234 A History ofMathematics
Intuitionist critique: we have no way of telling which of the two cases (1), (2) holds at each
point; so we cannot apply the law of the excluded middle.- The simple way of seeing this is that the right-hand knot is obtained from the left-hand one
by pulling the top loop out and lifting it downwards, so as to surround the whole diagram.
If we want to use Reidemeister moves, begin with knot 2 and tilt the diagram slightly. The long
edge at the bottom can then be taken up through the diagram to the top (say above it) using
Reidemeister move 2 for the two bottom loops and move 3 for each of the four crossings in
turn. When this is finished, you find that you have not quite knot 1, but one which differs by
two loops which can be removed by Reidemeister move 1. The total number of moves (I think)
is then 2+ 4 + 2 =8.