My answer to a question on Quora: **Do mathematicians feel embarrassed when a conjecture they claim is disproved by counter-example?**

I do not remember seeing a mathematicians who was embarrassed by their conjectures disproved.

Why? Because making conjectures and refuting them is a normal cycle of mathematics. I think 90% of conjectures die on the same writing desk where they were born, being killed by the same mathematicians who formulated them. In mathematics, it is a daily routine. **Refutations** are as important as proofs. There is a famous book about the role of refutations in mathematics, Imre Lakatos’ *Proofs and Refutations*.

And the famous Lewis Carroll’s lines in *Through the Looking-Glass *capture the spirit:

“I can’t believe that!” said Alice.

“Can’t you?” the Queen said in a pitying tone. “Try again: draw a long breath, and shut your eyes.”

Alice laughed. “There’s no use trying,” she said: “onecan’tbelieve impossible things.”

“I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for half-an-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.”

Proofs and refutations co-exist in the most natural way. Mathematical problems are conjectures. To solve a problem means to prove this conjecture or refute it.

Proofs are frequently done by constructing, in parallel, a counter-example: when a mathematician identifies obstacles for a proof, he/she may wish to try to use them to construct a counterexample; when this attempt at refutation encounters its own difficulties, a mathematician may try to isolate these difficulties and understand their nature – for use in the proof. In this zig-zag movement the aims — to prove a conjecture and refute it — alternate. In a happy outcome , the process converges on a definite answer: either proof or refutation.

But, if you look back at that zig-zag prowl in search of a kill, you may say that **half of the time the mathematician believed impossible**. Even worse, it is like lions in hunt: ten chases result in one kill; a mathematician normally solves about one problem out of ten that he or she tries.

There is one extreme case of the proof/refutation balance: the original proof of the Classification of finite simple groups. I quote Wikipedia:

The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004.

As a rule, almost each of these “*several hundred journal articles” *contains a proof of a particular theorem, a special case or an intermediate step of the “global” statement. Since all that is about finite objects, proofs frequently use mathematical induction in a specific form: proof of **non-existence of a minimal counterexample** to the theorem. As a result, it makes thousands of pages of arguments about non-existent objects. At a first glance, it gives an answer to another question on Quora: What are some aspects of mathematics that are nonsense? But these arguments about eventually non-existent minimal counterexamples are **not nonsense** — for example, they can be re-used in proving theorems in other branches of group theory.