Where do numbers come from?

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

  1. TheFrogger Banned Valued Senior Member

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    Write4u. I believe because songs contain, "verses" does not mean, "verse" means song. I believe it is more related to, "versus," or, "version."

    It could be un-iverse (un-inverse?)
     
    Last edited: Jan 6, 2019
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  3. Write4U Valued Senior Member

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    No, "verse" comes from "vertere"

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    https://www.merriam-webster.com/dictionary/verse

    If the bible can use the term, science can use the term.........

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  5. arfa brane call me arf Valued Senior Member

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    Two rocks. You have to hold one steady, with one hand, and strike the other with . . . the other rock in the other hand.
    If that is, you want to make a more useful rock.

    Two different things, in this case different because one is "fixed", the other is "moved". We seem to have a kind of fundamental requirement, in almost any kind of logic, to be able to distinguish between two things--two vectors, two colors, two numbers, and some operation on two things.

    It seems to me that there aren't many meaningful questions about less than two different things, or questions about how different two things are, such as two rocks.

    Or say, a question like, what's the difference between a vector space, and a module?
     
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  7. Write4U Valued Senior Member

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    They each have their mathematical configuration, a specific "pattern" or "geometry", which can be described with the language of mathematics.

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    https://www.merriam-webster.com/dictionary/module
     
  8. arfa brane call me arf Valued Senior Member

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    --https://en.wikipedia.org/wiki/Module_(mathematics)

    Ok, what's the difference between a defect in an algebra and a physical defect in say, a crystal lattice?
    Given that you use the algebra to describe the physics?
     
  9. DaveC426913 Valued Senior Member

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    What is "an algebra"?
     
  10. Write4U Valued Senior Member

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    I think I understand what you're asking.
    Let me put it this way; the algebra is the abstract concept of equation. In this respect, the concept of a mathematical type accounting of physical values (potentials), and orderly functions (expressions) in nature as specific recurring emergent patterns, including crystal lattices all the way down......

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    The word "pattern" is the given name to any repeating "regularity", which by its very abstract nature implies a form of mathematical regularity.

    The subjective recognition of patterns does not just go back to the dawn of man, but all the way back to single celled organisms, such as the "slime-mold". Patterns (mathematical regularities) guide our lives.
     
    Last edited: Jan 9, 2019
  11. arfa brane call me arf Valued Senior Member

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    A crystal lattice has long-range order. A defect is a region in the lattice where the order breaks down (the lattice symmetry is broken).

    In semiconductor technology, defects in a lattice are introduced, atoms or ions are implanted. Doping otherwise non-conducting lattices with defect atoms is why semiconductors conduct electrons.

    One visualisation I have is of two separated lattices, growing toward each other but remaining symmetric, then they meet up and the symmetry breaks because they are not aligned perfectly at the join. So there's a defect region between two symmetric regions in the finished lattice.
     
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  12. Write4U Valued Senior Member

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    And is that objectionable in a probabilistic world? Instead of a flaw, it seems to have given crystal growth a certain advantage in the formation of independent disparate fractal patterns.

    Is that not the fundamental concept of evolutionary growth and natural selection of fitness?
     
  13. arfa brane call me arf Valued Senior Member

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    I'm not too sure about crystals having a growth advantage. But certainly the types of controlled growth of pure and impure (i.e. doped) silicon and silicon dioxide is advantageous, though not really for the crystals.

    Crystallisation, like in a test tube say, is a spontaneous change in phase, right? It involves a measurable flow of heat energy.
     
  14. arfa brane call me arf Valued Senior Member

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    But mathematically I'm interested in the connections between planar diagram monoids and symmetric groups, in particular the symmetric group on four letters, or \( S_4 \).

    I have a tentative model of a certain permutation group which has \( S_4 \) as a boundary, and \( S_8 \) as another boundary.
    In between these the boundary has a broken symmetry. It's because of a restriction, but otherwise the whole structure (the graph of the permutation group) is a tree graph, it "grows" symmetrically up to where the \( S_4 \) symmetry breaks spontaneously because of a restriction.

    I'm interested in seeing the restriction as a defect in an algebraic model. A virtual permutation group, if you will, in the sense of a virtual machine.
     
  15. Write4U Valued Senior Member

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    Ah, I see. You are talking about a refinement in the human mathematics to account for an apparent anomaly in nature. Well, sure, if we do not quite understand all potentials of a specific growth pattern, it is reasonable to assume that the human perspective is incorrect or incomplete. The same thing happened to Newton, no?
    As Einstein said; "God (the universe) doesn't play dice". If it appears it does, the observation is flawed.

    Seems there are a lot of modules.
    https://en.wikipedia.org/wiki/Module_(mathematics)#Submodules_and_homomorphisms
     
    Last edited: Jan 9, 2019
  16. arfa brane call me arf Valued Senior Member

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    It isn't surprising that modules are common; vector spaces are everywhere and modules are generalised vector spaces.

    You get to avoid the word "space"

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    (mathematics still works with this economy of terms).
     
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  17. arfa brane call me arf Valued Senior Member

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  18. arfa brane call me arf Valued Senior Member

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    My initial response to this was to ignore it because it's obvious what an algebra is.
    Formal algebra is symbolic, but that's just convenient for us, why not use symbols?

    There are many everyday examples, but for instance, communication can have several algebras defined in that space. When you buy something from a store, maybe get change, maybe use a card, whatever, . . ., algebra.
     
  19. DaveC426913 Valued Senior Member

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    I know what algebra is; I wasn't sure if you meant something different by "an" algebra. As far as I am aware, that is not the correct use of the word. It's like saying "I'll pay for my lunch with a cash".

    I suspect what you're looking for is "an algebraic formula".
     
  20. QuarkHead Remedial Math Student Valued Senior Member

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    Actually, on this particular point, arfa is correct - there are many different algebras - Boolean, Clifford, Grassmann, Lie.........

    ........each with it's own domain of applicability.
     
  21. arfa brane call me arf Valued Senior Member

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    Algebra is named after a mathematician who was called Al-Jabr.

    That aside, and given there is a history to consult, my understanding of "what is it", is vaguely: a space with a set of elements in it that also has "addition" and "multiplication" defined on pairs of elements from the set.
    The elements of the set need not be numeric, the operations just have to look like addition, type of thing, and like multiplication as a form of repeated addition (or just a scaling, multiplication in vector spaces is "scalar").


    In high school, algebra is initially about polynomials, simultaneous equations, etc. The set of elements is the reals or the integers.

    What I've posted, say the images of the trefoil knot, it's an algebraic space because you can easily see a pair of linked tori has a pair of common boundaries, both circles which are "minimal" and have a set of lines intersecting them.
    Knot algebra is topological . . .
     
    Last edited: Jan 10, 2019
  22. arfa brane call me arf Valued Senior Member

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    I suppose I should add that music is obviously algebraic, but how easily can addition and multiplication be defined?
    On my Roland I can have 64 simultaneous notes (a sum or a product of notes?), it has 64-note polyphony.

    On a guitar you don't usually get that amount of freedom of, ah, movement.
    So I might just assert that I can choose to hear one of the 64 possible held notes (I can keep them in the play buffer by standing on a foot pedal), so it's an operation of disjoint union, although the result might be notes interfering in a way that doesn't sound harmonic or melodic (more like rubbish).

    Scale a tone by hitting a key harder (with more force, that is), if your keyboard can respond to this action, the note is amplified. This is a clash of terminology because it's a scalar multiplication, a music scale is, well, an algebraic structure, a word of some kind, abstractly speaking.

    You can scale everything by increasing the output volume (not usually included with guitars or pianos). Amplification is the scalar multiplication of combinations of musical notes, musical because there are scales of notes defined by . . . an algebra.
     
    Last edited: Jan 10, 2019
  23. DaveC426913 Valued Senior Member

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    Like there are many different cashes?

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    OK, if arfabrane meant it like "a defect in a whole type or class of algebra" that would make sense. But I'm not sure that's what he meant.

    Anyway, no need to derail with nitpicks. Carry on.
     

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