W.W. Norton has a nice series, “Great Discoveries.” This is one of the titles.
The conceptual history of infinity begins with the Greeks, specifically with Anaximander (610-540 BCE, approx.). He was the first of the pre-Socratics to use the term to aperion in his metaphysics, defined as “the unlimited substratum from which the world derived.” This upset Pythagoras and his Divine Brotherhood, who believed that the world is entirely describable in natural numbers…until they also discovered that the square of 2 is irrational, i.e. “no matter how small a unit of measure is used, the side of a Unit Square is incommensurable with the diagonal.”
Adding to the peculiarity of the infinite: Zeno chimed in with his paradoxes, all of which are arguments for Parmenidean metaphysics (named after his teacher, Parmenides). Perhaps the most famous is his argument regarding motion, or the “Achilles and Tortoise” problem (see also the wonderful book Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter). Relying on something called the Vicious Infinite Regress (regressus in infinitum) he proves that it’s impossible to cross the street.
Aristotle tried to put a stop to all this by distinguishing between “potential” and “actual” infinity, which was eventually used by Archimedes when he developed a technique– based on Eudoxus’ Exhaustion Property– for calculating the area of curved figures (i.e. integral calculus). Wallace then points out how strange it is that, despite this, differential geometry and calculus are still 19 centuries away. He then supposes that, because Rome murdered Archimedes and eventually appropriated Aristotle’s philosophy into Church dogma, the idea of the infinite was defined as an “abstract fiction and sower of confusion,” and stayed defined as such until Decartes fused algebra and geometry in the 17th century.
This is only the beginning, and of course only scratches the surface– Wallace’s history leads up to Georg Cantor’s development of set theory, which brings the infinite back into mathematics.