QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: Traden4AlphaQuoteOriginally posted by: CuchulainnHypothesis: Nonlinear problems have no closed solution. True/false?What say you to the good old quadratic formula? Or do you have a restricted definition of "problem"?You have a special/restricted example of "nonlinear'. To prove something true it must be true for all problems. One swallow makes a summer not.Sometimes you get lucky, e.g. reducing Bernoulli ODE to a linear ODE. This is called serendipity.1) Linear and quadratic polynomial equations are mildly nonlinear. We got lucky. Cardano and Leonardo spent years trying to solve cubics. Besides, you get complex roots but we only want real roots..2) Non-low-order polynomial equations have no closed solution. e.g. solve factorial(x) = x! = y for given y.3) Find closed solution for 1-factor early exercise option problem.Ok, I misinterpreted the quantifier. You are saying "there exist non-linear problems for which no closed form solutions exist." My first reaction is that you are right and it goes beyond examples of problems where we are not sure if a closed form solution might exist if only we were lucky or clever enough to find it.A good example of the non-existence of a closed form solution can be found in certain cellular automata (a nonlinear discrete-valued differential equation on a grid) which can be shown to be capable of universal computation (i.e., one can configure the automata such that it simulates the logic gates of a computer and the system can do any computation based on a program encoded in the initial conditions or coefficients in the system). That implies that at least some non-linear problems map to the halting problem. And I bet there are ways to tie non-linear problems to Godelian undecidability, too. If the system's outcome is undecidable, how can it have a closed form solution?Then there's Stephen Wolfram's contention that there are some systems in which the system itself is the most compact representation of the system and the fastest solution process is to turn the crank on the system rather than look for some magic equation that expresses the end state.But my second reaction is to question what constitutes a valid closed-form solution. It seems to be a compact representation of the solution -- a short string of function symbols that define the values or set of values of the unknowns that solve the original equation. But what are the constraints on the set of function symbols that we might use to create a closed form solution? Is it limited to arithmetic operations? Can we also use square roots, logs, exponentials, and trigonometric functions? What about hyperbolic functions or maybe Bessel functions? I would hypothesize that the history of closed form solutions has been one of creating new symbols and much as it is one of finding ways to write a solution with the set of known function symbols. And yet those symbols encode increasingly complex sets of underlying numerical operations.If one restricted every closed form solutions to the four arithmetic functions(-,+,*,/), then one might be inclined to conclude that almost no nonlinear problems have closed form solutions (e.g., can one write a nice closed form solution for the quadratic formula without using square root or any other function beyond simple arithmetic?). But if one is allowed to construct new symbols (square root, ln, sin, sinh, Bessel, etc.), then perhaps every nonlinear problem has a closed form solution if we could just invent the right function symbol that encodes key elements of that solution. There might even be a generalized halting problem function that lets one write the solution to a universal computation cellular automata in more compact symbol.Thus, if the definition of a closed form solution is not closed, the question of existence such solutions may be open.