"A model is an interpretation of the sentences generated by a language. A model is a structure which assigns a truth value to each sentence generated by some language under some logic."

I think this phrasing will be very misleading to anyone who tries to learn model theory from these posts. This is one thing a model DOES, but it isn't what a model IS. As far as I can tell, you nowhere say what a model is, even approximately. Writing out precisely what a model is takes a lot of space (like in the book you're reading!) so let me give an example.

Our alphabet will be the symbols of first order logic, plus as many variable names as we need, and the symbols +, =, 0.

Our axioms are

∀ x : x+0=0+x=x

∀ x,y: x+y=y+x

∀ x,y,z: (x+y)+z=x+(y+z)

∀ x ∃ y : x+y=y+x=0

Our THEORY is the set of all logical consequences of these statements, where "logical consequence" means "obtainable by the formal rules of first order logic . A MODEL of our theory is a specific set G, a specific element of G called 0 and a specific operation + taking two elements of G and returning a third element of G, such that all of these statements are true about G. In other words, a model of this theory is an abelian group.

One thing an abelian group can do is give us a way to assign a true or false value to any statement in our language. For example, consider the statement ∀ x ∃ y : y+y+y=x. This statement is true in the group of rational numbers, but false in the group of integers. If we choose a particular abelian group, that will force a specific choice as to whether this statement is true or false.

However, you shouldn't identify an abelian group with a way of assigning truth values to statements about abelian groups. For example, the rational numbers and the real numbers are both abelian groups and, as it turns out, there is no statement using only +, 0, = and logical connectives whose truth value is different in these two groups. Nonetheless, they are different models.

I need some advice. I recently moved to a city and I don't know how to stop myself from giving money to strangers! I consider this charity to be questionable and, at the very least, inefficient. But when someone gets my attention and asks me specifically for a certain amount of money and tells me about themselves, I won't refuse. I don't even feel annoyed that it happened, but I do want to have it not happen again. What can I do?

The obvious precommitment to make is to never carry cash. I am strongly considering this and could probably do so, but it is nice to be able to have at least enough for a bus trip, a quick lunch or for some emergency. I have tried to give myself a running tally of number of people refused and when that gets to, say, 20, I would donate something to a known legitimate charity. While doing so makes me feel better about passing beggars by, it doesn't help once someone gets me one-on-one. So I've never gotten to that tally without resetting it first by succumbing to someone. Is there some way to not look like an easy mark? Are there any good standard pieces of advice and resources for this?

However, I always find these exchanges to be really fascinating from the point of view of the Dark Arts used. The most recent time this happened, I was stopped and asked for the time which he promptly ignored. Then he told me that he had seen me around before - this is entirely plausible since I walk by there most days but is also likely to be true of a randomly selected person so could just be a shot in the dark. He shook my hand multiple times. Gave me his name and told me to call him by his nickname. He told me about being a veteran, talked to me about any veterans I knew. Tried to guess my current job and messed up in a way that implied I was younger than I am which was probably his only significant mistake as that could have annoyed some people. He then acted impressed when I corrected him. Asked where I was from and then said he had an acquaintance from nearby. Then of course he asked for train ride money which started at 8 dollars and ended up being 23.

I could practically check off the chapters of Cialdini's Influence one-by-one on this list and noticed at least two of these tactics while they were being used. Unfortunately, Cialdini's book has laughable excuses for sections on "Defense Against" said dark arts, rarely saying anything more than "just use the fact that they're using these tricks against them since now you know better!" So, here I am, knowing the nature of my foe and yet still being utterly dragged in by it.

Here is an attempt to create a roadmap to the amplituhedron work. My relevant background and disclaimers: I am a mathematician with interests in particle physics who has been trying to learn about Arkani-Hamed and collaborators' ideas for the last two years. The specific result which is getting press now is one that has not been public for most of that time; my goal had been to understand the story of scattering amplitudes as described in his prior 154 page paper. I have been meeting regularly with a group of mathematicians and physicists here at the University of Michigan in pursuit of this goal.

So, what should you learn first:

You should be completely comfortable with quantum mechanics and special relativity. I would point out that Less Wrong will give you great ideas about the philosophy of QM but is very short on computing any actual examples; you should understand how to actually use QM to solve problems.

Mathematically, I found my familiarity with representation theory and Lie groups extremely useful. However, a lot of the physicists in our group didn't have this background and compensated for it with strengths of their own.

You should understand the material of a first graduate course in Quantum Field Theory, through the computation of tree-level amplitudes. To learn this, I audited a course taught out of Srednicki's book, and also read on my own in Peskin-Schroeder and Zee. I can't claim to have a great understanding of this material, and if anyone has advice as to how to learn it better, I'd love to hear some. However, I feel confident in saying that, had I been enrolled in that class, I would have gotten an A, and I think you should at least be at that level. A second course in QFT certainly wouldn't hurt -- the fact that I had never worked through any loop integrals in detail handicapped me -- but I am managing without it.

If you get this far, I strongly recommend you next read Henriette Elvang and Yu-Tin Huang's notes on scattering amplitudes http://arxiv.org/abs/1308.1697 . As the abstract says, "The purpose of this review is to bridge the gap between a standard course in quantum field theory and recent fascinating developments in the studies of on-shell scattering amplitudes." I have found this extremely helpful. (Of course, being able to knock on Henriette's door and get her to explain something to me is even more valuable :).)

After that, I'd look at "Scattering Amplitudes and the Positive Grassmannian" http://arxiv.org/abs/1212.5605 . This is long and hard, but has the advantage that it is written down in full detail, unlike the current subject which only exists in lecture notes.

At this point, you will have caught up to me, so I'm not sure I can advise you how to go further. However, I will suggest that I find Arkani-Hamed's co-author, Trnka, much more understandable than Arkani-Hamed. These lecture notes http://wwwth.mpp.mpg.de/members/strings/amplitudes2013/amplitudes_files/program/Talks/WE/Trnka.pdf are the clearest presentation of the amplituhedron material I have found yet.

"What hidden obstacle could be causing my failures?"

My mental shorthand for this is the following experience: I try to pull open the silverware drawer. It jams at an inch open. I push it shut and try again, same result. I pull harder, it opens a tiny bit more before stopping.

Reflection: Some physical object is getting in the way of the motion. Something could be on the drawer track, but more likely it is inside the drawer. It is a rigid object, because I always stop at the same place, although slightly squashable because I was able to yank and pull a little harder. It is probably striking the inner wall of the cabinet in which the drawer is mounted. It is on an angle because I can't see it when I look through the inch gap. There is a fork or knife angled up and poking against the inner wall. Digging around with my finger quickly finds a fork.

Since then, I've brought up this question by asking myself "what is the fork in the drawer"?

For example, my linear algebra students generally seem smart and attentive, but they become confused whenever I do a detailed computation with inner products. After some thought about which computations confuse them, hypothesize that whoever taught them basic matrix manipulations didn't teach the "transpose" operator, and particularly didn't teach the rule (AB)^T = B^T A^T. Fixed very quickly. (Of course, I also try to encourage them to ask questions about what confuses them, but I think that it is impossible to ever get a class comfortable enough questioning you to not need to think on your own about what is the underlying difficulty causing confusion.)

So, what do you all think is Voldemort's goal here? In canon, he was a power hungry sadist, so conquering the world while torturing his minions made sense. But MOR!Voldemort seems to find people tiresome and is happiest as an immortal in lifeless space. In that case, why not Horcrux Pioneer 11, kill his earthly body and be done with it?

At the moment, he has a plausible motivation -- provoke Harry into discovering a better form of immortality than Horcruxes, and use it for himself. But it seems implausible that this was his goal until Harry came to Hogwarts this year since he had no reason to expect that his murder of James and Lilly would lead to a combined scientific genius/wizard. What was he trying to accomplish before that?

For example, what if I told you that in reality, I don't speak a word of English ? Whom are you going to believe -- me, or your lying eyes ?

http://www.youtube.com/watch?v=dd0tTl0nxU0

In the proof of Theorem 2, you write "Clearly is convex." This isn't clear to me; could you explain what I am missing?

More specifically, let be the subset of obeying . So . If were convex, then would be as well.

But is not convex. Project onto in the coordinates corresponding to the sentences and . The image is . This is not convex.

Of course, the fact that the $X(\phi,a,b)$ are not convex doesn't mean that their intersection isn't, but it makes it non-obvious to me why the intersection should be convex. Thanks!

In the proof of Theorem 2, you write "Clearly is convex." This isn't clear to me; could you explain what I am missing?

More specifically, let be the subset of obeying . So . If were convex, then would be as well.

But is not convex. Project onto in the coordinates corresponding to the sentences and . The image is . This is not convex.

Of course, the fact that the $X(\phi,a,b)$ are not convex doesn't mean that their intersection isn't, but it makes it non-obvious to me why the intersection should be convex. Thanks!

I've seen some (old) arguments about the meaning of axiomatizing which did not resolve in the answer, "Because otherwise you can't talk about numbers as opposed to something else," so AFAIK it's theoretically possible that I'm the first to spell out that idea in exactly that way, but it's an obvious-enough idea and there's been enough debate by philosophically inclined mathematicians that I would be genuinely surprised to find this was the case.

If memory serves, Hofstadter uses roughly this explanation in GEB.