Does mathematics really exist in nature or is math just a human construct?

Discussion in 'Physics & Math' started by pluto2, Dec 2, 2015.

  1. James R Just this guy, you know? Staff Member

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    Moderator note: Waiter_2001 has been warned (again) for posting nonsense to the Science subforums. Due to accumulated warning points, he has been automatically temporarily banned. He is also on the path to being excluded from posting in the Science sections of sciforums.
     
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  3. exchemist Valued Senior Member

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    Yes, we do see it differently. I see things much as Dave does, apparently. To me, the order arises from the forces at work in nature. We model these using human mathematics, generally by making simplifications and approximations to do so.

    Your picture is of a single sheet of the graphite structure of carbon, known as "graphene". The shape arises from "sp2 hybridisation" of the electron orbitals in the valence shell of carbon atoms. sp2 involves three 2-electron bonds, at angles of 120 degrees, with the remaining electron in a p orbital, perpendicular to the sheet. It is very common in carbon chemistry.

    sp2 hybridisation is one of a number of electron configurations for the carbon atom which are of low energy because they minimise the electrostatic repulsion between pairs of electrons. Because low energy makes a chemical compound less reactive, stable compounds tend to employ such electron configurations.

    The resulting elegant geometry arises from this stability. In other words, it arises from minimising electrostatic forces; both between electrons and the nuclei of the atoms, and between electrons and other electrons. We can model this in molecular quantum mechanics, so it may look as if it has a mathematical basis. However, it is not possible to solve Schroedinger's equation exactly for any system more complex than the hydrogen molecule ion (which has 2 nuclei and only one electron) due to the intrinsic insolubility of the interactions between more than 3 bodies in motion. Furthermore, we can only use Schroedinger's equation at all in chemistry if we make what is called the "Born-Oppenheimer Approximation". This assumes that nuclei move so slowly compared to electrons that we can take them as static when working out the wave function of the electrons. This allows us to define the quantum states the electrons can occupy, separately from the various quantum states involving motion of the nuclei (vibration, rotation , translation). It is a pretty good approximation in that it gives good results in almost all cases ( though there are a few scenarios in which the Born-Oppenheimer Approximation breaks down).

    Why am I telling you all this? It is to point out that the mathematical models we have are not exact. What is happening in nature seems to be due to the interactions of certain forces, but the the resulting behaviour actually does not fit, exactly, any mathematics that we have. I sometimes think that we chemists are more aware of the limitations of mathematical models than our physicist colleagues, due to the untidy complexity of the multi-electron atoms, and the multi-atom molecules, that we study.

    Anyway perhaps, from the foregoing, you can see why a chemist may be a bit sceptical about grandiose claims that nature either is, or conducts, mathematical operations.
     
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  5. pluto2 Registered Senior Member

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    In my opinion humans are too primitive to answer questions of this kind.

    These kind of questions are on the same difficulty as a question like: can something come from nothing?

    Our brains are just too primitive to ever be able to answer such questions.

    I think that some things will forever be beyond the grasping ability of humanity and then again that is because we are just too primitive.

    For gods sake we can't even master interstellar travel and we can't even build a Dyson sphere, for example.

    Our society needs to be much more advanced technologically and socially than it is now to be able to answer this kind of questions but for that to even happen humans need to stop behaving so primitively.
     
    Last edited: Jan 20, 2016
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  7. Write4U Valued Senior Member

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    Actually no, you only need Scotch tape to arrive at a single atomic layer of graphene. A child can do it.
     
  8. Write4U Valued Senior Member

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    Tank you exchemist for clarifying the "variables" when working at that small level. It sounds reasonable that human mathematics are mostly an *approximation*, although I believe that we have identified and symbolized a few universal constant mathematical functions. But I also realize there are definitely pockets of chaos, where mathematics are overwhelmed by conflicting forces..
    However, IMO, that does not negate that the universe functions in some sort of mathematical way (by any means). I just cannot see how it could be otherwise. There has to be some essential ordering processes associated with the laws of the universal geometry. And I cannot think of a better model.
     
  9. exchemist Valued Senior Member

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    To be accurate these are not "variables". That is not the correct mathematical term. These are, rather, indeterminacies - things we cannot, even in principle, work out exactly. And by the way, I have not even mentioned Heisenberg's Principle of Indeterminacy, which is the most famous example of all. But yes indeed, there are "pockets of chaos", if you like.

    However I think it is not right to say these correspond to realms where mathematics is "overwhelmed by conflicting forces". Chaos, or rather randomness, seems to be an intrinsic feature of the world, but there is also order, brought about by the operation of the 4 fundamental forces. What I find so wonderful about QM and Statistical Thermodynamics is they are disciplines that show, between them, how large scale order arises, perfectly naturally, from randomness at the atomic level, due - directly or indirectly - to the order imposed on randomness by these forces (and mass, I suppose).

    To me (and I confess it had never quite occurred to me before in such terms, so thank you for the idea), that is the "conflict", if you like. Intrinsic randomness of nature on the one hand, versus the 4 forces of nature acting on mass. The result is something we can usually model with maths, so long as we are prepared to neglect some effects in order to to simplify the system enough to do it.
     
  10. PhysBang Valued Senior Member

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    And how does that single layer look?

    You can't appeal to an idealized model of something and then say say, "See how perfect the real thing is!" That's just wrong.
     
  11. DaveC426913 Valued Senior Member

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    And perhaps that is the crux of the matter. Mathematics is an idealized description of the forces and actions in nature.
     
  12. exchemist Valued Senior Member

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    Erratum: just realised I said something not quite right in 2nd sentence of 4th para. I should have said "In other words, it arises from minimising the energy due to electrostatic forces; both between electrons and the nuclei of the atoms, and between electrons and other electrons."

    Minimising the energy of the system in fact involves minimising the repulsive forces (electron-electron) and maximising the attractive forces (electron - nucleus).

    Didn't spot this until after the edit period had expired - and I don't want to risk the wrath of some of the physicists on this forum

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    .
     
  13. rpenner Fully Wired Staff Member

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    Yeah, well, that explanation works acceptably for the hydrogen atom but gets messy for multi-electron atoms and molecules. While Hamiltonian mechanics and Lagrangian mechanics are both about finding the equations of motion of particles, the underlying principle is about finding extremal values of the action of trajectories of the system which is a different beast than minimum energy but generalizes to quantum mechanics quite well.

    Simply minimizing the potential energy in neglect of other aspects of physics leads to a catastrophe as classically both electrostatic and gravitational potential wells are infinitely deep. You need the Heisenberg uncertainty principle to have atoms of non-zero volume and the Pauli exclusion principle for chemistry as we know it.

    https://en.wikipedia.org/wiki/Principle_of_least_action
    https://en.wikipedia.org/wiki/Lagra...Lagrange_equations_and_Hamilton.27s_principle
     
  14. exchemist Valued Senior Member

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    Thanks, fair enough. I should really have mentioned the kinetic and potential energies.....I am aware of the principle of least action but that would take a page to explain in itself. And I deliberately did not want to go into the wavelike nature of matter and hence into HUP, ground states that do not end in collapse, and all the rest. I wanted to keep this just about comprehensible for non-expert readers, even if there are a few shortcuts here and there. Risky for a chemist posting here with blokes like you around.......

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  15. Write4U Valued Senior Member

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    A very light gray. And that illustration is not an *idealized* model, it is the real configuration of Graphene at single layers of atoms. This is why today AI is very interested in Graphene, because it an extremely efficient transmission medium.

    No, this is the real thing. Just think of writing with a pencil.
    http://www.msn.com/en-us/news/techn...tance-ever-discovered/ar-BBh7yYQ?ocid=U220DHP

    and a demonstration of how to use simpe tape can be found here;
    (start at 18:15 for graphene)!!!!!!! Watch at least that part.
     
    Last edited: Jan 21, 2016
  16. river Valued Senior Member

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  17. exchemist Valued Senior Member

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    Acc. the video, you need a silicon chip as well as tape, but indeed that seems to be all. However, what PhysBang may be getting at is what this video does not say about "defects".

    In most real crystalline materials composed of so- called "giant" structures (as opposed to crystals formed by simple stacks of molecules, weakly bound by Van der Waals forces), you do not get an indefinitely large expanse of perfect structure. Impurities and physical dislocations introduce discontinuities in the structure in certain places. These are defects in the structure. The presence of a few of these may not matter for the electrical properties this guy on the video is talking about. However, once again, you need to be aware that the intrinsic, untidy randomness of nature raises its head, even in lovely structures like this.

    Your picture was a graphic, not an electron micrograph of graphene, I think, so it did not show any of these imperfections.
     
  18. iceaura Valued Senior Member

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    My own take is to view math applied to the physical world as the construction and operation of jury-rigged sensory organs. It resembles other sensory apparatus in that it 1) translates certain differences in the perceived into differences within the workings of the construction 2) integrates this record or arrangement of differences with the rest of the brain's workings, available thereby to consciousness.

    In this take, asking if it's something we made or something found in nature is like asking whether a color is constructed or discovered, a sound, a touch.

    So the color red, say, is created within the human brain, even so it is not a human creation only but a way of perceiving certain wavelengths of light. Wavelengths we cannot see we perceive via mathematical structures, wherein we organize certain differences in the one realm into patterns of differences we can handle in the other.
     
  19. PhysBang Valued Senior Member

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    Are you high? Seriously?

    Your picture was not a picture of actual graphene, it was a computer generated model given some pretty idealized circumstances.

    Please lie to us all you want, but don't lie to yourself.
    The image on that link is an artists illustration! Again, you seem to be confusing an idealization of graphene with the actual substance and using that idealization as evidence that the substance is ideal.

    and a demonstration of how to use simpe tape can be found here;
    youtube: gjx5y9lrtwU (start at 18:15 for graphene)!!!!!!! Watch at least that part.[/QUOTE]
    Still no picture of graphene at the molecular level.

    Your argument for the immense precision of this element relies on an artist's rendering.
     
  20. Write4U Valued Senior Member

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    Well, here is an actual photograph of Graphene.

    Ok

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    Scanning probe microscopy image of Graphene

    Graphene can self-repair holes in its sheets, when exposed to molecules containing carbon, such as hydrocarbons. Bombarded with pure carbon atoms, the atoms perfectly align into hexagons, completely filling the holes.
    https://en.wikipedia.org/wiki/Graphene#Structure
     
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  21. Write4U Valued Senior Member

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    We may not be able to see wavelengths directly, but we CAN measure them. All the different *names* we have assigned to wavelengths are man-made, but does that mean wavelengths exist only because we identified and named them?
     
    Last edited: Jan 21, 2016
  22. sweetpea Registered Senior Member

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    That happens most days on this science forum. A crank starts a thread which waffles on for lots of pages. Some cranks taking the number of pages to mean their waffling shows their right in their opinion. Something from nothing. It's especially found at Sciforums com
     
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  23. Write4U Valued Senior Member

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    Is this related to the theory of "necessity and sufficiency"?
    And I found this interesting in context of the OP.
     
    Last edited: Jan 21, 2016

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