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That is hard to estimate, but I think I need the same or less time for studying them. But of course the issue is how one reads them and how much one spends into extracting the hidden ideas and translating them into one's own mental structures (instead of turning one's mind into an emulation of the author's one) . One can study and understand very advanced papers fast and well without a productive "translation", and the more one knows, the easier it is to restrict reading that way.
Why do you think, the arxiv article is more precise than classical astronomy? Actually, it is about "vague" philosophical interpretations of QM, in this case leading it back to classical, newtonean concepts. Whereas the classical physics was free of such issues.
A related experiment with mice: http://www.wired.com/wiredscience/2011/01/circadian-disruption/
Once upon a time, there was a prisoner in solitary confinement, a former public enemy number one of France, aside the wards alone in the prison and allowed only to read science books. When he came across an astronomy textbook by Lagrange, he suddenly had the same idea as you express. http://ideafoundlings.blogspot.com/2009/10/nemo-eternal-returning.html
What do you think I would not undestand? Hinton's Cubes share since ages a bad reputation of disturbing the minds of his followers, fitting nicely to contemporary theories of learning and habit-development of the brain. Only two mathematicians seem to have profited from an exposition to them in their childhood. Ans the one who played around with constructing 3D/4D-analogues to Penrose/Escher 2D/3D-"impossible figure" doubted that such an endeavor woud threaten his health. The info on Talmud etc. came from a well known scholar. Both examples fit to the questions above.
I find it interesting that the cofounder of the Singularity Institute now expresses so sarcastic about attempted work on AI the past decades. Has there been any related discussion on this or similar sites?
A (real)I's should find the universe pretty boring, too boring to be (un)friendly. At talk touching that at the IAS by Wilczek: "Quantum theory radically transforms our fundamental understanding of physical reality. It reveals that the world contains a hidden richness of structure that we have barely begun to control and exploit. In this lecture, Frank Wilczek indicates the extraordinary potential ofquantum engineering (the size and nature of Hilbert space); reviews one important ongoing effort to harness it (topological quantum computing); and speculates on its ultimate prospects (quantum minds).": http://video.ias.edu/wilczek I just looked it and found his remarks, that supersmart aliens may just have lost their interest in the universe, a nice confirmation of my idea, which came from an other route to the same conclusion: http://ideafoundlings.blogspot.com/2009/09/seti.html
Martin Gardner reported a case of No. 3 on people becoming mad after mistreating their minds wih "Hinton Cubes". That are simple mechanical tools for developing a various parts of 4-dim. visual imagination (actually, when reading on regular solids etc., I can imagine such things without toys tools...). The unlucky practitioneers of a cult Hinton made out of that got mad because the trained ways of modified perception started working automatically, they could not stop that any more. Now imagine, someone would have confronted them with suitable analogues of impossible figures etc....
An other case of No. 3 was reported by a jewish scholar on the Talmud this way: It's 63 volumes are not only 'passionate disputes': By demanding the reader to follow them, they provide a training in the type of thinking used by those medieval, hermetic scholars. ... Then a clever designed 'mental trick lock', an 'intellectual vortex' comes. The emotional and motivational aspect of all the training with repetitions, variations, rythms, starts working like a maelstrom along the previous levels of learning. The first 'trick lock' the student meets is the barrier of his own intellectual strength, a barrier artificially made thicker than necessary. Only then comes the real 'trick lock' for the student who thinks he has the trouble behind him: The now since long dragged-in mental concepts and thinking-ways are set into conflict. A Talmudist described it: "... it puts the mind at war with itself; the more powerful the mind, the more destructive the conflict."
I mean that substantial innovations came the past ca. 3 decades much rarer than one should have expected. Kasparov and Thiel say that in view of AI and communication technology, whereas my impression comes from science.
Not quite so. The n-Lab contains a page on it: http://ncatlab.org/nlab/show/hyperstructure , but that is not that new. The usual deficiency of such constructs (and the many attempted definitions of n-categories) is their reliance on set theory. Grothendieck seems to have been the first to suggest to forget set theory as foundations, and Voevodsky's way to build a homotopy-theoretic foundation of mathematics on some sort of computer language (leading to entirely new approaches to artificial theorem proving/checking): http://ncatlab.org/nlab/show/hyperstructure may be interesting for Baas' ideas too. Interestingly too, homotopy theory, n-category were caused by attempts to deal with topology, and Baas' concepts come from the same background. He was apparently motivated by Charles Ehresmann's ctitique that n-categories should be insufficient.
A weird version: http://www.vetscite.org/publish/items/005309/index.html
Some relates Q&A's from a math student discussion group: http://mathoverflow.net/questions/28158/a-learning-roadmap-request-from-high-school-to-mid-undergraduate-studies http://mathoverflow.net/questions/1812/learning-new-mathematics http://mathoverflow.net/questions/28438/mathematics-and-autodidactism http://mathoverflow.net/questions/4580/pacing-for-learning-new-material
What do yuo mean with "Omega"? E.g. Chaitin's Omega or large cardinals do not fit the remarks.
The talk you link to is below the level of 1960's and 1970's discussions of that issue. Exist no better contemporary discussions of such issues?
Concerning "Guns, Germs, and Steel ": Murray Gell-Mann is involved in some interesting research on general patterns of civilization. But his and Diamond's schemes are just about some general and indirect indicators, not about what the essence of "civilization" is. To get an idea of that, I am curious about instances where "civilization" went down quickly. This puts e.g. "Berlin Diary: The Journal of a Foreign Correspondent 1934-1941", and "Human Smoke: The Beginnings of World War II, the End of Civilization" on my desk. A very remarkable case is told about in Simone Weil's "L'agonie d'une civilisation". I scanned two essay by Zbigniew Herbert on decivilization from this - someone curious to get them? "The Dream of Scipio" by Ian Pears sketches (in typical french history-through-novels manner) a thrilling pattern of several critical moments of the last 1500 years of European history.
Just take "desk archeology" as metapher to be filled as one likes in that context.
A funny idea. My goal is to work further into these issues. Of course I have a precise idea what to do in which order, but as amateur-math reader my attitude is generic lazyness which only occasional deforms into real studying. Unfortunately some friends with whom I discuss those issues are away for more than a month, so there is not much of external motivation.
You may be curious about this information collecting project. Concerning "skeptical that we can do anything ... through policy": Just a few months before people teared down the Berlin wall, even the most respected researchers in sociology and economy estimated that East-Germany would last at least one hundret years more. Like cold war, which was generally extected to be solvable only by politics, but that this should be extremly complicated. Actually, it was easy. (And even more urgent than everyone had guessed, as an aquaintance had researched.
There is something which one could call the "Pirx paradigm", coming from Stanislav Lem:
The complexity of really nontrivial questions surpasses that of the formalized methods used by the conscious part of the scientist's mind. Only the whole mind's complexity meets the questions in view of complexity and flexibility. Therefore, a great mind/scientist works with his complete personality, which e.g. expresses in the observable unique and personal way big scientists write their work. What one perceives as "humbleness", puzzlement, irrational curiosity, unsecurity or absentmindedness are then actually the marks of essential parts of the personality outside the narrow frame of conscious procedures (personal feelings, memories, associations, Lem stressed explicitely "honesty" too, because that is the absence of conscious "trickyness").
Lem discussed in some of his stories the contrasting side too - the deformation and degeneration of attempted, but conscious and therefore subcomplex, "rationality" into crackpot-science and crackpot-engineering. There, seemingly rational approaches gradually exchange the issues to be tackled (and the parts of nature in which they are embedded) by misfigured echos of the researcher/engineer's neurosis and mental entropy. I recommend to take a look into Lem's stories: The Inquest, Ananke, Test.
BTW, this MIT reserach program looks very much like one of Lem's jokes...
Thanks!
Concerning "predict(ing) future success of high school students in research mathematicians by spending a few hours talking": From my experience by private tutoring a wide variety of (university and other) students is that one develops an intuitive sensitivity for that. I wonder if others experience that too as quite unpleaseant: one has the feeling of an inappropriate intrusion into the personality of others, a violation of privacy, and because such intuitive guess comes very quickly, one feels to be very unjust. The obvious cause is that the human mind is less complex than usually estimated.
I just remember a very nice online docu on the Chudnovsky brothers, an old NY'er article , the "one mathematician in two brains".
Felix Klein may be seen as a "bird/beaver" hybrid, in view of the calculational view on complex multiplication and class fields in the 19th century, leading to modular equations etc. Klein's "icosahedron" and Weber's 3 vol. "Algebra" (the best until v.d. Waerden's book and E. Noether's school). link A more modern example may be this, but I know only a part of the story.
Thanks for your answer. The html link button did not work when I posted it. As far as AI etc., the possible relevance of homotopy theory is a theme since the 1970's, so it should be not too alien to anyone interested in pattern recogn. and related fields. It is similar unlikely that renormalization from QFT should be an entirely unknown theme, as that sort of dealing with generating series even in cases where they are seemingly ill-defined is a bit issue since long in e.g. combinatorics.
My previous post on QFT, Homotopy Theory and Ai etc. fits very good to this one, as it is about a special case related to computer science and AI (acc. to their selfdescriptions both a core part of "Singularity"-discussions and fitting to the forum member's areas of specialization/interests), and at the same time part of research programs in which the mathem. mentioned above are active. So I wonder about the strange reactions? (Princeton IAS and Fance's IHES surely are not below the intellectual level here, even if the selfreported IQ's acc. to the survey are above those of Feynman or Grothendieck...)
The feedback by several of the mathematicians of "bird" type told me that this summary of some exchanges fits their mentality quite good. Actually, some of the remarks on poetry I made a few days ago in some other discussion here come directly from those feedbacks.
However, the distinction you draw between "birds" and procedure- or algorithm-mindedness is not so strict: You see this in the way they deal with QFT, leading to (among others) Kontsevich's, Manin's and Voevodsky's previously posted thoughts comes directly from their acceptance that e.g. Feynman integrals are nice ideas, justifyed by their computational power, and that their deficient consistency a topic of minor importance (i.e. you don't need a logical consistent th. physics, because your point of reference is the nature itself, whose consistency is not a reasonable issue; similar with the platonic world of mathematics). A close look to e.g. Manin's and Kontsevich's work shows that they are largely determined by computational issues.
Grothendieck is among those "giants of 20th century mathematics" a very special case, as one sees from the astounded admiration which one senses among those "giants" who met him personally. It is not surprising that Grothendieck thought of himself as "mutant" (after analysing his work and comparing it with that of others). And, as others around him with medical expertise observed, he was different, strangely similar to the novel figure Odd John. There is a very good talk by Yves Andre online and here an other great article by Herreman. A video of Scharlau's talk (in english) at the IHES is here. It may be of interest that acc. to those who knew her, Grothendieck's sister was a genius of comparable power.
And: Somehow my upvote of your post didn't work, some of my links do not too in my QFT-AI post.
Those who can provide information. What is "OB"? Have you data on the level of knowledge in sciences and mathematics? "Libertarians" - is that not an other name for right-extremists? Then, "Singularity" is the 21st century libertarian's "Wunderwaffe"-myth?
A friend recommended it (and to look at the posts of some of his friends here). But I would like to know more about how the participants are, how many and what backgrounds. Many contributions look to me like from male teenagers with deficient education, but being hobby comuter users with interest in popular science. Correct guess?
I find it hard to believe that there are no better solutions, esp. in London - do you really think London offers it's inhabitants so little? By far less than remote districts in Germany or the US?
Conc. books: A good way to orient is to define the field of one's interests and to look at the websites of seminars and workshops in good universities on those and related topics. This helps to formulate a few possible learning routes and with some luck you find the sources free online. But if you want to avoid to crash (because low altitude flights of learning always crash into dead ends) , you need to follow Ravi Vakil's advise: "(mathematics) is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards". (Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)" BTW, here is a good collection of math related reading tips.
In mathematics, I would call "advanced" roughly what is above the average of "Springer Graduate Text" book series level. Of course, I do not say that one finds nothing, e.g. the Bourbaki seminar series is very good for autodidacts, or this site. It is a bit like the difference between a complex organism and the bunch of isolated cells.
I never had a problem with free access as non-student to a university library (and usually to it's computing center). I would suggest to contact the people and to try it. And are there no good other libraries in London?
University libraries are usually very good and have good long distance services. Have you looked for one? I would not buy books, as most of them one reads only once and science books are expensive. But one can suggest university libraries to buy them, I use those opportunities as the cheapest and fastest way to get them.
No, for several reasons (drawn from experiences in math, I am sure in physics and other sciences it is even worse):
Even if one includes hidden download-sites and special access by university subscriptions, only sources at the low or medium levels are available in a sufficient amount. The contents of an advanced level are only insufficient there, even some of the decades old and basic ones.
Suitable and really good existing texts on the web can be found only if one knows very precisely what one is looking for. But someone who wants to learn needs to find the better stuff, which is in part outside his/her mental frame. In contrast, good texts, even if found, still become hard to detect because of the noise by shallow pseudo-substitutes.
Browsing a real library makes your brain detect very quickly much more information and orientation, e.g. experiments (by friends who tutor at the local university) with beginning university students who were grouped and then asked to look for literature (in physics) by web/library only, for an hour, showed a very huge difference. An other experiment with students showed that students using a library have much better learning techniques that the other, but the later don't notice. Maybe it is a special case of this. On the interesting ways of how subconscious learning and "active" memory gets connected by seemingly irrelevant sensory inputs may play a role then, a famous extreme case here.
A good collection of hints, fits to my experiences in autodidactism. But you need a very good library at hand for such browsing (I used as teenager what a mathematician had designed as "Bibliothekskontinuum" ). "Don’t ... study ... as if you had to pass a test on it" is IMO very good too. Skimming connects with the subconscious procedures of memory and learning and works much better that one usually expects. In a similar way, I would suggest to take interesting looking books at home, laying them aside your bed and browsing like you please each evening before sleeping. That helps very much in identifying fitting themes and texts. But then one should stick on one issue and dig deeper, learn and think about a specific topic. After that, the cycle starts again, but on a higher level. Making notices is simplified by having a small paper notebook all the time around.
And, of course, it helps not to be frightened by sensing how small one's knowledge is. When I first entered the math library, I took a book by chance, and understood nothing, not even what the author told about. Then I tried an other one, with the same result. This continued until I came across a "Basic Number Theory" (by A. Weil) and could not believe that I even did not understand the first page. That was a bit too much for my taste - I stopped browsing then, sat down and worked systematically until I found out what to understand in which order for digesting that "Basic Number Theory" - a really tough job for me then. Some more than a few months later I happily finished it's last page and asked some professor lurking in the library for "the 2nd volume". He barely could supress laughter, as there was no "2nd volume", I already was in some sense at the top of the mountain.
"novels/poems/etc. help you understand your own motivation and more easily put you in the shoes of others" That is only a very late and somewhat restricted idea. E.g. ancient greek science of history used novels etc. as epistomological tool, because the core of the things, that what really happened shows not in the surface of the facts, but has to be found and by poetic/artistic work (re)constructed. That was the reason, why their statues were colored like pop art, and why Thukydides' history book contains poetic inventions as quotes . It is a bit as if in a documentary on e.g. the cold war, suddenly Thatcher, Reagan and Gorbatchev would sing an opera. In contemporary american literature you have this e.g. in Tim O'Brien's The Things They Carried . Or in Reed's "Naming of Parts.". A friend (a great mathematician AND great intellectual ) allowed to illustrate that point by the poem there, scroll down.
News on "intelligence" issues link: "But there are few more controversial areas of science than the study of intelligence and, in reality, there's not even agreement among researchers about what this word actually means. ... Now my team at the UK Medical Research Council's Cognition and Brain Sciences Unit in Cambridge has come up with the ultimate test of intelligence. ... Click here to test your own brain now:"
No, that is wrong. E.g. Proust, Flaubert, Balzac, the Mann's, etc. had a very strong focus on the cognitive content of their writings. Weil, Grothendieck, B. Mazur, Y. Manin and many other science writers (I am pretty sure that it fits to Dirac too, but lack precise infos) spend much thoughts on literature, language and poetics. The idea you express fit only to low level texts of both sorts (lit/sci). But the question was about good texts which help to improve the reader's mind.
Did I write that you said that you are confused? The books I recommended were written for the general readership, and I do not see how your remarks should apply to them. Which books would you recommend?
How is the educational level of the participants of this forum, by the way?
Just to continue your list of spectacular infos from math history: Newton probably suffered from microcephalia (as was speculated upon at Leibniz) by alcohol abuse of his mother during pregnancy.
If someone wants to walk in the footsteps of Ramanujan, here the textbook he used as teenager for autodidactism. Unfortunately I do not know if anyone tried that book with teenagers. Here someone's collection of basic math texts by which Gauß, Euler and other math geniusses learned to make their first steps.
Dear jimr..., your confusion could be cured by "reading" and "thinking". Books and other texts should be taken with respect of their content, nor their age, cover design, typeset, or other features. However, if you want a more recent one, I'd recommend this, as a kind of emergency aid in cases of acute confusion.
My recommendations are entirely caused by the quality of the texts and by their fitting to the question above. That some were written between the beginning and the middle of the 20th century is not quite an accident: That was the time of the science and university revolution in the US: The huge and sudden growth of science education then caused a strong demand on easy to read, unpretentious, well written and high quality introductory texts for the many students from rural, underdeveloped regions and equally undereducated kids from (then, when few and simple machines were in use) unskilled industrial workers.
As you are already inclined to read texts you expect to broaden your mental scope, I would recommend to move to real literature, e.g. a few novels by Dostoyevsky. Or this excellent and beautifull to read history of civilization. This, a study-in-a-novel used like a textbook in french history seminars, on the mindsets and times when modern science was born, may be a bit complex, but recommendable. Russel's History of Western Philosophy is a very good and usefull book too. Among the smaller texts, I found some of Putnam's Essays good here. Plato's dialogues are directly focussed on your question, as they are much more about igniting a process of thinking in the reader, than specific contents or statements they discuss. I would suggest his "Parmenides". But perhaps Descartes is more accessible for you. As you mention sci-tech interests, here finally a very good book relating to that.
A very excellent recent book, with fascinating new ideas and superior readable intros into many themes, is the new edition of Manin's "course in mathematical logic". So I'd recommend that. But: Why "foundations"? Like "foundational themes" in th. physics, "foundations" are not an appropriate place to start, they are a bundle of very advanced research areas whose intuitions and ideas come from core fields of research. "Foundations" in the sense of "what is it, really?" can be exprerienced probably much better by studying a good piece of core math, like number theory. Cox' "Primes of the form x^2 +n*y^2" or Khinchin's "Three Pearls of Number Theory" is what I would suggest. If your mind prefers geometry, I'd suggest to browse a good library for some of the great projective geometry textbooks from the early 20th century.
My experiences, as a kind of outsider who is just curious about some themes in math too and asks around for infos, explanations and preprints/slides, is that mathematicians are by far the easiest science community to communicate with. I conclude that status is of little relevance.
Thanks for the nice article! Cox' book is really very beautifull on some of the most beautifull themes in mathematics! Here is the link to an old text on related issues. Conc. the comment below, I'd say that it does not relate to individual theorems or definitions, but to global ideas (which may allow several, different expressions, like Grothendieck's "Dessins d'Enfant").