Mathematical phantoms

Question

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 ?

Answer 1

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.)