Why don't the poles of a magnet meet in the middle? If opposite poles of magnets attract each other, then why are the poles of a magnet at opposites ends of the magnet, rather than meeting in the middle and canceling out each other?
Edit:
What got me thinking was this quote from "mystic philosopher" Walter Russel:

"We see the positive pole of a compass needle pointing toward the
  negative pole of a magnet and the negative pole of the magnet pointing
  toward the positive pole. This evidence is one of the bases of our
  conclusion. That is what our eyes see. What actually is happening is
  that they are voiding each other's unbalanced condition to seek
  balance through each other.
Opposite poles get as far away from each other as they can, until
  their opposition is voided by balance in their fulcrum and they cease
  to be.
When the positive pole of a magnet is brought into contact with the
  negative pole of another magnet, that effect which we think of as
  attraction is one of voidance. It is a cessation of opposition or
  power to manifest anything. Polarity utterly ceases at that point and
  each opposite extends to each opposite end, each getting away from the
  other and through the other, spirally, as far as it can.
If opposite poles attracted each other, they would be together in the
  middle of a magnet instead of at its ends."

 A: You must not think that a magnet is kind of "a source of Northern flux" plus "a source of Southern flux" put together. Magnetic field lines are unique, and they go from one pole to the other one.
Think of a magnet as a ring. The north pole is the source of field lines. The south pole is the drain.

As you can see, there is continuity in field lines, even in the joining section (the "middle"). Magnetic field lines are closed lines (as far as we know), so this is what happens.
You cannot divide a magnet because of this either.
A: Welcome to Physics SE, Kashmir Shiva!
Your question is an interesting one, partially because one could actually experimentally look into it.
At home I have a bar magnet which I accidentally broke in half. Now, when you break a magnet in half they form two new magnets. When you place the two broken parts together, the two parts do cancel out, and it acts as a single bar magnet.
The reason why all of this is true is that you can think of a single macroscopic magnet as being made out of loads of microscopic magnets (or magnetic dipoles). These dipoles affect the magnetic response within the material, whilst they also generate an external field, when they reach the edge and don't counter-act each other within the material.
Hopefully that begins to answer your question - but in summary, basically the two halves of the magnet do actually all cancel out, until they reach then end, at which point there is nothing else which they can cancel out, and an external magnetic field is generated.
