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    1. #1
      Xei
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      Sexy Science

      In this thread we collect and talk about random bits of science that make us horny.

      Make sure it's bizarre but also try to make it accessible!

      I'll go first with some interrelated bits of set theory:

      • It's possible to make a one-to-one mapping between the points of n dimensional space and m dimensional space. For example, using this idea, you can fill a 3D room with an infinitely thin 1D string, with no gaps.


      • The set of everything, or the set of all possible sets, cannot exist, as it creates a paradox. This is because, in general, a set is smaller than the set of all of its subsets. However, the set of everything must contain, among other things, the set of all its subsets (if it didn't it wouldn't be the set of everything!), hence it is LARGER than the set of its subsets at the same time as being SMALLER.


      • There are different sizes of infinity. The smallest type is the infinity that represents all ordinary counting numbers; 1, 2, 3, ... . This is the same size as the infinity of positive and negative numbers together, and also fractions.

        The awesome fact is that the infinity that counts the continuous numbers (all of the decimals) is BIGGER than the previous infinity.

        However, it is impossible to determine whether there is an infinity between the two. This means that maths has split into two different sets of truths: one where you can assume that there is a middling infinity, and one where you assume there isn't.

        Above the continuous infinity, there are even bigger infinities. In fact, they go on infinitely.

        If you try to count how big the infinity of infinities is, though, you get another paradox.
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      Dionysian stormcrow's Avatar
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      Someone has been reading Godel Esher Bach...
      Maybe I'm still just a layperson in science because I cant think of any theories that make me sexually aroused but I deeply enjoy paradoxes.

      Banach–Tarski paradox
      A solid ball in three dimensional space can be split up into an infinite number of non-overlapping pieces, which can be arranged back together(by moving an rotating pieces) in a different way to produce two identical copies of the original ball. If Philospher876 was still around he would be pissed and go on a tirade about Euclidean geometry. I miss him.

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      Xei
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      This is all stuff from the first year of my degree, actually! It's certainly some of the most awesome stuff I've learned so far, although totally useless, as far as I can tell. I don't remember seeing it in GEB... actually, Hofstadter does briefly talk about the diagonal argument and I think even a hierarchy of infinities, but that's as far as he goes.

      Banach-Tarski is not as clear cut as it seems: it is actually one of those truths which belong in one of the bifurcations of mathematics that I alluded to, but not in the other. The bifurcation in particular is the axiom of choice, which basically refers to the idea that, given an infinite collection of sets, you can hand pick an element from each set. This in fact related to the bifurcation I mentioned before (whether or not there is a middling infinity): the stronger version of this basically says that there are no 'middling infinities' between any of the members of the hierarchy I alluded to; this assertion in turn actually implies the axiom of choice.

      Oh, and I think you may have characterised it a bit incorrectly; I think you can actually do it with five (not infinite) pieces, but the pieces are more like scattered fuzz, or electron clouds if you know what one of them is.

      Also, this is all done within Euclidean space... non-Euclidean stuff is hella trippy, I was going to post some stuff about that later.

      Incidentally, do you know much about Kant? People kept telling me to read him because he is supposed to have elucidated how some truths are a priori (which is something I'm fundamentally opposed to), so I started browsing last night... turns out his entire reconciliation is based on the premise that humans have a priori knowledge of space and time. Unless I'm doing it wrong, I found it reassuring that I could basically discount Kant as nonsense.

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      Think of a transistor as used in a digital circuit. Two simple switches, one that is ON when the gate is ON, and one that's OFF when the gate is ON (CMOS). And think of how you can combine these to get simple elements like adders, multipliers, latches, memory elements etc. And think of how these can be connected together to form complex structures, and in the end a processor. And imagine all the things this processor can do. And how all it does is really defined by a long string of ones and zeros. Now that's hot.
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      Dionysian stormcrow's Avatar
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      Quote Originally Posted by Xei View Post
      This is all stuff from the first year of my degree, actually! It's certainly some of the most awesome stuff I've learned so far, although totally useless, as far as I can tell. I don't remember seeing it in GEB... actually, Hofstadter does briefly talk about the diagonal argument and I think even a hierarchy of infinities, but that's as far as he goes.
      What is your degree in? Sorry I just assumed, he talks about set theory sporadically and Cantors Theorem occasionally as well. I’ve just been obsessed with understanding this book for about a month so I just notice references to it everywhere I go.

      Quote Originally Posted by Xei View Post
      Banach-Tarski is not as clear cut as it seems: it is actually one of those truths which belong in one of the bifurcations of mathematics that I alluded to, but not in the other. The bifurcation in particular is the axiom of choice, which basically refers to the idea that, given an infinite collection of sets, you can hand pick an element from each set. This in fact related to the bifurcation I mentioned before (whether or not there is a middling infinity): the stronger version of this basically says that there are no 'middling infinities' between any of the members of the hierarchy I alluded to; this assertion in turn actually implies the axiom of choice.
      Ha this reminds me of Hilberts Hotel paradox: If a hotel has an infinite number of rooms which are occupied by an infinite number of guests. To fit a new customer the hotel staff must move personA from room 1 to room 2 where personB from room 2 moves to room 3 and on into infinity. Actually I don’t even know why this is a paradox.

      Quote Originally Posted by Xei View Post
      Oh, and I think you may have characterised it a bit incorrectly; I think you can actually do it with five (not infinite) pieces, but the pieces are more like scattered fuzz, or electron clouds if you know what one of them is.
      Ah shit my bad.


      Quote Originally Posted by Xei View Post
      Incidentally, do you know much about Kant? People kept telling me to read him because he is supposed to have elucidated how some truths are a priori (which is something I'm fundamentally opposed to), so I started browsing last night... turns out his entire reconciliation is based on the premise that humans have a priori knowledge of space and time. Unless I'm doing it wrong, I found it reassuring that I could basically discount Kant as nonsense.
      I know a bit about Kant, honestly Im new to the whole philosophy/science thing so I have a little bit of trouble understanding Kant at the moment, I had to stop halfway through Critique of Pure Reason because I got to a point where I wasn’t even comprehending it. But Ill give it a go. You had it right, Kant says geometry is synthetic a priori because space (as well as time) are the prerequisites for understanding reality, they are the conditions of knowledge so to speak and we impose them onto reality so according to him they are a priori.

      This idea (that geometry is a priori) has been refuted by the existence of non-Euclidean geometry and relativity because relativity states that space is non-Euclidean because is not flat but curved which was discovered empirically so it is posteriori.

      I wouldn’t say Kant is nonsense but he did live a couple hundred years ago so it is inevitable that some of his ideas will be refuted especially in the light of modern science. I would give Kant a read he is probably the most influential philosopher since Plato but I find him a tad obscure but then again most continental philosophy is that way. I find that I agree with most of Kants basic ideas, he synthesizes empiricism and rationalism and I find his system more agreeable than those previous branches of thought taken alone.
      Last edited by stormcrow; 05-25-2011 at 11:35 PM.

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      Quote Originally Posted by stormcrow View Post
      Someone has been reading Godel Esher Bach...
      Maybe I'm still just a layperson in science because I cant think of any theories that make me sexually aroused but I deeply enjoy paradoxes.

      Banach–Tarski paradox
      A solid ball in three dimensional space can be split up into an infinite number of non-overlapping pieces, which can be arranged back together(by moving an rotating pieces) in a different way to produce two identical copies of the original ball. If Philospher876 was still around he would be pissed and go on a tirade about Euclidean geometry. I miss him.
      I am so angry about Euclidean geometry! Identical copies of the original ball enfuriate me. Your face!
      I hope that helps.
      ---o--- my DCs say I'm dreamy.

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      Quote Originally Posted by Xei View Post
      • There are different sizes of infinity. The smallest type is the infinity that represents all ordinary counting numbers; 1, 2, 3, ... . This is the same size as the infinity of positive and negative numbers together, and also fractions.

        The awesome fact is that the infinity that counts the continuous numbers (all of the decimals) is BIGGER than the previous infinity.

        However, it is impossible to determine whether there is an infinity between the two. This means that maths has split into two different sets of truths: one where you can assume that there is a middling infinity, and one where you assume there isn't.

        Above the continuous infinity, there are even bigger infinities. In fact, they go on infinitely.

        If you try to count how big the infinity of infinities is, though, you get another paradox.
      ∞^∞

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      Quote Originally Posted by Oneironaut View Post
      Two words:
      Kari Byron
      Yes. But they need to do more myths that involve her being in a bikini.


      You can take two balls, chop them up into infinity pieces, and put back together to form two perfect copies of the original. I forget the name of the paradox, but it's true.
      Last edited by ninja9578; 05-25-2011 at 11:46 PM.

    10. #10
      Xei
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      You are allowed to post stuff other than set theoretic paradoxes, yaknow.

      Anybody want an Euler disk? Beautiful piece of applied math / physics. It's the optimal solid for coin spinning:



      Also screening tonight: frikkin' math jigging pendulums.

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      Quote Originally Posted by Xei View Post
      You are allowed to post stuff other than set theoretic paradoxes, yaknow.

      Anybody want an Euler disk? Beautiful piece of applied math / physics. It's the optimal solid for coin spinning:
      In coin spinning the object is supposed to spin around while standing up. That one just falls over right away.

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    12. #12
      Dionysian stormcrow's Avatar
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      Quote Originally Posted by Xei View Post
      You are allowed to post stuff other than set theoretic paradoxes, yaknow.
      Ya I know I guess I was playing off your OP.

      Something I find extremely interesting is the existence of mathematical entities in nature. I believe that the universe is structured mathematically as well as understood mathematically. This idea has of course been heavily opposed since Plato for a number of reasons but I think one reason is because if mathematics do exist in nature it kinda implies that the universe was created with intent. Hence the euphemism for god "The Great Mathematician". I don't think the universe was created with intent but I am open to the idea if the evidence presents itself.

    13. #13
      Xei
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      Sorry, I was actually referring to ninja, who posted Banach-Tarski again.

      I used to have the same opinion as you, but it changed as I aged. The main source for my belief in Platonism as a teen was basically listening to Roger Penrose in a rather submissive manner, who talks about it a lot in his books; but as I became more accustomed to and knowledgeable about maths I changed my mind. I now think that mathematics is as empirical an endeavour as any other (that is to say, completely), and the only reason for its huge success is the same reason that science has been so successful. The fact that there exist multiple, contradictory mathematicses is one of the more perturbing arguments against their inherent truth. I'm reading Penrose on and off again at the moment, and now that I feel a bit more enlightened, and independently critical, the errors in his arguments are glaring.

      Perhaps as I learn more about theoretical physics I'll change my mind again, but I'm yet to see anything that'd sway me. My tentative belief at the moment is that intelligence can only emerge in the first place in universes with a high degree of pattern and structure (so, an elaboration of the anthropic principle); this explains the success of induction and the beauty of what we may mistake for an underlying reality.

      With respects to Kant; of course, he is not to blame (although it does show that at least some of his substantial arguments were actually assertions). I've mentioned before, in fact, how philosophy seems to diffuse through the mass consciousness, which explains why I have so many original thoughts, without overt external reference, which turn out to have been had before, and moreover made their originators very famous. Only last night I discovered a chief idea of mine (briefly that much of philosophy is based upon conflation of words) was the one of the major ideas that made Wittgenstein famous and has apparently been a basis for most contemporary philosophy. My awareness of non-Euclidean geometry was one of the major things that I'm conscious of having shaped my opinion, and probably shaped my thought processes systematically; if Kant were alive in modern times, perhaps his philosophy would be much the same as mine.

      And my degree is in maths.

      Quote Originally Posted by Marvo View Post
      In coin spinning the object is supposed to spin around while standing up. That one just falls over right away.
      I have no clue what you're talking about.
      Last edited by Xei; 05-26-2011 at 12:42 AM.

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      Superconductor levitation



      Also

      He's the man.
      Last edited by ninja9578; 05-26-2011 at 12:24 AM.
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      Quote Originally Posted by ninja9578 View Post

      He's the man.
      ^ This.

      I'd hit that like freight train.
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      Chapter 2, "The Replicators," from Dawkins's The Selfish Gene (the chapter is available in its entirety online here as a sample of the book) was definitely a "holy shit" moment for me when I first read it. In it he gives a discussion of how the principles of imperfect replication and survival of the most stable could have given rise to the first forms of "life," if we want to call it that. It's a relatively brief treatment of the topic, and not an area in which Dawkins himself is an expert, but it's a pretty amazing introduction nonetheless both to the ideas of abiogenesis and to evolution.

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      Non-newtonian fluids

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      I bite and blood turns me on. Science!

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      THIS QUESTION IS FALSE!

      just trying to add another paradox.

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    20. #20
      Xei
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      I didn't know about non-Newtonian fluids until the day I found my neighbour at Cambridge filling our sink with corn starch, haha... I was very nearly late to a tutorial and needed a change of clothes before I went to it.

      Quote Originally Posted by DuB View Post
      Chapter 2, "The Replicators," from Dawkins's The Selfish Gene (the chapter is available in its entirety online here as a sample of the book) was definitely a "holy shit" moment for me when I first read it. In it he gives a discussion of how the principles of imperfect replication and survival of the most stable could have given rise to the first forms of "life," if we want to call it that. It's a relatively brief treatment of the topic, and not an area in which Dawkins himself is an expert, but it's a pretty amazing introduction nonetheless both to the ideas of abiogenesis and to evolution.
      Thanks DuB, I'll read that. I've been meaning to read Dawkins for a while; in particular I'd like to know about evolution in a bit more depth, but I don't know how objective, relevant, or how far beyond layman's stuff Dawkins is..?

      The realisation of the extremely simple idea, 'if something is good at making copies of itself, it will become numerous', especially in the context of abiogenesis, is quite a moment. Are you referring to this, partially?

      It's really quite astonishing that nobody even considered natural selection before the mid 19th century; it really is that simple and obvious. Again this shows how methods of thought in general tend to grow gradually among populations as a whole (I suppose this is memetics); I'd very much like to see what'd happen if an intelligent person today were asked to try to figure out a mechanism by which life could have developed. I wouldn't be surprised if it a very large proportion got it without much trouble.

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      Dionysian stormcrow's Avatar
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      Quote Originally Posted by Xei View Post
      It's really quite astonishing that nobody even considered natural selection before the mid 19th century; it really is that simple and obvious. Again this shows how methods of thought in general tend to grow gradually among populations as a whole (I suppose this is memetics); I'd very much like to see what'd happen if an intelligent person today were asked to try to figure out a mechanism by which life could have developed. I wouldn't be surprised if it a very large proportion got it without much trouble.
      The pre-Socratic philosopher Anaximander had a theory of evolution which stated humans evolved from a lower life form and that all life came originally from the sea and humans used to be fish

      The self-replication principle is also used in chaos theory to explain (one way) in which a simple system can become a complex system through emergence. Thanks for the link Dub Ill definitely check it out.

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      DuB
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      The epiphany for me was that the principles that drive biological evolution, with which I was already somewhat familiar, could just as easily be applied to abiogenesis. My view at the time was along the lines of "this evolution by natural selection stuff seems great once life has already appeared, but it's still a pretty big mystery how that happened in the first place." The passage I referred to opened my eyes to the possibility that it could just as easily bridge the gap between life and non-life. It really expanded the sheer breadth and significance of the principles for me.

      The Selfish Gene is well within the grasp of an intelligent layman. It's a superb exposition of both the general idea of gene-level evolution and of the ideas of what was then known as sociobiology but what is now referred to as evolutionary psychology. The chapter I singled out above is really little more than a footnote in the overall thesis of the book, the rest of which is also well worth reading. Dawkins's writing style has a clarity and wit that I really admire. I highly recommend the 30th anniversary edition of the book, which features the original 11 chapters, unchanged but well-annotated, as well as two new chapters.

      The Extended Phenotype, from what I've read and heard, is a sequel of sorts to The Selfish Gene. It defends and expands on many of the ideas in the latter. Dawkins is on record as saying that it's the book he's most proud of writing. It's been on my reading list for a long time, but I don't have as much leisure reading time as I used to, so I haven't gotten to it.

      If you can put up with his long-winded and somewhat roundabout writing style, Dennett's Darwin's Dangerous Idea further expands on these ideas, with a broader philosophical focus and considerable forays into artificial intelligence, morality, linguistics, etc. There's a lot of great stuff in this book, although brevity is apparently not a virtue for Dennett.

    23. #23
      Xei
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      Quote Originally Posted by stormcrow View Post
      The pre-Socratic philosopher Anaximander had a theory of evolution which stated humans evolved from a lower life form and that all life came originally from the sea and humans used to be fish

      The self-replication principle is also used in chaos theory to explain (one way) in which a simple system can become a complex system through emergence. Thanks for the link Dub Ill definitely check it out.
      I was really referring to natural selection though; without such a concept, Anaximander's views were actually pretty loony.

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      Not sexy, but it relates to abiogenesis. I have a hard time believing that aristotelian abiogenesis was taken seriously...maggots just popped out of rotting meat and stuff. KSDJhghfgjk
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      Aristotle was pretty much wrong about everything.

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