Indispensable counterexamples
Just as a small drawing is worth more than a long speech, in the realm of mathematics, nothing beats a counterexample to refute a false intuition or an erroneous conjecture. While it is fascinating (or sometimes amusing) for enthusiasts, the search for counterexamples is not a mere distraction. It is fundamental, both in the reasoning process and as a pedagogical tool. It has punctuated the development of mathematics, allowing for example the theory of functions to take shape. More recently, programs using artificial intelligence have highlighted sophisticated counterexamples, refuting numerous conjectures, particularly in graph theory.
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The exception that does not prove the rule
A persistent belief holds that counterexamples are primarily a source of amusement. Yet they play a fundamental role in several areas, both as mathematical proofs and as teaching tools—not to mention their entertaining, or even artistic, side, depending on how one looks at them.
From intuition to rigor:
From Cauchy to Weierstrass, rigor steadily took hold in analysis throughout the 19th century. A variety of counterexamples swept away mistaken beliefs and forced mathematicians to define the relevant concepts more precisely. Some "monstrous" functions were introduced, fascinating to some and repellent to others.
Artificial intelligence to the rescue
Using computers to solve mathematical problems is nothing new. Adam Zsolt Wagner of Tel Aviv University (Israel) has now shown how artificial intelligence can uncover counterexamples to several previously open conjectures. A promising path for the future?
Schwarz's theorem
From Euler to Cauchy, via Clairaut, no one doubted that reversing the order of partial differentiation left the result unchanged. Hermann Schwarz overturned this belief with a superb counterexample.
Hypercube: a mathematical conjecture is disproved | Tangente
The challenge: covering sets of points with as few hyperplanes as possible, particularly sets consisting of certain vertices of the hypercube.