New mathematical objects are introduced for practical considerations, primarily as a solution to existing difficulties in our given mathematical system. Hilbert explains this process in terms of Ideal Elements and the unique notions of existence attached to these creations. This can be done by constructing new objects and sewing them in (as in complex numbers to the reals) or by postulating new axioms (and showing that they are consistent with the rest of the system). [MancosuFromBouwertoHilbert]

Since Hilbert believed that it was only possible to conceive of mathematical existence with respect to a particular system, the introduction of new mathematical objects must only be possible when it can be done as a consistent extension of an original system. [The only exception being Hilbert's own declaration that proofs were themselves mathematical objects, however, this really just deals with definitions]. [MancosuFromBouwertoHilbert] For example, in geometry lines and points are the only actual objects, yet the ideal elements such as points at infinity exist within the system. [HilbertOnTheInfinite] As the Fregean Context Principle would render such notions meaningless apart from the sentence in which they are contained, Hilbert believes that they are meaningless apart from the system.

-- BenjaminZamzow - 23 Oct 2006