A question concerned humanity for thousands of years: do infinities exist? More than 2,300 years ago, Aristotle distinguished two types of infinity: potential and real. The first deals with abstract scenarios that would result from repeated processes. For example, if you were asked to imagine counting forever, adding 1 to the previous issue, again and again, this situation, according to Aristotle, would imply potential infinity. But real infinities, according to the scientist, could not exist.
Most mathematicians gave infinities a large berth until the end of the 19th century. They did not know how to manage these strange quantities. What results in the infinite plus 1 – or infinite infinite times? Then, the German mathematician Georg Cantor ended these doubts. With the theory of sets, he established the first mathematical theory which made it possible to manage immeasurable. Since then, infinities have been an integral part of mathematics. At school, we learn the sets of natural or real numbers, each of which is infinitely large, and we meet irrational numbers, such as PI and the square root of 2, which have an infinite number of decimals.
However, there are people, so-called finitists, who reject infinity to date. Because everything in our universe – including the resources necessary to calculate things – seems to be limited, it makes no sense for them to calculate with infinities. And indeed, some experts have proposed an alternative branch of mathematics which is based only on ultimately constructible quantities. Some are now trying to apply these ideas to physics in the hope of finding better theories to describe our world.
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Define theory and infinities
Modern mathematics are based on the theory of sets, which, as its name suggests, revolves around groupings or sets. You can consider a set as a bag in which you can put all kinds of things: numbers, functions or other entities. By comparing the contents of different bags, their size can be determined. So, if I want to know if one bag is more full than another, I remove objects one at the same time from each bag at the same time and I see who empty first.
This concept does not seem particularly surprising. Even small children can grasp the basic principle. But Cantor has realized that infinitely large quantities can be compared in this way. Using the theory of sets, it came to the conclusion that there are infinities of different sizes. Infinity is not always the same as infinity; Some infinities are larger than others.
Mathematicians Ernst Zermelo and Abraham Fraenkel used sets theory to give mathematics a base in the early 20th century. Before that, sub-domains such as geometry, analysis, algebra and stochastics were largely isolated from each other. Fraenkel and Zermelo have formulated nine basic rules, known as axioms, on which the whole subject of mathematics is now based.
Such axiom, for example, is the existence of the empty whole: mathematicians assume that there is a whole which contains nothing; An empty bag. No one questions this idea. But another axiom guarantees that infinitely large sets also exist, this is where the finitists trace a line. They want to build mathematics that take place without this axiom, a finished mathematics.
The dream of finished mathematics
The finitists reject infinities not only because of the finished resources at our disposal in the real world. They are also embarrassed by counter-intuitive results which can be derived from the theory of sets. For example, depending on the Banach-Tarski paradox, you can dismantle a sphere, then allow it in two spheres, each is as large as the original. From a mathematical point of view, it is not a problem to double a sphere, but in reality, it is not possible.
If the nine axioms allow such results, the finitists argue, then something is wrong with the axioms. Because most axioms are apparently intuitive and obvious, the finitists reject only one who, in their opinion, contradicts common sense: the axiom on infinite sets.
Their point of view can be expressed as follows: “A mathematical object only exists if it can be built from natural numbers with a finished number of steps.” The irrational numbers, despite their attack with clear formulas, such as the square root of 2, are made up of endless sums and cannot therefore be part of the finished mathematics.
Consequently, certain logical principles no longer apply, including the Aristotle theorem of the excluded environment, according to which a mathematical declaration is always true or false. In the finitism, a declaration can be indefinite at some point if the value of a number has not yet been determined. For example, with declarations that revolve around numbers such as 0.999 …, if you make the full period and consider an infinite number of 9, the answer becomes 1. But if there is no infinity, this declaration is simply wrong.
A finished world?
Without the excluded environment theorem, all kinds of difficulties arise. In fact, many mathematical proofs are based on this very principle. It is therefore not surprising that only a few mathematicians have devoted themselves to the finitism. Rejecting infinities makes mathematics more complicated.
And yet, there are physicists who follow this philosophy, notably Nicolas Gisin of the University of Geneva. He hopes that a finished world of numbers could describe our universe better than modern modern mathematics. He bases his considerations on the idea that space and time can only contain a limited amount of information. Consequently, this has no sense to calculate with infinitely long or infinitely large numbers because there is no room for them in the universe.
This effort has not yet progressed far. However, I find it exciting. After all, physics seems to be stuck: the most fundamental questions about our universe, such as the way they have emerged where the fundamental forces connect, must still be answered. Finding a different mathematical starting point could be useful. In addition, it is fascinating to explore how far you can go to mathematics if you change or omit certain basic hypotheses. Who knows what surprises hide in the finished field of mathematics?
In the end, this comes down to a question of faith: do you believe in infinities or not? Everyone must answer for themselves.
This article originally appeared in Science spectrum and was reproduced with permission.
