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Alexander Kruel (M, 35)
Gütersloh, DE
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Transhumanist, atheist, vegetarian who's interested in science fiction, science, philosophy, math, language, consciousness, reality...
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    Mathematics as a way of knowing?
    Follow-up to: Knowing beyond science and mathematics?

    I think I have to clarify why I mentioned mathematics in my recent post — Knowing beyond science and mathematics? — as a way of knowing in addition to science and separated from other logical systems.

    First of all, I'm only at the beginning of my long journey towards a decent education, as I've only started as recently as a year ago to seriously get into mathematics and other important fields. I'm still struggling with the basics.

    My posts here at SpaceCollective.org are merely mental outpourings, NOT so well-thought-out musings and problems that haunt my mind. What I write does not necessarily reflect my opinion.

    So let me explain...

    There seem to be quite a few arguments in favor of the Mathematical universe hypothesis and that math is timeless and being discovered, not invented.

    …there’s still sometimes a tendency to think as though turning on a sufficiently advanced calculator causes something to mysteriously blink into existence or awareness, when all it is doing is reporting facts about some very large numbers that would be true one way or the other.
    The mathematical universe: the map that is the territory

    I recently started to read The Big Questions, a book by Steven Landsburg. The basic tenet seems to be that mind is biology, biology is chemistry, chemistry is physics, physics being math. Mind perceives math, thus the universe exists physically. Erase the “baggage” and all that’s left is math.

    If we can feel real inside a non-magical computer simulation, then our feeling of reality must be due to necessary properties of the information being computed, because such properties do not exist in the abstract process of computing, and those properties will not cease to be true about the underlying information if the simulation is stopped or is never created in the first place. This is identically true about every other possible reality.
    The mathematical universe: the map that is the territory

    There has been a really nice plain English description of this idea being posted on lesswrong.com recently. Though everybody who has read Greg Egan’s Permutation City might already be unknowingly familiar with it.

    Existence is what mathematical possibility feels like from the inside. Turn off G.O.D., and we’ll go on with our lives, not noticing that anything has changed. Because the only thing that has changed is that the people who were running the simulation won’t get to find out what happens next.
    The mathematical universe: the map that is the territory

    Mathematics is the basis of all of science and even seems to play a large role in our thinking, as it may be probabilistic. When it comes to quantum mechanics it all seems to be about probability as well, and therefore math. But in what sense do mathematical structures exist? Well, as Steven Landsburg puts it, mathematics is a kind of extrasensory perception. It's all out there, 1+1=2 has always been there and true, even before anybody ever thought about it. And yet you won't find perfect circles anywhere "out" there. Does that mean math has no influence, that it is not tangible, that math doesn't exist? It is subject to inquiry. I think you can make predictions that are falsifiable. It even bears fruit by providing accurate descriptions of the physical world that can be tested. So yes, I believe mathematics, as science, is a way of knowing.

    There is something almost mystical about this: any sequence of digits, for example, randomly conceived in the mind, must correspond to a sequence of digits in the unknowable expansion of Pi (in that realm over 10^1000 digits into the expansion), based on the laws of probability.
    — Garth Kroeker, Irrational Numbers Metaphor

    What about other logical structures? They are ultimately part of mathematics, as they are describable by pure math, and in the case of a Mathematical universe are timeless structures. But the difference between World of Warcraft or a programming language like Haskell is that these systems are not subject to induction, as you cannot arrive at general, much less specific conclusions and facts about reality by a sole examination of these structures. They do not imply everything that exists. Whereas mathematics can be used to deduce even the most abstract fact about our world, about reality. Mathematics does specify everything that is possible and thus includes everything that exists.

    Anyway, I'll likely need a long time to see if this makes sense, if the idea of a mathematical universe might be likely, or even worthy of consideration. Thus I just put it out there into that post, for that otherwise I couldn't have posted it for a long time. And I believe it might be a good idea to first make your own thoughts, come up with your own ideas and conclusions about a subject, before you go listen and learn what other people have to say about it. What I think may or may not turn out to be completely wrong, but this way I might be able to learn how to think, how to be less wrong the next time I encounter something new that I'll have to reason about myself.

    Sun, Apr 4, 2010  Permanent link

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    johnrod     Sun, Apr 4, 2010  Permanent link
    >What about logical structures?
    Some have claimed that computational models of reality allow induction and experiment, though efforts have been limited in scale so far, e.g. Lloyd, Wolfram or Turkle, also discussed by Kurzweil. Thanks.
    XiXiDu     Thu, Apr 8, 2010  Permanent link
    Further reading (will constantly add content via comments as it arrives):

    SUPPOSE we had a theory that could explain everything. Not just atoms and quarks but aspects of our everyday lives too. Sound impossible? Perhaps not.

    It's all part of the recent explosion of work in an area of physics known as random matrix theory. Originally developed more than 50 years ago to describe the energy levels of atomic nuclei, the theory is turning up in everything from inflation rates to the behaviour of solids. So much so that many researchers believe that it points to some kind of deep pattern in nature that we don't yet understand. "It really does feel like the ideas of random matrix theory are somehow buried deep in the heart of nature," says electrical engineer Raj Nadakuditi of the University of Michigan, Ann Arbor.

    [...]

    The matrix was a neat way to express the many connections between the different rungs. It also allowed Wigner to exploit the powerful mathematics of matrices in order to make predictions about the energy levels.

    Bizarrely, he found this simple approach enabled him to work out the likelihood that any one level would have others nearby, in the absence of any real knowledge. Wigner's results, worked out in a few lines of algebra, were far more useful than anyone could have expected, and experiments over the next few years showed a remarkably close fit to his predictions. Why they work, though, remains a mystery even today.

    More: Enter the matrix: the deep law that shapes our reality

    From all of this I am forced to conclude both that mathematics is unreasonably effective and that all of the explanations I have given when added together simply are not enough to explain what I set out to account for. I think that we-meaning you, mainly-must continue to try to explain why the logical side of science-meaning mathematics, mainly-is the proper tool for exploring the universe as we perceive it at present. I suspect that my explanations are hardly as good as those of the early Greeks, who said for the material side of the question that the nature of the universe is earth, fire, water, and air. The logical side of the nature of the universe requires further exploration.

    More: The Unreasonable Effectiveness of Mathematics
    XiXiDu     Fri, Apr 9, 2010  Permanent link
    The Unreasonable Effectiveness of Mathematics in the Natural Sciences

    by Eugene Wigner

    The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.
    XiXiDu     Sat, Apr 17, 2010  Permanent link
    Jellyfish Math | Is mathematics invented or discovered?
    Take prime numbers, for example, which as far as I’m concerned, constitute a more stable reality than the physical reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world…We run up against a reality every bit as uncontestable as physical reality.
    XiXiDu     Thu, May 6, 2010  Permanent link
    Illusions of the Mathematical Imagination
    The question about the nature of mathematics is not new, and the number of possible answers is small. I think that everyone would agree that mathematics has something to do with the physical world, since it has been so successfully applied in science. Regarding its precise relationship to nature, three answers are generally given, and two of these are wrong.

    The first wrong answer identifies nature and mathematics. Mathematical space is the same thing as physical space. Physical quantity is the same as mathematical quantity. We could call this the Pythagorean answer.

    The second wrong answer begins by assuming no relationship between mathematics and nature, mathematics being simply a human construct. Mathematics is then pasted onto nature, and provides models for understanding nature. It is a language for describing nature, and so on. We could call this the positivist error. Positivists can never explain why mathematics actually works.

    The third and correct answer is found in Aristotle. It is the hardest to express precisely because it is correct, and it contains aspects of each of the wrong answers. According to this view, mathematical objects are "beings of reason" that is to say human concepts (as in answer 2) but they have been derived by abstraction from physical nature, and so bear an intrinsic relation to nature (as in answer 1).

    So now we know what mathematics is; at least I have applied some labels to the issue. But the errors themselves typically have a history in the development of the student. For instance, the Pythagorean error is frequently committed by beginners with genuine mathematical aptitude. They get carried away by the equations, and then see the mathematics instead of nature. The second error is a sort of undergraduate, pseudo-sophisticated response to the discovery that nature and mathematics are not identical, typically occurring when one first meets up with non euclidean geometry. In anyone older than that, it's an intellectual cop-out; regretably a cop-out that you often find in those introductory chapters in textbooks all about the so-called scientific method: math is a language pasted onto nature, and we paste a different equation every hundred years or so when we have a paradigm shift. The third answer comes with
    experience. Whether experience can be hastened by good teaching is something I don't know. It can sometimes be retarded through bad teaching.
    XiXiDu     Fri, May 14, 2010  Permanent link
    Nature by Numbers

    http://vimeo.com/9953368

    A SHORT MOVIE INSPIRED ON NUMBERS, GEOMETRY AND NATURE.
    XiXiDu     Fri, Jun 18, 2010  Permanent link
    Math is Subjunctively Objective
    I am quite confident that the statement 2 + 3 = 5 is true; I am far less confident of what it means for a mathematical statement to be true.

    In "The Simple Truth" I defined a pebble-and-bucket system for tracking sheep, and defined a condition for whether a bucket's pebble level is "true" in terms of the sheep. The bucket is the belief, the sheep are the reality. I believe 2 + 3 = 5. Not just that two sheep plus three sheep equal five sheep, but that 2 + 3 = 5. That is my belief, but where is the reality?
     
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