Westfield State College Writer's Guide

Like other sections of this site, this page remains under construction and will continue to expand.

• Deductive Reasoning - Deductive reasoning is based strictly on the rules of logic. In deductive reasoning conclusions are validated using only previously, deductively established results together with the rules of logic. While other forms of reasoning are utilized in mathematics to develop theories and make conjectures, a statement is not considered a mathematical truth until it has been proven deductively.
• Inductive Reasoning - Inductive reasoning is the process of drawing general conclusions from limited observations. Inductive reasoning uses particular observations to draw conclusions about all future observations. Although it is the dominant method of reasoning in the sciences, and in life, it is not always valid. Inductive reasoning is important for the development of mathematical ideas, but is not capable of the definitive validation of a given result that deductive reasoning is.
• Inferential Reasoning - Inferential reasoning is the process of deriving generalizations from statistical data. It is strongly related to inductive reasoning.
• Conjecture - A conjecture is statement that is arrived at without reliance on deductive reasoning. That is, it is largely an educated guess. The word conjecture is approximately the mathematical equivalent of the word hypothesis that is used throughout the sciences. The distinction is that in the sciences a hypothesis is generally a statement that is designed to be tested empirically while a conjecture is often based little evidence and is rarely able to be tested directly.
• Parameter - A parameter is a "constant" that is allowed to vary from application to application, but that remains fixed for a given application. I.e. a parameter is a "variable constant." For example, in describing the free-fall of objects near a planet the acceleration due to gravity is a critical parameter. Near the surface of the earth it takes on one value, near the surface of the moon another, and so forth. For any given problem near the surface of a given planet the acceleration due to gravity is constant, but over the whole of free-fall problems near any planet this constant will take on different values.
• Theorem - A theorem is the usual heading for an important mathematical statement of fact.
• Lemma - A lemma is the usual heading for a brief, technical result. It stands well below a theorem in importance and is often used to supply justification for mundane details needed as part of the proof of a theorem or for a result that will be used repeatedly in a given work.
• Proposition - A proposition is similar to a theorem, but often a proposition refers to a necessary but more matter of fact result than the grandiose theorem. For example, in differential calculus one states the addition rule for derivatives (i.e. (f+g)'=f'+g') as a proposition rather than a theorem as it follows in a very mechanical way from the definition of the derivative.
• Corollary - A corollary is a consequence that follows almost immediately from a preceding theorem, proposition, or lemma. Example: Suppose one proves that the sum of any two even integers is even. A direct corollary of this is that the sum of three even integers is even. For in adding the first two of three even integers we arrive at an even integer using the previously established result. But if we then add the third even number to this sum we are again adding two even numbers, so we can use the previously established result again to conclude that the sum of the three even numbers is even.

Copyright