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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    7,832
    What does the trefoil knot have to do with quantum physics?

    The path to beginning to understand the connection, which is just one of them, can take in as I may have mentioned, a shitload of mathematics.

    What is all that, really? A way to formulate things like curves and sections of curves. You find a set of points on a curve and move them around, asserting that the part you leave alone is left invariant by all the moves, as an example.

    The universe has one way of representing objects, our mathematics has more than one. As they say, the map isn't the territory, and we have a lot of maps to prove it. So the question, where do numbers come from might be better framed as, why does the universe allow multiple maps to the same territory, perhaps an unknown, erm, number of them?
    Or is that just anthropomorphism looking at itself, or something?
     
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  3. Write4U Valued Senior Member

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    20,069
    Roger Antonsen observes that mathematics allows us to look at the territory from different perspectives.
    He claims that is the key to understanding.
     
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  5. arfa brane call me arf Valued Senior Member

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    7,832
    Suppose you're looking for interesting ways to invert pairs of points, maybe you'd like to be able to extend this inversion to any number of points.

    You think there might be something worth looking at in the domain of minimal surfaces, which come "equipped" as it were, with a boundary (even if the boundary is at infinity). A minimal surface is one that minimises its area everywhere, the obvious example is the Euclidean plane--it has no bumps or hollows, cannot be stretched or folded etc, anywhere = everywhere.

    Here's a nice piccy of a finitely bounded minimal surface (actual physics!), in three dimensions (degrees of freedom the surface has to minimise itself--a spontaneous action).

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    If you look closely you can see the edge of the soap film connects to the loops at a minimal "1-surface" which is a boundary of each wire loop (physically a pair of tori, somewhat contracted in one dimension).
     
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  7. Write4U Valued Senior Member

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    20,069
    This contraction intrigues me. Why are the walls not straight? Is there a pressure differential between inside and outside the wall? A Casimir effect?

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    https://en.wikipedia.org/wiki/Casimir_effect
     
  8. arfa brane call me arf Valued Senior Member

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    7,832
    You mean why isn't the surface a nice flat cylinder? Topologically it is a cylinder because you can straighten it into a nice regular shape (with completely abstract, not necessarily physical topological 'moves').

    Geometrically the surface is a catenary, it has negative curvature so looks suspiciously like an hyperboloid of one sheet. You should be able to find saddle points on it, if you want to get all analytical.

    I'm not aware of any connection between soap films and the Casimir effect. The surface spontaneously 'rearranges' so the tension is equal everywhere; if you will, the shape of the soap film is then a 'solution' of a system of differential equations.

    As to the contracted tori, I meant the wire loops. These are physically each a "skinny" torus with a handle, and the soap film attaches to each torus along a line of longitude which is also minimal (on the inside surface of the torus), except where the handle is, but the surface stays minimal there too.
     
    Last edited: Dec 31, 2018
  9. Write4U Valued Senior Member

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    20,069
    Thanks for responding. I'll need to do a little study on these explanations.......

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  10. arfa brane call me arf Valued Senior Member

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    7,832
    An easy algebra to introduce (from nowhere in particular), is that of sets of points along the upper boundary of the surface, and along the lower boundary.

    Start with say a set of n = 2 points on each boundary, connected by continuous lines that don't intersect, so they might as well be vertical and something you can represent with a planar diagram--a rectangle with two points on its top edge, and two points on its bottom edge, connected by vertical lines (of longitude).

    So if you now hold one of the loops fixed and rotate the other, keeping it horizontal, the lines twist around the "cylinder". But what happens in the planar diagram? If you rotate the upper loop through 2π, the points are back in the "box", but with trailing strands either side (since the box represents a cylinder sliced along a line of longitude).
     
  11. Write4U Valued Senior Member

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    20,069
    I believe this may be pertinent to the question raised by Tegmark regarding mathematics.
     
  12. Michael 345 New year. PRESENT is 72 years oldl Valued Senior Member

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    13,077
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  13. Michael 345 New year. PRESENT is 72 years oldl Valued Senior Member

    Messages:
    13,077
    Also concerning numbers

    From a friend's email December 2018

    ***
    December calendar. (This will be the only time you will see this phenomenon in your life.)

    S M T W T F S

    1

    2 3 4 5 6 7 8

    9 10 11 12 13 14 15

    16 17 18 19 20 21 22

    23 24 25 26 27 28 29

    30 31

    The month of December 2018 will have 5 Saturdays 5 Sundays and 5 Mondays. It only happens once every 823 years.

    The Chinese call it "BAG FULL OF MONEY". Send this message to all your friends and within 4 days the money will surprise you. Based on Chinese Feng Shui, the one who does not transmit this message can lose this great opportunity ... I do my part, you never know

    *****
    Don't think the last part is accurate

    And on January 1st 2019 the PRESENT is 69 years old

    **""*
    Because the "present" time changes, standard practice is to use 1 January 1950 as the commencement date of the age scale

    https://en.m.wikipedia.org/wiki/Before_Present#Radiocarbon_dating

    ******

    Bonus

    I hope in 2019 you all are
    12 months happy
    52 weeks funny
    365 days successful
    8760 hours healthy
    52600 minutes lucky
    3153600 seconds joyful

    Happy 2019

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  14. arfa brane call me arf Valued Senior Member

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    7,832
    Calendars have been around for a while, but days and nights have been around a lot longer. We number the days with calendars, but really it's just a day-counting method so we can also count sidereal years.

    About twisting a cylinder: when you number some points and connect them, the connections also twist or wrap around the cylinder. You can draw a diagram of this, and show that the diagram is a composition of two "smaller" diagrams, corresponding to shifting each point by 1 (in a chosen direction); the composition does this twice so it's a 2-point shift.

    So by extension an n-point shift is a composition of n 1-point shifts. Likewise shifting two points is a composition (tensor product) of shifting one point.
    So you can represent this axial symmetry of a cylinder by rotating a boundary around the axis, and the planar diagrams give you an algebraic representation.

    What about a rotation that takes a boundary through the axis? rotating say, the upper loop about an "x axis" will make the boundary intersect itself. But what does the minimal surface do and can this rotation be mapped to a planar diagram?
     
  15. TheFrogger Banned Valued Senior Member

    Messages:
    2,175
    A billion (to me) is twelve-zeros.

    10 (ten)
    100 (a hundred)
    1000 (a thousand)
    10,000 (ten-thousand)
    100,000 (a hundred-thousand)
    1,000,000 (a million (a thousand-thousands))
    1,000,000,000,000 (a billion)

    American's use nine-zeros for a billion, but to me that's a thousand million.

    10,000,000,000 (ten-thousand-million)
    100,000,000,000 (a hundred-thousand-million)
     
  16. arfa brane call me arf Valued Senior Member

    Messages:
    7,832
    You can define an hyperboloid of revolution algebraically, as a composition of 1-point shifts over n points.

    This diagram represents part of the continuous surface as a set of intersecting lines, in fact each intersection is a saddle point.


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    But if you remove all the intersections by removing half the lines, the result is a cylinder twisted by an angle (for really large n). This has a planar diagram as a shift by k points between a top row and a bottom row, of n points.
    The shift operators look like this:

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    Last edited: Jan 2, 2019
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  17. arfa brane call me arf Valued Senior Member

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    7,832
    One caveat with the above is that the lines can't be stretched, and have to remain straight. So you make the line length an invariant; if each is part of a minimal 2-surface then it's a minimal 1-surface and so length is fixed by the distance between the two boundaries.
     
  18. Write4U Valued Senior Member

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    20,069
    Question (which may reveal my ignorance): regarding fixed lengths lines.
    In theory is it possible that Einstein's "man in the elevator" where a light beam may curve and cover a greater distance than a straight line, yet reach the target at the same time. i.e. @ SOL, all distances become variable?
     
  19. arfa brane call me arf Valued Senior Member

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    7,832
    In SR, distances are stretched because of Lorentz invariance, roughly. I recall a post by rpenner about Minkowski diagrams and the stretching of straight lines, it might be in the archives.

    But that's the frame of relative velocities (all relative to the speed of light, or the velocity of light in one dimension), not accelerations which bend straight lines, so to speak. The bending of a light beam inside the elevator as it propagates from one side to the other, is a local effect.
     
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  20. arfa brane call me arf Valued Senior Member

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    7,832
    So back on that topic: stretching and bending of lines.

    In the above animation, the lines do stretch (linearly!), because both boundaries are kept at a fixed distance. Change this so the upper boundary moves vertically, such that all the lines have a constant length. Which you say more formally as, the rotation as a transformation leaves lengths (bounded by pairs of points) along the surface invariant.

    Bending the lines is already excluded because each line is a minimal surface = a straight line in the plane. So the animation would become rotating lines (with a symmetry) passing through each other with the upper boundary oscillating up and down.

    But there's even more (it literally falls off the back of the truck), as the surface is 'twisted', it contracts around a centre, so all the lines intersect (but of course are left invariant); at the point in time they intersect, the surface almost pinches itself apart into an hyperboloid of two sheets.

    If you pulled the boundaries apart when this happens, that's what you (maybe) transform the surface into. Or you can think about the surface oscillating between a stable one-sheeted surface, and an "unstable" two-sheeted one, the unstable state is not a spontaneous break.
     
    Last edited: Jan 2, 2019
  21. arfa brane call me arf Valued Senior Member

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    7,832
    But, you can define a two sheeted surface in terms of a rank-0 diagram in TLn, for n points on each boundary. TLn is the free monoid, and if you parametrise it with what's called the loop representation, each closed loop in a composed diagram is given the same value, usually a complex value. So TLn(x) is the algebra that uses TLn as a basis.

    The thing about the rank-0 diagrams is that they are a vector space on their own. Indeed, in TL4 there are four of these, isomorphic to a vector algebra over the following matrix basis:

    \( \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix},\; \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix},\; \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix},\; \begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix} \),

    \( \begin{pmatrix} a & b \\ c & d \end{pmatrix}\; =\; a\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\;+\; b\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}\;+\;c\begin{pmatrix} 0 & 0 \\ 1 & 0 \end{pmatrix}\;+\;d\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} \).
     
    Last edited: Jan 2, 2019
  22. Write4U Valued Senior Member

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    20,069
    Four superposed states? Qubits? Just trying to find common denominators...

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    Last edited: Jan 2, 2019
  23. arfa brane call me arf Valued Senior Member

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    7,832
    No it's "just an algebra". You can first of all show that the matrix basis, under 'composition' is a semigroup. The identity matrix doesn't exist, unless you also allow matrix addition, then for instance, when a = d = 1, and b = c = 0, the sum shown above is the identity matrix. So again, the algebra is a semiring.

    Allow a,b,c,d to be complex and a lot more mathematics descends onto the stage.
     

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