Produced by Charles Wells.  Home    Website TOC    Website Index   Blog

Back to top of Understanding Math chapter

Posted 12 April 2009

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.

Example: Derivatives

Conceptually, the derivative of a function f is another function  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

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:

Conceptual proof

This diagram shows why the identity is true geometrically (for b < a).  Note that the two rectangles marked “  ” are congruent. 

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

, then either

 or

. 

Geometric proof

The number y has to be in one of three intervals in this picture of (part of) the real line:

If y is in the left interval or the middle interval, then .  If it is in the middle or right interval, then . 

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 .  So, rewording the contrapositive and using one of the DeMorgan laws, we get: if , then either  or , 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: 

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.

The conceptual proof given above provides a geometric visualization of the situation required by the hypothesis of the theorem, and this visualization makes the truth of the theorem obvious. But there is a sense of "conceptual", related to the idea of conceptual definition given under elementary, that does not have a geometric component. This is the idea that a proof is conceptual if it appeals to concepts and theorems at a high level of abstraction.

To a person familiar with the elementary rules of first order logic, the symbolic proof just given becomes a conceptual proof (this happened to me): "Why, in a totally ordered set that statement is nothing but the contrapositive of transitivity!" Although this statement is merely a summary of the symbolic proof, it is enough to enable anyone conversant with simple logic to generate the symbolic proof. Furthermore, in my case at least, it provides an aha experience. Citations: Rub89421, BieGro86425.