shou4577

I used to live in a set of townhouses that did exactly your "drop box" idea. It worked really well, and I thought that it was quite common.

Yes, it's much easier for humans to prove complex theorems than computers. Just as it is much easier for humans to build complex programs than computers.

Often times, a proof is (much) more complicated than just mixing theorems A, B, and C together to get result D. Many times proofs will require developing entirely new mathematical structures, or can be done only in steps.

As an example, a friend of mine just proved a great theorem about whether or not Hessenberg varieties can be paved by affines. Ignoring what this actually means, the proof involved firstly recognizing that a Hessenberg variety actually fits within a spectrum - it is somehow "between" other types of well- (or better-) understood varieties. These are paved by affines in (somewhat) obvious ways, and so it is natural to try to use the "between-ness" of the Hessenberg variety to construct a "between-ish" paving. This isn't a proof, but it is a direction (one that turned out to be fruitful). It would be very hard for a computer to have made this intuitive leap.

And it almost works. Except now you can't reduce things like you would like, and you need to construct multi-level fiber bundles in order to get the triviality that you want. And wait - this only works if the Hessenberg variety that we started with has such-and-such property.

So she's reduced the problem to a simpler problem. How does she know it is simpler? Intuition. A computer would have a very hard time recognizing whether or not this is actually simpler (even if it could get there). It turns out that this property holds when our Hessenberg variety is constructed from an element whose semisimple part under the Jordan decomposition is regular. So she has proved it for a huge class of varieties, but not all. Such a partial result would also be very difficult for a computer, since it isn't exactly what she set out to do. She's only partly there. But even this part is huge.

And there isn't anything hand-wavy about her argument, either. It's perfectly grounded in well-established theorems, all of which she understands the proof for. It's just that the original concept was shakey - the "between-ness". She had to work to ground it. It would have been exceptionally hard for a computer to make several of the intuitive leaps necessary to come up with the ideas for the proof in the first place.

I'm very sorry to hear about your experience, thank you so much for sharing it.

Thanks, you don't see any concrete experience because there isn't anything worth showing. I'm working on some stuff right now to do just that. It's actually a webapp to allow teachers and students to communicate math effectively without having to know LaTeX and email pdfs.

From what I've heard so far, it sounds like there is definitely opportunities for me, and that means that I can justify spending time working on these types of projects over the next couple of years.

Awesome! I went to check out your blog post, only to discover that I've already read it (and it contains a lot of helpful advice, thanks!).

Yeah, I've heard about the Wall Street programming options for mathematicians, but it is not a field that I'm interested in.

Actually, the CS dept at my uni has a big computer vision research program. I should go talk to them.

Thanks

Thanks for the warning. I'm working on a project right now that I should be releasing soon to help with the "real project" problem.

This is another thing I'm afraid of - being overqualified/people thinking that they are actually my backup job. Is there any way to combat this?

Thanks, that's really good to hear (I love big problems!)

My research is in abstract algebra (specifically Lie theory and algebraic geometry), which is pretty far removed from statistics or applied math in general.

I can try:

1. Matrix multiplication is indeed defined oddly. To understand why, it can help to think of matrices as "linear transformations". For example, the matrix

| 0 -1 |

| 1 0 |

corresponds to counterclockwise rotation of vectors by 90 degrees (go ahead, try multiplying the vector <1,0> by this matrix - the answer should be <0,1>). Let's call this matrix A. Also, the matrix

|-1 0 |

| 0 1 |

corresponds to reflection over the y-axis (call this matrix B). If we begin with a vector v = <1,1>, then rotate twice by 90 degrees, then reflect over the y-axis, we should get the vector <1,-1> (it helps to draw a picture, but I am limited in these comments). We can figure this out by repeatedly multiplying by matrices: Av = <-1,1>, A(Av) = <-1, -1>, and B(A(Av)) = <1,-1> (you may want to multiply these out to check, then check by evaluating the operations geometrically).

But wait - there is also a matrix which represents this transformation (that is, the transformation which is rotation counterclockwise by 90 degrees twice then reflection over the y axis). It's the matrix

| 1 0 |

| 0 -1 |

which we'll call C (go ahead and check that this matrix works the way that I just said). Since multiplying a vector by C is the same as multiplying the vector by A then A then B, we have mathematically that

Cv = B(A(Av))

Now, it sure would be nice if there was a way to "multiply" matrices together so that the product B(AA) = C. In abstract terms, we want multiplication of matrices to correspond to composition of functions, since we can use matrices to represent functions. If we define multiplication of matrices in the usual "weird" way, then this property holds!

This lets us examine composing functions by multiplying matrices, which is a very powerful tool indeed. Another interesting thing to notice is that this explains why multiplication of matrices is not commutative (that is, order matters). If we rotate counterclockwise by 90 degrees, then reflect over the y axis, that's different than reflecting over the y axis then rotating counterclockwise by 90 degrees. So BA is not equal to AB.

2. I'm not sure that I understand your question here.

3. This question is beyond my ability to answer, so I'll leave it to someone else.

4. On the one hand, there are tricks to see this (e.g. 0.333... = 1/3, so 0.9999... = 3 * 0.3333... = 3 * (1/3) = 1), but I expect that you have seen them and been unsatisfied by them (I don't believe that these are satisfying explanations, personally, because I have simply replaced the unknown fact 0.999... = 1 with the equally unfounded fact 0.333.... = (1/3)).

In order to really understand what's going on, we have to understand what we really mean by 0.999... What is meant by 0.9? This is equal to 9/10. How about 0.99? This is 9/10 + 9/100. 0.999 = 9/10 + 9/100 + 9/1000. Following this reasoning, 0.999... is equal to the infinite sum 9/10 + 9/100 + 9/1000 + ... + 9/10^k + ...

Since there are infinitely many 9s in the decimal, we must add an infinite number of fractions of the form 9/10^k.

We've opened up a can of worms here - how do we add up infinitely many things? Also, if we add up infinitely many things, how can the result possibly be finite?

A Calculus II course will answer these questions rigorously, but let me attempt to make an explanation here.

Think of it as a game of wits. You start. Choose a number less than 1 - any number at all, just so long as it is less than 1. I'll try to find some number k such that 0.9999...999 (with k 9s in it) is closer to 1 than your number.

Suppose that you pick 0.97. That's pretty close to 1, only 0.03 away. But 0.01 < 0.03, and 1 - 0.99 = 0.01, so 0.99 is closer to 1 than 0.97. I win.

You try again, this time picking 0.999878787. That's much closer to 1, but still 0.000121213. Since 0.0001 < 0.000121213, 0.9999 is closer to 1 than 0.999878787. I win again.

In fact, let's suppose that you pick the number x. Since x is less than 1, 1 - x is some positive (but very small) number. No matter how small it is, though, I can find some k such that 1/(10^k) is even smaller (this fact is called the Archimedean principle, and it is one of the fundamental theorems of the real numbers). Another way to think about it is that 1/(1-x) may be very big, but I can find some k such that 10^k is even bigger. But then 0.999...99 (with k 9s) is going to be closer to 1 than your number, and I'll win again.

Now, what does this mean? Since 0.999... is bigger than 0.999...99 (with any finite number of 9s), it is bigger than every number which is less than 1 (by the game we just played). The smallest number which is bigger than every number less than 1 is just 1. Anything smaller just won't do (since there would be numbers between it and 1). This shows that 0.999... is at least 1.

I hope that helps. It isn't as elegant as the featured answer, but maybe it can be a little bit helpful to you.

I have mixed feelings about this explanation.

On the one hand, the mathematician inside of me is raging "this is neither specific, nor rigorous!" In my opinion, this definition is as close to useless as a mathematical definition can be, since the person who has this (and only this) will be hard pressed to answer any other questions involving complex numbers. For example, what is multiplication by i+1? Without further understanding, I don't think such a question can be answered.

On the other hand, the teacher in my head is celebrating. This person managed to take a concept normally feared and hated by the population at large and made it interesting and understandable. The response is well thought out, and is much more engaging to the reader than any response that I could come up with. In short, this explanation is more likely to make the person interested in mathematics, while a more technical definition would probably just make them zone out.

What do other people here think about this? Is the clarity and interest worth the loss of utility? Am I totally off base?

While it depends on what you mean by "incredibly difficult", I would say that if you are doing any work with 3d modelling, it would be necessary to understand basic linear algebra. These concepts will transfer directly - if you are representing your model with points in 3-dimensional space, then displaying those points on a screen is a projection map, and can be described (and quickly calculated) with a matrix. This is a typical example.

Another common example would be a course in number theory. A common topic in number theory is public-key encryption, much more in-depth than is sometimes covered in other courses. In addition, the techniques that you learn in such a course can easily be applied to any "number-crunching" you would need to do. Here is an example:

Problem 1 from Project Euler (http://projecteuler.net/problem=1): Find the sum of all the multiples of 3 or 5 below 1000. A brute-force program for this is simple, but it is much faster if we first notice some patterns: (3 + 6 + 9 + ... + 999) = 3 * (1 + 2 + ... + 333). But then (1 + 2 + ... + 333) = 334333/2 (this would be easily recognized by someone who has taken a number theory class), and so the sum of all of the multiples of 3 below 1000 is 3 333 * 334 / 2. Similarly, for 5, there are 199 multiples, so the sum of all of these is 5 * 199 * 200 / 2. But then we must subtract the multiples of 15, since we double counted them: there are floor(1000/15) = 66, with sum 15 * 66 * 67 / 2. So the answer is (3 * 333 * 334 + 5 * 199 * 200 + 15 * 66 * 67)/2 = 299498.

That was a particularly trivial example - number theory will give you a lot more than some parlor tricks like the above. But this example took a loop through n numbers and reduced it to a single line of simple arithmetic. And for someone who understands number theory, this algorithm is as quick to think of as it is for a typical CS student to think of the brute-force algorithm.

I would also venture that a course or two in "Abstract Algebra" would be both beneficial and accessible to CS majors. This is less immediately practical than linear algebra in the above example, but closer in its style of thought to programming than, say, calculus, topology, or differential equations. The abstraction (which many math majors complain about) would be fairly straight forward for a CS major, I would think.

In addition to the other comments, it should be noted that this happens after the fact. You don't really know what your bugs will do until they turn up, which you can only know after you run the program.

Similarly, they didn't know what their results would be until after they ran the experiment. At that point, they can either: share the results, or hide the results. Hiding the results is the absolute worst thing that any scientist can do. The only time the option not to share the results would be acceptable is when they can be discredited or discounted (and sometimes they should still be shared). Since they couldn't discredit their results (they tried), they took the only responsible option remaining.

Indeed, they didn't just publish the results, they asked others to look for the bug. Like others have said, this is the best possible outcome.

I think that you are right, in that this would probably prepare students for jobs very well.

However, I think that a Bachelor's degree is supposed to be about something other than job training. Sure, there is some overlapping material between "prepare for a job" and "get a Bachelor's degree," but not very much. People who have a Bachelor's are expected to have knowledge that is both broad and deep. That is, they are expected to know far more than they need to know, and have skills outside of their area of expertise.

Not that there is anything wrong with your approach, I just think that it disagrees with the overall goals of a standard university degree program, which is not terribly concerned about job preparation.

If you can just copy and paste answers, with source, then you haven't actually learned anything (technically, you might have, but there is nothing in your work to support this conclusion - unknowledgeable until proven knowledgeable). The grade you receive should be a measure of the skills you have obtained, and the things that you have learned, rather than the results you produce. Sure, you can do this in the real world, but in the real world you are not trying to show what skills you have obtained, you are only asked to produce results - this is the major difference between the education environment and the work environment.

Immediately afterward, though, the parent does step in and help out - a small amount. It sounds to me like the parent here is trying to subtly remind the employee that this is a learning experience for Guy. The entire exchange seems, in my mind, to take no more than 30 seconds away from the busy customer service representative, which is about 20 more seconds than it would have taken for the parent to ask in the first place.

There are, of course, some things wrong here. The child should not interrupt another customer to get help - that is rude. But part of learning is making mistakes, and I think it totally agrees with the spirit of the rest of the story to let the child make the mistake, and point out afterwards what improvements could be made (to be clear, the story does not say that this happens, I am only suggesting the hypothetical that it could happen, and I claim that it would justify the inconvenience). Again, this is slightly inconvenient for the people involved, but a great life lesson for the kid. By making these social mistakes, he gains experience with the types of subtle cues people let on when you are performing a social faux pas. A short explanation after the fact from the father can help the child not to make the same mistakes again.

Of course, this particular father may have done none of those things. He may be a sadistic bastard who pointed out the most busy, overly worked employee in the store simply because he likes bothering people, then congratulated his kid on making a spectacle. But the spirit of the story is lighthearted, so I interpret his actions as well-meaning (and probably slightly exaggerated, for effect).

I don't think that people should have to put up with any obnoxious thing a kid does. In fact, I think it would have been a great thing for the other people in the story to treat him like an adult, just like the dad wanted them to. If the clerk had said "hang on, I'll be able to help you in one minute", that wouldn't have been a problem. Or if the other customer had said, "pardon me, but I am in the middle of a question here - can you wait?", then the obnoxiousness of the situation could have been avoided entirely and the kid would have gotten a great lesson out of it.

People just need to relax. I know kids can be obnoxious. It bothers me to hell when I see children running around, bumping into people and sword-fighting in the aisles of the supermarket while the parent seems totally oblivious. But these sort of minor inconveniences need to be viewed as they are - minor. Total time wasted of other people in the store on this day - maybe about 45 seconds. Hardly a big enough deal for you to label the daddy a "self-entitled asshole".

By the way, I didn't down-vote you. I don't agree with you, but I think that you made a perfectly valid point - children are annoying. In fact, I up-voted you for it.

Speaking from personal experience, I think the positives of this outweigh the negatives. I've had many customer-interactive part-time jobs in my life, and I think that it can be great fun when adults have their children do the interacting. So long as the store is not too busy, and the parents provide intervention when it is necessary (as in this particular story), I think that most people will not be inconvenienced significantly.

More importantly, I think that forcing your kids to do these types of interactions from a young age is very instrumental in their growth and development. While I am not well-versed in childhood development, I think that I rarely (if ever) was encouraged to do these sorts of things on my own when I was young, and that (possibly due in part to this) I had to break an irrational phobia of these types of social interactions at around the age of 18.

I was literally afraid to ask for help in a store, or to call a business for almost any reason. Mentally preparing for phone calls with strangers, even something as innocuous as finding out store hours, could leave me shaking.

Fortunately, I've managed to overcome this anxiety. I can't know for certain whether more of these types of interaction at a young age would have prevented this problem, but I think that it is reasonable to believe that it could have. The slight inconvenience is a small price to pay for this sort of developmental skill-building.

That's not to say that some people don't need to reign in their kids now and then, just that an appropriate amount of interaction is healthy and really not very annoying at all to others.

In fact, it says that 6174 is more than just a fixed point, it is also the only fixed point, and an attracting fixed point. That is, every four digit number with transform to 6174 in finite (indeed, less than 7) steps under Kaprekar's operation.

This is significantly stronger than just being a fixed point. As an analogy, when modeling population dynamics, one often finds both stable and non-stable equilibrium solutions. The stable ones are the only solutions that can occur in nature, and so their stability is an important thing to note. This stability often (I hesitate to say always) is the result of the solution having the same attractive nature as 6174 does.

I think that I'm one of those elitist attitude people. I apologize for this, I don't mean to sound elitist (I certainly don't think of myself as elite). I agree with you that we are just skilled laborers.

What I mean to say is that graduate students are adults. Let us decide for ourselves what is important. I agree that hyper-specialization is probably not best for humanity, and that interdisciplinary work can really crack open some unsolved problems. And there are jobs that want people to have these skills. I think that a grad student is much more hire-able (for many positions) if he/she has some broad skills. But why should getting a PhD require this? Shouldn't it be up to the person doing the work to decide whether or not the work they are doing will help themselves/the world?

I simply don't think that we should alter our degrees to match requirements for job openings. We certainly should alter our skill sets, but a PhD should be a certification that you are knowledgeable and capable of research. It's not (and I don't think that it should be) a certification that you are ready for a job (regardless of what that job is).

In my opinion, "Getting a PhD" and "Getting a Job" should be as disjoint as possible. Otherwise people who want one will do the other simply because they have to, and this is good for nobody.

I think that a Master's degree is more the type of degree that this is suited for. Most people that I know who are getting Master's degrees are getting them for the express purpose of getting a specific job. This is fine. Most people I know who are getting PhD's are getting them for the purpose of learning their subject, not to get a job. Sure, they need a job after they graduate, but this is tangential in their motivations, and should therefore be tangential in degree program as well. At least, that's my opinion.

Thanks for pointing this out. I agree with you that having managerial skills would be important to someone who manages people, and that many faculty in academia do need to manage people.

My point is not so much that nobody needs management skills, it's that not everybody needs management skills. Let me decide whether or not I will need those skills for myself. Perhaps I already have some management skills from an undergraduate education, or perhaps I simply have a natural talent for it. If I think that I need management skills, I will learn them - I have plenty of opportunities to take such courses or seminars, if I want. I'm an adult, you don't have to force me to do something because I might need it later, just tell me the facts and I will decide for myself.

But if I think that it would be a better use of my time to do something else, like collaborate with a researcher at another university on a new problem, let me decide that. I hope that whatever job I apply for decides whether or not to hire me based on my specific skill set.

I guess what I'm trying to say is this: a PhD is not (or should not be) a certificate declaring you qualified for a job. Not for an academic job, nor an industry job, nor any other job. It is a certificate declaring that I am very knowledgeable in a specific subject, and that people better and wiser than me believe that I can do meaningful research in the future.

If I want to show people that I have other job skills, I must show them some other way. A PhD only verifies the above, it doesn't verify that I am a good teacher, a good manager, a good communicator, a friendly guy, or that I am in any other way competent. If a job wants skills that I don't have, they shouldn't hire me.

For your other point, I totally agree that this attitude keeps the salaries low. Unfortunately I don't know what should be done about it. Either we can lie, and pretend that we only enjoy our research to the extent that it pays the bills, and otherwise do not enjoy it, or we can actually try to stop enjoying our work. Either seems silly. You can't hide what you love, and if other people try to take advantage of the fact, well, so be it. In fact, I can understand the counter-argument: just because work is important, doesn't mean we should pay people more than the bare minimum they will accept. It's not fair, and it isn't right, but it is rather true.

From the article: "Imagine bright young things entering a new kind of science PhD, in which both they and their supervisors embrace from the start the idea that graduates will go on to an array of demanding careers — government, business, non-profit and education — and work towards that goal (see page 381). The students meet supervisors from a range of disciplines; they acquire management, communication, leadership and other transferable skills alongside traditional academic development of critical thinking and analysis; and they spend six months to a year abroad."

This sounds a lot like my undergrad degree, and not at all what I expect from a PhD.

In my opinion, people get bachelors degrees in order to get a job. It used to be that was what trade school was for, but now it seems that you need a degree just to qualify for an interview in a lot of fields. In my opinion, though, you get a PhD because you are interested in the subject - not because you need it for a job.

I think PhDs are an end in themselves, not a means to an end. Please don't water down my PhD by forcing me to do cross-discipline studies that do not interest me, or by teaching me management skills - I'm not in business school.

Now, a smart PhD student would realize that they might be getting a job in the industry, and they might not be able to get the job that they are hoping for, and plan accordingly. For example, while I hope to get a job at a research institution when I graduate, I realize that it is possible that I may not, so I take extra-curricular courses and seminars to prepare me for other jobs. But I do this because I choose to, not because some people somewhere else think that I might not get the job they think that I want.

Frankly, it may be rude, but if there are more PhDs than job openings, that's a good thing. It creates a lot of competition. It keeps mediocre people (or at least it helps to) out of these critical positions. It prevents people (or at least helps to) from trying to get a PhD solely for the purpose of getting a job, which cheapens the degree all around.

So long as PhD students are aware of their future prospects (and I don't know any who aren't - this stuff is crammed down our throats), I think that this is fine. Entrepreneurs do this all the time. They know that starting a business is not guaranteed - indeed, they might go bankrupt. But they might hit it big. This isn't a problem, and we don't start government initiatives to help entrepreneurs who fail get a good job anyway. We don't make entrepreneurs take back-up courses in plumbing and construction, so that when some of them fail they have a fall-back job. They don't want that job anyway. If they did, they would prepare for it on their own, because they are smart people.

In science it is the same thing. The people who love science for science's sake - these are the people we want to be scientists. These are the people we can trust with tenure, because they would do research even if we didn't pay them at all. They are smart, and they know that they might not make it into their dream job, so they can prepare themselves for alternatives if they want. Or they can shoot for broke, spend years of their lives doing fantastic research and focusing on doing only the things they love (which, by coincidence, help the rest of the world). But forcing them to prepare for alternatives because some of them might benefit from it is just a turn-off.

I think the mentality that most of the research done is useless crap is a common mindset that can hurt the entire field of science if too many people believe it. There is already a feeling among the general populace that most research is useless, and that we should be investing more money elsewhere.

Research (as a whole) is a lot like mining. The human race can put in so many hours of work researching new areas, and the majority of it does not produce immediately fruitful work. Only a very small percentage of the work done results in "breakthrough" technologies that are the main reason money is invested into research anyway. But I don't understand why people think that if we put less time into it, we will cut out all of the garbage and only bring out the really good stuff. You have to wade through a lot of dirt to find a diamond, and cutting your crew down won't help that. It helps to look in the right places, but at the end of the day, you really don't know what will end up producing magnificent results.

I agree completely. I don't think that recreational mathematics should replace calculus. I think that there are some definite changes that should be made to mathematics education, but that the majority of these should take place at a much younger age, even grade school. Early education is the perfect place for recreational mathematics. Fostering an interest at a young age (rather than force memorization of formulas, algorithms, and tables) can help a lot to keep people engaged throughout their lives.

I think that by the time students reach the college level (which I think starts with Calculus, even if they reach this level in high school), they should be able to deal with some of the more boring parts of mathematics, sustained only by nuggets of usefulness and beauty. After all, not all mathematics is wondrous joy.

I also agree that many of these problems could be fixed simply by teaching better. In calculus, for example, the beautiful thing is how closely it mirrors nature. Calculus states that if we know the position of an object at all times, we also know the velocity of the object at all times. This is a beautiful and fundamental result, one that I feel is not stressed enough during a first-year calculus course. I think that it is amazing not just that we know (in theory) the velocity, but that we can actually calculate it explicitly (in most circumstances) or approximate it numerically (in most circumstances). The parts of calculus on how to do that are somewhat less beautiful and paradigm-shifting, but are necessary, and adult students ought to be able to deal with that.

While I would agree that a basic programming course might be helpful as a required course for all schools, I don't understand why it should replace Calculus? There is room for both, and they have different goals and advantages/disadvantages. You say yourself that calculus is awesome for several reasons - why do you think we should replace it?

If it is only because a different course might be more practical, you have entered a slippery slope. College is not the same as trade school - we don't learn things because they are practical, but because there is an advantage simply to learning. I believe that all basic college courses should aim to promote critical thinking above all else, as the practicability of any given subject will vary widely between students.

In mathematics?

Primary sources are way above the level of most college mathematics students, much less high school or younger students. Almost anything written in the last 50 years requires very advanced mathematical ideas, while anything written before that uses outdated notation which is very difficult to understand. Moreover, the primary sources for common ideas are often spread out over huge periods of refinements - going from a small idea to a general notion over the course of years, perhaps involving dozens of papers by multiple authors. One notable exception to this: Euler's papers. These might be readable by some students, yet when I see them presented in college courses, they are often presented by a secondary source for clarification.

The other thing I don't agree with is your use of the word "interpret." I don't think that it is common to find a mathematician who believes that mathematics is open to interpretation.

Lastly, original research in mathematics is hard. It's not the interpretation of previous ideas (although it is occasionally relating concepts previously thought to be unrelated - some consider this the most important type of mathematical result, but it is much harder to do than you might think), but the formulation of new ideas. Even to know if an idea is new requires a great deal of mathematical training.

Perhaps I'm wrong about this, but I think that for young mathematics students your ideas are unfeasible. Remember that the original sources for a lot of basic mathematics are hundreds of years old. Even Euler's famous writings on geometry (the exception that I said might be workable) were largely a rewriting of previous ideas into a coherent whole - basically a textbook - and could not really be considered an original source.

Edit: my apologies, I misunderstood. I did not realize that we were talking about teaching history, in which case I defer to someone of greater experience.

I think a lot of people have problems with teachers unions because they think of teaching like any other job. It's not - here are the important differences:

Pay raises. It's a fantastic idea for a company to increase the pay of individuals based on performance. If individuals do well, the company does well and makes more money, and can therefore afford to pay the employees more, so this reward system is great.

In schools, there is no profit. So if teachers do better, their students perform better, and the teachers should get paid more. But they don't, since this has not brought any more money into the school. In fact, if schools are doing well they are less likely to have their budgets increased, since people will see no need to increase the budget of something that is functioning well.

This fixed-budget kind of thing is the root of the difference between the education system and private enterprise. In a business, if you are not making enough money (that is, if you have a spending problem), you cut back on things: often you may lay off employees. You will have fewer employees now, but since your business isn't doing well you probably have less work to do, so it can even out. Or, if it doesn't even out, your business closes down.

Neither of these is true of education. Except for rare cases, it is unthinkable to close down a school. Obviously closing would bad for students and education in general, so it is avoided.

More importantly, workload doesn't flex as easily in schools. That is, if you run into budget problems and have to cut teachers, the other teachers must work harder to pick up their slack, since the student body population remains unchanged. Thus it is in the interest of both the teachers and the students to keep budgetary layoffs to an absolute minimum.

So what do you do with a tight budget? Who knows? Removing teachers decreases the effectiveness of education severely. So does the numerous other things I've seen schools do - stop providing textbooks, make the quality of food go down, shorten the school day, remove planning time of teachers, stop giving teachers their own offices, stop plowing the parking lot, etc. In other words, make schools suck in every way possible.

I don't have a good solution for this, beyond the obvious "spend more on education". Personally, I think this is our best bet, but that money would have to come from somewhere else, and people are not willing to pay more taxes for it.

Maybe a bit off-topic, but this scam has always intrigued me. You can, in fact, do better than just "Money back guarantee!" You can do as much as, say, "We'll give you your money back, plus an extra 50% of the fee!" So your "customers" pay $50, and if you are wrong you give them back $75. You still make $12.50 on average per customer by guessing randomly, so long as you scam sufficiently many people.

I've got one problem, but I haven't really figured out how it fits in exactly.

Assuming a lot of things in how this plays out, I don't find it terribly unlikely that some subscription services will have to raise their rates to maintain a profit. Maybe not a big problem for iOS users, since they get the benefit of using the app store for purchases.

What about non-iOS users? I subscribe to Netflix, but I don't use any apple products to access that subscription. According to Apple's rules, Netflix can't have the subscription available at different prices, and so my price goes up because of Apple's policy.

There may be some ways around this, like Netflix offering two plans - one that includes iOS use, one that doesn't - without hiking the latter. But they might not.

I haven't yet worked out what the "fairness" of this all is. As a consumer, I certainly don't feel that I am entitled to never having price hikes, and I realize at the same time that some of the money from products I pay for is used to support products that I don't pay for. But this feels wrong to me - maybe it's just the directness of it. Apple says "pay more" and even people who don't buy Apple products have to pay more.

I own this book, and I use it occasionally, but it really requires the right audience.

Firstly, it has several parts. Some of these are historical, philosophical, or otherwise interesting to a general audience, but some of them are technical (I would put most of pages 157-729, nearly 60% of the book, in this category).

For people that are interested in mathematics, but do not have a fairly extensive background (I would estimate 2-3 courses beyond the calculus series), these technical sections are probably not very useful (and certainly not entertaining).

For people (like me), who are still pursuing an education in mathematics, I would say this book is indispensable. It gives a great overview of individual branches of mathematics, including fairly rigorous explanations of important results and conjectures.

The exposition portions of the book are good, but don't warrant the high price of the book on their own. Overall, though, it is a tremendous reference.

Edit: I forgot to mention that the folks who write the articles are all outstanding mathematicians - many of them Fields medalists or other award winners. That makes for some very good reading.

I've wanted one for a while. As a student, I take notes all day long. Digitizing them by hand is a pain, but if they were immediately put into digital format, that would be great. I could carry around years of notes on a flash drive, conveniently organized by subject. With a little work, you can even add keywords/subject searchability to notes.

Currently, I could take notes on a laptop, but in mathematics this is extremely difficult (you have to be very very fast with LaTeX), and realtime diagram entering/editing is basically impossible with a laptop.