A Dictionary For Reading Proofs
What those phrases really mean
Eduardo Tengan is a mathematician at the Institute for Mathematical Sciences and Computation in Sao Paulo, Brazil. He has written a delightful set of notes titled, “An Invitation to Local Fields.” He also has a great sense of humor. Go check his home page for a proof.
Today I thought I would quote some of his cool comments and add some of our own.
This is not a heavy theory discussion. In fact we are leaving the discussion of local fields themselves to our later agenda. But it is not all fun. I must confess to have written some of these phrases in my own papers and notes. In any event I hope that you enjoy these.
Tengan in his paper on Local Fields has an early section heading that poses the question,
Should I read these Notes?
His answer is:
Well, the answer to this question is of course up to you. But here are some of the “lollipops” that you may miss if you decide not to.
Some Commonly Used Terms
Here are Tengan’s proof terms and his own definitions of them:
- CLEARLY: I don’t want to write down all the “in-between” steps.
- RECALL: I shouldn’t have to tell you this, but
- WLOG (Without Loss Of Generality): I’m not about to do all the possible cases, so I’ll do one and let you figure out the rest.
- CHECK or CHECK FOR YOURSELF: This is the boring part of the proof, so you can do it on your own time.
- SKETCH OF A PROOF: I couldn’t verify all the details, so I’ll break it down into the parts I couldn’t prove.
- HINT: The hardest of several possible ways to do a proof.
- SIMILARLY: At least one line of the proof of this case is the same as before.
- BY A PREVIOUS THEOREM: I don’t remember how it goes (come to think of it I’m not really sure we did this at all), but if I stated it right (or at all), then the rest of this follows.
- PROOF OMITTED: Trust me, it’s true.
- UNFORTUNATELY, DUE TO RESTRICTIONS OF TIME AND SPACE: The author is lazy.
Ken recalls from the mid-1980’s a meeting of a regular weekly seminar with Peter Neumann in his rooms at Queen’s College, Oxford, in which Peter talked about the life and ideas of ‘Evariste Galois. He showed a replica of manuscript pages of Galois’ famous testamentary letter. Near the top in plain French it has the dreaded phrase:
Il É de voir que …
(It is easy to see that )
Especially in lecture notes, unless you really mean to set a check-for-yourself exercise, this is perhaps the most important one to avoid.
Galois of course had an excuse in 1832. He was writing in great haste the night before the duel in which he would perish. The letter was published a few months after his death in its entirety. But we don’t have that excuse, or hopefully you don’t.
What are some other examples?