↪Baden Maybe I should move my money into a stronger currency, like the Vietnamese Dong. — Michael

*Puts 'Vietnamese Dong' into Google... thinks a second... *puts Vietnamese currency into Google*...huh, is that so.

A held opinion is a belief, X knows that P iff X has a justified true belief that P is a reasonable approximation to knowledge that. There are ambiguities; see fallibilism and and epistemic luck. Also see hinge propositions for another complexity. Accounts here look like accounts of truths, justifications, beliefs and relationships between them.

Sociologically/anthropologically, what matters is what is usually treated or presumed as true. Another wrinkle here is that knowledge might not be just a relationship between an agent and a proposition, it might be a relationship between an agent and an interdependent system of other agents, texts and interpretive norms; knowledge institutionally is a collaborative endeavour whose production requires trust of sources and a commitment to the truth from all involved. Accounts here look like (historical) descriptions of scientific and institutional practices.

Also, knowledge-that is under half of knowledge, knowledge-how or competence/skill plays a big role and requires a much different approach to analyse. Knowledge-that may require knowledge-how to get going.

You might like doing a project on propositional and predicate logic then. Depending on the length. It's very mathsy.

The problem is we do see where you're coming from. You just don't see where we are coming from. Imagine yourself back in time before you attained your current political ideology. Someone comes up to you in the street and says "Disabled access ramps are a slippery slope to Naziism", no matter what you say to them they keep defending the position. What would you think of them, honestly?

(1) “Let us suppose a variable piece of a manifoldness of one dimension” - I’m not sure what work the word ‘variable’ does here in ‘variable piece of a manifoldness’. Can one take an invariable piece of a manifold? And what would this distinction mean? — StreetlightX

I read that as saying variable piece of a manifoldness might be a connected chunk of a manifoldness. But when we take 'a' variable piece of a manifoldness Riemann intends us to be discussing an arbitrary one. The other bit 'of one dimension', connotes that the manifoldness he's considering is just a 1 dimensional curve.

So you might imagine cutting a cylinder down the middle really finely to produce a circle.

(2) “Let us take a continuous function of position within the given manifoldness, which, moreover, is not constant throughout any part of that manifoldness.” - Here, I’m not sure what work ‘not constant’ is doing. Is it the variation of position on the manifoldness we are asked to think of is ‘not constant’? — StreetlightX

This continuous function describes what point you are on on the previously considered curve. If the function was something like f(x)=x for x<1 and f(x)=1 for all x>=1, this makes the entire region [1,infinity) map to 1, so it can't be used to uniquely specify the position. More specifically, in this example, if you know the function is f(x) = x, you can take an output of this function f(x) and directly map it to an input x, allowing you to translate between the position described using the function and the position on the manifold. However, when this x becomes greater than 1, all this function tells you is that it's equal to 1. Which means the input which caused the function to be 1 could be anywhere between 1 and infinity - so we can't invert the function to uniquely specify the point on the curve.

Phlogiston? Aether?

Survival without cooperation, ethics without normativity, morality without obligation, politics without groups, people without identities, freedom without the expansion of autonomy, justice without law or fairness, order without appeal. A recipe for a world bereft of anything resembling human life; yet still apparently a philosophy for living.

Have you read the thread so far? We've discussed things relevant to your questions.

When I refer to "winning" what I'm saying is that the winners have the government on their side. The government shouldn't be on anyone's side. — AppLeo

Hey, you're right, that's why people take up progressive politics.

I'm going to stop responding to you now. If you take that as a victory, hurrah for you, you win!

Minorities who have affirmative action. The poor who demand welfare benefits. Employees who use government to force businesses to give them "living" wages. Employers who use government to pass unfair regulations against their competitors. Gays who use government to make a christian baker bake a cake he doesn't want to bake. Christians using the state to enforce their religious policies on people who don't believe in Christianity. — AppLeo

So you agree that disabled people organising together to push the introduction of disabled access ramps is fine? And that it secures their individual rights? If not, why? And why does it go against individual rights?

And you agree that slave revolts and humanitarians back home organising to push the abolishing slaving was good? And that it secures individual rights? If not, why does the abolition of slavery go in the face of individual rights?

Why is group of disabled people needing to intervene on the level of building construction norms to allow wheelchair access to shops against individual rights?

What in hell is the difference between what disability activists did to procure access to places and the procurement of some individual rights for disabled people?

These are groups that that try to win at the expense of other groups. And it completely ignores the individual. — AppLeo

Do you think disabled people wanting disabled access groups are 'trying to win at the expense' of non disabled people? Slaves revolting and humanitarians back home also definitely were 'trying to win at the expense of other people' - they wanted the fucking slave owners not to remain in possession of some of their assets. This is completely incoherent, and I believe you know this because you're always presenting more trivial reasons people might organised to solve their collective problems.

In this is the incredible equivocation that the abolition of slavery was the same as forcing a baker to make a gay couple a wedding cake.

You see it as a problem, but I don't. — AppLeo

You see disabled people not having access to the same places as people who can walk as not a problem. Of course you don't, you don't have to care about the problem[/u]. You're a bloke who doesn't need a wheelchair. People who need wheelchair access see it as a problem because it is a problem for them.

Also, why should what you see as a problem matter? Lack of disabled access really is a problem for people who need wheelchairs! You would deny them access to spaces because you believe them raising their voices together to gain access is disabled people 'winning' over the non-disabled. The reason they would want to do this is because non-disabled people already win over disabled people due to the established norms and expectations of society.

Even this whole framing of winners vs losers is stupid, what you should be considering is a cost/benefit trade off. Cost - people who generally have enough money to do this must spend some money to introduce a ramp (which through a quick google apparently costs about 1900 dollars). Benefit - everybody can come in and spend money, the architecture is no longer exclusive. The right approach is to assess whether the benefit is worth the cost, rather than declare all collective action wrong by fiat.

A move which does not allow you any purchase on any real world issue, by and by. This fiat declaration of the immorality of all collective action does not allow you to distinguish between just movements and unjust ones. ISIS starts to look like Amnesty International.

If you're so concerned about the veterans, start your own charitable group for them. — AppLeo

Red herring. Nothing about whether I'm engaged in charitable activity for US war vets has any bearing on this argument.

You're getting really lazy now.

That works. Your statement of the logic is imprecise though: you need to change:

Every even number ends on 0, 2, 4, 6, or 8.

To

If a number ends in 0, 2, 4, 6, or 8. then it is even

your implication was the wrong way round, but you used it correctly (the right way round).

Do you think you can prove:

If a number ends in 0, 2, 4, 6, or 8. then it is even.

?

What's the difference. Not being able to walk and not being able to socialize are both hard things to deal with. The question is, are you going to turn the government to give you special privileges or are you going to solve your own problems? — AppLeo

Because the government is giving you something for free and forcing all shop-owners to cater to your needs. How is that not a privilege? — AppLeo

It isn't a privilege to be able to access the shop. This is because non-disabled people have access to the shop. It's a limitation to be unable to access the shop. Now, the disabled person has their freedom limited; they can't go in the shop, they can't live up stairs easily or without disabled access; so what do you have to do to enhance their freedom to the level of a non-disabled person? Ensure that places have disabled access whenever possible.

It is. — AppLeo

It frightens me that you'd employ a metaphor to score rhetorical points when not actually believing in it, then. It makes me believe you're not actually being sincere or caring about the truth of what you say, it makes me suspect that we're in a pointless pissing match and I'm wasting my time trying to show you what I think are errors.

No, these people had their individual rights stolen. And they have every right to fight for those rights. — AppLeo

Do you support that they banded together to fight for those rights? That they... made a group... to ...force society... to ...treat them fairly.

What about... Vietnam veterans with post traumatic stress disorder banding together for healthcare aid?

It's not. It's placing the individual above the group. Individuals banding together for individual rights is fine because fighting for individual rights is equal and just. It's the groups that are creating by mindless individuals that sacrifice their individuality, for a group interest that demands special privileges from the government. — AppLeo

Right. Ok. So you agree that gay people banding together for equal treatment, the Haitians, slaves and so on banding together for their individual rights are fine. I'm curious what you think remains of your original position at this point. All the examples I gave of people banding together were for their individual rights, and seemingly you thought towards the start of the thread that they were banding together for special privilege. They were not, it's mostly for an expansion of individual freedom; a removal of unfair limitations on their conduct.

What you're saying doesn't make any sense. There are no loser groups. There are only individuals and what matters is that all individuals have human rights. — AppLeo

There are no loser groups? This is crazy talk man. You actually introduced this talk of winning and loser groups into the thread, I was only borrowing your vocabulary. Look here:

Capitalism without regulation is what we need. Capitalism without regulation is capitalism that is for the individual. For everrybody. As soon as you add government you start picking winners and losers; it becomes an unfair game. When state and church was the same, one religion controlled everything and made the state unfair and unjustified in its actions. When church and state were separated you had a free coexistence of religions. The same applies in economics. Want people to be free and prosperous economically? You get the government out.

Want an example of capitalism with minimal to no regulation? 19th century America. Largest increase in quality of life that ever happened. True economic freedom. There were no wars, the government wasn't in the way, people were free to buy and sell what they wanted. That is what America needs to return to. Because the government respected the individual. It didn't pick winners and losers like it does today in our economy. — AppLeo

There is a default state assumed by a society. This default state consists of norms of conduct and expectations of capacities. If we live in a world where the default state is considered to be a human who can walk, this limits the freedom of people who cannot. If we live in a world where the default state is considered to be a human who can hear, this limits the freedoms of the deaf. What we can do, in such a world, is to try and accommodate these differences by placing requirements on society that allow these people to function as normally as possible. How you could possibly do this without a legal interface or some amount of legislative power is beyond me. I have no idea how this works in your Randian paradise. I don't think you do either, I don't think you know how people resolve disputes, ensure freedoms long term, and grow freedoms by tackling common problems in your ideal world. I think you stopped thinking at everyone freely associates and obeys the trader principle, I don't think you got your hands dirty by interfacing your abstraction with the real world.

Your world view has lead you to equate the approval of laws which require the construction of disabled access ramps with Naziism. — fdrake

you respond with:

It's all the same. There is no difference between them except time. When you start giving out privileges, eventually people will want and expect more until the government will have all the power. — AppLeo

It was at this point that Appleo explicitly endorsed the idea that requiring disabled access ramps is Naziism. So entrenched in his position while beginning to realise its sheer absurdity and underdevelopment, he decided that behaving as if disabled access ramps were Naziism was the optimal face saving play. Disabled access ramps are a slippery slope to Naziism.

Though I do commend you for your steadfastness, usually people with the same talking points as you don't actually bite the bullet when they call such societal adjustments Naziism; they backtrack because they know how fucking absurd it is, and how bad it looks to any onlooker for their position.

And immediately after you give us this gem:

You are equating having your legs blown off by a landmine with being socially awkward. The veteran can't magic their legs back on, they can try to organise politically to receive equal priority to people who can walk. They're going to be more of a man than you, they've been to war and not let it fuck them up for life, they want to make life better for veterans. — fdrake

you respond with:

What's the difference. Not being able to walk and not being able to socialize are both hard things to deal with. The question is, are you going to turn the government to give you special privileges or are you going to solve your own problems? — AppLeo

You actually believe that having your legs blown off in a minefield is equivalent to having mild social anxiety.* You actually believe that the poor veteran shouldn't attempt to lobby for the introduction of disabled access ramps - what, should they have just gone up to the owners in the shop they couldn't get inside and asked them to fit something? How would that solve the problem in general? It wouldn't! That's the point. The only way you're going to be able to solve this problem is through collective action, and it's easiest to achieve by influencing the creation of a law which binds the construction of buildings. I think you're starting to realise this though, since you say:

I thought you were talking about groups that weren't interested in individual rights, but propelling their own groups at the expense of everybody else. — AppLeo

you're starting to realise that people organise precisely to ensure individual rights; to remove limitations society itself places on them through its structure; not to ask to be brought above the people, but to take their place beside them.

*I don't mean to say your social anxiety is easy to deal with, that's a low blow. What I'm stressing is that the kind of options available to you to try and fix it just aren't available to the vet with no fucking legs - they need to address things at the level of building regulations, not at the level of themselves, they can't get their legs back. Though, I'm sure if they could get those cybernetic lower leg implants that are possible nowadays they would, but they're probably way too pricy to get esp. if you're out of work due to the disability of having no fucking legs. You can address social anxiety by doing normal people stuff, there are free counsellors online and so on, no amount of personal change will get the vets their legs back.

Well I agree with both the dog is in the box and the dog should be in the box. I think Ayn Rand is right, anyway — AppLeo

The dog being in the box would be that we're already in a state of unregulated capitalism. The dog should be in the box would be that we should be in a state of unregulated capitalism. You clearly believe that we are not in a state of unregulated capitalism, but you also believe that we should be in a state of unregulated capitalism. Should $\neq$ is

AHAHAHA

In a free country people have guns pointed to their heads. Unbelievable. — AppLeo

You actually introduced the metaphor in the thread.

The government is a giant gun — AppLeo

and there are governments in countries you consider free. This is very inconsistent.

With your logic, anyone would have a gun pointed to their head for anything. And then of course, everyone would have the ability to get government privileges because everyone's a poor little victim of their own lives.

Why do you believe that a law which requires disabled access ramps for building access if at all possible is a special privilege? It's actually a special privilege to enter the building without using a wheelchair - it is a capacity which some humans lack.

Though, I'm glad that you picked up on that I was trivialising the gun metaphor. It is very silly. But, there was a good reason for me applying it out of the context. The logic of the gun metaphor is that people are prohibited from doing things due to threat of force, this applied to black people who were caught disobeying white people being punished, gay people who were caught having sex and so on. I would prefer if the metaphor were more generalised, that a person has a gun to their head whenever the norms of the actions of others impinge upon their freedoms - just like when construction norms for buildings did not require disabled access ramps or elevators. These are all limitations on freedom that people deserve.

I thought you'd be down with things that improve the freedoms of individuals, but apparently you don't write as many blank cheques in this area as you say you do.

I don't know about you, but I want to live in a free country with productive, hard-working, independent, and responsible people. Not a mindless mob ruled by a Hitler or Stalin. — AppLeo

You know, at some point I'd have thought someone equating the claim that there should be laws which require buildings to have disabled access with being a Nazi or Stalinist was ridiculous. Unfortunately I've been having this kind of conversation for too long for me to skip a beat whenever someone does it. Though I will repeat this for special emphasis:

Your world view has lead you to equate the approval of laws which require the construction of disabled access ramps with Naziism.

What's wrong with profits? — AppLeo

Nothing has to be wrong with profits in general. What I'm against is profiting from slavery, because I think slavery is wrong. What unsettles me is that slavery, the slave trade as it was called, is consistent with unregulated capitalism. It's part of what makes me suspicious of unregulated capitalism.

Women's rights just want their lives to be easier than men's lives. Black lives matter just want their lives to be easier than white people's. Gays just want their lives to easier

All these groups perceive themselves as victims, but they're not they're just being victims by choice. — AppLeo

Would you extend this to the slaves? Who had literal guns and other weapons pointed to their heads. And if they disobeyed their masters they would be tortured, sometimes to death.

Were the black towns in the US after abolition victims of the KKK and other hate groups by choice?

Were the Jews victims of the Holocaust by choice?

Are veterans who have their legs blown off due to mines privileged whiners who want their life to be easier than others'?

Placing the group above the individual is good for the individual? Makes perfect sense. Wow, Ayn Rand is so dumb for point out the opposite. — AppLeo

The Haitian rebels did not want to be slaves. So they banded together so that no person would have to be a slave, fighting their masters. This improved the rights of individual slaves. The motive for banding together was so that no one had to be a slave - improving the lot of the individual. Please explain to me how this is placing the group above the individual, when its goal is literally the freedom of all individuals in the group.

The smallest group is the individual. So if you want to help minority groups, you did it with individual rights. When you support a group that consists of more than one individual, you are picking a winner group and shunning all the loser groups. — AppLeo

Actually it doesn't always pan out like this. If you pick the default, you often pick a winner by default. The default position in the time of slavery was more slavery, the default position before disabled access legislation was no disabled access, the default position for treating acute depression in women was confinement to an asylum. You pick a winner by picking the default. What collective action attempts to address is that a winner has already been picked, and it isn't fair. Life already shuns all the loser groups, that's why there are differential advantages and specific problems that groups organise to tackle.

I chose to learn to how to socialize rather whining and complaining about how no one around me cares to help me. — AppLeo

You are equating having your legs blown off by a landmine with being socially awkward. The veteran can't magic their legs back on, they can try to organise politically to receive equal priority to people who can walk. They're going to be more of a man than you, they've been to war and not let it fuck them up for life, they want to make life better for veterans.

They shouldn't concern groups. Only individuals. That's why I said there are only individuals because there are only individuals and to act in a way that doesn't is bad. — AppLeo

The two claims are inequivalent. "There are only individuals" vs "politics (in some vague sense) should concern only individuals.". This is the same difference as the difference between "the dog is in his box" and "the dog should be in his box" - see? Huge. Biiig difference. That you're not particularly attuned to the distinction between normative and descriptive claims isn't really your fault though, Rand herself notoriously has a deaf ear for it - google Ayn Rand 'is ought problem' and you'll find loads of literature. Some of it supportive of her, of course, so you can maybe learn your way out of this objection for the next time someone highlights it to you.

Being disabled shouldn't grant you extra privileges or handouts. Just like being gay, black, woman or a rich white man doesn't. I don't care who you are, you are treated equally under the law like everybody else. To say otherwise creates a sense of tribalism where everyone wants a piece of the government (the giant gun) to force people to obey to their standards. It is not empathy. It's using empathy to mask victimhood and then to use that victimhood as an excuse to use force against free people. I utterly disapprove. — AppLeo

You're being dense here. Not being able to get in the shop isn't a right or a privilege for the wheelchair user, what they actually want is equal standing with other people who can enter the shop. They want to remove an arbitrary limitation on their lives placed there due to planning oversights. They want to enter the shop. They can't. They need to buy shit. What to do? Maybe try to change it so that in the future people who need to use wheelchairs can access shops. Simple.

The same thing applies to your gay rights example, collectively organising to exert political pressure is how they got their rights. These are rights for individuals, the collective organisation concerned obtaining and then ensuring the rights of gay individuals.

The wheelchair users and the gay people already have a giant gun pointed at their heads all the time, it's called being a wheelchair user in a world designed for walkers or a gay in a world designed for straights. They're forced to act in ways healthy/straight people don't, and can't act in ways healthy/straight people do. What they want is to be able to go in the shops or have civil partnerships (for example). How should they go about getting it?

She means unregulated in the sense that individuals are free to make whatever transactions they want to make. This doesn't mean that people are allowed to force people to be slaves. If that were the case, people wouldn't be free to make the transactions they wanted. — AppLeo

It's irrational to want to have a slave. You want people to be free and prosperous because their freedom benefits you. — AppLeo

Not as much as having a free worker, sex slave, and tradeable asset. If all you care about is your profits, you don't give a damn... What world are we talking about again?

We're not talking about a world that resembles Ayn Rand's fantasy claptrap at all. Whether some Russian bint threw a book at the slave owner and the slave has no fucking relevance here. Nothing in this entire fantasy of how things should be is telling us anything about the real world. As soon as you switched to how the world should be, you switched to a realm of your imagination. Now the world is being measured by how it fails to live up to your imagination, and phenomena within it are being predicted with respect to deviations from your imaginary fantasy land.

You suggest organisation along group lines is bad because it's not individual, but it demonstrably advances individual freedoms and can bring a more just, equitable and free world. You switched the discussion explicitly to a normative one, how things should be, then gave this amateur hour horse shit to justify it as a principle:

Which basically means, if you take a group of people like LGBT. You can break that group up into two groups. And break those two groups in two 4 groups. And then 8. Until you're left with every gay person standing as an island. If you want to help gay people, you help them according to individual rights. This is fair and just because everybody else from every other group, even groups that have nothing to do with gay people, is an individual, so you'll also be helping them, the gays, and basically everybody as a whole by standing for individual rights. — AppLeo

So, right, I take 8 individuals, and they pair themselves in groups of 2 voluntarily, then the pairs pair, giving groups of 4, then the groups of 4 pair and look! We have constructed the number 8 out of 8 copies of the number one! Amazing. Yes. The ability to group a collection of people together into different sub groups which sum to the original number is totally something related to how politics works.

You're doing this instead of focussing on the easy reality that people organise along group lines to address common problems, that this organisation is done to attempt to give the individuals in the organisations a life without those problems, and that this is how political actions are taken.

You're doing this instead of focussing on how people are effected by stuff, like the wheelchair user and the stairs, and forming groups based on the stuff people have to suffer.

You remember in my first post to you when I said:

The weakest point of Randian political theory in my view is precisely that it explains political and economic phenomena with reference to deficiencies from an ideal state, an unregulated free market system, which would emerge save the interventions of corrupt government officials. — fdrake

when you noticed that Rand's account of unregulated capitalism and the ideas about how things work are easily refuted; descriptive claims about how reality is; you shifted ground to defend her ideas as how reality should work; normative claims about how reality should be. Don't treat what should be as what is, nor shift between these two registers, because this means you're changing the point of the conversation.

That's not identity politics. Fighting for your freedom is something that all individuals agree on. — AppLeo

I forgot to respond to this bit, sorry. People want themselves to be free, people don't always want other people to be free. We don't agree on this. Slaves, prisons etc. We don't want the prisoners to fight for their freedom from prison.

I’m not deluded at all. And assuming that I’m already deluded makes me think that whatever I have to say will fall flat to you. Why listen to a deluded person? And the idea of there only being individuals is not silly at all. It’s an idea that should be taken seriously. — AppLeo

I take silly ideas seriously all the time. That's part of why I enjoy philosophy and learning more generally. I would not have attempted to rebut your ideas if I didn't think they were worth rebutting - see the difference in approaches in my response to your different claims: On the one hand I put a bit of effort in explaining why I thought you were wrong during most of the post, but just linked you to a introduction to climate change video course for your climate change denial. I was hoping that since you've enjoyed studying objectivism so much you'd spend some time, maybe in the future, actually looking into climate change analysis from reputable/well cited climatological sources.

Well I understand that there are groups of people and that we have words for these groups of people, and that laws and treaties depend on acknowledging people in groups rather than as individuals. — AppLeo

Good, then you believe in the constitutive entities of politics, and the claim 'There are only individuals' is reduced to nothing more than a metaphor.

What I don’t like about it is that these groups have taken on identities of their own when they shouldn’t have. There is no collective stomach. There is no collective mind. The groups that form together form based on individuals and their values. But even if you have a group of like-minded individuals in a group, all those individual minds are still very very different with their own goals and are their own person.

And the people that over identify with their groups are essentially sacrificing their own individuality and livelihood for a group or cause that will only fulfill the one interest they have that even made them join or be apart of the group in the first place. A black person’s blackness is one small and very pointless detail to everything about them. A poor person’s bank account is a small and pointless detail compared to everything else that makes them an individual.

Firstly, differences of opinion over what to do, or differences in taste, still do usually persist within political subjects. For example, citizens of Britain that own passports and have no criminal record have (at least nominally) the same rights for travel within Europe. That there is no collective mind for all of Britain does absolutely nothing to change how the political subjectivity works. That pseudoproblems such as 'it is problematic that political actions are done by and effect groups' and 'since individuals are not a hive mind politics done along group lines is problematic' arise is a function of your flawed framing rather than a problem of the world.

Note that in spite of this you are actually conceding the point that political actions are done by and effect groups. Your claim has now morphed into the claim that political actions should not concern groups, something much different from the original descriptive claim of 'there are only individuals'.

Caring about these groups It creates identity politics. It’s not about your responsibility, your work ethic, or who you are as a person. It’s about what group you belong to and who’s group beats the others. Your group defines your identity, not you. If you are in a group that is perceived to be good, you are a good person regardless if you are actually good. If you are in a group that is perceived as bad, you are a bad person regardless if you are actually bad. And the worst part about it is when the government sees these groups as actual entities with rights of their own – that groups of people can have rights that trump individual rights… You will get an unfair and unjustly system. The government can pick winners and losers among individuals depending on which individuals are in which groups. The government is a giant gun. And every group wants control over it. Republicans, democrats, liberals, conservatives, blacks, whites, gays, straights, christians, atheists, environmentalists, women, men, the rich, the poor, employer, employee.. it goes on and on…. All these groups are minorities in a sense.

This is quite strange. You seem to be under the impression, at least for the purposes of your post, that people choose their political subjectivity like they choose what they have for dinner or what bars they go to. A wheelchair bound person does not identify as a wheelchair bound person because they want to park closer to the supermarket, they get to park closer to the supermarket because they need a wheelchair. Now imagine that the supermarket has steep steps, and you have one supermarket within range of access. Now this poor sod has to get someone else to do their shopping for them. The thing about this that induces a group identity is that for some people, they need to enter the shop but can't because of the steps. Then it makes sense to organise along those lines to exert political or economic pressure to get such things changed, for the betterment of your group. This 'betters the group' because they were already aligned by an identity that was ascribed to them usually without their volition.

The same thing applies to what country you're born in, whether you're LGBT, whether you're white or black or hispanic or Asian, male, female, trans, whether you're a steel worker or a telephone salesperson and so on. People do not choose the effects of these things, it just so happens that they have common life problems to organise around. The same thing even applies for big businesses, they want to make lots of money and so organise around issues that either prevent them from losing money or allow them to get more money. It's really simple: alliances of people form from shared problems, not through arbitrary associations. Groups of people form alliances to attempt to solve common problems for the individuals which constitute them. You get all of this out of the effective application of self interest, you don't even have to be a nice person, just pragmatic.

On top of these coalitions of common problems, we also can also add empathy and solidarity; an attempt to aid those who suffer from problems we might not for the betterment of everyone.

So, why shouldn't individuals organise to tackle problems common to them?

Blaming capitalism or finding faults in capitalism for people having slaves is ridiculous. — AppLeo

So you're quite happy to admit that slavery is perfectly consistent with unregulated capitalism? Great! We have some common ground.

First of all, objectivists do not advocate for anarchy, they advocate for a limited government that protects individual rights. Second, Rand's philosophy preaches rational self-interest. A slave owner isn’t rationally self-interested, and neither is a slave. Why would Ayn Rand tell slaves to continue being slaves? Her entire message was to fight for your life and happiness and to treat others as desiring their own life and happiness as well. — AppLeo

Please note that I never said Ayn Rand was an anarchist, and also note that in the original post you quoted I said Ayn Rand reserves a place for government in her political theory; it protects the sanctity of freeform contracts, which is taken to be as equivalent to protecting the trader principle.

The issue isn't whether Ayn Rand would tell the slaves to be slaves; for all her failings she did have some human dignity, the issue is whether unregulated capitalism is consistent with its idealisation. Since we already have that unregulated capitalism is consistent with slavery, surely you must see that it isn't.

I do however find this a bit sickening:

A slave owner isn’t rationally self-interested, and neither is a slave.

The slave owner doesn't give a damn about the slave, they're an asset. All that matters is maximising the return from them; perfectly calculated, just immoral. If the slave doesn't want to be tortured to death, if they want to survive (remember Rand's ethics has survival as a cornerstone) they will usually benefit most from behaving like a slave.

Unless of course they banded together to break their chains, but we wouldn't want any identity politics coming in here would we.

This isn't from the paper, it uses mostly external ideas to what's been presented in the paper so far.


If I can make an analogy, imagine yourself as a point rolling down a hill. When the hill changes shape, so does your acceleration. A 'straight line' on the surface of the hill isn't a 'straight line' in the embedding space. To be sure, if we have that the manifold is locally flat, 'straight lines' of tiny extent on the manifold will look like straight lines in a Euclidean embedding space. But they don't actually have to be straight (in the sense of the embedding space) because of the possibility of curvature.

Though, I think some of your intuition about the derivative is correct. Imagine if we place two points A, B on the hill really close together and draw a smooth path between them, like a piece of string bound tightly to the surface.

A-----B

Imagine that we parametrise this path so that s=0 at point A and s=k at point B - this uniquely specifies every point on the path. The average change in the function (per unit length on the path) that this path corresponds to would be:

$\frac{f(k)-f(0)}{k-0}=\frac{k}{k}=1$

so the second derivative would be 0, since the first would be a constant (except if the points A and B coincided). But what does this calculation actually mean? All the function f does is take the arclength along the curve between A and B and spit it back out. IE f(s)=s, with f(A)=0 and f(B)=k. The rate of change of the arclength with respect to itself is always 1. More generally, the rate of change of any function with respect to itself is always 1.

Another thing to note is that the tangent vector to a manifold at a point - a line that is visualised in the embedding space - agrees with the first derivative of the manifold in its direction, but the second derivative of the tangent vector is 0 - whereas the manifold itself 'curves away' from it, showing the presence of a nonzero second derivative of the manifold (with respect to the coordinate system we're using) - curvature.

The situation that we usually have on the manifold is more like the form:

$x(s),y(s)$

where there is more than one coordinate required in the specification of the path, if it's a one dimensional curve we also have that y(s) = f(x(s)). This means that we can consider how the curve bends over the coordinate system x(s), y(s). The curve itself, like the length of wiring, can be straightened out, so the curvature it has isn't intrinsic to its shape. This ability to straighten out something precisely means that there is a smooth transformation from distances within the shape to distances like they behave in a Euclidean (flat) space. You can also visualise this as the curvature of spaces rendering the linear approximation to their surface (like the tangent plane or vector) worse and worse when you go away from the point of approximation.

So it might be that we can bend the wire, but we don't introduce any irremovable/intrinsic curvature. What intrinsic curvature actually measures is how movements of oriented objects constrained within the surface change the orientation of those objects when moving around closed paths - paths with the same start and end point.

This even applies to forming a circle out of it, circle boundaries don't have intrinsic curvature whereas the surfaces of spheres do! A tangent vector to a circle at one point, transported around the circle while remaining tangent, is still in the same direction as it was when you started! You can check this by rotating, say, your phone around the top of a coffee mug. So let's talk about the surface of a sphere.

Another task we might imagine is taking a perfectly taut bit of copper wiring and trying to wrap it onto the surface of a sphere. You can press it onto a point, and it'll touch the point. But, if the sphere is absolutely huge relative to the size of the perfectly taut bit of string, the taut bit of wire might resemble the surface of the sphere very well, it wraps away slowly from the wire with respect to spatial changes (points flowing away from the wire). If the sphere is tiny compared to the bit of string, it just glances off it and the sphere quickly wraps away from it. The acceleration with which the sphere wraps away from the taut bit of string is its curvature (no acceleration = flat!) - in the small sphere case you'd have to bend the wire a lot to fit the surface, in the large sphere case you'd have to bend the wire a tiny amount to fit the surface. The curvature of the surface of a sphere is not something that can be removed by these smooth transformations.

The presence of curvature, like on the surface of a sphere, does something pretty strange to straight lines from the embedding space. If you take the iron wire, press it onto a point on the surface of the sphere, and you hold it onto the sphere, what happens when you take the iron wire, still taut and straight, from around a closed path on the sphere? This requirement that it's still tight equates to that the iron wire must be tangent to the sphere. If you move it along a path, where the start and end points of the path are the same (say carving out a quarter of the sphere). By the time it's back to its start, the direction you're holding the iron wire in actually changes. The presence of these changes signals intrinsic curvature.

You can check this with your fist and your mobile phone. Clench your fist with your thumb toward you, take your phone and press it hard onto the knuckle of your index finger. Push the phone away from your body along the line of your knuckles. When it gets past your final knuckle, still keeping it pressed onto your hand, force the phone towards you from the first finger joint on your pinky to the first finger joint on your index finger. Finally, bring it back from the first joint on your index finger to the base of your thumb, and back up again to the first knuckle. You should see that the phone has inverted. Contrast this to the mug and phone thing from earlier.

Have you watched Adam Curtis' mini documentary Nonlinear Warfare and longer one Hypernormalisation (second one has NSFW images)?

This might be surprising but I didn't actually have Trump in mind while writing the OP. The alt right views him as a useful idiot, but his rhetoric also pandered to them. I more had in mind the political discourse that arises from or is rooted in the various chan boards. Not to say Trump's rhetoric doesn't have a similar effect of normalising cruelty and (probably intentionally) dominating the news cycles, but pulling this kind of crap as an explicit political strategy of (genuinely) fascist and prejudicial politics predates him.

They're more relaxing, yeah.



Scientists hate him, see this one trick that makes reactionaries apologise!

Some of you probably thought I was paranoid writing this, but I found this which is a much better example of the things I was talking about, it's an explicit style guide/mission statement from a popular white supremacist website.

'Packing our message inside of already existing cultural memes and humour can be viewed as a delivery method. Something like adding cherry flavour to children's medicine'.

Alright, well first of all I don't see myself like Galt at all. I am not a hero versus the world. I am not gifted, nor rich, nor an engineer. I'm studying software engineering, so kind of similar actually... But I do plan on being self-made and independent and being confident for it. No one benefits from society as a common good. There is no society or common good. There are only individuals. And your life is determined by your own efforts and choices (if you live in a free country of course). — AppLeo

Well I'm glad you're not that deluded, and being determined to succeed is usually a good thing, so grats. This idea that there are only individuals is rather silly though.

If we take the statement that there are only individuals literally, this would mean that no aggregates of individuals exist - which is quickly undermined by group nouns like people, sheep and so on. So we definitely have the capacity to refer to groups, and it's useful to be able to do so.

If you strengthen the idea to a more metaphysical principle, that there are no groups except for the individuals which constitute them, this is true in a literal extensional sense; the posters on this philosophy forum now, say, applies to each and every poster, and without each poster the collective would be different. What this highlights though is that aggregates don't have to markedly change their properties with the addition or subtraction of members.

Take another example of a group, a football team, what would the claim that there are no groups except for the individuals which constitute them say about the team? Well, it could only say that the football team is equivalent to its constitutive members. This misses a lot though, because changing members can change team dynamics. What this highlights is that aggregates can markedly change with the addition or subtraction of members, and moreover that the playstyle of the team is a property of the aggregate, the group of players, and not a property of the individuals. IE, there are groups, and they can have distinct properties or even types of properties from their members.

A more mathematical example, certain concepts like median wage, GDP, and so on; statistical properties; apply first and foremost to aggregates/populations.

So, what remains of the concept that there are no groups except for the individuals that constitute them? What about these things that look like groups of people, say innovative capitalists, captains of industry, the poor, charity workers and so on. In what sense are they not groups? Why of course because the group doesn't exist, only the individuals do!

Except this misses a lot, a lot of our social reality is founded on inter group relations and laws which concern groups. EG, treaties between countries, affirmative action in hiring. Current legal systems adjoined to capitalism actually treat things on the level of the aggregate - we can have interventions to bring needle exchanges into heroin addled areas, increase literacy in poor areas, confine immigrant children to cages and so on. This is to say nothing of corporate personhood, in which a corporation itself has certain rights and responsibilities similar to but distinct from its constituent members. Even the idea of regulatory capture which Randians are so against still requires two groups - corrupt capitalists and corrupt government workers - to get going.

It's debatable if what they're doing is actually in their self-interest and whether or not they're doing a good things. If the spice manufacturer replaces the content of his spices to cheap salt, he's being a liar. If his customers find out, he'll probably go out of business. Or maybe it simply doesn't matter and the customers won't care. — AppLeo

I certainly trust that these people know how to maximise their profits and run their businesses over you, after all, you are not a skilled captain of industry, you're a parasite like I am. The oil company and the spice company I mentioned still function with impunity by the way. They show no signs of going out of business, even though it absolutely matters that an oil company acknowledges that climate change is likely to raise sea levels and seeks to cover the cost to minimise their exposure on the back of the taxpayer in one breath, then propagandises against climate change's existence in another. It also matters that when people buy something, they know what they're getting and what's involved in it. Let no one ever squander the opportunities of deceit.

Lobbying to a government is not a thing in the Randian world. The government would have no power. And this example doesn't work anyway because climate change is a bunch of nonsense to take down capitalism. — AppLeo

Oh dear oh dear. Please watch this series through.

The only way to protect the people is to limit the government. — AppLeo

Firstly, please note that nowhere in my previous post did I speak about giving governments more power, or less power, what I specifically stated was that collective bargaining strategies are required to make governments (and other institutions like corporations) serve the interests of the people. A people without a government that represents their interests are vulnerable to opportunism on the part of corporations and special interest groups; just as with regulatory capture and tax evasion. You want to benefit from the commons of education, family rearing and health care? Pay those taxes you greedy corporate cunt.

Well yes, government intervention always leads to socialism and crony capitalism. That's not what we want. But in the "Randian" world, the state doesn't represent the powerful capitalists. That couldn't be further from the truth... The state represents everyone's individual rights. Rand was an individualist. Everyone matters individually, not what group they belong to. In Atlas Shrugged, the rich were not being treated as individuals with their own rights. They had their rights stolen by everybody else and that is why they fled. — AppLeo

Except when it leads to fascism, public revolt or increased prosperity. And yes, I know Rand is an individualist, this is precisely why I said the only possible legitimate function Rand imagines for a government is to enforce the contracts made between individuals, and even then preferably not have that power granted to a state.

Anyway, let's take an example of people who actually did have their rights stolen - slaves in the Atlantic slave trade. The trade of slaves wasn't regulated by a government, it was rich colonialists with guns stealing people and rich collaborators corralling candidate slaves, the slaves were traded for profit and were extremely efficient in producing returns for their owners. Laws allowed slavery at the time of course, and when the laws changed through the work of rebels in the colonies and the work of humanitarians at home, history rejoiced. This is a good example of a government changing their tune for justice, and is strikingly opposed to the unfettered capitalism of the slave trade.

You will probably say; a true Randian doesn't believe in the slave trade! Yes, maybe so, but a true Randian wouldn't believe in legislation to end it either - a wrong government cannot legislate rightly, so to speak. Will you join me, against Rand, and say that sometimes governments can do right? And sometimes unfettered capitalism can be systemically wrong?

Want an example of capitalism with minimal to no regulation? 19th century America. Largest increase in quality of life that ever happened. True economic freedom. There were no wars, the government wasn't in the way, people were free to buy and sell what they wanted. That is what America needs to return to. Because the government respected the individual. It didn't pick winners and losers like it does today in our economy. — AppLeo

Lastly, your example of 19th century America as a time of prosperity and opportunity for all is incredibly misguided and historically inaccurate. Most factory workers did not make enough to live on, worked impossibly long hours, children worked in the factories, and the working conditions lead to long term sickness and death - with no sick pay or medical insurance of course. It was only under pressure from disgruntled factory workers that eventually child labour laws were put in place, with similar humanitarian developments on workplace safety and an attempt to provide a living wage following from later efforts of unified workers.

Don't complain about the length of my response, you've read Atlas Shrugged for Christ's sake.

I'm looking forward to your reply, then. :grin:

Ayn Rand isn't particularly popular among the regular posters. It's very common that lots of people engage a Randian at once.

In other news, this.

That's fine. You can leave me in my ignorance, it won't hurt.

They were condescending first. You were condescending first. If I'm already summed up according to my age and my favorite philosopher, especially by people who think they understand Ayn Rand when they clearly do not... And on top of it say that I shouldn't be taken seriously. They don't deserve the respect of me getting to know them. — AppLeo

Yeah, some of it really isn't your fault. If you go back through the forum over the years, we get about a few Randians per year. They usually come, having solved all the problems of philosophy, preaching the virtues of freedom and the market (is there really any difference?) and of non-aggression (people should relate to each other as individuals and form contracts thereby, rather than having them interposed by a government which has monopoly over force). They also usually come with the attitude that everyone's an idiot.

I do feel genuinely surprised that people identify with Galt more than the dregs of society though, considering that seeing yourself as a hero like Galt or the captains of industry and innovation should require feeling like you have a lot of power and influence and that you're a self made person. It's frustrating to me to see people who have the freedom and opportunity to study, typically students at universities, biting the hand that feeds them; as if they were not benefitting from what society (at least attempts to treat) as a common good.

Of course the usual Randian rejoinder is that all the ills of the university system, like our current debt peonage, is as a result of government intervention ensuring education monopolies or power concentration, so they start charging through the roof for a premium good. This follows the general pattern of economic power concentration being equated to 'crony capitalism' - which is where capitalists are allowed regulatory capture by governments. In the ideal Randian world, such regulatory capture would not be possible as it requires a state to represent the interests of powerful capitalists rather than the interests of general people (which, apparently, is always aggressive and thus immoral).

However, Rand does not draw much of a distinction between the interests of powerful capitalists and the interests of general people. Her ethics focuses on heroic individuals associating freely with each other, and a state is ethical just when it enforces individual contracts between them - if the state oversteps those bounds it is forcing people to do things, which goes against a non-aggression principle that's central to Randian ethics. What this misses is that political negotiation doesn't actually occur in a sphere of individuals freely associating with each other, there are power differentials everywhere, and what's needed to get a good deal in the presence of a big power differential is collective bargaining strategies; an inverse of regulatory capture where the government is forced to serve the interest of its people.

The weakest point of Randian political theory in my view is precisely that it explains political and economic phenomena with reference to deficiencies from an ideal state, an unregulated free market system, which would emerge save the interventions of corrupt government officials. A not-so minor point here is that the capitalists are not being corrupt by attempting regulatory capture, propagandising and so on, they're actually acting in their own best interests. They are acting in their own best interests when say an oil company propagandises against the existence of climate change while lobbying government for construction of levees to protect low altitude oil fields, or when a spice manufacturer does something more minor by replacing content of spices at supermarkets with cheaply available salt, or when leveraging a rent gap and making long term denizens homeless. They were acting in their own best interests when opposing the creation of the NHS in Britain.

Really what this shows is a big misalignment between the short term profit motive that makes good business and the long term welfare motive that makes good politics. There's no special emphasis in Randian theory on protecting the commons from powerful corporate interests or the requirements of collective bargaining strategies for those subject to power differentials to get a fair deal; it's a theory tailored to the short-term interest of capitalists and shareholders rather than the long-term interest of humanity and stakeholders. The world it speaks about doesn't exist, and the closest historical analogues we have to capitalism without regulation took a huge toll on the people and, eventually, the planet.

Why are the people around me so stupid? — AppLeo

Maybe because you've not let yourself get to know them better. Acting from such a position of condescension confines others to fit your already established opinions. Which presumably means you = smart and we = stupid.

I'm always surprised by how many people read Ayn Rand and imagine themselves as similar to her hero, rather than the societal dregs that depend upon heroes' invention. A quick test to see whether you are a hero or a parasite, do you have a job? If so, go to next question, if not, you are a parasite. Would the economy at large be effected by you leaving your job? If not, you are a parasite, if so - congratulations, you might be a Randian ethical hero! Praise Galt!

I'm finding the same thing, what I'm benefitting most from I think is trying to integrate the imaginative background from the first section with the mechanical mathsy bits in the second. I imagine philosophically the first section and the final section would do, so if others feel like this isn't progressing quick enough to the philosophical juicy bits I could summarise the maths so far and then we could move on to §3.

It's really hard going! Spending hours on paragraphs means you know it's hard.

I wanna highlight something in this post because it's cool, and I just grokked a connection. Jon suggested writing the general quadratic equation (without linear terms) as:

$\sum_{i} p_{ii} x_i ^2 + \sum_{i}x_i \sum_{i\neq j} p_{ij} x_{j}=xPx^T$

see here for a worked example. The usual way we assign a size, called a norm, to a vector is through Pythagoras' theorem: the distance of the hypotenuse (squared) $s^2$ is the sum of all the squared components of displacements $\sum x_i ^2$, so we write

$s^2=\sum x_i ^2$

this sum is equal to $x x^{T}$, and is the squared (Euclidean) norm of $x$.

now imagine that instead of it just being orthogonal directions and terms involving $x_i ^2$ alone, we instead replace the expression for $s^2$ by an arbitrary quadratic in the same variables:

$s^2=\sum_{i} p_{ii} x_i ^2 + \sum_{i}x_i \sum_{i\neq j} p_{ij} x_{j}=xPx^T$

this, then, is the condition Riemann plays with when he says:

This differential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coefficients are continuous functions of the quantities x

the Taylor series analogy of transforming coordinates x to coordinates y before links in at this point, now including the quadratic terms, then looks something like:

$y=f(x_0)+Ax+xPx^T+O(|x|^3)$

if we can zoom in close enough - shrinking the norm $|x|$ towards zero, the terms above the quadratic term $xPx^T$, denoted O(|x|^3) will be the first to go, giving the local approximation of the coordinate transformation as:

$y=f(x_0)+Ax+xPx^T$

we can think of the successive terms as 'wrapping' the coordinate system of $y$ onto the coordinate system $x$ in the region around $x_0$ with greater degrees of flexibility, more flexible means more adaptive to the local topography and thus more accurate. The approximation to the transformation around $x_0$ is, in turn, just the raw function evaluation (where $f$ is the function mapping $x$-space to $y$space) at the point, then 'a bit further out' the matrix $A$ allows correction for linear variations around the point, which we can think of as fitting a tangent plane to the manifold at the point $x_0$ (this information is encoded in the first derivatives of $f$ with respect to each coordinate), then after we've got as far as the tangent plane will work, we start seeing the influences of the curvature of the manifold crop up - we need to bend the y coordinate system onto the x one!

Now, what's the connection between this bending and the norm $s^2$? Riemann is noting that any localised information about the curvature is precisely given by the behaviour of a quadratic $xPx^T$ function around the point; so the matrix $P$ gives precisely how to translate the lengths of lines in the $x$ coordinate system to those in the $y$ coordinate system. So the bending of a line (1 direction in a coordinate system/a simply extended manifoldness) can be thought of as its wrapping onto the manifold over a region. Comparing how they both wrap into each other lets us derive information about the local curvature.

So, the quadratic terms are simultaneously 'corrections' in translating one coordinate system to another over a slightly larger, though still infinitesimally small, neighbourhood; and they are also a transformation of distance notions between the two coordinate descriptions localised to a point on the manifold. This is why curvature changes the behaviour of 'straight lines' within the manifold, the curvature of the manifold encoded in $P$ tells us what 'additional ingredients' it takes to get from the flat space metric $xx^T$ to the curved space metric $xPx^T$. Just as $A$ translates straight bits to straight bits, $P$ (locally) translates curved bits to curved bits and thus encodes information about the curvature the coordinate systems both represent.

But yes, I think this gives the correct expression, and you can force the matrix to be either a lower or upper triangle at your whim. Engineer's proof for n=2:

$(x_1,x_2)\begin{bmatrix}p_{11} & p_{12}\\p_{21} & p_{22}\end{bmatrix}(x_1,x_2)^T$
equals
$\begin{bmatrix}p_{11} x_1 + p_{21} x_2 & p_{12} x_1 + p_22 x_2\end{bmatrix}(x_1,x_2)^T$
equals
$p_{11} x_1^2 + p_{21} x_2 x_1 + p_{12} x_1 x_2 + p_{22} x_2 ^2$

we can zero out $p_{12}$ without losing any expressive power, as you said (and as I abused by letting myself double count). Which means we have:

$p_{11} x_1^2 + p_{21} x_2 x_1 + p_{22} x_2 ^2$

an arbitrary quadratic in two variables $x_1,x_2$

Edit: For some reason I thought we were discussing the matrix A above rather than the curvature, apologies for confusions. I've changed this comment to describe the curvature rather than the translation of the linear bits of the coordinate systems to each other.

The P there generalises the relationship away from flat space, for an arbitrary invertible P this would present a quadratic with all the cross terms like $x_1 x_2$ thrown in - which I'm thinking of as the two directions 'interacting' in the surface, so that change in the local topography can't be neatly partitioned into independent directions. For the case where P is the identity matrix (the 1 in matrix algebra, multiplying by it changes nothing), we end up with $xx^t$, which through the usual inner product/norm rules is just $\sum x_i ^2$, if we treat this as an infinitesimal displacement we end up with Riemann's characterisation of flat space $ds^2 = \sum dx_i ^2$.

Yeah. I screwed up the formulas a few times and have been editing them since. That post's very much a work in progress. It should read something like:

$\sum_{i}a_i x_i+ \sum_{i} b_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} c_{ij} x_{j}$

the reason I was struggling with it was because I wanted to present the overall expression as a sum of linear and quadratic terms, with one sum/sigma-notation for each group. I also glossed over the double counting because if we double count a term its coefficient will be the sum of two others. Really all these issues go away if I gave up on trying to represent it as two sums. I could achieve the same effect by just grouping the sum of sums using brackets!

Edit: I've updated the previous post to use the three sums and provided a comment below its first instance to highlight which parts are which.

Finishing §1 in section 2, Riemann starts to introduce the idea of changing coordinate systems - more specifically, using different coordinate systems to express the same local information.

Such an expression (a quadratic in differentials) we can transform into another similar one if we substitute for the n independent variables functions of n new independent variables. In this way, however, we cannot transform any expression into any other; since the expression contains ½ n (n + 1) coefficients which are arbitrary functions of the independent variables; now by the introduction of new variables we can only satisfy n conditions, and therefore make no more than n of the coefficients equal to given quantities. The remaining ½ n (n - 1) are then entirely determined by the nature of the continuum to be represented, and consequently ½ n (n - 1) functions of positions are required for the determination of its measure-relations.

The numbers seem like they're coming from mid air, but they actually just come from the combinatoric structure of quadratic equations in n variables. If we have 2 variables x and y, there are 3 possible quadratic terms. x^2, y^2, xy. This is 2*3/2, ie 0.5 n(n+1) with n=2. If we have 3 variables x y z, there are 6 possible quadratic terms, x^2, y^2, z^2, xy, xz, yz, and so on. What this is saying is if we take some set of coordinates:

$x_1, x_2, ... , x_n$

and consider the possible quadratic functions possible from this set:

$\sum_{i}a_i x_i \sum_{i} b_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} c_{ij} x_{j}$

where the things besides x's and y's are just numbers. We have more required coefficients than just a simple linear relation would require for a new coordinate system $y_1,...,y_n$. In particular, comparing coefficients between the equation of the two quadratics:

$\sum_{i}a_i x_i \sum_{i} b_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} c_{ij} x_{j}=\sum_{i}d_i y_i \sum_{i} e_i x_i ^2 + \sum_{i}x_i \sum_{i\neq j} f_{ij} x_{j}$

the first sum deals with the linear terms, the second two deal with all the quadratic terms. The terms without x or y in are just constants. Comparing coefficients of the raw coordinates here only fixes the linear terms. The remaining 0.5(n+1)n-n=0.5n(n-1) terms, then, must be determined entirely from the local structure of the manifold as given by continuous functions of position. The situation here is analogous to looking at a Taylor expansion of the y coordinates in terms of the x coordinates:

$y=f(x_0)+Ax+O(\text{quadratic terms in the x variables})$

this higher dimensionality - needing more coefficients to specify - attained by the curvature is why curvature is associated with a tensor! It needs more information than the linear terms and their associated square matrix to specify. .

the x and y are vectors (the complete coordinate specification for the same point in two different systems), the capital A denotes an invertible matrix, and the quadratic terms determine the curvature. Emphasising this point, the matrix A only codifies the linear relation of the coordinate systems y and x to each other - the first derivatives/tangent vectors -, the remaining parts 'spread out' along local topography of the manifold and encode its curvature . When the space is flat everywhere, the quadratic terms vanish - meaning flat spaces have no intrinsic curvature. When the curvature vanishes at a point, the neighbourhood around that point is locally flat. The remaining part of §1 is a preparatory remark to set up for Riemann's study of the more general spaces with constant (nonzero) curvature in §2.

Manifoldnesses in which, as in the Plane and in Space, the line-element may be reduced to the form \sqrt{ \sum dx^2 }, are therefore only a particular case of the manifoldnesses to be here investigated; they require a special name, and therefore these manifoldnesses in which the square of the line-element may be expressed as the sum of the squares of complete differentials I will call flat. In order now to review the true varieties of all the continua which may be represented in the assumed form, it is necessary to get rid of difficulties arising from the mode of representation, which is accomplished by choosing the variables in accordance with a certain principle.

A bad OP which has already generated a decent conversation is more likely to remain. If we catch a bad OP before any discussion starts on it, we're more likely to trim it; either deleting the thread or closing the discussion.

That this happens more frequently to new posters is less about favouritism and more about new posters being less experienced in writing OPs. Posters that have stuck around for awhile have more experience writing OPs, and indeed regular posts, that conform to site guidelines sufficiently well to not require moderation.

Which is not to say we all act as a monolith, we all have different standards. EG, I imagine I'm more lax on moderating content than some others, but I'm more likely to think punishing aggressive or otherwise unpleasant behaviour is required.

I don't think there's much extra philosophy in that bit from what we've already covered, it's just detail on how to construct localised distance measures.

Starting §1 in section 2. It's very likely that I have some misconceptions and falsehoods in my presentation since it's outside of my comfort zone. So take what I say with a pinch of salt.

§ 1. Measure-determinations require that quantity should be independent of position, which may happen in various ways. The hypothesis which first presents itself, and which I shall here develop, is that according to which the length of lines is independent of their position, and consequently every line is measurable by means of every other.

Riemann is constraining his discussion to metrics, means of measuring distances in continuous manifoldnesses, which ascribe distances independent of the location on the manifoldness. Note that this is a way of assigning a notion of size to a notion of geometry, rather than measuring a specific shape. This notion is what sets up the meaning of length in a geometry, rather than an instance of measuring any particular distance within it. To be sure, objects (sub-manifoldnesses, neighbhourhoods etc) will have their sizes expressible through this notion of size, but the notion of size itself is a characteriser of the geometry rather than of any particular shape.

When you say the length of lines is independent of their position, what this means is that the distance notion applies the same everywhere in the space - there are no partitions acting on the size notion that create regions of distinct size ascriptions. To make this clear, consider two notions of interpoint distances in our usual 1 dimensional Cartesian coordinates, the real line:

$(A): d(x,y)=\sqrt{(x-y)^2}$
the usual distance notion
and:
$(B): d_2 (x,y)=\begin{cases} 0 \leftrightarrow x^2+y^2\lt1 \\ d_2 (x,y)=d(x,y) \leftrightarrow x^2+y^2 \geq 1 \end{cases}$

(A) computes the distance between the number 2 and the number 1, d(2,1) by sqrt (2-1)^2 = sqrt(1)=1, which is the usual distance between the numbers, and behaves exactly the same over the entire real line. (B) computes distances as 0 if x^2+y^2<1, and computes them exactly as in (A) if x^2 + y^2 is greater than or equal to 1. The picture here is that if we pick two numbers x,y that give a coordinate within the unit circle centred at the origin in the plane, the distance between them is 0, if we pick two numbers that give a coordinate outside of the unit circle, the distance between them is the usual distance on the real line. (A) is a metric in which the size of a line is independent of the position, (B) is a metric in which the size of a line is dependent upon the position..

However, the distinction between this 'global sense' of the metric is that (A) operates on the entire embedding space whereas what Riemann's after is a localised version. In order to set up this localised version, however, we still need to have a localised coordinate system (n-ply extended magnitude) of appropriate dimension for the manifold (of n dimensions).

Position-fixing being reduced to quantity-fixings, and the position of a point in the n-dimensioned manifoldness being consequently expressed by means of n variables x1, x2, x3,..., xn, the determination of a line comes to the giving of these quantities as functions of one variable.

The idea here is that if we take a collection of coordinates $x_1 (p), x_2 (p), ... , x_n (p)$, we determine a line (1 dimensional manifold) on the overall manifoldness by making all the coordinates a function of a single variable - like the arc length example above shows. We can imagine p as an arc-length along a curve, and all the x's are translations of the arc length to the n-dimensional coordinate system used to chart the (localisations of the) manifold. The problem then is to find a localised/differential expression for the arc-length $ds$ in terms of the infinitesimal changes (localised changes) in the (local) coordinate system. To do this we consider an infinitesimal increment along the curve, which associates the differentials $dx_1(p),dx_2(p),..,dx_n(p)$ to it - this can be thought of as a tangent to the curve at the point p, and a localised metric will take these infinitesimal changes; the infinitesimal tangent vectors; and relate them to the infinitesimal arc-length $ds$. As Riemann puts it:

The problem consists then in establishing a mathematical expression for the length of a line, and to this end we must consider the quantities x as expressible in terms of certain units. I shall treat this problem only under certain restrictions, and I shall confine myself in the first place to lines in which the ratios of the increments dx of the respective variables vary continuously. We may then conceive these lines broken up into elements, within which the ratios of the quantities dx may be regarded as constant; and the problem is then reduced to establishing for each point a general expression for the linear element ds starting from that point, an expression which will thus contain the quantities x and the quantities dx.

the task of finding a localised metric (for a continuous space) is solved by finding an appropriate expression of the localised arc-length $ds$ in terms of the localised changes $dx_1,...,dx_n$ - IE setting up the arc-length as a function of these infinitesimal changes. Riemann begins this task by noting various constraints on the functions which can count as localised metrics.

I shall suppose, secondly, that the length of the linear element, to the first order, is unaltered when all the points of this element undergo the same infinitesimal displacement, which implies at the same time that if all the quantities dx are increased in the same ratio, the linear element will vary also in the same ratio (1). On these suppositions, the linear element may be any homogeneous function of the first degree of the quantities dx, which is unchanged when we change the signs of all the dx (2), and in which the arbitrary constants are continuous functions of the quantities x.

(1) The length $ds$ at p and the length $ds'$ at the infinitesimally displaced p' only differ by a function of the variables $x_1,...,x_n$ and differentials $dx_1,...,dx_n$ which have an infinitesimally vanishing non-linear component above the quadratic terms; this is to say that the curve is locally linear with constant curvature, so scaling the changes proportionally scales the arc length in infinitesimal regions.

(2) $ds$ should not depend on the sign of the changes, IE if we replaced $dx_1$ with $-dx_1$ in whatever function we have, the function should be unchanged. An example here is the function f(x)=x^2, we have that f(-x) = (-x)^2=x^2 (which is the case Riemann will actually use).

Riemann then takes these two conditions and finds the simplest possible set of examples.

To find the simplest cases, I shall seek first an expression for manifoldnesses of n - 1 dimensions which are everywhere equidistant from the origin of the linear element; that is, I shall seek a continuous function of position whose values distinguish them from one another.

If n=3, we have a 2 dimensional manifold which is everywhere equally distant from the origin of the space - the surface of a sphere. If n=2, we have a 1 dimensional manifold with the same condition - the boundary of a circle. We imagine wrapping such a boundary of constant distance around a manifold - and then we increment out infinitesimally from the origin, each increment gives an n-1 dimensional sphere surface (of constant infinitesimal distance from the origin of the curve). If we're going out from the origin in all directions, this means that the increments must all be increasing (getting more positive) or that the increments are decreasing (getting more negative), either way they are getting further away from 0 uniformly. From (2) we have that this is a symmetry of the problem, so Riemann can deal just with the case where all the differentials are increasing.

In going outwards from the origin, this must either increase in all directions or decrease in all directions; I assume that it increases in all directions, and therefore has a minimum at that point. If, then, the first and second differential coefficients of this function are finite, its first differential must vanish, and the second differential cannot become negative; I assume that it is always positive.

Since the arc-length increases going away from the origin in both directions, the arc-length must have a minimum at this point, which from basic calculus means the first derivative of the arc-length with respect to the point vanishes. So long as we assume that the differentials are bounded, anyway (like we're not going into a region with infinite curvature).

This differential expression, of the second order remains constant when ds remains constant, and increases in the duplicate ratio when the dx, and therefore also ds, increase in the same ratio; it must therefore be ds2 multiplied by a constant, and consequently ds is the square root of an always positive integral homogeneous function of the second order of the quantities dx, in which the coefficients are continuous functions of the quantities x.

The vanishing behaviour ensures that the the second order differential of s, the curvature, remains constant when the infinitesimal increment in the arc length remains constant; thus we have constant curvature at a point on the manifold, which ensures that the lengths of lines within this infinitesimal region of constant curvature do not depend on their position! Combining this with (1) ensures that the arc-length, the localised distance measure, has the following properties (restating the first two):


(1) The length $ds$ at p and the length $ds'$ at the infinitesimally displaced p' only differ by a function of the variables $x_1,...,x_n$ and differentials $dx_1,...,dx_n$ which have an infinitesimally vanishing non-linear component above the quadratic terms ; this is to say that the curve is locally linear, so scaling the changes proportionally scales the arc length.

(2) $ds$ should not depend on the sign of the changes, IE if we replaced $dx_1$ with $-dx_1$ in whatever function we have, the function should be unchanged. An example here is the function f(x)=x^2, we have that f(-x) = (-x)^2=x^2 (which is the case Riemann will actually use).

(3) if we map $x_1(p),x_2 (p),...,x_n (p)$ to $ax_1(p),ax_2(p),...,ax_n (p)$, scaling by a positive constant a, this maps the arc length $ds(p)$ to $ads(p)$.

(4) $ds\geq 0$

The simplest example of this is the usual distance measure (A), which is characteristic of flat Euclidean space. Riemann restricts his discussion to manifolds which can be locally geometrically represented - namely those whose arc-length element is the square root of a quadratic function of the coordinate system differentials. As he puts it:

The next case in simplicity includes those manifoldnesses in which the line-element may be expressed as the fourth root of a quartic differential expression. The investigation of this more general kind would require no really different principles, but would take considerable time and throw little new light on the theory of space, especially as the results cannot be geometrically expressed; I restrict myself, therefore, to those manifoldnesses in which the line element is expressed as the square root of a quadric differential expression.

I'm thinking of making a post that describes the flow of the argument without going into the mathematical detail. Similar to @John Doe's post earlier in the thread.

Will take some time to digest the argument in §2.