- Opinion
- 30 de January de 2025
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- 5 minutes read
What are the masterpieces of mathematics?

What are the masterpieces of mathematics?
Mathematical proofs are sequences of logical thought. They are, in a sense, a bridge to philosophy and, more importantly, the link between mathematics and the humanities

In 1990, William Dunham published the book Journey through Genius: The Great Theorems of Mathematics. This book is based on the premise that almost all artistic disciplines have catalogued their masterpieces, which can be admired in museums. The humanities also have a clear sense of which books or essays should be prioritised to better understand the human condition, and these are promoted in museums, libraries, or exhibitions. I could continue drawing parallels with music, architecture, and so on. For someone visiting one of these spaces, it is useful to know that, in order to fully grasp that discipline, they must study these works of art. But mathematics lacks such a selection of its masterpieces.
It is true that mathematics museums exist, but they are usually spaces designed to spark an interest in mathematics and logic. Many of the exhibits are aimed at children and teenagers, as if a painting museum were filled not with great works of art but with brushes, canvases, and other tools to help visitors begin painting a portrait. This type of museum is certainly necessary, in part to counteract the influence over children of those who proudly display their mathematical ignorance—an attitude they would likely try to conceal in any other discipline.
However, what mathematics museums lack is precisely what Dunham proposed in his exquisite book: a selection of the great mathematicians, their theorems, and their proofs.
His selection provides a historically contextualised insight into how the ideas of the greatest mathematical minds have shaped our understanding of the world. It reveals the true intrinsic beauty of mathematics: the proofs. Logic and precision are required to deduce any of the theorems we know today, starting from Euclid’s well-defined axioms. And yes, beauty is required too.
The proof as a work of art. In classical Greece, they understood that simply stating what seemed intuitive was not enough—it had to be verified, deduced step by step from what was already certain.
For example, they proved that the diagonal of a square with side length one is not a rational number, contradicting the supposed perfection of the “commensurable” world as they understood it. This proof uses reductio ad absurdum, a method that involves assuming a hypothesis and logically deriving consequences until a contradiction is reached. The error, therefore, lies in the hypothesis itself. This discovery was a shock for the Pythagoreans, who regarded whole numbers as supreme, and, as legend has it, they threw Hippasus into the sea as punishment. Humanity has not always been ready to accept proofs that dismantle preconceived ideas… or has it ever been?
Archimedes used iterations to approximate the value of pi, inventing an entirely new approach to numerical exploration. Euler applied the method of induction to prove Fermat’s Little Theorem. Cantor, through bijections, demonstrated that there are as many natural numbers as rational numbers, but that the set of real numbers belongs to a higher order of infinity. His proof invites us to ponder the paradox of the infinite hotel.
All these proofs, and many more that I have not listed, provide us with methods of reasoning—logical structures for chaining deductions together, which many people have never trained in, and which took humanity centuries to conceive and apply.
Thus, mathematical proofs are sequences of logical thought. They are, in a sense, a bridge to philosophy and, more importantly, the link between mathematics and the humanities.
At a time when certain populist narratives—ones that would not withstand even a reductio ad absurdum—are gaining followers, should we not give mathematical proofs the importance they deserve? Should we not ensure that everyone can appreciate these mathematical works of art, which form an essential part of universal thought?
And let us not forget the most important point: great mathematical proofs are beautiful in their own right, and they compel us to question some of our deepest intuitions.
Source: educational EVIDENCE
Rights: Creative Commons