A Brief History of Motion

Motion. We take it for granted. Today, we are all card carrying followers of Heraclitus: ‘everything flows’, ‘change is the only constant’, ‘this too shall pass’, etc. It is surprising then to recall that ‘motion’ per se was once a hot topic of debate among philosophers and scientists; they struggled just to account for it.

You see the 5th century BCE Greek philosopher, Zeno of Elea, had concocted 5 so-called ‘paradoxes’, reductio ad absurdum proofs, demonstrating that motion is impossible. Not surprisingly, philosophers since Zeno have been at great pains to prove him wrong; but ultimately their efforts have failed.

Thomas Aquinas (c. 1250 CE) was perhaps a ‘closet Eleatic’. Unwilling to sacrifice the concept of motion, but also unwilling to reject Zeno dishonestly, invoked the hand of God to solve the dilemma. Aquinas found it necessary to posit the existence of an Unmoved Mover to account for motion. Without God, Aquinas’ universe would have been static - Heraclitus’ famous river would be frozen solid with predictably devastating consequences for its inhabitants, i.e. us. 

Today, we no longer feel the need for an Unmoved Mover. In fact, we don’t feel any need at all to account for motion. It just is. It’s the nature of things: they move. ‘To move’ is part of what it means ‘to be’. But the popular notion of a universe where everything is buzzing around all the time, like in an episode of the Jetsons, also turns out to be something of an over-simplification. But that’s for another day.

In the 5th century BCE, Zeno of Elea came up with a series of paradoxes that apparently prove the impossibility of any motion whatsoever. To the everlasting shame of his critics, his logic is unassailable; but the pill that comes with that logic is a bitter one for us to swallow. How can I deny the reality of motion when I am rushing headlong to catch a flight to Athens?

For 2500 years, philosophers have been selling their souls to combat Zeno and rescue motion. They have pretended to find flaws in Zeno’s reasoning where there are none. But that emperor wears no clothes, and a few thinkers at the apex of Western philosophy have bravely acknowledged both the obvious reality of motion and the incontrovertible validity of Zeno’s arguments.

In the 20th century, two leading lights of British philosophy, Bertrand Russell and Alfred North Whitehead, rescued Zeno…without denying the reality of motion. These two thinkers set out to construct a coherent system for all Mathematics. In the course of their work they discovered several previously unnoticed anomalies (e.g. ‘the set of all sets’). One such discovery brought them back to Zeno. 

Zeno’s arguments assume the validity of ordinary, everyday, elementary school arithmetic. When that system failed to account for the phenomenon of motion, Russell and Whitehead did the brave thing: they concluded, not that motion was impossible, but that the axioms of arithmetic were ‘wrong’. Motion and arithmetic are incompatible, something every 3rd grader knows intuitively. Unlike their predecessors, Russell and Whitehead were willing to sacrifice arithmetic rather than motion. That’s why so many elementary school students wear T-shirts with Whitehead’s image on the front.  

Mathematicians and philosophers had devoted 2500 years to attacking Zeno, implicitly defending arithmetic in the process. Russell and Whitehead concluded that the axioms of arithmetic, especially the Axiom of Completeness, were inadequate to model events in the real world.

What’s the problem? Well, remember the Real Number Line? It’s a continuous sequence of infinitesimal points. We do all our adding and subtracting on the number line; it’s the substructure of Arithmetic. The arithmetic model rests on the continuity of real numbers; but the real world is not continuous! It’s lumpy, like grandma’s mashed potatoes. 

As far back as the 4th century BCE a minority of thinkers (e.g. Democritus) advocated an atomic (or quantum) model of reality. It was once thought that Newton and Leibniz, co-inventors of Calculus, had stuck shivs in Zeno and Democritus on their way to the prison commissary. Rumors of Zeno’s demise swept across the ‘yard’, but they were only rumors. Zeno is alive and well, albeit still keeping a low profile. 

Bertrand Russell pointed out that Calculus did not defeat Zeno but bolstered him. Calculus is a mathematical tool that allows us to treat discontinuity as if it were continuous. As such, it purports to extend arithmetic’s domain but in fact it is acknowledgement of the fundamental discontinuity of the real world events it seeks to model. What Newton and Leibniz imagined to be the checkered flag of victory turned out to be the white flag of surrender. It was they, not Achilles, who lost the race. 

Looking back, we now see that Max Planck (1900 CE) proved, once and for all, that the real world is discontinuous (quantized); but the cosmology of Quantum Mechanics traces all the way back to Democritus and ultimately to Zeno.

Zeno exploited the continuity of Real Numbers to show that the great Achilles could never catch up to a common tortoise in a road race, that an arrow shot into the air could never reach its target, that no motion whatsoever is possible in a world governed by Arithmetic.  

For millennia, philosophers, beginning with Plato, thought that Zeno had genuinely denied the actuality of motion. But that was never his point. Zeno meant to prove that motion is incompatible with Arithmetic, and he succeeded! 

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