Good mathematical technique and the case for mathematical insight

Good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, one can often pass from a single insight to the solution of a family of problems, and in some cases, to the solution of the general question itself. … Good mathematical technique has built within it the mathematical insight of the best of previous generations.


Good Mathematical Technique and the Case for Mathematical Insight

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: the development of good mathematical technique, accompanied by the judicious selection of symbols, extends the capability to perform chains of complicated manipulations reasonably efficiently in the derivation of a useful result.

It is in this way that good mathematical technique can bring the solution to certain mathematical questions within reach. By a proper formulation (one that is both tractable and that generalizes readily) and the use of mechanical techniques, 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

The value of a mathematical computing toolset

Having a decent mathematical computing toolset provides a similar advantage: 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 general results are often not required and where specific or semi-general results are needed “now”, 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 illustration of this is a general problem from discrete mathematics: the finite summation of integer powers S_p(n) = \sum_{k=1}^N k^p. This problem does not require extensive machinery, yet is meaty enough to show how Maxima and Ruby can be effective tools in the development and validation of general mathematical results. Though the case p=1 is likely to be familiar to every mathematician, the specific cases p>1 and the general case for all p are unlikely to be obvious to all but specialists. In addition, the development for small p is arranged so that an interested non-specialist should be able to follow the arguments and understand the methods being developed. The problem is thus ideal for illustrating the three main points of this series of articles:

  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).

For those who do not work in this area, the motivated development of the solution approach may additionally hold some useful techniques.

Introduction to the Problem

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

\displaystyle \sum_{k=1}^{N} k^p

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

(1) \displaystyle \sum_{k=1}^{N} k = 1 + 2 + 3 + \ldots + (N-1) + N

(2) \displaystyle \sum_{k=1}^{N} k^2 = 1^2 + 2^2 + 3^2 + \ldots + (N-1)^2 + N^2

(3) \displaystyle \sum_{k=1}^{N} k^3

and, generally for every p:

(P) \displaystyle \sum_{k=1}^{N} k^p = 1^p + 2^p + 3^p + \ldots + (N-1)^p + N^p

Mathematical Solutions: Fundamental Insight? or Good Technique?

If you’re nine year old Gauss, the story goes that you knock of (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 N+1. There are N 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

\frac{1}{2} (N+1)N = \frac{1}{2}[N^2 + N].

But what if you’re not Gauss?

I claim you need a little insight for this first problem in the series, but with a closed form formula for the first problem and appropriate technique (recurrences and induction), you can obtain a recurrence solution for the general case without requiring any further fundamental insight.

A direct (non-iterative) solution can be obtained using a linear independence argument from linear algebra. Here the insight is recognizing the appropriate technique.

So, though the province of mathematical insight and good technique blur, it is undoubtedly the case that the development of good technique brings the solution of mathematical problems within reach.

Good mathematical technique has built within it the mathematical insight of the best of previous generations.


>> Continue reading: Finite Summations of Integer Powers x^p, Part 1

>> Continue reading: Finite Summations of Integer Powers x^p, Part 2

>> Continue reading: Finite Summations of Integer Powers x^p, Part 3

You can also download the PDF articles (TeX = easier to read) from here.


Footnotes

  1. Whitehead was the major collaborator with Bertrand Russell in the strenuous 10 year attempt, ultimately unsuccessful, at driving through the logicist program in Mathematics, i.e. reducing the entire body of mathematics to a fixed system of logic. The program of logicial reductivism, of which this was perhaps the last major attempt, and certainly one of the best known and most influential, was put to rest by Godel’s discovery of the essential incompleteness of every sufficiently strong logical system (proved in his Incompleteness Theorem). In this, he establishes that any logical system sufficiently strong to obtain arithmetic will be able to generate statements that the system cannot prove.
  2. Whitehead claimed in the original that it is Civilization that advances in this way. I have reduced the claim for the purpose of this article.
  3. Fields Medalist Terence Tao has written a short piece that describes the role of rigor and the value of mathematical technique in the training of a mathematician. In the online discussion of this article, he adds two particularly interesting remarks: the first concerns the difference between the training pathways of physicists and engineers versus mathematicians that acknowledges that the final destination is the same, but the training route is different (pre-rigorous, post-rigrous). He then speculates on the observation that the two pathways are not the same, and that the order in which one traverses them influences the final outcome, and he makes the analogy with the order of learning languages.
  4. Maxima is a system for the manipulation of symbolic and numerical expressions, including differentiation, integration, Taylor series, Laplace transforms, ordinary differential equations, systems of linear equations, polynomials, and sets, lists, vectors, matrices, and tensors. Maxima yields high precision numeric results by using exact fractions, arbitrary precision integers, and variable precision floating point numbers. Maxima can plot functions and data in two and three dimensions.
  5. Ruby is a dynamic, open source programming language with a focus on simplicity and productivity. It has an elegant syntax that is natural to read and easy to write.

Comments are closed.

Articles by Topic

Professional Tools

Mathematics-Applications

Mathematics-Technical

Engineering-Software

Engineering-Systems

Mathematical Education

Mathematics-Phenomenology