep talking about mathematics as if you forget that you're speaking to someone who has been observing for over five years that you don't know anything about mathematics. You are just making up strawman to have an argument that only you care about.

There is quite a lot of mathematical thought about what "sameness" is in math. I'm thinking of the work in Univalent foundations, in which there's a univalent axiom that sort of says that "things that are isomorphic are the same." It's based on intuitionist math and the denial of LEM. It's all the rage in proof assistants and the formalization of math. A lot of philosophically inclined mathematicians have worked n that area.


— Metaphysician Undercover

This appears to be the substance of our difference, or disagreement. If you do not like long posts, we could just focus on this specific issue. The issue is whether "the real numbers" refers to a conceptual structure, or whether it refers to a group of things, numbers. I believe the former, and the fact that "numbers" is plural is just a relic of ancient tradition. From my perspective, "5", in the context of "a real number" is just a specific part of that conception. Then the relations are purely intensional, and there is nothing extensional here. If however, you apprehend "the real numbers" as referring to a group of things called "numbers", then "5" refers to one of those things, and there is the premises required for extensionality. — Metaphysician Undercover

If I'm understanding you, I agree. I don't think the mathematical real numbers refer to anything in the world at all. They describe the idealized continuum, something that we have no evidence can exist.

I agree. "Truth" is negotiable it seems. The word should be avoided in mathematical discussions. — jgill

But of course you yourself know that's not true. I assume you think of your research as discovering truths about abstract mathematical structures that have some Platonic existence in the conceptual realm. You surely feel that the things you study are true. Do you not?

Tarski's Undefinability Theorem says (Wiki): — jgill

What made you quote that? Not sure of the relevance. It's another diagonal argument.

In any event, my sense is that most mathematicians are at heart Platonists. The things they study are real. The number 5 is prime, and there is no possible world in which it isn't. The number 5 is prime even when there are no intelligent minds in the universe to comprehend it. The fact that 5 is prime is True even before mathematics exists. There is indeed truth about the things mathematicians study.

Hasn't this been your experience?

Chaos theory has already been brought up twice, which he ignored, like he does everytime his incorrigible nonsense is challenged. — Lionino

I try to keep an open mind and take the good with the bad of all, say, a bit eccentric posters. I hope that is not too uncharitable to @Tarskian. Am I being fair?

Yes. Is that a definition or an axiom? Whatever it is, it isn't just another assignment of a probability because it enables the actual assignments to the outcomes to be made. But I don't see that anything is wrong with representing them as percentages, in which case the probability of the entire event space is 100. Meteorologists seem to be very fond of this. — Ludwig V

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.

— Ludwig V

Yes. Most of the discussions I get involved in are at the applied level. But I have seen some posts that are completely abstract. So I think I understand what "event space" means. It is a metaphor to describe a formulation that doesn't identify actual outcomes, but only gives, for example, E(1), E(2)... - variables whose domain is events. In particular applications, that domain is limited by, for example, the rules of the game. That's not a complaint - just an observation. — Ludwig V

Yes. 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. — Ludwig V

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. — Ludwig V

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. — Ludwig V

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, — Ludwig V

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. — Ludwig V

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.

Deep down humans could also be deterministic. — Tarskian

I stipulate that:

1. This is a very hip and TED-talky idea going around; and

2. I personally disagree strenuously; but I concede that I can't prove it.

But given that, my original point stands. That programs can't have free will. And I hope you agree that humans being deterministic would not contradict that point.

As long as the theory of humans is incomplete, humans would still have free will. — Tarskian

We all have moral choice.

It is bizarre to suggest there's any arguing the point, when the point has been so profusely documented. Your retraction and your offer to retract the bizarre qualifier in the retraction are a self-serving and sneaky way to put the ball back in my court where it doesn't belong. — TonesInDeepFreeze

If you reject my retraction and apology, that is your right.

Yeah no I ain't assigning random variables to generic mathematical expressions. — fdrake

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.

Gift link to today’s 5,000 word NY Times editorial, Trump is Unfit to Lead. — Wayfarer

Last week they said Biden is unfit to lead. Who are they for, Harold Stassen?

What's the point of the Times wasting 5000 words to yammer about Orange Hitler, as if they have anything new to say on that subject. Maybe they should just lie us into another war.

I have no clue why you started talking about cages. Maybe you have more in common with Biden than you think? — Benkei

Did I get my convos crossed? My bad. No matter. It's all the same, really. Whatever it was, I can let it go.