8 ms·

Great Ideas in Theoretical Computer Science

290 points by tu7001 9y ago

75dvtwin 9y ago
reading one of the PDFs referenced http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec05.pdf http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec0... Why other mathematicians at the time, were so nasty towards Cantor's ideas? The quotes noted in the PDF -- just sound nasty and dismissive (or may be this is not the whole story)
- Nomentatus 9y ago
  People don't self-select themselves to study and teach mathematics because they like surprises. This is true of academia in general, but very true re math.
  - Ar-Curunir 9y ago
    What? Science and mathematics requires you to be open to new ideas, beause your ideas can be proven wrong easily.
    - Nomentatus 9y ago
      That's the ideal. But if you've spent much time in academia, you'd know that the personality types that choose science usually do so to wallow in certainty. High School doesn't tell you much about the real history and philosophy of science. I trust everyone who added a downvote has actually spent some time inside a math dept of a University (but I highly doubt that.)
- voidhorse 9y ago
  I think, as with many paradigm changing and major ideas, its novelty required a change in the prevailing thinking of the epoch, which always makes people uncomfortable--according to wikipedia most objections were to Cantor's notion of transfinite numbers: https://en.wikipedia.org/wiki/Georg_Cantor https://en.wikipedia.org/wiki/Georg_Cantor There's even a separate article on objections to his theory: https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th... Seems like the notion of "completed infinity" which sort of gives infinity ontological status (? I'm no mathematician) did not sit well with some--I guess I could understand why--it definitely isn't an intuitive notion: "Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.[18] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[19] Carl Friedrich Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.[20] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets." I guess this shows that the overall pragmatic value of your work may eventually win out against formal refutations.
  - chriswarbo 9y ago
    It may be that a more deep-seated change has taken place within mathematics: the rise of formalism, with Frege, Russell, etc. separated mathematics from intuitions about 'reality'; a formal theorem can't be refuted by saying it's 'unrealistic', since it's a watertight consequence of the chosen axioms and deduction rules. We can attack those axioms and deduction rules as being 'unrealistic', but since Goedel shot down the attempts to define "one true" formalism, it seems that mathematicians have come to accept axioms and logics based on their own merits (e.g. whether they lead to interesting, non-trivial consequences) rather than whether they're accurate models of 'reality'. In this relaxed view, Cantor's arguments make sense from his premises, and it's interesting to entertain the idea of his premises because they lead to such interesting results. Whether or not "completed infinities" are "real" is delegated to the philosophers, while mathematicians 'play the proof game', and change the rules if they make it more fun.
- addicted 9y ago
  You are highly underestimating the mental leaps required to understand infinity pre Cantor. It's not surprising at all that people should find it ridiculous and get especially angry about it because they couldn't really explain why it was wrong, despite it just seeming like complete BS to them. It's like telling a lay person (or even many mathematics students for that matter) that 0.999999.... = 1. They understand the proofs, but it's so against their intuition there is a visceral negative reaction to it.
  - pacala 9y ago
    There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and 1, for example 0.9[k]5. Your alleged proof is wrong :p
    - alangpierce 9y ago
      Usually 0.9... is defined as the limit (from calculus) of the sequence 0.9, 0.99, 0.999, 0.9999, etc. So it's the unique number that the sequence gets arbitrarily close to (even if it never reaches it), which is exactly 1. With the proper definition in place, the statement becomes a lot less interesting, in my opinion, but regardless, I'm pretty sure the statement "0.9... = 1" is well-accepted in the math community. I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.
      - igravious 9y ago
        I may be out of my depth but I disagree. When I see … I read “and so on” so my brain when it sees 0.9… writes a function which first produces 0.9, then 0.99, then 0.999 and keeps going and never stops – it's return type is the bottom type to use type theoretic jargon. I say 'produces' but the function never returns so if there was some way to sample where the function was at or … (heh, that's an ordinary ellipsis). Or rather say it is a function where the first time it returns 0.9, the next time it returns 0.99 at a higher resolution, then 0.999, and so on. If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“. If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way). Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery. [0] http://www.sophiaperennis.com/books/rene-guenon-series/the-metaphysical-principles-of-the-infinitesimal-calculus/ http://www.sophiaperennis.com/books/rene-guenon-series/the-m... Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...
z1mm32m4n 9y ago
Some context for those who don't go/didn't go CMU: 15-251 Great Theoretical Ideas in CS is the freshman spring semester discrete math & theoretical CS (a follow up to the discrete math course they take freshman fall). The course itself goes for breadth over depth. In this aspect, it serves as a solid foundation for all theory classes students take going forward. It's a class where students learn proof techniques, collaboration, and problem solving. The course is often praised for its rigor. The problem sets require a fair amount of work to complete, and the course staff sets a high standard for responses. On the other hand, it commonly receives criticism for being quite intense as a freshman class. Many CMU students look back on it as one of the hardest classes they took. I personally feel quite lucky to have taken 251. It gave me a foundation to appreciate a lot of deeper CS topics, and taking it freshman year meant I had 3 years more to put it to use.
- kutkloon7 9y ago
  I was just about to comment that the topics seem really intense. I got the impression that Carnegie Mellon is an excellent university with very high standards (much better than Harvard, Stanford, Berkeley, etc., which are prestigious and have lots of professors which do great research, but the actual scientific education is good, but not extraordinary, and moreover, the courses are not very hard from what I've heard). It happened multiple times that I googled some relatively obscure topic, and I found multiple excellent sources from Carnegie Mellon professors. For example, there is this excellent document [1] by Jonathan Shewchuk, about the conjugate gradient method, and you will found as many as four lectures about DRAM memory on youtube. [1] https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf https://www.cs.cmu.edu/~quake-papers/painless-conjugate-grad...
  - laughinghan 9y ago
    My friends had him and I hear Jonathan Shewchuck is indeed an excellent teacher. However, he has never been a CMU professor. In 1994, when your link was dated, he was getting his PhD at CMU, and since 1998 he's been a professor at UC Berkeley (where I went): https://people.eecs.berkeley.edu/~jrs/ https://people.eecs.berkeley.edu/~jrs/ I have no doubts about CMU's excellence, but I found Berkeley's undergrad CS education to be quite good. There certainly were research professors who were bad teachers, but many like Shewchuck were outstanding teachers, and there were also several Teaching Professors whose job description to focus is more on teaching than research, like Dan Garcia, Paul Hilfinger, John DeNero, and Brian Harvey (since retired): https://www2.eecs.berkeley.edu/Faculty/Lists/faculty.html https://www2.eecs.berkeley.edu/Faculty/Lists/faculty.html I too have heard that private schools like Harvard and Stanford are not very hard once you get in, just very hard to get into (though I can't speak from experience). I haven't heard people say that about Berkeley, which is a public school so it doesn't have shareholders where alumni can form a controlling majority, and it's funded more by the state than by tuition. That also certainly was not the experience of myself or anyone I know.
rawnlq 9y ago
Recording of the lectures for Spring 2016: https://www.youtube.com/watch?list=PLm3J0oaFux3aafQm568blS9blxtA_EWQv&v=uaAvVNWvi4A https://www.youtube.com/watch?list=PLm3J0oaFux3aafQm568blS9b... Slides: http://www.cs.cmu.edu/~aada/courses/15251s16/www/schedule.html http://www.cs.cmu.edu/~aada/courses/15251s16/www/schedule.ht...
- tu7001 9y ago
  Those YouTube lectures are prof O'Donnell course. I don't know if they match.
westurner 9y ago
List of important publications in computer science https://en.wikipedia.org/wiki/List_of_important_publications_in_computer_science https://en.wikipedia.org/wiki/List_of_important_publications... http://paperswelove.org/ http://paperswelove.org/ https://github.com/papers-we-love/papers-we-love#other-good-places-to-find-papers https://github.com/papers-we-love/papers-we-love#other-good-...
sidusknight 9y ago
Most of the links are behind a login. Why even post this here?
- kutkloon7 9y ago
  The slides are not, and there are so many links that are directly behind a paywall on hacker news... Those really bother me.
  - dmix 9y ago
    WSJ style paywall articles are bypassable via the 'web' link. Not quite the same at all.
    - kuschku 9y ago
      If you are in the US, that is. Their paywall bypass doesn’t work for all countries.
anaccountwow 9y ago
Why such an old version ;-;
dnautics 9y ago
no love for coding theory!
raister 9y ago
The Markov Chains lecture has a huge 'meh!' feeling attached to it, I'd be better of not clicking on that PDF in the first place.
Chathamization 9y ago
I wonder what kind of retention there is. Is one day of class studying quantum computation going to be particularly useful to people when they study it 6 months or 2 years later? I've lately studied a bit of program design when it comes to working out. A lot of effort goes into designing workout programs, making the programs efficient, trying to get the maximum results from the least amount of effort. There's a clear progression and a clear way of measuring progress. Every exercise, stretch, and movement is supposed to a particular function that best moves you toward your goal. Particular muscle sets are targeted at particular intensities for particular intervals on particular days. There's a reason for every specific decision that gets made. You can contrast this with the aimless way a lot of people workout. They go to the gym, they do whatever exercises or work on whatever machines seem like a good idea to them at the moment, and hope it increases their fitness in some non-specific way. Lacking good metrics for measuring effectiveness, they often judge a workout by how sore it makes them the next day (an extremely poor way to measure effectiveness). It's not that this approach can't improve your health, it's just that it's extremely inefficient compared to the more goal orientated and progression based system. In my experience most universities follow the aimless second approach - throwing what they can at the students, hoping some of it sticks, and judging the courses based on difficulty (this usually becomes even more obvious when professors explain their courses). I don't think I've come across any study measuring the effectiveness and metrics of different programs and progression paths.
- Arcten 9y ago
  Probably, one day of lecture on the material would be useless. But the weekly homework assignments force you to develop a better understanding. I frequently remember topics, assignments, and even specific questions from this course when they are relevant to something I'm working on now.
- bichiliad 9y ago
  It's a bit more than it seems, on the surface. You're not just expected to pick up what the professors lecture about, but to take it and apply it to a sampling of fairly advanced problems. You end up doing a lot of learning on your own (or in a small group) while completing the homework. To use your gym analogy, it's like focusing on one muscle group per week. On Monday, you're shown proper form and technique for a squat, but by the following Monday, you'd better be able to squat significantly more than you could a week ago (and you'll get graded on it). Are you going to become a competitive squat weight lifter? Absolutely not. Do you now know proper form, and understand what it takes to train properly? Almost certainly. I'd also note two things: - Most topics covered were useful in later courses. - A lot of the credit for this course's success can be given as much to really, really dedicated TAs as it can to the professors. (Note: I took this course my freshman year, it was blisteringly hard, and I wouldn't have traded anything for it.)

290 points by tu7001 9y ago

reading one of the PDFs referenced http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec05.pdf http://www.cs.cmu.edu/~aada/courses/15251f15/www/slides/lec0... Why other mathematicians at the time, were so nasty towards Cantor's ideas? The quotes noted in the PDF -- just sound nasty and dismissive (or may be this is not the whole story)

Nomentatus 9y ago
People don't self-select themselves to study and teach mathematics because they like surprises. This is true of academia in general, but very true re math.
- Ar-Curunir 9y ago
  What? Science and mathematics requires you to be open to new ideas, beause your ideas can be proven wrong easily.
  - Nomentatus 9y ago
    That's the ideal. But if you've spent much time in academia, you'd know that the personality types that choose science usually do so to wallow in certainty. High School doesn't tell you much about the real history and philosophy of science. I trust everyone who added a downvote has actually spent some time inside a math dept of a University (but I highly doubt that.)
voidhorse 9y ago
I think, as with many paradigm changing and major ideas, its novelty required a change in the prevailing thinking of the epoch, which always makes people uncomfortable--according to wikipedia most objections were to Cantor's notion of transfinite numbers: https://en.wikipedia.org/wiki/Georg_Cantor https://en.wikipedia.org/wiki/Georg_Cantor There's even a separate article on objections to his theory: https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_th... Seems like the notion of "completed infinity" which sort of gives infinity ontological status (? I'm no mathematician) did not sit well with some--I guess I could understand why--it definitely isn't an intuitive notion: "Before Cantor, the notion of infinity was often taken as a useful abstraction which helped mathematicians reason about the finite world; for example the use of infinite limit cases in calculus. The infinite was deemed to have at most a potential existence, rather than an actual existence.[18] "Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already".[19] Carl Friedrich Gauss's views on the subject can be paraphrased as: 'Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn't belong in mathematics'.[20] In other words, the only access we have to the infinite is through the notion of limits, and hence, we must not treat infinite sets as if they have an existence exactly comparable to the existence of finite sets." I guess this shows that the overall pragmatic value of your work may eventually win out against formal refutations.
- chriswarbo 9y ago
  It may be that a more deep-seated change has taken place within mathematics: the rise of formalism, with Frege, Russell, etc. separated mathematics from intuitions about 'reality'; a formal theorem can't be refuted by saying it's 'unrealistic', since it's a watertight consequence of the chosen axioms and deduction rules. We can attack those axioms and deduction rules as being 'unrealistic', but since Goedel shot down the attempts to define "one true" formalism, it seems that mathematicians have come to accept axioms and logics based on their own merits (e.g. whether they lead to interesting, non-trivial consequences) rather than whether they're accurate models of 'reality'. In this relaxed view, Cantor's arguments make sense from his premises, and it's interesting to entertain the idea of his premises because they lead to such interesting results. Whether or not "completed infinities" are "real" is delegated to the philosophers, while mathematicians 'play the proof game', and change the rules if they make it more fun.
addicted 9y ago
You are highly underestimating the mental leaps required to understand infinity pre Cantor. It's not surprising at all that people should find it ridiculous and get especially angry about it because they couldn't really explain why it was wrong, despite it just seeming like complete BS to them. It's like telling a lay person (or even many mathematics students for that matter) that 0.999999.... = 1. They understand the proofs, but it's so against their intuition there is a visceral negative reaction to it.
- pacala 9y ago
  There is no such thing as 0.9... There is 0.9[k], for k as large as you'd like. However, no matter which k you pick, I can squirrel a number between 0.9[k] and 1, for example 0.9[k]5. Your alleged proof is wrong :p
  - alangpierce 9y ago
    Usually 0.9... is defined as the limit (from calculus) of the sequence 0.9, 0.99, 0.999, 0.9999, etc. So it's the unique number that the sequence gets arbitrarily close to (even if it never reaches it), which is exactly 1. With the proper definition in place, the statement becomes a lot less interesting, in my opinion, but regardless, I'm pretty sure the statement "0.9... = 1" is well-accepted in the math community. I agree that all of the proofs you normally see in high school are flawed in one way or another because they don't properly define what "..." means in the first place.
    - igravious 9y ago
      I may be out of my depth but I disagree. When I see … I read “and so on” so my brain when it sees 0.9… writes a function which first produces 0.9, then 0.99, then 0.999 and keeps going and never stops – it's return type is the bottom type to use type theoretic jargon. I say 'produces' but the function never returns so if there was some way to sample where the function was at or … (heh, that's an ordinary ellipsis). Or rather say it is a function where the first time it returns 0.9, the next time it returns 0.99 at a higher resolution, then 0.999, and so on. If I'm wrong then I'm wrong for this reason. Mathematicians are using the symbol … ambiguously. Because in some places in math it really does just mean “and so on“ and nothing more. Or more pedantically “and keep going in such a manner following that pattern“. If, as you say, in instances like this mathematicians take … to mean “the limit (from calculus) of the demonstrated sequence” this is news to me. Well, maybe it's not news to me but I'm not exactly buying it. What I'm saying is that the expression could be taken to be one of any number of things (not sure if I ought to qualify that with the phrase 'mutually exclusive'): (1) the (symbolic representation of the) sequence, (2) the most recently generated term, (3) the generator (4) the limit (from calculus) of the sequence. That you say the convention is that it most assuredly is 4 is not something I'm buying into. (I'd nearly go a step further and assert that it ought not be defined this way). Where we get into problems, I believe, is when people say nonsensical things like the sequence reaches 1. Or if they start talking about the actually completed sequence rather than a symbolic representation of it. And this, by the way, is the gripe that people have with Cantor's completed infinities. It really is an ontological claim, not a mathematical one that such things exist. It's a claim whose refutation I support. When this topic comes up on HN I invariably point[0] people towards the absolutely best book I've read on the subject. René Guénon's The Metaphysical Principles of the Infinitesimal Calculus. This book is lucid, compelling, precise. I urge anyone with even the slightest bit of interest in this topic to read it. The notion of the continuum is a very very slippery beast and I know of no other book that deals with the topic so well. Category theorists have a much better handle on the smoothness of the continuum than set theorists because they have far better machinery. [0] http://www.sophiaperennis.com/books/rene-guenon-series/the-metaphysical-principles-of-the-infinitesimal-calculus/ http://www.sophiaperennis.com/books/rene-guenon-series/the-m... Here's a link to a copy in various formats on Archive.org https://archive.org/stream/reneguenon/1946%20-%20The%20Metaphysical%20Principles%20of%20the%20Infinitesimal%20Calculus https://archive.org/stream/reneguenon/1946%20-%20The%20Metap...

z1mm32m4n 9y ago

Some context for those who don't go/didn't go CMU: 15-251 Great Theoretical Ideas in CS is the freshman spring semester discrete math & theoretical CS (a follow up to the discrete math course they take freshman fall). The course itself goes for breadth over depth. In this aspect, it serves as a solid foundation for all theory classes students take going forward. It's a class where students learn proof techniques, collaboration, and problem solving. The course is often praised for its rigor. The problem sets require a fair amount of work to complete, and the course staff sets a high standard for responses. On the other hand, it commonly receives criticism for being quite intense as a freshman class. Many CMU students look back on it as one of the hardest classes they took. I personally feel quite lucky to have taken 251. It gave me a foundation to appreciate a lot of deeper CS topics, and taking it freshman year meant I had 3 years more to put it to use.

kutkloon7 9y ago
I was just about to comment that the topics seem really intense. I got the impression that Carnegie Mellon is an excellent university with very high standards (much better than Harvard, Stanford, Berkeley, etc., which are prestigious and have lots of professors which do great research, but the actual scientific education is good, but not extraordinary, and moreover, the courses are not very hard from what I've heard). It happened multiple times that I googled some relatively obscure topic, and I found multiple excellent sources from Carnegie Mellon professors. For example, there is this excellent document [1] by Jonathan Shewchuk, about the conjugate gradient method, and you will found as many as four lectures about DRAM memory on youtube. [1] https://www.cs.cmu.edu/~quake-papers/painless-conjugate-gradient.pdf https://www.cs.cmu.edu/~quake-papers/painless-conjugate-grad...
- laughinghan 9y ago
  My friends had him and I hear Jonathan Shewchuck is indeed an excellent teacher. However, he has never been a CMU professor. In 1994, when your link was dated, he was getting his PhD at CMU, and since 1998 he's been a professor at UC Berkeley (where I went): https://people.eecs.berkeley.edu/~jrs/ https://people.eecs.berkeley.edu/~jrs/ I have no doubts about CMU's excellence, but I found Berkeley's undergrad CS education to be quite good. There certainly were research professors who were bad teachers, but many like Shewchuck were outstanding teachers, and there were also several Teaching Professors whose job description to focus is more on teaching than research, like Dan Garcia, Paul Hilfinger, John DeNero, and Brian Harvey (since retired): https://www2.eecs.berkeley.edu/Faculty/Lists/faculty.html https://www2.eecs.berkeley.edu/Faculty/Lists/faculty.html I too have heard that private schools like Harvard and Stanford are not very hard once you get in, just very hard to get into (though I can't speak from experience). I haven't heard people say that about Berkeley, which is a public school so it doesn't have shareholders where alumni can form a controlling majority, and it's funded more by the state than by tuition. That also certainly was not the experience of myself or anyone I know.

rawnlq 9y ago

Recording of the lectures for Spring 2016: https://www.youtube.com/watch?list=PLm3J0oaFux3aafQm568blS9blxtA_EWQv&v=uaAvVNWvi4A https://www.youtube.com/watch?list=PLm3J0oaFux3aafQm568blS9b... Slides: http://www.cs.cmu.edu/~aada/courses/15251s16/www/schedule.html http://www.cs.cmu.edu/~aada/courses/15251s16/www/schedule.ht...

tu7001 9y ago
Those YouTube lectures are prof O'Donnell course. I don't know if they match.

westurner 9y ago

List of important publications in computer science https://en.wikipedia.org/wiki/List_of_important_publications_in_computer_science https://en.wikipedia.org/wiki/List_of_important_publications... http://paperswelove.org/ http://paperswelove.org/ https://github.com/papers-we-love/papers-we-love#other-good-places-to-find-papers https://github.com/papers-we-love/papers-we-love#other-good-...

sidusknight 9y ago

Most of the links are behind a login. Why even post this here?

kutkloon7 9y ago
The slides are not, and there are so many links that are directly behind a paywall on hacker news... Those really bother me.
- dmix 9y ago
  WSJ style paywall articles are bypassable via the 'web' link. Not quite the same at all.
  - kuschku 9y ago
    If you are in the US, that is. Their paywall bypass doesn’t work for all countries.

anaccountwow 9y ago

Why such an old version ;-;

dnautics 9y ago

no love for coding theory!

raister 9y ago

The Markov Chains lecture has a huge 'meh!' feeling attached to it, I'd be better of not clicking on that PDF in the first place.

Chathamization 9y ago

I wonder what kind of retention there is. Is one day of class studying quantum computation going to be particularly useful to people when they study it 6 months or 2 years later? I've lately studied a bit of program design when it comes to working out. A lot of effort goes into designing workout programs, making the programs efficient, trying to get the maximum results from the least amount of effort. There's a clear progression and a clear way of measuring progress. Every exercise, stretch, and movement is supposed to a particular function that best moves you toward your goal. Particular muscle sets are targeted at particular intensities for particular intervals on particular days. There's a reason for every specific decision that gets made. You can contrast this with the aimless way a lot of people workout. They go to the gym, they do whatever exercises or work on whatever machines seem like a good idea to them at the moment, and hope it increases their fitness in some non-specific way. Lacking good metrics for measuring effectiveness, they often judge a workout by how sore it makes them the next day (an extremely poor way to measure effectiveness). It's not that this approach can't improve your health, it's just that it's extremely inefficient compared to the more goal orientated and progression based system. In my experience most universities follow the aimless second approach - throwing what they can at the students, hoping some of it sticks, and judging the courses based on difficulty (this usually becomes even more obvious when professors explain their courses). I don't think I've come across any study measuring the effectiveness and metrics of different programs and progression paths.

Arcten 9y ago
Probably, one day of lecture on the material would be useless. But the weekly homework assignments force you to develop a better understanding. I frequently remember topics, assignments, and even specific questions from this course when they are relevant to something I'm working on now.
bichiliad 9y ago
It's a bit more than it seems, on the surface. You're not just expected to pick up what the professors lecture about, but to take it and apply it to a sampling of fairly advanced problems. You end up doing a lot of learning on your own (or in a small group) while completing the homework. To use your gym analogy, it's like focusing on one muscle group per week. On Monday, you're shown proper form and technique for a squat, but by the following Monday, you'd better be able to squat significantly more than you could a week ago (and you'll get graded on it). Are you going to become a competitive squat weight lifter? Absolutely not. Do you now know proper form, and understand what it takes to train properly? Almost certainly. I'd also note two things: - Most topics covered were useful in later courses. - A lot of the credit for this course's success can be given as much to really, really dedicated TAs as it can to the professors. (Note: I took this course my freshman year, it was blisteringly hard, and I wouldn't have traded anything for it.)