Posted 28 April 2009
This page is intended to raise your consciousness about the many ways there are to understand “understand” in math. A link to each chapter about understanding math is given here with some comments about the ideas expressed in it.
¨ One piece of information you must have about a math concept is its definition.
¨ Every proof of a fact about the concept must be based on a logical chain of reasoning starting with the definition.
The definition is one path to understanding the concept but not the only one.
¨ Mathematical objects are what mathematics is “about”.
¨ The number 42 is a math object.
¨ The set of even positive integers is a math object. Even though it is infinite, it is a single math object.
¨ The function is a math object. Its value can be computed at many different numbers but it is a single, static math object.
¨ We talk and think about math objects in some of the same ways that we talk about physical objects, but in other ways they are not like physical objects.
A mathematical structure is a special kind of math object defined as a set with some associated objects called structure. Equivalence relations, partitions and topological spaces are examples of mathematical structures.
¨ “Representation” and “model” have several related meanings.
¨ We can represent the number 42 in binary notation as 101010 and in hex as 2A.
¨ We can represent a function by its formula or by its graph, or sometimes by a listing of its values.
¨ When we think about some math objects we may have a mental representation of them.
¨ We can model the trajectory of a ball by a function.
¨ A group is a model of the group axioms.
¨ We think of a function that has a positive derivative as “going up”. But the function isn’t really going up, it just sits there like your pet rock.
¨ We think of the parabola as like a bowl whose sides go up forever. But “forever” is a metaphor. There is no time involved.
¨ Besides, we may forget that the sides go out in the positive and negative x direction “forever”, too (every vertical line cuts the graph!), even though the picture looks like they are too steep to do that.
¨ As these examples illustrate, we think about math objects in terms of images and metaphors that we have developed out of our experience with them. These are valuable insights but they generally cannot be used to prove theorems about them.
¨ There is a way of thinking about any math object that helps when you are trying to prove something: this is the rigorous view of math objects, also discussed in this chapter.
When mathematicians consider a different aspects of it:they are typically interested in two
¨ What is it? What properties does it have? How is it different from other math objects? I want a conceptual understanding of the object.
¨ How do I compute with it? How do I find a value of the object (if that makes sense)? How to I tell how big it is (in some sense of big)? How do I determine in an efficient way what properties it has?
Proofs can have a conceptual side and a computational side too.
¨ A conceptual proof helps you understand why the statement is true.
¨ A computational proof may be easier to check systematically to see if it is correct, and to automate using some suitable computer program.
This catch-all chapter talks about several special phenomena that are involved in understanding math. I expect to add to this list from time to time.