Mathematical phantoms The field with one element or characteristic one ($\mathbb{F_1}$ or $\mathbb{F_{un}}$) is a mathematical phantom, which can defined as a beast who  clearly (i.e. within the current mathematical framework) does not exist, but there are many pointers in a direction that it should.
Are there other examples of mathematical phantoms ?
 A: I agree with Dietrich (in the comments) that the meaning of "mathematical phantom" is not entirely clear. In particular, the demarcation from merely very fruitful abstractions is blurry; perhaps useful criteria are:


*

*There should be a statement which was held to be obviously true before the discovery of the phantom, but which is false in view of the new concept.

*It should have required a nontrivial effort to make it precise (to "help it come into being").

*It should have great explanatory power and vast consequences (Gavin Wraith: "which [...] obtrudes its effects so convincingly that one is forced to concede a broader notion of existence", sorry for linking to http).


Phantoms in this stricter sense could include:


*

*The irrational numbers. (Running counter to the basic tenet "all is number" by the Pythagorean school, where by "number" they meant "rational number".)

*The complex numbers. (The idea of a number squaring to $-1$ was surely held to be obviously nonsensically.)

*The $p$-adic numbers.

*Actual infinity, together with the flexible notion of sets we have nowadays (vastly surpassing recursive subsets of $\mathbb{N}$) and the axiom of choice.

*Sobolev function spaces.

*Infinitesimal numbers.


Phantoms in a broader sense (where I can't think of any held-to-be-obviously-true statement falsified by them) could include:


*

*Symmetries of zeros of polynomials, or more generally groups.

*The field with one element.

*Ideals in number theory.

*Motives.

*$\infty$-categories.

*Toposes. (Generalizing and unifying various cohomology theories.)

*Nonclassical logics. (Born in the foundational crisis, nowadays with lots of applications in mainstream mathematics.)

