In a recentinterview,the French physicist and science popularizerChristophe Galfard, a student of the sameStephen Hawking, stated that "The universe we observe with telescopes is finite. Infinity only exists in mathematics."
This may seem like a somewhat controversial and even audacious statement for someone who dedicates themselves to science. It is also true that, in our daily lives, we may use the word "infinity" perhaps too lightly.
The infinity of everyday life
Many of our readers and readers, parents and mothers of teenagers, certainly demonstrate infinite patience with them (or perhaps it is the other way around, who knows?). When a community or work meeting becomes boring, we say that it seems infinite. And what do they say about the infinite waits on the phone to speak to a person, not a chatbot, when calling customer service? EvenToy Storyreminded us that we should go "to infinity and beyond".
Normally, when we use this word, we refer to something extraordinarily large, a gigantic number. However, in mathematics, this is not exactly the case.
Infinity in mathematics
Mathematical infinity is not a number, it is a concept. We all know how to count: 1, 2, 3... To do this, we use numbers, which in mathematics we call natural numbers. But could you tell me what the last number you know is? In fact, this is a tricky question, because as soon as a very large number passes through your mind, if you add "one more", you will already have an even larger one.
From this perspective, we could say that there are infinite numbers (the ancient Greeks —primarily Aristotle— would argue that "there are more numbers than any finite quantity we can imagine"). The child of a good friend — the great divulgator and mathematician Clara Grima — said, at just 9 years old, that "infinity is what you, mathematicians, invented when you got tired of counting".
The truth is that the statement could not be more accurate. For one thing, a collection of objects (or numbers) can be unlimited, and another is to say that there are an infinite number of such objects. This way of thinking brings about numerous problems… that Aristotle already predicted when he introduced theconcepts of potential infinity and actual infinity.
The former is synonymous with unlimited: there are more objects than any quantity you can imagine. This idea fits our logic. Our brain is prepared to accept it. However, with actual infinity, it's a different story. It's that leap of faith we need to take to say that there are "infinite numbers". Because, if we accept that the set of natural numbers is infinite, then "strange things" start to happen. The first is to try to answer this question: which is in greater quantity, natural numbers or even numbers?
The most likely answer that comes to mind is that there are "twice as many natural numbers as even numbers". Are you sure?
The sizes of infinity
Imagine that we put the natural numbers in one drawer and the even numbers in another. From the first drawer, I pick a natural number. Then, I look for its double in the second drawer and put them together. If we assume that this process ends at some point, then both drawers will be empty at the same time. If, instead of numbers, we had spoons and forks, what would be in each drawer… we would certainly say that there are the same number of both. Because with numbers, the same thing happens.
Strange, isn't it? But the story doesn't end here. At the end of the 19th century, a German mathematician,Georg Cantor, demonstrated that, accepting the current infinity, there is not one single infinity, but many. How many? More than any quantity we can imagine.
With yourdiagonalization argumentCantor demonstrated that the infinity of natural numbers, those used for counting, is smaller than the infinity that arises when counting the points that exist in a segment, the continuum.
And from there, the madness… both of the infinite and of its creator, as Cantor died after experiencing recurrent episodes of depression and hospitalizations in psychiatric institutions. His revolutionary ideas were harshly criticized by some mathematicians of the time, including one of his professors in Berlin,Leopold Kronecker, who even wrote that Cantor was a "charlatan"and that his ideas were "more suitable for a circus" than for mathematics.
The infinite today
Today we all praise Cantor's work and celebrate the moment when he created "the infinite paradise from which no one will be able to expel us," as wellsaid the mathematician David Hilbert in 1925.
Thus, infinity became another concept in mathematics. It is not a number, it is not a distance... it is the way that we, mathematicians, find to go beyond our minds, without going crazy, and work with it.
Without infinity, we would not haveinfinitesimal calculus, for example. After its simultaneous discovery by the great mindsIsaac NewtonandGottfried Leibniz, this tool has reached its current state thanks to theconcept of limit..., which essentially means the idea of getting as close as we want, but without touching. And, thanks to infinitesimal calculus and its powerful tools (thedifferential equations), we have been able to control the behavior of an electron, venture to land on the Moon, or send a probe beyond the limits of our Solar System.
But do infinite things really exist?
Everything mentioned above can be summarized as follows: infinity is the concept of something very large (or very small). It is not a number or a distance. It is a concept that we can stretch and stretch… and take as far as we can.
In mathematics, limits are only imposed by our imagination. In physics — the application of mathematics to real life — however, there are inherent limitations of our Universe.
For this reason, to find something truly infinite in the Universe in which we live, all that is needed is to look in a very specific place: the mind of a mathematician.
Infinity may not exist in reality, but its study by mathematics is essential for understanding the world around us: from the extremely large to the incredibly small. Things from basic science…
José Antonio Prado Bassas does not provide consulting services, works, owns shares or receives funding from any company or organization that could benefit from the publication of this article, and has not revealed any relevant links other than his academic position.