All the issues involved with the question of infinite divisibility are involved here. The beginning of the 20th century was occasioned by a huge controversy over the Axiom of Choice. Most upsetting was Cantor’s definition of a countable transfinite set: any set whose elements can be put into one-to-one correspondence with the elements of one of its proper subsets. This definition prescribed a certain kind of identicalness between the whole and the part -- which are supposed to be distinct, the one being larger than the other because the one contains the other! Eighty years later, the hologram would be found to provide a physical model of this transparent whole-part identity relation. If you break a holographic plate and then project the image stored upon it using only a small piece, you get the whole image, if a bit fuzzy like the new logic the Japanese have recently applied to refrigerators. The part contains all the information of the whole, in a fashion analogous to the definition given by Cantor! Selfsameness, self-identicalness, being the same-as-itself was apparently violated by the mathematical objects receiving Cantor’s consideration: transfinite sets.
Properties of structural generation of such sets gave rise to what was eventually called “Cantor dust”: an infinite set of points. This idea appeared about the same time as Impressionism in painting and music: pointillism and chromaticism, both expressions of a breakdown of clear distinctness in identity relations. And Cantor made his definition of a countable transfinite set very concrete with the famous diagonal proof. Many mathematicians, however, called: Foul! The identicalness between part and whole was regarded by some as a logical fallacy. Russell and Whitehead wrote their PRINCIPIA MATHEMATICA largely to address this reservation concerning what Cantor had done. Others disparaged the method used to prove the existence of the set, because it involved all at once choosing a transfinite number of elements, which they maintained could not, in principle, be practically accomplished. Controversy around this issue centered upon the Axiom of Choice and the idea of an “existence proof”. By the turn of the century, it would be fair to say, alarm in higher mathematics had turned to virtual hysteria. And this, at the very time in music that the gravitational pull of tonality was evaporating into the “fog on fog” -- to use Herman Weyl’s pejorative characterization of Cantor’s mathematics -- of atonality, and in physics when Einstein was formulating the ideas which would lead to developments slaying the notion of “force” as a fundamental in physics. The basic issue at stake in all these cases was how identity as a property of existence was to be conceived. Alarm turned to hysteria in mathematics, not only because the simple selfsameness of the mathematical object was under assault, but also because there was a growing awareness that the simple-identity of the mathematician, himself, was, by direct implication, being called into question. If choosing an infinite number of elements was to be allowed as practical of accomplishment, clearly this was not something a selfsame mathematician could actually do in a finite period of time; it would require some other kind of mathematician, one perhaps with non-simple identity, with multiple selves, with collective properties. My God! Such choosing might even be a collective occasion of experience. The spiritual aspect of the mathematician, itself -- his unconscious mind -- might in some way involve transfinite sets! Which was precisely what Cantor had explicitly been saying all along in his theological commentaries. Spirit entities, inner voices, channeling messages from higher realities: all of this was possibly implied. The violation of binary Aristotelean-Baconian logic and the simultaneous invocation of m-valued logics involved in Cantor’s definition of a denumerable transfinte set were neatly skirted, leading to tragic developments. (For a fuller elaboration on the extra-mathematical factors which determined the attitude of the mathematical establishment toward the Axiom of Choice, see “Echo of the Mockingbird”, in The Santa Fe Papers, posted at http://members.brandx.net/user/autopoy/sfepapers/echo.html)