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Posted 28 April
2009

UNDERSTANDING MATH

**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 **math
object** they are
typically interested in two different aspects of it:

**¨
****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.

**Other aspects of understanding
math**

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.

¨
**Abstraction and the axiomatic method**

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