{"id":476,"date":"2010-01-31T19:16:14","date_gmt":"2010-01-31T19:16:14","guid":{"rendered":"http:\/\/mathscitech.org\/articles\/?p=476"},"modified":"2024-01-01T11:59:52","modified_gmt":"2024-01-01T11:59:52","slug":"good-math-technique","status":"publish","type":"post","link":"https:\/\/mathscitech.org\/articles\/good-math-technique","title":{"rendered":"The Place of Insight, Technique, and Computing in Mathematics"},"content":{"rendered":"

<\/p>\n

The mathematician Alfred North Whitehead1<\/a><\/sup> observed that “[Advancement occurs] by extending the number of important operations which we can perform without thinking of them.” (Introduction to Mathematics<\/em>, 1911 2<\/a><\/sup>) 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<\/a><\/sup> <\/p>\n

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.<\/p>\n

<\/p>\n

A good example of this is a problem from discrete mathematics: the finite summation of integer powers<\/a> S_p(n) = \\sum_{k=1}^N k^p, for whole number powers p. <\/p>\n

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<\/a> 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<\/a> and Ruby<\/a> can be effective tools in the development and validation of general mathematical results. <\/p>\n

The problem is ideal for illustrating three main points:<\/p>\n

    \n
  1. the value of mathematical technique, in this case the manipulation of summations, as covered in greater detail in Concrete Mathematics<\/em> 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.\n
  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,\n
  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<\/a> 4<\/a><\/sup>, and the interactive object oriented programming language, Ruby<\/a>5<\/a><\/sup>).\n <\/ol>\n

    Let’s take a closer look: <\/p>\n

    We are looking for closed form formulas for a family of summations of powers of integers:<\/p>\n

    \\displaystyle \\sum_{k=1}^{N} k^p<\/p>\n

    In particular, we want closed form solutions (formulas in N) for the following finite summations:<\/p>\n

    (1) \\displaystyle \\sum_{k=1}^{N} k = 1 + 2 + 3 + \\ldots + (N-1) + N<\/p>\n

    (2) \\displaystyle \\sum_{k=1}^{N} k^2 = 1^2 + 2^2 + 3^2 + \\ldots + (N-1)^2 + N^2<\/p>\n

    (3) \\displaystyle \\sum_{k=1}^{N} k^3<\/p>\n

    and, generally for every p:<\/p>\n

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

    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 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 <\/p>\n

    \\frac{1}{2} (N+1)N = \\frac{1}{2}[N^2 + N].<\/p>\n

    For the general problem, some insight is also needed, in this case recognizing that a direct (non-iterative) solution<\/a> 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<\/a> mechanically, i.e. without further fundamental insight. <\/p>\n

    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.<\/p>\n

    Continue reading:<\/h4>\n

    Finite Summations of Integer Powers x^p<\/strong>
    \n    Chapter 1<\/a>, Chapter 2<\/a>, Chapter 3<\/a><\/p>\n

    \n

    Footnotes<\/h4>\n

    <\/p>\n

    \n
    <\/div>\n
      \n
    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. ↩<\/a><\/span><\/li>\n
    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. ↩<\/a><\/span><\/li>\n
    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.<\/a> 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. ↩<\/a><\/span><\/li>\n
    4. Maxima<\/a> 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. ↩<\/a><\/span><\/li>\n
    5. Ruby<\/a> 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. ↩<\/a><\/span><\/li>\n<\/ol>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<\/p>\n

      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 [Read More…]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"nf_dc_page":"","footnotes":""},"categories":[7,4],"tags":[25,26,28,23,29,27,24],"coauthors":[112],"class_list":["post-476","post","type-post","status-publish","format-standard","hentry","category-education","category-mathematics-phenomenology","tag-discrete-mathematics","tag-finite-summation","tag-heuristics","tag-maxima","tag-problem-solving","tag-results","tag-ruby","odd"],"views":9796,"_links":{"self":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/476","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/comments?post=476"}],"version-history":[{"count":69,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/476\/revisions"}],"predecessor-version":[{"id":531,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/posts\/476\/revisions\/531"}],"wp:attachment":[{"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/media?parent=476"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/categories?post=476"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/tags?post=476"},{"taxonomy":"author","embeddable":true,"href":"https:\/\/mathscitech.org\/articles\/wp-json\/wp\/v2\/coauthors?post=476"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}