HilbertSpace

> I think you misread and/or misinterpreted OP's point

His "point" was not clear: What was more important in the article, that there was a Google+ community discussing some "problems" or the "problems" he listed of concern to that community?

The leading problem listed was difficulty in getting published, and I suggested a way to get published: Do well on "new, correct, and significant" and, in particular, 'mathematize' the field.

Responding to that "problem" was fully appropriate.

In more detail, one of the best ways to mathematize CS is more in probability, especially as it connects with 'information'. It does connect; trust me on that one.

That was a good and helpful suggestion and should have been welcome.

> Later came these repeated and almost manic allegations of some secret cabal of MODs (using your capitalization) who are after you.

There's a LOT of evidence that some MODs are angry with me and for the past month or so have been attacking me personally. For several reasons I've listed on this thread, much of the downvoting had to be from MODs. Apparently HN is a PG PC sandbox, and MODs will downvote anything they feel violates the PC norms of HN and do so without responding.

PG won't respond to deny this.

I would have been a fool to ignore the clear evidence that I was being attacked and a wuss not to defend myself. It's a scandal for HN, and PG has not responded.

On another user ID I was 'hell banned' here at HN.

Today I finally responded in plain terms to defend myself. If by defending myself I get banned from HN, then so be it. But for now the HN community will see that HN attacks some users personally.

I tried to help CS researchers get published. My claim, easy enough to understand, is that the research paradigm of CS is largely bankrupt, and my recommendation is to further mathematize the field. They are in a "wretched mess" if only from their statement of their "problem".

> puerile delusion of grandeur on display was mildly amusing and continues to amuse me

Insulting nonsense. There is no delusion involved. I was quite apparently attacked and have been several times for about a month now. It's personal, not technical or anything else.

> if I may put it in my own words

My own words for my own statement were much more appropriate: To repeat, CS is close to 'information technology', and there 'information' should be taken seriously mathematically. The R-N theorem is one important approach.

The "lack of background" of the Editors in Chief and chaired professors of computer science was surprising and shocking. I used some group theory and probability based on measure theory and got a new family of statistical hypothesis tests both multidimensional and distribution-free and applied them to ASAP detection of anomalies in server farms and networks, a good CS problem, and bluntly too much of the best of the CS community couldn't handle the math.

So, there is a big, huge gap between some of the best of current CS and what it would take to do at all well applying some 'modern probability' to some CS problems. Due to this gap, my suggestion to learn the math is appropriate and should be seen as helpful.

In simple terms, if a student wants a research career in CS, then as an undergraduate it is much more important for them to major in math than CS. And likely similarly at the Master's level. That is a surprising point but potentially quite helpful and should be welcome.

"On other hand if you can show that your favorite Neveu calculus can be proved sound according to some computable logic, that would indeed be helpful."

Nonsense. Neveu is based just on set theory, axiomatic set theory if you wish, the foundations as in Bourbaki if you wish, or P. Suppes, essentially the same as all of math for the past 100 years or so. For the importance of that material, there is nothing anyone should have to "prove". 'Computability' is not directly relevant.

> For example, to follow up on your own example, ... such a submission will just get dropped

I never had a submission dropped in the sense of 'rejected'. For the paper in question, I sent copies to some journals and just asked if they would like a formal submission. Some journal Editors in Chief said that their journal couldn't review the math. With one such I wrote tutorials before he gave up. So, I didn't make a formal submission and, thus, never got rejected or "dropped".

One journal welcomed a formal submission, and we went forward. They had problems getting reviewers who could read the paper, and reluctantly I suggested a qualified friend who did a good review. Eventually the Editor gave up, and for more reviews apparently the Editor in Chief walked the paper around his campus, had the CS guys say the problem was good and had some math guys say the math was good, and the paper was accepted, in an archival journal. I was invited to present the paper at a conference but declined. I just wanted to publish the thing and be done with it; I had no desire to go to a conference.

It's clear: In being mathematized, the CS community is very short on the needed math. I'm sure it's happened before, e.g., Hamming and the start of coding theory based on finite field theory, likely poorly known in the CS community then. For a researcher with the math, there are important CS problems that can be solved fairly easily just sitting there and, thus, are good research opportunities.

E.g., in Feller II is renewal theory, and it has some obvious applications to a lot that goes on in a server farm or network. For more, in Neveu is martingale theory, and there is one of the strongest inequalities in math (knock off the strong law of large numbers in one line) and more, and it's easy to find martingales in nearly any stochastic system. Should be able to get some nice, new, strong inequalities in many algorithms and processes important in CS. There's plenty that can be done with stochastic optimal control. There are the applications of 'machine learning', that is, statistics done very badly, and can solve the problems much better with statistics done well.

For the paper I wrote, there's much more that could be done; one could run off a dozen or so papers as a 'stream' in roughly the same direction by changing some of the assumptions better to fit various real situations.

So, sure, I'm suggesting some 'field crossing', long known to be a good approach to 'innovation'.

Moreover I'm suggesting exploiting some of the best, rock solid math of the last century. So this is a very sound, conservative suggestion.

That the CS community is very short on this math is a big point and a good research direction for anyone wanting to publish in CS.

Indeed, as is painfully obvious, CS has been taking intuitive and heuristic approaches far too seriously and neglecting solid math approaches. This situation makes CS look dumb, but the flip side is a terrific opportunity.

Again, with the right math from the last century, now should be able to knock off important CS problems by the dozens like shooting fish in a barrel.

HN and you don't like this remark. Fine with me.

I tried to offer some help, but you don't want to hear it and want to criticize me for offering. Fine: I won't offer. I've already deleted the post.

With your remark on 'computability', there's no more reason to respond to you.

I wrote a good post. The CS community lost out.

Can lead a horse to water but can't make him drink.

WHAT a MESS.

I wrote:

> The main criteria for research work are "new, correct, and significant". Do well on all three of these and have NO PROBLEMS getting reviewed and published.

So, I said that have to "do well".

So your

> You couldn't be more wrong.

is a bit strong.

I've published maybe a dozen papers, and I've never had a paper rejected or needing significant revision.

I've never been interested in publishing, and maybe the situation has changed a lot in the last few years while I've being doing other things.

Whatever, my original post to help CS profs get published was not wanted, and I've deleted it. The CS profs can struggle on their own.

As I explained in

http://news.ycombinator.com/item?id=3421033

early in the article was that the 'community' had a "serious problem" which was getting published. So I outlined how to get published, and that should have been a welcome contribution.

It wasn't very clear in the article if the article was mostly about the community or the 'problems' of the community. Whatever, I gave a solution to what seemed in the article to be the most important "problem" of the community.

But my solution wasn't welcome, so I deleted it. CS profs can struggle on their own to get published without my help.

Because there is a pattern that would not apply to the relatively random users and needs a very persistent user, that is, a MOD. Because the downvotes happen QUICKLY, as if from a MOD. Because the downvotes don't come with replies. Because only a small fraction of users can downvote. Because part of the pattern of what gets downvoted is material that is relatively advanced technically and, really, a challenge to CS.

My original post was a nice contribution to CS people having trouble getting published. But, if people don't like such a contribution, then they won't have it. So, I deleted it. It's gone. The CS profs can continue to struggle to publish without my help.

It's become clear that HN is PG's PC sandbox.

My original post was downvoted within minutes of when I posted it, and there was no reply. This is NOT nearly the first time. There is a pattern, and I'd be a fool not to conclude that one or more HN MODS are personally angry at me.

"schizophrenic"? Total nonsense; look up the meaning of the word, a very specific medical term. Not even paranoid. I'm just defending myself against a pattern over the past few months at HN. There is a pattern of my being attacked, and this one post is just the most recent example.

As far as I can tell, I'm responding to every response that can use a response.

I've more than once asked PG to correct this. We will see what he does. This situation is a scandal at HN.

This is NOT nearly the first time. There is a pattern, and the only reasonable explanation is that one or more HN MODs just do NOT like a lot of my posts and jump to downvote ASAP with anger.

A guess that fits the facts of what they don't like is anything critical of any other media outlet, anything about CS, or anything about venture capital. But it is necessary to guess since the downvoters essentially never respond to the substance of my posts. The situation has clearly become personal for them and, then, me.

Problems getting published in CS was a big point early in your article, and that is what I responded to. For more on this point, see

http://news.ycombinator.com/item?id=3421033

below.

The article had, early on:

"I can’t speak for other disciplines, but within computer science, I’ve always felt that these meta discussions were inadequate — in terms of volume, vigor, and format. They happened mostly at conferences were largely limited to more senior/well-connected researchers, and lagged behind some of the serious problems that have accumulated such as restrictive publisher copyrights and very low acceptance rates at journals and conferences."

So, the claim here I noticed was:

"some of the serious problems that have accumulated such as ... very low acceptance rates at journals and conferences."

So, a big topic in that community was a "serious" problem getting published. So, I outlined how to get published. Simple.

In a word, my outline said to 'mathematize' CS. That suggestion is potentially earth shaking, likely not already in the community, and should be a welcome contribution.

"Rant" is an insult.

"Far cry" in what sense? The AAAI IAAI conference was about the best could expect from the AAAI.

I published in JIS because it has a wider audience than, say, another Elsevier journal on 'theoretical computer scince' or some such, which I did consider.

For JOTA, I contend that such applied math is closer to the important future of CS than nearly any journal in CS. If I am correct here, and my theme in my post was that CS needs to 'mathematize', then I have been helpful.

There's no "Big Shot", another insult, involved. I just listed where I got published. From the article, one of the leading topics on the Google+ community was the difficulty of getting CS published, and I indicated that I had essentially been successful even though I wasn't much interested.

For the academic 'pecking order' of the journals, I have never cared since I've never had any academic aspirations.

But for someone having trouble getting published, the journals I listed are a big step up from no publications. So, if they can do what I did, then they would be ahead. So, my advice should have been seen as helpful, which was clearly the intention.

That lament was one of the leading topics in the Google+ community that was the the subject of the article.

Yes, the article was mostly about the community and not the issues of the community. But the article did list some of the leading issues of the community, and I responded to one of them. Uh, the issues of that community are more important than just the existence of the community! So it should have been appropriate for me to have commented on one of the issues of the community.

I was attacked, downvoted, for my main comment, and then defended myself. This is NOT nearly the first time for such. Usually the result of the downvote is to send my comment to the end of the thread where it is effectively lost, which is clearly the intention of the MOD.

Why a MOD? Because only a small fraction of the users can downvote, and users instead of the MOD attacking me personally write rebuttals. So far I have not a single rebuttal to any of the content of my original post. I did a great service to some lost CS profs -- outlined for them how to publish and, then, got attacked by a MOD.

Then I defended myself. You found my defense offensive. It was, but not nearly as much as that of the attack of the MOD.

The attacks are from downvoting, not upvoting. Only a small fraction of HN users can downvote. With so much downvoting, it has to be the MODs who are doing it. It's so consistent that it's a personal attack on me.

That's what PG's HN MODs do also, and then they attack, usually without any comments. It's PERSONAL. It's a SCANDAL.

There is STILL not a single, meaningful comment in rebuttal to anything I wrote. Not a one.

I outlined for CS profs how to get published, and the HN MODs got their bowels in an uproar.

PG: STOP these GD personal MOD attacks on me. STOP IT.

"Credibility"? Look up the Radon-Nikodym theorem. I suggest Neveu, 'Mathematical Foundations of the Calculus of Probability'.

"Rant" is an INSULT. The 'rant' was in the article complaining that they could not get their paper published. I got them back on track.

In short, HN has a chicken shit MOD that routinely attacks my posts but makes no comments. This is PERSONAL and is CRAP.

And the situation and the attacks really SUCK, much more than anything I've written.

Okay chicken-sh*t, mod, downvoter, if you have a tummy ache, then upchuck so we can all see what you can't take.

PG: I'm being personally attacked by your mods again. This is a HN scandal.

Yup. But in this case I know all too well just what the heck I'm talking about, unfortunately. Hint: I live in Dutchess County, NY and for years worked in Westchester Country, NY.

Besides, the NYT wrote just a puff piece. Now just why would the highly self-esteemed, formerly highly revered, long pseudo-objective NYT do that?

Besides, what details did the NYT piece provide?

Wise up and appreciate a gift when you get it!

On close examination, you may find that their Research Division is mostly just a patent shop.

Likely their 'services' work is mostly just 'help desk'.

Likely much of what they are doing is what they did decades ago called 'facilities management' or 'service bureau'.

Likely for new products, they find a small company with a good customer list, buy the company, and have the IBM sales force include that company's products in the list of products available. That is, they don't really try to 'develop' such products on their own.

So, in many companies, the CIO can 'bring in IBM' and go play golf.

Likely a good CIO could give faster results, more innovative and valuable results, at lower cost. But not all CIOs are good.

> I just want you to be clear on the fact that you are deliberately choosing to leave science behind and have entered the realm of science fiction.

Not really: I'm guessing. Or in more erudite terms, I'm conjecturing.

You seem to be saying that science is what is all wrapped up and solid and that anything else is "science fiction". That's a bit rigid! Also that view would keep us from ever discussing the unknown before it becomes known, and science is a long march through millions of small and dozens of huge cases of the unknown becoming known.

Indeed, if read Einstein's special relativity paper, it reads like conjecture or guessing. That is, when he wrote the paper it was not at all clear that the Lorentz transformation had any physical reality. That paper became accepted as the real physics only later.

There's no solid proof that we can't go faster than light (FTL), and I have much more than enough math and plenty of science to have seen the arguments.

Sure, for any particle with ordinary mass as we know it, as it goes faster through space and approaches light speed, its mass increases and at the speed of light would be infinitely large. Right. So, for that case, FTL would be impossible.

But we don't really know the deeper mechanism for this mass increase. Thus we are like someone in the 18th century saying that travel faster than 60 MPH would be impossible because no horse could move its legs fast enough. That is, once we understand the deeper mechanism that establishes the speed of light speed limit, maybe we could find a way around the mechanism and the limit. The mechanism seems likely connected with the Higgs field, and we don't understand that very well yet.

E.g., we know so little about dark matter we can't be very sure it can't go faster than the speed of light.

For dark energy, we are assuming conservation of energy much as we understand it, estimating the energy of dark energy, assuming that E = mc^2 also applies there, assuming that how matter and energy curve space in general relativity continues to apply, and then concluding the huge mass of dark energy. That's a lot of assumptions from extrapolations. We're assuming that what we see for ordinary matter in accelerators applies to dark energy; that's a GUESS.

So, in science we need to be able to talk about possibilities not yet established. Such "talk" is not the best science, but it's also not "science fiction".

> "visible from our telescopes today."

Our nearest other star is 4 light years away. 1000 light years away may still not cover many planets good for ET to colonize. But for us now to see evidence of a colony on a planet 1000 light years away or more would be TOUGH.

Note: The planet hunters mostly don't actually see the planets but just shadows or evidence in star wobbles, etc.

But we are putting up some new telescopes with some astounding resolution, etc.

Yup, we both might be correct!

I had a 'qualification': "Usually the CEEB, SAT, and GRE", and with that we both can be right!

For those tests, at the level of detail of the distribution of the scores, it can be tough to get solid data.

But it is easy and common in educational testing to scale 'raw' scores so that the 'scaled scores' are Gaussian.

Also in educational testing, it is easy and common to have enough data on individual questions so that the distribution will be known fairly accurately for a test made of such questions -- my father did that for years as the main 'educational architect' at one of the world's largest and most important technical schools, with 40,000 students there at any one time.

Having those tests be accurately Gaussian more than 2.5 standard deviations away from the mean is likely challenging. I've often suspected that on some of the SAT tests they didn't give any 800 scores.

There have been suggestions that those testing companies have not always been very open about just what they were doing in their details!

That we don't know everything is scientifically rock solid!

Your point is mostly that the stuff we don't know is bad science; I agree.

Still, this thread is to try to answer the question, where is ET?

Historically it would have been better to assume that there was stuff unknown that was better as an explanation than the stuff we did know, going WAY back: The Earth is a ball riding on the back of a turtle? We understood balls and turtles but not gravity of spheres! The earth is a ball held up by Hercules? Similar. To keep the sun moving across the sky we have to pour blood on this special rock? The planets are from wheels rolling in wheels? The stars are light coming through holes in ths sky? The sun is a fire based on coal? The Milky Way is all of the universe? The universe is expanding so there are just three cases, (1) keep expanding but more and more slowly, (2) stop expanding and reach 'steady state', (3) quit expanding and contract into a big crunch. It was ALL wrong! And in all cases the right answer was from things we didn't know yet!

For "We're running out of places for such putatively better mechanisms to hide". Ah, it's always been such! Where was Newton going to look for a solution to the orbit of Mercury? Where was Newton going to look for why he couldn't make gold with chemical reactions? The big bang seems to have things moving faster than the speed of light; where to look? We're not getting the right flow rate of neutrinos from the sun; where to look? Essentially all life on earth is just from DNA; so where to look for the reason there is no other mechanism?

But just now we have at least two biggie places to look: Dark matter and dark energy. About both, we don't have hardly a clue. We are unsure about the Higgs field. We haven't unified gravity. We haven't detected gravitational waves. We're unsure about why galaxies have black holes at their centers. There remain questiona about the 'size' of our universe, especially given the evidence about the flatness of the geometry. We're really mixed up about EPR.

But not all is lost! Some of the new telescopes on the way into orbit are amazing, maybe count the hairs on the back of ETs head! Maybe!

So, why haven't we heard from ET? My guess, stated as just a guess, is that ET communicates by means we don't understand yet. Is this answer solid science? Nope! Might it be correct? Yup! Is there some historical, circumstantial evidence for it? Yup. Are we still in the dark? Yup.

Let's see: We've been okay with radio for less than 200 years so far. The earth was formed ballpark 4.5 billion years ago. The big bang was about 13.7 billion years ago.

Assume that to get a civilization need a planet and, thus, elements heavier than hydrogen and helium. Those heavier elements are created in big stars that supernova, that is, have their cores collapse under gravity and blow off their outer layers where during the explosion all the heavier elements in the periodic table are created by fusing together lighter elements. So to get a planet, have to have a star form, burn out, and supernova. So, assume that are getting candidate planets 3.7 billion years after the big bang. Then have been forming candidate planets for 10 billion years.

Now take the planets that have achieved mastry of radio (i.e., working with photons from 60 Hz up to gamma rays). Take the date when they first made this progress with radio. Take the distribution of those dates over the planets. That distribution is 'concentrated' on the last 10 billion years. Now in that distribution spread out over 10 billion years, what is the probability mass of the last million years? It's TINY. Maybe it's 0.01%. Then 99.99% of the civilizations that mastered radio did it over one million years ago and, thus, are a million years or more ahead of us. Done.

Okay?

The question was, could a student without a Bachelor's just show up at a grad school, offering to take the Ph.D. qualifying exams and believing that they are well prepared, AND, without a Bachelor's degree, be permitted to do so? In particular, to take the qualifying exams, would they have to be 'enrolled' and would that be possible without a Bachelor's degree? And, although not said, maybe the student needs financial aid and hopes to get it based on their good qualifying exam performance.

So, they are ready to take the exams. But they have no Bachelor's, are not enrolled, without a Bachelor's would likely not be accepted or enrolled, and need financial aid. So, can they take the quals? If so, then how? That was the question. I suggested maybe an Associates degree, some really good GRE scores, and offering to take the quals BEFORE applying for admission.

I meant both tests but definitely the subject tests.

Usually the CEEB, SAT, and GRE tests are designed to have Gaussian distribution with mean 500 and standard deviation 100. So, can get a table of the cumulative Gaussian distribution and see what percentile 4 standard deviations above the mean is; I would guess less than 1%.

I got 800 on the GRE subject math test, the only 800 I got on any of those tests, and that 800 always intimidated my wife, MUCH smarter than I am, PBK, etc.

Why especially the subject tests? Because the question was how to skip a Bachelor's degree and do not pass GO, do not collect $100, and go directly to grad school. There the grad school may still want a Bachelor's but for anything less really good scores on the relevant GRE subject tests may be the difference. Show up with 750 or better on GRE subject math, physics, and computer science, and offer to take the Ph.D. quals right away, and may, just MAY, be permitted to 'enroll'. Blow away the quals, publish a paper or two, even in a conference, maybe knock out some code just to prove are not all theory, and may be regarded as a good student. Then will be closer to the front of the line for various kinds of financial aid.

How to skip a Bachelor's is chancy. For the importance and potential of good independent work, at the best research universities that's rock solid. Read the story of the guy who gave the name a 'good' algorithm, Jack Edmonds. Read what Feynman did at Princeton. Read what Gleason did at Harvard. Independent work was just crucial in how I got my Ph.D.: It helped that I did the research for my dissertation independently in my first summer and worte a 50 page manuscript. Then it helped that took a 'reading course', selected a long outstanding problem, and found a solution which also solved a problem in a famous paper by Arrow, Hurwicz, and Uzawa (poor Uzawa was left out of the prize). It helped that the department chair taught a flunk out course, an advanced, second, course in linear algebra and I took it as my first course in linear algebra and blew everyone else away. How? I'd done a LOT in linear algebra independently and in my career in 'scientific programming' before grad school. It even helped that I was the only student that year who showed that there are no countably infinite sigma algebras. That's the kind of stuff that can help one get through grad school.

In the US, for Web 2.0 projects, mostly can boil down to just one question: Do you have traction that is significantly high, say, over 100,000 unique visitors a month, and growing rapidly?

A point in my guess is that an old civilization won't be trying to use radio to communicate with us and will be using something better than radio we don't know about. Next, for communications over 1000 light years with just radio, there can be some issues of signal to noise ratio!

Sure, we can see a star 1000 light years away, a galaxy 1 billion light years away, and a quasar 10 billion light years away, but a signal from a planet over 1000 light years away? Besides, 1000 light years may not be far enough to cover many planets transmitting. Our galaxy is, what, 100,000 light years across? So, just for our galaxy, might want to think about distances of 50,000 light years or so. The nearest ordinary galaxy is the one in Andromeda, and that's about 1.5 million light years away.

Those are long distances for a radio signal from a planet.

I didn't cover getting "enrolled", but usually that won't be much of a problem: Even the top graduate programs are hungry for good students. If you are ready for their qualifying exams when you show up, then SAY SO. And/or submit a stack of nicely done, hard exercises from a well known, challenging text. Then being enrolled and able to take the exams should be routine.

For an alternative to the Bachelor's, mostly graduate schools will expect a student to have that degree. But, a graduate school doesn't have to care very much about where a student got a Bachelor's. That is, nearly everyone in serious academics knows that a student who shows up ready for the qualifying exams deserves the credit themselves, that is, the school didn't deserve the credit. The US is awash in relatively inexpensive Bachelor's programs, coast to coast, border to border.

There is a hidden point going for a graduate student who can do research: At the top schools, especially Harvard, Princeton, Johns Hopkins, the emphasis in graduate school, for both the profs and the students, is just RESEARCH. In particular, commonly there is no coursework requirement. Princeton has been known to state on their Web site that grad students are expected to prepare for the qualifying exams on their own, that the courses given are for introductions to research, and that there are no courses for preparation for the quals. So, this stuff about independent work to get ready for the quals is mostly what have to do anyway. Some schools publish copies of their old qualifying exams and have lists of references for preparation for the quals. So if you are good with the references and the old exams, then you have a good shot.

All those things said, ugrad school is usually more strict about courses, credits, and grades. Still, there may be some full or partial ways around the time and expense of a full ugrad program: First, it may be possible to impress a grad school just with GRE scores. So, if you are well prepared, then see if you can take the GRE without a Bachelor's. Second, the US is also just awash in junior or community colleges. Since their student quality is usually low, they would be thrilled to have a good student with a shot at getting a Ph.D. These colleges are usually just for two years and give only an Associates degree. These colleges are uaually quite inexpensive. But, then, the last two years of a four year program are mostly just in the major subject, anyway. So, a grad school MIGHT be willing to accept just an Associates degree. Third, there is a fairly strong but rarely written principle in US higher education: Really good students are wanted and go for free or nearly so! If you are ready for Ph.D. quals, then the money for a Bachelor's should be no big problem, even at Harvard. And once you get at a really good school, say, Harvard, you mostly won't be held back. E.g., I knew a bright student who at Harvard as a sophmore took a reading course with A. Gleason, right, who solved one of Hilbert's problems. And Gleason never got his Ph.D.: His work was so good Harvard made him a Harvard Fellow and put him on the faculty. James Simons was an ugrad at MIT and in his senior year took a reading course from Singer, right, as in the Atiyah-Singer index theorem -- watch the video of Simon's talk on YouTube or some such. Simons? Right, he is as in the Chern-Simons result (try Wikipedia), was Chair at Stony Brook, started a hedge fund, for some years paid himself $2 billion a year, and, net, is likely the most successful hedge fund guy.

Yes, the field where this independent study works best is math. In physics, chemistry, or biology, a ugrad program will typically have a lot of lab work can't do independently and some lecture courses with material difficult to get independently. But math remains the most powerful major, maybe even for physics and biology. Even biology? Look up Eric Lander! But, again, for a really good student, ugrad can be for free or nearly so.

> "But what would be better? How can we create a better system? And if there's such an obvious better answer, why doesn't someone do it?"

Here's a simple answer: The better system exists, but people overlook it.

As a student, first get a Bachelor's degree.

Second, pick a field and be sure have learned it well at at least the Bachelor's level. Do this learning independently if necessary. A Bachelor's degree is supposed to teach you how to do at least this much learning independently.

Third, from that learning about the field you selected, learn some more, to 'the next level'. Likely do this independently.

Fourth, show up at any one of the better research universities and take the Ph.D. qualifying exams based on what you learned.

Fifth, stick around that university and attend some seminars and courses that are introductions to research given by experts in research. Here your work is largely independent.

Sixth, pick a research problem and get some good results, independently or nearly so. If there is any doubt about the significance of your research, then publish it.

Seventh, submit your research as your Ph.D. dissertation.

Congratulations: You are now out of school; you went all the way; you are educated. Done.