The explanation is no explanation at all. It is true that water has inertia, and it is true that water has to go somewhere. It is not true that this is sufficient to explain why the water goes up.
In particular, the statement "because of inertia and the fact that water is not compresible, it has to keep moving" is false.
The fact that water is not compressible means that the water moving in one direction is capable of exerting an extremely large force on the water moving in the other direction when they run into each other, which would bring the water to a halt as the hole closes. The interesting question is: why doesn't this happen?
One possibility is that the free surface at the bottom of the hole travels upward due to what amounts to a buoyancy force (think of the hole as a bubble with no top). The water pressure at the bottom of the hole is the same as it is everywhere at that depth, and so the bottom move upwards, and this pressure force creates upward momentum that is not met by any opposing force when the bottom of the hole reaches the water's surface, and therefore creates an upward-moving column of water.
One way of testing this would be by inserting an empty pipe into the water against a flat disk of the same diameter, and then rapidly pulling the pipe out. The flat disk would impede water from moving upward into the hole, which would--on my account--dramatically reduce the size of the spike.
As to the "unreasonable effectiveness of mathematics", the argument is bogus: http://www.tjradcliffe.com/?p=381 Our mathematical descriptions of reality almost all admit of extraneous solutions that have to be discarded based on physical--not mathematical--considerations. This is what you would expect if mathematics was a human-made tool for describing reality. It is not at all what you would expect if mathematics was somehow the pre-existing armature upon which physical reality was constructed.
I think that you're responding to an extreme interpretation of the "unreasonable effectiveness of mathematics". I don't think you have to believe that nature follows perfectly simple mathematical rules to be surprised to the extent that mathematics is effective at describing physical reality.
You make a good point, that mathematics is often designed and invented to describe reality. You need to take into account the fact that a lot of what you "get out" of mathematics has already been "put in". But it is my opinion that even accounting for this, it is quite frequent that you "get out" more than you "put in" to mathematics.
I think a strong set of examples of this sort of behavior is when attempts to address one problem in a mathematical theory often end up addressing other seemingly unrelated problem. The introduction of quantized energy states to explain the blackbody radiation distribution then goes on to explain the photoelectric effect. (I'm not too familiar with what physical phenomena were observed before QM but needed QM to explain them to give more examples).
Now you might say "well of course this theory solves both issues, because they are essentially the same phenomenon, and so there's nothing surprising about your theory explaining what it was designed to explain". But that's precisely the point, that mathematics gives us the tools to see relations between things that on the face of them are different. And that there's no reason that this /has/ to be true (without appealing to mathematics), makes the fact that it is true seem surprising.
Of course, being surprising, or unreasonably effective, etc is a relative statement. Certainly to someone who expected the universe to be described mathematically to precisely the degree that it is in reality, nothing would be surprising. But I think to most people who haven't been introduced to mathematics (kids), the power it gives you certainly is surprising.
The fact that water is not compressible means that the water moving in one direction is capable of exerting an extremely large force on the water moving in the other direction when they run into each other, which would bring the water to a halt as the hole closes. The interesting question is: why doesn't this happen?
Why would you expect that to happen? The hole in the water is an area of low pressure that the water is going to rush in to fill. Now that water has a lot of kinetic energy and momentum. Why would it suddenly lose that momentum? And how does water's incompressible nature help that? Surely, if anything, this contributes to the difference in resistance between the water and the air and help explain the spike that you initially see.
In particular, the statement "because of inertia and the fact that water is not compresible, it has to keep moving" is false.
The fact that water is not compressible means that the water moving in one direction is capable of exerting an extremely large force on the water moving in the other direction when they run into each other, which would bring the water to a halt as the hole closes. The interesting question is: why doesn't this happen?
One possibility is that the free surface at the bottom of the hole travels upward due to what amounts to a buoyancy force (think of the hole as a bubble with no top). The water pressure at the bottom of the hole is the same as it is everywhere at that depth, and so the bottom move upwards, and this pressure force creates upward momentum that is not met by any opposing force when the bottom of the hole reaches the water's surface, and therefore creates an upward-moving column of water.
One way of testing this would be by inserting an empty pipe into the water against a flat disk of the same diameter, and then rapidly pulling the pipe out. The flat disk would impede water from moving upward into the hole, which would--on my account--dramatically reduce the size of the spike.
As to the "unreasonable effectiveness of mathematics", the argument is bogus: http://www.tjradcliffe.com/?p=381 Our mathematical descriptions of reality almost all admit of extraneous solutions that have to be discarded based on physical--not mathematical--considerations. This is what you would expect if mathematics was a human-made tool for describing reality. It is not at all what you would expect if mathematics was somehow the pre-existing armature upon which physical reality was constructed.