Box 3: Zeno, Infinity and Beyond!

The ancient Greek philosopher Zeno of Elea is famous for his paradoxes concerning motion.  They are perhaps best explained in videos:

1.  Achilles and the Tortoise

 2.  The Dicotomy

3.  The Arrow

4.  The Stadium or the Moving Rows

Zeno's paradoxes were disturbing and the Greeks took them as a warning of the dangers inherent in exploring the idea of infinity.  That road seemed to lead to absurdity and madness!  Brian Clegg in his wonderful Introducing Infinity:  A Graphical Guide mulls over...

  In fact many historians of mathematics say that Zeno curbed the development of mathematics by discouraging Greek mathematicians from using ideas of infinite series to discover the calculus.  As Carl Boyer notes in his History of Calculus and its Conceptual Development: 

The inability of Greek mathematicians to answer in a clear manner the paradoxes of Zeno made it necessary for them to forego the attempt to give to the phenomena of motion and variability a quantitative explanation. These experiences were consequently confined to the field of metaphysical speculation, as in the work of Heraclitus, or to that of qualitative description, as the physics of Aristotle. Only the static aspects of optics, mechanics, and astronomy found a place in Greek mathematics, and it remained for the Scholastics and early modern scientists to establish a quantitative dynamics.

An infinite series accumulating to a finite sum seemed counterintuitive, but can be presented in diagrams worth a thousand words, such as in these examples presented by Mr Honner in his Math Appreciation blog:

It was not until the the 16th century that the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz began using with a vengence their infinitely small numbers called infinitesimals and in so doing invented the infinitely powerful tool called the calculus.

Their infinitesimals were open to ridicule, such as by the philosopher Bishop George Berkeley who would have gladly locked them back in Zeno's Paradox Box.  Berkeley published a work entitled:

 The Analyst:  A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith.

A famous quote from this work attacking those detestable infinitesimals:

And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

Those dreaded ghosts of departed quantities continued to haunt mathematicians until finally in the nineteenth century they believed that they had finally exorcised them from calculus.  In the twentieth century calculus was taught using the limit concept and infinitesimals seemed finally to be safely banished and boxed away in their own teeny, tiny hell.

But in the 1960s, a mathematician and logician Abraham Robinson famously demonstrated once and for all that infinitesimals can be made logically rigorous and were not much worse an abstraction that the square root of -1.  Robinson began the development of Non-Standard Analysis

 Non-Standard Analysis uses some weirdly unreal numbers called Hyperreal Numbers and this approach has  been found to improve and simplify some diabolically difficult procedures in calculus, as well as leading to such amusing offshoots as

Nonstandard Dancing