How To Read Proofs Peter Suber, Philosophy Department, Earlham College

1. Know the formalism (notation) intimately. Use it in practice, and make it as transparent to you as English.

2. Understand the justification for each step. Skip obscure ones on first reading, but eventually go back and master them. When you appreciate the validity of each step, you may still not understand the whole proof as a single act, but you will have laid the necessary foundation.

3. Restate or summarize proofs in prose of your own words.

4. Translate notational wffs into English. Translate English into notation or into new English. Restate proofs until you "own" each step.

5. In your restatement, describe "the thread" of the proof, the gist or general movement of thought, even if this means dropping detail. If you can be clear in a few sentences, then you might be able to grasp this "thread" in one act of consciousness.

6. Chunk long proofs into a series of short "stanzas" or sub-proofs. If there are major sub-conclusions with their own sub-proofs, then consider them lemmas and remove them from the larger proofs.

7. Number the steps or use indentation to show sequence and subordination.

8. Isolate and identify the ingenious, obscure, or important steps.

9. Visualize relations or draw them graphically.

10. Write down your questions about obscure or fascinating proofs. State them precisely in full sentences. (Sometimes making our questions fully explicit helps us to see the answers.)

11. Make your conjectures explicit; write them down. As the course goes on, try to prove or disprove them.

12. Go to the library. Look up other formal or informal renderings of the same proofs.

13. Relate parts to the whole and each other, or to some field or application familiar to you (e.g. arithmetic, set theory, computer programming, philosophy...).

14. Explain the material to peers, over beers.

15. Play; explore; wonder; experiment.

This file is an electronic hand-out for the course, Logical Systems.