Where do numbers come from?

Discussion in 'Physics & Math' started by arfa brane, Dec 17, 2018.

  1. arfa brane call me arf Valued Senior Member

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    I have this idea that in order to "have" numbers, there has to be a symmetry, and then this symmetry has to "break", or become locally at least, less symmetric than one you assert exists a priori.
    Numbers arrive when you compare the "globally asserted" symmetry, with the local one which is something you, erm, distinguish locally . . .

    So, a question about what symmetry is.

    Is a bar made of iron more or less symmetric than a similar bar which is magnetised? What about say, an iron ball in either case? What about the appearance of numbers?
     
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  3. Beer w/Straw Transcendental Ignorance! Valued Senior Member

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    Is it about quantum mechanics or mathematics?

    There are boundless transcendental numbers between 1 and 0.
     
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  5. arfa brane call me arf Valued Senior Member

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    It's about numbers, more or less. Perhaps keep it in the domain of those numbers we can represent as data types in some computer language, such as Ruby, or C++.

    But, in my recent look at "logic on a boundary", I thought about something connected to how much information can be stored in a given volume.

    Basically, framed like a question it's, "Is there more information in a closed curve which isn't smooth (i.e. is deformed) than in a closed curve which is (i.e. a circle)?"

    Or, forgetting about what we think information is or might be, is there more of "anything", apart from extra curvature (a deformed, closed curve "meanders around), in the above case?
     
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  7. Beer w/Straw Transcendental Ignorance! Valued Senior Member

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    I think it's odd to talk about numbers, curves and information in a given volume like this.

    Just for fun, what is the exact volume of a sphere that has the diameter of a Planck length?? This isn't a question to be answered exactly because you run into the transcendental number Pi -but you could say it at least would fit inside a cube with length, height and width of a Planck length...
     
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  8. Write4U Valued Senior Member

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    IMO, numbers are arbitrary symbolic representations of relative positive values inherent in measurable patterns
    (symmetries)?
    Every culture has/had it's own symbolic numbers representing values. Fortunately, science has a standardized symbolic system, which negates the need to retranslate the translated relative values....

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  9. Confused2 Registered Senior Member

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    On reading the OP I'm immediately thinking of symmetry breaking in the Higgs field - the ingredients are there (symmetry breaking and magnetic field) to be fishing for the nature and origin of (perhaps) a toy field or mechanism that breaks symmetry. Can you put numbers in a field that 'breaks symmetry'? - What? How? and much more - I know nothing but would love to know more (or even something).
     
  10. Write4U Valued Senior Member

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    Forget doing anything to nature with numbers. Numbers are human inventions.

    The universe works with relative values and algebraic functions. IMO, symmetry breaking is a quantum superposition function.

    "symmetry breaking"

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    A ball is initially located at the top of the central hill (C). This position is an unstable equilibrium: a very small perturbation will cause it to fall to one of the two stable wells left (L) or (R). Even if the hill is symmetric and there is no reason for the ball to fall on either side, the observed final state is not symmetric.
    https://en.wikipedia.org/wiki/Symmetry_breaking
     
  11. arfa brane call me arf Valued Senior Member

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    Humans are Nature's inventions.
    According to . . . humans.
     
  12. Write4U Valued Senior Member

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    But we are not artifacts.
    Yes we make and use artifacts.
    1. Google
    IMO, numbers are artifacts to identify and symbolically represent universal relative values.

    That way we can use the mathematics of natural functions for human endeavors, apart from the greater environment, but (hopefully) follow natural directives attached to Universal mathematical functions.

    This lecture by Roger Antonsen reveals the natural beauty of abstract mathematical patterns afforded by the logical functions of relative values and the results of logical functions to represent the abstract pattern of 4/3. It's really good stuff....

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    In a very down to earth manner he presents some extremely profound implications of universal mathematics, as expressed from different artificial perspectives

    https://www.ted.com/talks/roger_antonsen_math_is_the_hidden_secret_to_understanding_the_world
     
    Last edited: Dec 19, 2018
  13. arfa brane call me arf Valued Senior Member

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    In physics there are three fundamental kinds of symmetry: time symmetry, charge symmetry, and parity.

    These are global symmetries; time is symmetric according to "laws of physics", a particle moving forwards in time is equivalent to one moving backwards in time (changing the direction of time is an identity), charge is symmetric since changing its sign everywhere (a global transformation) is an identity.
    The time symmetry clearly isn't what we perceive, but physics here shrugs its shoulders.

    What about parity symmetry, and/or breaking a global symmetry -- the emergence of numbers?

    Moreover, what of the symmetries of the different known phases of matter? In what sense are solid or crystalline phases more or less symmetric than liquid or gas phases?
     
  14. arfa brane call me arf Valued Senior Member

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    Another example from topology:

    This is a diagram of K(1,2), the torus knot. The numbers mean the knot wraps once around the minor axis of symmetry (of the 2-torus, natch), and twice around the major axis.

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    If you rearrange the internal part of the knot by making it smaller and closer to the outer part, you don't change the "going around a central point of symmetry twice" rule.

    You can break the symmetry by making a second loop (with the same parity) in the interior somewhere along the outer part of the curve, so there are two centres to go around.

    But you can restore everything by making the two internal loops go around the minor axis twice, so the rest of the curve now goes around the major axis once, and you in fact have the knot K(2,1). But K(2,1) = K(1,2) in the context of topology.

    Although equivalent, deforming one into the other on an actual 2-torus is not trivial . . . there's a shitload of mathematical "information" encoded in this transformation. You could kick off with the notion of a minimal surface lying inside a boundary, which then gets twisted over itself and folded together.
     
    Last edited: Dec 19, 2018
  15. Confused2 Registered Senior Member

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    @arfa,
    I've abandoned a post suggesting chaos being what undoes numbers in physics - you've gone for symmetry which is much more interesting.

    Looking at your K1. To be 'asymmetric' I'd expect a car racing round the track (adding energy) to be able to accelerate K1 to (say) the left. Like F=ma^2 would do the trick but it doesn't. Don't you have to put asymmetry in (somehow) to get asymmetry out?
     
  16. arfa brane call me arf Valued Senior Member

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    Moving along a track is a way to consider motion globally (from a distance) and locally, from the frame of a moving point. You can do this without introducing physics, or energy. You can even leave metrics out of the "frame", just say distance is an affine construction--you move around a centre along a path which is "some" distance from this arbitrary fixed point.

    Leaving aside the problem of where the point is, of course. This isn't probably an easy thing to spot if you're a point moving along a one-dimensional path. From a distance, maybe.
     
  17. TheFrogger Valued Senior Member

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    We use a decimal system because we have ten fingers.
     
  18. arfa brane call me arf Valued Senior Member

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    A clue about "going from" K(p,q) to K(q,p). In the case we fix one of p,q, we then have K(q,2) = K(2,q) say.

    So here are the first four planar diagrams:

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    Such that (sorry if it seems pedantic) Kj = K(j,2).

    Looking at K1, it doesn't seem to be possible that the region inside the inner part can have a minimal surface, it must be a hole. The surface must extend to the left and right of the single crossing.

    But suppose you can grab the boundary near the crossing and pull part of it up out of the page, along with the surface it bounds. Keep this "new" loop at a right angle and rearrange the loops still in the plane so they are much closer together. Then lift one of them out of plane slightly.

    Ok, so if you then resize the loops so they're all the same, you have a "symmetrised" frame of reference which generates a surface of revolution, or a solid of revolution since you rotate a surface; you can rotate the frame about either centre, so you can generate a pair of linked tori. Whether you have K(p,q) or K(q,p) is a matter of which centre you choose to rotate the loops around.

    As this image shows for K(3,2):

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    There are 3 + 2 loops (from our global perspective), and there are a number (how many) of lines lying across a common boundary of the linked tori. Since we can imagine the minimal surface and the surfaces/solids of revolution, we don't need to include them other than to prove continuity or something. A convenient arrangement.
     
    Last edited: Dec 20, 2018
  19. arfa brane call me arf Valued Senior Member

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    So do numbers show up because we "arrange" for them to?

    You're a sheep farmer and there's a flock of several hundred sheep in a big paddock. You can estimate the number, or you can arrange things so you can count them more accurately, which involves getting them all into the same yard, then moving them through a race, single file.

    Or you own a farm but you're not a farmer, you want to know how many sheep you have, you arrange for the farmer running your farm to find out. Etc.
     
  20. Michael 345 New year. PRESENT is 69 years old Valued Senior Member

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    And to lazy to take off shoes and socks to make a Duo-decimal system

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  21. Write4U Valued Senior Member

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    Does it matter that only K1 and K3 consist of a single continuous line, whereas K2 and K4 use two separate lines?

    I have another question about where numbers come from. How many different numerical systems have existed through history and still exist locally, where each system used different symbolic representations for the same values.

    Seems to me that all numerical systems use(d) arbitrary symbols, but that the decimal system was agreed on by the scientific community for clarity and general referential convenience.

    But in non-scientific endeavors there are still several remnants of old local forms of "counting values", some of which are not even translatable from one to another without an actual explanatory cross reference.

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    Chinese decimal system
     
    Last edited: Dec 21, 2018
  22. TheFrogger Valued Senior Member

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    "ToO lazy," Michael345. And numbers come from the person doing the calculating.

    "Beauty is in the eye of the beholder."-Unknown.
     
  23. arfa brane call me arf Valued Senior Member

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    It means you've spotted something, a pattern. Can you explain why Kj where j is odd, means you have a single link, but where j is even there are two links knotted on the torus?

    Could you conjecture anything about more than two links, given the above?
     

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