There was a popular anecdote about mathematician G. H. Hardy (1877 – 1947) that goes like this: While conducting a lecture on Number Theory, he said that a certain mathematical notion was trivial. But after a little while, he hesitated and asked, “Is it trivial?” He then excused himself and went to his office. After thinking about it for half an hour, he went back to the lecture hall and finally concluded, “Yes, it is trivial.”

When Hardy was asked about the validity of the story, he repudiated it. He said that at some point in the past, he might have said, “This is trivial,” hesitated for a moment asked, “Is it trivial?”, and then after a pause, said, “Yes, it is trivial.” However, he never left the lecture hall to ponder about it.

“All of which goes to show that it doesn’t pay to look too closely into the truth of many an anecdote,” mathematical writer Howard Eves remarked, “if one does not wish to lose the story.”

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In my graduate level algebra course, we were allowed to use “…and the rest of the proof is trivial” if it really was. Some tried to use it to cover up the fact that they didn’t really know how to finish the proof; but, my professor nearly always caught on. When I tried it, I ended with “and the rest is trivial”

He wrote below it, “I don’t see how”

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Any new bloggers??

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When you read this comment you will understand that it is trivial.

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I enjoyed reading this as it reminded me of teaching and times when I would do a similar thing! I guess it stems from always having an inquiring mind and what I call my ‘popcorn’ brain. Thanks for sharing and making me pause and reflect this morning.

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So there was an interesting story, and then someone asked: “is this trivial?”. And then they found that the answer as “yes, this was actually trivial.”

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So much for trivialities…

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Good to hear Hardy’s version of events! Sometimes you do have to check if something’s trivial, but if the checking takes a while, it isn’t.

You may already know about it, but you’d probably enjoy Steven G. Krantz’s book “Mathematical Apocrypha.” I read some of it years ago, and keep meaning to track down a copy and read the rest.

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Funny you should say that. In pure mathematics literature, it’s easy to find lines like “It is obvious”, “It is trivial” and the classic “The proof is left as an exercise to the reader.” as if it’s really the case. However, in several instances, it’s not really. There are many notable examples of this. Krantz’s book related one example involving Shizuo Kakutani and a supposedly “obvious” lemma.

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I’m glad you know about this book already! In my experience, the moments in a proof when I’m most tempted to say “clearly…” or “it is obvious that…” are the moments when I’m most likely to be leaving a hole in my proof. The fact that I’m having trouble expressing why something’s true should be a clue that it might not be!

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