If you glance at my answers to problem sets 4, 5, and 8 then you'll get the general idea of what the course involves. The problem sets are really fun because you're forced to play with the numbers and really understand the material. It might not be as hard as a traditional physics course, but it's every bit as rigorous in the real sense of the word. It's brilliance is that it really stresses the fundamentals, probably more so than any other physics material in existence. This is important because, as one professional Go player said after observing a Japanese baseball game,
"In every confrontation with a real American professional team it seems that what we need to learn from them, besides their technique of course, is how uniformly faithful their players are to the fundamentals. Faithfulness to the fundamentals seems to be a common thread linking professionalism in all areas." (T. Kageyama)
It starts assuming nothing more than basic arithmetic, and builds up to differntial equations, tensors, string theory, quantum relativity, etc.
I'm warning you up front that it's very math heavy. It takes some work to get through, but its worth it.
In this context, the tens or thousands of experiments are generalized to support some "theory"
Every time i touch ice it is cold. By induction, all ice is cold. that's not "proof" but it's a great theory of ice.
I'll agree that we can set up a game we play with symbols and rules that doesn't permit inductive reasoning as valid. I can't agree that "what people do" has anything to do with logic.
On the other hand E Yudkowsky claims Poppers philosophy is a subset of Bayes Theorem(or wait dont take my words for that I might have misinterpretad), here is the link and quote:
The Bayesian revolution in the sciences is fueled, not only by more and more cognitive scientists suddenly noticing that mental phenomena have Bayesian structure in them; not only by scientists in every field learning to judge their statistical methods by comparison with the Bayesian method; but also by the idea that science itself is a special case of Bayes' Theorem; experimental evidence is Bayesian evidence. The Bayesian revolutionaries hold that when you perform an experiment and get evidence that "confirms" or "disconfirms" your theory, this confirmation and disconfirmation is governed by the Bayesian rules. For example, you have to take into account, not only whether your theory predicts the phenomenon, but whether other possible explanations also predict the phenomenon. Previously, the most popular philosophy of science was probably Karl Popper's falsificationism - this is the old philosophy that the Bayesian revolution is currently dethroning. Karl Popper's idea that theories can be definitely falsified, but never definitely confirmed, is yet another special case of the Bayesian rules; if p(X|A) ~ 1 - if the theory makes a definite prediction - then observing ~X very strongly falsifies A. On the other hand, if p(X|A) ~ 1, and we observe X, this doesn't definitely confirm the theory; there might be some other condition B such that p(X|B) ~ 1, in which case observing X doesn't favor A over B. For observing X to definitely confirm A, we would have to know, not that p(X|A) ~ 1, but that p(X|~A) ~ 0, which is something that we can't know because we can't range over all possible alternative explanations. For example, when Einstein's theory of General Relativity toppled Newton's incredibly well-confirmed theory of gravity, it turned out that all of Newton's predictions were just a special case of Einstein's predictions.
You can even formalize Popper's philosophy mathematically. The likelihood ratio for X, p(X|A)/p(X|~A), determines how much observing X slides the probability for A; the likelihood ratio is what says how strong X is as evidence. Well, in your theory A, you can predict X with probability 1, if you like; but you can't control the denominator of the likelihood ratio, p(X|~A) - there will always be some alternative theories that also predict X, and while we go with the simplest theory that fits the current evidence, you may someday encounter some evidence that an alternative theory predicts but your theory does not. That's the hidden gotcha that toppled Newton's theory of gravity. So there's a limit on how much mileage you can get from successful predictions; there's a limit on how high the likelihood ratio goes for confirmatory evidence.
On the other hand, if you encounter some piece of evidence Y that is definitely not predicted by your theory, this is enormously strong evidence against your theory. If p(Y|A) is infinitesimal, then the likelihood ratio will also be infinitesimal. For example, if p(Y|A) is 0.0001%, and p(Y|~A) is 1%, then the likelihood ratio p(Y|A)/p(Y|~A) will be 1:10000. -40 decibels of evidence! Or flipping the likelihood ratio, if p(Y|A) is very small, then p(Y|~A)/p(Y|A) will be very large, meaning that observing Y greatly favors ~A over A. Falsification is much stronger than confirmation. This is a consequence of the earlier point that very strong evidence is not the product of a very high probability that A leads to X, but the product of a very low probability that not-A could have led to X. This is the precise Bayesian rule that underlies the heuristic value of Popper's falsificationism.
Similarly, Popper's dictum that an idea must be falsifiable can be interpreted as a manifestation of the Bayesian conservation-of-probability rule; if a result X is positive evidence for the theory, then the result ~X would have disconfirmed the theory to some extent. If you try to interpret both X and ~X as "confirming" the theory, the Bayesian rules say this is impossible! To increase the probability of a theory you must expose it to tests that can potentially decrease its probability; this is not just a rule for detecting would-be cheaters in the social process of science, but a consequence of Bayesian probability theory. On the other hand, Popper's idea that there is only falsification and no such thing as confirmation turns out to be incorrect. Bayes' Theorem shows that falsification is very strong evidence compared to confirmation, but falsification is still probabilistic in nature; it is not governed by fundamentally different rules from confirmation, as Popper argued.
So we find that many phenomena in the cognitive sciences, plus the statistical methods used by scientists, plus the scientific method itself, are all turning out to be special cases of Bayes' Theorem. Hence the Bayesian revolution.
The most apparent problem with Yudkowsky's position is: where do probability estimates and hypotheses come from in the first place? That is an issue which Popper's philosophy provides answers for, and Bayesian Inference does not.
"Oh," said I, "that's interesting. What does it mean for a form to be transcendent or immanent?"
"I don't know, I just know that's what they thought."
We then had a discussion about whether or not he really knew that, and whether or not he was OK with paying over $25k a year to learn how to be a parrot.
The sad thing: he was OK with it, and probably rightly so. He realized that mostly what people want are parrots and you can get paid quite a decent amount of money for being a good parrot. Not a bad gig if you're fine being a cog. There are certain comforts it provides.
High school science competitions (school science fairs, all the way upto Siemens-Westinghouse and Intel STS) particularly guilty of this.
I remember I just -couldn't- accept that we can't take square roots of negative numbers. I mean, sure, two negatives multiplied together is a positive, but...there HAS to be a way! Numbers, to my 6th grade mind, weren't arbitrarily going to deny your ability to manipulate them.
Fortunately for me, I happened to be reading a book on Mandlebrot and in the course of discussing his work the author introduces the concept of imaginary numbers. So the next day in class I asked: "Are you suuuure?"
"Yes, I'm sure."
"Well, I was reading in this book and so-and-so says you can take the square root of negative numbers, you get 'i' for the root of -1 and..."
"Why don't you go wait outside."
Where I got told that although I move to the beat of a different drummer, when I'm in her class, I really just need to be quiet. That's when public education and I had a falling out from which our relationship has never recovered.
All of that to say: I think part of the 'problem' is that kids who do try to think on their own are often in a situation where they're challenging accepted rules without the social grace to do so constructively. But rather than being taught the social skills, they're just shut down. Very sad, actually. They could be taught how to 'wander well'.
Am I wrong there?
i is defined as sqrt(-1), if that answers your question.
My point is that sqrt(-1) = i is an extra assumption that you don't need to make in order to derive the math you do in school. If you don't make that assumption, you can still derive a lot of math. If you do make that assumption, then you can also derive the theorems of complex analysis.
I think that's true, anyway. I could be wrong.
It just seems strange to me that someone would be bothered by the lack of negative square roots, since their existence is never derived, only assumed. But then again, people's minds work very differently, especially in Mathematics.
I was the opposite in Math class. I fought accepting "i" when they tried to teach it to us since it seemed so arbitrary and contrived to me. This was before I discovered that all Math is arbitrary and that there is no "real math", anyway.
sqrt(-1) = sqrt(e^Pi) = e^(Pi/2) = i
So the new concept was really a 2D plane for numbers, not a new definition for sqrt of negative integers.
e^(i)(pi/2) = i.
The simple assumption that sqrt(-1) = i is problematic because you can get something like
i^2 = (sqrt(-1))^2 = sqrt((-1)^2) = 1.
The wonder and surprise of complex numbers is that assuming seemingly arbitrary properties of a constant like i lead to a number of deep and beautiful results.
I hadn't reached my correct answers the "right" way.
I would also lose points for "not showing my work", but if you can't factor, say, x^2-9 in your head, maybe you have deeper issues.
I realized that math in school was more like one of those simon games where you just mindlessly press the arbitrary sequence of lights you just saw, than actual mathematics. This analogy helped me suffer through "school math" while I perused real math on my own.
I also once got an answer marked wrong because the teacher couldn't understand my intermediate steps, even though the answer (and steps) were actually correct. When I showed him, he refused to change my grade. That was at University.