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The Turing award winner and father of LaTeX thinks the proofs you and everyone else are writing are sloppy, non-rigorous and quite likely flat-out wrong. Sir Michael Atiyah is not quite so sure. He also has a paper explaining the same ideas. He imagines a possible future of proofs in hypertext, with collapsible layers of explanation, optional sketches and other extras to help guide you through a dense proof.
Sir Michael Atiyah was not wholly convinced either. Later in the conference, I asked Atiyah about his recent work on the Feit-Thompson Theorem a result from my area of maths, finite group theory. He has a proposed new proof, an order of magnitude shorter than the current page behemoth.
Perhaps both ways of looking at these things are necessary. Understanding the high-level structure behind a theorem is bound to be necessary to place the result in its context and learn about the wider field.
Checking the nitty-gritty details will always be necessary, and as long as published proofs exist that help with neither of these very successfully, advances in either area are to be welcomed.How to Write a 21st Century Proof Leslie Lamport 23 November Minor change on 15 January Abstract A method of writing proofs is described that makes it harder to prove things that are not true.
The method, based on hierarchical structuring, is simple and practical.
The author’s twenty years of experience writing such proofs. TLA+ is a specification language based on standard set theory and temporal logic that has constructs for hierarchical proofs. We describe how to write TLA+ proofs and check them with TLAPS, the. I began writing proofs the way I and all mathematicians and computer scientists had learned to write them, using a sequence of lemmas whose proofs were a mixture of prose and formulas.
I quickly discovered that this approach collapsed under the weight of the complexity of any nontrivial proof. How to write a 21st century proof Leslie Lamport To D. Palais Abstract. A method of writing proofs is described that makes it harder to prove things that are not true. How to Write a Proof Leslie Lamport Microsoft Research 1 Proof Suppose that f is a one-to-one map of A into B and g is one to one on B to A.
It may be supposed that A and B are disjoint. The proof of the thoerem is accomplished by decomposing A and B into classes which are most easily.
May 14, · How to Do Math Proofs. Mathematical proofs can be difficult, but can be conquered with the proper background knowledge of both mathematics and the format of a proof.