The Place of Insight, Technique, and Computing in Mathematics

The mathematician Alfred North Whitehead¹ observed that “[Advancement occurs] by extending the number of important operations which we can perform without thinking of them.” (Introduction to Mathematics, 1911 ²) This is certainly true in mathematics where the development of judicious notation, accompanied by good mathematical technique, extends the capability to perform chains of complex reasoning accurately and efficiently. Through proper problem formulation (tractable yet generalizable), one can sometimes pass from a single insight to the solution of a family of problems, and in some cases, to the solution to the general question itself.³

Here, mathematical computing can provide a useful benefit: helping to efficiently explore conjectures, dispatch with false directions, and save time during the development, error-checking and validation stages of obtaining general results. In industry, where specific or semi-general results are needed fast, such tools allow rapidly working up the required material and providing the necessary certainty before the fully general results or complete proof are ready.

A good example of this is a problem from discrete mathematics: the finite summation of integer powers , for whole number powers p.

The case p=1 is familiar to every mathematician and is accessible enough so that an interested non-specialist should be able to follow the arguments and understand the methods being developed for small p. The the general case for all p is likely only to be familiar to specialists. While the general problem does not require extensive background to understand, it is substantive enough to illustrate how Maxima and Ruby can be effective tools in the development and validation of general mathematical results.

The problem is ideal for illustrating three main points:

1. the value of mathematical technique, in this case the manipulation of summations, as covered in greater detail in Concrete Mathematics by mathematicians Donald Knuth, Ron Graham, and Oren Patashnik, as well as linear algebra, in particular linear independence of monomials in a polynomial vector space.
2. the value of using judicious symbolism (in this case, a symbolism that allows the development of the recurrences and passing readily to the general case), and,
3. the value of mathematical computing tools that assist in the exploration, development, and validation of general results (in this case, the symbolic computation package Maxima ⁴, and the interactive object oriented programming language, Ruby⁵).

Let’s take a closer look:

We are looking for closed form formulas for a family of summations of powers of integers:

In particular, we want closed form solutions (formulas in ) for the following finite summations:

(1)

(2)

(3)

and, generally for every :

(P)

If you’re nine year old Gauss, the story goes that you knock off (1) with an elegant insight: write the expansion twice, once in forward order and once in reverse order. Now add the terms column-wise, and observe that the column sums are uniformly . There are columns, but you introduced the second column thereby doubling the sum. Taking all this into account, the closed form formula for the sum (1) is

.

For the general problem, some insight is also needed, in this case recognizing that a direct (non-iterative) solution can be obtained using a linear independence argument from linear algebra. Armed with a closed form formula for the first problem and some manipulative ability (recurrences and induction), you can obtain a recurrence solution for the general case mechanically, i.e. without further fundamental insight.

The point is that there is insight and technique in mathematics. While technique facilitates insight, it should not be mistaken for it. And it is fundamentally the insights that should be identified, catalogued, and acquired by those wishing to improve their abilities in mathematics.

Continue reading:

Finite Summations of Integer Powers
Chapter 1, Chapter 2, Chapter 3

Footnotes