incompleteness of a result, in terms of say proving all of its particular moving parts goes with the territory as what Kurt Goedel was trying to present. I.e. in the case of boundary value problems of physics, there are certain situations in which one has multiple solutions to a set of equations
quote
I’ll assume for now that by “answer on a math problem” you mean the solution to an equation or a set of equations. In this case, there are certainly instances where there are multiple solutions to a set of equations. For example, if I asked you to find the solutions of x2−4=0
, you would say x
could be either 2
or −2
, and both of these would be correct if I put no more restrictions on the problem. However, even with this problem there is a single, well-defined solution set. In this case, the solution set of x2−4=0
is given by {2,−2}.
For a similar example from calculus, consider finding the anti-derivatives of f(x)=x.
The solution set is any function of the form x2+C
where C
can be an arbitrary constant. Although this shows that infinitely many solutions to the problem exist, there is still a single solution set.
There are also many examples of problems which have no solutions. For example, find all real numbers, x,
such that x+5=3
and x+5=4
. The first equation shows us that x=−2
, and the second that x=−1
. Since these are not equal to one another, this set of equations has no solution. However, it still has a single, well-defined solution set. In this case, the solution set is the empty set, which is typically denoted {}
or ∅
.
For a problem to be considered “well-posed” there should be a single solution set, but this set could contain many solutions, a single solution, or no solution at all. Many algebra or calculus textbook examples are constructed so that there is exactly one, or only a few answers. However, with more interesting problems it’s often a challenge just to prove that a single solution exists, even if you can’t explicitly write that solution down.
In fact, many of the biggest unsolved problems in mathematics come down to questions about the existence of solutions to certain equations. For example, the Riemann hypothesis can be phrased as a question about whether or not any zeros of a (complicated) function exist which are away from a certain line in the plane. The Navier-Stokes problem can be phrased as a question about the existence (or non-existence) of badly behaved solutions to a set of partial differential equations. P vs NP can similarly be written as an existence question. In all cases, a single solution set should (hopefully) exist, but whether it is empty, filled by one element, or filled by many elements is a very difficult question. It just so happens that solving any of these problems can win someone a one million dollar prize.
end of quote
In a word there are many times in which evolution equations have MULTIPLE solutions but due to physics rules, assumed, we only take a few of them . I .e. I saw a liquid Helium Laboratory where only a few of the known Navier Stokes equations were implimented
We simply do not have the time in a lot of cases to generate ALL known solutions, and if several equally plausible solutions exist, we will focus on ONE particular solution, and that is the best we can do