Dangerous Knowledge
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Dangerous Knowledge

2007, Science  -   125 Comments
Ratings: 8.54/10 from 218 users.

In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity.

Ludwig Boltzmann's struggle to prove the existence of atoms and probability eventually drove him to suicide. Kurt Gödel, the introverted confidant of Einstein, proved that there would always be problems which were outside human logic. His life ended in a sanatorium where he starved himself to death.

Finally, Alan Turing, the great Bletchley Park code breaker, father of computer science and homosexual, died trying to prove that some things are fundamentally unprovable.

The film also talks to the latest in the line of thinkers who have continued to pursue the question of whether there are things that mathematics and the human mind cannot know. They include Greg Chaitin, mathematician at the IBM TJ Watson Research Center, New York, and Roger Penrose.

Dangerous Knowledge tackles some of the profound questions about the true nature of reality that mathematical thinkers are still trying to answer today.

Directed by: David Malone

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125 Comments / User Reviews

  1. Alan Turing probably committed suicide because of the homophobia he suffered in that era.

  2. The theme inside the opening music is from the film Threads

    I felt that fear in my teeth again....

  3. Intellect and emotion would, ideally, work together in peace or the creative cannot be free but remain caged by confusion, indecision and depression.

  4. Boltzman, Cantor, Godel and Turing help us understand that there is a cultural domain that operates in prior unity or serves the sociopolitical or economic domain of humankind.

    Mathematics, 'mind' or attention itself, is only one of four faculties that make up the cultural domain of humankind.

    Body or service
    Mind or attention
    Emotion or feeling
    Breathe or energy

    What Boltzman, Cantor, Godel, Turing help us to do is understand that there are other faculties operating prior to mind or attention such as feeling or energy.


    1. I'd like to know more about what you say here. Mind sharing some references? Thank you!

  5. Bottom line we invented language and numbers to represent everything that can be observed . so how could there be something we cant prove, regardless of medium its told through...? We made up the rules observable to us through sight....with somthing that is not found in nature. (words and numbers) to represent the splendor..of course we can make up some new words if we need to or plug in some numbers in a graph. We invented and tweaked these things based on what happened physically first so of course theres always an ability to clarify mistakes or make revalations using language or numbers.... we invented them! I havent even wwatcheds this yet lol but as far as peoples minds, the ability to interperate the system thats been forged by the geniuses before us is the adaptation of todays great minds that is being tested. We have to evolve with the graph or system because it seems to me like gravity pressure speee force what have you....pretty muchc links up to the point where we caan control those things to such accuracy it cant be wrong or we wouldnntbave been able to create the internet or put people in space with massive rockets.

  6. Reading the above review of the doc doesn't seem to indicate exploring how society treated them might have played a larger role in their demise other than their endeavors to solve and understand the mysteries of life.

    1. yeah i find it rather offensive that they choose to ignore the significance of the role persecution that many of these individuals endured because of who they were or what they believed played in pushing them toward ending their lives. revisionist to say the least.

  7. Probably the best documentary in mathematics. Watched in many times. Unfortunately haven't seen any more such by David Malone.

  8. "He who knows does not speak, and he who speaks does not know" . . . or something like that . . . from ancient Chinese wisdom . . . .

  9. silkop....I hope you will one day take a course in something like discrete mathematics and get an opportunity to realize how there can be different levels of infinities(of course it will appear silly if we do not have prerequisite knowledge), one above the other. Those guys are all geniuses....don't try to judge them without proper knowledge...
    Consider the infinities of natural numbers and real numbers...it is very easy to show that they are different levels of infinities....you should properly understand how infinities are defined and measured...it might seem upsurd but it's fun ones you start learning

  10. I'm not sure if math is even real. I think they figured out it's all a hoax and off'd themselves because they wanted to relive with the astral gods above.

  11. The way to judge this documentary in proper context is to realize how
    outside the box thinking these great men were. Consider how modern
    mathematical physicists, like religious zealots who take for granted a
    perfection in creation, also seek a perfection in proving a calculus based order in the universe that does not exist. The irony of these tragic figures is that when their contrary views were rejected by a
    mechanical materialist establishment, they had breakdowns or went mad like so many computers that crash from problem solving. What to take away from this is that great thinkers need not be one track minded and must be in touch with heart, soul and intuition to solve existential questions beyond our comprehension. And whether or not infinity or eternity can answer the unanswerable, the concept of God is not a math equation.

    1. well said

  12. This is a terrific documentary that needs to be viewed by everyone, especially those who can only accept that which can be proven. Einstein, and Gödel were both correct, but this thinking can only be embraced by those who possess creativity, and intuition. Definitely a good watch. I leave the readers with a quote from Einstein:
    "For me, it is enough, just to wonder at the possibilities of existance".

  13. This is a fascinating documentary that taught me about people and things I never knew about; but I agree with "Guest" who dismisses Malone's premise that "what these men saw" through their unorthodox thinking drove them into madness. I agree with his/her statement that "Sensitively balanced minds have been known to go over the edge." These were brilliant men who happened to have either concurrent psychiatric problems or in Turing's case, a lifestyle that was taboo in 1950s Britain. The psychologist whom Malone interviews himself contradicts Malone's premise in the film when he remarks that in his opinion, it's not so much that these men "could not look away" from the reality they perceived with their innovative thoughts, but that it takes a certain type of personality to go that way in the first place. Malone seems not to have realized that this may well be inconsistent with his premise. There have been lots of brilliant physicists and mathematicians who were psychiatrically more or less normal, e.g. Christian Gauss and Richard Feynman to name but two. Isaac Newton started out peculiar and then seemed to get MORE normal later in life after he'd published his Principia.

    1. G'day JMD, welcome to TDF mate.
      I liked your first post, well thought through and written. Good point about Malone hearing only what he wanted from the psychologist. I'm looking forward to reading more of your thoughts on TDF. I also agree with your conclusion about their mental health issues. Another well known case that supposedly went mad because of his 'insights' that came to my mind was Friedrich Nietzsche. Seems likely in hindsight he had a brain tumor that caused his mental issues, not the 'insight' of his philosophy or the syphilis diagnosis he received.

  14. Bottom line we invented language and numbers to represent everything that can be observed . so how could there be something we cant prove, regardless of medium its told through...? We made up the rules observable to us through sight....with somthing that is not found in nature. (words and numbers) to represent the splendor..of course we can make up some new words if we need to or plug in some numbers in a graph. We invented and tweaked these things based on what happened physically first so of course theres always an ability to clarify mistakes or make revalations using language or numbers.... we invented them! I havent even wwatcheds this yet lol but as far as peoples minds, the ability to interperate the system thats been forged by the geniuses before us is the adaptation of todays great minds that is being tested. We have to evolve with the graph or system because it seems to me like gravity pressure speee force what have you....pretty muchc links up to the point where we caan control those things to such accuracy it cant be wrong or we wouldnntbave been able to create the internet or put people in space with massive rockets.

  15. This is the most beautiful science documentary I have ever seen(and I have seen many hundreds) True, the emphasis on the sociological aspects may have been overly dramatic, nonetheless, it points out the very deep truth that the need for an anchor or security blanket is one of the primary forms of attachment spoken of by Nagarjuna in his three poisons. It is very interesting to me that Einstein saw Schopenhauer as one of his guiding lights and that Schopenhauer and Nietzsche were students of Buddhism(certainly a long ways from being arhats). Godel and Einstein could imagine mathematical intuition but unfortunately they did not have a Bodhi tree to discover Bodhicitta. Martin Heidegger, a contemporary, conceived of emptiness but failed miserably to realize the Bodhisattva vow. This all goes to show the ethical sterility of Western philosophy. It has only been the mystics like Meister Eckhart who had the vision(intuition?) to look beyond the slavish devotion to the written word(Logos) to see a larger reality. Rupert Sheldrake exposes this hypocrisy in his book "Science Set Free: 10 Paths to New Discovery" ISBN 978-0770436728. Hopefully, we will all awaken from this multi-millenial sleep to embrace the truly infinite potential of being human and the exquisite beauty of living with a spirit of love and compassion.

  16. Is it absolutely true there are no absolute truth?

    YES or NO

    1. 'Yes (and) No"! Which is the (real) side of a coin? Heads or Tails?

  17. Why would someone want to solve a problem that made people go berserk? On the other hand if that was my purpose in life I will go for it.

  18. wow Amazing!

  19. I get tired of the sentimental thesis that genius spurs this romantic thing called madness.

    Sensitively balanced minds have been known to go over the edge. So-called madness also results from bipolar disorder, which is found disproportionately among keenly intelligent and creative people. It is a severe illness that can extend to psychosis and puts its sufferers at high risk of suicide.

    Did much learning make these men mad? Doubtful.

    For the record, Alan Turing died a martyr not to mathematical theory but to British anti-sodomy laws. I note that the posts here are from mathematicians arguing mathematics, not making psychiatric speculations. That suggests to me that they weren't too impressed with the film's thesis.

  20. Beautiful!

  21. ignorance is bliss but only for the weak willed.

  22. George Cantor theory of trans-infinities is right and wrong simultaneously. this calls on the flaw of our basic logic that a statement is either true or false but not both. with infinities and infinitesimals our logic is doomed to failure. in fact it is our logic that is aleph-null and there are higher logics; aleph-one aleph-two e.t.c

  23. 1.George Cantor was wrong: The flaw in his theory of trans-infinities is within the ideal of one to one correspondence comparison.
    Your monthly salary is twice mine and we are to work forever and ever adinfinitum. the pension is proportional to salary. Who earns more pension?
    Very sorry, the question itself is contradictory; there is no pension to infinity employees.
    2. Me and you are to race and you can run twice my speed. The race is adinfinitum in time. who wins? 'wins' means there is an end at which we are to be compared who has moved longer distance at that time which endless.

    1. Isn't the infinity problem solved with Calculus? i.e. Lim 1/x as x approaches infinity = 0. Infinity/x as x approaches 0 = 0 as x approaches infinity = 1?

  24. Mathematics is not for finding certainty, but for proving that there are no boundaries for the human brain, when it comes to possibilities.

  25. Certain things are better left unknown it could be too much for the mind to handle.

    1. i disagree. but even if it is too much, evolution will take its place and in time we will be beyond this idea.

  26. the beginning is the end. the end the beginning. life deaf infinite circle

  27. There are many pieces of the jigsaw of everything still waiting to be put in place by all too few with the ability to do so.

  28. @Crypticfreak You speak of an infinitely big circle.
    But an infinite sized circle does not and can not exist.

    As a Circle gets bigger it's curvature decreases.
    So as a circle tends to infinity it's curvature tends to 0. A staright line has curvature of 0 and nothing else does. So therefore an infinitely large circle cannot exist.

    1. There is no such thing as a perfect straight line or circle in nature, everything no matter how small is composed of mandelbrot sets. As for circles getting bigger and bigger to infinity could be, spacetime itself may be curved, and we may live in an infinite universe.

      And then our universe may be part of a multiverse, either quilted or inflationary.

      So without any empirical evidence to the contrary, pro and con, may or may not be infinite sizes to circles.

    2. The trajectory of a beam of photons in a vacuum perfectly models a straight line, just as a drop of water falling due to gravity, with negligible air resistance models a perfect sphere. Granted these conditions are not natural as you pointed out.

      However... Diamond is found naturally on earth and the carbon atoms in diamond are actually aligned in perfect straight line to one another (I'm sure this therefore is true for other allotropes of carbon).

      Beside the point, at least theoretically an infinite circle cannot exist, maybe in nature it could.
      Not a clue what a mandelbrot set is however :’)

    3. @10jwalton
      I thought as a child, if i could pin down a worm to have it's mouth at it's tail, it would eventually decide to eat itself. And since worms can grown tail back, it would create a doughnut shape worm.
      Now i was very young when i thought of that, before i heard about the snake biting it's own tail of spirituality....and this discussion brought it all back, i had almost forgotten.

    4. What if that circle is a torus in motion? It could pulse in/out of itself on it's way to eternity knowing it MAY or MAY not ever reach it.

    5. But will it ever reach zero? If I draw the smallest circle I can, then draw another circle around that one, then draw another circle around that one, etc, etc, ad infinitum, will I ever get a straight line?

    6. what about a circle with infinite diameter ?

    7. that's logically impossible by definition, a priori. (see Wittgenstein) :(

  29. This doc evade too many question! For example: If you mentioned Cantor's work then you must explain the work first before rambling about its significance, and similarly with Boltzman's work. The only thing that excite me in this doc is the Pythgorean's idea of infinity: which the host can explain himself (other stuff is just too bland)

  30. the film was brilliant. the academia making the commentary underneath the screen section is perfect example of intellectual incompleteness.

  31. @Robert Allen
    Now I am more acquainted with your background, I will work in accordance. Meanwhile, indulge in no-nonsense math history. I think you will find doing so not only edifying, but toughly enjoyable, as well. Fortunately, in this area, there are many fine publication to choose amongst. I also encourage a re-read of Hardy's 'Apology'. It's a quick read, most especially for one with your remarkable literary acumen (yes, it does show!).

    1. Pyrrhus, Robert

      I don't know if you will ever see this, it has been 8 years, but I want you to know that this conversation and the texts provided has been an inspiration to a young engineering student, and I hope that you are both still well and continue to learn. Thank you


  32. I have NOT received your "missives."

    Did you prefix the subject line with 'XYZ'? (I get so much junk.)

    Would you please re-encrypt for me, so that I can be sure I got it right in my posting to you? All this word play is so silly, I know; but it's the only way I can think of. Thanks, Robert.

  33. @Pyrrhus

    To the general and the lion:

    Not only did I take the time yesterday to decrypt your post, but have also written you there. Please let me know if you have received my missives.

    I have ordered "The World of Mathematics" and eagerly await its arrival. I also plan to read the Penrose introduction.

    In college, I started off as a math/music major and came to realize that at that time I lacked the maturity, and perhaps the discipline, to pursue math. After dropping the math part of my major (I graduated with a major in music/comparative literature), I did not look at a math book, much less think about math, for well over 30 years--then I saw the movie "Pi," which though I realized it was total nonsense, for some strange reason rekindled my interest in math. As a result, I have gone or tried to go through texts on calculus (first and second year), linear algebra,abstract algebra, analysis and differential equations and finally determined that although the willingness and effort were there, the talent was not--the biggest hurdle was trying to follow the textbooks and working all the problems. All I can say about the level of my mathematical expertise is that I am dissatisfied with it and disappointed with my understanding of the subject. So, therefore, you have no competition from me as far as mastery. This is as clean as I can come.

    I don't mean to cause you further pain by alluding to the distasteful, but I would still like to find out if your opinion of Dembski is the same as mine.

    P.S. I think it's terrible that Cauchy "misplaced" Abel's paper. While Cauchy was certainly a fine mathematician, everything I've read about him suggests--well, you know. Anyway the mathematics comes first.

    P.P.S. T.S. Eliot is one of the few poets in the English language I like; the others are Blake, Dryden and for the sake of doggerel, Robert Service and Samuel Hoffenstein. I have no use for either the English or the American romantics.

  34. I hope you realize, Robert, that our discussion is taking away ALL time intended for the math's outline. I do not mind, not at all; but please forgive, in advance, "the Shadow" (as T.S. Eliot would have it) which "Falls:
    "Between the idea
    "And the reality
    "Between the motion
    "And the act"
    Too, for me, at any rate:
    "Life is [not] very long."

    As to viewing 'Jesus Camp', I have not, nor intend to. I know enough to fear coming away angry, frustrated, and depressed. Doubtless, 'Jesus Camp' would present me additional information, at a cost: additional depression, elevated blood pressure, heaped atop depressive pathologies in whose caste-iron grip I find myself. Avoidance of 'Jesus Camp' type exposures, in exchange for maintenance of whatever fragments of mental stability are left me, is a chosen trade-off.

    With all respect that can be mustered, I request you make no further mention of potential 'shocks-to-the-system'. Amplifications of the 'horror' are intolerable. I am confident you will honor my request, and am grateful.

    Now, the mundane:

    1) I urge you to read the Penrose introduction, in its entirety. (Upon doing so, one is often tempted to read on :)

    2) 'The world of Mathematics' is a comprehensive, historical overview of mathematics' evolution from zero B.C.E. to the late forties. Within its class, I know of no superior program. Its very existence is a tribute to all who contributed. Naturally, one is free to read, linear-fashion, covers to covers; though highly recommended, this is by no means necessary. Any readings in the collections invariably enrich. I love it.

    3) I've yet to locate anything, at all, remotely accessible concerning groups' isomorphic nature vis-à-vis groups of permutations; but I've by no means given up trying.

    4) Robert, if you can find the time, and are so inclined, would be so kind as to describe for me your level of mathematical expertise. At times I am of the impression you are a beginner. At other times, I come away vaguely suspicious you are far more advanced than I. Structure layouts facilitating endeavors useless to you would waste both your time, and mine. (With no reference to Cauchy intended), 'couched' in the vulgate vernacular, would you kindly: 'come clean'?

    I think I like you, Robert, though we each know so very little of the other. Have you taken time to decrypt the post of mine, meant for your eyes, to be seen here, four up preceding this posting? In closing: Pyhrrus' name is Leon.

  35. @Pyrrhus

    Another question more in your field, although I believe I know the answer. What do you think of William Dembski?

    Talk about religioius ravings. Here's someone in academia who lies and misrepresents just for his church and who, to boot, resorts to a faith healer to cure his autistic son.

    Even I who cannot begin to qualify as an amateur immediately saw through his ostensibly probabilistic refutation of evolution.

  36. @Pyrrhus

    Amen! And we don't need to know anything about biology to appreciate what Dawkins stands for. It's not so much what he says about reason; it's his presentation. The man is a born teacher of sterling eloquence and an inspiration, especially to the young.

    Religious rant--I like a good hell-fire, brimstone tirade a la Jonathan Edwards as well as the next man, but I treat it like what it is, a diversion. In this regard, at least, Peter Popoff is creative. In one of his perorations, he exclaimed that he had lent Jesus $500.00 from his credit card and exhorted his flock to do likewise. (To raise the hackles, see "Jesus Camp" which is on this site.)

    It is chilling when this type of non-thinking (intelligent design) essays to insinuate itelf into the classroom as it did in Kitzmiller v. Dover Area School District. While ID is another view, it is not a scientific one and has no place in a science class any more than numerology has in a mathematics course. To take this digression further, why do its proponents have to lie, misrepresent, etc. to "prove" it.
    So your arrogance is as justified as Dr. Dawkins'.

    To answer your question, I asked you about Dr. Dawkins to ascertain if we were in the same camp.

  37. I like Richard Dawkins (in fact, he's my kind of people!); it's just that my knowledge of biology is not 'next to nothing', it IS nothing (if 'nothing' can 'exist', that is). Dawkins is an evolutionary biologist and his work veers to the theoretical. I am completely ill-equipped to appreciate his scholarship. That is sad, but it also is true.

    In these, his retirement years, Dr. Dawkins is fighting the 'good fight' on behalf of rational thought and debate, in opposition to what, at times, appears as an all-pervasive resurgence of superstition and willful ignorance in politics, culture -- public discourse, at large. The likes of Dr. Dawkins, and our poor friend, Christopher Hitchens, are all too few in number. I wish I were able to lend a hand, but I am not.

    You ask what I think of Richard Dawkins. Concerning the merits of his scholarship, I am unable to comment. As to his valiant strivings to stop this world returning to superstitions it has taken our species tens of thousands of years to conquer, I am, needless to say, in concurrence; but, frankly, there really is not much Dr. Dawkins says on behalf of reason that I do not already know. That sounds arrogant. But I think the same might be true for you and countless others who are sickened by religious ravings.

    Well, I have spoken my mind. May I ask you, Robert, why you asked me? And, please, share your thoughts, if you wish.

  38. @Pyrrhus

    I don't quite understand your comment on Richard Dawkins and would appreciate your view even if it differs from mine, but like him (and obviously you) I'm no theist either.

    I'm sorry, I do not speak or read German, but I have read a mountain of German literature. Again, I do not speak or read Russian, but I have read an equal mountain of German literature. Which makes me wonder why most Americans have not read Tom Sawyer, for just about every German and Russian I have met has.

  39. What do I think of Richard Dawkins? I don't. However, I do share one thing with the man: I am not a theist, nor have I been since leaving a minor seminary, aged thirteen.

    By the way, do you happen to speak or read German?

  40. @Pyrrhus

    As I wrote, if you could find a lucid presentation of Cayley's theorem which seems to be basic to group theory, I would appreciate it.

    What do you think of Richard Dawkins? I realize he is not a mathematician, but all the same, I am curious.

  41. @Pyrrhus

    Congratulations. We're contemporaries.

    I will take a look at "Road to Reality." Thank you for the reference.

  42. Cayley’s Theorem. Got it.

  43. I turned 63 four days ago!

    1. haha!

  44. P.S.:
    Rather than attempting a full print-out, it is probably best to just save the Roger Penrose, 'Road to Reality', PDF file to your hard drive for viewing, as printing it would require 1,094 pieces of paper!

  45. @Pyrrhus

    Thanks for trying, but what I meant by Cayley's theorem is the one which states, "Every group is isomorphic to a group of permutations."

    As far as being a young stallion, I am 63 years old.

  46. re: Arthur Cayley, am looking for expository intro to graph theory

    Meanwhile, do search on "Road to Reality" "Roger Penrose"
    It's a PDF file which you can print upon download at:

    student 'dot' fizika 'dot' org 'forwardslsh' 'tilde' dzoljom 'forwardslsh' Roger_Penrose_ 'hyphen' _The_Road_to_Reality 'dot' pdf

    Decided to start out 'Serial Fashion' and have something in the works. Coming soon!

    Robert, you're like a young stallion galloping in all directions at once (lol)!

  47. @Pyrrhus

    Thank you. I have ordered it.

    Do you know of any good, clear expositions of Cayley's theory?

  48. World of Mathematics 1956

    auntiem ecrater c o m "forwardslash" p "forwardslash" 6655682 "forwardslash" 1956-world-of-mathematics-box-set-simon#

  49. @Pyrrhus

    I knew it was some sport.

    Wasn't there a professor, I believe his name was Aiken and I believe he taught at Princeton, who, among other things, could recite Pi to several hundred places, both the correct and incorrect versions? He seems to be just about the only mathematical idiot savant who became a true mathematician.

  50. Second only to mathematics, stood Hardy's passion, not for rugby but, rather, for cricket, as spectator. He was in the habit of carrying an umbrella to matches born of the conviction that God would have it rain only were he to come unprepared.

  51. Quote is the concluding sentence from Hardy’s ‘A Mathematician’s Apology’.

  52. @Pyrrhus

    In the words of the old song, "Wishing will make it so."

    Nice quote. Where does it come from?

  53. From memory:

    "The case for my life, then, or for that of any one else who has been a mathematician, in the same sense in which I have been one, is this: that I have added something to knowledge, and helped others to add more; and that these 'somethings' have a value which differs in degree only, and not in kind, from that of the creations of the great mathematicians, or of any artist, great or small, who has left as a memorial something behind."

  54. In appreciating it very much, your "wish" has come true.

  55. @Phrrhus

    I would appreciate it very much. I wish there were something I could do for you in return.


  56. Robert, your permitting, I have decided to offer an outline for your consideration, a structured approach of means for arriving at a place where you will be able to make hard choices as to which discipline(s) within 'The Discipline' you feel might best suit both your talents and your passions. This will take some little time; you may not hear from me for a few days. If you'd rather I not, be assured, no offence will be taken; but please be honest and let me know. Thanks.

  57. @Pyrrhus

    Yes, I have and I remember the closing passage. It seems that there were only two things in life which Hardy liked: mathematics and rugby or some such sport.

    As for utility, what's useless today is . . ., e.g., Newton's view of imaginary numbers as worthless. Besides, I am a firm believer in learning (scholarsip) for learning's sake. Like the Great Circus of Oklahoma in Kafka's America, don't worry, we'll find a use for it.

  58. Robert (utterly off topic), if you haven't already, read Hardy's 'A Mathematician's Apology'; at the very least, the closing passage:

    "I have never done anything 'useful'. No discovery of mine has made, or is likely to make, ... the least difference to the amenity of the world. The case for my life, then, ... is this: ..."

  59. @Pyrrhus

    No apology necessary. I'm just thankful that you've taken the time to try to answer me.

    In discussing Cantor, you write of infinity(ies.) I am fascinated by the history of the concept of the inifinitely large (also small). Apparently it was unknown to the ancient mathematical civilizations (although it was suspected)and with its eventual acceptance, the established church(es) began to fear for their lives, for it militated against the dogma of an omnipotent deity (read omnipotent church/religion)--and we know what happens when that happens and it did. Finally when the church grudgingly accepted the infinitely large and with it, the infinitely small, along comes Cantor, a Protestant if ever there was one, with the notion of a number of infinities, some larger (smaller) than others. One thing I like about math is the creativity, especially in the hands of its geniuses who relegate it to an art form.

    In your last paragraph, you touched on specialty. At this point, all I can say is I'm scouting the field(s). I know that my mathematical interest lies more in the abstract than in the practical and there is so much with which I desire to acquaint myself that you can imagine how frustrating it is to fight with a textbook rather than learn from it, much less be inspired. In addition, as you can see, I'm also interested in the pedagogical aspects of the discipline.

    You ask what I'm looking for--clear, informative, readable perspicuous texts, especially in analysis (advanced calculus), group theory and differential geometry.

    Once again, thank you for taking the time to write to me.

  60. Robert, (typically) I failed in explaining:

    The purpose of the diagonals thing was not to indicate the rationals' countability, but to point you in the direction of Cantor's exposition as to the ASYMPTOTIC nature of the non-rationals' density within the rational continuum, which lies at the heart of Cantor's demonstration that the irrationals' infinity is of a cardinality larger than that of the rationals. (Cantor is reported to have said: "Though I see it, I still cannot believe it.")

    I only did this because you'd indicated having had a problem with some of the logic involved. To explain Cantor's reasoning would be wordy, so I suggested you find a pictorial rendering. The crime is no different from blog-professors directing you to alternate sources. I am sorry. Please accept my apology.

    As to Rainville's textbooks, I've no direct experience. My understanding is that each is comprehensive. I think a former colleague (now dead) used one (something about special functions, if memory serves). But Rainville's work is dedicated almost exclusively to differential analysis, not group theory, which is what you're looking for(?).

    I'll have a look around for you, but please do tell me what I'm looking for.

  61. @pyrrhus

    My apology. I meant Earl Rainville. What are your thoughts?

  62. One last comment, and then I'm going to bed.

    I am extremely heartened that people like you still exist.

  63. I think our dear moderator must have blacklisted anything contain the letters "f r e a k" (without spaces). It seems free speech is headed in the same direction as education.

  64. When authors of mathematics textbooks do not state in the introductions to their publications that their text is intended for classroom use, it is probably because they take it to be a given.

    I agree, the self-study 'industry' is a wreckage, and it will only get worse(*).

    Claims that a textbook is ideal both for the classroom as well as for individual study are quite simply lies put their by the publisher or, more likely, by the marketing corporation to which the publisher outsourced the job of lying. Disregard any such claims and hold in grave suspicion any publishers who promotes such fabrications of the truth.

    If, as a student, you expect to be spoon-fed, you had better first be one. Then expect to pay exorbitant tuition fees.

    (*) Don't know where you live, but here, in the United States, the popular value placed on education is zero.

    What 'should be' and what 'is' live on different planets. Those who were self taught, like Ramanujan, struggled (and how!) in addition to them having been exceptionally special people with talent bordering on the f r e a k i s h.

    Robert, it is a struggle, and that won't change, ever.

  65. i'll try resubmitting

  66. I answered you questions and got:
    "Your comment is awaiting moderation."

  67. I don't anything about Earl Raintree. Do you mean Earl Rainville?

    When authors of mathematics textbooks do not state in the introductions to their publications that their text is intended for classroom use, it is probably because they take it to be a given.

    I agree, the self-study 'industry' is a wreckage, and it will only get worse(*).

    Claims that a textbook is ideal both for the classroom as well as for individual study are quite simply lies put their by the publisher or, more likely, by the marketing corporation to which the publisher outsourced the job of lying. Disregard any such claims and hold in grave suspicion any publishers who promotes such fabrications of the truth.

    If, as a student, you expect to be spoon-fed, you had better first be one. Then expect to pay exorbitant tuition fees.

    (*) Don't know where you live, but here, in the United States, the popular value placed on education is zero.

    What 'should be' and what 'is' live on different planets. Those who were self taught, like Ramanujan, struggled (and how!) in addition to them having been exceptionally special people with talent bordering on the freakish.

    Robert, it is a struggle, and that won't change, ever.

  68. @Pyrrhus

    If math textbooks are intended to be used by teachers in the context of classroom study, why don't the authors state that in their introductions . . .

    . . . which leads me to your comment on go-it-alone textbooks. You're right, they vary in quality--but from bad to abysmal--and one of the problems is that too many of the introductions indicate an ambidextrous nature which simply does not exist, i.e., that the book is ideal either for the classroom or for individual study.

    Perhaps it's a difference in philosophy. At the level of mathematics I'm discussing, the author has a duty to be clear and direct. As a student, I expect to be spoonfed until I mature. I have neither the time nor the inclination to try to make sense out of a convoluted exposition.

    Why shouldn't one be able to learn mathematics pretty much on his own? Some of our greatest mathematicians have done it.

    As for canvassing bookstores, been there, done that and as you say, came up empty-handed. I have found one series of texts which seem to be closer to what I am looking for although they are far from ideal. Are you familiar with Earl Raintree's books on calculus and differential equations?

    Once again, I would like to thank you for spending time to read my e-mails and respond to them.

  69. There are myriads of 'go-it-alone' textbooks for any subject matter you care to name. But they vary wildly in quality of structure and general readability. An afternoon in a good bookstore might prove useful. Don't be too disappointed, however, if you come away empty-handed. Just keep searching.

    Can you sign up for classes or perhaps take classes on a not-for-credit basis, or does time not permit?

  70. One could be cynical in commenting on the whys-&-wherefores of textbooks, math textbooks included. That aside, the underlying cause for so many mathematics textbooks being what they are is to be found in their intended function: use by teachers in the context of classroom study of the text's content.

  71. Glenn Gould

  72. @Pyrrhus

    Thank you for taking the time to write to me and no, your pep talk did not cross me, quite the contrary.

    Yes, I knew about the Pathagoreans who were, I believe, contemporaries of Homer.

    But what I didn't know was that you were a teacher (I assume a math teacher) which means that you are the best one to answer my basic question: Why are math textbooks, especially in abstract algebra, analysis and differential equations, written with so little thought to the person who tries to learn from them. Is there perhaps a metamethematical reason for this--like quick money for publication and required reading? I am a far better writer than a mathematician and one of my pet ideas has been to team up with a mathematician to create a superior set of textbooks.

    As for non-rationals, I like Archimedes' proof of the irrationality of sq. rt. 2 (or at least I think it was Archimedes). I enjoy negative proofs which is one of the reasons I like Wiles', that is as much of it as is accessible to my limited understanding.

    I guess we disagree about the documentary. Perhaps it would have been better if it had been shorter.

    P.S. Perhaps I missed something, but who is Glenn?

  73. Robert,

    Didn't like docu? Fine (was made for lawyers).
    Chopstx, no prob? Fine.
    Can't get eg's/counter-eg's? Not fine.

    You come across as genuinely cross.
    Sorry if peptalk crossed you.
    Didn't mean to be condescending.

    Am an old man, haven't taught in decades.
    Shouldn't be, and am not, in classroom.
    Can't help you.

    Find study-bud.

    Good luck.

    Imagine a single piece of string with a starting point at (1,1) looping continuously, without end, across all points on the diagonals. God then pulls string tight into straight line. Voilà! All wholes and fractions are accounted for (infinitely many times).

    "Accounted for" means 'counted' (put into one-to-one correspondence with the natural numbers (or, 'counting numbers': 1, 2, 3,...). That this can be done in infinitely many ways ought to be viewed as overkill, not confusion. The wholes and fractions are counted. That's the point.

    This countable compact (i.e., continuous-- no holes) number-line is defined as the set of rational numbers. (Some texts use the exposition to show that rationals are 'rational'. No wonder people get confused.)

    As would seem correct (intuitively), the Pythagoreans believed that the number-line consisted of rational numbers, exclusively. The Pythagoreans' slogan was "All is Number."

    Inadvertently, the Pythagoreans themselves showed the hypotenuse of the unit right-triangle to be 'incommensurable', thus demonstrating the existence of a numerically inexpressible number, radical-2. (This was done geometrically, so here the word 'number' is used only as a metaphor).

    All is Number?

    The Pythagoreans were simply horrified and made a pact, swearing an oath to never tell anybody, not nobody, not ever, subsequent to which some poor chump squealed, and they drowned him (but his ghost lives on).

    The infinite array of combinations for putting the naturals and their ratios into one-to-one correspondence with the naturals can be drawn upon as a means for constructing a non-rational number.

    Expositions for the existence of non-rationals are easy to construct, but attempting to put one into words would mean tedious writing for me and tedious reading for you. Search the internet. Stay away from blogs and forums and Wikipedia. Look for GRAPHICAL expositions (the fewer words/symbols, the better). Look for those expositions derived from a matrix of the rationals. I'll look, too.

    The 'Real Numbers' are nothing more than the union of the Rationals and Irrationals. The set of Reals is symbolized by the double-barred capital 'R'.

    Some definitions of 'R' include the set of Complex Numbers. As far as I know, this is a relatively recent perversion. Don't like it. Won't use it. (Btw, the symbol for the 'Complex Numbers', which subsumes the Reals, is a capital 'C' with an inner vertical bar.) That the Complex Numbers subsume the Reals is why I reject the Reals as including the set of Complex Numbers (be nice to 'C').

    If all this is but a rehash for you, please let me know.

    One last plea for the documentary: 'Dangerous Knowledge' is not about mathematics and did not claim to be. 'Dangerous Knowledge' was made to pay homage to four great people, all four of whom I respect, profoundly; two of whom I love, truly.

    I love Andrew, too.

    PPPPS: Two months = 'forever' when you don't know it's going to be two months.)

    PPPPPS: Be nice to Glenn.

  74. @Pyrrhus

    I can play chopsticks (in various styles, mostly bravura)and on my better days, Fauré, Griffes, Glazanov, Porter, Gershwin and Warren--as a matter of fact I practice three to four hours a day. I find Arrau too cold and pedal-heavy and Glenn Gould too percussive.

    It's not that I'm freightened off by the blackboard scribble; it's not that I don't realize that learning mathematics is a stepwise process, it's just that I'm fed up with math textbooks in general. I skimmed briefly through the site you recommended and while I realize that it's just an outline (like many I've seen), found it lacking in samples and countersamples which are just as important. All too often, expositions are either unclear or deliberatively evasive and too many authors have failed in their duty to teach, not bewilder. One egregious example comes to mind, a textbook in analysis (advanced calculus) provided only a verbal description of a Cauchy series (you can imagine how it was couched) and expected the student to come up with examples!!!

    Sometime ago, I e-mailed a mathematics professor who had written a blog on Cantor to ask about a certain aspect(I believe it had to do with fractions vs. whole numbers--I could not understand the logic behind the diagonals). Rather than providing me with an explanation, even a partial one, the professor merely referred me to another source. This was insulting. If I was concerned and interested enough to ask him a question, he should have been diligent and delighted enough to provide me with an answer. Besides, why didn't he explain the logic behind the diagonals in his article? Quite frankly, with that attitude, I do not feel he should be teaching.

    Everything you write about Dr. Wiles is everything that I admire about him and the fortitude that he showed represents the acme of scholarship, no matter the discipline. By the way, didn't it take him just about two months to fix the gap? A nanosecond in the realm of serious study.

    P.S. I am being nice to Alan. It's the documentary that I'm caviling at. Anyway, thanks for the encouragement.

  75. "...and lack the ability to become one..."

    Please, please don't say that.

    We are, all of us (Andrew Wiles, included) embarked upon the same road.

    Some people make millions at golf, but that didn't stop my Dad at age 70 from thoroughly enjoying a day out on the course. The same with the piano: there are the Claudio Arraus and Glenn Goulds of this world, while some can play only 'chop-sticks'. But the chop-sticks player is, nonetheless, a pianist; and tomorrow (s)he'll be struggling (successfully) through Beethoven's 'Für Elise'.

    All you require is honesty, will, and, from time to time, a modicum of friendly guidance.

    Robert, do not be frightened off by all the fancy blackboard scribble. It's really easy, just like reading (if you know how); and ANYBODY can learn to read. It takes time and patience-- the ability to take it one step at a time.

    Interestingly, each tiny step not only renders the next step more accessible but, in addition, opens up possibilities for a subsequent steps to be not only less inaccessible, but richer in scope, too. The staircase is not linear. The staircase is an exponential flight.

    Even backsliding can be turned round into a critically valuable learning experience.

    Just think how alone, lonely, and heartbroken Andrew must have felt when he 'had it', only to find that he hadn't. And think how he must have felt knowing that the day before his failure was to be announced, he had, erroneously, been heralded as the man who, against all odds, had finally DONE IT (on the front page of the New York Times, no less).

    Despite the embracement (devastatingly difficult even for an 'alpha-male' to withstand, never mind for a shy, unassuming Andrew), our Andrew, somehow, sought and found within himself the courage to forge on completely in the dark as to whether subsequent effort might not make him look even more ridiculous and, this time, not swaddled within the cocoon of privacy, as before, but rather under the full, intense glare of every spotlight on earth peering DOWN upon him (not to mention 'friends', shielded by their anonymity, using his inroads in hopes of toppling him so as to usurp the glory that should, if there be a god, be his).

    If Andrew could shoulder all that, any of us can take heart.

    I do no know what you work day looks like, but if circumstances permit, request of a professor at your alma mater permission to sit in on classes, serious, introductory classes, designed for serious, freshmen mathematics students who will go on to become the Andrew Wiles of tomorrow. I promise you, you WILL find professors delighted to take you under there wing. I know. It happened to me.

    And finally, Robert, please try to be nice to Alan, okay? (for me).

  76. @Pyrrhus

    Thank you for the information.

    Even though I am not a mathematician and lack the ability to become one, this does not militate against my admiration of a wonderful discipline, except for one thing--THE TEXTBOOKS! Why the authors write as they do is beyond me. Most of them do not seek to elucidate, but rather to bewilder (i.e., show off their brilliance). The appalling lack of clarity and continuity which I have too often encountered are enough to discourage future mathematicians and deplete their ranks--and I'm certain you agree that there is always room for a fine mathematician.

    So much for one of my pet peeves. I won't try your patience with further details.

    I have not experienced your problems with the site and suggest that you contact Vlad (his E-mail is in the contacts section). If you have any problems, I will try to assist you.

    One more thing, I often find it easier to understand a theorem if the proof comes first i.e., leads up to it. I'm not saying this works all the time; however, I would appreciate your thoughts.

    Thank you once again.

  77. Course 311 - Abstract Algebra
    Lecture Notes for the Academic Year 2007-08

    Abstract Algebra has evolved into a blanket code for group theory, and group theory, in turn, has subsumed much of number theory and is very important to both the theoretical study of QCD (quantum chromodynamics) and string theory.

    I am afraid to post a link.

    It seems, of late, that practically everything I post is instantly removed from the bulletin board, much of it never to return.

    I have found that even referring directly by name to a fellow contributor results all too often in removal or, at the very least, 'review'. I do not understand what is going on at this site, I really don't. I can't even key in the name of the former Vice President of the United States without it being immediately censored.

    Therefore, I hope googling the above course info from Trinity College, Dublin, will result in your finding excellent, full course instruction.

    Hope you see this.

  78. The scene with Greg Chatham rambling about philosophy is simply beautiful. Great stuff. The mountains just get higher...

  79. @gero2006
    Finally, a fellow mathematician with whom I can discuss.

    Gero, would you be so kind as to direct me to the journal in which you have published your ground-breaking 'Mathematics is Trivia' theorem. I am barely able to contain my curiosity about the full nature of your discovery and about some of its possible implications.

    I look forward to your response with excitement and anticipation.

    I am at once both surprised at, and intrigued by, your conviction that some of your colleagues are somehow able to garner enthralling and fascinating illusions of control pursuant to their respective mathematical fields of interest. My experience has been just the opposite.

    That many of your fellow mathematicians, hopefully yourself included, are 'addicted' to finding things out is not only uplifting but also a source of inspiration. Thank you for sharing.

    Naturally I know that upon reflection you now remember that, of the four freaks to whom the documentary paid homage, only Ludwig Boltzmann ended his own life.

    Cantor, a deeply religious soul, died of natural causes in the German sanatorium at Halle.

    Alan Turing, far from killing himself, was forced, and much against his will, to undergo a chemical alteration of the Self into a thing quite estranged from, and otherworldly to, the Self who had once been Alan, a thing hardly to be held responsible for its actions, and, thus, for all intents and purposes, was killed by the very government he had helped save from the psychological pathology of Adolf Hitler. What a tragic waste of life.

    And Kurt Gödel, as I am confident you now recall, died as did Cantor, from natural causes. In Kurt's case, it was anorexia nervosa.

    Details, dtails!

    As have most of us mathematicians, from time to time I dabble in areas within our profession not particularly akin to my own field of research (number theory). I have toyed with game theory and chaos theory, to name but two. But the brave new world of 'Trivia Theory' you are currently embarked upon is tantalizing and, as already intimated, I am simply dying (metaphorically speaking) to learn from you all I can.

    I implore you, cease at once in casting your intellectual passion as a disease of "mathematics’ allure" to which you feel yourself strangely "susceptible." Know that your genius is no psychological flaw.

    I think you may be overworked.

    Allow yourself time away from all the pondering. Go for walks. Get out in Nature.

    I, not being a genius, do not know if this can help you, but I do believe it might be worth a try. In any case, whatever you do, don't kill yourself.


    And, please! do not blame yourself for having been suckered into mathematics. It is not your fault. Fault lies with those dastardly devils promoting mathematics in sinister ways for the purpose of seducing innocent graduate students, as were you upon a time, into some sick mythological cult mentality.

    Gero, it need not end in tragedy.

    Think of Cantor, who never let go of his belief in God and kept on striving 'till his dying day. Stand up for yourself. Stand up! Don't let the Devil win. Serve as a beacon of hope. Prove to your young graduate students, through the example you set, that mathematics is not a psychological pathology to be feared.

    You may already be well upon the road towards reforming mathematics with the new Trivia Theory on which you are now working.

    Pay no attention to foolish critics who cannot see beyond the horizon. You are ahead of your time. Look to the young; they will understand you. Show them that a brilliant mathematician, such as yourself, really can wield the power of his art, not as an icon of death but, rather, in order to fling open all the ways to Life's great Glory! Glamorize Freedom, Hope, and Life! They will listen to you. The young, they will listen.

    I truly believe that Trivia Theory may be the way out from the hell of secularism's false and dangerous faith.

    With Trivia Theory, you can do it!

    Trivializing mathematics, rigorously, by way of air-tight proofs of salient theorems will place mathematics where it rightly belongs, at the bottom of the cultural heap. Prove to the world that the 'New Trivia Mathematics' rests upon a foundation far below the heights of madness flaunted so disgracefully by the religious fanatics.

    The challenge to the fantasy of mathematics as some sort of empowering key to the universe will arise through 'Trivia Theory'.

    This I know, because I have faith.

    And never forget, Gero:
    There is nothing 'good' about grief!

  80. The subject of maths has always fascinated me, but alas, ain't my forte.

    Will look forward to read all the posters that are far more knowledgeable than I.

  81. @Pyrrhus

    First of all, my apology. I meant to type "continuum hypothesis."

    As to the treatment of Alan Turing, one of the most important of 20th century mathematicians, the emphasis is on the word "iterative." The expatiation on Mr. Turin's state of mind was boring and unnecessary--once would have been enough--then the rest of the time could have been spent on Mr.Turing's accmplishments which are far more important. If this is pitiful, sobeit.

    I have not only watched "Fermat's Last Theorem," a documentary with little philosophy and psychology, thank God, but have read two books on it: one highly-intelligent, accessible account by Simon Singh and the other an ignorant Boeotian effort by Marilyn Vos Savant who believes her high IQ allows her to judge Dr. Wiles (She even criticizes Dr. Wiles for using hyperbolic functions!) Dr. Wiles is as much a hero to me as Turing and Godel are to you.

    As your knowledge of mathematics far exceeds mine, could you please recommend a good, clear textbook on abstract algebra. The ones I've tried to use are abysmal. A professor who teaches this subject once informed me that it can be learned only in the classroom--I find this impossible to believe!

  82. @Robert Allen

    1. The documentary does indeed imply that the set of irrationals is of a magnitude of infinity 'higher' than that of the rationals. This is done at the outset when explaining that Cantor could see that, although the rationals (decimally expressible numbers) are compact, within any pair of infinitely close rational infinitesimals lie an infinitude of decimally INEXPRESSIBLE numbers. This is easy to prove, but the documentary is intended for lay people and not for mathematicians (who know all that, anyhow).

    2. There is no "continuum theory."

    Cantor's continuum 'hypothesis' is the first in a program of twenty-three problems set out in 1900 by German mathematician David Hilbert for 20th century mathematicians to solve.

    The documentary did indeed fail to explain what the Continuum Hypothesis is all about. That is a pity, because it is not that difficult (to state). Still, it does boarder on what the average lay person might be expected to grasp, and I guess that is why the decision was made to leave it out.

    Put simply:
    If one thinks of an order of magnitudes of infinity, like one thinks of the counting numbers, one naturally asks whether there might not be something in-between. Is aleph sub1, sub2, sub3,... the ONLY continuous order possible?

    Do 'infinities' progress continuously as a 'continuum', or do they occur more as do 'quanta'? And, if they do progress as a continuum, do they progress countably, like the rationals, or do they progress uncountably, like the irrationals? The question is easy to 'sate', but attempts to settle the dilemma have sent more than one man mad (maybe that is why the piece is beset by so much "psychology").

    The Continuum Hypothesis remains to this day an 'hypothesis' because in 1963 an American mathematician by the name of Paul Cohen showed that it can be neither proven nor disproven. That is to say, taken as an axiom (either way), the two resulting Zermelo-Fraenkel set theoretical extensions are, both, independent from each other, as well being free from any respective inherent self-inconsistencies. So the world is now at liberty to 'mathematize' merrily away to its heart's content in either, or both, mathematical systems.

    Whether Cohen's proof amounts to a 'theorem' is questionable. The quandary immediately arises: theorem of what?!

    (PS: The Axiom of Choice is non-relevant to the proof.)

    3. As to Alan Turing's invention of mathematical biology, you know as well as I that to do it even cursory justice would require an entire season of documentaries. And, anyhow, no matter how much of Turing's work were covered, an infinity of uncovered work would still remain.

    You speak disparagingly of Alan's "state of mind" being included into the documentary. Presumably you refer to his suicide (murder).

    Whatever the subject matter, any documentary film maker who dares to even mention Alan's name has an absolute moral obligation to make clear to any and all exactly what the British government did to its favorite son, especially considering that, were it not for Alan, we'd all be speaking German! That I feel very strongly about this is obvious; that you feel a need to be spared "desultory and iterative comments anent his state of mind" is pitiful. In September of last year, Prime Minister Gordon Brown publicly understated:

    "...on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better."

    That is a lie!

    Alan deserved the BEST.

    4. Along with Alan Turing, Kurt Gödel is one of my favorite people in this whole wide world. The significance of his Incompleteness Theorem cannot be overstated yet is so subtle in substance as to be almost inexpressible (in words). Perhaps the wisest approach would be not to have mentioned the theorem at all.

    Robert, might I float the suggestion that, if you are not getting enough mathematics, but too much philosophy and too much psychology, you view fewer documentaries in favor of more time in the mathematics section of your local library? Alternately, why not watch 'Fermat’s Last Theorem', also available for viewing in the Science section of this site?

  83. In these lives lies the epitome of our humanity and God stands explained, the "infinite" and unexplained...

  84. 1. First of all, this documentary fails to describe Cantor's accomplishments i.e., proving which set is greater, the number of fractions or the number of irrationals, two infinite sets of different sizes.

    2. Although the documentary mentioned the continuum theory over and over again, it failed to state what the theory (which to my knowledge is still unproved) stands for--certainly this can be done, however superficially, in laymen's terms. Skip the high-minded quotations and meandering philosophy and get to the facts!

    3. Omitted mentioning Alan Turin's forays into mathematical biology, concentrating instead on his research, although of considerable importance, into computers. Spare me the desultory and iterative comments anent his state of mind and serve the results.

    4. As for Kurt Goedel, pretty much the same comments. I realize that Goedel is abstruse, but some simple explanation, coupled with examples of mathematical instability, would have been far more interesting than what was presented.

    In short, too much psychology and philosophy and not enough mathematics.

  85. I agree with J.Miller, these people had to think about something nobody else was willing to think or even wanted to hear about. But the thing is, I don't believe it was all because of the others and the 'burden' of this knowledge as if nobody else could think about it after or as if nobody else had ever thought about it... I happen to understand a little about the feeling the 'theory of infinites' can give to your life and is not that hard to get there if you ever read Jean-Paul Sartre... and if you are interested in what is this all about. I came to the conclusion that Infinity is just one, and of course you can divide that into any amount of said infinities you want, but I feel that underneath what was told in the documentary it was most of all hard for them to accept the consequences of such a concept, Im sure they knew it. Just tell people infinity is just one, and a very rare few will understand, care, or even try to find out what precisely could that mean.

  86. Mathematics is really trivial but it gives some people an illusion of control and thus fascinates and enthralls them. They become, so to speak, addicted. To be fixated on mathematics one must have a psychological need to feel in control. These four men were, it seems, control freaks and killed themselves when they could not have the control they wanted. What a tragic waste of life.

    These men had psychological flaws which made them susceptible to mathematics' allure but the way mathematics was promoted to them suckered them in. This means that the 'cult' of mathematics and how it was mythologised was partly responsible for these four tragedies.

    Since mathematics continues to be presented in the same way, I predict that other people, those who have the right psychological pathology, will be suckered in too and there will be more suicides amongst mathematicians in future.

    I also predict that these suicides will not be used to critique mathematics and reform it but will be used to add to its glamour - suicide is what happens when a mathematical genius, a mathematical 'Icarus', gets too close to the glare of the mathematical 'sun'. The death of a 'brilliant' mathematician just proves (a) how brilliant they were and (b) how powerful mathematics is. Such suicides are a form of sacrifice, a sacrifice to the cult of mathematics.

    I think mathematics is morally corrupt and a sort of secular 'false religion'. I regret the place it has in our culture. I hope it reforms itself but I suspect that mathematicians would rather have a few suicides than challenge the fantasy of mathematics as some sort of empowering key to the universe.

    Good grief! I've just read what I have written! A 'key to unlock the secrets of the universe'! How mad is that?! What a megalomaniacal delusion! But there are mathematicians who do genuinely think that is what they are doing. I do not know whether to laugh or cry. It is totally infantile and it makes the claims of religious fanatics sound modest by comparison. What a world!

  87. J.Miller, i´m really sorry about your attitide towards the faith and religion, especially after all what the history of science have learned us concerning the "unumbiguous truth" and the "absolute certainty".

  88. pt2 finis! makes me wonder if Jobs named his co. per Turing rather than a variety. great doc Vlatko..a most impressive site.

  89. Just finished pt1..and am doing 2 ..good doc on these unfortunates..anybody ever look into the Chudnovsky brothers? The 2 Russkies computing pi out to 2 billion digits on a supercomputer they've built from mail order parts in their NY apt?....strange days indeed!

  90. Maths is a formal system for understanding nature but it is a product of the Human Mind. The universe is not the product of the Human mind only to the extent that any individual observes it. The universe all matter and all systems cannot be defined absolutely by maths as such things exist unaware of these notions and have no necessity to conform to them, we seek to make the universe conform to our understanding this does not mean that our conclusions are correct. Science and mathematics can only offer us an ever more refined and predictable method of observing and 'predicting' Natural phenomena. It is curious that the notion of infinity still stands unresolved at the subatomic and larger cosmological scales. It is still in my mind linked closely with the notion of God. My belief is if all things are at the subatomic scale infinite, then there is a god if not the the opposite is the case, cant rationally explain that one but its just a hunch. NASA are 99% certain that the Universe is flat but we have been wrong about such notions before! I'd like to believe there was no God and that we could prove it but I think we will never be free of the Notion and perhaps there is a reason for this.

  91. Thought is Telepathy. The first language or Original form of communication. The original sin was to use words to try to describe infinity. Words all being containments or finite in their nature. The tree of knowlege was a devloved form of communication incapable of describing The tree of life because it is infinite. If you ever get a glimpse of it, you will be at a loss for words. We can never know anything, The minute we do it changes. The veil between this Tree of Knowlege side and that Tree of Life side is only a smoke screen which we can step through by accepting more and more of what we are being consatntly fed by the telepathic stillness which lies between the finite words. Like the spaces between the notes in music, there in lies Life. All the words, notes and numbers are the confusion, the insanity, the smoke screen. Be still and merge with god.

  92. Absolute Perfection, Certainty and Infinity can only be percieved at the cost of time, evolution and entropy. They are contradictions. Although time, evolution and entropy are simply concepts born of agreement as are Perfection, Certainty and Infinity. There is only Perpetual now.
    The part which makes the great thinkers a little tilted is Through our agreements on these concepts, The Mind can grow in its assessment of existance and accumulated understanding while everything around it is in a constant state of creative evolutionary renewal, devolution and Thermodynamic entropy. If this were not true, we would be able to build pyramids equal to those of Giza and we would have taken care of the ones already there using them for their original intent. What ever that might have been. WhoaHaHaHa
    This is why most questions on subjects of agreed upon existance can only be answered with more questions begging for new or continued agreements. The more extended and powerful the agreement, the more effective words of agreement are on the Light which fills the void and agrees to condense from light matrix, Mind, into material existance. Everything is Alive, There is only Now. And down the rabbit hole we go HaHa.

  93. So, things are uncertain. Since we have no authority or certainty, what is wrong with claiming divine inspiration? Often when science "proves" something beyond a doubt, they find it actually isn't true; the equation is of a higher order. Yet, if there is a God, then He understands everything and can make it clear. If Cantor was inpired by God, then Cantor might have actually known some real truths.

  94. I would also like to say that even though there is uncertainty - theories that describe the uncertainty can be true. Mathematics and Science are collectively a framework paradigm that is attempting to describe as best as possible through probabilities what has happen what is and what will happen. Perhaps instead of talked about the truth it is better to talk about the best truth right now. Remember though that a paradox can be described and by doing so will become another unit of truth. We will gain certainty in the foundations being uncertain - therefore our solutions will be more realistic. It is good to say what we no, better to admit and classify what we do not no, so that eventually we may understand what we do not know we do not know.

  95. I love the conclusion of part 2, truly insightful it is.

    "Are we grown up enough to live with uncertainty or will we again pledge blind allegiance to another certainty" - repeating the horrors of history. Just watching this is infinitely more educational than spending a million sabbaths listing to the ranting preacher say the same conjecture based on ancient myths and call it somehow perfect and logical (in this case my father). That is the great fallacy so many people have - faith - the truly great have ideas that swing back and forth like pendulums while retaining an anchor (often a family) in the ocean of the abstract.

  96. Just because Infinity may or may not appear in nature does not mean that it should not be described and/or used in mathematics.
    There are many examples in the history of math where seemingly crazy ideas are brought into the subject for the simple reason that they work (examples include complex numbers and infinity) where at a later date some application has been found. Furthermore if you ask most pure mathematitions you will find that they could't give two hoots about applications, just that there ideas are consistent within the axiomatic system. Someone will invariably find and apllication 500 years down the line!

  97. "these men i guess had a hard time being the bearers of truth." - J. Miller

    J, you forgot - there IS NO truth.

    Remember, from your rant - you "absolutely realized" the progression of history is merely the selling of fairy tales by snake oil salesmen to a retarded populace.

    Drink up.
    Science is no different.
    The temple has been relocated to MIT.
    The gowns now include pocket protectors.

    Still, the (new?) clergy dabbles with the same old Strange Fire.

    Or is the Big Bang theory not a creation myth?

    What can science and a clip board prove True?

    Please stop confusing the activity of compiling statistical probability, with Truth.

  98. >>"It is thus again silly to talk about one infinity being bigger than the other AS IF they had dimensions"
    If you work with sets it's very easy to show two sets A and B
    each with infinite elements where A is a subset of B
    and at the same time B has elements in it that are not elements of A
    thus showing that B has more elements than A even if both have infinite elements.

    Similar example as mentioned in this docu:
    Think the two sets of natural numbers and integers.

    You can even show that that #Z = 2*#N - 1 since for every natural number there is a negative number with equal numbers between itself and zero f.e. -10 and 10.
    If you accept that -0 is diffrent from 0: #Z = 2*#N.

    So you can not only "rank" infinities but also show some exact relations.

  99. silkop, like your description! "idealized, thought up model = a description of something that we wish to describe." Sadly, "Dangerous Knowledge" does seem to describe four unhappy lives that thoughtful peers were unable to ameliorate.

    The date is late so you have my apologies for taking exception to the "bollocks" remark. This may indicate the encumbered nature of my condition. Unable to complete the Sunday Crossword yours truly has learned to just put it down and move on. This makes me humble.

    Trapped in a fragile and imperfect Allegory, dominated by the infinitely divisible Continuum, the Entropic, the Incomplete, the Uncertain ... my life is in Chaos. How is a mere human (hobbled by constant Change) able to assume that languages like Physics, Mathematics or English may be able to provide answers to the chaotic puzzle?

    Is what follows not the mathematical definition of "a perfect circle"? "A set of points all exactly equidistant from a single center point?" In grade school Cantor must have learned that since that single center point is infinitely divisible, the circle cannot in reality be perfect.

    Why didn't these great men just put down the puzzle? What's that? Don't wish to preach my dear silkop but here goes. Surely I'm not putting words in your mouth when it is said that good direction dictates we should still be proud of those who refuse to believe in our imperfection and entrapment? Those who, having a simple proof of imperfection, are willing to again take up the chaotic puzzle, following tangent after tangent in an innocent search for the completion of our mathematically perfect circle?

    Now this is bollocks!

    Beautiful documentary!


  100. Do unto others as you would have them do unto you.

    Is this not an equation that explains infinity as;
    infinity sustaining itself by keeping other infinities alive therefore, to exist there must be other infinities that don’t take care of others as unto themselves and thus wither away into themselves and create the difference necessary to hold an opposing view and thus LIFE.(...that thinks about itself)?

  101. J Miller, i'm 100% agree
    very well said

  102. excellent J. Miller - couldn't have said it better myself. I tend to think that the human lacking of quantitative knowledge and certainty lead us into abstraction (pain does this as well). It is then that we create our own one word answers to explain what we do not understand, thereby putting our minds at ease. In my case this answer was always God. Mathematics is a tool for human understanding of the relationships in the universe and is imperfect as physics has failed to find the smallest parts - the discovery of atoms was supposed to satisfy this requirement but there are yet smaller pieces.

    I also think human tenancies of pattern finding can often cause us to end up heading down the wrong path. Especially if one holds preconceptions. The opposite pitfall is to label anything that doesn't as an anomaly, coincidence, and fail to follow your instincts even if it is based off circumstantial evidence. Every theory must be looked into. "Lets get messy" as the yellow school bus teacher miss frizzle always say. Yep the truth is messy.

    This being said I haven't watched the documentary yet. ^^

  103. Isn't this show supposed to be about good mathematicians who go insane? In that case, it seems to me that telling these stories with a romantic yet melancholy tone is what they are trying to teach us. Of course the math is important too, and i'm learning lots as i watch this, but i kind of like the humanity aspect of these people. I remember when i finally, absolutely realized that religion was all just fairy tales made up by uneducated people from the distant past in order for them to feel like there was something worth living for so that they could face the hard lives they had to endure. In creating a fantasy they could live out "after death" made them feel like their lives were worth living. Of course religion eventually evolved into something that quit helping humanity and began to harm it, but people didn't, and still don't want to give up the fantasy. After I got away from all the religion of my childhood and could finally think clearly, even though i was very willing to shed that religion because of how horrible it had been to endure, that religion had been in my head since i was a baby, crammed in day after day. So, when i was able to get away from that religion and let it go, i was surprised at how difficult it was to still let go of the concepts i'd been taught. Even though i knew none of it was true, even when i was a kid it was obvious religion, all religion, is false, yet when i finally had to clear my mind and let reality seep in, i was still sad to see my fantasy go away. knowing i would die and that was it wasn't easy to let myself accept. today i've had time to become used to the truth and i no longer feel like i did, nor am sad that lies are not true true. What I think happened to these math men was that they had a view of how the universe was, and when they learned it wasn't going to be the way they once believed, nor had it ever been the way they once thought, these men suffered more than most when they realized that they not only had to let go of old concepts, they actually were the ones who had to bring the change to the world. And they coudln't help knowing they themselves wrought the change, and that everyone on earth would eventually have to suffer the loss of their original thinking, or fantasy, in order to know the truth. And i suppose these men suffered more because they knew of the suffering they were delivering to the world. Change is difficult to accept. Men of science, i suppose, must bear it first and bear it well if it is to be accepted by others. these men i guess had a hard time being the bearers of truth.

  104. I really liked this documentary.
    I am no Math person, but still feel a resemblance that 'Life = Continuum Hypothesis'

    You have reached one goal in life......and you will find that there is still a bigger goal that you want to achieve !!

  105. The first mistake,Cantor made was trying to define an infinitesimally small point. The only
    way to define such a point is to factor out,infinity.
    The second mistake,obsessing about,"Pi" on the number line,with infinite digits.
    Doesn't the next number after 1.000...(infinite zeros)0,also have infinite points?

    1.000...(infinite zeros)1

    Thank you

  106. silkop - you sound like a second year student in some vaguely related discipline. Very pretentious and very shallow. Read more, think harder and type less.

  107. When I posted the above comment, I was unaware that Poincaré and Gauss essentially stood by my intuitive interpretation of infinity, as did Wittgenstein later on. Having informed myself, I'm even more confident that Cantor's treatment of "infinite sets" is bollocks and that it's not ME who is a looney having trouble to accept the obvious. How reassuring.

    1. George Cantor was wrong: The flaw in his theory of trans-infinities is within the ideal of one to one correspondence comparison.
      Your monthly salary is twice mine and we are to work forever and ever adinfinitum. the pension is proportional to salary. Who earns more pension?
      Very sorry, the question itself is contradictory; there is no pension to infinity employees.
      2. Me and you are to race and you can run twice my speed. The race is adinfinitum in time. who wins? 'wins' means there is an end at which we are to be compared who has moved longer distance at that time which endless.
      Edit Reply

  108. To Part 1: to treat mathematical infinities as a "mythical something" and to be baffled about them is silly from the get go. Confusion and paradoxes disappear when you treat every "infinity" as a target of a process with a finite description, a target which has no empirical equivalent whatsoever in Nature (and is just an idealized, thought up model = a description of something that we wish to describe). The "amazing" different infinities simply correspond to different finite generating processes. It is thus again silly to talk about one infinity being bigger than the other AS IF they had dimensions (an empirical concept). Rather, we should just admit that the different generating processes are, well, different.