Hmm. I don't think it's not useful to practice looking at the truth even when it hurts. For instance with the paperwork situation, it could be that not fixing the paperwork even if you recognize errors in it is something you would see as a moral failing in yourself, something you would be averse to recognizing even if you allowed yourself to not go through the arduous task of fixing those mistakes. Because sometimes the terminal result of a self-evaluation is reducing one's opinion of oneself, being able to see painful truths is a necessary tool to make this method work properly.
That said, I do think this is a much more actionable ritual than just "look at the painful thing". It also serves better as a description of reality, encompassing not just why certain truths are painful, but also how they become painful. It establishes not just a method for coping with painful truths and forcing confrontation with them, but also for establishing mental housekeeping routines which can prevent truths from becoming painful in the first place.
This has been a topic I started thinking about on my own some months ago (I even started with the same basic observation about children and why they sometimes violently reject seemingly benign statements). But I think my progress will be much improved with a written document from someone else's perspective which I can look at and evaluate. Thank you very much for writing this up. I really appreciate it.
Just so you all know, Clifford Algebra derivations of quantized field theory show why the Born Probabilities are a squared proportion. I'm not sure there's an intuitively satisfying explanation I can give you for why this is that uses words and not math, but here's my best try.
In mathematical systems with maximal algebraic complexity for their given dimensionality, the multiplication of an object by its dual provides an invariant of the system, a quantity which cannot be altered. (And all physical field theories (except gravity, at this time) can be derived in full from the assumption of maximal algebraic complexity for 1 positive dimension and 3 negative dimensions). [Object refers to a mathematical quantity, in the case of the field theories we're concerned with, mostly bivectors].
The quantity describing time evolution then (complex phase amplitudes) must have a corresponding invariant quantity that is the mod squared of the complex phase. This mod squared quantity, being the system invariant whose sum describes 'benchmark' by which one judges relative values, is then the relevant value for evaluating the physical meaning of time evolutions. So the physical reality one would expect to observe in probability distributions is then the mod squared of the underlying quantity (complex phase amplitudes) rather than the quantity itself.
To explain it in a different way, because I suspect the one way is not adequate without an understanding of the math.
Clifford Algebra objects (i.e. the actual constructs the universe works with, as best we can tell) do not in of themselves contain information. In fact, they contain no uniquely identifiable information. All objects can be modified with an arbitrary global phase factor, turning them into any one of an infinite set of objects. As such, actual measurement/observation of an object is impossible. You can't distinguish between the object being A or Ae^ib, because those are literally indistinguishable quantities. The object which could be those quantities lacks sufficient unique information to actually be one quantity or the other. So you're shit out of luck when it comes to measuring it. But though an object may not contain unique information, the object's mod squared does (and if this violates your intuition of how information works, may I remind you that your classic-world intuition of information counts for absolutely nothing at the group theory level). This mod squared is the lowest level of reality which contains uniquely identifiable information.
So the lowest level of reality at which you can meaningfully identify time evolution probabilities is going to be described as a square quantity.
Because the math says so.
By the way, we're really, really certain about this math. Unless the universe has additional spatial-temporal dimensions we don't know about (and I kind of doubt that) and only contains partial algebraic complexity in that space (and I really, really doubt that), this is it. There is no possible additional mathematical structure with which one could describe our universe that is not contained within the Cl_13 algebra. There is literally no mathematical way to describe our universe which adequately contains all of the structure we have observed in electromagnetism (and weak force and strong force and Higgs force) which does not imply this mod squared invariant property as a consequence.
Furthermore, even before this mod squared property was understood as a consequence of full algebraic complexity, Emmy Noether had described and rigorously proved this relationship as the eponymous Noether's theorem, confirmed its validity against known theories, and used it to predict future results in field theory. So this notion is pretty well backed up by a century of experimental evidence too.
Tl;DR: We (physicists who work with both differential geometries and quantum field theory and whom find an interest in group theory fundamentals beyond what is needed to do conventional experimental or theory work) have known about why the Born Probabilities are a squared proportion since, oh, probably the 1930s? Right after Dirac first published the Dirac Equation? It's a pretty simple thing to conclude from the observation that quantum amplitudes are a bivector quantity. But you'll still see physics textbooks describe it as a mystery and hear it pondered over philosophically, because propagation of the concept would require a base of people educated in Clifford Algebras to propagate through. And such a cohesive group of people just does not exist.
I saw the path Frozen's plot took as well done.
I liked the fact that Anna's relationship with Hans didn't work out. Disney went out of its way to poke holes in the traditional 'love at first sight' meme, something I think is a huge improvement on how Disney portrays most relationships. Furthermore, they showed Anna and Kristoff's relationship to grow on a solid foundation over time, and to be mutually pursued, as opposed to being a one-sided chase. Whereas Anna wanted her relationship with Hans to miraculously change her life, her relationship with Kristoff is an important part of her life without being her reason d'etre. All of this, to me, seems much better than the stereotypical fairytale romance.
Yes, Elsa doesn't end up with a relationship. Which isn't really a problem to me. She has personal problems she needs to work out, and she doesn't show any interest in a relationship. So a relationship is unnecessary.
You made a rather big deal out of the trade deal being broken and the ramifications thereof. But honestly, I think it was the right decision. The mayor of Wesseltown shows a clear desire to exploit the resources of Elsa's kingdom (in classic Disney fashion, he says so out loud). He bears them no good will. When Elsa's power broke loose, a potentially salvageable situation was ruined by his hostile reaction. And when they attempted to capture Elsa, his orders to kill her almost got people killed and came dangerously close to permanently ruining any hope of resolving the eternal winter. He showed a clear disregard for their kingdom's well-being, demonstrated an inability to see past his own prejudices, and tried to KILL THEIR QUEEN. Any one of these would be good reason to break off trade. In particular, the political ramifications of a show of weakness on the order of ignoring an attempted assassination are probably much worse than loss of trade.
Fairytale stories have a habit of setting up female protagonists as damsels (who am I kidding, stories in general have this habit). Time after time after time we see female characters put in situations where their only hope is for the strong male to save them. This trope could see some time without use, which is what we saw in Frozen. Disney played on our expectations of Anna being saved by her newly minted boyfriend's love so they could violate that expectation. Instead they showed a selfless act, reconciliation, and a long-term bond as the ingredients for an act of true love. I think that's a good thing.
Yes, in the end things were not perfect. Elsa still needs to learn how to deal with people. Anna is a bit more idealistic and naive than is healthy. Kristoff still needs to learn how to deal with people. The trade repercussions with Wesseltown are going to suck. Which I find a nice change of pace from the neat and tidy "happily ever after" endings. Life goes on, still imperfect but better than before.
(Can't say anything about Sleeping Beauty. I haven't really been paying attention to it.)
There was a time when I was very rude to religious people because I thought that made me wise. Then there was a time when I was very polite because I thought equity in consideration was wise.
Now I'm just curt because I have science to do and no time to deal with fools.
This ought to be embedded deeply in the minds of everyone involved in education. Most regrettably, it is not.
That last part is the most important.
We can't answer every question.
No, but I think we can answer any question.
There are cases in which you can relate dimensionless units. For instance, moles is a dimensionless unit, it just means times 6.022*10^23. But you can relate moles to moles in some cases, for instance with electrolysis. If you know how many electrons are being pumped into a reaction and you want to know how much Fe(II) becomes Fe, then you can compare moles of electrons to moles of Iron, even though neither moles, elements, or electrons can be related directly to one another in the conventional sense of m/s. In the same way one can relate dollars of one thing to dollars of another and get a meaningful answer.
You are right to point this out though, it is skirting very close to the gray areas of dimensional analysis without being explicitly mentioned as doing so.
"Not if they change their minds when confronted with the evidence."
"Would you do that?"
This is where I think the chain of logic makes a misstep. It is assumed that you will be able to distinguish evidence which should change your mind from evidence that is not sufficient to change your mind. But doing so is not trivial. Especially in complicated fields, simply being able to understand new evidence enough to update on it is a task that can require significant education.
I would not encourage a layperson to have an opinion on the quantization of gravity, regardless of how willing they might be to update based on new evidence, because they're not going to be able to understand new evidence. And that's assuming they can even understand the issue well enough to have a coherent opinion at all. I do work pretty adjacent to the field of quantized gravity and I barely understand the issue well enough to grasp the different positions. I wouldn't trust myself to meaningfully update based on new papers (beyond how the authors of the papers tell me to update), let alone a layperson.
The capacity to change a wrong belief is more than just the will to do so. And in cases where one cannot reliably interpret data well enough to reject wrong beliefs, it is incredibly important to not hold beliefs. Instead cultivate good criteria for trusting relevant authority figures or, lacking trusted authority figures, simply acknowledge your ignorance and that any decision you make will be rooted in loose guesswork.