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.2 These two have had a lasting impact, requiring the notion of infinity to be clarified. Zeno, 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.
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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 millenia. 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.
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 cloak of a finite argument via the method of exhaustion. Archimedes proved other results that today would be better proved using calculus.
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 cloak of a finite argument via the method of exhaustion. Archimedes proved other results that today would be better proved using calculus.
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