EDIT: source: took logic as undergrad + wrote on the tractatus which required a lot of pre-reqs to understand. 0 chance a course at undergrad level ever assigns principia mathematica. I don't care if you went to yale or oxford or ecole normale ... 0 chance. Most charitable interepretation: some pages of it + was on a bibliography. not required reading.
if feel embarrassed, that is the consequence for lieing. There is such a thing as intellectual honesty.
Thanks for calling out these sort of posers and charlatans on HN. We should not tolerate these people if we are to discuss/argue/motivate interesting/hard subjects productively.
I automatically discount anybody on HN (until i have looked at their profile/comment history/any personal bio websites etc.) who claim they have read/studied a) Euclid's Elements b) Newton's Principia c) Maxwell's Treatise on Electricity and Magnetism d) Einstein's 1905 Annus Mirabilis papers. e) Principia Mathematica by Russell/WhiteHead f) Godel's Theorem g) Bourbaki's mathematics books etc. etc. They might have browsed it out of curiosity but that is not the same as reading/studying it.
Actual conceptual mathematics/science is intrinsically hard even ignoring the archaic language/notations.
As a good example; the Nobel-prize winning physicist S.Chandrasekhar wrote Newton's Principia for the Common Reader where he explains a subset of the principia (only dealing with gravitation) using modern notation and language. He himself found it quite hard and thus the "common reader" in the title is somebody who has had a good course in calculus and has the motivation to put forth the effort in understanding it.
Einstein's 1905 Annus Mirabilis papers seem like they easiest of the bunch to just read through. I just pulled up 'Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen', the one about Brownian motion, and read the whole thing. It's only 12 pages and fairly accessible; more prose than equations.
(Of course, if you don't read German, you should get yourself a translation.)
I don't think I'm smart enough to casually read and understand original works on General relativity, but the Annus Mirabilis work seems much simpler. The famous E=MC2 paper is only three pages.
About Gödel: if you are interested in the theorems, and not necessarily their original presentation, you can get plenty of rigorous modern treatments. It's very common for mathematicians to work out simpler proofs and more appealing presentations of famous results over time.
See https://dn721807.ca.archive.org/0/items/uber-formal-unentsch... if you want to give one of Gödel's work a go. It's only 26 pages. Footnote 48a is especially interesting. Overall the prose is crisp, but the notation is rather archaic to modern eyes.
I agree with your general sentiment, and your heuristic in general.
My point is that each of the material above (and i forgot to add the original Quantum Mechanics papers) were major watershed moments in science/mathematics and hence are not easy to read/understand. You need a good background in the subject matter(and mathematics) and/or somebody guiding you through them.
So what i do is try and find books which are written for the "educated common reader" by an expert who guides you through the original paper/book. It is still difficult to understand if you do not have the necessary background but at least you have a good starting point.
Some books in my collection;
1) Newton's Principia for the Common Reader by S.Chandrasekhar
2) Maxwell on the Electromagnetic Field: A Guided Study (Masterworks of Discovery) by Thomas Simpson
3) Einstein's Miraculous Year by John Stachel.
4) The Annotated Turing by Charles Petzold.
What we need is for a group of professors to get together and start writing a series on explaining the original papers to the "educated common reader" i.e. not too trivial nor too overwhelming. I think there is a huge market for this since it humanizes how science is done in real life which is fundamental for motivation.
The Annotated Turing is great. Thanks for mentioning the others, I'll check them out. I've read a little of Einstein's original work (in longish excerpts) and it was a nice enough read I'd welcome more of it, especially with expert guidance.
You might also want to look at The Annotated Godel by Hal Prince which i mention in another comment.
We should read the original great works but with intellectual honesty to accept that some/most of it might be over our heads (depending on one's background) and not feel ashamed about it. The point is to not be completely clueless/ignorant but at least have some correct ideas even if it is a simplification/approximation. It is just one stage in the learning process and as we re-read little-by-little over time we begin to understand more and more. This is real education.
> My point is that each of the material above (and i forgot to add the original Quantum Mechanics papers) were major watershed moments in science/mathematics and hence are not easy to read/understand.
I must dissent. Einstein's Annus Mirabilis papers are really quite approachable. You don't need a book to guide you through.
Though if you are having fun with the book, more power to you! Enjoy!
We have to agree to disagree then. You definitely need a good background in Physics to understand them.
For a general reader, even though they might not get the whole thing they can still get an appreciation for the whole from the commentary/details added by the editor.
The book contains the English translations of the 5 papers with Einstein's original introduction and a short discussion on each, a informative introduction and a foreword by Roger Penrose.
No high school teaches you Electrodynamics at the level needed to understand his Relativity/Photoelectric-effect papers, the idea of Stochastic Randomness behind Kinetic theory/Brownian motion etc. You need both background physics theory and the mathematical tools.
It's a bit more involved than the one about Brownian motion, but it's still fairly accessible for the motivated layman.
I'm not sure why you are so convinced that this is so hard. Nor what you are trying to convince me that I should find it hard: I've literally had a look at the German originals. No amount of secondary or tertiary writing is going to overrule that.
Schools differ in what they teach, of course. And many of them aren't very good. But Wikipedia is always available and these days you can ask your favourite LLM for explanations, too.
Of course, as I said already, the books you are suggesting are probably great and fun to read. They just aren't necessary.
I am now beginning to suspect that you are validating the skepticism that i expressed in my original comment. You have been pretty cavalier about your claims (a high school student, really?) which cannot be accepted at face value.
As already mentioned, I already have the english translations of all the five papers, have spent some time with them, and they most certainly are not "accessible for the motivated layman" without the necessary physics/mathematics background. No amount of motivation/enthusiasm can make up for lack of necessary knowledge.
Since you kept mentioning the brownian motion paper as being "easy to read", the second section on "osmotic pressure from the standpoint of the moleculer-kinetic theory of heat" starts with the equations for entropy and free energy of the system which i would like to see a "motivated layman"/"high school student" understand, much less derive. They need knowledge of thermodynamics and statistical mechanics (assuming they are already comfortable with multivariable calculus/PDEs) to even get started.
Similar criticisms apply to the rest of your claims regarding other papers.
That is why you need guidance from teachers who do have knowledge of all the required sub-domains and can help the student navigate the paper. Of course one can also use all other available tools like LLMs etc. but their veracity again needs to be checked by a human in conceptually complex subjects like these.
> who claim they have read/studied a) Euclid's Elements
Of the lot, Elements feels misplaced.
Lots of people actually do read Elements as part of their course of study. It's niche but there's a whole cottage industry within academia for that sort of thing. There are probably over a dozen institutions that have either a degree program or a core curriculum that is organized around original texts, with Euclid usually serving as the math distribution of that sequence. So running into people who have read (big chunks of) Elements is not that uncommon. That's true even IRL outside of online discussions forums on thread topics that likely select for such people.
My impression is that this is not really true of the other examples. Except maybe Godel's proofs; I do think a sufficiently motivated instructor could pull a decent chunk of college students through the original text in a semester. Probably better ways to spend everyone's time, though.
Seconding that that one stood out to me. Actually studying at-least large portions of it is typical at a handful of liberal arts colleges that favor the "great books" approach, at least a couple of which have pretty good reputations and are likely to have turned out some folks who work in tech (maybe the programmer next to you... maybe your manager's manager), plus it's pushed in great books home learning programs that surely at least a fair number of people have credibly attempted, even if the overwhelming majority of those who start such programs don't complete them (and it's usually very early in those programs, so even those who gave it a real shot but abandoned it before getting far were likely exposed to quite a bit of Euclid, though maybe they dropped off before On Conic Sections or other texts common in those reading sequences).
... plus it's relatively approachable as such things go, and short enough that closely reading most or all of it isn't a crazy idea, and it was recently-enough widely used as an actual textbook that between that and ongoing modern interest in its use in that capacity, there are tons of study-oriented editions of it floating around and still being published. I mean hell "recreational mathematics" is a thing and lightly-annotated-and-updated Euclid's a pretty solid text for people with that kind of interest to noodle on, with bonus historical appeal since it's super-old and also is assumed background for all educated people into at least the early 20th century, so pops up all the time in historical writing and literature.
Now, Newton? That's more like it. Nobody reads a large amount of his mathematics unless they're some variety of mathematical historian.
As an Oxford undergrad I read a lot of Bourbaki on topology, to supplement the lectures; they were recommended by the lecturer. I loved their style of writing and the cleanliness of the approach. I have not been assigned Gödel's paper but have read them in classes where the proof was demonstrated. Maths is hard tho, I agree on that, and maths reading very hard, and reading these books outside of class where you are forced to keep going till you understanding is only rarely worth it outside of professional life.
I don't know the exact curriculum and I'm sure it's changed over the years, but one of my girlfriends from back in the day really did go to a school that had one like this. St. John's College, which has two campuses in Annapolis, MA and Santa Fe, NM. They had no majors and everyone learns by reading the classics directly. They also have to learn classical Latin and Greek and read many in their original languages. I don't know everything they assigned, but I remember at least they actually did learn geometry by reading Euclid and calculus by reading Newton.
Apparently, the history is that the school lost its accreditation and had to shut down during the Great Depression, so to attract investors and reopen, it adopted an extremely unique identity with no watering down of curriculum and commitment to western classics in an attempt to combat the rise of fascism.
They may have learned mechanics by studying Newton but they can’t have learned calculus. Principia includes geometric series and limits etc but given as geometric arguments so you don’t come out of newton’s principia knowing how to do calculus. If they learned the method of fluxions from Newton (which is equivalent to calculus) then I feel very sorry for them missing out on the far better modern presentation of Leibnitz’s calculus that they would get in studying say Spivak or Stewart or any other modern textbook. For the same reason everyone teaches Taylor series (which are fantastically useful) rather than Newton’s wildly inferior series derivation which he used because Taylor series hadn’t been (re)discovered yet.[1]
Euclid isn’t surprising. School children used to learn plane geometry from Euclid until the 1950s or so. I learned geometry at school using a syllabus from Euclid and we learned the modern form of Euclid’s postulates etc but we didn’t study Euclid itself.
[1] Taylor series were developed in the modern form by James Gregory who was trying to reverse engineer how Newton had come up with his series expansions. I say rediscovered above because they were first written down by Madhava of Sangamagrama who gave Taylor series expansions for the trigonometric functions and natural logarithms/exponential function in the 14th century.
I heard about St. John's from a twitter thread and find it deeply baffling.
It's as if a group of monks wanted to keep the quadrivium and trivium but their clock stopped at the 16th century. One of their faculty proudly said they study analysis by reading Descartes! Which I thought was a highbrow joke but nope, dead serious.
There's a reason that 'standing on the shoulders of giants' is a thing. Dive into the classics after you have gained the maturity from modern texts.
Take one of the easier problems from Rudin. Prove that a continuous function from the unit interval [0,1] to itself has a fixed point. I wonder how a student immersed in the 'classics' would even begin to tackle this.
It's not a Mathematics degree. It's not even a Philosophy of Mathematics degree. It's a particular type of Philosophy degree.
So, to be fair: most philosophy majors wouldn't have much luck with Rudin.
> Dive into the classics after you have gained the maturity from modern texts.
Diving into old texts is a skill unto itself. That's why a lot of institutions do the great books thing as a core curriculum (so, maybe 2-3 courses taught in this style, as an alternative to more conventional phil 101/history 101 style distribution requirements). Then a more conventional education from there onward. The theory is that this is a mid-point precisely because it provides lots of transferable skills for diving into the classics in your chosen field, while avoiding the "let's learn analysis from descarte" excesses.
When I was a pretentious high schooler with fantasies of being an intellectual, I considered going there. I ended up not even applying. It just seemed too far out of the norm.
I think their deal is taking seriously the "college is about learning how to learn" thing, and direct engagement with the output of people regarded as greats in their fields on the assumption that, when possible, that's a good idea for obvious reasons (whether that's true or not in some rigorously-provable way, I can't say, but the reasons one might suspect that it is seem clear enough)
Some report it's pretty damn effective at that and leads to an impressive breadth of intellectual confidence in tackling material of almost any sort, but IDK. Anecdotes.
IIRC (it's been a while since I looked into their programs) they do a lot of supplemental reading of newer papers, articles, and book excerpts, and tend to used updated notation when it makes sense. Plus all their classes are heavily discussion-oriented so the reading is potentially enhanced and brought "forward" by whatever their instructors and peers bring to class in their heads.
> Take one of the easier problems from Rudin. Prove that a continuous function from the unit interval [0,1] to itself has a fixed point. I wonder how a student immersed in the 'classics' would even begin to tackle this.
I don't think they tend to train mathematicians, and I think for most students (graduating from any college or university) they never, ever, ever touch the specifics of their more-advanced e.g. math classes (I think this is true even for most programmers or engineers or what have you) any time in the entire rest of their lives, to the point that entirely forgetting most of that stuff by a decade or so later and suffering for that not at all is utterly typical. How much does it matter for students who aren't going into extremely narrow vocations that they come out of them unable to perform this specific task, without first needing to study further?
The usual defense of this fact is "well it's about learning how to learn, expecting the actual content to ever matter for any but a teensy tiny proportion of the students is unreasonable" in which case... see the rest of the post.
I've certainly read and studied material explaining f), but I haven't read the original paper (besides, isn't that in German?). The proofs aren't that hard, as I remember them, maybe they're harder in original form? To be clear: I specialized in logic, formal verification and programming language theory at uni. This was a while ago, and I'm on new parent amounts of sleep, so pls b nice.
I have tried to read some articles and watch some videos "explaining" Godel but never really understood it. Everyone seems to be explaining the mechanics of what Godel did but explaining the Why is lacking i.e. What was it in mathematics that got him even thinking on these lines in the first place? Can this problem be demonstrated with a simple toy axiomatic formal system? How did he hit upon his approach? Answers to these sort of questions is what seems to me the most important thing to understand before following his arguments.
I recently came to know of The Annotated Godel: A Reader's Guide to his Classic Paper on Logic and Incompleteness by Hal Prince which i think i need to sit with :-)
And the reason for Hilbert's program? The problem of "Russel's Paradox" which is a contradiction in naive set theory - https://en.wikipedia.org/wiki/Russell%27s_paradox (Note that there were other paradoxes too).
Hilbert's idea was that by completely formalizing mathematics on a axiomatic/deductive basis, one can mechanically derive proofs so that you don't run into paradoxes/contradictions.
But then Godel showed such a formal system applied to basic mathematics can never be complete (if consistent) and never prove its own consistency.
To me the mechanics and the why are closely intertwined. If you feel like self-referentiality is a way to demonstrate a problem (this is the “why”), it is not a long step to the mechanics of encoding.
The work is in creating the theorem / contradiction from that point, but in the big picture, the approach doesn’t have to come from nowhere.
> the mechanics and the why are closely intertwined
But the "why and what" must always be explained first even if it is incomplete; since that is the problem we are trying to solve. With logic it is even more important since you can follow a proof from one step to the next but by the time you reach the conclusion you have lost the connecting thread to the starting point (for most general folks) i.e. "you have missed the forest for the trees". This is why many folks feel lost when doing mathematics as mere symbol-pushing.
Honestly, other than the length and tedious presentation, I don't really think the material in the Principia Mathematica is outside the reach of an advanced undergraduate. As a point of reference, MIT's capstone mathematical logic course[1] has a syllabus that requires at least as much mathematical maturity, and it wouldn't really surprise me that much to see it as an ancillary or excerpted text.
That said, even if the OP was assigned the text at some point as an undergraduate, I remain a bit doubtful it was actually read.
> As a point of reference, MIT's capstone mathematical logic course[1] has a syllabus that requires at least as much mathematical maturity,
The textbooks they use in that course are written in modern notation and are accessible to a knowledgeable reader; neither can be said of the Principia Mathematica. The archaic syntax is a serious issue.
The French president notoriously said a few weeks ago in June during the third (!!!) heat wave of this summer “who could have predicted [it]”… that’s the level, at the highest layer which decides on policy.
Governments have been working on this for more then 20 years now. 40 maybe. As long as I've been alive at least. It's just that all of the actions to try and deal with this have been continuously undermined by fossil fuel industries and right-wing politics.
Hansen is still going and has been ringing the alarm that we have deeply under estimated the relative carbon loading and heating potential on emissions. There is a real fear that we have locked in a lot more warming potential than the models have been predicting via several factors. Namely carbon warms slightly more than anticipated, more methane releases, great water vapor levels as a result of the previous two and reduced reflective materials decreasing our relative albedo.
There is a national election in Sweden this year. Which parties or blocks are against construction of thermal power plants, and which are against it? Thermal power plants are primarily fueled by oil or natural gas. Party platforms are public and posted on their websites.
A few years ago EU voted to define two technologies as green. One was natural gas, and the other was nuclear. Which parties voted for natural gas, and which parties voted for nuclear?
There is a EU decision to phase out ICE in cars. A good continuation would be to issue a similar decision to phase out ICE in the energy grid. It would be nice if both sides could just join together on such decision.
For me, it has degraded to the point where I simply presume that no significant promises are going to be kept after the election anyway. The respective parties will promise what they think they need to get the votes, and will then target whatever is relevant for the ever-evolving social-media driven zeitgeist.
"A completely fossil-free energy system" ("Ett helt fossilfritt energisystem" (1)) yesteryear and now lower fuel prices than 20 years ago (adjusted for inflation, yes).
Fuel prices is one of those things where people will complain about Iran war driving prices up, and then in the next sentence complain that low fuel taxes will encourage consumption. The reduction in taxes has basically removed half of the price on gasoline, but with the increase in global market price this has only resulted in a minor price reduction in total. For diesel the price is still above that of last year.
It will be interesting to see if researchers will find any significant behavioral change from having half the fuel prices compared to nearby countries. Personally I doubt it will be a significant statistical difference, but it is a perfect time to get some real data if people treat fuel as a necessity and pay what the market demands, or if its a luxury product where consumption goes up and down based on price.
Right-wing politics being opposed to environmental action is so weird when back in the day right-wing politicians didn't have such an issue with it (for example this speech by Margaret Thatcher: <https://www.margaretthatcher.org/document/107346>).
This is downvoted, but exactly right. Moreover, government is not a singular entity. Blaming the whole government just serves to take the blame from actually guilty and attributing it to those whonwere doing the right thing.
No, its that all the action consistently denies the fact that human societies can not keep i together if degrown. The policies produce regression aka facism.
Wow! that looks like a bit of software I have been dreaming about for awhile - will definately check out! You're at least doing something right in communicating the reasons why and appeal for starters! All the best!
