Object of the mind

An object of the mind is an object that exists in the imagination, but which, in the real world, can only be represented or modeled. Some such objects are abstractions, literary concepts, or fictional scenarios. Closely related are intentional objects, which are what thoughts and feelings are about, even if they are not about anything real (such as thoughts about unicorns, or feelings of apprehension about a dental appointment which is subsequently cancelled). However, intentional objects may coincide with real objects (as in thoughts about horses, or a feeling of regret about a missed appointment). Mathematics and geometry describe abstract objects that sometimes correspond to familiar shapes, and sometimes do not. Circles, triangles, rectangles, and so forth describe two-dimensional shapes that are often found in the real world. However, mathematical formulas do not describe individual physical circles, triangles, or rectangles. They describe ideal shapes that are objects of the mind. 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 is a one-sided, sealed surface with no inside or outside (in other words, it is the three-dimensional equivalent of the Möbius strip). Such objects can be represented by twisting and cutting or taping pieces of paper together, as well as by computer simulations. To hold them in the imagination, abstractions such as extra or fewer dimensions are necessary. If-then arguments posit logical sequences that sometimes include objects of the mind. For example, a counterfactual argument proposes a hypothetical or subjunctive possibility which could or would be true, but might not be false. Conditional sequences involving subjunctives use intensional language, which is studied by modal logic, whereas classical logic studies the extensional language of necessary and sufficient conditions.