The incredible precision of mathematical expression permits a vast applicability of mental abstractions to real life situations.
Many more mathematical formulas describe shapes that are unfamiliar, or do not necessarily correspond to objects in the real world.
For example, the Klein bottle[2] is a one-sided, sealed surface with no inside or outside (in other words, it is the three-dimensional equivalent of the Möbius strip).
[3]: 2029 Such objects can be represented by twisting and cutting or taping pieces of paper together, as well as by computer simulations.
For example, a counterfactual argument proposes a hypothetical or subjunctive possibility which could or would be true, but might not be false.
This is sometimes regarded as counterintuitive, but makes more sense when false conditions are understood as objects of the mind.
Examples include the contents of false documents, the origins of stand-alone phenomena, or the implications of loaded words.
As an example, the name of a team, a genre, or a nation is a collective term applied ex post facto to a group of distinct individuals.
Social reality is composed of many standards and inventions that facilitate communication, but which are ultimately objects of the mind.
Imaginary personalities and histories are sometimes invented to enhance the verisimilitude of fictional universes, and/or the immersion of role-playing games.
Calendar dates also represent objects of the mind, specifically, past and future times.
The theoretical posits of one era's scientific theories may be demoted to mere objects of the mind by subsequent discoveries: some standard examples include phlogiston and ptolemaic epicycles.
This raises questions, in the debate between scientific realism and instrumentalism about the status of current posits, such as black holes and quarks.
The situation is further complicated by the existence in scientific practice of entities which are explicitly held not to be real, but which nonetheless serve a purpose—convenient fictions.
Examples include field lines, centers of gravity, and electron holes in semiconductor theory.
In their respective imaginary worlds the Necronomicon, The Hitchhiker's Guide to the Galaxy, and the Red Book of Westmarch are realities, but only because they are referred to as real.