Object of The Mind - Mathematical Objects

Mathematical Objects

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.