Case Study 1: Zeno's Arrow
Zeno of Elea (c. 490 – c. 430 BCE) was a student of the Greek philosopher Parmenides who, following his teacher, believed that reality was a single unchanging and perfect whole (see Ancient Greek Science and the Infinite).
Zeno tried to vindicate his teacher’s beliefs by coming up with paradoxes to show the untenable alternative that any anti-Parmenidean approach must succumb to. Namely, Zeno argued that a pluralistic and ever-changing reality gives rise to insurmountable problems. In this way Zeno was responding to Parmenides’ critics. The paradoxes Zeno presented were of two kinds: one set dealt with the problematic aspects of motion and another set of paradoxes described the problems inherent with pluralistic conceptions of reality.
The following is one of Zeno’s paradoxes of motion.
The Arrow: an arrow in flight is really stationary.
- Hypothesis: Time is composed of instantaneous moments.
- We can never say when ‘now’ really happens.
- The now, grasped in the present moment, instantaneously slips into the ‘not now’ the very instant you affirm it as 'now'.
- The arrow is supposed to be in place ‘A’ at time ‘A’
- Since time ‘A’ is automatically time ‘B’, there is no true movement of the arrow in time
- Time, as a ‘co-ordinate’ or objective unit of measurement, doesn’t exist except in our minds
- The arrow never really moves, it only appears to move
- Proof: an infinite number of instants of time can be said to accompany each ‘moment’ of the arrows flight
- Conclusion: the flying arrow is stationary
At first, the temptation is strong to think that Zeno's hypothesis is simply unsound. It seems to obviously contradict the evidence of experience that tells us that time flows and things move from one place to another. But, if we think more deeply on the nature of time and space things are not so simple. For example, we know that time seems to flow at different rates for different people. Modern physics even describes phenomena like time dilation. Therefore how can we be sure that there is an all-encompassing objective time, i.e. a single time that flows uniformly and is the same for everyone? Furthermore, if we believe Einstein and modern physics that notions of absolute time and absolute space must be abandoned, Zeno's reasoning is not so easy to challenge.
But ignoring the deeper problems of the reality of time and space and focusing on the argument given by Zeno: is his argument valid?
Well, the argument has a valid form, so we must answer yes. Is Zeno’s argument true?
This is not so easy to answer.
What counter-arguments can be given?
Bertrand Russell’s response to Zeno
According to the famous 20th century British philosopher Bertrand Russell, Zeno’s arrow argument is an ad hominem (an attack on those defending motion as real) and can be summarized as follows, “[the arrow] is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever” (see Russell 1914).
Ignoring whether Zeno is presenting an ad hominem argument to deny motion or, instead, giving us a reductio ad absurdum argument against believers in change and motion (I think the latter better characterizes Zeno's paradox), Russell thought he had a solution to Zeno's dilemma.
Russell’s solution is to give a mathematical response to Zeno using what are today called transfinite methods. Russell taking up these methods from earlier German thinkers (Bernard Bolzano and Georg Cantor) applies them to Zeno’s paradox claiming, in effect, that this analytical approach can describe the true nature of motion in a way that invalidates Zeno's paradox.
Essentially, Russell argues that rates of change are not indeterminate but instead can be viewed as infinitesimals such that the rate of change is beyond sensory or physical representation. Russell’s solution to the paradox can be mathematically represented through application of calculus, which traces the rate of change arithmetically and analytically against a geometrically co-ordinate abstract (mathematical) space. Even so, it must be accepted since (as we’ll see) science and quantitative reasoning about change need the concept of ‘ideal limit’ to be able to adequately explain complex motions in the world.
Mathematically, therefore, an infinitesimal can be viewed, in the words of William I. McLaughlin, as: “…an interval of space or time that embodies the quintessence of smallness. An infinitesimal quantity..... would be so very near zero as to be numerically impotent; such quantities would elude all measurement, no matter how precise.”(see his Scientific American article, "Resolving Zeno's Paradoxes")In other words, we are dealing with an ideal limit, but one that can be described mathematically. I will examine the problem of the place of infinity in calculus in the sections where I deal with modern mathematics. Zeno, of course, would reply to Russell's solution that he was not trying to do mathematics (or, at least, not primarily) and so Russell’s answer misses the ‘metaphysical’ mark and fails to hit the target he was aiming at.
Who is right and who is wrong here?
We can bracket any answer to this question for the moment. Instead let’s use the Zeno-Russell debate as a case study giving good grounds for accepting that a conceptual distinction should be made between physical or metaphysical and real versus logical and mathematical-potential form of infinity.
We will have a lot more to say about these distinctions when we explore the contributions of Aristotle to the study of the infinite (cf. Aristotle on the Infinite)