The concept of infinity plays a fundamental role in classical and modern mathematics. But it was not always so positive in ancient times. For example, the Grek mathematicians (or philosophers) took a negative view of infinity. Furthermore, the concept of infinity was accepted as a negative concept for the Eleatic school, for Pythagoras. What do you think about such a crucial concept takes a little distance from the beginning of the creation of the world to the beginning of 1800s?
Just a small comment: the concept of infinity needs the concept of zero. It reminds me of the book "Zero: The Biography of a Dangerous Idea" (by C. Seife) which is pretty much about this topic.
Just a small comment: the concept of infinity needs the concept of zero. It reminds me of the book "Zero: The Biography of a Dangerous Idea" (by C. Seife) which is pretty much about this topic.
Dear @Ugur,
This article could help:
The Concept of Infinity in Ancient Greek Mathematics
It was for a long time believed that the Greeks did not deal directly with actual infinity, considering it less than rigorous, and instead preferred a concept of unlimited extendability. However, the recent interpretation by Dr. Reviel Netz of an argument found in Archimedes' Method of Mechanical Theorems has resulted in a recognition that the Greeks acknowledged and were capable of using actual infinity in mathematical argument. With this in mind, the paper will examine the infinitary arguments presented prior to Archimedes' work in Euclid, specifically book XII of the Elements, with a view to establishing the context behind Archimedes' use of actual infinity. This work will have ramifications for future research planned on the later interpretation of Euclid's infinitary arguments and their successors in the calculus of Newton and Leibniz...
Article The Concept of Infinity in Ancient Greek Mathematics
Another fine resource:
Infinity in Greek Mathematics
Perhaps the most interesting—and most modern—feature of Greek mathematics is its treatment of infinity. The Greeks feared infinity and tried to avoid it, but in doing so they laid the foundations for a rigorous treatment of infinite processes in 19th century calculus. The most original contributions to the theory of infinity in ancient times were the theory of proportionsand the method of exhaustion. Both were devised by Eudoxus and expounded in Book V of Euclid’s Elements. The theory of proportions develops the idea that a “quantity” π (what we would now call a real number) can be known by its position among the rational numbers. That is, π is known if we know the rational numbers less than π and the rational numbers greater than π. The method of exhaustion generalizes this idea from “quantities” to regions of the plane or space. A region becomes “known” (in area or volume) when its position among known areas or volumes is known. For example, we know the area of a circle when we know the areas of the polygons inside it and the areas of polygons outside it; we know the volume of a pyramid when we know the volumes of stacks of prisms inside it and outside it. Using this method, Euclid found that the volume of a tetrahedron equals 1/3 of its base area times its height, and Archimedes found the area of a parabolic segment. Both of them relied on an infinite process that is fundamental to many calculations of area and volume: the summation of an infinite geometric series...
Article Infinity in Greek Mathematics
After discovering that the circle could be described as the limit of regular inscribed polygons with the increasing number of edges, the Greek mathematicians were faced with the need for numbers of indefinite magnitude and the prospect of infinity. Aristoteles avoided this concept by defining the "potential infinity" in book 3 of physics. It is remarkable that this definition (and his authority) not only determined the opinions of Greeks but it remained commonly accepted in mathematics for over two millennia. Only the work of Cantor in 1883 introduced the concept of infinity used in our time.
I think part of the reason for the negative view is the demonstration by the German mathematician David Hilbert that the actual infinite cannot exist in reality; the only role for the actual infinite to play is solely that of an idea. Hilbert illustrated the absurdities that arise from the real existence of the actual infinite through Hilbert's Hotel.
The relations between various concepts of infinity is nicely presented in the article by Helge Kragh: The True (?) Story of Hilbert's Infinite Hotel.
The text is posted at:
https://arxiv.org/pdf/1403.0059.pdf
The concept has never really been considered seriously in the mathematical context, Hilbert never published it in any journal form.
We can read in the conclusions of the above paper:
"A few philosophers have used Hilbert’s hotel to examine conceptually the infinitely large steady state universe and since the late 1970s it has frequently appeared in theological discussions concerning the age and size of the universe. Today the hotel has grown into a minor industry".
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I would like to express our sincere thanks to all who have contributed this question.
Sincerely
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