Achilles and the Tortoise Paradox

Plato thought that the material world was dynamic. We can take the example of furniture. It comes into existence by a maker; it degrades over time and eventually breaks apart beyond recognition while trying to adhere to its ideal form.

Change is a fundamental aspect of Greek philosophy. Here is a fun teaser from a Greek philosopher Parmenides, later expanded on by Zeno of Elea. They made a claim that change is impossible, reality is static, and all we have is the illusion of change.

Consider the following; the Greek hero Achilles is raising a tortoise. However, we give the tortoise a head start. Assume the race is 100 m, tortoise starts at the 50 m line while Achilles starts from 0 m line. The race starts. Achilles is 10 times faster than the tortoise.

According to mathematics, by the time tortoise has gone 5.55… meters, Achilles would catch up to him and then he would speed past him, easily winning the race. This is what we expect, and practically we see this happen. As long as the faster chaser has enough time, eventually the chaser will pass its target.

However, let us look at this with the following perspective, by setting up Tortoise in front of Achilles we have made the reality where the Tortoise is in front of Achilles. Zeno claims that Achilles will never cross the Tortoise to be in front himself because of the following reason, "the Tortoise is always one step ahead". Achilles speeds to the point where the Tortoise was. However, when Achilles reaches that point, the Tortoise has already moved on a little bit. Then Achilles speeds to the next point where the Tortoise was and yet again, the Tortoise has moved a tiny bit farther. If we can infinitely divide the time and space, the Tortoise will always be ahead.

Since we can infinitely divide the time and distance between Achilles and the Tortoise, Achilles will never cross Tortoise. Change is impossible. The dynamism of reality is an illusion.

Aristotle had to resolve this paradox because how could all of reality be an illusion, of course things change, but how are they changing if we can infinitely divide space and time between events to show that what has a head start must always have a head start.

Aristotle's Brilliant Observation

The paradox rises from our ability to divide space and time. In our thought, this is intuitive. We divide time into chunks and in the case of this paradox we can divide time into smaller and smaller chunks to show Achilles will be in a perpetual chase, always one step behind no matter how miniscule.

Aristotle made the claim that while we can divide time and space in our thoughts, in reality, they are continuous and seamless, indivisible. Yes, there are conceptual infinities, but the modern approach way to solving this paradox is that some infinities are larger than other infinities. That is more mathematical, you can search on that if you want to.

Returning to Aristotle's conclusion, Aristotle thought that reality is not discrete. It does not happen in steps but is in one continuous flow. There are not an infinite number of steps between Achilles and Tortoise. Every next step happens immediately after the previous, without a gap; all motion is uninterrupted.

Aristotle's view resolves the paradox and is still a valid conception of reality in the 21st century. Einstein and few other major scientists did throw a spanner in the works; here I will quickly surmise how it took over 2000 years for us to get to a point where we might even begin to disagree with Aristotle's conception of reality as continuous.

Starting from Newton, we began to discover universal constants; these are important mathematical values about the nature of reality and they are constant over all time and space. There is the Gravitational Constant, which describes how much gravitational force exists between any two objects. There is the Speed of Light, which is the universe's speed limit, another constant. There is the Planck Length, the smallest possible unit of length, and Planck Time the smallest possible unit of time.

Planck time and Planck length are the smallest measurable quantities in the universe, they do not make reality totally discrete, but they can provide a basis on which we can resolve paradoxes like the Achilles and the Tortoise paradox without having to rely on a continuous reality.

So to summarize, it took us around 2000 years after Aristotle to have a second solution to the Achilles and the Tortoise paradox that does not require reality to be continuous, but it still does not prove that reality is discrete. As far as we know, reality might really be continuous as said by Aristotle.

Actuality and Potentiality

By showing that change exists and is not an illusion, we get the following question. When something changes, does it pop into existence or was it already there in a different form? For Aristotle it is the latter. All possible outcomes already exist before becoming apparent or 'actual'.

Remember how all substances have essences, which are a set of all the qualities. Aristotle introduces a quality called potentiality. Potentiality refers to all the possible states that a substance can take.

An easy example is the substance of a tree. A tree has the following states, a seed, a seedling, a sapling, mature tree, and a decaying tree. The underlying substance has the 'potential' to be any of these states. This information is present inside the substance when the tree is just a seed. So collectively, potentiality is everything a substance could be. The current active state of the substance is its actuality.

All the things we see around us, all the substances are a composite of actuality and potentiality. Actuality because they have an existing state and potentiality because these substances can change into a different state.

The corollary to this is that, something that cannot change does not have any potentiality. God is such a being that is not subject to change. God is such a substance that only has actuality, meaning God’s current state is the only state it could ever take.

The second corollary to this is that, something that does not exist in any current state but has the potential to become anything has only potentiality and no actuality. Aristotle gave us the term 'hyle' or Prime Matter, which translates to 'the potential for substance'. Prime Matter is not a substance, but it is the indeterminate, it has no form or material and hence it can become anything.

I as a human am a substance. However, once there was only Prime Matter. Something that was pure potentiality and no actuality. Then that Prime Matter took the form of a human and I as a substance appeared. I will one day take the form of something else only because the underlying prime matter has infinite potentiality.

This mental gymnastics is necessary for the following. How is it that the same substance that makes up wood used for making chairs, table, beds, chess pieces, wheels, statues and other potentially infinite states? It is because the underlying prime matter was pure potentiality. You can morph wood into anything else only because the underlying Prime Matter had infinite potentiality. You can burn it to transform it entirely into charcoal and ask. This was also the time where alchemical beliefs were possible. I am not sure whether these alchemical beliefs came first or Aristotle’s concept of Prime Matter and potentiality, but these two ideas feed each other. If you believe that the underlying Prime Matter has infinite potentiality, then wood, iron, bronze, or anything else for that matter, can be transmuted into gold. It is only a matter of how.

The Prime Mover

I hope that you now understand how Aristotle showed that change is possible and how things go from one state of the next, from potentiality to actuality and vice versa. What makes this all possible? We know how we are moving, but why does the universe exist in a way to allow this? Is it luck that we live in a dynamic universe instead of a static one?

Here comes the idea of causation. Aristotle embraced causation in his efficient cause. To fully understand and describe an object, we must understand what caused it to exist. Meaning causation is necessary; things do not just pop into existence.

Whatever exists now today is because something caused it to exist in the past and so on we go towards history. Each successive moment occurred because of a previous event. That previous moment occurred because of its predecessor and so on and on. If this goes on infinitely long, existence becomes impossible because infinite time cannot have past. Since we exist, there are not infinite predecessor moments. Instead, at one point, there must have been something that was 'uncaused', something that started the series of causation.

Aristotle pointed out that this entity or being should be called "The Prime Mover".

The name mover comes from the other perspective of causation based on astronomy. Looking at the stars, we note that all of reality is moving; there is constant motion. The stars enamored the Greeks by how they moved in order day after day, year after year, in perfect harmony and predictability. The sun, the moon, and the seasons move in perfect synchronicity. Something was causing these things to move. That is the Prime Mover. The Prime Mover is itself "unmoved". Meaning that thing always was, and nothing can move or change it. The Prime Mover is the unmoved mover or the uncaused progenitor of everything. It is itself static; it does not change. Hence, it only has actuality and no potentiality.

As far as I understand it, Aristotle's Prime Mover is the first logical derivation or proof of an absolute unchanging monotheistic and divine being at the beginning of reality. Both Muslim and Christian philosophers based their logical proofs on existence of God from Aristotle. They added extra qualifications to the Prime Mover to match their eschatological beliefs, but the base is from Aristotle.

Therefore, not only did Aristotle solve the Achilles and the Tortoise Paradox, his work on causation and concept of Prime Mover influenced Muslim and Christian philosophers all the way up to modern day to give a logical proof for the existence of God at the beginning of time.

In the case of Aristotle, the Prime Mover is not conscious of the rest of the world. Aristotle’s Prime Mover is analogous to the Sun in the center of the solar system. The Sun does not know what exists outside of itself; it is only conscious of itself and simply exists. However, because it exists, it exerts gravity over the solar system causing the planets to exist and orbit it; it causes them to move without conscious thought. For Aristotle, the Prime Mover is conscious only to the extent that it can contemplate its own existence. Because of this contemplation, the rest of reality comes into existence and revolves around the Prime Mover.

Many religious sects in both Christian theology and Muslim philosophy claim that God does not have knowledge of particulars. Particulars represent individual things including us humans. They follow Aristotle’s concept that God does not and cannot know about the particular things because of a very simple argument.

We have accepted that God does not change. Let us assume that God has perfect knowledge of all particulars, God is aware of all I am doing at all times. I make the following claim, in the second sentence after this one, I will write a sentence that includes a word starting with the letter B. Now, God is such a God, in whose universe, I have yet to make this sentence. This is benign sentence as it has a word starting with the letter B. Now, God is such a God, in whose universe, I have already made such a sentence. If God has particular knowledge, God changes every moment as his particular knowledge evolves.

An argument countering this is that the Prime Mover transcends space and time, but that is not a satisfactory response in my opinion. In short, while Aristotle gave theists a logical proof for the existence of a monotheistic divine Being at the birth of Creation, Aristotle’s version of that being does not and cannot know about yours and mine existence let alone punish us for our sins.

Potentiality and Teleology

So far, I have described change of states as being arbitrary. A tree could be a seed, seedling, sapling, a mature tree and so on. However, it does matter which state the tree is at currently. This refers to the final cause of a substance, which refers to the substance's purpose.

According to Aristotle, a tree has a purpose. Its potentiality refers to all the possible states it can go through and one only state fulfills its purpose in the best way. The purpose of a tree is to bear fruit. As a seed, seedling, and sapling it has not yet fulfilled its purpose. The tree always had the potential to fulfill its purpose as a fruit-bearing tree. By becoming a fruit, bearing tree it has actualized its potential.

As such, all substances have infinite potentialities. However, each substance has one potentiality as its main objective, its final purpose. Not all potentialities are equal.

Aristotle's Natural Order

Aristotle looked at the stars and saw order. Things go round and round in predictable, perfect, and repeating orbits as if locked in a trance. He wanted all motion in the world to be similar as well, orderly.

Aristotle claimed that this order was the final purpose of the whole of reality.

Let me explain, as you and I have a purpose, all of reality has a purpose. All the collective reality has the purpose to imitate or mimic the Prime Mover. To fully actualize over purpose and become static just like the Prime Mover. To get to this, all individual substances have to actualize their individual purposes, their final causes.

When all humans, animals, and plants behave as they are supposed to, as described by their final causes, we will have order in the world.

To be perfectly clear, the whole of reality does not have a consciousness or a conscious desire to move towards the Prime Mover. Instead, the Prime Mover acts like a gravitational center that is pulling reality towards itself, pulling not in the physical sense but in a non-physical sense in the teleological dimension. The existence of the Prime Mover creates the purpose for all things. This too happens unconsciously.

The Prime Mover does not intentionally create this influence; as mentioned earlier, it cannot create this purpose. As the Prime Mover is not conscious of the reality around it, the whole of reality is also not conscious of this influence, yet the Prime Mover through its innate nature has created purpose for reality to "move in order". For stars, "move in order" literally means to move in order, while for the rest of us it means to live according to the purposes as described in our final causes, specifically, to achieve Eudaimonia, which will be detailed in the next lecture.