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The technique of "mathematical induction" is a method of proof where you show some theorem is true for some starting integer and prove also that it holding at any arbitrary integer implies it must hold at the following integer. From this you conclude it must hold for all integers from the starting point on.

I think most people would consider this "obvious". So obvious in fact that I always considered it as dating back to the earliest history of mathematics, such as to the ancient Greeks. But I was teaching a course in Discrete Math and was surprised to read in the text that the first explicit explanation of the principle was from a mathematician in the 1500's named Francesco Maurolico: http://en.wikipedia.org/wiki/Francesco_Maurolico

How could it have taken so long for mathematicians to discover it? It makes you wonder if there are also other obvious principles out there that have not been yet explicitly enunciated.

Another point: European math and science in general was in deep decline during the Dark Ages. But Chinese and Indian and Arabic math continued to make great advances during this period. I wouldn't be surprised if we had greater records of the mathematicians in these other cultures during that period that they had stated the principle earlier.

Logan M
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Robert Clark
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    Dude, there are so many theorems and proofs that was lost through history and theorems that was credited to the wrong people ! –  May 29 '15 at 01:56
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    Well, the modern statement of induction is not that big of a deal. For instance, one can avoid it by showing that "there does not exist the first element disproving the therem". So, even when considering the mathematical content alone, it would not be so surprising that no one bothered formalising it for centuries. –  May 29 '15 at 02:17
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    I would have expected that idea to show up with the Greeks already. Isn't recursion an induction in the reverse direction? What about euclids proof on the existence of infinite many primes? –  May 29 '15 at 02:18
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    This question is already answered here http://hsm.stackexchange.com/questions/524/who-introduce-the-principle-of-mathematical-induction-for-the-first-time Greeks did use instances of what we call induction, but there would have been a major problem with modern formulation of it, because most of them rejected existence of actual infinity. – Conifold May 29 '15 at 03:41

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