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Getting Better at Proofs.md

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Introduction

  • I summarised the references that I found online regarding tips to get better at proofs

"Getting better at proofs" on math.stackexchange

  • One problem with the "practise, practise, practise" mantra

    • Practise what? Where are the lists of similar-but-not-quite-identical things to prove to practise on? I can find lists of integrals to do and lists of matrices to solve, but it's hard coming up with lists of things to prove.

  • Reading others proofs is not guaranteed to give you any insight as to how the proof was developed

    • A proof is meant to convince someone of a result, so a proof points to the theorem (or whatever) and knowing how the proof was constructed does not (or at least, should not) lend any extra weight to our confidence in the theorem.

  • One recommendation is that you take the statement that you want to prove and apply the following steps to it as often as you can:

    • Expand out unfamiliar terms

    • Replace generic statements by statements about generic objects

    • Include implicit information

    • Example is

      • Original statement

        The composition of linear transformations is again linear

      • Replace generic statements

        If S and T are two composable linear transformations then their composition, ST, is again linear

      • Include implicit info

        If S: V -> W and T: U -> V are linear transformations then ST: U -> W is again linear

      • Expand out definitions and Replace generic statements, and reorganise to bring choices to the core

        • At this point, what we need to prove should be clear!

Polya’s Problem Solving Techniques

  • Principles
    1. Understand the problem
      • Do you understand all the words used in stating the problem?
      • What are you asked to find or show?
      • Can you restate the problem in your own words?
      • Can you think of a picture or diagram that might help you understand the problem?
      • Is there enough information to enable you to find a solution?
    2. Devise a plan(Some of the ones mentioned in the Polyak's book)
      • Use direct reasoning or a formula
      • Look for a pattern
      • Draw a picture
      • Solve a simpler problem
      • Work backwards
      • Consider special cases
    3. Reflect and look back at what you have done

Problem solving strategies in Analysis by Terence Tao