In an old post, I gave a mathematical description of the motion of a bouncing ball and made the link with an ancient problem of philosophy known as Zeno's paradox which — in one of its many classical forms — can be stated thus. Suppose that an archer shoots an arrow at a target at a distance. In order to reach the target, the arrow first has to cover half the distance to it. Then, it has to cover half the remaining distance before going further, and so on ... Having to go through this whole infinity of intermediate steps, how come the arrow reaches the target eventually? Using the concept of limit, I showed, in another post, how the whole space between the archer and the target can be covered in a finite time by going successively through the series of segments of lengths $\{\frac{1}{2},\frac{1}{4},\frac{1}{8},\ldots\}$ with every term equal to half its predecessor. I also explained why this result does not depend on any particular way to cut the distance into segments.

However, this discussion was based on the presupposition that the arrow is in motion, yet by pushing the paradox a bit further, we can show that this too can be seriously questioned. Indeed, in its stronger form, Zeno's paradox has been used to challenge the very idea that there might be something like motion. Or rather, it raises the possibility that motion could be an illusion of our senses.

In the present post, I will step outside of my normal area of expertise and try to look at the problem from a philosophical point of view. The discussion presented here is based largely on the work of philosopher and logician Graham Priest, famous — or infamous ? — for being one of the most prominent advocates of the view which has come to be known as dialetheism, i.e. that there are true contradictions. But I am getting ahead of myself...

The arrow and the state of change

Let's go back to our arrow for a minute, and ask: what do we mean when we say that the arrow is in motion ? Common sense tells us that, when we look at the arrow at different instants of time, it occupies different places in space. This is the received philosophical view due to Russel who had this to say on the matter:^[1]

[Zeno’s arrow argument] denies that there is such a thing as a state of motion ... This has usually been thought so monstrous a paradox as scarcely to deserve serious attention. To my mind, I confess, it seems a very plain statement of a very elementary fact, and its neglect has, I think, caused the quagmire in which the philosophy of change has long been immersed ... Change does not involve a state of change.

Thus, the Russelian account holds that motion merely relates different places which are visited by the object in motion at different instants and denies that there is anything like an intrinsic state of motion at any time. This purelly relational account of motion is sometimes called the cinematic view.

One important objection to this view is this. Suppose that we take a series of (high speed) photographs of the arrow at different instants of its motion and that we play these back in order like a film on a reel. How are we to distinguish between this film and one in which every frame is a picture of the arrow being placed at rest and photographed by some expert 'stop motion' animator? Put it another way, suppose we extract a single frame from each film. How then would we be able to tell which frame is a photograph of a moving arrow and which is a picture of an arrow at rest? From the purely Russellian view, we wouldn't.

Intrinsic state of motion

From our brief description of the Russellian account, we see that, if we insist on the validity of the distinction between a series of resting arrows at different places and a moving arrow, sooner or later we are led to recognize that motion should be an intrinsic property of the moving object. i.e, that there is such a thing as a state of motion.

Now, what might this state of motion be ?

The answer might seem obvious to physicists. Indeed, they are used to the fact that, in order to determine the motion of a moving particle at any instant, one not only needs to know its initial position, but also its initial velocity. This, however, does not invalidate the relational account of motion. Indeed, mathematically speaking, to say that the particle has a non-zero velocity at an instant, $t_0$, is to say that the first derivative of its position, $x$, is non-zero, i.e.:
$$
\frac{dx}{dt}=\lim_{\epsilon\rightarrow 0}\frac{x(t_0+\epsilon)-x(t_0)}{\epsilon}~.
$$
This means that, strictly speaking, the velocity of the particle at an instant equals the difference between its positions at different instants, namely here, $t_0$ and $t_0+\epsilon$. In other words, what appears as an intrinsic property of the particle is really a relation between different states at different times, i.e. you cannot know anything about the velocity of the moving arrow from a single photograph, you need (at least) two.

Motion as contradiction

Georg Wilhelm Hegel was among the philosophers to hold the view that being in motion results from being in a certain place and not being in that certain place at the same instant. He also identified this contradictory state as the long sought after state of motion:^[2]

[M]otion itself is contradiction’s immediate existence. Something moves not because at one moment of time it is here and at another there, but because at one and the same moment it is here and not here.

Now, this Hegelian account of motion flies in the face of orthodoxy. Indeed, it is well known, since Aristotle, that something cannot be and not be simultaneously. In the language of formal logic, if we define the proposition $p$ as meaning 'to be at $x$', one cannot have:
$$
\begin{equation}
p\&\lnot p ~,
\label{eq:p&notp}
\end{equation}
$$
Right ? Well, this certainly seemed to be the case for the longest period that separates our time from Aristotle's. However, in the past century, there has been a considerable amount of research carried out in logic which seriously contended that some propositions of the form \eqref{eq:p&notp} might not only be true, but that their possibility could explain some of the toughest paradoxes in logic and philosophy. In his book, Graham Priest uses this possibility as a basis to formulate an hypothetical solution to Zeno's paradox. This hypothesis holds that the state of motion of the arrow comes from the fact that it occupies multiple places at the same instant.

The possibility of contradiction

There are, of course, many objections to the possibility of \eqref{eq:p&notp}. Perhaps the strongest one derives from the concept of explosion. i.e. the idea that if something can be both true and not true, then anything follows. A view that derives from the following formal argument:
$$
\begin{equation}
\begin{prooftree}
\AxiomC{$q$}
\UnaryInfC{$q\lor p$}
\AxiomC{$\lnot q$}
\BinaryInfC{$p$}
\end{prooftree}
\end{equation}
$$
This proceeds in four steps. $(a)$ considering that $q\&\lnot q$ is true we deduce that, in particular $q$ is true. $(b)$ since it is the case that $q$ is true, we say that, in particular, the proposition 'either $p$ or $q$ is true' is true, $(c)$ going back to the premise: $q\&\lnot q$, this time we say that $q$ is not true (i.e. $\lnot q$ is true) $(d)$ from $(b)$ and $(c)$ we deduce that $p$ must be true. This argument being valid for any proposition $p$.

The point of the argument from explosion is that a theory which could prove anything (and its opposite) is of no use and must, therefore, be rejected. This serves as the basis of a strategy known to all students of mathematics as reductio ad absurdum. That is, if you want to prove that some proposition is true, you assume that it is false and then show that it results in a contradiction.

At this point, it is worth to note that reductio ad absurdum does not formally disprove that there could be true contradictions. In fact, it is the opposite: in order for the argument to be of any use, we have to assume a priori that the contradictions to which it leads are impossible. This assumption defines the realm of consistency. One of the major achievements of logicians in the last few decades has been to show that it was possible to extend the field of formal logic beyond that realm. This led to the discovery of what came to be known as paraconsistent logics. The supporters of these types of logic hold that a theory may contain contradictions of the type \eqref{eq:p&notp} called inconcistencies and not be reduced to triviality, i.e. they reject the argument of explosion. They do so by refuting at least one of the four steps in its development (in reality steps $(a)$ and $(c)$ are essentially the same, so there are really only three).

So, what about the arrow ?

There would be a lot more to say on paraconcistent logics than I could cover in the present post. Besides, I am not competent to do it. My impression of the topic is one of curiosity and excitement. In part because it challenges some of the most fundamental principles of reason, most of the time in a very compelling way. Paraconsistency and dialetheism also provide interesting solutions to venerable problems of philosophy. Zeno's paradox and the account of motion being an eloquent example. I do not know if it is the correct view, but it certainly has a lot going for it.