John Earman - The Hole Argument

The Hole Argument

Earman has notably contributed to debate about the "hole argument". The hole argument was invented for slightly different purposes by Albert Einstein late in 1913 as part of his quest for the general theory of relativity (GTR). It was revived and reformulated in the modern context by John3 (a short form for the "three Johns": John Earman, John Stachel, and John Norton).

With the GTR, the traditional debate between absolutism and relationalism has been shifted to whether or not spacetime is a substance, since the GTR largely rules out the existence of, e.g., absolute positions. The "hole argument" offered by John Earman is a powerful argument against spacetime substantivalism.

This is a technical mathematical argument but can be paraphrased as follows:

Define a function as the identity function over all elements over the manifold, excepting a small neighbourhood (topology) belonging to . Over, comes to differ from identity by a smooth function.

With use of this function we can construct two mathematical models, where the second is generated by applying to proper elements of the first, such that the two models are identical prior to the time, where is a time function created by a foliation of spacetime, but differ after .

These considerations show that, since substantivalism allows the construction of holes, that the universe must, on that view, be indeterministic. Which, Earman argues, is a case against substantivalism, as the case between determinism or indeterminism should be a question of physics, not of our commitment to substantivalism.