The point of the Allais paradox is less about how humans violate the axiom of independence and more about how our utility functions are nonlinear, especially with respect to infinitesimal risk.
There is an existing Dutch Book for eliminating infinitesimal risk, and it's called insurance.
Yyyyes and no. Our utility functions are nonlinear, especially with respect to infinitesimal risk, but this is not inherently bad. There's no reason for our utility to be everywhere linear with wealth: in fact, it would be very strange for someone to equally value "Having $1 million" and "Having $2 million with 50% probability, and having no money at all (and starving on the street) otherwise".
Insurance does take advantage of this, and it's weird in that both the insurance salesman and the buyers of insurance end up better off in expected utility, but it's not a Dutch Book in the usual sense: it doesn't guarantee either side a profit.
The Allais paradox points out that people are not only averse to risk, but also inconsistent about how they are averse about it. The utility function U(X cents) = X is not risk-averse, and it picks gambles 1A and 2A (in Wikipedia's notation). The utility function U(X cents) = log X is extremely risk-averse, and it picks gambles 1B and 2B. Picking gambles 1A and 2B, on the other hand, cannot be described by any utility function.
There's a Dutch book for the Allais paradox in this post reading after "money pump".
In reality, the fact that parallel lines do not intersect does follow from the definition of the word "parallel". Therefore, the error results in several of the paragraphs in the original post being meaningless or untrue.
The trouble is that in 2-D Euclidean space, there are many equivalent definitions of "parallel". It just so happens that straight lines that don't intersect also have the same slope,will intersect any transverse line at congruent angles, and are always the same distance apart (and vice versa). However, these properties need not be equivalent in non-Euclidean geometry.
The OP's issue seems to be that defining parallel lines as those which do not intersect is artificial. It's a workaround Euclid developed to smooth over his presentation. He could not use local properties of lines and angles to prove parallel lines didn't intersect. So, he defined them as lines that don't intersect, introduced the parallel postulate, and then used those to prove the other properties of parallel lines. Later mathematicians found this to be rather inelegant and tried to prove parallel lines didn't intersect using only properties of lines and angles.
Sure, it's an error if you use Euclid's definition of parallel, but I wouldn't call the discussion meaningless. It touches on a very important issue of how to define things and what properties we want to retain when we generalize a notion.
When it comes to neutral geometry, nobody's ever defined "parallel lines" in any way other than "lines that don't intersect". You can talk about slopes in the context of the Cartesian model, but the assumptions you're making to get there are far too strong.
As a consequence, no mathematicians ever tried to "prove that parallel lines don't intersect". Instead, mathematicians tried to prove the parallel postulate in one of its equivalent forms, of which some of the more compelling or simple are:
The sum of the angles in a triangle is 180 degrees. (Defined to equal two right angles.)
There exists a quadrilateral with four right angles.
If two lines are parallel to the same line, they are parallel to each other.
It's also somewhat misleading to say that mathematicians were mainly motivated by the inelegance of the parallel postulate. Though this was true for some mathematicians, it's hard to say that the third form of the parallel postulate which I gave is any less elegant, as an axiom, than "If two line segments are congruent to the same line segment, then they are congruent to each other". Some form of the latter was assumed both by Euclid (his first Common Notion) and by all of his successors.
A stronger motivation for avoiding the parallel postulate is that so much can be done without it that one begins to suspect it might be unnecessary.
Your second sentence does not imply your first. (Nor is it true--ignoring the misphrasing of the axiom, the rest of the discussion is perfectly understandable.)
Understandable; perhaps. In mathematics, it is very easy to say understandable things that are simply false. In this case, those false things become nonsense when you realize that the meaning of "parallel lines" is "lines that do not intersect".
You might say that an explanation gets these facts completely wrong, then it is still a good explanation if it makes you think the right things. I say that such an explanation goes against the spirit of all mathematics. It is not enough that your argument is understandable, for many understandable arguments have later turned out to be incoherent. It is not enough that your argument is believable, for many believable arguments have later turned out to be false.
If you want to do good mathematics, the statements you make must be true.
I've always enjoyed Lewis Carroll's talk of maps:
"That's another thing we've learned from your Nation," said Mein Herr, "map-making. But we've carried it much further than you. What do you consider the largest map that would be really useful?"
"About six inches to the mile."
"Only six inches!" exclaimed Mein Herr. "We very soon got to six yards to the mile. Then we tried a hundred yards to the mile. And then came the grandest idea of all! We actually made a map of the country, on the scale of a mile to the mile!"
"Have you used it much?" I enquired.
"It has never been spread out, yet," said Mein Herr: "the farmers objected: they said it would cover the whole country, and shut out the sunlight! So we now use the country itself, as its own map, and I assure you it does nearly as well.
From Sylvie and Bruno Concluded by Lewis Carroll, first published in 1893.
Only a single mile to the mile? I've seen maps in biology textbooks that were much larger than that.
In morals, as in logic, you can't explain something by appealing to something else unless the chain terminates in an axiom.
The question "why is it bad to rape and murder?" can be rephrased as, "how can we determine if a thing is bad, in the case of rape and murder?"
The answer "rape and murder are bad by definition" may be unsatisfying, but at least it's a workable way: everything on the list is bad, everything else is not. But the answer "because they make others sad" assumes you can determine making others sad is bad. You substitute one question for another, and unless we keep asking why, we won't have answered the original question.
Okay, then interpret my answer as "rape and murder are bad because they make others sad, and making others sad is bad by definition".
I'm not sure why this comment is being downvoted
This:
This is bad and you should feel bad.
The tone is well-deserved. This is a serious mistake that renders all further discussion of geometry in the post nonsensical.
This doesn't answer the question. Why is doing things Joe doesn't like, or making his friends sad, bad? Consequentialism isn't a moral system by itself; you need axioms or goals.
You can always keep asking why. That's not particularly interesting.
It occurs to me that we can express this problem in the following isomorphic way:
Omega makes an identical copy of you.
One copy exists for a week. You get to pick whether that week is torture or nirvana.
The other copy continues to exist as normal, or maybe is unconscious for a week first, and depending on what you picked for step 2, it may lose or receive lots of money.
I'm not sure how enlightening this is. But we can now tie this to the following questions, which we also don't have answers to: is an existence of torture better than no existence at all? And is an existence of nirvana good when it does not have any effect on the universe?
Sometimes people will argue that if you would pay a lot to save your own life from a fatal illness, that means you don't value lives equally but prefer your own, and therefore you should sign up for cryonics. But this argument seems a bit problematic to me, because it assumes my preference to save my life in the case of the fatal illness is ideal. In reality it might not be ideal at all. I am certainly not Zachary Baumkletterer, but it's likely I would be a better person if I were. If this is the case, the problem is not that I am unwilling to sign up for cryonics, but that I would pay to save myself from the fatal illness instead of giving the money away. And this argument does not mean that if I don't want to sign up for cryonics, instead I have to start donating all my money to charity. It just means I am doing the best I feel that I can, and if I signed up for cryonics I would be doing even worse (by doing less for others.)
Yes, this, exactly.
I do nice things for myself not because I have deep-seated beliefs that doing nice things for myself is the right thing to do, but because I feel motivated to do nice things for myself.
I'm not sure that I could avoid doing those things for myself (it might require willpower I do not have) or that I should (it might make me less effective at doing other things), or that I would want to if I could and should (doing nice things for myself feels nice).
But if we invent a new nice thing to do for myself that I don't currently feel motivated to do, I don't see any reason to try to make myself do it. If it's instrumentally useful, then sure: learning to like playing chess means means that my brain gets exercise while I'm having fun.
With cryonics, though? I could try to convince myself that I want it, and then I will want it, and then I will spend money on it. I could also leave things as they are, and spend that money on things I currently want. Why should I want to want something I don't want?
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I didn't mean to imply nonlinear functions are bad. It's just how humans are.
Prospect Theory describes this and even has a post here on lesswrong. My understanding is that humans have both a non-linear utility function as well as a non-linear risk function. This seems like a useful safeguard against imperfect risk estimation.
If you setup your books correctly, then it is guaranteed. A dutch book doesn't need to work with only one participant, and in fact many dutch books only work with on populations rather than individuals, in the same way insurance only guarantees a profit when properly spread across groups.
Insurance makes a profit in expectation, but an insurance salesman does have some tiny chance of bankruptcy, though I agree that this is not important. What is important, however, is that an insurance buyer is not guaranteed a loss, which is what distinguishes it from other Dutch books for me.
Prospect theory and similar ideas are close to an explanation of why the Allais Paradox occurs. (That is, why humans pick gambles 1A and 2B, even though this is inconsistent.) But, to my knowledge, while utility theory is both a (bad) model of humans and a guide to how decisions should be made, prospect theory is a better model of humans but often describes errors in reasoning.
(That is, I'm sure it prevents people from doing really stupid things in some cases. But for small bets, it's probably a bad idea; Kahneman suggests teaching yourself out of it by making yourself think ahead to how many such bets you'll make over a lifetime. This is a frame of mind in which the risk thing is less of a factor.)