Whatever is the right definition of the proof (most of us feel it in the guts when something is doubtful while something else is certain, i.e. proven), every one agrees with the definition. The definition is immutable.
This is not true for several reasons. For example, there always were mathematicians who feel uncomfortable with the notion of infinity. Using the Axiom of Choice Tarski and Banach have shown that it's possible to split a tennis ball into a few parts that could be combined again only to produce a ball of the size of our Earth. Many feel that the axiom is not a reliable tool. However, most mathematicians would not hesitate to use it.
The notion of proof underwent very dramatic changes in the 19th century when, for many
reasons, mathematicians began questioning their intuition. Euclid's Elements define
a straight line as a breadthless length. In 1945, in a Russian math olympiad, an 8th grade boy who did not even attempt to solve but one problem received a first prize for a remark he submitted with an unfinished proof of that problem:
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I spent much time trying to prove that a straight line can't intersect three sides of a triangle in their interior points but failed for, to my consternation, I realized that I have no
notion of what a straight line is.
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In the 19th century mathematicians who questioned the 2300 year old intuition about straight lines discovered non-Euclidean geometries that, in the 20th century, were incorporated by Einstein into his General Theory of Relativity.
If intuition permeates any mathematical activity, how is it possible to build a theory that
prides itself in its rigor on top of something rather elusive and individual? A historic example will shed some light on how this happens.