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I know that starting from the 19th century the style of mathematical proof was more and more formal and inductive. However, I am trying to find reference (article or books) that analyze the way which mathematicians proved their claims during the 16th-18th century. I assume these methods were pretty different from what we know from the 19th and 20th century.... any reference would be appreciated!

Best, David

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You oversimplify what really happened to the standards of rigor. Rigorous proofs were invented by ancient Greeks. (According to their own tradition, by Thales, but his writings do not survive). At the mature stage of Greek mathematics (Euclid, Archimedes, Apollonius) a very high standard of rigor was established.

It changed little since then to the modern time, if one speaks of the mainstream mathematics (excluding the "new" areas of mathematical logic, set theory and foundations).

With the invention of calculus, this standard declined, because of the lack of foundations of calculus. Still at the time of Newton, best mathematicians perfectly understood this ancient standard; they just could not expose calculus on the same high standard as the Greek geometry. And I suppose this is the main reason why Newton preferred to use classical geometric methods (rather than calculus) in his Principia.

In 18th century people were more concerned with discovery of new things and in applications, then in the standards of rigor. So the standards declined. But in 19th century, by a long and difficult gradual process, this standard was finally achieved again. Rigorous foundation of calculus emerged.

In the end of 19th century, the attention to foundations increased. This followed by a crisis in foundations in the beginning of 20th century, until some satisfactory situation was finally established. Again, this crisis in foundation did not really affect much the work of most mathematicians.

Those mathematicians who are not interested in foundations write on approximately the same standard of rigor which was established by the Greeks, leaving the care about foundations to the specialists. (This is somewhat oversimplified picture: there is some important interaction between foundations and the rest of mathematics, but I cannot go into detail here).

In all epochs, mathematicians were sometimes more concerned in discovery of new things, then in rigorous justifications. Archimedes himself wrote a famous book with heuristic discoveries. He perfectly understood that the rigorous proofs were lacking. In the modern times, there is also plenty of such kind of research, I mean of the boundary of mathematics and physics. Where people use "semi-mathematical" reasoning to discover new things, and don't know how to prove these things rigorously.

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