What ought particle physicists look for?

The Banach-Tarski Paradox (https://www.youtube.com/watch?v=s86-Z-CbaHA) is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball.

(From Wikipedia)

The reassembly process involves only moving the pieces around and rotating them, without changing their shape (from Wikipedia).

(A stronger form of the theorem implies that given any two “reasonable” solid objects (such as a small ball and a huge ball), either one can be reassembled into the other. This is often stated informally as “a pea can be chopped up and reassembled into the Sun” and called the “pea and the Sun paradox” (from Wikipedia).)

The theorem is called a paradox because it contradicts our basic geometric intuition. “Doubling the ball” by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. It is thus called a “paradox” because it appears to contradict sense (as judged by intuition).

The proof of this result depends in a critical way on the choice of axioms for set theory. It can be proven using the axiom of choice, which allows for the construction of nonmeasurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices. (Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.)

It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another. (from Wikipedia)

Banach-Tarski Paradox thus suggests that new objects can be created “out of nothing” in reality. If this indeed is the case is difficult to find out empirically, but there are results from particle colliding that appears to support the reality of this mathematical finding.

This theorem is the counterpoint to particle physics’ idea of a “Higgs particle – if reality is comprehensible, then there must be a “Higgs particle”, whereas if the Banach-Tarski Paradox is true, then reality is incomprehensible. In a scientific sense, this contradiction means that particle physicists ought not look for a “Higgs particle”, but rather for a falsification of mathematics. If they can’t find such falsification, then there simply is no “Higgs particle” to find. Finding out what is fruitful to look for ought to precede an actual search, or …? Otherwise, particle physicists risk an endless search for something that can’t be found (or, an indefinite search to define the indefinable, as Charles Darwin expressed it). The fundamental question is if particle physicists even consider the possibility that reality is rationally incomprehensible.

Another contribution to understanding of conceptualization http://menvall.wordpress.com/