I am looking for concrete information on "induction" in the old sense of the word. Now it is used to refer to proof by "mathematical induction"; however, I am aware that it also means inductive reasoning, apparently a type of reasoning you can use to discover theorems.

My main concern is whether there is anything of substance in this. To illustrate what I mean, here is an example.

When I was reading an old geometry book, the author kept using words like "synthesis". At the time I thought he just fancied using this term to mean "combining the previous theorems, we can prove this", or perhaps it was just a word whose real meaning was lost in translation. The point is that I had no idea there was a deeper meaning to this word: The Greek method of synthesis. And after reading up on this in a 17th century textbook, I realized how powerful the method actually is, and I went on to use it to solve geometry yproblems that I literally could not solve before. (Proof, I missed one of them in a math competition I took last year, and still could not solve it on my own a year later, but I magically became able to solve it after reading about this method.)

In that context, my question is whether the old meaning of the word induction also has a deeper meaning to it than say the one Polya gives in his book "How to solve it". I am talking about the one that Gauss refers to when he says "we discovered this theorem by induction", referring to the Quadratic Reciprocity Law. What exactly is this "induction"? Is it a general method for discovering mathematical truths? Is it a long forgotten gem like the synthetic approach I talked about above?

Sorry if anything isn't clear; please ask and I'll try my best to clarify.

EDIT: William Whewell's words on the man who devoted his life to mastering the Greek method of synthesis (that is, Isaac Newton):

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I fear there are similar things to say about say Euler and induction, if it weren't for the fact that so many people do not know what induction means, or for the fact that the word induction has now practically been over-written by mathematical induction.

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    $\begingroup$ I added the terminology and math-history tags, because it seems to me that this is very much a question of what past terminology means. Feel free to remove them if you feel they don't belong. $\endgroup$ – Hayden Mar 5 '15 at 19:12
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    $\begingroup$ Another name for "induction" is "pattern recognition". Note the conclusions reached through such reasoning are often invalid, and need to be "tested" by some means. $\endgroup$ – David Wheeler Mar 5 '15 at 19:13
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    $\begingroup$ Well, in some areas, induction is a valid form of reasoning-for example, if a system is "periodic", one might "guess" it is so (or even develop the notion of "periodic") from a limited observation. Even a "wrong guess" may be useful-witness the usefulness of pendulum clocks, even though they don't keep "perfect time". Logically, "induction" means to "draw/lead on/in", from the Latin in- and -ducere (to lead, or draw), speculation of what is not yet known, from what is. In a sense, it is the foundation for the scientific method-things appear to follow rules, so we attempt to discover them. $\endgroup$ – David Wheeler Mar 5 '15 at 19:28
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    $\begingroup$ @Ark You might be interested in An Inquiry Concerning Human Understanding by David Hume. $\endgroup$ – David Wheeler Mar 5 '15 at 21:26
  • $\begingroup$ You can see The Problem of Induction for contemporary debates and Inductive Logic. You can see also : Ian Hacking, The Emergence of Probability (1975), id, The Taming of Chance (1990) and id., An Introduction to Probability and Inductive Logic (2001). $\endgroup$ – Mauro ALLEGRANZA Mar 6 '15 at 10:44

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