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Foundations of Mathematics: Reliability and Clarity: The Explanatory Role of Mathematical Induction

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Logic, Language, Information, and Computation (WoLLIC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9803))

Abstract

While studies in the philosophy of mathematics often emphasize reliability over clarity, much study of the explanatory power of proof errs in the other direction. We argue that Hanna’s distinction between ‘formal’ and ‘acceptable’ proof misunderstands the role of proof in Hilbert’s program. That program explicitly seeks the existence of a justification; the notion of proof is not intended to represent the notion of a ‘good’ proof. In particular, the studies reviewed here of mathematical induction miss the explanatory heart of such a proof; how to proceed from suggestive example to universal rule. We discuss the role of algebra in attaining the goal of generalizability and abstractness often taken as keys to being explanatory. In examining several proofs of the closed form for the sum of the first n natural numbers, we expose the hidden inductive definitions in the ‘immediate arguments’ such as Gauss’s proof. This connection with inductive definition leads to applications far beyond verifying numerical identities. We discuss some objections, which we find more basic than those in the literature, to Lange’s general argument that proofs by mathematical induction are not explanatory. We conclude by arguing that whether a proof is explanatory depends on a context of clear hypothesis and understanding what is supposedly explained to who.

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Notes

  1. 1.

    Page 26 of [Cof91].

  2. 2.

    Section 3 of [HM05] give several specific examples of mathematicians using ‘explain’ in various senses.

  3. 3.

    He and another U.I.C. mathematician, Neil Rickert, once held the record for the largest pair of twin primes.

  4. 4.

    I added the parenthetical descriptions. And, I modified the question because as originally phrased, the first question is not asked. But the explanations proffered by Hanna all deal with it.

  5. 5.

    There are various logics for studying this topic [Lin14]; but they are not considered in the papers under discussion.

  6. 6.

    In the paper Hanna draws on, Steiner [Ste78] refers to Quine’s set theory book [Qui69].

  7. 7.

    While this statement is appealing to students, a more formal version with the same proof is: How many edges are there in the complete symmetric graph on n vertices?.

  8. 8.

    This assertion of course depends on where I begin my arithmetic. There is no trace of arithmetic, if my assumption is that \((N,+,\times )\) is a semi-ring (ring without additive inverse). But there is if I go back one step further and define multiplication inductively from addition.

  9. 9.

    There is also a geometrical picture to understand the algebra in the numerator of this calculation. Represent \(n(n+1)\) by an n high by \(n+1\) wide rectangle. Then to add \(2(n+1)\), place two 1 by n strips on top of the rectangle.

  10. 10.

    Paul Sally presented this argument at a University of Chicago class for high school teachers on Aug. 3/4, 2012. Doubtless, the approach is old; the use of telescoping series dates at least to the Bernoulli’s, Euler and Goldbach [BVP06]. Sally was not only a distinguished researcher in p-adic analysis and representation theory but a national leader in Mathematics Education.

  11. 11.

    A set X is defined by generalized inductive definition if there is rule assigning to each finite subset \(X_0\) of X some larger set \(X_0'\) (the closure of \(X_0\)) and for each \(X_0 \subset X\), \(X_0' \subset X\). This notion is given a more inductive format if one starts with a set Y and successively closes it to obtain X [Sho67].

  12. 12.

    We take the version form [Mar02] but similar accounts can be found in any modern logic text.

  13. 13.

    Particularly relevant is the recognition by Wysocki [Wys] and Cariani [Car16] that the basic thrust of up versus down induction breaks down for argument based on generalized induction definition (such as induction of formulas or closing a subset of a group G to a subgroup of G).

  14. 14.

    This ignores, of course, the many uses of mathematical induction to prove P(n) and for each \(k\ge n\), \(P(k) \rightarrow P(k+1)\) then for all \(k\ge n\), P(k). His argument could be complicated to handle this case as well as it does the one it explicitly addresses, but he doesn’t even consider such situations.

  15. 15.

    Baker [Bak10] notes this objection but does not develop it.

  16. 16.

    If one were to develop this argument in first order logic, Q would be a formula. However, in the spirit of the general discussion of induction we describe here informal mathematical arguments.

  17. 17.

    Here, Q is the set of possible coordinatizations.

  18. 18.

    In fact, Hanna’s article in an educational journal reflects the common use of Gauss’ proof for future American teachers of middle school mathematics. The goal of the activity is not understanding the step from example to universal but just some notion of justification.

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  1. University of Illinois at Chicago, Chicago, USA

    John T. Baldwin

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  1. Dept of Mathematics & Statistics, Univ of Helsinki Dept of Mathematics & Statistics, Helsinki, Finland

    Jouko Väänänen

  2. Department of Mathematics and Statistics, University of Helsinki , Helsinki, Finland

    Åsa Hirvonen

  3. Centro de Informática , Recife, Pernambuco, Brazil

    Ruy de Queiroz

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Baldwin, J.T. (2016). Foundations of Mathematics: Reliability and Clarity: The Explanatory Role of Mathematical Induction. In: Väänänen, J., Hirvonen, Å., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2016. Lecture Notes in Computer Science(), vol 9803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52921-8_5

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