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Reformulating physics so more (or all) systems are 'solvable' Watch

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    In physics and applied mathematics, most problems can't be solved exactly i.e. we can't find a closed-form expression for the solution. An example of an equation that can be solved exactly is a quadratic equation in x like

    ax^2+bx+c=0.

    An example that can't be solved exactly is

    x=e^{-x}.

    As somebody (probably famous, I can't remember who) said, 'In Newtonian mechanics, the 3 body problem can't be solved exactly; in quantum theory, the 2 body problem can't be solved exactly; in general relativity, the 1 body problem can't be solved exactly'. In school we don't really notice this since the teachers, exam board etc. hide it by mostly giving us problems that can be solved exactly, but when we try to model real life, virtually all problems are impossible to solve this way.

    It seems to me like we treat this inability to solve problems exactly as something fundamental about real life, but I don't see why this has to be the case. Since the rise of computers, this hasn't been such a problem, so perhaps that's why - people just don't care. For a lot of problems that can't be solved 'exactly', we can still use computers to solve them to an arbitrarily high degree of precision- which is usually good enough. But this doesn't work for all problems, and can often be really fiddly.

    Is there some reason to suppose it is impossible to reformulate physics such that more (or even all) problems can be solved exactly? I can't see one. When we rearrange an equation to get an 'exact' formula for the solution, we aren't really gaining any new information. We are simply expressing the same information in a more convenient way. For example, the quadratic equation

    ax^2+bx+c=0

    contains the same information as its 'solution',

    x=\dfrac{-b \pm \sqrt{b^2-4ac}}{2a}.

    'Deriving' one from the other really is just writing down a tautology. So why shouldn't we be able to do the same for systems modelled by equations like

    x=e^{-x}?

    I have heard that being unable to solve a problem exactly is often related to symmetry and systems for which this is impossible are often asymmetrical and chaotic. But why shouldn't we be able to capture this asymmetry and chaos using some form of closed expression? I'm not suggesting that I'd personally be capable of constructing such a formalism - or even that it would be possible for an individual in a single lifetime - I'm just curious and want to know why this shouldn't be possible.

    I haven't posted this in the Physics forum because it's not study help, and it is philosophy. I guess I'll move it back to Physics if I don't get much response here!
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    This isn't philosophy, it's mathematics: to say that there is no closed form solution for x = e^{-x} is to say that there is no possible expression, at all, in terms of sensible functions, similar to the one that you give to the quadratic equation. It is literally impossible.


    Even where that isn't the case, the solutions get fundamentally awful and complicated ludicrously quickly: try looking up the general solution to a quartic.
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    (Original post by BlueSam3)
    This isn't philosophy, it's mathematics: to say that there is no closed form solution for x = e^{-x} is to say that there is no possible expression, at all, in terms of sensible functions, similar to the one that you give to the quadratic equation. It is literally impossible.


    Even where that isn't the case, the solutions get fundamentally awful and complicated ludicrously quickly: try looking up the general solution to a quartic.
    I understand what it means for there to be no closed form solution. I'm asking if there is a way to reformulate the physics and/or mathematics so that the same problem expressed in such a formalism would have a closed form solution. Or rather, I'm asking if there is any good reason to think such a formalism is impossible. I can't see a reason a priori why it shouldn't be possible, since an equation of motion itself (with appropriate boundary/initial) data should contain identical information to any closed form solution. I'm not asking if we can find the exact solution to x=e^{-x}!

    Reformulating areas of mathematical physics is nothing new. For example, Hartry Field in 1980 published a book called Science Without Numbers, which describes fully Newtonian gravitation without referencing - or quantifying over - numbers, functions and sets. It sounds batshit crazy, but he did it. On the less extreme end, it's common to hear people say things like 'we need complex numbers for quantum mechanics'. But on its face this is simply false, and quantum theory can be (and has been) formalised without complex numbers - it's just a bit messier.

    Either way, this falls well within the realm of philosophy (of mathematics)!
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    Thinking about it, perhaps whether or not a solution can be expressed 'analytically' or in 'closed-form' is completely arbitrary. I assume it depends on what functions we count as 'elementary' so that a solution can be expressed in terms of them...
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    Yes, but I'd argue it's non intuitive, and yes this is both philosophy and mathematics.

    So for example we are unable to find the anti derivative of e^{-x^2}, at least in terms of elementary functions, however suppose we now define it so that this anti derivative is equal to a function known as Ph. Then it is possible that we could solve an equation regarding this.

    With your example about e^-x -x = 0 we could easily 'create' a new function that directly solves equations such as this. This isn't too hard to imagine; for example equations such as e^x = 5 can be solved using logarithms, we could just create a function that is similar to a logarithm to resolve these sort of questions.

    Essentially, I imagine you are looking for a 'basis' of elementary functions such that every conceivable mathematical expression can be written in terms of. This way we would be able to find a closed form for any such equation to achieve a tangible solution. Ideally, we would also expect such a basis to be finite and relatively small if we want to have any practical application, but this is not strictly necessary.

    My interpretation is that you want to create a consistent structure with this basis as the set of functions of the structure. Then you want to be able to prove that for any conceivable expression in Mathematics written in terms of this basis, their exists another member of the basis that allows for a closed form.
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    (Original post by BankOfPigs)
    Yes, but I'd argue it's non intuitive, and yes this is both philosophy and mathematics.

    So for example we are unable to find the anti derivative of e^{-x^2}, at least in terms of elementary functions, however suppose we now define it so that this anti derivative is equal to a function known as Ph. Then it is possible that we could solve an equation regarding this.

    With your example about e^-x -x = 0 we could easily 'create' a new function that directly solves equations such as this. This isn't too hard to imagine; for example equations such as e^x = 5 can be solved using logarithms, we could just create a function that is similar to a logarithm to resolve these sort of questions.

    Essentially, I imagine you are looking for a 'basis' of elementary functions such that every conceivable mathematical expression can be written in terms of. This way we would be able to find a closed form for any such equation to achieve a tangible solution. Ideally, we would also expect such a basis to be finite and relatively small if we want to have any practical application, but this is not strictly necessary.

    My interpretation is that you want to create a consistent structure with this basis as the set of functions of the structure. Then you want to be able to prove that for any conceivable expression in Mathematics written in terms of this basis, their exists another member of the basis that allows for a closed form.
    I think this is the truth I was just beginning to stumble onto myself. Although rather than being able to write any conceivable expression in mathematics, I was after any expression that could conceivably appear in physics. But of course the former would probably be more useful.

    However, this leads me to an interesting conundrum. It seems now that whether or not a solution can be written down in closed form, or solved 'analytically', is in some sense arbitrary. It depends on what basis of functions we choose. However, in the physics of nonlinear systems, integrability is very strongly tied to the phenomenon of chaos. If a system is integrable (i.e. the solutions can be found analytically), it does not exhibit chaos. If a system is non-integrable, it exhibits either mixed or purely chaotic behaviour. It certainly doesn't seem that whether or a system actually exhibits chaos - which is a real, physical property - can depend upon what basis of functions we happen to have chosen. Is there something special about the 'standard' functions that we have chosen (e.g. algebraic operators, logarithms, trigonometry, hyperbolic functions etc. etc.) that creates this link? Or is integrability not as arbitrary as it seems?
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    I am not particularly familiar with Physics to be honest. By integrability, are you defining it as such, or at least stating that there is logical equivalence to having the closed form solution that we desire?

    Or are you referring to general integrability in the form of a definite or indefinite integral?

    Because if the former is the case then indeed there is a large problem whilst if it is the latter then this perhaps is just an issue with how we define the integral function.
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    (Original post by BankOfPigs)
    I am not particularly familiar with Physics to be honest. By integrability, are you defining it as such, or at least stating that there is logical equivalence to having the closed form solution that we desire?

    Or are you referring to general integrability in the form of a definite or indefinite integral?

    Because if the former is the case then indeed there is a large problem whilst if it is the latter then this perhaps is just an issue with how we define the integral function.
    I'm not referring to integrability alone as in the integral calculus, though the two are very closely related. I took a class on nonlinear dynamics last year and an integrable system was defined to be a system of equations that can be solved analytically. It was in this class that it was also claimed that integrable systems do not exhibit chaos and non-integrable ones do. I haven't studied these Cambridge notes at all, and so the answer to my question may well be contained in there somewhere, but the introduction at least seems to roughly coincide with the definition I was given:

    Integrable systems are nonlinear differential equations which ‘in principle’ can be solved analytically.This means that the solution can be reduced to a finite number of algebraic operations and integrations.
    However, the offering School for my class was Physics rather than Mathematical Sciences, and they do tend to use a lot of hand-wavy definitions and a lack of rigour that wouldn't be okay in a mathematics class! Perhaps I'll talk to my Professor and try to him pin him down on a rigorous mathematical definition of integrability.
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    That's really interesting, but I suspect this is much out of my depth as the majority of philosophy I study is focused around mathematical logic as opposed to applied Mathematics.

    I roughly understand what you mean though. I'll be using the definition of analytically solvable as solvable using a set of known rules (such as addition, multiplication, logarithm ect)

    So we have:

    1) Integrable iff analytically solvable iff solvable using a set of known rules (tautologies presumably).

    2) Integrable implies chaotic behaviour

    And that whilst the first set is presumably arbitrary the second set is not since it an actual physical phenomena. However, surely 2) only holds given that we are using the 'standard basis' essentially of known functions. If we happened to change this basis to include a new (possibly infinite) set of function surely integrable would no longer imply chaotic behaviour? Suppose we took a non linear system of dif equations that did not have chaotic behaviour and assigned it a new function as a solution, would this not change things?

    Sorry if this is messy and being pedantic.
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    (Original post by BankOfPigs)
    That's really interesting, but I suspect this is much out of my depth as the majority of philosophy I study is focused around mathematical logic as opposed to applied Mathematics.

    I roughly understand what you mean though. I'll be using the definition of analytically solvable as solvable using a set of known rules (such as addition, multiplication, logarithm ect)

    So we have:

    1) Integrable iff analytically solvable iff solvable using a set of known rules (tautologies presumably).

    2) Integrable implies chaotic behaviour

    And that whilst the first set is presumably arbitrary the second set is not since it an actual physical phenomena. However, surely 2) only holds given that we are using the 'standard basis' essentially of known functions. If we happened to change this basis to include a new (possibly infinite) set of function surely integrable would no longer imply chaotic behaviour? Suppose we took a non linear system of dif equations that did not have chaotic behaviour and assigned it a new function as a solution, would this not change things?

    Sorry if this is messy and being pedantic.
    Both of these are false: Many non-solvable things integrable, and many integrable things don't behave chaotically (take, for example, the zero function).
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    (Original post by BlueSam3)
    Both of these are false: Many non-solvable things integrable, and many integrable things don't behave chaotically (take, for example, the zero function).
    As per the above posts, in this context 'integrability' was defined as the property of being 'solveable', so this cannot be the case. I think there was some confusion with regard to chaos, as what I actually said was that chaos \Rightarrow non-integrable. No integrable systems exhibit chaos, and all non-integrable systems exhibit mixed or chaotic behaviour.
 
 
 
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