login
Home / Papers / What is Mathematics

What is Mathematics

88 Citations•2007•
J. Hoag
journal unavailable

No TL;DR found

Abstract

Mathematics is the manipulation of abstract symbols according to specific rules. As such, mathematics is a language, but it differs from other languages in its universal nature and its applicability to human endeavors. The symbols that are manipulated usually represent quantities or geometric objects. The quantities can be as specific as numbers, or they can be abstract and variable. The quantities manipulated often are the rules of manipulation themselves. A mathematician is a linguist who constructs symbols and words from their conceptual roots. These symbols represent an ever increasingly intricate web of thoughts woven from the fibers of primary ideas. In logic, axioms are chosen, consequences derived from them, and even though those axioms might initially appear reasonable, unreasonable consequences sometimes result, whence axioms are re-examined. Those axioms that result in unusual consequences delight mathematicians, who are attuned to the paradoxes inherent in the system. A paradox of the twentieth century is Godel’s theorem which states that any finite axiomatic system that includes the axioms of standard arithmetic is either inconsistent or incomplete. In the latter case, there are statements that are true but cannot be proven within the system. Mathematicians, other than some logicians, tend to smile at Godel’s paradox, but continue their own endeavors towards that objective truth that mathematics affords them. Pure reason allows the questioning and manipulation of the postulates to reflect the truth. Mathematics is the objective science of pure reason. Some might say that this ability to reason mathematically is a characteristic that humans have that is not readily apparent in other animals. Mathematics will be the first language of communication between us and other sentient beings when such communications occur. Art, music, and language also involve the manipulation of abstract symbols according to specific rules, but each has slightly differing purposes. Artists manipulate the icons of the predominant cultures. Even though art involves aesthetic objectives, it is neither right nor wrong. It can please us, it can disgust us, it can provoke us, it can cause us to think, but it ∗Supported in part by NSF Grants DMS #0301095, #0603926