Here's the lesson: if you have a point to make, please don't fit in extraneous claims, which are much debated themselves, into your argument or comment. It's not hard to see that they can be confused as being part of your point.

> This seems to be a recurring theme on HN, where people point out irrelevant nitpicks as if I couldn't do that myself.

I will say this again: there is nothing wrong with nitpicking. If you can't be technically correct, then you're not correct.

You said, with regards to memorization, in your OP:

"I'm convinced anyone can do this and the major thing stopping them is their belief that they can't do it."

If you want to now adjust that claim and say "some can do it", you better be able to tell us how to distinguish.

Oh you're just being so silly now. Here is my original comment:

I'm convinced anyone can do this and the major thing stopping them is their belief that they can't do it.

Here is how I am comfortable revising it for the offended sticklers in this thread:

I'm convinced anyone (of typical mental capacity) can do this and the major thing stopping them is their belief that they can't do it.

I'm asking you to be rigorous with your claims. And there's nothing wrong with being pedantic.

If you can't respond to questions without getting defensive, please grow up.

I'm just writing to let you know your insulting responses have earned you no further replies from me. I invite you to reconsider how you write to people on the internet and perhaps avoid writing things like, "grow up."

This thread has given me more karma than your entire account has received in its existence. I think it's clear there is a vocal minority of people who are posting a half-dozen comments each in this thread while the vast majority at least don't find what I'm saying as offensive as you apparently do.

>This thread has given me more karma than your entire account has received in its existence.

Oh my goodness! What was the point of saying that, really?

>I think it's clear there is a vocal minority of people who are posting a half-dozen comments each in this thread while the vast majority at least don't find what I'm saying as offensive as you apparently do.

I don't find this offensive. I'm just asking you to be technically correct with what you say, or to at least address the people who are confused by what you're saying. Apparently, that's too high of a standard for you to live up to.

Please just answer the question. You seem to be using the person (sillysaurus) as an excuse to avoid having to address problems in your own worldview.

In this short "debate", sillysaurus has called me "annoying", "smug", "arrogant", "biased" and referred to me repeatedly as "people like you".

He's not debating me. He's debating every person that ever insulted him and using me as the punching bag while he's doing it. He's not listening to what I'm saying. Every time I reply to him he hears the people who have been putting him down and becomes more angry. Why in the world would I type out another reply to him?

Anyway, I don't see the significance of his question as aimed toward me.

what the difference is between someone who has a mental handicap, and who just isn't very good at memorizing things

He used the word handicap - I was just quoting him. I don't know what a mental handicap that causes you not to be able to memorize might look like. Does he have a traumatic brain injury? That would be a handicap. But he says he was like this since birth. Did oxygen get cut off to him during delivery? I have no idea what he's talking about when he says he's handicapped. How in the world can I answer his question?

In any case, I was aiming my comment at normal people and to head off the next silly question of what normal is, I mean people of typical mental capacity.

In any event, I hope we all learned something from this. :)

I don't understand what you're saying. Pretend like I am idiot and explain it to me in simple terms.

1. Can you define consciousness? Or is consciousness = awareness? Is that what you meant?

2. You ask "What is the origin of awareness itself?". I really don't understand this question. What is awareness? I can't even comprehend what you mean when you ask "what is the origin of [it]?". Colloquially, physical things have "origins", which is to say that at some point some physical thing was "created" or "crafted". My intuitive idea of awareness is that it is an experience, it's not something physical.

1. Yes, consciousness is just awareness. Your mind is where all your thoughts and judgements are. Your body acts from your mind and your mind is generated by your consciousness.

2. The 'origin' of a phenomenon refers the causes that generate the phenomenon. In math, as well as philosophy, the question is the origin of the answer. I'll give you an example. Where does the light from a lightbulb come from? Does it come from the bulb or does it come from the electricity itself? The bulb is simply the object which converts the electrical energy to EM radiation, not the origin of the light itself. Where does the electricity come from? The activities of the generator. So, mind is like light and consciousness is like an electrical wave.

3. Awareness of a fact means that you see a fact. It means that you receive the fact as it is. The dichotomy between experiential and physical things is illusory. It's simply that you experience the physical things from your own perspective, which is formulated by everything that has happened to you and everything you have done in the past - all of which were originated by activities as well. If we compare to a physical object, your consciousness is like the driver of a car (where your body is like the car), and when you are driving a car you can see through the windshield and you can feel feedback through the steering wheel. Hope this makes sense. This is not so easy to understand at first listen so please feel free to ask me continuous questions as I can help you to make your understanding more concrete.

As far as I know, light is defined precisely in physics. The light from a light bulb comes from electricity traveling through a highly resistive wire. The particle interactions generate photons.

"The bulb is simply the object which converts the electrical energy to EM radiation, not the origin of the light itself."

If you were to ignore the particle interactions, then yes, the light originates from the light bulb. It certainly does not originate from electricity. Where the electricity comes from is irrelevant.

"It means that you receive the fact as it is."

I have no idea what that means.

"The dichotomy between experiential and physical things is illusory. It's simply that you experience the physical things from your own perspective,"

Oh, I know.

I still don't think you defined awareness precisely enough.

I tend to be dismissive of the philosophical conversation. I think what will resolve the question is scientific (biological) study. Maybe math/cs theorists (AI) will get there first. But I am almost certain that no philosophical conversation we can have will give me an insight about our cognition.

About point 2 and point 3: the quality of the undergraduate education is considered in applicants to top schools. Like I said before, there is definitely a standard of rigor that you don't experience at every school. As soon as you decide to optimize for financial cost (picking the cheapest school, doing community college), you've already placed on a burden on yourself.

pretty quick to dismiss community college. I took lower division classes at one, my math classes were taught by PhDs or people with three decades of math and statistics experience. The rigor was pretty good, I transferred to a very good school from the community college and was well prepared.

1. The point was "there is a standard of rigor that you don't experience at every school". I could have used anything as an example. It doesn't matter that I picked community college. If you agree with that statement, then do you disagree with the conclusion: that once you go down that path, you the student, have a burden to explain to everyone how the quality of education compares with the best?

Technically, yeah, I made a generalization: cheaper schooling means shittier quality education (not bad, just worse). Of course, I haven't stepped into every community college classroom. But this is the psychology of the people looking at your application.

Technically technically, what actually matters is all the things that go along with going to a cheaper school, not really the fact that it's cheap.

2. "The rigor was pretty good" - does that mean you kept taking math classes (real analysis, etc)? If not, then how can you make that claim?

The OP was asking for an engineering degree; there the math is powerful support but only support.

Sure, the first two years at Harvard can be something special, especially if the student uses AP or tests out of the standard first two years of ugrad school. E.g., Harvard's Math 55 taught to freshmen at least at one time used Halmos, FDVS, Rudin's 'Principles', and Spivak's 'Manifolds'. That's usually junior or senior level stuff.

I know one guy who went to Harvard and as a sophomore took a reading course from A. Gleason. So, if in the next two years he knocked off one of Hilbert's problems, like Gleason did, then he could also become a Harvard 'Fellow' and skip a Ph.D.! But, again, we're talking engineering, not being so advanced as a ugrad that really should end up with a Ph.D. in math instead of a Bachelor's.

What's so wrong about just taking the first two years of college as just the usual first two years? A CC can provide that. A CC can use a good calculus book; when I took calculus, I used the same book Harvard used; that is, Harvard was willing to teach a calculus course, just calculus from just a common book, to I don't know who took it. Given the book and anything like a competent CC teacher, the Harvard course doesn't have much room to be better than a CC course, especially if the student followed my recommendations on self study before the course.

I know; I know; maybe the freshman English lit course at Harvard is taught by a Nobel prize winner in literature and can provide some astoundingly profound insights into Henry James and, thus, change the lives of the students. I'm not impressed.

You seem eager to "pay a lot" for the first two years of college.

I'm not sure about "cheaper schooling, lower quality". In engineering and computer science there are state schools that compete, IMO, pretty handily with "the best". you might snigger and scoff, but hiring managers at good companies don't.

also, if you start your degree at a CC and transfer to a 4 year to finish, what does it says on your diploma? exactly what it would say if you went there for four years. I wouldn't recommend keeping secrets about your life, but if no one asks...

I did keep taking math classes! I have a math minor.

yeah, so, I was rushing. I edited the previous post: what matters is all the things that go along with cheaper schooling, not that fact that it's cheap.

Sounds that it is a lot about credentials. Sadly enough it is. My experience with my education in industrial engineering at "university of applied sciences" in germany (whatever that would be in the US...) was rather good even when compared to the big names around here as far as knowldge is concerned. Yet, the fact remained that it wasn't a big name.

A graduate school will be eager to forgive the first two years in a community college given what else I assumed: A four year college for the last two years, glowing recommendations from professors, and excellent GRE scores.

That a guy started at a community college and ended up with excellent GRE scores is an impressive 'trajectory', and mostly good graduate schools will be sufficiently impressed. The graduate schools are hungry for highly motivated, good students and will want to grab what I described.

I reread this branch and thought it sounded awful. What I meant to say is that if you optimize for low cost, you're limiting your choices to schools that lack the reputation. I don't know very many schools that have both (maybe some state schools). And yeah, this results in having a burden to defend yourself.

I wish I could delete the rest of this branch.

In the job market, reputation of your undergrad generally doesn't much doesn't matter after the first 5 years. Multiple studies show that students from different schooling backgrounds reach employment parity in most every situation.

Anecdotally, in my own case, I hit employment parity with my Ivy League/Top-10 school peers at 6 years, and have long since surpassed most of them in the job market and standard of living (due to an unbelievably smaller debt load).

The only people I know from my starting peer group who I could comfortably say have a higher standard of living than I do also went to regular 'ol state schools but just worked harder/smarter than I did.

I get fairly regular recruitment pitches from the Google's, Amazon's and Facebook's, and have worked for some of the biggest and best companies in their industries. (I've noticed an uptick in Google courtships over the last couple years as they finally brought their data crunching prowress to analyze hiring success and found that their traditional hiring pools did not necessarily provide the best employees)

All that being said, if I were so inclined, there are positions (high end consulting, finance, high-level politics) that I'm less likely to ever end up in due to fierce protectionism by "old guard". But I'm finding those hard kernels of school fashionistas are rapidly disappearing now as well.

Nice discussion, at least for me. So don't worry. Because you're right, having a big name like MIT on your CV opens doors that otherwise keep closed. That doesn't necessarily mean the education is really better, but the fact remains.

On the other hand, if you don't have the money there are not a lot of opportunities left. And I for my part prefer some who showed that kind of ambition over everyone else who hand it "the easy way" (not meaning an MIT degree is easy at all, just saying there are roads with more rocks than others).

You spent a significant chunk of your post suggesting self study. For something like math, this is really really really unrealistic. Most pure math textbooks don't have simple problems you can just check in the back. They're multistep proofs that can be done in a number of different ways. Oh, and you encounter plenty of material where you can easily trick yourself into thinking that you really understand it when you actually don't. Additionally, there is a standard of rigor that you don't experience at bad schools (or with no schooling). You could be writing complete nonsense solutions and not even know it.

Also: how are you going to self-study when you don't live at home? Studying is a full time job.

I self-studied some amount of pure mathematics, having only an undergraduate CS degree and while having a 40 hour a week day job, and while it's definitely harder and slower than learning it at a university, it's not unrealistic. If you look hard enough, there are books with proof-based exercises with answers in the back (example proofs) that you can use to check your understanding (Spivak's Calculus, books from Springer UTM series). For many other books, you can find course pages online with homeworks+solutions, e.g.

http://www.cs.berkeley.edu/~oholtz/teaching.html

has homeworks for Rudins "Principles of Mathematical Analysis", and Halmos "Finite-Dimensional Vector Spaces". Finally most problem books (again Springer has a nice selection) have very detailed solutions. Yes, there are proofs that can be done in a number of different ways, but in my experience diverging too far is not very common and in most cases some core ingredients have to make it in in the end anyway. It's impossible to go through a lot of exercises "writing complete nonsense solutions" like that, when you are precisely checking your solutions against the given ones. For many types of exercises there are also simple ways of validating your solution, for example, in probability theory you can often do a computer simulation. In the end that's what anyway has to be done in real world and in research work.

It's unrealistic because I really think you need the social aspect of it: collaboration and criticism. I didn't say you can't learn something on your own sometimes.

Also lots of people have convinced themselves of lots of silly things. You can probably find dozens of papers from people with bachelors in math (or no degree) claiming they have solved P=NP. A lot of these turn out to be completely bogus, but the authors nonetheless thought they were serious attempts.

What are you exactly claiming then, what would it mean that mathematics self-study is "unrealistic"? There is certainly a danger in not seeking external validation of your work (or denying it), and being at a university is very nice for getting that. But with some motivation I think you can get the knowledge equivalent of an undergraduate degree in mathematics by self-studying in maybe twice the time it would take at an university (I am speaking from personal experience and assuming full time job and having some life, and that you don't have kids yet). You can get feedback on the Internet as well nowadays, or seek university-level tutoring. And there is no shortage of people with degrees doing faulty P=NP proofs either.

Again, I agree doing it at the university is more effective way of doing it, and if you have the possibility to do it, good for you! But most of us can afford to dedicate at most 5 years to studying full time, and than other responsibilities kick in and you can't do it anymore. The majority of your life all the new knowledge you get will come from self-study. So you better learn to do it.

Having an undergrad CS degree is enough to self-study math, I believe. But having no formal education in the sciences/maths is not (unless you are a genius).

For me, merely pushing myself through a theoretical CS curriculum made me see (and write) hundreds of proofs, hear them explained by professors, and see non-trivial exercises solved during recitations. I don't think you can get the same kind of experience by just reading a textbook, even if it does offer full solutions to problems.

Maybe when there will be full video lectures for both lectures and recitations for the basic math (or theoretical CS) curriculum you could self-study by watching those and solving problem sets. Right now, the math courses offered by Coursera don't seem to match college level, and their platform doesn't really work for proof-based courses like Analysis, Linear Algebra (not the applied kind), etc...

I learnt only a bit of discrete math in my CS undergraduate degree, I had calculus and linear algebra but barely passed it by memorizing how to solve concrete problems and by having merciful professors - I was already working full time and had 4 or 5 courses going on, I just didn't manage to find enough time to study properly. If you have bigger gaps in your math knowledge and can't go to an university you just have to start at a lower level, there is a wide selection of "intro to higher mathematics" books meant for people like that, and if that level is still too high you might need to review high school math, e.g. Serge Lang has a good "Basic Mathematics" review book, or you can use Khan Academy videos etc. I myself had to review a lot of high school math when I was starting.

MIT OCW has excellent courses for discrete math, calculus and linear algebra and in some cases videos from the recitations are included.

http://ocw.mit.edu/courses/mathematics/18-01sc-single-variab...

I also stressed a few times already I don't think it is "the same kind of experience". But as long as you make an effort you will make progress and not everyone can manage to fit a university degree in their schedule.

Nice. A better response than I gave.

Halmos, Rudin, and Spivak 'Calculus on Manifulds' were at least at one time the main references in Harvard's Math 55 as at

     http://www.american.com/archive/2008/march-april-magazine-contents/why-can2019t-a-woman-be-more-like-a-man/?searchterm=Sommers

and are some of the best stuff for a ugrad math major.

Your last sentence is on the center of the target of reality.

You have a good point, but my post was already at the limit of 10,000 characters. Of course the solution to your point is partly a theorem proving course in high school plane geometry and then, finally, a theorem proving course as, say, a college junior in abstract algebra. For such a course, I did say that the last two years should be at a four year institution; at such a school, a good enough course should be available even if the first two years were in a community college where the calculus teaching was poor. Again your point is correct: To learn how to do proofs well enough to be self-sufficient, need at least one theorem proving course where can get homework and tests graded by a competent mathematician.

Don't worry: I've tried to show that P = NP and know that while I've had some candidate ideas I don't have a good idea or a proof. And, I've nearly never written a bad proof; once catch on to how proofs are done, they are surprisingly easy to check for correctness.

Studying is not a full time job -- I was heavily self taught in math and totally self taught in computing and nearly never studied full time. E.g., I read Nearing on linear algebra, Halmos 'Finite Dimensional Vector Spaces', Fleming 'Functions of Several Variables', yes, with the exterior algebra, and much more while working full time in mostly DoD work around DC. I did the research for my Ph.D. dissertation in stochastic optimal control independently in my first summer in graduate school.

Edit: There's a better answer in this thread in

     https://news.ycombinator.com/item?id=6177643

This is false. I studied Maths at Cambridge and learned some courses completely on my own.

Seconded. Math is one of the most easily self teachable subjects. The field of study is objects of mind (unless you're a platonist.) Literally no materials required except pen/paper, a brain and maybe a straightedge and compass. The point of math is not to do endless worked exercises. It's to understand mathematical objects and prove interesting things about them. You can generate unlimited problems for yourself by investigating some mathematical object at random.

Basically nothing you said actually made a case against anything I said, or for the thesis that it is easy to self teach. Just stating this for the record.

Well. There are no barriers to entry. If you have any inkling of logical ability, you should be able to tell when a proof is right. All it requires is critical thinking. Presumably humans come with that out of the box.

Could you explain briefly why this is the case? I don't understand.

Well, what was his argument? Based on his second reply, he seems to think that because it only depends on your ability to reason that it should be easy. But doesn't that trivialize the matter? As long as we have mathematical models, as we do in physics and in chemistry, then it should be just as easy to learn physics and chemistry.* So then what does he consider hard to learn? Are the social sciences hard to learn? The 'it is of the mind' is a non-argument to me. And I don't think he addressed anything I said.

Anyway, this is almost irrelevant to what I was saying. Even if you assume every person can reason well, I'm saying you could still be in error unless you seek validation and guidance. It's really easy to think you've given a solid argument for something, but actually be wrong. It happens to everyone.

*By the way, there is a definite trend in physics for math, instead of experimentation, to be leading the way towards discovery.

It's false that any of what I said can be true for some (or many) students? How do you determine if you're one of those students?

You're right, the above posters generalize way too much. They are either CS or Maths students and as such are already well-prepared for self-study! If you study Maths you already know what all the symbols like epsilon, e etc. mean, you can just gloss over a mathematical text and get the gist of it. Same goes for many (but not all!!!) CS-students, some unis lean a lot on algebra, some don't do much maths after the first two courses.

As such, I think it's ridiculous to go up to anyone and tell him/her to just "study by yourself", I could give a maths book to a biologist and that person would understand absolutely nothing without guidance.

Well, you could certainly do it when you start with "Maths 101" or something like that.

I was addressing the money issue and tacitly assuming that they could do the work. For just a Bachelor's and then just a Master's in engineering, they have a good chance of being able to judge correctly if they can do the work.

I think it is very reasonable.

There is plenty of free access to materials and papers.

It's incredibly cheap and fast to communicate with pretty much anyone in the world.

All that is really required is motivation, discipline, and curiosity.

When there wasn't cheap access to global communication networks and near-zero cost to duplicating data then it made sense to go to university because that was where everyone who you would be interested in talking to would be. That's where the libraries and books were. I don't think that is the only option anymore. And that's a good thing.

You can learn anything you want on your own and still have all of the benefits of a college (access to knowledgeable people, peer review, etc).

I was on a forum where there was a guy who had self-studied maths and physics and was an 'ideas man'. The problem was that he just didn't have the standardised nomenclature to get his ideas across - and also meant that he couldn't understand the reasons why other mathematicians debunked him.

One outstanding example was his method for a 'free energy' spacecraft movement system, that hinged on an arm throwing -foo- into a receiver. The argument was that there is a difference between throwing something and merely releasing it at speed, and he didn't have the understanding of physics to realise there is no difference.

Another one of his ideas was a system for finding prime numbers, which he couldn't articulate well enough for people to figure out whether it was a valuable predictor or merely a sieve.

He was a regular at the forum and respected, and I've never seen so many people patiently explaining physics and maths at such clear lengths before... and he just couldn't grok it, because he didn't have the standard language to get his ideas across.

This anecdata doesn't contradict the GP though, who is talking about doing self-study in parallel with formal study.

I am actually in a similar situation... except I'm in math. I can program though :/.

And by "similar" I mean that I may have to drop out because I cannot afford it. Loans from the government are not enough. Can't get private loans. Can't get money from the parents.

So what was the point?

To show that haskell and C can achieve equivalent performance on certain numeric calculations.