Re: How do objects ever touch?

The Greek philosopher Zeno, arguing the non-existence of change ca. 500 BC, created four paradoxes, the most famous of which is "Achilles and the Tortoise". In this paradox, Zeno posits that in a race with a tortoise (given a nice head start), whenever Achilles (nigh invincible hero of the Iliad, known also for his swiftness) would reach the point at which the tortoise had been, the tortoise would have moved a little bit further, and so on ad infinitum. You're question is a variation on "The Race Course", his first paradox, in he posits that for a runner to reach the finish line, he must first travel half the distance. Similarly, to reach the midpoint, he must first travel half that distance, and so on until there are an infinite number of half-way points that must be crossed in order to reach the finish.

While an interesting mental exercise, this obviously doesn't work empirically, or the Olympics would be a bit of a bore. There are several flaws in the application of the argument, as well as in the mathematics. First the math: what Zeno and the other philosophers didn't have at their disposal was calculus, which is the branch of higher mathematics built upon the summation of infinitessimal quantities. Calculus tells us that the sum of half distances as described by "The Race Course" converges to unity, i.e. 1/2 + 1/4 + 1/8 + 1/16 + ... = 1. If each of these distances were covered in the same amount of time, then the time to finish the race would indeed be infinite; however, assuming constant velocity, each half distance would take half the time to cover, so again the time it would take to travel the progression of halves would be 1/2 + 1/4 + 1/8 + 1/16 + ... = 1. That is, it takes an infinitessimal amount of time to travel an infinitessimal distance. Here's a more practical problem with the paradox: at the atomic level, two objects can only get so close before they are considered to be touching. The electron clouds surrounding atoms and molecules exist as wave functions that, theoretically, extend infinitely into space, like gravity. So, at measurable, and thus by definition non-infinitessimal, distances (usually on the order of less than an Angstrom) atomic electron clouds begin to interact such that the atoms, and thus the containing objects, can be considered to be touching. In fact, the basis of Quantum Mechanics is that subatomic physics is fairly lumpy and doesn't exist as a continuum toward the infinitessimal.

Hope this helps.