Abstract
In this paper we develop a case for introducing a new teaching tool to undergraduate mathematics. Lean is an interactive theorem prover that instantly checks the correctness of every step and provides immediate feedback. Teaching with Lean might present a challenge, in that students must write their proofs in a formal way using a specific syntax. Accordingly, this paper addresses the issue of formalism from both a theoretical and a practical point of view. First, we examine the nature of proof, referring to historical and contemporary debates on formalization, and then show that in mathematical practice there is a growing rapprochement between strictly formal proof and proofs-in-practice. Next, we look at selections from the mathematics education literature that discuss how and when students advance through higher levels of mathematical maturity to reach a point at which they can cope with the demands of rigorous formalism. To probe the integration of Lean into teaching from an empirical point of view, we conducted an exploratory study that investigated how three undergraduate students approached the proof of double negation with Lean. The findings suggest that the rigorous nature of Lean is not an obstacle for students and does not stifle students’ creativity in writing proofs. On the contrary, proving with Lean offers a great deal of flexibility, allowing students to follow different paths to creating a valid proof.
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Notes
Grenier & Payan (2006) explore the classroom use of generalisations of the chequerboard problem, namely whether a shape composed of squares of a given size can tile a larger shape composed of squares the same size, under various constraints. One might, for example, ask students to explore whether it’s possible to tile a ziggurat with an L-shape rather than a domino. Where a tiling is provably impossible, will the strategy of the original proof always work? Alekhnovich (2004) discussed the challenge that the chequerboard presents to automated reasoning systems. Tanswell (2015) argues that the various formalizations of the basic chequerboard proof are sufficiently different from each other that we cannot speak of the formalization of an informal proof.
Erdős said that you need not believe in God but, as a mathematician, you should believe in The Book (Aigner and Ziegler 2010, v).
This label is both appropriate (because Leibniz did indeed foresee the potential of formalized mathematics) and misleading (because Leibniz’s philosophy of mathematics was intimately related to his views about logic and metaphysics in ways that are not relevant to the present discussion).
“Die Mathematik ist ein BUNTES Gemisch von Beweistechniken.” (Capitals and italics in original). Hacking discusses the difficulty of translating ‘BUNTES Gemisch’ (Hacking 2014 pp. 57–58). Anscombe’s translation has ‘motley’. A more direct rendering might be a ‘BOUNTEOUS mixture’, that is, a mixture of wildly heterogenous items. A miscellany.
This is due to the Lean Mathematical Library https://leanprover-community.github.io/mathlib-overview.html, a community-driven collective effort to build a unified library of formalized mathematics. Over 300 contributors to the library have formalized more than 73K definitions and 134K theorems across a dozen mathematics topics in undergraduate mathematics, such as linear algebra, group theory, vector space, calculus, real analysis, and complex analysis.
P → (Q \(\to\) P) in Lean is written as P → Q \(\to\) P since implication is right-associative in Lean, meaning P implies that Q implies P.
How to prove it with Lean by Velleman.
The Natural Number Game is available in Lean 3 (https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_game/index2.html) and Lean 4 (https://adam.math.hhu.de/#/g/leanprover-community/nng4), developed by developed by Kevin Buzzard and Mohammad Pedramfar.
The Set theory Game: https://adam.math.hhu.de/#/g/djvelleman/stg4, developed by Daniel Velleman, Alexander Bentkamp, Jon Eugster, and Patrick Massot.
Lean for Teaching on ZulipChat; Formalization and mathematics by Matthew Ballard; Proving equalities in Lean by Heather Macbeth; Introduction aux mathématiques formalisées by Patrick Massot.
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We wish to acknowledge the generous support of the Social Sciences and Humanities Research Council of Canada. We also thank the anonymous reviewers for their valuable comments and suggestions.
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Appendix
Appendix
Tactics used and changes in goals in a proof of 0 + n = n
To prove that if n is a natural number, then 0 + n = n.
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The “induction” tactic proves the goal by induction on n, with the inductive assumption in the succ case (successor case) being “hd”. Specifically, “induction n with d hd” turns the goal into two goals, a base case with n = 0 and an inductive step where “hd” is a proof of the n = d case.
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The “norm_num” tactic normalized any numerical expressions.
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The “rw” (rewrite) tactic: given a hypothesis (h) of the form A = B, “rw h” replaces occurrences of A with B.
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Hanna, G., Larvor, B. & Yan, X.K. Using the proof assistant Lean in undergraduate mathematics classrooms. ZDM Mathematics Education (2024). https://doi.org/10.1007/s11858-024-01577-9
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DOI: https://doi.org/10.1007/s11858-024-01577-9