This is a great analogy, and one I'm surprised more people haven't raised. Instead all your hear are comments about mathematicians being sour losers, they should have know the model terms of service etc...
Sometimes reading these articles I can't help but think we've pushed ourselves into a local but sup-optimal maximum technologically speaking. i.e. If we didn't follow the path of trying to stick everything into an iPhone/Android (along with moving everything into the cloud), we'd have a more interesting technological landscape.
Everything has been optimized towards giving a small number of companies total control and surveillance over our media. That means forcing people to store their files on someone else's servers. Those companies don't really care about making better technology for others to use. Fewer options for the public, lack of innovation, and the increasing enshittification of everything is a price they are willing to pay for control, lock-in, and rent seeking.
I have no idea why these sorts of posts are popular. Past college you're not going to learn physics by trying to self-study an entire university course. The best way to learn is just to pick a small part of physics you'd like to learn (preferably related to your job), i.e. how GPS work or some fluid mechanics etc... Then learn the physics you need for that. Knowledge accumulation can't be organized in a straight line, it happens non-linearly and generally builds upon small wins that are useful for you.
I think this is unnecessarily pessimistic. I think there's actually a surprisingly large number of people who are interested in and have the discipline to study something on their own, and there's value in pursuing a whole course of study from beginning to end. Doing things piecemeal and incrementally has value too (especially in software development), but can obscure the whole shape of a discipline.
I'd disagree and have done exactly that. I started with a reasonable basis in physics, but self studying through an entire course dramatically expanded my understanding. For an example of what I mean this [1] is Feynman's lecture on conservation of energy. It's something every schoolkid learns -- I'm fairly sure I saw my first pendulum from the nose demo in middle school, yet Feynman will take this and make you completely rethink your understanding.
So for instance, what is energy? Somebody who knows a little would probably tell you something like the capacity to do work, and so it feels quite abstract. But the interesting thing is that energy really "exists" so to speak. If you paused the universe somehow, and then resumed it - you'd need to know exactly how much energy was and where in order to keep things moving as they were. Yet there is no known 'thing' that is energy - just a wide array of mathematical abstractions.
And then this thing is also perfectly conserved such that the amount in play will never change. It's completely bizarre to think about, and this is something you initially "learned" in grade school, and probably never even really though twice about.
--
And more generally I think the point of learning should not be to do something, but to expand your own mind and understanding of the world (and beyond). Outside of this being arguable alone as a philosophical point of view, I also think there's even a practical reason for it - unknown unknowns. There are things you can't even imagine that you don't know, and the only way you can reconcile this is trying to dive into things across a wide breadth.
> the point of learning should not be to do something, but to expand your own mind and understanding of the world (and beyond)
I agree with your comment overall, but people have different reasons for learning and it may not be productive to tell them they are learning for the wrong reason. Let's celebrate all learning for I fear we are heading in a direction where it will become increasingly uncommon.
Conservation of energy is simply the invariance of a system to translation in time: the system shows the same time evolution from a well defined starting state regardless of whether that state exists at a time t or a time t+t' with an arbitrary t'. This is one application of Noether's theorem.
This is an example of a topic that unstructured self study would likely skip. But it is included in every properly structured course because this theorem puts a fundamental explanation to all conserved quantities that occur in classical mechanics.
Ignoring that, saying "energy must exist because of Noether and time-translation" doesn't really tell what energy actually is, not a very satisfactory response
On the contrary, this is best description of what conservation of energy means: all externally generated forces acting on the system do vary with time. Im other words, this is a closed system as defined in thermodynamics.
The nature of energy is really just a result of this property.
The difficulties with energy in GR are related to the fact that those equations alone allow for space-times that are truly bizarre. Some that are technically solutions to the field equations are clearly nonphysical (e.g. Goedel's solutions). There is also a priori no reason why e.g. expanding metrics should actually yield proper conservation of energy.
Noether theorem is one of the nicest ones to derive (together with Maxwell equations). There is such an elegance to go from the axions to the result.
I still have a trauma associated with this, though, because it popped out in one exam and I could see the pages in my notes where it was derived but they were quite blank.
I suppose it depends on what you mean by unstructured (vs. the original claim of self-study). e.g. my method of self-study was generally to find course requirements for a degree program, find corresponding course descriptions and syllabi, find the relevant books, course notes, and/or video lectures and proceed through those.
e.g. I did formal study of basic physics in an engineering program, but Noether's theorem was never mentioned. I came to that through self-study, and can't really imagine how anyone would miss it unless "self-study" means "read random blogs and watch random youtube videos." I expect even browsing Stack Exchange and Wikipedia would expose you to most core topics fairly quickly.
You are following a structured course. I meant unstructured in the sense that topics are not chosen based on a properly designed curriculum or syllabus, but based on personal preferences alone.
I also don't like this focus on practical applications. It is true for me as an engineer, that in the end I want to build something. But when I force myself to focus only on the practical part, just as you said, the whole context is missing. There is this meme of a guy still living with his parents, and trying to acquire all knowledge. This is actually a dangerous mind virus. Learning is an idle activity, often confused with being a do no good. But for me at least this is the only way to really grok something. In university at the summer break, I could only really go through the math and enjoy it, without the stress of exams and cramming. And now it is the same, I learn things idly, and trust in the universe that the application will come, which it always does somehow.
This is new to me (thanks, I'll investigate) and it's the kind of things the books of Physics at University (Computer Science) was full of, completely detached from real world experience. Then there was the book of exercises to prepare for the exams. Sometimes finding the right way to go from theory to calculations was a puzzle in its own. I guess that they were almost the very same books they used at the Physics department and teachers are really interested only in students that can master the subject by themselves with no help. I had one of those students at high school. He was reading university books by 16 and he is a professor now.
Luckily the books of Physics at high school started from practical experience and then showed the equations. So, my advice to somebody willing to learn physics, with plenty of time, no ambition to become a researcher is: go through high school books to get the gist of the subject, then go deeper with one of those university books.
I break it down in reverse order to let you understand how I get a gist of that definition
wavefunction, for example light or any particle, this can't be argued.
the complex plane, because? But a complex number is only a pair of numbers and I know that in polar notation it's handy for:
rotation! And I remember that the electric and magnetic components are on orthogonal planes so we are back to complex numbers and maybe the components rotate, but how about neutral particles? But they are not neutral inside. I think that I'm already off track, very lost.
Energy = frequency, because I know that higher frequency photons are more energetic.
Finally, usually we learn about kinetic energy, then E = mc^2 and maybe about the relativistic effects, without the formula. Those things can be understood easily and are taught at school. The jump to angular rotation and the complex plane is something that few people are exposed to and, as I hope to have demonstrated, it's not easy to map to common experiences, which could be inevitable.
Hi, fair enough. I've thoroughly immersed myself in this stuff enough that I sometimes don't realise people aren't as familiar with the concepts and jargon. I'm going to write another article that I think should help about the Klein gordon equation, at some point.
> I have no idea why these sorts of posts are popular. Past college you're not going to learn physics by trying to self-study an entire university course.
You have no idea, because you have no motivation or interest to self-study.
But if you pay attention, you will notice that your personal motivation and interest does not match the ones from all the people engaging in this discussion.
The reason why this sort of topic is popular is because others have different interests, motivations, and ability to do it.
You should not spend time trying to convince others that they cannot, and instead you should look inside and see what you can do for yourself.
IMO: It's not learning that's the problem. It's learning and verifying you actually learned. I'm not saying "unless you can solve homework problems you haven't learned", but... kinda? With a complex generally-theory subject like Physics, the amount of effort you need to put into verification grows exponentially faster than the "learning" part; it's tough for many to get a "feel" like you could for chemistry or programming or electronics or mechanical stuff, difficult as those definitely can be.
Most of us who learn (and always learned) outside of schools and academia, do so for some purpose and with some intention beyond the learned thing. I didn't learn programming to learn programming, I learned it because that's how you built websites. I didn't learn woodworking just to learn woodworking, but in order to build my own furniture, and so on.
Same can be applied here I'd say, and probably must (at least for me) be applied that way, for the learning to even be engaging. So don't learn physics just to learn physics, but learn it in order to be able to execute on something else, then learn the related parts to that.
Then the verification becomes part of what you're doing, and fun as well, as you're progressing on other stuff :)
I think you're possibly generalising your own experience to the masses. That's absolutely not the case for me.
I love learning to learn. I learned programming because I found a quick basic compiler / IDE on my dad's computer, along with some shipped examples, and it was actually magic to me.
I read our encyclopedias as a kid because it was just really cool. Likewise, I did physics in university not for a job (the jobs are shit), but because it gives me that feeling of expansive possibility and wonder that Carl Sagan gave me when I watched Cosmos growing up.
Most of my most passionate self learnings came about because of that feeling of magic, not because I was chasing an end goal.
> I think you're possibly generalising your own experience to the masses
Yeah, that's fair and probably true, it's limited to my own experience and the experiences of the people around me, not gonna claim it's universal so you're right.
> I did physics in university [...]
My comment is also explicitly not about you and your type ;)
> Most of my most passionate self learnings came about because of that feeling of magic, not because I was chasing an end goal
One could argue that the end goal you were chasing was feeling that magic again :) Jest aside, I do understand what you mean.
I think there is more in this world than is dreamt of in your philosophy. The crowd on HN now is very different than it used to be rest assured there are many people self-teach themselves the equivalent of a university curriculum. I mean, nerds actually exists, they're not all humdrum corpo worker bees trying the maximize their employers' value and then just hiking or whatever.
Let's subject that claim to some basic arithmetic.
A university would expect maybe eight undergraduate level courses, four credit hours per course, and you are expected to do three hours of study per hour of credit per week. That's 1,500 hours.
How does that compare to the "hours played" on a typical strategy game Steam review?
much less dopamine, video games are realy good at stimulating that. Studying physics not so much.
Don't get me wrong Physics can be satisfying in its own ways but the amount of dopamine rushes you get from cracking a hard problem are few and far between. Often enough you just get frustrated and once you find the solution you think to yourself oh is that easy how didn't i get that before and feel absolutely stupid.
Somehow millions of people have learned physics in a "straight-line" at university. Most physics majors have a logical progression from the simplest to most complex ideas.
I think it's useful for learning about unknown unknowns. If you don't have a clear direction, it's entirely fine to start with a university course then stop when you get a feeling for what you really need.
I think self-studying a university curriculum is also helpful. I've studied physics at the undergraduate level myself.
Quantum mechanics, for example, is useful enough as a general background, and going through fluid dynamics and GPS problems is also helpful. But if you just follow the application problems, you can learn what you need, but the bigger risk is that you might miss the larger framework that defines those problems. That's not necessarily a bad thing. In fact, the entire university curriculum training is ultimately a process of translating complex phenomena into the Western scientific way of understanding things. It's a mental model that says 'this phenomenon can be interpreted with this kind of formula.'
In other words, it's about building mental models. In a formal university curriculum, you usually learn things like vectors, topological spaces, energy conservation, and how to map real-world phenomena onto these perspectives. It's about learning to simplify the world using mathematical tools. I didn't go to a top university, so I didn't learn things like tensors, but I hear they're taught now. I used tensors in grad school.
Of course, when I actually code and deliver factory equipment, I've done motor-related work under NDA, and the actual formulas aren't always perfectly accurate. There are corrections and adjustments. But the important thing is not just problem-solving itself, but building the mental model of 'how to approach the problem' before solving it. And I think the curriculum helps with that.
I have this incredible strong urge to try to understand how our world works. I keep finding new topics in physics I just need to understand to calm this urge. Back to grinding differential geometry for me. Im long past college
The analogy seems to be like learning classical music (like piano or violin) after as an adult.
You learn the basics like scales and chords to build and build to modern jazz.
But if you’re an adult, life is too short, just go straight to a few pieces you like. Get a simplified version and learn the bits you need from there from a teacher.
Not necessarily true. A lot of autodidacts have significant gaps. Going through fundamentals can unlock things applied across the board in the field and outside. Sometimes this approach is useful.
That's not what op is arguing. To use your example, coming up with singular examples of continuous non-differentiable functions is an example of "ugly" mathematics, whereas putting them into a nice framework where they can be analyzed as a whole (i.e. functional analysis, density of such functions, etc...) is an example "elegant and insightful" mathematics. The same with the monster group, on its own maybe nothing special, but then you have the connections with other branches of math. Tao seems so focused on the individual problems and not their connections/generalizations.
Well one does have to come up with continuous non-differentiable functions to begin with, right? Weierstrass had to shock the community with his weird series that's almost everywhere nondifferentiable before people could conceive of a nice framework that includes them. People do not invent whole encompassing abstractions out of nowhere
Education yes, research unfortunately no. I'm not saying research outside of academia is not possible, I'm just saying it's not taken seriously and this needs to change. We really do need to go back to the 19th century model of the researcher gentleman.
A real shock to academia is that top research increasingly takes place outside universities. On many areas universities are now 5-10 years behind what’s happening in the private sector. That’s causing a lot of panic within the system and a growing stream of departures as PhDs favor the private sector over academic tracts.
A few years ago, when I was actively involved with the academic world, I came to a similar realization. They're trying to do too many things at once. Universities need to acknowledge this reality and adjust.
After thinking about it, I came up with a straightforward solution (at least in STEM): offer more than one type of of doctoral degree. Every program will have at least two doctoral programs: a Doctor of Philosophy, and a Doctor of Science/Engineering/Mathematics/etc.
The Doctor of Philosophy (PhD) program is academia at its core, where the students in this doctoral program are explicitly seeking an academic teaching or research position as their career path. The coursework and educational activities are explicitly aligned for this area.
The Doctor of Science/Engineering/Mathematics is focused on creating a top-of-the-line researcher intended for industry or an FFRDC. Those students receive a different type of education which explicitly gives them the deeper research skills and connections needed to become an accomplished industry researcher.
The two programs are equally rigorous but have different end goals in mind. This specialization is overdue, and most departments already have a fuzzy line separating the "academics" from the "practitioners."
This just kind of sounds like a random idea that sounds good in your head but not based in reality; the point of a PhD has always been one thing, and one thing alone: train someone who can publish influential papers in top-tier venues.
Anyone who says otherwise is just either uninformed or selling a dream.
I've seen the ins-and-outs of academia within an R1 research institution from about as top-level as one can while remaining a student, and my idea is based on extensive interface with the reality of academia.
Your observation that the doctoral degree system has always been that way is precisely my point: the world has changed, and new forms of training are needed to complement the paper-publishers. The PhD system is broken in part because it's catering to multiple audiences when it should regain its focus on its core mission. That being said, many people want to do research but don't want to work in academia; in fact, I think their numbers are far greater than the academia-oriented. My idea caters to those people, and I think all parties (students, schools, industry, government, the general public) will benefit in this arrangement with almost no drawbacks.
From a degree-focused perspective, it's somewhat unusual that U.S. universities almost exclusively assign PhDs, save for the professional degrees (e.g., MD, PharmD, JD). Multiple types of bachelors and masters degrees exist, and those degrees are certainly differentiated from one another. In some European countries, the ScD is a terminal degree higher than a PhD.
That is how it should be, and how it has historically been.
There has been an unsustainable inflation of academic research on the last 150 years or so after governments decided to formalize research. But the thing about unsustainable stuff is that they always end.
The institutions that teach researchers also doing the majority of it necessarily turn into a Ponzi scheme.
Research is usually a collaborative effort nowadays. You’d need a League of Research Gentlemen. Not to mention that an important number of research fields require expensive research labs/equipment.
I'm sorry, but "they committed some debug code that made the program exit(0)" for me is pretty much equivalent to "they committed some debug code that made the program not work correctly".
reply