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On the Alleged Simplicity of Impure Proof

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Simplicity: Ideals of Practice in Mathematics and the Arts

Part of the book series: Mathematics, Culture, and the Arts ((MACUAR))

Abstract

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self-evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof-theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim.

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Notes

  1. 1.

    Cf. [14, p. 351], though statements of this sort are found throughout La géométrie. For more on the simplicity of the Cartesian method in geometry [3, Sect. 2], [41].

  2. 2.

    Cf. [39, p. 271]. For a detailed historical investigation of Lagrange’s views on purity in his algebraic work.

  3. 3.

    The prize question is described in [51, p. 4]. Otto Hölder showed in 1892 that there is no exact solution in finite terms to cubics in the casus irreducibilis that avoids imaginary numbers [32]. For a more thorough discussion of the casus irreducibilis in relation to purity [1].

  4. 4.

    More precisely, the prime number theorem states that \(\frac{\pi (x)} {x/\log (x)}\) approaches 1 in the limit, where π(x) is the number of primes less than or equal to x.

  5. 5.

    Cf. [24, p. 338]. For more on purity in arithmetic [2, 4].

  6. 6.

    Note that in [8] Avigad draws on work from automated reasoning, which is closely allied with proof theory; thus these approaches are not exclusive.

  7. 7.

    In [54], Simpson defines RCA0 (on p. 24) in a slightly different but equivalent way, using Σ 1 0-induction (with set parameters) but not primitive recursion. As he notes on p. 73, Friedman originally defined RCA0 in the way we have done here [23, pp. 557–558].

  8. 8.

    A significant remaining question is whether IΣ 1 is especially significant, as an arithmetical extension of PRA, for the thesis that impurity generally offers gains of efficiency; or whether a study of IΣ 2, for instance, would offer key additional insights. Toward this, Ignjatović has conjectured that further inductive strengthenings of PRA with respect to the quantifier-free theorems of PRA will yield a significant gain of efficiency, but to the best of our knowledge this is still open.

  9. 9.

    Polynomial speed-up may be more carefully defined as follows [10, pp. 4–5]. Let the length ℓ(π) of a proof π be the number of symbol occurrences in π. For any formula φ, let \(\pi _{T_{i}}^{<}(\varphi )\) be the shortest proof (in terms of number of symbol occurrences) of φ in T i . We say that T 1 is at most a polynomial speed-up of T 2 with respect to Φ if there is a polynomial p(x) with natural number coefficients such that for every φ provable in T 2

    $$\displaystyle{ \ell(\pi _{T_{2}}^{<}(\varphi )) <p(\ell(\pi _{ T_{1}}^{<}(\varphi ))). }$$
  10. 10.

    This can be defined precisely as follows. Firstly, a function f(x) eventually dominates a function g(x) if there is an m such that for all n > m, f(n) ≥ g(n). Secondly, let 2 m x be the function defined by: \(2_{0}^{n} = n, 2_{m+1}^{n} = 2^{2_{m}^{n}}\). For example, \(2_{1}^{n} = 2^{2_{0}^{n}} = 2^{n}\), \(2_{2}^{n} = 2^{2_{1}^{n}} = 2^{2^{n}}\), \(2_{3}^{n} = 2^{2_{2}^{n}} = 2^{2^{2^{n} }}\), and so on. A function f(x) has Kalmar elementary growth rate if there is an m such that 2 m x eventually dominates f(x). It turns out that 2 x x is the first function that dominates all Kalmar elementary functions. A function f(x) has roughly super-exponential growth rate if and only if (i) it does not have Kalmar elementary growth rate, but (ii) there is a polynomial p(x) with natural number coefficients such that p(2 x x) eventually dominates it.

  11. 11.

    Roughly super-exponential speed-up may be more carefully defined as follows [10, pp. 4–5]. T 1 has roughly super-exponential speed-up over T 2 if and only if

    1. 1.

      there is no function f(x) with Kalmar elementary growth rate such that for every φ provable in T 2, \(\ell(\pi _{T_{2}}^{<}(\varphi )) <f(\ell(\pi _{T_{1}}^{<}(\varphi )))\); and

    2. 2.

      there is a function g(x) with roughly super-exponential growth rate such that for every φ provable in T 2, \(\ell(\pi _{T_{2}}^{<}(\varphi )) <g(\ell(\pi _{T_{1}}^{<}(\varphi )))\).

    For Φ a set of formulas provable in T 2, T 1 has roughly super-exponential speed-up over T 2 with respect to Φ if and only if there is a sequence {φ i : i ∈ ω} of formulas from Φ such that

    1. 1.

      there is no function f(x) with Kalmar elementary growth rate such that for every φ n ∈ Φ, \(\ell(\pi _{T_{2}}^{<}(\varphi _{n})) <f(\ell(\pi _{T_{1}}^{<}(\varphi _{n})))\); and

    2. 2.

      there is a function g(x) with roughly super-exponential growth rate such that for every φ n ∈ Φ, \(\ell(\pi _{T_{2}}^{<}(\varphi _{n})) <g(\ell(\pi _{T_{1}}^{<}(\varphi _{n})))\).

  12. 12.

    He suggests as a possibility WKL0 + + COH, where COH asserts the existence of a cohesive set, having shown that WKL0 + + COH is a Π 2 1-axiomatizable Π 1 1-conservative extension of RCA0 (Corollary 2.5).

  13. 13.

    That WKL0 and WKL0 + are Π 2 1-axiomatizable can be seen by inspecting the logical form of their axioms. That they are not Π 1 1-axiomatizable follows, respectively, from Harrington’s result that WKL0 is Π 1 1-conservative over RCA0 and from Brown and Simpson’s result that WKL0 + is Π 1 1-conservative over RCA0. To see why for the case of WKL0, note that we can write WKL0 as RCA0 + φ. If WKL0 were Π 1 1-axiomatizable, then there would be a Π 1 1 theory T such that T is equivalent to WKL0. Since RCA0 is finitely axiomatizable, RCA0 + φ is equivalent to a single sentence that, by compactness, is provable in a finite subtheory of T that can be conjoined into a single sentence ψ. Hence RCA0 proves the equivalence of ψ and φ. Since WKL0 proves ψ, it follows by Harrington’s conservation result that RCA0 proves ψ, and thus that RCA0 proves φ, contradicting the fact that WKL0 is properly stronger than RCA0. For WKL0 + the argument is similar, using Brown and Simpson’s result instead.

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Acknowledgements

Thanks to Walter Dean, Michael Detlefsen, Sébastien Maronne, Mitsuhiro Okada, Marco Panza, and Sean Walsh for helpful discussions on these subjects.

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  1. Unité de Formation et de Recherche de Philosophie, Université de Paris-Sorbonne and Institut d’Histoire et de Philosophie des Sciences et des Techniques, Paris, France

    Andrew Arana

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  1. Department of Mathematics, The Graduate Center, City University of New York, New York, New York, USA

    Roman Kossak

  2. Department of Mathematics, Sarah Lawrence College, Bronxville, New York, USA

    Philip Ording

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Arana, A. (2017). On the Alleged Simplicity of Impure Proof. In: Kossak, R., Ording, P. (eds) Simplicity: Ideals of Practice in Mathematics and the Arts. Mathematics, Culture, and the Arts. Springer, Cham. https://doi.org/10.1007/978-3-319-53385-8_16

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