The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly *paradoxical description* which prevented them from existing at the time.

An iconic example is the imaginary number, $i$, which was originally described as $\sqrt{-1}$. However, further advancements in the field and exploring new perspectives towards the subject eventually led to a rigorous non-contradictory mathematical definition of $i$. A definition that serves as a *formal* introduction of the (already existing) concept to the mathematical literature while satisfying all of its required properties.

Similar attempts have been conducted along the lines of providing a solid formal basis for seemingly paradoxical concepts such as negative probability or sets with a negative number of elements (see also multisets):

*Loeb, D.*, **Sets with a negative number of elements**, Adv. Math. 91, No. 1, 64-74 (1992). ZBL0767.05005.

Another contemporary example of such paradoxical mathematical objects which are pending for a rigorous definition is the so-called "field with one element". It naturally appears in various occasions and relates different concepts together. For instance, a group can be seen as a Hopf algebra over the "field with one element". See also this related MO question for further information.

Question.I am looking for more examples of such deep and useful mathematical concepts with contradictory descriptions which are not rigorously defined yet (or at least have no widely accepted formal definition). References to their possible applications are also welcome.

**Remark.** I would like to emphasize the initial *paradoxical description* of the possible answers to this question. The important point is that the natural appearance of paradoxical objects often indicates the urgent need for a deep expansion of our mathematical worldview or an essential improvement of the foundations.