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The Shaggy Steed of Physics
The force on each body, whether gravitational or electric, is proportional to the square of the distance between the bodies. An isolated sun and planet form such a system, and a hydrogen atom, which is just a proton and electron, can be simplistically modeled as such. This may seem a trivial problem: you can sum it up in half a page in a physics book. But that's because all the detail work has been done for you. Furthermore, anything more complex than the two-body problem is chaotic and incapable of exact solution, so it's up to the two-body problem to carry us along. This is a complex problem, so this review is rather lengthy.
Let me warn you right off the bat that this is not a book for the faint of heart. It kicked my ass. The concepts are fast and furious, and the math is dense. Equations festoon the pages, daring you to ignore them. But you may not, they're fundamental to the discussion. Mr. Oliver opines that anyone with basic undergraduate math should be able to handle it. I had calculus, differential equations, and a good dose of physics in college and I still found the book tough going, mostly due to the whirlwind of notation and sheer number of variables introduced. I ended up keeping a cheat sheet of key definitions which ended up being four pages long, and took almost two weeks to process it. It reads like an advanced college physics book, except without extra examples or redundant explanation -- he expects you to be smart or motivated enough to keep up.
As an example: 'Using Hamilton's equations to eliminate p' and q', the total rate of change may be compactly expressed as df/dt = df/dt + [f,H] where [f,g] is the Poisson bracket of any two functions of the motion: [f,g] = (df/dqi*dg/dpi - dg/dqi * df/dpi)' I've reformatted this slightly for text limitations; he of course doesn't use * for multiplication, and you should read all 'i's as subscript i. This is fairly simple math in the context of the book.
So now that I've scared you off, what's the payoff? Well, unlike my college physics books which just lead me from factoid to factoid there are moments where the hard work pays off in big "oooh" moments. Your book might give you Kepler's second law: a planet sweeps out equal areas of its ellipse in equal times. But why? We'll just call it 'conservation of angular momentum'; that should hold you plebes. But in Shaggy Steed you'll find the equations like this that you might have thought were fundamental falling out of the woodwork, built up from the real fundamentals.
We start out by defining coordinate spaces and deciding that we're interested in Newtonian/Galilean rather than Einsteinian physics for the moment, since our subjects travel slowly enough and relativity makes things nastier. We start with a particle that has two vectors -- position and velocity. Turn this into two ensembles of rigid body particles exerting force upon each other. From this we build up the laws of motion, arriving at the total energy H of the system, and the 'gene of motion,' the Lagrangian: the difference between the kinetic and potential energy. 'Gene of motion' is a pretty bold claim, so we are shown how every mechanical quantity of the system may be derived from the Lagrangian. From there it's on to the 'action' principle, which is basically the integral of the Lagrangian over time - the key being that of any path the particles may take, they act in a way to minimize the action. Every other law of motion (including Newton's) follows from this, though to explain why it's the case we need general relativity. This was my first 'oooh' moment.
Chapter 3 really sets the pace for the rest of the book. If you're thrown off here, you're not going to make it out alive. To summarize: "Motion consists of the trajectory flow of particles in phase space. Each isolating invariant introduces a degeneracy into the motion in which the full phase space available to the trajectories degenerates into a submanifold. Increasing numbers of isolating invariants correspond to increasing degeneracies of the motion which restrict the trajectories to increasingly restricted submanifolds of phase space." This is more or less the programme of the entire book. Dig out as much complexity as required, then simplify to solvability.
Oliver introduces each new concept, so if you're following along carefully, you can follow along. This is all done half in equations, so we're diving so deep into math that you (okay, I) may be several pages in and forget where you were coming from and where you were going. Then suddenly you're out the back end and he nails it all with a beautiful concrete application or insight. For Chapter 3 it's Hooke motion, which you can think of as approximating two weights connected by a spring. Now if you've ever taken differential equations, or dynamics, you're probably uncomfortably familiar with this system. Now here it is all laid out for you, everything explained, and boy those resultant equations look mighty familiar. So that's where that all comes from, and why they use those particular symbols. The linear central force and the inverse-square forces of our two-body problem turn out to be closely related as well.
To be crushingly brief, Chapter 4 finally gets down to the (relatively) practical matter of classical planetary (Keplerian) mechanics, and why four dimensional spheres are special. Chapter 5 dives into quantum mechanics, and the hydrogen atom loosely simulated as a two body problem, since it has only the nucleus and one electron. And let's derive the fundamentals of quantum physics and the periodic table while we're here. Though I've neglected to mention it till now, Oliver doesn't neglect the human side of all this. He doesn't linger on it, but he does provide context. It's amusing to see how many of these inexorable equations were originally derived by geniuses like P. Dirac, only to be disowned because the implications were too outlandish.
In Chapter 6, it's time to step out of Newtonian/Galilean space and into Einsteinian space. We've made a lot of assumptions, such as the infinitely fast propagation of forces. This is no longer the case; time is no longer separate from space. In fact, we learn how to rotate space into time through imaginary rotation angles (known as 'boosts'). e=mc^2 falls out. But our shaggy steed eventually breaks down on the precession of Mercury. In the land of general relativity, even a simple two-body problem is really a many-body problem - forces are no longer instantaneous, they require force particles. The steed is of no more use.
But wait! Chapter 7, The Manifold Universe, takes on many-body motion like Don Quixote tilting bravely at a windmill, and tries to pull some order from the chaos. KAM theory is introduced and our many-body problem turns out to be not absolutely chaotic, but a mixture of regular and chaotic motion. You may have noticed that our many-body solar system doesn't just fly apart. We can model it more or less as a set of two-body problems with minor perturbations (minor being the key). And of course we can model fluids even though the internal motion is chaotic. Order emerges. Our shaggy steed is revived, transformed.
The back of the book contains the Notes, which are compact digressions into the hard (yes ...) math. I have to admit some of them completely lost me. But they're not required, just extra reading for those of you who eat this stuff up.
This all leaves me with a bit of a quandary. It's a beautiful book if you're a graduate-level student of math or physics, smarter than me (your best bet), or willing to put a lot of effort into it. Otherwise I can't recommend it -- the book is gibberish if you can't follow the math. I can't help but think that it would make a fantastic course in the hands of a skilled practical math teacher like Dr. Gary Sherman at RHIT; I certainly could have used his help with this. So, it's to teachers like him that I'd really suggest this book, for eventual dissemination to their students. Or if you dig physics and have the math skills, you might want to try riding "The Shaggy Steed of Physics" alone. If it throws you, there's no shame.
You can purchase The Shaggy Steed of Physics: Mathematical Beauty in the Physical World from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
Fair warning: the review is lengthy, because the book demands it.
Yes, I believe on the inside cover the book EULA stating "All reviews of this material must be over 3 pages in length"
Do really dense people warp space more than others?
Although there are some places that could be better, I give this book an 8 out of 10.
...is O'Reilly's Physics for Game Developers.
One of the chapters - on 'real world' projectile motion - is available for download at the above site, so you can get a feel for the writing and content.
The Army reading list
I know, it sounds like the beginnings of a bad porno.
Maybe I'm in a minority of 1, but that review didn't seem very long to me. Sure its longer than a jacket summery... but it hardly does as far enough to be in-depth let alone deserve a warning.
anyway... better than the usual 'contents table' affair we get on slashdot I suppose. Hardly Sunday paper review long though.
0daymeme.com: Great stuff.
Cheers,
Erick
http://www.busyweather.com/
Huzzah! Interest has been loosed upon the internet by phyl0x, the great emancipator. No longer shall interest be bound and forced to live and work as a slave to...well...ummm....
/petpeeve
Oh, I see... It looks like you meant to use the word "losing," as in "lose, losing, lost." Good luck with that next time.
But that's OK I guess. Sounds like a good book, albeit way over my head
Otherwise I can't recommend it -- the book is gibberish if you can't follow the math.
If you want it to make sense, you gotta accept the fact that the book, by itself, is not supposed to turn an interested laymen into a learned professor. Books like these, for me, spur me to go learn the basics instead. Even if I never get all the way through the book, I can at least use it to tell me what I need to know to be considered "learned" in the field.
I remember in college as a CS student, being spoon-fed the easy-to-learn computing theory and feeling like I was getting nowhere. I picked up the Hopcroft & Ullman automata book and was, at the time, completely inundated by the math (I went to a commuter college with a not-so-advanced math & CS dept.). But at least I knew what I really needed to learn next. I ignored the professor pretty much for the rest of the class (and never opened the textbook) and instead investigated only those things I required to understand the H&U book. I found that by the end of the class, though I was not yet a quarter of the way through the book, I knew a lot more than my classmates, who still struggled with the basic concepts of the field.
If the book seems too much for anyone other than an grad student, try using it instead as an index of things you need to learn first. Don't know those formulas? Look 'em up. Even if you don't grasp everything in your target book, you'll be smarter for it in the end.
Man, I was thinking this was an awesome book, but after scrolling through like 2 pages of the summary, I felt like I had been hit by a truck
Refer your friends, get an ipodThis is completely false. This is not a sig.
Pretty much sums up most physics books I've ever seen.
Do you mean to say you've unleashed your interest upon the world? Or did you mean to say that you've lost your interest, as in "losing?"
Perhaps I'm behind the times, but aren't gravitational force-carrying particles simply conjecture at this point in time? Yes, they're logical and fit nicely into our understanding of the three [four] fundamental forces, but they aren't scientific fact yet by any means of the term - perhaps at most a theory that makes sense, but we've found impossible to test. But like I said, maybe I'm just behind the times.
Anyone here care to explain to someone not yet finished with higher level maths.....?
Sometimes the majority just means all the morons are on the same side.
Kind of inappropriate if you ask me.
I glean from the review that the book is a detailed examination of the two-body model for object interactions, from femtoscopic to macroscopic, in the original, untranslated mathematical language. But what has all that got to do with a small, shaggy horse? I can guess from the Slashdot summary that the model is like the small Irish steed, guiding its rider to exciting, unknown places. But does that mean that the long review isn't as relevant to a synopsis as the Slashdot summary, and that the first line of this post is the capsule review proclaimed impossible by the reviewer?
--
make install -not war
Ohh, I am salivating already! Thanks for the reference!!!
1,082,811
The only acceptable defense of scientific results is to say that they were the product of the Scientific Method.
"I just stopped reading right there. What does this have to do with news for nerds?"
It is something for physics nerds!
We start out by defining coordinate spaces and deciding that we're interested in Newtonian/Galilean rather than Einsteinian physics for the moment, since our subjects travel slowly enough and relativity makes things nastier. We start with a particle that has two vectors -- position and velocity. Turn this into two ensembles of rigid body particles exerting force upon each other. From this we build up the laws of motion, arriving at the total energy H of the system, and the 'gene of motion,' the Lagrangian: the difference between the kinetic and potential energy. 'Gene of motion' is a pretty bold claim, so we are shown how every mechanical quantity of the system may be derived from the Lagrangian. From there it's on to the 'action' principle, which is basically the integral of the Lagrangian...
Online backup with Mozy, sounds like Ozzie, but more!
Equations festoon the pages, daring you to ignore them. But you may not, they're fundamental to the discussion. Mr. Oliver opines that anyone with basic undergraduate math should be able to handle it.
If you have to use math to explain something to someone else, it is because you do not truly understand it at its fundamamental level. Math does not explain anything. On the contrary, it is the math that cries for a physical explanation.
As an example, neither Newtonian's inverse square law nor Einstein's GR equations explain why things fall. They just describe the motion of massive bodies with respect to one another.
Mathematical Physicists tend to apply solutions to differential equations like the 2-body Shrodinger's equation as if they know how to solve an arbitrary differential equation. This type of posturing is probably the kind that you see in this book. The problem being described is actually found in just about every undergraduate modern physics textbook and every physical chemistry textbook. The way mathematical physics is delivered to an audience usually sends them as far away from the subject as possible. It is really possible for somebody to write the "kinder, gentler" textbook of mathematical physics. The subject is a pain in the ass, but not as much a pain in the ass as those who teach and practice it.
The exemplary equation does not make much ...
...
sense as it is and should really be:
df/dt = \partial f / \partial t + []
I am using LaTex notation where \partial t is the
partial derivative with respect to t and dt is
of course the total derivative.
By the way, the book seems to be a solid introductory text for physics students.
Nothing more nothing less
i'd like to be the third person to tell you that you used the wrong word. maybe this massive humiliation will help you remember?
Im really surprised at some of these responses... i would really have expected most of you folks out there to be able to understand the chain rule and poisson notation. Granted its a little lame to be in a fantasy book, but sounds pretty interesting and a good quality review. Thats my two bits.
...and it should be known by now
MIT Press blurb
The book is also online in html form. It sounds like you weren't used to the Lagrangian formulation of mechanics, which has been around for a long time but is usuually not taught in lower level undergrad physics courses (i.e. normal engineering physics). If you take an upper level class in classical mechanics, you'd cover it thoroughly. Sussman and Wisdom's book presents it in an interesting computer-inspired way. Note though that this is a textbook (with problem sets and all that), not a popularization.
Mr. Oliver opines that anyone with basic undergraduate math should be able to handle it.
What this really means is that the book is an entire treatise in the failure of the American educational system to produce people who know these things.
I took 3 semesters of calculus, DiffEq (twice), and Discrete Math, and another class that mostly consisted of proving things which I condidered more of a logic class but was in the math department anyway, and I don't recognize most of the things in the review. Poisson brackets? Hooke motion?
Furthermore, anything more complex than the two-body problem is chaotic and incapable of exact solution, so it's up to the two-body problem to carry us along.
Not quite; the restricted three-body problem, where one of the masses is infinitessimal compared to the other two, can be solved analytically. The solutions reveal the existence of five points where the net effective force on the massless third body vanishes -- these points being, of course, the Lagrange points familar to students of orbital mechanics.
I'm surprised that the reviewer found so much of the material new; do college physics courses these days not include classical mechanics and the like?
Tubal-Cain smokes the white owl.
I guess that in this book, before the prince rides away on his horse, that we start by assuming that it's a perfect sphere...
Basic problem with building a game physics engine: if you do all the obvious stuff, it sort of works. If you're competent, you should be to that point in a few months. Getting from "sort of works" to "works" is about 5x to 10x as hard as the first step. There are really only a few game physics engines out there that really work.
You'll find out more about stiff systems of nonlinear differential equations than you ever wanted to know, if you don't give up first.
It's interesting that the book talks about the problems that occur when you take into account the propagation delay of gravity. Game physics engines, having rather large time steps, have some similar problems. I'll have to read this and see if I get any new insights applicable to game engines.
There's a related book, an ACM prizewinner, on the N-body problem. There's a clever numerical solution to the N-body problem that works for large N (millions), so you can simulate galaxies forming and such. The basic idea is that you can treat a group of bodies as a single body if they're near to each other and far away from the body being affected. This can be quantified and safe limits computed for grouping. It's thus a numerical solution with a proveable upper bound on the error, which bound can be made arbitrarily small at the cost of more computation. This is effectively as good as a closed-form solution, although some older mathematicians deride it as inelegant.
Ok, a little layman summary:
There's a fairly easy problem in physics. It's called the two-body problem. In it, you model (or predict) the motion of two objects in space as dictated by the force of gravity.
It's based on the Newtonian equation for gravity, which is that the force of gravity acting on two objects is proportional to the square of their distances. To put this more simply, the force of gravity between two objects gets drastically weaker as they are moved farther away.
All that being said, the main thrust of the book is apparently related to the three-or-more body problem. In it, the same basic equation is used. But since every body is being influenced by every other body, which are in turn being influenced by every other body, it gets very messy. Well-nigh uncalculateable, at least by people. The calculus just becomes too complex.
Fortunately, the two-body problem establishes a good enough model, allowing for us to model the motion of planets in our solar system, so long as we take into account that there's some wobble we have to throw in.
Now, I know this didn't explicitly cover the math, but basically, the book takes all of what I just said and builds it up from very basic to very complex mathematics.
Or thats my understanding, anyway.
It's not what you know, or even who you know- It's how many people recognize your damn
Feel free to mod me as such, but the review reminded me how horribly mathematics is represented in a browser. Wouldn't it be great if one day we could simply type:
Quid festinatio swallonis est aetherfuga inonusti?
Africus aut Europaeus?
As long as you can't decently manipulate and measure the particle, I guess it up to your feeling of aesthics which model you follow.
I personally believe in: God isn't rolling dice, God is playing billard. Peter Schaefer
So no attractive particles for me, if feasible, although you could make up one if it makes calculations easier :-)
I'm still trying to figure out what people mean by 'social skills' here.
Ewww. There's just something with the combination "bad porno", "horse" and "shag" that's deeply disturbing...
This book is too nerdy even for nerds. I'm sure I'm not the only one who doesn't want to work post-graduate-level mathematics just to read a folk tale.
But isn't physics built on math? And isn't math built on philosophy? And don't we still not have a good understanding of how gravity and electromagnetic radiation act? (take dark energy for one example, or the gravitational anomalies dealing with eclipses)
So what we have is a complex system of symbology that is demonstrably incomplete? Why wouldn't I just want the easy version until somebody starts basing a calculus on something that may do better?
BTW, does anybody know of a computer system that can create a calculus randomly and see where it goes, a la genetic programming, for instance? Might make a neat science fair project.
...on top of the fact that his post is meaningless (see other responses), this is the 'genius' who believes in 'Artificial Intelligence from the Bible'. See, for instance, this Google post.
What a nutter!
Tubal-Cain smokes the white owl.
"I recommend reading some Wittgenstein for more insights and CLARITY[?!]"
yes, and when you're done with that, round it all off with Kant, Hegel, and Sartre to make everything just peachy, crystal, transparent.
or should I be merciful and shoot you first?
The prize-winning N-body book referred to in the parent is Leslie Greengard's 1987 PhD thesis, "The Rapid Evaluation of Potential Fields in Particle Systems".
-Tom Duff
"the fact that gravity does not act instantaneously has been observed"
wasn't it originally *deduced* before observation, by Einstein I believe?
seems to me I recently saw this on PBS (in a show about string theory) -- something about a gedanken experiment about the change in the Earth's path if the sun vanished instantaneously, and how instantaneous gravity would be contradictory with non-instantaneous light.
btw, it's "propAgation"
"prince" "kingdom" "druid" "great trials and tribulations"
How nerdy can you get?!?!?
OBSQ: "Symbology? What's the symbology there? I believe the word you are looking for is symbolism. What is the sym-bolism there?"
If other reasons we do lack, we swear no one will die when we attack
That your average engineer, chemist or other science-minded college-educated person is not at least comfortable with Lagrangian mechanics is a failure of physics education.
In physics we generally don't think in terms of Newtons Laws, but rather in terms of the action and fields.
In my view, the lower level college physics classes which teach 18th century physics are a complete waste of time (as the review points out, all those laws fall out of more fundamental principals). The engineering students who are forced to take physics are not even given the chance to learn "real" physics, and the physics and other science majors who take it will simply be told to forget it and learn a better way of thinking a year later.
I'm always asking people in my department (I'm a physics grad student) why in the world we teach these useless classes. Generally the defense is that people wouldn't learn the concepts if we taught them the real way, that the math would be too hard, and people would get caught up in it.
They forget what it was like as an undergrad. Physics can be hard, even old, 18th century physics. When I've taught physics, people always get caught up in the math. The best we can do is to at least teach the right way, and introduce the right concepts. The math can be taught, packaged or explained.
There has been very little effort that I have seen to put real physics concepts in a package which is understandable by your average freshman biology student. This book is obviously no exception. It does not have to be this hard, and physics does not have to be only for physicists. Why do we insist on complicated terminology and crazy sounding descriptions?
I know that a lot of engineers and others out there have had more modern classical physics classes. Were they any good? Was your education in physics enlightening or frusterating? These issues really bug me, and I hope some of you out there have had better than I've seen.
This sounds to me like a more advanced classical mechanics text. In my second year in college (physics major), we used Marion & Thornton's Classical Dynamics of Particles and Systems, which seems to be one of the standard texts at that level. I believe Symon's Mechanics is another book at about the same level. :D), he was working on another problem and noticed this. He then repeated his calculations and saw that tidal stresses on Io might be strong enough to give it a liquid interior. He had trouble getting the paper published in the short time before pictures started coming back from Voyager, but managed. As he told me, anyone can write a paper explaining why a moon is volcanic after the discovery of vulcanism on the moon, but he wanted to publish the prediction before the pictures came back.
In my first year in grad school, I took a great classical mechanics course taught by a guy who uses classical mechanics in his research on planetary systems. His name is Stanton Peale. He got semi-famous by publishing a paper just before Voyager arrived near Jupiter, saying that Io might be volcanic. He would have published it a lot sooner, but he didn't notice that orbital data on the Galilean moons are, for historical reasons, recorded differently than those for other moons in the solar system. He had therefore mistakenly calculated that none of the Galileans would be volcanic. By chance (if such a thing exists
But I digress... in Peale's class, we used the standard graduate text on Classical Mechanics, which is Goldstein's Classical Mechanics.
Both the Goldstein book and the Marion & Thornton book cover Lagrangian and Hamiltonian mechanics. Goldstein goes into more details about things like Poisson Brackets and canonical transformations.
The Landau & Lifshitz book Mechanics, the first volume of the "Course of Theoretical Physics," covers much of the same material, but is quite concise. For that reason, like most of the Landau/Lifshitz (and Lifshitz/Pitaevskii, after Landau died) books, it is pretty dense.
I'm not sure if Oliver intended to bring these things to folks other than physics majors, but who other than physics majors (and maybe the occasional math major or other science/engineering major) has enough interest in the subject to wade through the math? The math isn't all that complicated (for a physics or math major), but it's complicated enough to deter anyone not really interested in the subject. Peale's classical mechanics class was not quite a weed-out course, but it was one that a significant number of people dropped in their first year and were taking for the second time when I took it. I worked really hard in that class and ended up learning a lot. And it wasn't just the math that made it tough. But the point is that this material can be taught at a level that's challenging for grad students...
--Mark
"It is nice to know that the computer understands the problem. But I would like to understand it too." --Eugene Wigner
Sounds more like horror story instead of review:
College kid reads a white paper disguised as a book; learns stuff.
Bot Assisted Blogging
While I'm admittedly weak in this area, just reading this post has made me want to learn more about the two body problem. That can't be a bad thing.
(a) reading the Shaggy Steed of Physics
OR(b) sex with the Shaggy Steed of Physics
Coincidentally, I've been looking for a book on mechanics to read.
Why you ask?
Well, I've got a pretty good math background and I've read some (not all) of the Feynman lectures. So while the math of advanced physics doesn't scare me (okay, it scares me a little), I lack any physical intuition.
I wasn't quite prepared to plow through a dry 500 page book on mechanics. However, I was looking for an entertaining read.
The reason is that mechanics is the intuition of physics. Most mathematicians can run mathematical circles around their physicist counterparts. Ever ask a physicist to "prove" something? However, ask a mathematican to "calculate" anything complicated in physics and you'll usually stop them cold (with a few notable exceptions [Von Nuemann, Kolmogorov, etc]).
Mechanics is the practice of doing physics calculations. If axioms and proofs are the tools of mathematicians, then fundamental laws and calculations are the tools of physicists. All introductory physics books (including Feynman) dance around the calculations of physics. Sure, when you've finished reading the Feynman lectures, you can pontificate on basic E&M, QM, etc. You'll be able to describe all kinds of interesting phenonmena. You just can't calcuate anything.
I've haven't read this book, but I think I will. If it is as entertaining as the reviewer says it is, then I could imagine it might become quite the classic.
Of course, this is just the opinion of a stupid math major....
What do you mean my sig is repetitive? What do you mean my sig is repetitive? What do you mean....
Did anyone else notice that the Sliabh Mis mentioned in the Irish legend is the same mountain St. Patrick spent his time as a slave and swineherd?
Considering how many times people on /. are corrected on the proper spelling of the word "you're" and the correct form of "their", "there" and "they're", it is highly doubtful that this person will amend their ways.
We will bankrupt ourselves in the vain search for absolute security. -- Dwight D. Eisenhower
I own the first volume of his physics lectures and it doesn't use complex math.
What determines whether a smaller body collides with or falls into orbit around a larger body? I guess that the relevant parameters are the starting velocity (with vector - the vector implies what the closest distance would have been without gravity), radius of the objects, and mass of the objects.
Are all collision scenarios simply situations in which the orbit's closest approach (perigee) is less than the radius of the large object?
Geez, there's a name I haven't heard for a long time. Definitely an awesome math teacher.
Software sucks. Open Source sucks less.
How do we KNOW that the exponent in the inverse square law is ineger 2 and not 2.0000000001 or 1.999999999 ?
When the crystal falls into a clear stream, it will indeed sparkle. If it falls into a muddy pond, its light will dim as well.
Kant, Wittgenstein, Hegel, and Sartre are indeed clear and transparent. "Critique of Pure Reason" also specifically addresses the distinction between mathematical truths vs. material facts.
You should try some McLuhan or Foucault to really twist your head inside-out.