Chaitin concludes that, since there is no structure to , the very foundations of number theory are at risk to be washed away. Obviously there are well-formed relationships between some mathematical ideas, some things do follow from others, but it would seem that these relationships are the exception rather than the rule.
So, what does all of this mean? To paraphrase Horgan, are we reaching the "end of mathematics?" It seems certain that we're not reaching any sort of absolute limit, but perhaps it is time for a shift to a more empirical position. We're finally reaching the point in the field of computation where we can really start to tackle some very large problems. Of course, this situation will only get better over time, subject to some limit of Moore's Law. And who is to say that the foundations underlying this and other "laws" aren't random as well?
The paths of math and physics diverged over the past century but now seem to be converging again. The more empirical approach, already well established in physics, appears to be necessary for math as well. It seems like the thing to do is grab some of Chaitin's work, crank up Mathematica, and start doing experiments. Q.E.D.
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