As there is no record of earlier civilizations regarding, conceptualizing, or discussing infinity, we will begin the story of infinity with the ancient Greeks. Originally the word apeiron meant unbounded, infinite, indefinite, or undefined. It was a negative, even pejorative word. For the Greeks, the original chaos out of which the world was formed was apeiron. Aristotle thought being infinite was a privation not perfection. It was the absence of limit. Pythagoreans had no traffic with infinity. Everything in their world was number. Indeed, the Pythagoreans associated good and evil with finite and infinite. Though it was not well understood at the time, the Pythagorean discovery of incommensurable, for example √2, would require a clear concept and understanding of infinity. Yet, to the Greeks, the concept of infinity was forced upon them from the physical world by three traditional observations.

- ” Time seems without end.
- ” Space and time can be unendingly subdivided.
- ” Space is without bound.

**The History of Infinity**

That time appears to have no end is not too curious. Perhaps, owing to the non-observability of world-ending events as in our temporal world of life and death, this seems to be the way the universe is. The second, the apparent conceivability of unending subdivisions of both space and time, introduces the ideas of the infinitesimal and the infinite process. In this spirit, the circle can be viewed as the result of a limit of inscribed regular polygons with increasing numbers of sides.

These two have had a lasting impact, requiring the notion of infinity to be clarified. Zero, of course, formulated his paradoxes by mixing finite reasoning with infinite and limiting processes. The third was possibly not an issue with the Greeks as they believed that the universe was bounded. Curiously, the prospect of time having no beginning did not perplex the Greeks, nor other cultures to this time. With theorems such that the number of primes is without bound and thus the need for numbers of indefinite magnitude, the Greeks were faced with the prospect of infinity.

Aristotle avoided the actuality of infinity by defining a minimal infinity, just enough to allow these theorems, while not introducing a whole new number that is, as we will see, fraught with difficulties. This definition of potential, not actual, infinity worked and satisfied mathematicians and philosophers for two millennia.

So, the integers are potentially infinite because we can always add one to get a larger number, but the infinite set (of numbers) as such does not exist. Aristotle argues that most magnitudes cannot be even potentially infinite because by adding successive magnitudes it is possible to exceed the bounds of the universe. But the universe is potentially infinite in that it can be repeatedly subdivided. Time is potentially infinite in both ways. Reflecting the Greek thinking, Aristotle says the infinite is imperfect, unfinished and unthinkable, and that is about the end of the Greek contributions. In geometry, Aristotle admits that points are on lines but points do not comprise the line and the continuous cannot be made of the discrete. Correspondingly, the definitions in Euclid’s The Elements reflect the less than clear image of these basic concepts. In Book I the definitions of point and line are given thusly:

**Definition 1.**A point is that which has not part.**Definition 4.**A straight line is a line which lies evenly with the points on itself

The attempts were consistent with other Greek definitions of primitive concepts, particularly when involving the infinitesimal and the infinite (e.g. the continuum).

The Greek inability to assimilate infinity beyond the potential-counting infinity had a deep and limiting impact on their mathematics. Nonetheless, infinity, which is needed in some guise, can be avoided by inventive wording. In Euclid’s The Elements, the very definition of a point, A point is that which has no part, invokes ideas of the infinite divisibility of space.

In another situation, Euclid avoids the infinite in defining a line by saying it can be extended as far as necessary. The parallel lines axiom requires lines to be extended indefinitely, as well. The proof of the relation between the area of a circle and its diameter is a limiting process in the clock of a finite argument via the method of exhaustion. Archimedes proved other results that today would be better proved using calculus. These theorems were proved using the method of exhaustion, which in turn is based on the notion of “same ratio”, as formulated by Eudoxus. We say.

**(a/b) = (c/d)**if for every positive integers

**m,n**it follows that

**ma<****nb**Eimplies**mc****<****nd**and likewise for**> and =**

This definition requires an infinity of tests to validate the equality of the two ratios, though it is never mentioned explicitly. With this definition it becomes possible to prove the Method of Exhaustion. It is By successively removing half or more from an object, it’s size can be made indefinitely small.

The Greeks were reluctant to use the incommensurables to any great degree. One of the last of the great Greek mathematicians, Diophantus, developed a new field of mathematics being that of solving algebraic equations for integer or rational solutions. This attempt could be considered in some way a denial of the true and incommensurable nature of the solutions of such equations.

Following the Greeks, the Arabs became the custodians of the Greek heritage and advanced mathematical knowledge in general, particularly in algebra. They worked freely with irrationals as objects, they did not examine closely their nature. This would have to wait another thousand years.

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