In physics and applied mathematics, most problems can't be solved exactly i.e. we can't find a closedform expression for the solution. An example of an equation that can be solved exactly is a quadratic equation in x like
.
An example that can't be solved exactly is
.
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
contains the same information as its 'solution',
.
'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
?
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!
Reformulating physics so more (or all) systems are 'solvable'
Announcements  Posted on  

TSR's new app is coming! Sign up here to try it first >>  17102016 

 Follow
 1
 14042016 13:03

 Follow
 2
 15042016 00:08
This isn't philosophy, it's mathematics: to say that there is no closed form solution for 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. 
 Follow
 3
 15042016 10:27
(Original post by BlueSam3)
This isn't philosophy, it's mathematics: to say that there is no closed form solution for 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.
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)!Last edited by Implication; 15042016 at 10:42. 
 Follow
 4
 15042016 13:55
Thinking about it, perhaps whether or not a solution can be expressed 'analytically' or in 'closedform' 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...

 Follow
 5
 15042016 14:24
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.Last edited by BankOfPigs; 15042016 at 14:28.Post rating:1 
 Follow
 6
 15042016 17:34
(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.
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 nonintegrable, 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? 
 Follow
 7
 15042016 19:37
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.Last edited by BankOfPigs; 15042016 at 20:01. 
 Follow
 8
 15042016 22:51
(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.
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. 
 Follow
 9
 16042016 01:10
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.Last edited by BankOfPigs; 16042016 at 01:13. 
 Follow
 10
 16042016 22:41
(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. 
 Follow
 11
 22042016 10:36
(Original post by BlueSam3)
Both of these are false: Many nonsolvable things integrable, and many integrable things don't behave chaotically (take, for example, the zero function).
Write a reply…
Reply
Submit reply
Register
Thanks for posting! You just need to create an account in order to submit the post Already a member? Sign in
Oops, something wasn't right
please check the following:
Sign in
Not got an account? Sign up now
Updated: April 22, 2016
Share this discussion:
Tweet
TSR Support Team
We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.
This forum is supported by: