An essential requirement to be able to enjoy playing chess – or backgammon, go or checkers – is to internalize the rules and basic movements of the pieces. Saving the distances, something similar happens with mathematics, whose basic rules are called axioms or postulates. These are fundamental principles that allow the development of mathematics, within a coherent ontological framework. The fundamental difference between the axioms and the results –or theorems– is that the latter are obtained from the axioms using a finite number of logical deductions, while their truthfulness is implicitly assumed from the axioms, without trying to prove them.
Contrary to the common view that everything is prescribed in mathematics, different choices of axioms can lead to different mathematics. So how do we choose those foundations on which we will build mathematical thinking? Is there freedom in choosing them? The truth is that the selection is neither fortuitous nor random; There are two fundamental characteristics that stand out from a good choice. On the one hand, they must be natural and intuitive laws that represent the universal character of mathematics, but, on the other, they must also be malleable and generic enough to allow us to deal with unimaginable mathematical universes. In particular, if one axiom can be proved from the others, it is redundant and unnecessary.
Like any rule or law, the axioms are not exempt from being modified and even rejected by the mathematical community
However, like any rule or law, the axioms are not exempt from being modified and even rejected by the mathematical community. Good example of it was what happened with the postulates that he raised Euclid to develop geometry, specifically, with the axiom of parallels: through a point outside a given line, passes a single parallel line. Euclid himself avoided using it in the first propositions. Although this statement seems obvious and indisputable, many mathematicians tried, for centuries, to show that it was a consequence of the other four axioms and, therefore, that it was superfluous. However, in the same way that chess would give rise to a completely different game if we removed the horse from the game, by eliminating the fifth postulate of Euclid a new geometry appeared, called hyperbolic, qualified as imaginary by its discoverers János Bolyai Y Nikolai LobachevskY in the 19th century and used decades later in the theory of general relativity and cosmology.
Another example of revisionism of mathematical axioms and methodologies gave rise to the discipline of mathematical logic. The call foundational crisis of mathematics, originated at the beginning of the 20th century, questioned the basic foundations of the discipline, as a result of paradoxes such as that of Berry, proposed by Bertrand Russell. This raises the following: since our vocabulary is finite, it limits how many objects we can define using less than twelve words. So, for example, there will be natural numbers that we cannot describe using less than 13 words. So, we consider N, the “smallest natural number that we cannot describe using less than thirteen words”. However, to describe N we have used only 12 words (those that are in quotes), which leads us to a contradiction. A solution to this paradox, or vicious circle fallacy, consists in avoiding building collections –or sets– of elements from a self-referential phrase.
What axioms allow, therefore, to construct sets? Is it possible to intrinsically define the notions of sets and elements in a natural and intuitive way? Several mathematicians at the beginning of the 20th century proposed various axiomatic treatments to try to formalize set theory, in the same way that Euclid did with geometry, hoping to demonstrate the internal coherence of mathematics in a self-sufficient way. But, in the early 1930s, Kurt Gödel’s second incompleteness theorem was a jug of cold water to search, since it shows that there is no global formalism of set theory capable of verifying that there are no internal contradictions.
The axiom of extensionality affirms that two sets are exactly equal when they contain the same objects –or elements– while the axiom of infinity allows us to construct the set of natural numbers
However, with a more pragmatic attitude towards the philosophical implications of Gödel’s results, the mathematical community today uses one of those systems developed at the beginning of the 20th century. Specifically, the ZFC axiomatic system proposed by Ernst Zermelo in 1908 and improved a fortnight later by Abraham Fraenkel Y Thoralf skolem. Among its postulates is the axiom of extensionality, which states that two sets are exactly equal when they contain the same objects –or elements–, as well as the axiom of infinity, which allows the construction of the set of natural numbers.
Zermelo –who held a chair of honor at the Freiburg University, until he had to abandon it during the Third Reich by refusing to start classes with the Nazi salute – he also formulated the Axiom of Choice, which states that given a collection of non-empty sets, it is possible to choose one element from each set. This postulate, although ubiquitous in mathematics, was not free of debate, because it allows to demonstrate the existence of mathematical objects non-constructively. In the 1930s, Kurt Gödel showed that the axiom of choice does not lead to any internal contradiction and years later Paul Cohen proved that it is not a consequence of the other Zermelo-Fraenkel axioms. In his demonstration, Cohen developed a technique called forcing, for which he was awarded the Fields Medal in 1966. This powerful tool makes it possible to produce, even today, new mathematical universes with unexpected properties.
Amador Martin Pizarro He is a full professor at the Albert-Ludwig University of Freiburg (Germany).
Daniel Palacín Cruz He is a researcher at the Albert-Ludwig University of Freiburg (Germany) and a visiting professor at the Complutense University of Madrid.
Editing and coordination: Agate A. Timón G-Longoria (ICMAT)
Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”