The themes of scientific knowledge are very varied. Perhaps the result of an infinite curiosity, we study mathematics, physics, medicine and physiology, and even the social history of jazz. Some sciences end up being called exact; others, biological. And there are also applied social ones, such as law and economics. But what classification is this? Why is mathematics said to be exact? And who said math is science?
The debate about mathematics as a science is profound, involving several ideas. It is undeniable that it gathers relevant knowledge. However, it is not evident that research in mathematics depends on empirical evidence. More subtle is that mathematical theses are not falsifiable, in the sense proposed by the Austro-British philosopher Karl Popper in 1934.
Popper suggests that no experiment (or set of experiments) is capable of demonstrating a scientific thesis. But evidence to the contrary is enough for the theory to be discarded. Even though all the flamingos I’ve seen are pink, that doesn’t guarantee the validity of the thesis that all will be. If there’s a single gray flamingo out there, we’ll know the thesis is false.
Now, once demonstrated, a mathematical proposition cannot be falsified. Let’s think about the Poincaré Conjecture. Formulated by Henri Poincaré in 1904, it claims that any three-dimensional object with certain topological properties is a sphere: a soccer ball, even if it withers, is a ball. Earlier in this century, Grigori Perelman demonstrated that the conjecture is true. And its demonstration is independent of empirical observations. It is impossible for a research group to emerge with results that invalidate the conjecture.
Overcoming this first layer of debate, although far from a conclusion, comes another question: mathematics as an exact science. Now, a science is exact if it produces precise conclusions and results. An example is the sensational “two plus two”. The climax of popular wisdom, the expression “as sure as two plus two equals four” has a serious problem. It’s just that two plus two can be, say, zero. And it all depends on where “we are” or the initial premises.
But that’s okay: a mathematical argument depends on assumptions and axioms. Be it those of Euclid (there’s the point!), or those of set theory—as the fundamental Axiom of Choice. In other words, once the initial definitions are established, the mathematical procedures are precise and unambiguous. But establishing the premises is a free exercise. A friend says that in mathematics you can define anything: it is difficult to prove interesting facts about it.
Now what is an interesting fact in mathematics? Who decides if a question is interesting? In the early 20th century, David Hilbert announced a list of 23 problems he thought were the most important in mathematics. Known as Hilbert’s Problems, they immediately became questions that interest all mathematicians—and some of them remain without a complete answer. On a more rarefied (and current) scale, the influence of the editorial system – and of project financing – is also important. Mainly because it reflects the paradigm of the relevant scientific community. In other words: it is possible that the decision about which problems are important in mathematics is, above all, a social phenomenon.
But perhaps the most curious thing about this conversation is the mistaken impression that mathematics is about mathematics: nothing more inexact. In 1947, Kurt Gödel appeared in Trenton for an audience about his US citizenship—it is even said that he traveled from Princeton to Trenton with friends Oskar Morgenstern and Albert Einstein. Knowing that he would be examined at this hearing, Gödel studied the country’s constitution and noticed logical contradictions: one of the model constitutions of modern democracy would admit the emergence of a dictator and an authoritarian regime. The story goes further: when announcing his discovery to the judge who was examining him, he would have changed the subject and closed the hearing.
And if the subject is to design a voting system, mathematics has important considerations. Suppose a society needs to choose between some alternatives. Is there a rule that makes individual preferences a social choice and still meets reasonable criteria? Arrow’s Impossibility Theorem is emphatic that such a rule simply does not exist.
Exact, formal, or just fun: maybe it’s not even important to classify math. Regardless of the label, the problems that bring light to our eyes are still around: Hilbert’s, Emilia’s, mine, yours. Exactly must be our courage to attack them.
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Edgard Pimentel is a researcher at the Center for Mathematics of the University of Coimbra and a professor at PUC-Rio.
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