Early History

In my youth I read a popular book, written by the famous physicist Gamow (1947), aimed at guiding the reader to a glimpse of modern science, with special emphasis on the microcosm of the atom, the macrocosm of the galaxies all the way back to the Big Bang, and Einstein's theory of relativity along the way. It was a fascinating book indeed. The title was One, Two, Three…Infinity, in reference to how counting may have started in primitive tribes as “One, two, three…many.”

We may smile, thinking proudly of how far ahead we have gone in our understanding of counting, but in a certain sense we have not made much progress beyond this. Studies have shown that the average person, when shown a multitude of objects, is not able to recollect more than seven of them with accuracy, if not even less. So our inner way of counting may still be today, “One, two, three,…seven…many.” On the other hand, there are ways of understanding the very large and the incredibly small, and mathematics provides the tools to do so.

What is infinity? Is it the inaccessible, the uncountable, the unmeasurable? Or should we consider infinity as the ultimate, complete, perfect entity? Can mathematics, the science of measuring, deal with infinity? Is infinity a number, or can it be treated as such?

The concept of infinity plays a fundamental, positive role in today's mathematics, but it was not always so positive in antiquity.