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Finiteness has to do with the existence of boundaries. Intuitively, we feel that where there is a separation, a border, a threshold ? there is bound to be at least one thing finite out of a minimum of two. This, of course, is not true. Two infinite things can share a boundary. Infinity does not imply symmetry, let alone isotropy. An entity can be infinite to its ?left? ? and bounded on its right. Moreover, finiteness can exist where no boundaries can. Take a sphere: it is finite, yet we can continue to draw a line on its surface infinitely. The ?boundary?, in this case, is conceptual and arbitrary: if a line drawn on the surface of a sphere were to reach its starting point ? then it is finite. Its starting point is the boundary, arbitrarily determined to be so by us.

This arbitrariness is bound to appear whenever the finiteness of something is determined by us, rather than ?objectively, by nature?. A finite series of numbers is a fine example. WE limit the series, we make it finite by imposing boundaries on it and by instituting ?rules of membership?: ?A series of all the real numbers up to and including 1000? . Such a series has no continuation (after the number 1000). But, then, the very concept of continuation is arbitrary. Any point can qualify as an end (or as a beginning). Are the statements: ?There is an end?, ?There is no continuation? and ?There is a beginning? ? equivalent? Is there a beginning where there is an end ? And is there no continuation wherever there is an end? It all depends on the laws that we set. Change the law and an end-point becomes a starting point. Change it once more and a continuation is available. Legal age limits display such flexible properties.

Finiteness is also implied in a series of relationships in the physical world : containment, reduction, stoppage. But, these, of course, are, again, wrong intuitions. They are at least as wrong as the intuitive connection between boundaries and finiteness.

If something is halted (spatially or temporally) ? it is not necessarily finite. An obstacle is the physical equivalent of a conceptual boundary. An infinite expansion can be checked and yet remain infinite (by expanding in other directions, for instance). If it is reduced ? it is smaller than before, but not necessarily finite. If it is contained ? it must be smaller than the container but, again, not necessarily finite.

It would seem, therefore, that the very notion of finiteness has to do with wrong intuitions regarding relationships between entities, real, or conceptual. Geometrical finiteness and numerical finiteness relate to our mundane, very real, experiences. This is why we find it difficult to digest mathematical entities such as a singularity (both finite and infinite, in some respects). We prefer the fiction of finiteness (temporal, spatial, logical) ? over the reality of the infinite.

Millennia of logical paradoxes conditioned us to adopt Kant?s view that the infinite is beyond logic and only leads to the creation of unsolvable antinomies. Antinomies made it necessary to reject the principle of the excluded middle (?yes? or ?no? and nothing in between). One of his antinomies ?proved? that the world was not infinite, nor was it finite. The antinomies were disputed (Kant?s answers were not the ONLY ways to tackle them). But one contribution stuck : the world is not a perfect whole. Both the sentences that the whole world is finite and that it is infinite are false, simply because there is no such thing as a completed, whole world. This is commensurate with the law that for every proposition, itself or its negation must be true. The negation of: ?The world as a perfect whole is finite? is not ?The world as a perfect whole is infinite?. Rather, it is: ?Either there is no perfectly whole world, or, if there is, it is not finite.? In the ?Critique of Pure Reason?, Kant discovered four pairs of propositions, each comprised of a thesis and an antithesis, both compellingly plausible. The thesis of the first antinomy is that the world had a temporal beginning and is spatially bounded. The second thesis is that every substance is made up of simpler substances. The two mathematical antinomies relate to the infinite. The answer to the first is: ?Since the