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Posted 26 June 2017
UNDERSTANDING MATH:
CONCEPT AND COMPUTATION
Introduction
When mathematicians consider a mathematical object, they are typically interested in two different
aspects of it:
What is it?
I want a conceptual understanding of the object.
 How do I think about it?
 What properties does it have?
 How is it different from other math objects?
 How can I understand it so that I can see possible applications?
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
Proofs can have a conceptual side and a computational side too.
 A conceptual proof helps you understand why the statement is true.
 A computational or symbolic proof may be easier to check systematically to see if it is correct,
and to automate using some suitable computer program.
Conceptual and computational
Below I give some examples of what people call conceptual or computational. Those are not well
defined ideas. Conceptual presentations are commonly geometric, but they don’t have to be. And if you
understand some technicalities really well, you might call something conceptual that looks like a horrible
bunch of symbolic manipulations to someone else.
Whether something is conceptual or not depends on what concepts you understand!
Example: Derivatives
Conceptually, the derivative of a function f is another function f ' whose value at a is the slope of the
tangent line to f at a. (See here for discussion and examples.) In calculus class you may have learned
many specific formulas for computing the derivative of various functions. Those formulas are all proved
using the epsilon-delta definition of derivative. The epsilon-delta definition has a (rather subtle) conceptual basis. Epsilon-delta proofs are usually longish chains of symbolic transformations.
Example: An algebraic identity
Here is a simple example that shows the distinction between concept and computation. You are no
doubt familiar with the identity
a2 - b2 = (a - b)(a + b)
which holds for all real numbers. (In fact it holds in any commutative ring.)
Computational proof
This proof shows an explicit series of steps that verify the identity using basic laws of algebra:
(a - b)(a + b) =­ (a - b)a + (a - b)b =­ a 2 - ba + ab - b2 =­ a 2 - ab + ab - b 2 =­ a 2 - b 2
distributive law
distributive law
commutative law
cancellation
Conceptual proof
This diagram shows why the identity is true geometrically (for b < a). Note that the two rectangles
marked “ * ” are congruent.
a
b
a–b
*
b
*
a–b
b
This conceptual proof requires no algebraic laws or computations at all. On the other hand, it is not
easy to see how to generalize it to a commutative ring.
Example: Property of the ordering of the reals
The idea of “conceptual proof” depends on your experience. If you are not familiar with basic geometric facts, the preceding geometric proof may not be conceptual!
Here is another example that shows how “conceptual” depends on what you know about.
Theorem:
For all real numbers x, y, and z, if x £ z , then either x £ y or y £ z .
Geometric proof
The number y has to be in one of three intervals in this picture of (part of) the real line:


x
z
If y is in the left interval or the middle interval, then y £ z . If it is in the middle or right interval, then
x£ y .
Logical Proof
We know that for real numbers, if x > y and y > z, then x > z (this is the transitive law). The contrapositive of this statement is: If x is not greater than y and y is not greater than z then x is not greater than
z. But for any real numbers r and s, saying that r is not greater than s is the same as saying that r £ s .
So, rewording the contrapositive and using one of the DeMorgan laws, we get: if x £ z , then either
x £ y or y £ z , as was to be proved.
If you have some experience with mathematical logic, you might react to the logical proof (as I did)
this way:
Why, the statement in the theorem is nothing but the contrapositive of the transitive law!
When I realized that, I felt that I had acquired a new insight, so I would call this statement conceptual. The geometric proof gives you a different insight.
Another point: The logical proof works in a much more general setting: The statement is true in any
totally ordered set. The geometric proof gives you no clue that that is true.