While there are many wonderful things that can be said for the study of mathematics, one of the best is that it trains the individual the art of being not wrong. By this I do not mean that mathematicians are right while others are wrong. Rather the study of mathematics trains us to completely understand the assumptions that we are making in the course of a logical argument, thereby allowing us to argue from a stronger foundation.

On the first day of any good geometry class we learn that there are certain things in mathematics which we cannot prove, yet we need them to be true in order to begin any meaningful conversation. These "facts" are taken as postulates (also known as axioms), meaning they are assumed to be true at the outset of the conversation so that we can build our understanding of the system that would result if they were true. We assume the most basic statements to be true and then spend the rest of our lives exploring the logical implications of these basic assumptions.

Two civilized mathematicians will never disagree with one another for long. If they both start with the same set of assumptions (which they usually do) and they both use sound logic in their arguments, they will end up with the same conclusions. Upon realizing that they disagree with one another, they have exactly two ways in which they could have gone wrong. First they begin the process of analyzing each others' arguments, looking for calculational errors or logical missteps. If they find errors in logic, the "wrong" mathematician finds it easy to admit that a mistake was made, rework the logic, and conclude that they agree after all.

If no logical errors were made by either, then the two mathematicians will turn to comparing their assumptions. If they disagree about the postulates taken at the outset of the conversation, then a curious thing happens. The two mathematicians draw different conclusions, and yet neither of them is "wrong" in any way. They're just working from different sets of assumptions. They can respectfully go about their business knowing that they are both right, even though they came to different conclusions.

Although rare, controversy occasionally arises among mathematicians over the appropriateness of taking certain postulates to be true. One of the most famous examples is known as the Axiom of Choice, which states that if we are given a countably infinite number of non-empty sets then we can form a new set by picking one element from each of the old sets. This statement may seem mildly uncomfortable because if involves doing something an infinite number of times, but it is certainly reasonable to devise a strategy for picking the element from each set. Perhaps we decide that we will always pick the first element listed in each set. Our finite lifespan might prohibit us from actually completing the selection, but we understand perfectly well how we would do it if we were not limited by time. So it seems reasonable to assume the Axiom of Choice.

If we assume it is true, however, there are several very troubling results that follow. One resulting theorem from topology tells us that it would then be possible to take an soccer ball, cut it into little pieces, and then glue the pieces back together to form two new soccer balls, each the exact size and shape of the original one. This clearly violates some natural law of conservation of something. So perhaps assuming the Axiom of Choice is not such a good idea. Or perhaps we just need to be sure that if we are going to assume the axiom of choice, we do so consciously and honestly.

Another famous example lies at the heart of geometry. The Euclidean Parallel Postulate, which shows up in the first few pages of any good Euclidean geometry book, states that given a line L and a point P not on L, there exists a unique line M through P that is parallel to L. More simply, the Euclidean Parallel Postulate asserts that parallel lines look like railroad tracks. When taken together with the other axioms of geometry we get the entire body of geometric knowledge that we all learn in high school. For thousands of years mathematicians understood there was something wrong with this postulate. It seemed to reflect their perceptions of parallel lines in the real world, but they couldn't justify this as being the way parallel lines "had to behave".

It wasn't until the 1830's that mathematicians discovered why their uncertainty was well founded. Lobachevsky and Janos Bolyai independently discovered that you could change the statement of the parallel postulate without contradicting the other axioms of geometry. In fact, if you replace "a unique parallel line" with either "no parallel lines" or "infinitely many parallel lines" you get two completely different geometric theories. Changing these four little words gives rise to systems where crazy things happen: angles in triangles no longer add up to 180 degrees, the plane can be tiled by regular polygons of any number of sides, and our area and volume formulas fall completely to pieces. Hyperbolic and Elliptical Geometry were born by changing one simple postulate in the foundation of geometry. As a result of this simple change in assumptions I can honestly claim that my favorite triangle has angles that add up to 270 degrees, and I am not wrong.

On the first day of any good geometry class we learn that there are certain things in mathematics which we cannot prove, yet we need them to be true in order to begin any meaningful conversation. These "facts" are taken as postulates (also known as axioms), meaning they are assumed to be true at the outset of the conversation so that we can build our understanding of the system that would result if they were true. We assume the most basic statements to be true and then spend the rest of our lives exploring the logical implications of these basic assumptions.

Two civilized mathematicians will never disagree with one another for long. If they both start with the same set of assumptions (which they usually do) and they both use sound logic in their arguments, they will end up with the same conclusions. Upon realizing that they disagree with one another, they have exactly two ways in which they could have gone wrong. First they begin the process of analyzing each others' arguments, looking for calculational errors or logical missteps. If they find errors in logic, the "wrong" mathematician finds it easy to admit that a mistake was made, rework the logic, and conclude that they agree after all.

If no logical errors were made by either, then the two mathematicians will turn to comparing their assumptions. If they disagree about the postulates taken at the outset of the conversation, then a curious thing happens. The two mathematicians draw different conclusions, and yet neither of them is "wrong" in any way. They're just working from different sets of assumptions. They can respectfully go about their business knowing that they are both right, even though they came to different conclusions.

Although rare, controversy occasionally arises among mathematicians over the appropriateness of taking certain postulates to be true. One of the most famous examples is known as the Axiom of Choice, which states that if we are given a countably infinite number of non-empty sets then we can form a new set by picking one element from each of the old sets. This statement may seem mildly uncomfortable because if involves doing something an infinite number of times, but it is certainly reasonable to devise a strategy for picking the element from each set. Perhaps we decide that we will always pick the first element listed in each set. Our finite lifespan might prohibit us from actually completing the selection, but we understand perfectly well how we would do it if we were not limited by time. So it seems reasonable to assume the Axiom of Choice.

If we assume it is true, however, there are several very troubling results that follow. One resulting theorem from topology tells us that it would then be possible to take an soccer ball, cut it into little pieces, and then glue the pieces back together to form two new soccer balls, each the exact size and shape of the original one. This clearly violates some natural law of conservation of something. So perhaps assuming the Axiom of Choice is not such a good idea. Or perhaps we just need to be sure that if we are going to assume the axiom of choice, we do so consciously and honestly.

Another famous example lies at the heart of geometry. The Euclidean Parallel Postulate, which shows up in the first few pages of any good Euclidean geometry book, states that given a line L and a point P not on L, there exists a unique line M through P that is parallel to L. More simply, the Euclidean Parallel Postulate asserts that parallel lines look like railroad tracks. When taken together with the other axioms of geometry we get the entire body of geometric knowledge that we all learn in high school. For thousands of years mathematicians understood there was something wrong with this postulate. It seemed to reflect their perceptions of parallel lines in the real world, but they couldn't justify this as being the way parallel lines "had to behave".

It wasn't until the 1830's that mathematicians discovered why their uncertainty was well founded. Lobachevsky and Janos Bolyai independently discovered that you could change the statement of the parallel postulate without contradicting the other axioms of geometry. In fact, if you replace "a unique parallel line" with either "no parallel lines" or "infinitely many parallel lines" you get two completely different geometric theories. Changing these four little words gives rise to systems where crazy things happen: angles in triangles no longer add up to 180 degrees, the plane can be tiled by regular polygons of any number of sides, and our area and volume formulas fall completely to pieces. Hyperbolic and Elliptical Geometry were born by changing one simple postulate in the foundation of geometry. As a result of this simple change in assumptions I can honestly claim that my favorite triangle has angles that add up to 270 degrees, and I am not wrong.