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A Dictionary For Reading Proofs

September 27, 2013

What those phrases really mean {\dots}


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 {\dots}
  • 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.



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 est aisé de voir que …

(It is easy to see that \dots )

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.

Open Problems

What are some other examples?

24 Comments leave one →
  1. September 27, 2013 5:42 pm

    Nice notes. Did you know there is no version Rolle’s theorem in padic local fields? Perhaps this is the main reason for their scarcity in TCS. How can you use local fields in TCS?

  2. September 27, 2013 5:55 pm

    I regularly tell my students to read “clearly” and “obviously” as synonyms for “probably false”.

    • rjlipton permalink*
      September 29, 2013 10:04 am


      Good advice. An early proof by a star of a key theorem used “obviously”.

  3. September 27, 2013 6:50 pm

    THE PROOF DOES NOT FIT IN THE MARGIN: I’m lazy, probably wrong, and I know who Fermat was.

  4. Russell Impagliazzo permalink
    September 27, 2013 7:09 pm

    My personal definition of “clearly” in a proof is “This is where the mistake is”

    • Russell Impagliazzo permalink
      September 27, 2013 7:10 pm

      Sorry, I didn’t see that Jeff E. beat me to this one.

    • rjlipton permalink*
      September 29, 2013 10:03 am


      I love this one.

    • October 5, 2013 8:59 am

      David Preiss used to present this in a sort of “converse” form: if you are suspicious of a purported proof and want to find the mistake as quickly as possible, then look for the words “clearly” or “it is easy to see that”.

  5. September 27, 2013 7:27 pm

    Elementary, my dear Watson.

  6. September 27, 2013 8:34 pm

    on jugera

  7. September 27, 2013 8:36 pm

    “Read my ellipsis …”

  8. September 27, 2013 11:14 pm

    “Left to the reader”

    I would never say this if its proof is directly related to the main body of the proof.

  9. John Sidles permalink
    September 28, 2013 9:05 am

    Vladimir Arnold’s Contact geometry: the geometrical method of Gibbs’s thermodynamics (1990) begins

    “Every mathematician knows that it is impossible to understand any elementary course in thermodynamics.”

    This translates as:

    Thermodynamicists will find it similarly impossible to understand the elementary geometric framework that my article will now present.

    As Kurt Vonnegut’s novels remind us: So it goes.”

  10. September 28, 2013 10:43 am

    You could find the proof to $P=NP$ at this link here whose loading time is undecidable.

  11. September 28, 2013 11:03 am

    Not sure if it’s just me, but the links to both the notes and home page of eduardo tengan lead to a page telling me I’m not allowed to view them in Brazilian (Acesso proibido)

    • September 28, 2013 12:49 pm

      This seems to be true of the Math Dept. page on the whole, as linked from the Institute (which is still working). Of course we mouse-copy links into our source text—this causes an occasional problem with subscription-access links, but we don’t see fit to warn when we can’t find an appropriate free link.

  12. Tommy Pensyl permalink
    September 28, 2013 8:49 pm

    BY ABUSING NOTATION: I’m lazy, but at least I’m forthright about it.

    This excuses the offending notation about as much as shouting “LEFT ON RED!” excuses a driver from obeying basic traffic laws.

  13. September 29, 2013 3:27 pm

    “In the fifties, it was said in Princeton that there were four definitions of the word “obvious”.
    If something was obvious in the sense of Beckenbach, then it is true and you can see it immediately. If something is obvious in the sense of Chevalley, then it is true and it will take you several weeks to see it. If something is obvious in the sense of Bochner, then it is false and it will take you several weeks to see it. If something is obvious in the sense of Lefschetz, then it is false and you can see it immediately.” (S.G.Krantz, Mathematical anecdotes. – Math. Int., 1990, vol.12, N.4, p.32-38.)

    • Serge permalink
      September 30, 2013 10:35 am

      I’d say that Bourbaki members such as Jean Dieudonné, Jacques Dixmier or André Weil were of a similar type as Edwin Beckenbach, whereas Jean-Pierre Serre or Alexander Grothendieck were more like Claude Chevalley…

      I think it all depends on whether the text is intended at students or at researchers. At elementary level, lazy practices are very unpleasant too but not too harmful anyway, since the full proofs can be found in several other sources. However, some teachers are more honest than others and when they say that something is clear or is left as an exercise, they actually mean it. And sometimes they simply write “proof omitted”, which means the requirements for the proof exceed the level of the course.

  14. September 30, 2013 3:18 am

    Well-known computer sci. problem: very often in non-trivial cases, source code of correct program looks like “obvious”, but it is insufficient for human understanding of used algorithm. To describe new algorithm we prefer a text in English and pseudocode rather than listing in programming language.

  15. Serge permalink
    September 30, 2013 10:20 am

    Galois actually wrote the following:

    Il est aisé de voir que quand le groupe d’une équation n’est susceptible d’aucune décomposition propre, on aura beau transformer cette équation, les groupes des équations transformées auront toujours le même nombre de permutations.

    In English:

    It is easy to see that when the group of an equation is not liable to any proper decomposition, in whichever way this equation is being transformed the groups of the transformed equations will always have the same number of permutations.


  1. A Dictionary For Reading Proofs | Rocketboom
  2. Details Left To The Reader… | Gödel's Lost Letter and P=NP
  3. Peter M. Neumann, 1940–2020 | Gödel's Lost Letter and P=NP

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