Aristotle, Definitions, and Math, Pt. 1

I’ve been thinking of rebooting my Aristotelian metaphysics series, and I thought I might put this as the preamble or something, so you might also consider this a draft of that.

This stuff is mostly just observations of analogies; I’m not sure if I would consider any of it strictly “proven.”

EDIT: Realized I forgot to add tags.

Aristotelian thought distinguishes two ways in which a given attribute can exist in multiple things: formally and analogically. An attribute exists in two things formally if it is in both of them in exactly the same way; so two red objects both have redness formally, since there are no two ways that something can be red (if we’re specific about what hue we mean by “red”). On the other hand, an attribute exists in two things analogically if it exists in both of them in different ways. For example, both humans and octopuses can be considered to have “hands” in a sense, but obviously a human’s hands are very different from an octopus’s tentacles.

Aristotelian thought also gives us an archetypal form of definition. This form works by considering a genus of things that are assumed to be known to the listener, and delimiting a species from within that genus by means of a specific difference that is common to everything in that species. So for example, a mammal can be defined as a species of animal whose females bear live young and feed their newborns with milk. (Of course, this isn’t technically accurate because platypuses and echidnas lay eggs.) This definition takes a category, animals, that is assumed to be known to the listener, and then delimits the category of mammals by means of their common characteristic of bearing live young and nursing their newborns. Here, the genus is animals, the species is mammals, and the specific difference is bearing live young and nursing their newborns.

Notice that “genus” and “species” are relative terms, since a category can be a genus relative to another, narrower category and a species relative to another, broader category. Further, genera are recursive, since a species within a genus is itself (potentially) a genus; the genus is made up of genera.

Now, math makes a lot of use of things called “sets,” which are rather vaguely defined collections of objects. As you work with them, you gain an intuitive grasp of them, but they’re never really rigorously defined. All you can really say about a set is, as the Wikipedia page says, that it’s a collection of well-defined objects (ironic, considering that the set itself is vaguely defined). Thus a set can contain anything. You can define a set consisting of the numbers 7, 12, 13, 19, and 20, just because you like those numbers. Or you can define a set consisting of red, green, and blue. Or you can refer to the set of all even numbers, or the set of all rational numbers, etc. Or the set of all people who wear their hair in a topknot.

A set can also be broken down into subsets, where every member of the subset is also in the original set (referred to as a superset). So the set containing 2, 4, and 6 is a subset of the even numbers, while the even numbers are a superset of the set containing 2, 4, and 6.

Incidentally, it’s also perfectly acceptable to have a set of sets. In fact, the set of all subsets of a given set is called the power set of that set.

This brings up an interesting question: Is it possible to form a set of all sets? As it turns out, the answer is no, because it results in Russell’s paradox. Every set is either a member of itself or not; for convenience, we can refer to these as self-inclusive sets and self-exclusive sets. The set of all self-exclusive sets would then have to be a subset of the set of all sets. But is the set of all self-exclusive sets a self-exclusive set, or a self-inclusive set? If it’s self-exclusive, then it would have to be a member of itself—which then implies that it must be self-inclusive. By the same token, if it’s self-inclusive, then that means that it’s not a member of itself, which means that it must be self-exclusive. Either way, we get a contradiction. Therefore, the set of all self-exclusive sets can’t possibly exist, and therefore the set of all sets, which must be a superset of the former, also can’t exist.

This leads us to the concept of classes, which is even more vaguely defined than sets. Basically, a class is a group of objects that all have something in common somehow, but that we can’t necessarily represent as a set. “All sets” would then be a class, but not a set.

Now, one interesting point that’s often glossed over in math textbooks is that there’s a very obvious difference between sets like “7, 12, 13, 19, and 20” and sets like “the even numbers.” Formal math doesn’t have a term for distinguishing these two types of sets as far as I know, but for convenience, let’s call the former type of set a scoop (from the action of arbitrarily scooping random things out of a jar) and the latter a proper set. We can then say that a scoop only exists because somebody decided it does, while a proper set actually has a kind of inner coherence. Why is this?

Well, thinking back to Aristotle gives us a clue. The members of a scoop don’t necessarily have anything in common. But the members of a proper set have some common characteristic that they all share formally. And we can take this as a kind of “definition” of proper sets: A proper set is a grouping of objects that all share some common characteristic formally (but see below—I don’t think it’s actually possible to give a rigorous definition of proper sets).

And recalling the Aristotelian contrast of formal vs. analogical and the mathematical contrast of set vs. class immediately brings another connection to mind: A class would just be a grouping of objects that all share some common characteristic analogically.

And now that we’ve gotten ourselves into a math-and-Aristotle-y sort of mood, we might as well go a bit further. Recall how genera and species behave recursively—any species within a genus can potentially be a genus itself, and any genus can potentially be a species of another genus. Well, notice that the relation of supersets and subsets behaves in exactly the same way—any subset can potentially be a superset of another set, and any superset can potentially be a subset of another set. And further notice that a species is delimited by some characteristic that all its members share formally. In other words, a species is a proper set. So a definition is nothing other than a delimitation of one proper set from within another.

And this shows why it’s not possible to define proper sets—we would have to delimit the proper set of all proper sets from some other proper set, which is impossible because, as shown above, there is no proper set of proper sets. But the collection of all sets is a class, which tells us that proper sets are an analogical concept.

Against the Ontological Argument

I’ve always thought the ontological argument was invalid, and I’ve known what the problem is, but I’ve never really been able to express it. Today, I found a book at the library that took the problem right out of my head and expressed it perfectly:

“St. Anselm says: If the most perfect being that can be conceived did not exist, it would be possible to conceive of a being which has all the qualifications of the former, plus existence, so that this latter being would then be more perfect than the most perfect being that can be conceived. I admit that if this being did not exist, and was not conceived as self-existing, it would be possible to conceive one more perfect. But I deny the assertion that if it did not exist, though it was at the same time conceived as self-existing, then it would be possible to conceive of a more perfect being. Hence it is not logical to conclude: ‘Therefore God exists’; all that can be logically concluded is: Therefore, God must be conceived as self-existing, and in truth does so exist, and is entirely independent of any other being, if He exists.” (Reginald Garrigou-Lagrange, God: His Existence and His Nature, vol. 1, ch. 2)

The problem is that the argument fails to distinguish between whether God is conceptualized as existing and whether God actually exists. The argument shows that God conceptualized as actually existing is greater than God conceptualized as not existing. But just because one must conceptualize God as existing if one is to conceptualize Him at all, doesn’t mean God has to exist independently of a person’s concept of Him.

Aristotelian Metaphysics, Pt. 2: Efficient and Final Causes

N.B.: You will not understand this post unless you know about the concepts of act and potency; for my attempt at explaining them, you can see Aristotelian Metaphysics, Pt. 1: Act and Potency. Note that my earlier post on per se vs. per accidens causation (available here) was intended as a supplement, NOT officially part of this series.

Now it’s time to derive the implications of act and potency.

The most obvious implication of act and potency is the efficient cause. Recall that, as mentioned in the first part, the outcome of any change exists potentially before the change. Well, how does that potency become actual? It can’t make itself actual, because nothing in potency can do anything. If it could, then the potential fire in all of our unlit matches would be wreaking havoc.

So a potency cannot raise itself to act. Nor can any other potency raise another potency to act. Therefore, a potency must be actualized by something else that is already actual. This we call the efficient cause. This is the cause we usually have in mind when we use the unqualified word “cause” in modern English. So e.g. the front car of a train is the efficient cause of the whole train moving, electricity running through a filament is the efficient cause of a light bulb giving off light, air blowing through the vocal folds is the efficient cause of a person’s voice, and a sculptor’s chiseling is the efficient cause of a statue coming into being.

Incidentally, that last step in the derivation of the efficient cause is an important presupposition in the First Mover argument: Any potency that is raised to act, must be actualized by something that is already in act. Most people get this wrong; you usually encounter this axiom phrased as, “Everything has a cause.” People then use this misformulation to argue for the existence of God, which leads to a blatant contradiction between the conclusion and the premise. Neither Aristotle nor Thomas Aquinas ever made such an argument. As Michael “TOF” Flynn says in his series of posts on the First Mover argument, “The Argument from Motion may be wrong, but it is not stupidly wrong.”

The next, less obvious implication is the final cause. We see that the things around us have many different potencies. When an efficient cause actualizes a potency of something, it does not actualize all of the latter’s potencies; that would be impossible, because any given thing will have contrary potencies, e.g. the potential to be blue and the potential to be red, or the potential to be freezing cold and the potential to be boiling hot.

Nor is the potency to be actualized selected at random. Rather, we see that the potency actualized depends largely on the efficient cause in question, and to a lesser extent on the subject that is being changed. Thus e.g. taking a fruit and putting it in a blender makes one result (a smoothie), while taking the same fruit, pulverizing it into mush, and putting it in a candy gives another result (Skittles (TM)). Here, we start with the same object, but end up with different results because the action done to it was different.

So an efficient cause “chooses,” so to speak, a single potency to actualize (or, more commonly, a single range of potencies to actualize, with the final outcome being determined by the thing that is being changed). This particular outcome (or range of outcomes) that a cause is naturally directed toward is what we call the final cause. So e.g. the final cause of striking a match is the starting of a fire, the final cause of bowing a violin is the generation of music, the final cause of gravity pulling on something is the downward acceleration of that thing, etc.

Notice that our examples of final causality and efficient causality are interchangeable, because every instance of efficient causation is also an instance of final causation, and as far as physical processes are concerned, every instance of final causation is also an instance of efficient causation. So the final cause of a train car pulling on the other train cars is the motion of the entire train, the final cause of electricity running through a filament is the generation of light, the final cause of air blowing through the vocal folds is the generation of the voice, and the final cause of a sculptor’s chiseling is the coming into being of a statue. Similarly, the efficient cause of the starting of a fire is the striking of the match, the efficient cause of the generation of music is the bowing of the violin, and the efficient cause of the downward acceleration of objects is gravity pulling on those objects.

Note as well that there is a close relationship between final cause and purpose. For one thing, it is because we are capable of knowing the final causes of natural objects that we are then able to take advantage of them and order them to our own ends; thus e.g. we can take advantage of the wind’s tendency to push on things in order to generate electricity. For another thing, since our conscious purpose is the result that we order our actions to achieve, conscious purpose is essentially a subset of final causes; our goal is the final cause of our actions. In this sense, I guess not every instance of final causation has to be an instance of efficient causation. However, much, perhaps most final causation is unconscious. Gravity does not pull on things because it (he?) wants them to accelerate.

It might seem counterintuitive to think of the final cause as a “cause,” since on the face of it it looks more like a result than a cause. But think of it this way. What would it mean for something to be changing, but not into anything in particular? Clearly this is absurd; if a thing wasn’t changing into anything in particular, then it wouldn’t be changing at all. It would be like a thing existing without being anything in particular. To change is to change into something, just as to be is to be something. In this sense, change depends on its endpoint if it is to occur at all. The endpoint is what gives the change its “identity,” so to speak.

Or again, the only kinds of changes I can think of that might credibly be called changing into nothing in particular would be changing into every possible thing, which is absurd, and fading out of existence. And even these changes have a kind of goal: in the former case it would be everything, and in the latter it would be non-existence. So change can’t even be conceived of without being “aimed” at some particular goal.

Possibly the best evidence that people actually do (consciously or not) think in terms of final causes is that we can still tell what the final cause of a given thing is even if it fails to achieve that final cause. So e.g. if you strike a match and no fire comes out, you wouldn’t say that, oh, I guess that particular thing I just tried to strike wasn’t actually a match, then. No, the natural inference is that it was a match, and it was supposed to make fire, but some kind of deficiency or interference prevented the normal result―maybe you didn’t strike it hard enough, or maybe the match was wet, for example. This tells us that the final cause is an intrinsic aspect of things, regardless of whether they actually achieve that result or not. It also rules out the “law of nature” interpretation of the regularity of causes. There’s no law that every match must catch fire when struck, nor does that actually happen in practice. And yet we still have this notion that striking a match is “supposed” to make it catch fire. If our understanding of causality were based on laws of regularity, then the moment we witnessed a match fail to catch fire for the first time, either A) we would have to conclude that the object we just attempted to strike was not a match at all, or B) our entire understanding of matches that we had built up until then would fall apart. We would have to change our conception from “Matches catch fire” to “Some matches catch fire,” or “Matches catch fire 90% of the time.” But neither of these options is what happens in practice; nobody has any difficulty coping with duds, and even after encountering them, people still say “Matches catch fire” without any qualms. This is because “Matches catch fire” is not a statement of regularity, but a statement of directedness―in other words, final causality.

By contrast, if you use a computer mouse and it doesn’t catch fire, well… now that just makes sense, doesn’t it? That’s because using a mouse doesn’t have catching fire as a final cause. This case is fundamentally different from the match case; here there is no expectation for the mouse to catch fire to begin with, and therefore no existential crisis when the mouse fails to catch fire.

Final causes are particularly evident in physics, where we talk about how e.g. so many newtons of force would cause a mass of so many kilograms to accelerate at a rate of so many meters per second per second. It doesn’t matter whether the force ever does accelerate that much mass at that rate; simply having that magnitude means that by nature that force would cause such a mass to accelerate at that rate. Here, the acceleration is the final cause of the force. Even if the force is counteracted so that it never actually accelerates something, or it accelerates something at a slower rate than it would otherwise, we still understand that the force is “supposed” to accelerate something at a certain rate. So if you dismissed the match example above as being a pragmatic aspect of the way we think, not a true part of reality, then try taking the above paragraph and changing it so that instead of matches catching fire, it talked about forces causing things to accelerate; I’ll wager that the argument still makes sense. Since forces and acceleration are definitely not something humans just invented, the fact that it’s possible for us to think about forces the same way we do about matches, and that this way of thinking still works in that context, seems to me like strong evidence for real, not merely conceptual or pragmatic, final causality.

In any case, whether you like to call it a cause or not, it’s impossible to make sense of both the regularity of efficient causes, and our ability to tell apart cases of deficiency (like the match dud) from cases of simply not happening (like the computer mouse), without either implicitly or explicitly assuming that every cause inherently “leads” or “points” toward a definite result regardless of whether it actually achieves that result or not.

There are two other causes left to discuss, but I think it might be easier to explain them after going over substance and accident. So rather than go over those here, I’ll leave them for part 3.

Aristotelian Metaphysics: Per Se vs. Per Accidens Causation (With Guest Appearances from A Certain Magical Index)

I really like A Certain Magical Index. I’ve watched the anime, or at least the first season of it, three times (the second season didn’t quite sit as well with me), as well as the spinoff, A Certain Scientific Railgun, twice. I have two of its opening themes, “Masterpiece” and “No Buts!”, on my iPod, and I would have “PSI Missing” too if it was on iTunes. And lately I’ve started collecting and translating the original light novels.

But one thing I don’t like about Index is the main character, Kamijou Touma.


I like the concept behind him; a protagonist with the ability to cancel out other people’s superpowers rather than having powers of his own is a pretty cool idea. But Touma just doesn’t have any personality. He has no goals of his own; all he ever does is prevent the villains from achieving their goals. Nor does he have any funny quirks to make him endearing. The only really distinctive thing about him is his bad luck, and that isn’t a feature of him so much as of the world around him. By the end of the show, I felt like the only real incentive I had for caring about him was that Misaka liked him, and Misaka is actually interesting.

Actually, now that I think about it, that’s kind of fitting in a way. Instead of having his own superpowers he just cancels out other people’s superpowers, so instead of having his own goals he just cancels out other people’s goals. But still, it doesn’t make for an interesting character.

However, what I do like about Touma is that he is a perfect illustration of the Aristotelian distinction between per se causation and per accidens causation.

So what are per se and per accidens causation?

Well, go and ask any random person on the street what the temporal relationship between cause and effect is, and they will probably tell you that the cause comes before the effect. But according to Aristotle―and Aristotle is right, of course―cause and effect, in the truest sense, are simultaneous. So why do so many people say otherwise? That’s because they’re failing to distinguish between per se causation and per accidens causation.

A per se cause is one that causes its effect by virtue of itself, whereas a per accidens cause is one that causes its effect by virtue of something that belongs to itself. A per accidens cause generally exists before its effect, but a per se cause exists simultaneously with its effect, and once it ceases to exist its effect ceases to exist as well.

Notice that by definition, per accidens causes are not causes in the truest sense of the word; it isn’t so much the per accidens cause itself as a particular part of the per accidens cause that is a cause in the truest sense. It still makes sense to call it a cause, but this isn’t so much because it is the cause as because it contains the cause.

To see the difference, observe the effects of Touma’s superpower-canceling ability, called the “Imagine Breaker,” on Misaka Mikoto and Accelerator’s powers.

As was mentioned briefly above, Touma is unique among anime protagonists in that he has no superpower of his own. Instead, he has the ability to cancel out other people’s superpowers, on the condition that he touches the location where the power is operating with his right hand.

So let’s start with how Touma’s right hand interacts with Misaka’s ability.


Misaka has the power to control electricity. As you can see in the anime, when she launches lightning bolts at Touma, Touma is able to make them disappear by touching them with his right hand.


Nor does it make a difference if she doesn’t attack using lightning bolts directly. In the spinoff series, A Certain Scientific Railgun, she tries going for a different approach, using her electricity to generate magnetism and then using that magnetism to manipulate iron particles in the ground, which she uses to attack Touma. But here again, Touma is able to cancel out her ability by touching the iron particles. This is crucial because in the first case, there was at least a possibility that Misaka is generating electricity rather than manipulating electricity. The fact that Touma can cancel out her ability by touching the iron particles, which are definitely not generated by Misaka’s power, shows that the thing he touches need not be something generated by a superpower; it can also be something manipulated by a superpower.


Now compare this to Accelerator.


Accelerator’s ability is “vector manipulation.” A vector in physics and math is anything with both magnitude and direction. Accelerator has the power to arbitrarily change the magnitude and/or direction of any vector quantity associated with any object that touches his body, be it momentum, electric current, sound, or anything else. He can also set his ability to automatically reflect the vectors of objects the moment they touch him. This makes him practically invincible, since most attacks cause damage through momentum, which is a vector; all he has to do is set his power to automatically reflect the momentum vectors of anything he comes into contact with, and he has no need to worry about being shot, punched, or cut by anything. (Ok, I think bullets and bladed weapons technically use pressure to cut things, and pressure isn’t a vector. But still, in order to exert pressure a sharp object has to be pushing on something, which would require it to be moving, so he can always reflect it before it does any damage.) He can also use his power to create projectiles, as he can touch any old object and increase its velocity vector to send it flying.

Now, if you watch the fight between Touma and Accelerator, Accelerator uses his power to send a lot of objects flying at Touma. What’s interesting here is that this time, unlike in his fights with Misaka, Touma never seems to use his Imagine Breaker on Accelerator’s projectiles.

accelerator-projectiles(Not sure if you can see it well, but those things sticking out of the ground in the background are parts of train tracks that Accelerator shot at Touma.)

Why is that? We saw from the iron particle situation that Touma can cancel out the effect of a power by touching the affected object, so why can’t he just stop Accelerator’s power by touching the things he sends flying at him?

Well, look back at the condition on Accelerator’s power: Accelerator can only manipulate a vector if it’s touching his body. In other words, once an object leaves contact with his body, his power stops acting on it. So why do his projectiles keep on flying after they leave his body? The reason is Newton’s First Law: “A body in motion tends to stay in motion.” So Touma couldn’t stop Accelerator’s projectiles with his right hand even if he tried; once they’re in the air, they’re moving by their own momentum, not from the operation of any superpower.

Now let’s look at these situations in terms of what’s causing what. What is the cause of the electricity in the case of Misaka and of the motion of objects in the case of Accelerator? On the one hand, it would be perfectly reasonable to say that the respective causes are Misaka and Accelerator. But clearly Misaka doesn’t generate electricity simply because she is Misaka; in that case, there’s no reason why she wouldn’t be generating electricity constantly. Similarly, Accelerator doesn’t cause motion simply because he is Accelerator; in that case everything he touched would be flying around.

So what if we say that the causes are Misaka and Accelerator’s powers? This doesn’t work either, for the same reason as before; Misaka and Accelerator have their powers constantly, so there’s no reason why they shouldn’t be constantly generating electricity and constantly launching things around.

So what is there that is always present while the effects of their powers are observed and absent when they aren’t? The answer is the operation of their powers, as distinct from the powers themselves. This is the per se cause of Misaka’s electricity and Accelerator’s vector manipulation. The powers themselves, and the people who use them, are per accidens causes.

So now that we know the causes we’re concerned with, let’s go back the other way and consider the effects of the operations of their powers. In the case of Misaka, the answer is fairly obvious. The effect of the operation of her power is electricity, as well as whatever it is that the electricity does, since she manipulates the electricity as well as generating it.

But Accelerator is a little different. It might seem at first glance that the effect of his power is the new vector that it generates. But remember, the vector that Accelerator creates persists after his power ceases to operate. This means that it is not the per se effect of the operation of his power. What does begin and end with the operation of his power is the acquisition or alteration of a vector. This is why the nickname “Accelerator” fits him so well; his power is not to make things move, but to make them change their motion.

So this is why Touma’s Imagine Breaker works on Misaka’s electricity but not on Accelerator’s projectiles; it only cancels out the per se effects of the operation of a power, not the per accidens effects either of the power itself or of its operation.

Incidentally, this is also the reason why Touma is unable to heal the amnesia caused by the damage to his brain from the Dragon Breath spell; the brain damage is only caused by the Dragon Breath spell per accidens, not per se. On the other hand, the reason he is able to convince Index that he did heal the damage is that since the damage was caused by the spell per accidens, it seems believable to her that he was able to cancel it out. If Index knew the difference between per se and per accidens causation, she would have caught Touma’s lie and figured out that he really did have amnesia.

This idea that causes and effects are simultaneous is critical to our view of causality; if we think of effects as coming after their causes, then the connection between cause and effect begins to seem tenuous, and we may end up thinking that there is no necessary connection between them at all. Hume famously argued that causes and effects are not connected, and the only reason we think they are related is because we see the effect come after the cause over and over again. But if we have a clear picture of causality, these propositions immediately seem ridiculous. To borrow an example from Edward Feser’s The Last Superstition, when someone throws a brick at a window, the act of the brick pushing through the window and the act of the window giving way to the brick are one and the same act. Though we can separate them conceptually, in reality, there is no separation between them either temporally or ontologically. So to say that the window need not give way to the brick when the brick pushes through it is like saying that we need not get four when we add two and two. The obvious answer to that is, “But two and two just is four.” Similarly, the answer for someone who claims that an effect need not follow upon a cause is, “But the operation of the cause just is the generation of the effect”; these are just two different ways of looking at the same event.

The distinction between per se and per accidens causation is also one of the number one reasons people misunderstand Thomas Aquinas’s Unmoved Mover argument for the existence of God. Most people who read the argument envision the series of causes Aquinas discusses as a series of per accidens causes, but what he is actually talking about is a series of per se causes, which is completely different.

… So yeah. At least Touma is good for something.

Aristotelian Metaphysics, Pt. 1: Act and Potency

Alright, now I’ve FINALLY gotten that post on Aristotle off my chest. Well, okay, since I’m dividing this into parts I still have more to do. But still, it’s progress.

Now, the word “metaphysics” has a pretty bad connotation nowadays. People apply the word “metaphysical” to things like psychic energies and whatnot. But in the context of philosophy, this word has a different meaning. In fact, the philosophical meaning is the original one. It is the study of existence itself, its prerequisites, the rules governing it, and how we make sense of it. So really, everyone engages in metaphysical thought at some point, because everyone has some idea of what can and can’t be, and how existence is supposed to work.

However, Aristotle’s metaphysical worldview is fairly different from most people’s nowadays. If I had to boil the difference down to one easy-to-understand point, it would be their stance on the statement, “The whole is greater than the sum of its parts.” Aristotelians would affirm this, while most people nowadays would say that nothing is any more or less than the sum of its parts. But this is ultimately an oversimplification. So without further ado, let’s jump right in to the gory details.

The starting point of Aristotelian metaphysics is the question of how change is possible. For example, consider an apple changing from green to red. The question here is: Where does the redness come from?

This might seem like a stupid question, but the idea behind it is pivotal to our understanding of the world. Consider, for example, the way Parmenides answered this question: The redness that appears in the apple can come either from being or from non-being. It doesn’t come from being (it’s not as though the redness was just stored away in a compartment somewhere and the apple just brought it out when the time was right; the redness actually did not exist before the change happened). Nor does it come from non-being, because nothing can come from non-being. Therefore, there is nothing the redness could have come from, and therefore the apple cannot have actually turned red; the change must be illusory. And since the same argument can be applied to any sort of change, all change must be illusory.

Many ancient Greek rhetoric teachers appealed to Parmenides in justifying their teaching practices. They claimed that because this world is illusory, what we do doesn’t matter, and therefore there is no need to worry about whether one’s arguments are correct; all that matters is that the audience is persuaded. These rhetoric teachers were called “sophists,” and this is where our pejorative use of the words “sophist” and “sophistry” comes from.

So this question actually has implications for what exactly we think the world is and how exactly we think it works, and our conception of the world can have implications for our views on morality. So this question might actually be a big deal after all.

Now getting back to the apple: Aristotle came up with an alternative response to Parmenides’s. That response was that the redness existed potentially before the change occurred, and was brought into actuality through the change. And the same can be said for any kind of change: the result of a change exists potentially before the change occurs, and is brought into actuality through the change.

Notice, first of all, the difference between this answer and Parmenides’s. Parmenides thought that the redness did not exist at all before the color change, while Aristotle says that it it did really exist before the change, though only potentially.

And note also that this answer is phrased in terms of concepts more fundamental than atoms, chemicals, etc. It’s similar to the practice in math where you assume as little as possible about your variable so that whatever you prove about that variable is as general as possible. A proof that starts with a generic odd number and shows that it can be written in the form 2k+1 shows that all odd numbers can be written in the form 2k+1. In the same way, Aristotle says that the results of a change exist potentially before the change occurs, not specifying what sorts of change this might apply to. So, as Aristotle claims, this statement applies to all kinds of change, be it on a macroscopic scale, like with the apple turning red, or on the microscopic scale, like in the interactions of atoms.

Therefore―and this is where I’ll probably lose what few readers I have―science cannot prove or disprove this axiom. I can’t stress this point enough. It’s a pretty popular practice nowadays to say that Aristotle is outdated; I mean, come on, who believes in the four elements anymore? And the four humors? And that the sun floats in the sky because it’s light, while the earth is below because it’s heavy? I mean, sheesh! But, while Aristotle did make some scientific claims, the most important parts of his philosophy are about existence and reason themselves, and therefore are actually logically prior to science. In fact, far from disproving the idea of potentiality, science actually presupposes it!

To see why, consider how a chemist would respond to the question of how apples turn red. He would say that there was some sort of green chemical in the skin of the apple, and the atoms in that chemical became reconfigured to turn into a red chemical. (Notice how I’m using general terms here.) But here our old question just pops up in a new form: Where did the red chemical come from? Well, it’s not like it really “came” from anywhere. The red chemical was just one more form that the green chemical, with its atoms mixed and matched with the atoms of a few other chemicals, was capable of taking.

But notice the language of potency in this answer: “one more form that the green chemical was capable of taking.” In other words, when making this answer we are presupposing that the way a thing exists in the present determines how it might possibly exist in the future; that a given thing is predisposed toward acting or changing in certain ways, to the exclusion of others, by what it is and what processes it undergoes in the present. This simply is what Aristotle means by potency―the possible outcomes “pointed to” by actual things. Any time you say “I can do that” or “That’s impossible,” you are making use of this concept, and therefore thinking in terms of what Aristotle means by potency.

Now, at this point, you might be thinking, “Just because I believe that the way a thing is now affects the ways it can change, doesn’t mean I have to believe in some kind of vague, shadowy middle state in between existing and not existing.” Well, I don’t know about you, but that’s the objection I would feel inclined to raise if I were the reader and not the writer here. The idea that the way a thing is now affects the ways it can change seems pretty obvious and intuitive, but the idea of potency being a real thing sounds absurd.

So let’s try to elucidate this idea. First of all, there’s the question of how there can be a state in between existence and non-existence. The answer to this question is actually pretty simple: There is no middle state between existence and non-existence. Aristotle himself explicitly stated this in what we now refer to as the Law of Excluded Middle: A thing must either be or not be; there is no way it can do both, neither, or a mixture of the two. This is one of the three laws that are presupposed by all rational thought, and therefore can neither be proven nor disproven―any proof or disproof would have to presuppose these laws and would therefore be presupposing what it was trying to prove/disprove. (The other two are the Law of Non-Contradiction―nothing can both be and not be at the same time and in the same respect; and the Law of Identity―any given thing is itself and nothing else. Just try to prove or disprove any of these, and you’ll catch yourself presupposing it every time.)

It’s easy to make the mistake of thinking that potency is in between existence and non-existence; I’ve made that mistake myself, which is why I guessed that that’s probably the objection that people will bring up. But Aristotle’s thesis here isn’t that there’s a third state besides existence and non-existence; rather, his claim is that existence is not a single monolithic category―it is divided into actual and potential existence.

To see that potency is a kind of existence, consider a block of wood and a pool of water. The block of wood is capable of being formed into a statue; the water is not. Now, this capability is something that is in the wood here and now, but that the water lacks here and now. In other words, this capability is a real feature of things that any given thing can either have or not have.

Again, appealing to science doesn’t change anything. Even if someone said, “The wood’s molecules are packed closely together and therefore retain their shape, whereas the water’s molecules are only loosely connected and therefore slip out of any shape they’re put into,” this would only be moving the problem somewhere else. In this case, the wood molecules are capable of forming a tight, firm bond with other wood molecules, whereas the water molecules do not have the capability to do this with other water molecules, unless you lower their temperature. Again, we are brought back to the idea that the capabilities of things are real features that can either be or not be in any given subject. Taking things even further and talking about individual electrons and protons will just bring up the same thing again.

And so, if all of that was convincing to you, then we have finally arrived at the idea of act and potency, the two words that sum up all of Aristotelian metaphysics just as supply and demand sum up all of economics. The usage of these words in this sense is pretty archaic, but as you can probably guess, “act” refers to actual existence (being “in act”) while “potency” refers to potential existence (being “in potency”). “Act” might seem like a strange name for “actuality”―although they’re from the same root, their meanings aren’t really considered to be related in modern English―but it makes sense on a certain level because one of the key differences between actuality and potentiality is that only actual things can be causes. If a thing’s cause is only potential and not actual, then it won’t occur. So potential things can’t “act.”

Now, this theory has a lot of implications, not all of which are exactly popular. In fact, I would argue that it’s because of these unpopular implications, and not because the theory is actually defective, that scholars have abandoned it. But those are topics for another time. If you want to read more about Aristotelian metaphysics, I would highly recommend Edward Feser’s Scholastic Metaphysics: A Contemporary Introduction. Pretty much everything I know about metaphysics, I learned from Feser. Feser’s blog also has a lot of posts on individual questions of metaphysics. This free online book (starting at chapter 4) is also a great resource.

EDIT: Just realized, in the second paragraph, instead of writing “The whole is greater than the sum of its parts,” I accidentally wrote “The whole is greater than the part.” And I never noticed for a whole year. Well, that’s embarrassing.

Nothing New Under the Sun

I started reading Thomas Aquinas’s commentary on Aristotle’s Physics recently. The Physics isn’t actually about what we would call physics; it’s about what in modern terms would be called philosophy of nature. Anyways, Aristotle devotes large parts of this book to rebutting the opinions of other philosophers, and every time I see one of these rebutting sections I’m always amazed at how similar his contemporaries sound to modern philosophers. Here are a few parts I was reading over the past couple of days that particularly stood out to me. The Latin text is from the Corpus Thomisticum, and the translation is by yours truly.

Thomas Aquinas, Commentary on Aristotle’s Physics, book 2, reading 2

Postquam philosophus ostendit quid est natura, hic ostendit quot modis natura dicitur. Et primo ostendit quod natura dicitur de materia; secundo quod dicitur de forma, ibi: alio autem modo et cetera. Circa primum, sciendum est quod antiqui philosophi naturales, non valentes usque ad primam materiam pervenire, ut supra dictum est, aliquod corpus sensibile primam materiam omnium rerum ponebant, ut ignem vel aerem vel aquam: et sic sequebatur quod omnes formae advenirent materiae tanquam in actu existenti, ut contingit in artificialibus; nam forma cultelli advenit ferro iam existenti in actu. Et ideo similem opinionem accipiebant de formis naturalibus, sicut de formis artificialibus. Dicit ergo primo quod quibusdam videtur quod hoc sit substantia et natura rerum naturalium, quod primo inest unicuique, quod secundum se consideratum est informe: ut si dicamus quod natura lecti est lignum, et natura statuae est aes; nam lignum est in lecto, et secundum se consideratum non est formatum. Et huius signum dicebat Antiphon esse hoc, quod si aliquis proiiciat lectum ad terram, et ligna putrescendo accipiant potentiam ut aliquid ex eis germinet, illud quod generatur non erit lectus, sed lignum. Et quia substantia est quae permanet, et naturae est sibi simile generare, concludebat quod omnis dispositio quae est secundum quamcumque legem rationis vel artem, sit accidens: et illud quod permanet sit substantia, quae continue patitur huiusmodi dispositionum immutationem. Supposito igitur quod rerum artificialium formae sint accidentia, et materia sit substantia, assumebat aliam propositionem, quod sicut se habent lectus et statua ad aes et lignum, ita et quodlibet horum se habet ad aliquid aliud quod est materia ipsorum; ut aes et aurum ad aquam (quia omnium liquefactibilium materia videtur esse aqua), et ossa et ligna ad terram, et similiter est de quolibet aliorum naturalium. Unde concludebat quod illa materialia subsistentia formis naturalibus, sint natura et substantia eorum. Et propter hoc quidam dixerunt terram esse naturam et substantiam omnium rerum, scilicet primi poetae theologizantes; posteriores vero philosophi vel ignem vel aerem vel aquam, vel quaedam horum, vel omnia haec, ut ex superioribus patet. Quia tot de numero eorum dicebant esse substantiam omnium rerum, quot accipiebant esse principia materialia; et omnia alia dicebant esse accidentia horum, idest materialium principiorum, vel per modum passionis vel per modum habitus vel per modum dispositionis, vel cuiuslibet alterius quod reducatur ad genus accidentis. Et haec est una differentia quam ponebant inter principia materialia et formalia, quia dicebant ea differre secundum substantiam et accidens. Alia autem differentia est, quia dicebant ea differre secundum perpetuum et corruptibile. Nam quodcumque praemissorum corporum simplicium ponebant esse materiale principium, dicebant illud esse perpetuum: non enim dicebant quod transmutarentur invicem. Sed omnia alia dicebant fieri et corrumpi infinities: ut puta, si aqua sit principium materiale, dicebant aquam nunquam corrumpi, sed manere eam in omnibus sicut substantiam eorum; sed aes et aurum et alia huiusmodi dicebant corrumpi et generari infinities.


After the Philosopher has shown what nature is, he here shows how many meanings “nature” has. And first he shows that nature is said of matter; second that it is said of form, there: “But in another way,” etc. Concerning the first, it must be noted that the ancient natural philosophers, unable to arrive at prime matter, as was said above, held some sensible body to be the first matter of all things, like fire, or air, or water: and thus it followed that all forms came to matter as to something already existing in act, as occurs with artificial things; for the form of a knife comes to iron that already exists in act. And therefore they held a similar opinion on natural forms and artificial forms. He [Aristotle, citing objections?] says then first that to some it seems that this is the substance and nature of all natural things, that which is in each of them first, which considered in itself has no form: as if we should say that the nature of a bed is wood, and the nature of a statue is bronze; for wood is in the bed, and considered in itself is not formed. And Antiphon said that a sign of this fact is this, that if someone throws a bed to the ground, and the wood by rotting actualizes [?] its potency to have something germinate out of it, that which is formed will not be a bed, but wood. And because substance is that which remains, and it is characteristic of nature to create something similar to itself, he concluded that every arrangement that follows some law of reason or some skill, is accidental: and that which remains is substance, which uninterruptedly undergoes such arrangements and changes. Given then that the forms of artificial things are accidents, and their matter is their substance, he adopted another proposition, that as a bed and a statue is to wood and bronze, so also any of these things is to any other thing that is its matter; as bronze and gold are to water (for the matter of all liquefiable things seems to be water), and bones and wood to earth, and it is similar with any other natural thing. From this he concluded that those materials underlying natural forms, are their nature and substance. And because of this some have said that the earth is the nature and substance of all things, namely the first theologizing [?] poets; philosophers afterward said that fire or air or water, or some of these, or all of them [were the nature and substance of all things], as is clear from what was said above. For they said as many of those were the substance of all things, as they held to be material principles*; and everything else they said were accidents of these, that is of the material principles, either in the manner of passion [“passion” in the sense of receiving rather than causing change; opposed to action] or in the manner of a habit* or in the manner of an arrangement, or of whatever other thing that falls under the genus of accident. And this, they said, is one difference between material and formal principles, for they said they differed as substance and accident. And another difference is, that they said they differed as permanent and corruptible. For whichever of the aforementioned simple bodies they said was a material principle, they said that that was permanent: for they said that they [the principles] were not changed into each other. But all other things they said came into being and passed away an infinite number of times: as for example, if water was a material principle, they said water never passed away, but remained in all things as their substance; but bronze and gold and other such things they said passed away and came into being an infinite number of times.

(*I’m not entirely sure how to translate “principium”; literally it just means “beginning,” but it often carries the connotation that it’s a source rather than just a beginning. Translations of Aquinas usually just translate “principium” as “principle” everywhere, so since I’m not entirely sure what Aquinas means, I’m just falling back on that. Similarly, I’m also not sure how to translate “habitus”; literally it just means “a[n act of] having,” which could mean any one of a number of things. But most translations of Aquinas just render this as “habit,” so I’m falling back on that.)

It might be hard to understand this if you don’t know all the technical terminology Aquinas uses, but the basic idea is that the ancient philosophers thought that any complex thing exists only as a feature or arrangement of the parts that make it up, rather than the whole being a thing in itself. They thought that the elements were never created or destroyed, only rearranged to form different things. They also thought that there was no ontological difference between living things and artificial machines. All of these are positions taken for granted by modern philosophers and scientists, though we have different elements now. This contrasts with the position Aristotle came up with; he held that natural things (“natural” being defined as “having within itself a principle of motion and rest, which is in it primarily and essentially and not accidentally” (book 2, reading 1)) are things in their own right rather than simply sums of their parts.

ibid., book 2, reading 7

… sicut Empedocles, qui dixit quod aer non semper adunatur superius supra terram quasi hoc ei sit naturale, sed quia ita accidit a casu. Dicit enim quod quando mundus est factus, lite distinguente elementa, accidit quod aer se collegit in istum locum, et sicut tunc cucurrit, ita semper stante isto mundo cursum habebit: sed multoties in aliis mundis, quos ponebat infinities fieri et corrumpi, ut supra dictum est, aer aliter ordinatur inter partes universi. Et similiter dicebat quod plurimae partes animalium fiunt a fortuna; sicut quod in prima constitutione mundi fiebant capita sine cervice.


… like Empedocles, who said that air does not always rise upward above the eath as if this is natural to it, but because it occurs thus by chance. For he says that when the world was made, with strife [incompatibility? repulsion?] separating the elements, it happened that air went to that place, and as when it ran there then, so will it always take that course as long as the world stands thus: but often in other worlds, which he thought were created and passed away an infinite number of times, as was said above, air is otherwise ordered among the parts of the universe. And similarly he said that most parts of animals occur by chance; as when in the first creation of the world heads were made without necks.

And here we have the multiverse with the laws of nature changed at each iteration.

ibid., book 2, reading 12

Circa primum sciendum quod ponentes naturam non agere propter aliquid, hoc confirmare nitebantur removentes id ex quo natura praecipue videtur propter aliquid operari. Hoc autem est quod maxime demonstrat naturam propter aliquid operari, quod ex operatione naturae semper invenitur aliquid fieri quanto melius et commodius esse potest, sicut pes hoc modo est factus a natura, secundum quod est aptus ad gradiendum; unde si recedat a naturali dispositione, non est aptus ad hunc usum; et simile est in ceteris. Et quia contra hoc praecipue opponere nitebantur, ideo dicit quod potest opponi quod nihil prohibet naturam non facere propter aliquid, neque facere semper quod melius est. Invenimus enim quandoque quod ex aliqua operatione naturae provenit aliqua utilitas, quae tamen non est finis illius naturalis operationis, sed contingit sic evenire; sicut si dicamus quod Iupiter pluit, idest Deus vel natura universalis, non propter hunc finem, ut frumentum augmentet, sed pluvia provenit ex necessitate materiae. Oportet enim, inferioribus partibus ex propinquitate solis calefactis, resolvi vapores ex aquis; quibus sursum ascendentibus propter calorem, cum pervenerint ad locum ubi deficit calor propter distantiam a loco ubi reverberantur radii solis, necesse est quod aqua vaporabiliter ascendens congeletur ibidem, et congelatione facta, vapores vertantur in aquam; et cum aqua fuerit generata, necesse est quod cadat deorsum propter gravitatem: et cum hoc fit, accidit ut frumentum augeatur. Non tamen propter hoc pluit ut augeatur; quia similiter in aliquo loco frumentum destruitur propter pluviam, ut cum est collectum in area. Non tamen propter hoc pluit, ut destruatur frumentum, sed hoc casu accidit, pluvia cadente; et eodem modo videtur casu accidere quod frumentum crescat per accidens, pluvia cadente. Unde videtur quod nihil prohibeat sic etiam esse in partibus animalium, quae videntur esse sic dispositae propter aliquem finem: utpote quod aliquis dicat quod ex necessitate materiae contingit quod quidam dentes, anteriores scilicet, sint acuti et apti ad dividendum cibum, et maxillares sint lati et utiles ad conterendum cibum. Non tamen ita quod propter istas utilitates natura fecerit dentes tales vel tales: sed quia dentibus sic factis a natura propter necessitatem materiae sic decurrentis, accidit ut talem formam consequerentur, qua forma existente sequitur talis utilitas. Et similiter potest dici de omnibus aliis partibus, quae videntur habere aliquam determinatam formam propter aliquem finem.

Et quia posset aliquis dicere quod semper vel ut in pluribus tales utilitates consequuntur; quod autem est semper vel frequenter, conveniens est esse a natura: ideo ad hanc obiectionem excludendam, dicunt quod a principio constitutionis mundi, quatuor elementa convenerunt ad constitutionem rerum naturalium, et factae sunt multae et variae dispositiones rerum naturalium: et in quibuscumque omnia sic acciderunt apta ad aliquam utilitatem, sicut si propter hoc facta essent, illa tantum conservata sunt, eo quod habuerunt dispositionem aptam ad conservationem, non ab aliquo agente intendente finem, sed ab eo quod est per se vanum, idest a casu. Quaecumque vero non habuerunt talem dispositionem sunt destructa, et quotidie destruuntur; sicut Empedocles dixit a principio fuisse quosdam generatos, qui ex una parte erant boves, et ex alia parte erant homines.


Concerning the first, it must be noted that those who held that nature does not act for an end, tried to prove this by taking away the cases because of which nature especially seems to work for an end. And this is what most seems to show that nature works for an end, that out of the operation of nature a thing is always found such that it can be so much the better and more convenient, as the foot is made in this way by nature, so that it is fit for stepping; so that if it departs from its natural arrangement, it is not fit for this use; and it is similar in everything else. And because they wanted to argue against this especially, therefore he [Aristotle, citing objections?] says that it can be objected that nothing prevents nature from not acting for an end and always making what is better. For we find all the time that out of some work of nature some useful thing comes forth, which is still not the end of that natural operation, but happens to occur this way; as if we say that Jupiter rains [?]*, that is God or all of nature [?]*, it is not for this end, that it should make fruit grow, but the rain comes forth out of the necessity of matter. For it is necessary, when lower areas have been warmed by their closeness to the sun, for vapors to be released out of water; and these ascending because of their heat, when they have arrived at a place where heat is insufficient because of its distance from the place where the rays of the sun are rebounded, it is necessary that the water, ascending as a vapor, be congealed there, and once it has congealed, that the vapors be turned into water; and when the water has been formed, it is necessary that it fall downward because of its weight: and when this occurs, it so happens that fruit grows. But it is not for the sake of this, so that the fruits should grow, that it rains; because similarly in another place fruit is destroyed because of rain, as when it is gathered together in an area. But it is not for this purpose, that the fruits should be destroyed, that it rained, but this happened by accident, when the rain was falling; and in the same way it seems to occur by chance that the fruit grows accidentally, while the rain is falling. From which it seems that nothing prevents things from also being this way with the parts of animals, which seem to be so arranged for some end: namely that someone should say that out of the necessity of matter it so happens that some teeth, that is the front teeth, are sharp and fit for cutting food, and molars are broad and useful for grinding food. But it is not that nature made teeth in such or such a way for these uses: but that, when teeth had been so made by nature because of the necessity of matter running down [?] in this way, it happened that it followed such a form, and once such a form existed, such a use followed. And it can similarly be said of all other parts, which seem to have some definite form for some end.

(*I think this might be one of those cases where the ancient Greeks and Romans used the name of a god metonymically to stand for the thing that that god is a god of. So in Virgil, you often find lines where he says “Venus” but means “love.” Here, I’m guessing Aristotle is saying “Zeus” to mean “the sky,” or “all of nature” as Aquinas says, and since Aquinas is using a Latin translation (he didn’t know Greek), that’s rendered as “Jupiter.”)

And because someone could say that such uses follow always or for the most part; that is, always or frequently, it is fitting that this be by nature: therefore to preclude this objection, they say that from the beginning of the formation of the world, the four elements came together to the formation of natural things, and the many and varied arrangements of natural things were made: and whichever things happened to be fit for some use, as if they had been made for this, only those were preserved, because they had an arrangement fit for preservation, not from some agent intending that end, but from that which in itself is empty, that is by chance. On the other hand whatever did not have such an arrangement was destroyed, and is destroyed every day; as Empedocles said some people were formed in the beginning, who in one part were bulls, and in the other part were men.


Now, is it just me, or does this whole section, and particularly the underlined part (the underline is mine, by the way), sound exactly like evolution and natural selection?

After Aristotle devoted so much of this book to rebutting these philosophers, it looks like nothing has changed.